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+{"text":"\\section*{\\refname}} \n\n\n\\begin{document}\n\n\\title{Verticalization of bacterial biofilms}\n\n\\author{Farzan Beroz, Jing Yan, Yigal Meir, Benedikt Sabass, Howard A. Stone, Bonnie L. Bassler, and Ned S. Wingreen*}\n\n\\begin{abstract}\nBiofilms are communities of bacteria adhered to surfaces. Recently, biofilms of rod-shaped bacteria were observed at single-cell resolution and shown to develop from a disordered, two-dimensional layer of founder cells into a three-dimensional structure with a vertically-aligned core. Here, we elucidate the physical mechanism underpinning this transition using a combination of agent-based and continuum modeling. We find that verticalization proceeds through a series of localized mechanical instabilities on the cellular scale. For short cells, these instabilities are primarily triggered by cell division, whereas long cells are more likely to be peeled off the surface by nearby vertical cells, creating an ``inverse domino effect''. The interplay between cell growth and cell verticalization gives rise to an exotic mechanical state in which the effective surface pressure becomes constant throughout the growing core of the biofilm surface layer. This dynamical isobaricity determines the expansion speed of a biofilm cluster and thereby governs how cells access the third dimension. In particular, theory predicts that a longer average cell length yields more rapidly expanding, flatter biofilms. We experimentally show that such changes in biofilm development occur by exploiting chemicals that modulate cell length.\n\\end{abstract}\n\n\\maketitle\n\n\n\\renewcommand{\\figurename}{Figure}\n\n\\titleformat{\\section}    \n       {\\normalfont\\fontfamily{cmr}\\fontsize{12}{17}\\bfseries}{\\thesection}{1em}{}\n\\titleformat{\\subsection}[runin]\n{\\normalfont\\fontfamily{cmr}\\bfseries}{}{1em}{}\n\n\\renewcommand\\thefigure{\\arabic{figure}}    \n\n\n\\setcounter{section}{0}\n\\setcounter{figure}{0}\n\\setcounter{equation}{0}\n\n\\titlespacing*{\\subsection}\n{0pt}{3ex plus 1ex minus .2ex}{10ex plus .2ex}\n\n\nBiofilms are groups of bacteria adhered to surfaces \\cite{R22, R21, VG}. These bacterial communities are common in nature, and foster the survival and growth of their constituent cells. A deep understanding of biofilm structure and development promises important health and industrial applications \\cite{R20, R23}. Unfortunately, little is known about the microstructural features of biofilms due to difficulties encountered in imaging individual cells inside large assemblies of densely-packed cells. Recently, however, advances in imaging technology have made it possible to observe growing, three-dimensional biofilms at single-cell resolution\\cite{R16,R1, R4}.\n\nIn the case of \\emph{Vibrio cholerae}, the rod-shaped bacterium responsible for the pandemic disease cholera\\cite{R24, R5}, high-resolution imaging revealed a surprisingly complex biofilm developmental program\\cite{R1, R4}. Over the course of 12-24 hours of growth, an individual founder cell gives rise to a dome-shaped biofilm cluster, containing thousands of cells that are strongly vertically ordered, especially at the cluster core. Notably, this ordering is an intrinsically non-equilibrium phenomenon, as it is driven by growth, not by thermal fluctuations. Indeed, the striking extent of ordering cannot be explained by Onsager's theory for the equilibrium ordering of rod-shaped objects\\cite{R1,R31}.\n\nAn important clue to understanding the emergence of vertical order in \\emph{V. cholerae} biofilms comes from genetic analyses that established the biological components relevant for biofilm development\\cite{R25, R1, R4}. To facilitate their growth as biofilms, \\emph{V. cholerae} cells secrete adhesive matrix components: Vibrio polysaccharide (VPS), a polymer that expands to fill gaps between cells, and cell-to-cell and cell-to-surface adhesion proteins. Cell-to-surface interactions enable vertical ordering by breaking overall rotational symmetry. However, despite previous work on the orientational dynamics of bacterial cells\\cite{R6,R9,R27,AR1,AR2,R3,R17,R8,R7,GS}, the nature of this physical process remains unclear.\n\nIn this work, we establish the biophysical mechanisms controlling \\emph{V. cholerae} biofilm development. We show that the observed structural and dynamical features of growing biofilms can be reproduced by a simple, agent-based model. Our model treats individual cells as growing and dividing rods with cell-to-cell and cell-to-surface interactions, and thus serves as a minimal model for a wide range of biofilm-forming bacterial species. By examining individual cell verticalization events, we show that reorientation is driven by localized mechanical instabilities occurring in regions of surface cells subject to high in-plane compression. These threshold instabilities explain the tendency of surface-adhered cells to reorient rapidly following cell division. We incorporate these verticalization instabilities into a continuum theory, which allows us to predict the expansion speed of biofilms as well as overall biofilm morphology as a function of cell-scale properties. We verify these predictions in experiments in which we use chemicals that alter cell length. Our model thus elucidates how the mechanical and geometrical features of individual cells control the emergent features of the biofilm, which are relevant to the survival of the collective.\n\n\\begin{figure*}[t!]\n\\includegraphics[width=1.6\\columnwidth]{fig1.jpg}\n\\caption{\\label{fig:FG0}\nDevelopment of experimental and modeled biofilms. (\\textbf{a}, \\textbf{b}) Top-down and perspective visualizations of the surface layer of (\\textbf{a}) experimental and (\\textbf{b}) modeled biofilms, showing positions and orientations of horizontal (blue) and vertical (red) surface-adhered cells as spherocylinders of radius $R=\\SI{0.8}{\\micrometer}$, with the surface shown at height $z=\\SI{0}{\\micrometer}$ (brown). Cells with $n_z < 0.5$ ($>0.5$) are considered horizontal (vertical), where $\\boldsymbol{\\hat{n}}$ is the orientation vector. The upper-left panel of (\\textbf{a}) shows a confocal fluorescence microscopy image, and the upper-right panel shows the corresponding reconstructed central cluster using the positions and orientations of surface cells. The upper-left panel of (\\textbf{b}) shows a schematic representation of modeled cell-cell (orange) and cell-surface (yellow) interactions, which depend, respectively, on the cell-cell overlap $\\delta_{ij}$ (purple) and cell-surface overlap $\\delta_i$ (red) (see Methods for details). Scale bars: $\\SI{5}{\\micrometer}$. (\\textbf{c}, \\textbf{d}) 2D growth of biofilm surface layer for (\\textbf{c}) experimental biofilm (same as shown in (\\textbf{a})) and (\\textbf{d}) modeled biofilms. The color of each spatiotemporal bin indicates the fraction of vertical cells at a given radius from the biofilm center, averaged over the angular coordinates of the biofilm (gray regions contain no cells). In (\\textbf{d}), each spatiotemporal bin is averaged over ten simulated biofilms. In (\\textbf{c},\\textbf{d}), the horizontal dashed pink lines show the onset of verticalization. The black dashed lines show the edge of the biofilm. Insets show the distribution of cell orientations at time $t=300$ minutes, with color highlighting horizontal and vertical orientations.\n}\n\\end{figure*}\n\n\\subsection*{Biofilm radius and vertical ordering spread linearly over time} ~\n\nHow do cells in \\emph{V. cholerae} biofilms become vertical? Biofilms grown from a single, surface-adhered founder cell initially expand along the surface (Fig. 1a, Supplementary Video 1). This horizontal expansion occurs because cells grow and divide along their long axes, which remain parallel to the surface due to cell-to-surface adhesion\\cite{R1}. After about three hours, progeny near the biofilm center begin to reorient away from the surface (Fig. 1c). Reorientation events typically involve a sharp change in a cell's verticality $n_z$, defined as the component of the cell-orientation vector $\\boldsymbol{\\hat{n}}$ normal to the surface (inset Fig. 1c). At later times, the locations of the reorientation events spread outward, and eventually the biofilm develops a roughly circular region of vertical cells surrounded by an annular region of horizontal cells. Both of these regions subsequently expand outward with approximately equal, fixed velocities. The radial profile of verticality versus time shows that the local transition of cells from horizontal to vertical occurs rapidly, taking $10$-$30$ minutes for cell-sized regions to develop a vertical majority.\n\n\\subsection*{Agent-based model captures spreading of biofilm and vertical ordering}~\n\nTo understand how the behavior of living biofilms arises from local interactions, we developed an agent-based model for biofilm growth (left inset Fig. 1b, Methods, Supplementary Fig. 1). The model extends existing agent-based models\\cite{R18, R15, R3} by incorporating the viscoelastic cell-to-surface adhesion\\cite{R11,R19} that is crucial for \\emph{V. cholerae} biofilm formation\\cite{R24, R25}. Specifically, we treat the cells as soft spherocylinders that grow, divide, and adhere to the surface. We simulated biofilms by numerically integrating the equations of motion starting from a single, surface-adhered founder cell (Supplementary Video 2, Methods). To make the computations more tractable for systematic studies, we simulated only the surface layer of cells by removing from the simulation cells that become detached from the surface. This quasi-3D model is a reasonable approximation of the full 3D model at early times (Supplementary Fig. 2), and closely matches the dynamics and orientational order observed over the full duration of the experiment, including an inner region of vertical cells surrounded by an annular periphery of horizontal cells, both of which expand outward at a fixed rate (Fig. 1b,d). As in the experimental biofilm, the horizontal and vertical orientations of the modeled cells are sharply distinct (inset Fig. 1d), and the conversion from horizontal to vertical occurs rapidly (Fig. 1d). The emergence of this distinctive orientational-temporal pattern demonstrates that our simple agent-based model is sufficient to capture the physical interactions that underpin the observed ``verticalization'' transition of experimental biofilms.\n\n\\subsection*{Mechanical instabilities cause cell verticalization}~\n\nWhy do the experimental and modeled cells segregate into horizontal and vertical orientations, with transitions from horizontal to vertical proceeding rapidly? To investigate the local mechanics that drive verticalization, we considered the dynamics of a single model cell of cylinder length $\\ell$ adhered to the surface. The surface provides a combination of attractive and repulsive forces that, in the absence of external forces, maintain the cell at a stable fixed point with elevation angle $\\theta=0$ (i.e. horizontal) and penetration into the surface $\\delta_0$. However, when additional forces are applied to the cell, the cell may become unstable to vertical reorientation.\n\nWe determined the onset of this instability by performing a linear stability analysis for a cell under constant external forces (inset Fig. 2a). For simplicity, we took the external forces to be applied by a continuum of rigid, spherical pistons that are distributed uniformly around the cell perimeter. The pistons compress the cell in the $xy$ plane with an applied surface pressure $p$. For values of $p$ larger than a threshold surface pressure $p_{t}$, the cell becomes linearly unstable to spontaneous reorientation (Supplementary Figs. 3-4). Our model yields a value of $p_{t}$ that increases with $\\ell$. In particular, over a broad range of $\\ell$, we find a simple linear increase of $p_{t}$ with $\\ell$ (Fig. 2a). Intuitively, this increase occurs because the surface adhesion of the model cell scales with its contact area, creating an energy barrier to reorientation that increases with cell length.\n\nTo determine whether this simple model can predict verticalization events in the biofilm surface-layer simulations, we examined the forces acting on individual modeled cells throughout the development of a biofilm. Specifically, we computed the reorientation ``surface pressure'' $p_{r}$, defined as the sum of the magnitudes of the in-plane cell-cell contact forces on a cell, normalized by the perimeter of its footprint, at the instant it begins to reorient. We determined the instant of reorientation as the time of the peak of the total in-plane force on a cell immediately prior to it becoming vertical (Supplementary Fig. 5). We found that the average reorientation surface pressure $\\langle p_{r} \\rangle$ increases with $\\ell$, as expected from the compressive instability model (Fig. 2a). Furthermore, the predicted value $p_t$ is in good agreement with the observed $\\langle p_{r} \\rangle$ for short cells. However, for long cells, $\\langle p_{r} \\rangle$ saturates more rapidly than $p_{t}$.\n\n\\begin{figure*}[t!]\n\\includegraphics[width=1.6\\columnwidth]{fig2.jpg}\n\\caption{\\label{fig:FG0} Mechanics of cell reorientation in modeled biofilms. (\\textbf{a}-\\textbf{b}) Properties of individual cells at the time $t_{r}$ of reorientation, defined as the time of the peak of total force on the cell prior to it becoming vertical. Analyses are shown for all reorientation events among different biofilms simulated for a range of initial cell lengths $\\ell_0$. (\\textbf{a}) Distributions of reorientation ``surface pressure'' $p_{r}$, defined as the total contact force in the $xy$ plane acting on a cell at time $t_{r}$, normalized by the cell's perimeter, versus cell cylinder length $\\ell$. The white dashed curve shows the average reorientation surface pressure $\\langle p_{r} \\rangle$ as a function of $\\ell$. The magenta dashed curve shows the theoretical prediction for $\\langle p_{r} \\rangle$ from linear stability analysis for a modeled cell under uniform pressure, depicted schematically in the inset. (\\textbf{b}) Distributions of the logarithm of reorientation torque $\\tau_{r}$, defined as the magnitude of the torque on a cell due to cell-cell contact forces in the $z$ direction at time $t_{r}$, for different cell cylinder lengths $\\ell$. The white dashed curve shows the average values $\\langle \\log \\tau_{r} \\rangle$ as a function of $\\ell$. The orange dashed curve shows the scaling prediction $\\tau_{r} \\sim \\ell^2$ from linear stability analysis for a modeled cell under torque, depicted schematically in the inset. (\\textbf{c}) Mean reorientation length $\\langle \\ell_{r} \\rangle$ (red), defined as the average value of cell length at $t_{r}$, and mean cell cylinder length $\\langle \\ell \\rangle$ (gray), defined as the average length of all horizontal cells over all times of biofilm growth, averaged over ten simulated biofilms, each with initial cell cylinder length $\\ell_0$, plotted versus $\\ell_0$. The inset shows the distribution of reorientation lengths (red) and horizontal surface-cell lengths (gray) for $\\ell_0 = \\SI{1}{\\micrometer}$. (\\textbf{d}) Mean avalanche size $\\langle N \\rangle$, defined as the average size of a cluster of reorienting cells that are proximal in space and time (Supplementary Figs. 8-10), versus initial cell length $\\ell_0$ for the experimental biofilm (red triangle) and the modeled biofilm (red circles). Open gray triangle and circles indicate the corresponding mean avalanche sizes for a null model. Inset shows a side view of cell configurations in the $xy$ plane at times $t_{r}$ for all reorientation events in a simulated biofilm with $\\ell_0 = \\SI{2.5}{\\micrometer}$. Reorientation events are colored alike if they belong to the same avalanche. Scale bars: $\\SI{10}{\\micrometer}$ and $1$ hour.\n}\n\\end{figure*}\n\n\\subsection*{The dominant mechanism of verticalization depends on cell length}~\n\nHow can longer cells become vertical at surface pressures much lower than the threshold values predicted by the compressive instability model? The large extent of the discrepancy suggests that for long cells, the in-plane forces alone are insufficient to cause the instabilities. Indeed, incorporating the numerically-observed distribution of in-plane forces acting on cells into the compressive instability model does not significantly improve the prediction for $\\langle p_r \\rangle$ (Supplementary Fig. 6). Thus, we hypothesized that in the case of long cells, forces acting in the $z$ direction might play an important role in triggering verticalization.\n\nTo explore this idea, we returned to the single-cell model and considered the effect of forces in the $z$ direction (inset Fig. 2b). Applying small forces in the $z$ direction to a cell with fixed center-of-mass height shifts the equilibrium elevation angle of the cell to a small finite value of $\\theta$, proportional to the net torque. Under large enough torque, a single end of the cell becomes free of the surface, at which point the cell becomes unstable to further rotation and effectively ``peels'' off the surface. This nonlinearity, inherent in the geometry of contact, competes with the compressive instability, and under specific conditions can initiate reorientation at much smaller values of surface pressure. Specifically, for a fixed center-of-mass penetration depth $\\delta_0 \\ll \\ell$, the threshold torque for peeling a cell off the surface due to forces in the $z$ direction scales as $\\tau_t \\sim \\ell^2$ (Supplementary Fig. 4). For the whole-biofilm surface-layer simulation, the average reorientation torque, defined as the total torque due to forces in the $z$ direction at the instant of reorientation, closely obeys the predicted $\\ell^2$ scaling for long cells (Fig. 2b). Taken together, our predictions from the compressive and peeling instabilities explain the verticalization of cells over the entire range of cell lengths studied.\n\n\\subsection*{Cell division can trigger verticalization}~\n\nFor both compressive and peeling instabilities, the presence of an energy barrier to reorientation explains the sharp distinction we observed between horizontal and vertical cell orientations. Furthermore, both mechanisms predict larger reorientation thresholds for longer cell lengths. Hence, the model suggests that shorter cells should reorient more readily. To confirm this effect in our simulated biofilms, we compared the distribution of reorientation lengths $\\ell_{r}$, defined as the cell cylinder length at the instant of reorientation, to the full distribution of horizontal cell lengths for a series of simulations with different values of the initial cell cylinder length $\\ell_0$ (Fig. 2c). For all values of $\\ell_0$, we found that the mean reorientation cell length $\\langle \\ell_{r} \\rangle$ is substantially smaller than the mean horizontal cell length. In addition, for simulations with average cell lengths comparable to those in our experiments, most reorientation events occur immediately after cell division. The limited time resolution of the experiments precludes a quantitative analysis of division-induced verticalization; nevertheless, the propensity for cell division to trigger reorientation is clearly observed in the experimental biofilm (Supplementary Video 1, Supplementary Fig. 7).\n\n\\begin{figure*}[t!]\n\\includegraphics[width=1.5\\columnwidth]{fig3.jpg}\n\\caption{\\label{fig:FG0}\nTwo-component fluid model for verticalizing cells in biofilms. (\\textbf{a}) Schematic illustration of the two-component continuum model. Horizontal cells (blue) and vertical cells (red) are modeled, respectively, by densities $\\rho_h$ and $\\rho_v$ in two spatial dimensions. The total cell density $\\tilde{\\rho}_{\\mathrm{tot}}$ is defined as $\\rho_h + \\xi \\rho_v$, where $\\xi$ is the ratio of vertical to horizontal cell footprints. (\\textbf{b}) Radial densities $\\rho$ of vertical cells ($\\rho_v$, red), horizontal cells ($\\rho_h$, blue), and total density ($\\tilde{\\rho}_{\\mathrm{tot}}$, black), versus shifted radial coordinate $\\tilde{r}$, defined as the radial position relative to the boundary between the mixed interior and the horizontal cell periphery. Results are shown for the continuum model (left; radial cell density in units of $\\SI{}{\\micrometer}^{-2}$), the experimental biofilm (middle; radial cell density in each $\\SI{}{\\micrometer}$-sized bin averaged over an observation window of 50 minutes), and the agent-based model biofilm (right; radial cell density in each $\\SI{}{\\micrometer}$-sized bin averaged for ten biofilms over an observation window of $6$ minutes). For the continuum model and the agent-based model biofilms the parameters were chosen to match those obtained from the experiment (Supplementary Figs. 12-13).\n}\n\\end{figure*}\n\n\\subsection*{Verticalization is localized}~\n\nWe next investigated how the surface compression and peeling instabilities influence the propagation of reorientation through a biofilm. First, we generalized our model for the surface compression instability to the multi-cell level. A linear stability analysis of the model suggests that reorientation events should be independent and spatially localized for short cell lengths (Supplementary Fig. 3). By contrast, for long cell lengths, the tendency of neighboring vertical cells to trigger reorientation suggests an (inverse) domino-like effect in which one cell standing up can induce its neighbors to stand up.\n\nTo quantify the extent of cooperative verticalization, we computed the size of reorientation ``avalanches'', defined as groups of verticalization events that are proximal in space and time\\cite{ava} (inset Fig. 2d, Supplementary Fig. 8). We found that the mean avalanche size increases with cell length, consistent with the prediction of the inverse domino effect for long cells (Fig. 2d). Interestingly, however, the distribution of avalanche sizes decays roughly exponentially for all values of cell length we studied (Supplementary Fig. 9), with only a modest number of cells ($N \\sim 1-3$) involved in typical avalanches (Fig. 2d). Our results indicate that the sizes of reorientation avalanches are limited by an emergent spatiotemporal scale governed by cell geometrical and mechanical properties, rather than by the growing total supply of horizontal cells.\n\nA natural explanation for the small mean avalanche size comes from the reduction in cell footprint that occurs upon reorientation, which rapidly alleviates the local surface pressure responsible for verticalization (Supplementary Fig. 5). This effect combines with the disorder of the contact geometries and forces throughout the biofilm, which separates horizontal cells near the verticalization threshold into disconnected groups (Supplementary Video 3, Supplementary Fig. 10). Thus, although the inverse domino effect transiently increases verticalization cooperativity, avalanches quickly exhaust the supply of nearby horizontal cells that are susceptible to becoming vertical. Consequently, verticalization occurs throughout the biofilm in scattered, localized regions.\n\n\\subsection*{Two-fluid model describes propagation of verticalization}~\n\nTo understand how localized cell verticalization gives rise to the global patterning dynamics of the biofilm, we developed a two-dimensional continuum model that treats horizontal and vertical cell densities as two coupled fluids (Fig. 3a, Methods). The local horizontal cell density $\\rho_h$ grows in the plane at a rate $\\alpha$ and converts to vertical cell density $\\rho_v$ in regions of high surface pressure or, equivalently in our model, high total 2D cell density $\\tilde{\\rho}_{\\mathrm{tot}} \\equiv \\rho_h + \\xi \\rho_v$, where $\\xi$ is the ratio of vertical to horizontal cell footprints. These interactions yield the following equation for the change in $\\tilde{\\rho}_{\\mathrm{tot}}$ in regions of nonzero surface pressure:\n\\setlength{\\abovedisplayskip}{12pt}\n\\setlength{\\belowdisplayskip}{12pt}\n\\begin{equation}\n\\dot{\\tilde{\\rho}}_{\\mathrm{tot}} = \\gamma \\nabla^2 \\tilde{\\rho}_{\\mathrm{tot}} + \\alpha  \\rho_h - (1 - \\xi) \\beta \\Theta(\\tilde{\\rho}_{\\mathrm{tot}} - \\tilde{\\rho}_t) \\rho_h,\n\\end{equation}\n\n\\noindent where $\\gamma$ is the ratio of the Young's modulus of the biofilm to the surface drag coefficient, $\\Theta$ is the Heaviside step function, and $\\tilde{\\rho}_t$ is the threshold surface density for verticalization. We simulated this continuum model, and found that for $\\alpha < \\beta$, the biofilm generically develops into a circular region containing both horizontal and vertical cells (``Mixed interior'') surrounded by an annular region containing horizontal cells (``Horizontal cell periphery''), closely matching both the experimental biofilm and the agent-based model biofilms (Fig. 3b, Supplementary Video 4, Supplementary Note). In this regime, the biofilm front spreads linearly in time at a fixed expansion speed $c^*$. Furthermore, the total cell density and the surface pressure are constant in the mixed interior. This constancy is stabilized by the competing effects of cell growth and cell verticalization, and occurs provided that $\\alpha < \\beta (1 - \\xi)$ (Supplementary Fig. 11). When this condition is satisfied, verticalization can reduce cell density faster than cell density can be replenished by cell growth and cell transport due to gradients in surface pressure. Physically, this results in the cell density rapidly fluctuating around the verticalization threshold. This rapid alternation effectively tunes the verticalization rate down to $\\alpha = \\beta (1 - \\xi)$, and thereby ensures a constant total cell density and surface pressure in the mixed interior. The resulting ``dynamical isobaricity'' (constancy of pressure) provides the boundary condition for the horizontal cell periphery that determines the horizontal expansion speed $c^*$, independent of $\\beta$ and $\\xi$. In the limit of slow expansion $c^* \\ll \\sqrt { \\alpha \\gamma } $, $c^*$ is given by:\n\\setlength{\\abovedisplayskip}{12pt}\n\\setlength{\\belowdisplayskip}{12pt}\n\\begin{equation}\nc^* \\simeq c^*_0 \\sqrt{  1 - \\frac{\\tilde{\\rho}_0}{\\tilde{\\rho}_{t}}  },\n\\end{equation}\n\\noindent where $c^*_0 = \\sqrt{2 \\alpha \\gamma}$ and $\\tilde{\\rho}_0$ is the close-packed, but uncompressed, cell density. Thus, we find that $c^*$ increases with $\\tilde{\\rho}_{t}$ until it saturates to a maximum speed $c^*_0$ for $\\tilde{\\rho}_{t} \\gg \\tilde{\\rho}_0$. Intuitively, higher values of the verticalization threshold density $\\tilde{\\rho}_{t}$ sustain a wider periphery of horizontal cells, which results in a higher rate of increase in the total number of surface cells. Thus, our continuum model reveals how the geometrical and mechanical properties of individual cells influence the global morphology of the growing biofilm.\n\n\\begin{figure}[t!]\n\\includegraphics[width=1\\columnwidth]{fig4.jpg}\n\\caption{\\label{fig:FG0}\nGlobal morphological properties of experimental and modeled biofilms. (\\textbf{a}) Top-down (upper row) and side views (lower row) of experimental biofilms grown with $\\SI{0.4}{\\microgram\/\\milli\\liter}$ A22 (magenta), without treatment (yellow), and with $\\SI{4}{\\microgram\/\\milli\\liter}$ Cefalexin (cyan), following overnight growth (upper row) and 7 hours after inoculation (lower row). Scale bar: $\\SI{10}{\\micrometer}$. (\\textbf{b}) Expansion speed $c^*$, defined as the speed of the biofilm edge along the surface, versus the initial cell cylinder length $\\ell_0$ for experimental biofilms (A22, magenta; no treatment, yellow; Cefalexin, cyan), agent-based model biofilms (black circles), and continuum model (dashed black curve). Expansion velocities were determined from a linear fit of the basal radius $R_{B}$ of the biofilm versus time, where $R_{B}$ is defined at each time point as the radius of a circle with area equal to that of the biofilm base. For experimental biofilms, the boundary was extracted from the normalized fluorescence data (see Methods for details). For each treatment, the vertical error bars show the standard error of the mean of the expansion speed and the horizontal error bars bound the measured initial cell cylinder length (Supplementary Fig. 1). Inset: model cells with lengths and radii corresponding to the averages for different treatments. (\\textbf{c}) Biofilm aspect ratio $H\/R_{B}$ for experimental biofilms grown under different treatments, where the biofilm height is defined as $H=3V\/2R_{B}^2$, the height of a semi-ellipsoid with a circular base of radius $R_{B}$ and volume $V$ equal to that of the biofilm. Color designations and treatments same as in panel (\\textbf{a}). \n}\n\\end{figure}\n\n\\subsection*{Increasing cell length yields more rapidly expanding, flatter biofilms}~\n\nBecause the threshold surface density for verticalization $\\tilde{\\rho}_{t}$ increases with cell length, we expect biofilms composed of longer cells to maintain a wider periphery of horizontal cells and to thereby expand faster along the surface than biofilms composed of shorter cells. To test this notion in our agent-based model, we computed the expansion speed of the modeled biofilms for a range of initial cell cylinder lengths $\\ell_0$ (Fig. 4b). Upon fitting the continuum model parameters to those of the agent-based model (Supplementary Figs. 12-13), we found that the expansion velocities of the two models were equal to within a few percent. In living, experimental biofilms, we can increase or decrease the average cell length using chemical treatments\\cite{R5,T7} (Fig. 4a, top row). Similar to the agent-based model biofilms, in experimental biofilms, the surface expansion speed increases with increasing cell length (Fig. 4b, Supplementary Video 5). The experimentally observed speed appears to saturate as cell length is increased, as occurs in the modeled biofilms. Furthermore, when the model biofilm parameters are fitted to experiment (Supplementary Fig. 1), the experimental and model biofilm speeds agree to within twenty percent or better. Taken together, these observations support the conclusion that self-organized dynamical isobaricity governs the observed expansion of \\emph{V. cholerae} biofilms.\n\nHow do different surface expansion speeds influence the ensuing biofilm development into the $z$ direction? After a few hours, living biofilms grow into roughly semi-ellipsoidal shapes with volume $V=(2\/3)R_{B}^2 H$, where $R_{B}$ is the basal radius and $H$ is the height. For equal rates of total volume growth, we expect a biofilm that expands more rapidly along the surface to develop a lower aspect ratio $H\/R_{B}$ than a biofilm that expands less rapidly along the surface. We verified that this trend holds for the experimental biofilms (Fig. 4a, bottom row). In particular, the measured aspect ratio $H\/R_{B}$ increases with cell length over a wide range of volumes (Fig. 4c). Thus, our results show how the elongated geometries of individual cells govern the global morphology of the collective.\n\n\\subsection*{Discussion}~\n\nBacterial biofilms are pervasive lifeforms that significantly influence health and industry\\cite{R22, R21, R20, R23,F1,KKLC}. An important step towards control over biofilms was achieved when the molecular building blocks of \\emph{V. cholerae} biofilms were identified\\cite{R25}. In particular, cell-to-surface adhesion factors were found to be necessary to generate vertically-ordered biofilm clusters\\cite{R4}. Despite this progress, the dynamical process by which cells in biofilms become vertical has remained mysterious. Here, we showed that cell verticalization begins to occur when the local effective surface pressures that arise from cell growth become large enough to overcome the cell-to-surface adhesion that otherwise favors a horizontal orientation. Subsequently, the reduction in cell footprint that occurs upon cell verticalization, which acts to reduce the effective surface pressure, provides a mechanical feedback that controls the rate of biofilm expansion. Our continuum and agent-based models quantitatively capture the rate of horizontal expansion of experimental biofilms, and also predict the observed changes in the height-to-radius aspect ratio that occur with varying average cell length. \\\\\n\\indent Our results suggest that bacteria have harnessed the physics of mechanical instabilities to enable the generation of complex architectures. We expect that individual cell parameters have evolved in response to selective pressures on global biofilm morphology, e.g. during resource competition\\cite{R12, R13,R16,T1,KCH}. Since optimal morphology may be condition dependent, cells may also have evolved adaptive strategies that alter biofilm architecture, which could be investigated experimentally by screening for environmental influences on cell size, shape, and surface adhesion\\cite{T9}. \\\\\n\\indent For simplicity, we focused on flat surfaces, nutrient-rich conditions, and \\emph{V. cholerae} strains that have been engineered to have simpler interactions than those in wild type biofilms (Methods). Moreover, our agent-based model does not explicitly incorporate the VPS matrix secreted by cells\\cite{R21, osm, R15}. Understanding the modifying effects of the VPS matrix, cell and surface curvature (Supplementary Fig. 14), cell-to-cell adhesion (Supplementary Fig. 15), and chemical feedback\\cite{T2} will be important directions for future studies. More broadly, we must develop a systematic method to account for the diversity of architectures that can be produced by local mechanical interactions (Supplementary Discussion). \\\\\n\\indent Our study of a two-fluid model for verticalizing biofilms led us to discover a novel type of front propagation. Interestingly, in the biofilm surface layer, the front profile of cell density is precisely uniform starting at some finite distance from the edge, whereas previous models of front propagation saturate asymptotically toward uniformity \\cite{T4, T5, T6, BS}. The self-organized nature of this process yields a universal dependence of the expansion speed on the cell geometrical and mechanical parameters that is robust to details of the mechanical feedback. We have focused on the mean-field behavior of biofilms, but an open question is to understand the role of fluctuations in the ``pressure'' acting on cells, e.g. either from a jamming perspective\\cite{R29}, a fluctuating hydrodynamical perspective\\cite{T3, MLM}, or a combination of approaches. \\\\\n\\indent In summary, we have elucidated the physical mechanism underlying a complex developmental program observed at the cellular scale in bacterial biofilms. The relative biochemical and biophysical simplicity of this prokaryotic system allowed us to quantitatively understand the developmental pathway from the scale of a single cell to the scale of a large community assembly. Going forward, we expect bacterial biofilms will take on increasingly important roles as tractable models that can be used to understand how living systems generate and maintain their structures.\n\n\n\\part*{\\centerline{Supplementary Figures}}\n\n\\section{\\textbf{Schematic illustration of agent-based model}}\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{fig_s1.jpg}\n\\caption{\\label{fig:FG0}\nCell model. Schematic of model cells, showing a horizontal cell (elevation angle $\\theta = 0$, left), a cell that is angled with respect to the surface (middle), and a cell that is vertical ($\\theta = \\pi\/2$, right). Cells are modeled as cylinders of length $\\ell$ with two hemispherical endcaps of radius $R$. The cell orientation is specified by the unit vector $\\boldsymbol{\\hat{n}}$. The direction normal to the surface is specified by the unit vector $\\boldsymbol{\\hat{z}}$. The distance along the cell cylinder is parameterized by the coordinate $s$, which is zero at the cell's center of mass.\n}\n\\end{figure}\n\n\n\\subsection*{Cell model}\n\nWe model each cell as a cylinder of length $\\ell$ with hemispherical endcaps of radius $R$ (``spherocylinders'', Supplementary Fig. 1). In total, the volume $V$ of a model cell is therefore:\n\n\\begin{equation}\nV = \\frac{4}{3}\\pi R^3 + \\pi R^2 \\ell.\n\\end{equation}\n\n\\noindent We treat cell growth as an increase in $\\ell$ at a fixed radius $R$ with total volume growing at a rate $\\alpha$:\n\n\\begin{equation}\n\\frac{dV}{dt} = \\alpha V.\n\\end{equation}\n\n\\noindent Thus, the rate of increase of cylinder length $\\ell$ is given by:\n\n\\begin{equation}\n\\frac{d\\ell}{dt} = \\alpha \\left( \\frac{4R}{3} + \\ell \\right),\n\\end{equation}\n\n\\noindent which results in the following growth equation for $\\ell$:\n\n\\begin{equation}\n\\ell = e^{\\alpha t} \\left( \\ell_0 + \\frac{4R}{3} \\right) - \\frac{4R}{3},\n\\end{equation}\n\n\\noindent where $\\ell_0$ is the initial cell cylinder length.\n\nWe treat cell division as an instantaneous conversion of the mother cell into two daughter cells that occupy the same total cell length. Specifically, the mother cell at position $\\boldsymbol{r}$ with orientation $\\boldsymbol{n}$ is replaced by two daughter cells of cylinder length $\\ell_0$ at positions $\\boldsymbol{r} \\pm (R+\\ell_{0}\/2)\\boldsymbol{n}$ with the same orientations $\\boldsymbol{n}$. The condition that the daughter cells occupy the same total cell length as the mother cell requires division to occur at a final cell cylinder length $\\ell = 2\\ell_0 + 2R$. This treatment results in a doubling time $t_{\\mathrm{double}}$:\n\n\\begin{equation} \\label{eq:td}\nt_{\\mathrm{double}} = \\frac{1}{\\alpha} \\log \\left( \\frac{10R + 6\\ell_0}{4R + 3\\ell_0} \\right).\n\\end{equation}\n\n\\noindent Replacing the mother cell with two daughter cells in this manner results in a modest loss of total cell volume, but importantly, it does not increase the overlap with any neighboring cells. This division protocol was chosen to avoid introducing non-physical impulses that might alter reorientation dynamics.\n\n\\subsection*{Cell-to-cell repulsion}\n\nBacterial cells maintain their shape due to the presence of the cell wall. Although the cell wall itself is rigid, it is coated by soft materials such as cell-bound extracellular polysaccharides (EPS)\\cite{S3}. These extracellular bio-components can deform elastically when cells encounter obstacles such as other cells or external surfaces, and these deformations produce repulsive pushing forces. To treat this elastic interaction, we employ the Hertzian theory of mechanical contact\\cite{S1}. The elastic interaction between two cells, $i$ and $j$, has an energy that scales with the cell-cell overlap $\\delta_{ij}$, defined for our model cells as $2R$ minus the smallest distance between the centerlines of the cell cylinders. For generic contact geometries of two spherocylinders, the contact energy is given by:\n\n\\begin{equation}\nE_{\\mathrm{cell-cell},ij} =E_0 R^{1\/2} \\delta_{ij}^{5\/2},\n\\end{equation}\n\n\\noindent for $\\delta_{ij}>0$ and $0$ for $\\delta_{ij}<0$ (i.e. $0$ for cells not in contact), where $E_0$ is the cell stiffness.\n\n\\subsection*{Cell-to-surface interactions}\n\nDuring biofilm growth, cells may interact with the surface. When a cell presses against the surface, the surface exerts a repulsive force against the cell. On the other hand, cells can secrete surface adhesion proteins Bap1\/RbmC that coat the surface\\cite{S3} and produce attractive forces. We therefore model cell-to-surface contact as a combination of repulsive and attractive interactions. To match the experimental surface geometry, which consists of a relatively flat and homogeneous surface, we model the surface as an infinite, two-dimensional plane located at $z=0$. We take the normal vector of the surface to point along the $z$-direction, which defines the vertical direction (Supplementary Fig. 1).\n\nWe treat the pushing interaction between cells and the surface analogously to the cell-to-cell interactions described above, but with a contact interaction that acts along the entire length of the cell. Specifically, the elastic contribution to the cell-to-surface contact energy is given by the integral of the elastic contact energy density along the centerline of the cell cylinder. In what follows, we parameterize the centerline as the set of points given by $\\boldsymbol{r} + s \\boldsymbol{n}$, where $r$ is the position of the cell center, $\\boldsymbol{n}$ is a unit vector that specifies the cell orientation, and $s$ is a coordinate that runs from $-\\ell\/2$ to $\\ell\/2$ (Supplementary Fig. 1). Thus, the overlap $\\delta(s)$ of each infinitesimal segment at $s$ with the surface is given by:\n\n\\begin{equation}\n\\delta(s)=R- (z+s \\boldsymbol{n} \\cdot \\boldsymbol{\\hat{z}}),\n\\end{equation}\n\n\\noindent for $\\delta(s) >0$ and 0 otherwise (i.e. $0$ for points not in contact), where $z$ is the height of the cell center.\n\nFurthermore, we also account for changes in the cell-to-surface contact geometry as the cell is reoriented (Supplementary Fig. 1). In the limit that the cells are completely horizontal, i.e., when $\\boldsymbol{n} \\cdot \\boldsymbol{\\hat{z}}=0$, we treat the contact geometry of the integrated surface interaction as that of the contact between a horizontal cylinder and a plane. For completely vertical cells, we treat the contact geometry of the integrated surface interaction as the contact between a sphere and a plane. For generic values of the cell orientation, the contact geometry is given by a sum of both cylindrical and spherical contributions weighted by a smooth crossover function that depends on the cell orientation $\\boldsymbol{n}$, or equivalently, on the angle $\\theta = \\sin^{-1}(\\boldsymbol{n} \\cdot \\boldsymbol{\\hat{z}})$ between the cell and the surface. The crossover functions are chosen to be sinusoidal in $\\theta$, as these are the simplest functions that preserve the scaling of contact energies with contact penetration for linear deviations around the horizontal and vertical orientations of the cell. Taken together, the contribution to the energy of cell $i$ due to its elastic cell-to-surface interactions is given by:\n\n\\begin{equation}\nE_{\\mathrm{el},i} = E_0 R^{1\/2} \\delta_{i}^{5\/2},\n\\end{equation}\n\n\\noindent where $\\delta_{i}^{5\/2}$ is given by:\n\n\\begin{equation}\n\\delta_i^{5\/2} = \\int^{s\/2}_{-s\/2}  \\left[  R^{-1\/2} \\cos^2 (\\theta) \\delta^2(s) + \\frac{4}{3} \\sin^2 (\\theta) \\delta^{3\/2}(s) \\right] ds.\n\\end{equation}\n\nTo model the cell-to-surface adhesion interaction\\cite{S3}, we assume that each infinitesimal segment in contact with the surface provides a constant energy $-\\Sigma_0$ per unit of contact area, according to the Derjaguin approximation\\cite{S2}. The total contribution to the energy of cell $i$ due to cell-to-surface adhesion is given by:\n\n\\begin{equation} \nE_{\\mathrm{ad},i} = -\\Sigma_0 A_i,\n\\end{equation}\n\n\\noindent where $A_i = \\int^{s\/2}_{-s\/2} a(s) ds$ is the total contact area as a function of the contact area density $a(s)$ given by:\n\n\\begin{equation}\na(s) = R^{1\/2} \\cos^2 (\\theta) \\delta^{1\/2}(s) + \\pi R \\sin^2 (\\theta) \\Theta( \\delta(s)) ,\n\\end{equation}\n\n\\noindent where $\\Theta$ is the Heaviside step function. In the above expressions for $\\delta_i^{5\/2}$ and $a(s)$, we have incorporated the appropriate geometrical factors and scaling exponents for spherical and cylindrical Hertzian contacts. Thus, the total cell-to-surface energy $E_{{s},i}$ is given by:\n\n\\begin{equation} \\label{eq:s}\nE_{{s},i} = E_{\\mathrm{el},i} + E_{\\mathrm{ad},i}\n\\end{equation}\n\nWhen all points of the cell's centerline are separated from the surface by distances larger than $R$, i.e. when the cell is detached from the surface, $E_{{s},i}$ is zero and the surface does not exert any force on the cell. In contrast, when the cell is in contact with the surface, the surface exerts both repulsive and attractive forces on the cell. In the absence of external forces, the competition between these opposing forces results in a stable fixed point at $\\theta_0=0$, i.e., the cell is horizontal, and the penetration $\\delta_0$ is given by:\n\n\\begin{equation}\n\\delta_0 = \\frac{1}{R}\\left( \\frac{R^2 \\Sigma_0}{4 E_0} \\right)^{2\/3}.\n\\end{equation}\n\n\\subsection*{Viscosity}\n\nCell motion is strongly opposed by drag from both its three-dimensional environment, including the surrounding ambient fluid and the polymer matrix, as well as by friction from the surface. For simplicity, we treat both of these effects via Stoke's drag terms that oppose the motion of each infinitesimal segment of the cell cylinder's centerline. For the ambient fluid, the density of the drag force along the centerline is taken to be proportional to $\\eta_0 \\boldsymbol{v}(s)$, where $\\eta_0$ is the ambient viscosity and $\\boldsymbol{v}(s)$ is the velocity of the segment at centerline position $s$:\n\n\\begin{equation}\n\\boldsymbol{v}(s) = \\dot{\\boldsymbol{r}} + s \\dot{  \\boldsymbol{\\hat{n}}  },\n\\end{equation}\n\n\\noindent where the dot indicates the time derivative. To model the effect of the surface drag, we take the drag force provided by the surface to oppose the segment's motion tangential to the surface. Furthermore, we assume that the surface viscosity of a contacting segment is proportional to its contact area density $a(s)$ as given by the Hertzian contact geometry detailed above. The combination of ambient drag and surface drag corresponds to the following dissipation function:\n\n\\begin{equation} \\label{eq:sd}\nP_i = \\frac{1}{2} \\int_{-\\ell \/2}^{\\ell \/ 2} \\left( \\eta_0 \\boldsymbol{v}^2(s) + \\frac{\\eta_1 a(s)}{R} \\left[ \\boldsymbol{v}(s) - (\\boldsymbol{v}(s) \\cdot \\hat{z}) \\hat{z} \\right]^2 \\right) ds,\n\\end{equation}\n\n\\noindent where $\\eta_1$ is the surface drag coefficient.\n\n\\subsection*{Equations of motion}\n\nTaken together, the above interactions determine the equations of motion for the model cells. For a collection of cells, we define the total energy $E$ as follows:\n\n\\begin{equation}\nE( \\{ \\boldsymbol{q}_i \\} ) = \\sum_i \\left( E_{\\mathrm{el},i} + E_{\\mathrm{ad},i} \\right)  + \\sum_{ i \\neq j } \\left( E_{\\mathrm{cell-cell},ij} \\right),\n\\end{equation}\n\n\\noindent where $\\boldsymbol{q}_i = \\{\\boldsymbol{r}_i, \\boldsymbol{\\hat{n}}_i \\}$ is the generalized coordinate vector of cell $i$. We compute the equations of motion for cell $i$ using Lagrangian mechanics as follows:\n\n\\begin{equation}\n\\frac{\\delta P_i}{\\delta \\dot{\\boldsymbol{q}_i}} = -\\frac{\\delta E_i}{\\delta \\boldsymbol{q}_i} + \\lambda \\boldsymbol{\\hat{n}}_i \\frac{d\\boldsymbol{\\hat{n}}_i}{d\\boldsymbol{q}_i},\n\\end{equation}\n\n\\noindent where $\\lambda$ is a Lagrange multiplier introduced to account for the constraint $\\boldsymbol{\\hat{n}} \\cdot \\boldsymbol{\\hat{n}} = 1$ on the cell orientation vector.\n\n\\subsection*{Choice of parameters}\n\nWe determined the parameters in our agent-based model by fitting them to experimental data:\n\n\n\\begin{itemize}\n\\item {Initial cell cylinder length $\\ell_0$: In our cell model, the region enclosed by the spherocylinder represents the portion of the cell enclosed by the cell membrane and cell wall as well as the biopolymer coating that surrounds the cell. Since our experiment does not image the cell wall or coating directly, we determine the cell shape parameters $\\ell_0$ and $R$ by fitting to simulations of the agent-based model. To do so, we identify the length of the rigid cell cylinder of the model cell with the length of the cell cylinder of the model cell plus an offset due to the presence of the biopolymer coating. Physically reasonable values of the offset lie between $0$ and $R$. Therefore, we report the cell cylinder length as the value in the center of this range, with the full range giving the error bars. In practice, we measure the initial cell cylinder length by first recording the average cell length from the experimental images, e.g. top row of Fig. 4a. We then determine the average cell length of the modeled cells by computing the average cell length as a function of the initial cell length $\\ell_0$ (Fig. 2c). This function provides a mapping from the average experimental cell length to the initial cell length of the modeled cells.}\n\\item{ Cell spherocylinder radius $R$: We determine the radius $R$ from the experimental cell density near the edge of the biofilm (where cells are close-packed but under negligible compression). The cell radial density near the edge of the experimental biofilm in Fig. 1 is roughly $0.16$ cells per square micron (Fig. 3b), which is achieved by an agent-based model with $\\ell_0=1.25R$ and $R=\\SI{0.8}{\\micrometer}$. The cell radius does not change significantly for the different drug conditions.}\n\\item{ Cell stiffness $E_0$: The cell stiffness is approximated as $E_0 = Y \/ (2-2\\nu^2)$, where $Y$ is the Young's modulus, and $\\nu$ is the Poisson ratio, in accordance with contact mechanics. These elastic parameters correspond to the effective material properties of the cell, which is a composite of the hard core of the cell starting at the cell wall and the soft biopolymer coating surrounding the cell. Since the cell wall is very rigid compared to the biopolymer coating, the elastic properties of cell interactions are primarily determined by the latter. The Young's modulus and Poisson ratio were measured using bulk rheology to be $Y\\simeq \\SI{450}{\\pascal}$ and $\\nu \\simeq 0.49$.}\n\\item{ Cell growth rate $\\alpha$: To model the noise in cell growth rate, we assign a random value of $\\alpha$ to each cell upon birth. Specifically, we take $\\alpha$ to be a random variable drawn from a Gaussian distribution. The mean value of $\\alpha$ is determined from experiment by first measuring the average doubling time. The average doubling time for all the experimental colonies, (including those treated with drugs) is roughly $t_{\\mathrm{double}} \\sim 35$ minutes. From this value, we determine $\\langle \\alpha \\rangle$ using Supplementary Eq. \\ref{eq:td} above. The standard deviation of $\\alpha$ is chosen to be $0.2 \\langle \\alpha \\rangle$, to ensure that cell division events throughout the biofilm become desynchronized over times comparable to those observed in experiment.}\n\\item { Cell ambient viscosity $\\eta_0$: We set the ambient viscosity $\\eta_0$ equal to the biofilm viscosity measured from bulk rheology, which yields $\\eta_0 \\simeq \\SI{20}{\\pascal\\second}$.}\n\\item { Cell surface drag coefficient $\\eta_1$: We estimated $\\eta_1$ using microfluidics. To do so, we inoculated cells in a microfluidic chamber at a low density, which allowed us to image isolated, individual cells adhered to the surface using high resolution microscopy. We subsequently gradually increased the flow rate until we observed cell motion. From the observed motion, we estimated the surface drag as $\\eta_1 \\sim F_{{s}} \\Delta t \/ \\Delta x$, where $F_{{s}}$ is the estimated force on the cell due to the effect of shear flow, $\\Delta t$ is the duration of a short observation window, and $\\Delta x$ is the distance traveled by the cell during the window. We estimate the force as $F_{{s}} = \\eta_0 A (du\/dz)$, $A$ is the cell footprint assuming a horizontal configuration and $du\/dz$ is the derivative of the flow velocity in the direction along the channel, with the derivative calculated along the direction transverse to the surface and evaluated at the cell midline. This estimate yields a value $\\eta_1 \\simeq 2\\cdot 10^5\\ \\SI{}{\\pascal \\second}$. }\n\\item { Cell adhesion $\\Sigma_0$: The contact adhesion energy is chosen to match the onset time of verticalization for the modeled biofilm to that of the experimental biofilm (Fig. 1c,d). The corresponding value of $\\Sigma_0$ yields a penetration depth $\\delta_0 \\simeq 0.04 R$. }\n\\end{itemize}\n\n\n\\newpage\n\n\\section{\\textbf{Validation of quasi-3D approximation of the agent-based model}}\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{fig_s2.jpg}\n\\caption{\\label{fig:FG2}\nVarying the ratio of ambient viscosity to surface viscosity for modeled biofilms. (\\textbf{a-d}) visualizations of full 3D agent-based model biofilms at time $t=250$ minutes, showing horizontal (blue) and vertical (red) cells and the surface (brown) for ambient viscosity $\\eta_0 = \\SI{20}{\\pascal\\second}$ and ratios $\\eta_0 \/ \\eta_1$ of ambient viscosity to surface viscosity (\\textbf{a}) $10^{-5}$, (\\textbf{b}) $10^{-4}$, (\\textbf{c}) $10^{-3}$, and (\\textbf{d}) $10^{-2}$. (\\textbf{e-h}) Distributions $P(n_z)$ of cell verticality $n_z$ at time $t=250$ minutes for surface-adhered cells in the full 3D agent-based model biofilms in (\\textbf{a-d}, black). The green curve in (\\textbf{f}) shows $P(n_z)$ for the quasi-3D agent-based model biofilm reported in the main text. The inset of (\\textbf{f}) shows visualization of the quasi-3D agent-based model biofilm at time $t=250$ minutes, showing horizontal (blue) and vertical (red) cells and the surface (brown) for ambient viscosity $\\eta_0 = \\SI{20}{\\pascal\\second}$ and ratio of ambient viscosity to surface viscosity $\\eta_0\/\\eta_1 = 10^{-2}$.}\n\\end{figure}\n\nSimulating mature three-dimensional biofilms requires prohibitively large amounts of computational time for systematic studies, due to the exponential growth of cells combined with the large separation of scales between the ambient and surface drag. However, for the purposes of describing the verticalization transition, it is sufficient to consider only the dynamics of the surface layer (see Results). Therefore, we developed a quasi-3D simplification of the full 3D agent-based model to make the computations tractable. This quasi-3D model exploits the large separation of scales between the ambient and surface drag by removing from the simulation cells that become detached from the surface. Since the cells on the surface are all subject to the large surface drag, the overall variation in forces throughout the biofilm is substantially reduced, which significantly lowers the computational time required to grow a biofilm surface layer of a given size. Provided the number of layers of cells above the surface layer is small, the forces exerted by these cells on the surface cells are negligible compared to the forces on surface cells produced by other surface cells, and so we expect this approximation to be accurate at early times.\n\nTo verify that this quasi-3D model is a reasonable simplification of the full 3D model, we directly compared the orientation patterning of both models for small biofilm sizes (Supplementary Fig. 2). We found that removing the detached cells results in a slightly narrower peak of the vertical cell orientation distribution. A simple explanation for this effect is that modest deviations of the cell orientation from a completely vertical orientation are not strongly constrained by the surface pressure, and thus the small forces exerted by detached cells in the full 3D simulations are enough to cause such deviations. To account for this feature, in simulations in which we remove surface-detached cells, we employ a larger effective value of the ambient viscosity $\\eta_0 = 10^{-2} \\eta_1$ to match the orientational distribution observed in the full, 3D simulations (Supplementary Fig. 2). This variant model can reproduce the orientational patterning of the surface layer in the full 3D model for smaller biofilms, as well as the orientational patterning of the surface layer in the experimental biofilm throughout the full duration of the experiment.\n\n\\newpage\n\n\\section*{\\textbf{Supplementary Figures 3 and 4: Models for cell instabilities}}\n\nCell verticalization events are triggered by mechanical instabilities. In this section, we elucidate the physical mechanisms underpinning cell instabilities by investigating a series of minimal models. We first consider a line of cells under compression and show that instabilities are localized. Next, we study two different classes of dynamical instabilities that can occur at the cell-scale: surface compression and peeling. For each of these classes, we present a simplified rod-spring model followed by a more detailed model that includes the cell and surface geometries. Our detailed models describe the reorientation thresholds of the agent-based model biofilms (Fig. 2a,b).\n\n\\subsection*{Cell line instability}\n\nCells become unstable to verticalization under large enough compressive forces. This threshold effect is analogous to Euler buckling. In contrast to Euler buckling, however, we observed that cell instabilities are spatially localized within a biofilm cluster. A key feature underlying this discrepancy is the role played by the restoring potential, defined as the interaction that stabilizes the system in the absence of external forces. In Euler buckling, the restoring potential is provided by the rod's internal bending rigidity, whereas for cell instabilities, the restoring potential is provided by the external surface. Therefore, to understand why verticalization is localized, we start by briefly reviewing the conventional scenario of Euler buckling before going on to study the verticalization instability of a biofilm.\\newline\n\n\\textbf{Case I: Euler buckling}\\newline\n\nThe energy of an inextensible elastic rod under uniform compression is approximated by\\cite{S1}:\n\n\\begin{equation}\nE_{\\mathrm{rod}} = \\frac{1}{2} \\int dx \\left[ \\kappa \\left( \\frac{d^2h(x)}{dx^2} \\right)^2 - F \\left( \\frac{dh(x)}{dx} \\right)^2   \\right],\n\\end{equation}\n\n\\noindent where $h(x)$ is the height field (transverse to the rod's axis), $\\kappa$ is the bending rigidity, and $F$ is the externally applied compressive load. Here, the first term is the restoring potential and the second term corresponds to the work performed by the external force, which is proportional to the end-to-end contraction $\\Delta x \\simeq \\int (dh\/dx)^2$. The Fourier decomposition of the height field is given by:\n\n\\begin{equation}\nh(x) = \\sum_q h_q \\sin(q x),\n\\end{equation}\n\n\\noindent where $h_q$ is the amplitude of a mode with frequency $q$. Modes with $F>\\kappa q^2$ provide a negative contribution to the energy and thus are unstable. If the force $F$ is increased from zero, the first mode to become unstable is the lowest spatial frequency mode. In this case, the lowest mode $q_1$ is an extended deformation limited by the length $\\ell_{\\mathrm{tot}}$ and it corresponds to $q_1 = \\pi \/ \\ell_{\\mathrm{tot}}$ (Supplementary Fig. 3a).\\newline\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{fig_s3.jpg}\n\\caption{\\label{fig:FG3}\nCompressive instabilities of one-dimensional media. (\\textbf{a}) Schematic of Euler buckling of an elastic rod under uniform compression. As the equal-and-opposite compressive forces are increased from zero, the rod first becomes unstable to an extended deformation given by the lowest spatial frequency mode. (\\textbf{b}) Schematic of the verticalization instability of a line of cells under uniform compression. As the compressive forces are increased from zero, the line of cells first becomes unstable to a combination of vertical motions and rotations, with a large mode number.\n}\n\\end{figure}\n\n\\textbf{Case II: Cell line verticalization}\\newline\n\nWe now consider a cluster of cells interacting with a surface. For brevity, we focus on a line of cells in one dimension. Thus, each cell $i$ can undergo center of mass motion transverse to the line, as well as rotate. For stiff cells ($E_0 \\rightarrow \\infty$), the cell-cell contact distance remains fixed. In the continuum limit of small cells, the end-to-end contraction $\\Delta x$ of the biofilm is given by:\n\n\\begin{equation}\n\\Delta x \\simeq \\frac{1}{2} \\int dx \\left[ c_1 \\left(  \\frac{dh(x)}{dx}\\right)^2 + c_2 \\theta(x)^2 - c_3 \\theta(x) \\frac{dh(x)}{dx}  \\right],\n\\end{equation}\n\n\\noindent to second order in the height field $h(x)$ and the orientation field $\\theta(x)$, where $c_1$, $c_2$, and $c_3$ are geometrical factors on the order of the cell length. Intuitively, these terms arise because both differential changes in cell heights as well as cell rotations (first and second terms) free up space along the surface and allow the cluster to pack more densely. However, coupled center of mass motions and rotations can either increase or decrease the contraction depending on their signs (third term).\n\nThe continuum limit of the surface energy (Supplementary Eq. \\ref{eq:s}) is given by:\n\n\\begin{equation}\nE_{\\mathrm{ad}} = \\frac{1}{2} \\int dx \\left[ \\lambda_1 h^2(x) + \\lambda_2 \\theta^2(x)   \\right],\n\\end{equation}\n\n\\noindent where $\\lambda_1$ and $\\lambda_2$ are elastic parameters proportional to the cell stiffness $E_0$ in the limit of small penetration $E_0 \\gg \\Sigma_0 R^{-1}$. Thus, for a biofilm under a uniform compressive load $F$, the total energy is given by:\n\n\\begin{equation}\nE_{\\mathrm{col}} = \\frac{1}{2} \\int dx \\left[ \\lambda_1 h^2(x) + \\lambda_2 \\theta^2(x)  - F c_1 \\left(  \\frac{dh(x)}{dx}\\right)^2 + F c_2 \\theta(x) \\frac{dh(x)}{dx} - F c_3 \\theta(x)^2   \\right].\n\\end{equation}\n\n\\noindent To understand how this line of cells becomes unstable, it is instructive to first consider what happens when either rotations or center of mass motions are forbidden. The former scenario corresponds to a flexible chain. Here, modes with $F>\\lambda_1 \/ (c_1 q^2)$ are unstable, so the instability first occurs through the \\emph{highest} spatial frequency mode. On the other hand, when center of mass motions are forbidden, the biofilm first becomes unstable when $F> \\lambda_2 \/ c_2$, independent of the mode number.\n\nWhen both center of mass motions and rotations are allowed, the coupling between the height field and the orientation field can facilitate the instability. In particular, when both fields are completely out of phase, the negative contribution to the energy due to the coupling is maximized. As the force is increased from zero, the first unstable mode is therefore a combination of center of mass motions and rotations at a large mode number (Supplementary Fig. 3b). Thus, in contrast to Euler buckling, cells first become unstable to verticalization on length scales comparable to the cell length.\n\nHow do these results apply to growing biofilms? In a growing biofilm, cells are subject to a spatially non-uniform distribution of forces. Since verticalization instabilities can proceed on wavelengths comparable to cell length, any region in which growth-derived forces overcome the restoring potential will become locally unstable. Thus, the propensity for verticalization to occur at high mode number explains why we observed cell reorientations to occur locally in regions of large forces (Fig. 2d).\n\n\\subsection*{Toy model for compression instability}\n\nOur results for the line of cells implies that verticalization instabilities occur at the single-cell scale. Therefore, to understand the onset of reorientation, we now turn to models for the instabilities of individual cells under applied forces. We first explore a minimal toy model that consists of a rigid rod of length $\\ell$ attached to an elastic foundation. The elastic foundation is comprised of a large number of identical Hookean springs spread evenly over the length of the rod (Supplementary Fig. 4a). The ends of the springs are fixed to lie at the same height, which allows for an unstretched reference configuration with elevation angle $\\theta=0$. We consider motions for which the rod is free to rotate about its center, i.e. to finite values of $\\theta$. Thus, the energy $E_{\\mathrm{ef}}$ of the elastic foundation is given by:\n\n\\begin{equation}\nE_{\\mathrm{ef}} = \\frac{k \\ell^3}{2} \\sin^2 \\theta,\n\\end{equation}\n\n\\noindent to leading order in $\\theta$ in the limit of a continuous foundation, where $k$ is an elastic parameter with the same units as the cell stiffness $E_0$. To represent the cell-cell interactions, we apply equal-and-opposite forces of magnitude $F$ to both ends of the rod. The forces act to squeeze the rod and always point along the initial direction of the rod. Thus, the total energy of the system $E_{\\mathrm{tot}}$ is given by:\n\n\\begin{equation}\nE_{\\mathrm{tot}} = \\frac{k \\ell^3}{2} \\sin^2 \\theta - F \\ell \\cos \\theta,\n\\end{equation}\n\n\\noindent For $F=0$, the rod rests on the foundation at an elevation angle $\\theta=0$. For motion around this configuration opposed by friction, the elastic and compressive forces must balance the drag force $F_{d}$, which is proportional to the rate of change of the elevation angle:\n\n\\begin{equation}\nF_d \\sim \\dot{\\theta},\n\\end{equation}\n\n\\noindent assuming that the friction is provided by Stoke's drag terms that act along the length of the rod, as in Supplementary Eq. \\ref{eq:sd}. Thus, for small elevation angles, the rate of change of $\\theta$ is given by:\n\n\\begin{equation}\n\\dot{\\theta} \\sim (F - k \\ell^2) \\theta.\n\\end{equation}\n\n\\noindent For small forces, $\\theta=0$ is a stable fixed point of the system. However, when the force $F$ becomes large, the rod becomes unstable to reorientation. This bifurcation instability occurs at a threshold force $F_t = k\\ell^2$. Intuitively, the dependence of this threshold force on cell length arises because the elastic foundation provides a fixed restoring energy per unit length of the rod.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{fig_s4.jpg}\n\\caption{\\label{fig:FG4}\nToy models for cell-scale mechanical instabilities. (\\textbf{a}) Schematic depiction of the toy model for surface compression instability, showing a rigid rod of length $\\ell$ on an elastic foundation with a vertical stiffness modulus $k$. The rod is compressed by external, equal-and-opposite forces $F$ oriented horizontally (green arrows). For small values of force $F<F_{t} \\sim k\\ell^2$ (top), the rod remains stationary. For large values of force $F>F_{t}$ (bottom), the rod becomes unstable to reorientation. (\\textbf{b}) Schematic depiction of the toy model for peeling instability, showing the system from (\\textbf{a}) with the rod initially embedded a distance $\\delta_0$ below the surface (brown). The rod is under an external torque $\\tau$ (green arrows). For small values of torque $\\tau<\\tau_{t} \\sim k\\ell^2$ (top), the equilibrium position of the rod shifts to finite elevation angles. For large values of torque $\\tau>\\tau_{t}$ (bottom), one end of the rod moves above the surface. If we assume that the elastic foundation detaches from the segment of the rod above the surface, the rod will be unstable to further reorientation.\n}\n\\end{figure}\n\n\\subsection*{Instability of model cell to compression}\n\nThe squeezed rod model shows how horizontal forces can result in the vertical motion of a rigid object. However, we expect this instability mechanism to depend on the details of the cell geometry, as well as the cell-to-surface interactions. Therefore, we now perform an analogous stability analysis on our spherocylindrical model cells in the presence of the surface potential given by $E_{{s},i}$ in Supplementary Eq. \\ref{eq:s}.\n\nIn what follows, we allow the cell to undergo vertical center of mass motion in addition to rotation. In the absence of forces, the cell is embedded in the surface at its stable fixed point ($\\theta=0$ and $\\delta = \\delta_0$). To mimic the average distribution of forces acting on cells in a biofilm, we now consider applying a uniform distribution of forces around the cell perimeter and acting in the $xy$ plane. For simplicity, we take the force at each point along the perimeter to be applied by a spherical piston of radius $R$ (equal to the cell radius) that is rigid with respect to the cell. The centers of the pistons are fixed to lie at the same height $R-\\delta_0$ as the midline of the cell in the initial configuration, and their motions in the $xy$ plane are constrained to occur entirely along the direction of the shortest line connecting their centers to the cell cylinder's centerline in the initial configuration. For concreteness, we take the initial cell orientation vector to point in the $\\boldsymbol { \\hat { x}} $ direction. For pistons applied to the endcaps of this cell, the vector $\\boldsymbol{d}_{\\mathrm{end},\\pm}$ between the piston's center and the cell cylinder's centerline is given by:\n\n\\begin{equation}\n\\boldsymbol{d}_{\\mathrm{end},\\pm} =  ( R - \\delta_0 - z ) \\boldsymbol{\\hat{z} } + (\\ell\/2) \\boldsymbol{\\hat{n}} \\pm (2R - \\Delta d_{\\mathrm{end},\\pm} ) \\boldsymbol{\\hat{m}} \\pm (\\ell\/2) \\boldsymbol{\\hat{x}}  ,\n\\end{equation}\n\n\\noindent where $z$ is the cell height above the surface, $ \\boldsymbol{\\hat{n}} = (\\cos \\theta, 0, \\sin \\theta)$ is the cell orientation vector, $ \\boldsymbol{\\hat{m}} = (\\sin \\phi, 0, \\cos \\phi)$ is the piston's angle of attack, and $ \\Delta \\boldsymbol{d}_{\\mathrm{end},\\pm} $ is the piston's displacement. As the cell moves, the piston is assumed to stay in contact with the cell. This constraint corresponds to the following equation:\n\n\\begin{equation}\n|\\boldsymbol{d}_{\\mathrm{end},\\pm}| = 2R,\n\\end{equation}\n\n\\noindent which determines $ \\Delta d_{\\mathrm{end},\\pm} $ as a function of the cell configuration. For a piston applied to the cylindrical portion of the cell, the vector $\\boldsymbol{d}_{\\mathrm{side},\\pm}$ between the piston's center and the cell cylinder's centerline is given by:\n\n\\begin{equation}\nd_{\\mathrm{side},\\pm} = ( R - \\delta_0 - z ) \\boldsymbol{\\hat{z} } - (2R - \\Delta d_{\\mathrm{side},\\pm} ) \\boldsymbol{ \\hat{ y }}  + s \\boldsymbol{ \\hat{ x }} .\n\\end{equation}\n\n\\noindent The constraint that the piston remains in contact with the cell is specified by the following equation:\n\n\\begin{equation}\n|\\boldsymbol{d}_{\\mathrm{side},\\pm} - (\\boldsymbol{d}_{\\mathrm{side},\\pm} \\cdot \\boldsymbol{\\hat{n}} ) \\boldsymbol{\\hat{n}} | = 2R,\n\\end{equation}\n\n\\noindent which determines the piston displacement $\\Delta d_{\\mathrm{side},\\pm}$ as a function of the cell configuration. The total work $W_p$ performed by the pistons on the cell is obtained by integrating the contributions from pistons around the perimeter of the cell:\n\n\\begin{equation}\nW_p = p \\int_0^{\\pi} d\\phi ( \\Delta d_{\\mathrm{end},+} +   \\Delta d_{\\mathrm{end},-}  ) +  p \\int_{-\\ell\/2}^{\\ell\/2} ds ( \\Delta d_{\\mathrm{side},+} +   \\Delta d_{\\mathrm{side},-}  ),\n\\end{equation}\n\n\\noindent where $p$ is the applied ``pressure''. The total energy $E_{\\mathrm{cp}}$ of the cell-piston system, i.e. the cell-to-surface energy minus the work done by the pistons on the cell, is given by:\n\n\\begin{equation}\nE_{\\mathrm{cp}} = E_{{s,i}} - W_p.\n\\end{equation}\n\n\\noindent For motion around this configuration opposed by friction, the equations of motion are given by:\n\n\\begin{equation}\n\\dot{z} = -\\frac{1}{\\eta_0 \\ell} \\frac{\\partial E_{\\mathrm{cp}}}{\\partial z},\n\\end{equation}\n\n\\begin{equation}\n\\dot{\\theta} = -\\frac{12}{\\eta_0 \\ell^3} \\frac{\\partial E_{\\mathrm{cp}}}{\\partial \\theta},\n\\end{equation}\n\n\\noindent to leading order in $z$ and $\\theta$, where we have assumed that the friction is determined according to the dissipation function Supplementary Eq. \\ref{eq:sd}. To determine the behavior of the system as a function of the applied surface pressure, we perform a linear stability analysis around the initial configuration. We first construct the stiffness matrix $\\mathcal{D}$ as follows:\n\n\\begin{equation}\n\\mathcal{D} = \\begin{bmatrix}\n     -\\frac{1}{\\eta_0 \\ell} \\frac{\\partial^2E_{\\mathrm{cp}}}{\\partial z^2}       &  -\\frac{1}{\\eta_0 \\ell} \\frac{\\partial^2E_{\\mathrm{cp}}}{\\partial z \\partial \\theta} \\\\\n    -\\frac{12}{\\eta_{0} \\ell^3} \\frac{\\partial^2E_{\\mathrm{cp}}}{\\partial \\theta \\partial z}       & -\\frac{12}{\\eta_{0} \\ell^3} \\frac{\\partial^2E_{\\mathrm{cp}}}{\\partial \\theta^2}\n  \\\\\n\\end{bmatrix}.\n\\end{equation}\n\n\\noindent For our model cell under surface pressure from the pistons, the off-diagonal terms of this matrix are zero, which indicates that vertical motion is decoupled from rotation. Therefore, the signs of the diagonal terms determine whether the cell is stable to infinitesimal perturbations. For our choice of parameters above (Supplementary Fig. 1) and for small values of force, both eigenvalues are negative and the system is stable. However, as the force is increased, the cell first becomes unstable to reorientation. The threshold value of surface pressure $p_{t}$ for which this instability occurs is given by:\n\n\\begin{equation}\np_{t} = \\frac{3 E_0 \\ell^2 R - 8 (3 \\pi - 1) \\Sigma_0 + 9 \\sqrt[3]{2} R^{-1} \\Sigma_0^{4\/3} E_0^{-1\/3} }{\\ell^2+3 \\pi \\ell R+24 R^2} .\n\\end{equation}\n\n\\noindent For physiologically-relevant parameters, this surface pressure is roughly linear as a function of $\\ell$ over a large range around $\\ell = R$. For $\\ell \\sim R$, the threshold surface pressure is approximately:\n\n\\begin{equation}\np_{t} \\sim E_0 (b_1 \\ell - b_2 R),\n\\end{equation}\n\n\\noindent in the limit of small penetration (cell stiffness $E_0 \\gg \\Sigma_0 R^{-1}$), where $b_1 = (144+9\\pi)\/(25+3\\pi)^2$ and $b_2 = 69R\/(25+3\\pi)^2$. The dependence of the threshold surface pressure on cell length arises in this regime because the total forces acting on the cell endcaps are comparable to the total forces acting on the cell cylinder. For longer cell lengths, however, the forces acting on the cell cylinder dominate and the threshold surface pressure saturates to $p_{t} \\sim 3 E_1 R$. Intuitively, this saturation occurs because the pistons provide a fixed surface energy per unit length that balance the fixed surface energy per unit length of the model cell.\n\nIn the intermediate cell length regime, the scaling $p_{t} \\sim R$ for $\\ell \\sim R$ implies that $F_{t} \\sim R^2$, as in the toy model. However, the spherocylindrical cell model deviates from the toy model in two compensating ways. First, the work performed by the pistons to rotate the spherocylindrical cell scales more rapidly with cell length than the work performed by the purely horizontal forces in the toy model. For the case of forces applied to the end of the cell, the piston yields $W_{p} \\sim \\ell^2 \\theta^2$ whereas the horizontal forces yield $W_{p} \\sim \\ell \\theta^2$. Second, for a fixed amount of total force, spreading the pistons around the entire perimeter of the cell yields a smaller amount of in-plane torque than if the forces were concentrated entirely at the ends, as in the toy model. Thus, our spherocylindrical cell model demonstrates that it is important to consider the full effects of the cell-cell contact geometry together with the cell-cell contact distribution to fully capture the surface compression instability.\n\n\\subsection*{Toy model for peeling instability}\n\nOur agent-based simulations suggest that for long cell lengths, forces in the $z$ direction play an important role in triggering verticalization. To describe this effect, we now return to the toy model of a rod on an elastic foundation discussed above and we consider the effect of external forces in the $z$ direction (Supplementary Fig. 4b). For simplicity, we take the rod's center of mass to be fixed. In this case, the configuration of the rod depends on the net torque $\\tau$ provided by the external forces. The total energy $E_{z} $ of the system becomes:\n\n\\begin{equation}\nE_{z} = \\frac{k \\ell^3}{2} \\sin^2 \\theta - \\tau \\theta.\n\\end{equation}\n\n\\noindent Upon minimizing this energy, we find that the applied torque shifts the stable configuration of the cell to a finite elevation angle $\\theta_0 = \\tau \/ k \\ell^3$. How would this finite elevation angle influence the contact between a cell and the surface? For large elevation angles, the bonds between a cell and the surface must eventually break. When this occurs, continued peeling of the cell from the surface requires decreasing amounts of external torque. We can incorporate this mechanism into the torqued rod model in a simple manner by assuming the springs of the elastic foundation break when they are stretched more than a small distance $\\delta_{t}$. For $\\delta_{t} \\ll \\ell$, this distance is reached by one end of the cell when $\\theta_0 \\simeq \\delta_{t} \/ \\ell$. Therefore, the threshold torque for peeling scales as $\\tau_{t} \\sim \\ell^2$.\n\n\\subsection*{Instability of model cell to peeling}\n\nTo determine the threshold verticalization torque for the model cell, we consider the spherocylindrical model cell in the presence of the surface potential. For simplicity, we take the cell center to remain fixed. For a small constant torque $\\tau$, the stable angle $\\theta_0$ is obtained by solving the following equation:\n\n\\begin{equation}\n\\tau = \\frac{\\partial   E_{s} } {\\partial \\theta} \\Bigm| _{\\theta_0}.\n\\end{equation}\n\n\\noindent For $\\theta_0 \\ll 1$, we find that $\\theta_0 = \\tau \/ b_3$, where $b_3$ is given by:\n\n\\begin{equation}\nb_3 = (3 E_0 \\ell^3 - 8 (3 \\pi - 1) \\Sigma_0 R^{-1} \\ell + 9 \\sqrt[3]{2} R^{-2} \\ell \\Sigma_0^{4\/3} E_0^{-1\/3}) \/ 12 .\n\\end{equation}\n\n\\noindent The critical angle $\\theta_c$ for peeling a cell from the surface is reached when one end of the cell begins to leave the surface, i.e.:\n\n\\begin{equation}\n\\delta_0 \\simeq \\frac{\\ell}{2} \\theta_{t}.\n\\end{equation}\n\n\\noindent Setting $\\theta_0 = \\theta_{t}$ yields a threshold torque $\\tau_{t} = 2 b_3 \\delta_0 \/ \\ell$. In the limit of small penetration (cell stiffness $E_0 \\gg \\Sigma_0 R^{-1}$), $\\tau_{t}$ is given by:\n\n\\begin{equation}\n\\tau_{t} = \\delta_0 E_0 \\ell^2 \/ 2 .\n\\end{equation}\n\n\\noindent Thus, in this regime we find that $\\tau_{t} \\sim \\ell^2$, in agreement with the scaling found for the torqued rod.\n\n\\newpage\n\n\\section*{\\textbf{Supplementary Figures 5 to 7: Verticalization events (in experiment and simulation)}}\n\n\\textbf{Tracking verticalization events} To probe the local conditions driving verticalization, we tracked the cell-to-cell contact forces acting on individual cells in the agent-based model around the time when cells start to become vertical (Supplementary Fig. 5). We found that as a cell becomes vertical, the total surface force acting on it reaches a local maximum before decaying rapidly. The trend arises due to the nonlinearity of the cell-to-surface contact geometry, combined with the reduced footprint taken up by a vertical cell relative to a horizontal cell. That is, before the cell starts to become vertical, the surface forces increase due to cell growth, which increases cell-cell overlaps more rapidly than the overlaps can be resolved by rearrangements of cells. As the cell becomes vertical, it requires progressively lower amounts of force to induce further reorientation due to the peeling of the cell (see \\textbf{Peeling instability of model cell} above). Reorientation frees up space along the surface for local rearrangements that reduce cell-cell overlaps and thereby alleviate the accumulating forces. These complex dynamics are readily apparent from visualizations of the force chains throughout a biofilm (Supplementary Video 3).\n\nAs a result of this behavior, the forces acting on a cell provide a characteristic signature of its transition from horizontal to vertical. Specifically, we identified the moment $t_{{r}}$ of the verticalization transition as the time of the peak force prior to the cell exceeding a critical orientation, which we took to be $n_z > 0.25$. In the main text, we showed that the values of the peak forces are consistent with the instability models we presented above (Fig. 2, Supplementary Fig. 3,4). \\\\\n\n\\textbf{The effect of the cell-cell contact distribution on the predicted reorientation pressure} In the main text and in a section above, we presented a theoretical prediction for the average reorientation pressure $\\langle p_r \\rangle$ in the agent-based model obtained by performing a linear stability analysis for a modeled cell under uniform surface pressure (see \\textbf{Instability of model cell to compression} above). We found a large discrepancy between the calculated threshold reorientation pressure $p_t$ and the observed average reorientation pressure $\\langle p_r \\rangle$ (Fig. 2a). To eliminate the possibility that the discrepancy could arise from heterogeneity in the contact forces in the agent-based model, we made a separate theoretical prediction for $\\langle p_r \\rangle$ that incorporates the numerically-observed distribution of cell-cell contact forces. Specifically, for each reorientation event, we first recorded the distribution of cell-cell contact forces, i.e. the magnitudes, directions, and points of application of forces in the $xy$ plane applied to the reorienting cell by neighboring cells. For each set of cell-cell contact forces, we determined the threshold surface pressure via linear stability analysis by uniformly rescaling the magnitudes of the forces until the onset of an instability. Incorporating the numerically-observed distribution of cell-cell contact forces in this manner did not yield a substantially different prediction for $\\langle p_r \\rangle$ compared to the prediction assuming a uniform surface pressure (Supplementary Fig. 6). Based on the substantial discrepancy between $p_t$ and $\\langle p_r \\rangle$, we hypothesized that cell-cell forces acting along the $xy$ plane alone do not account for the verticalization of long cells.\n\n\\newpage\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{fig_s5.jpg}\n\\caption{\\label{fig:FG9002}\nVerticalization of individual cells in the agent-based model. Representative examples of the total surface force on a cell (green), defined as the total contact force in the $xy$ plane acting on a cell, and cell verticality $n_z$ (purple) versus the time since the cell starts to become vertical (red vertical dashed line), for cell cylinder lengths $\\ell_0 = \\SI{0.4}{\\micrometer}$ (left column), $\\ell_0 = \\SI{1.2}{\\micrometer}$ (middle column), and $\\ell_0 = \\SI{2}{\\micrometer}$ (right column). For $\\ell_0 = \\SI{0.4}{\\micrometer}$ and $\\ell_0 = \\SI{1.2}{\\micrometer}$, the traces begin at the moment of cell birth, whereas only partial traces are shown for $\\ell_0 = \\SI{2}{\\micrometer}$.\n}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.4\\columnwidth]{fig_s6.jpg}\n\\caption{\\label{fig:FG9002}\nIncorporating the distribution of cell-cell contact forces into the compression instability model. Distributions of reorientation surface pressure $p_{r}$, defined as the total contact force in the $xy$ plane acting on a cell at time $t_{r}$ of verticalization, normalized by the cell's perimeter, versus cell cylinder length $\\ell$. White dashed curve shows the average reorientation surface pressure $\\langle p_{r} \\rangle$ as a function of $\\ell$. Magenta dashed curve shows theoretical prediction for $\\langle p_{r} \\rangle$ from linear stability analysis for a modeled cell under uniform surface pressure, and purple dashed curve shows average of the predicted distribution of $p_{r}$ from linear stability analyses for a sample of reorienting modeled cells under the numerically-observed cell-cell contact forces. The numerical data for $p_r$ and the distribution of contact forces is obtained from all reorientation events among different biofilms simulated for a range of initial cell lengths $\\ell_0$. Insets show schematic depictions of example cell-cell contact geometries considered in the linear stability analysis.\n}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{fig_s7.jpg}\n\\caption{\\label{fig:FG9003}\nDivision-triggered reorientation events. Confocal fluorescence microscopy images of living, growing biofilm under standard conditions at approximately $t=300$ minutes (left) and $40$ minutes later (right). Red circles indicate mother cells immediately prior to division (left) with either one or both daughter cells becoming vertical following division (right). Scale bar: $\\SI{5}{\\micrometer}$.\n}\n\\end{figure}\n\n\n\n\\newpage\n\n\\section*{\\textbf{Supplementary Figures 8 to 10: Cell avalanches}}\n\nHow are cell verticalization events correlated in space and time? To quantify such correlations, we computed the joint radial distribution $P(\\Delta r_{ij} , \\Delta t_{{r},ij} ) \/ \\Delta r_{ij}$. Here, $P(\\Delta r_{ij} , \\Delta t_{{r},ij} )$ is the joint distribution of spatial separations $\\Delta r_{ij}$ and temporal separations $\\Delta t_{{r},ij}$, where $\\Delta r_{ij} = | \\boldsymbol{r}_i - \\boldsymbol{r}_j |$, where $\\boldsymbol{r}_i$ is the position of cell $i$ at the time $t_{{r},i}$ of the peak of total force on the cell prior to it becoming vertical, and $\\Delta t_{{r},ij} = |t_{{r},i} - t_{{r},j}|$ (Supplementary Fig. 8). For all values of average cell length we studied, this distribution of separations displayed a characteristic peak for small spatial and temporal separations, followed by a rapid decay in both distance and time.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{fig_s8.jpg}\n\\caption{\\label{fig:FG6}\nCorrelations among verticalization events for agent-based model. (\\textbf{a-b}) Joint radial distribution $P(\\Delta r_{ij} , \\Delta t_{r,ij} ) \/ \\Delta r_{ij}$ of spatial separations $\\Delta r_{ij} = | \\boldsymbol{r}_i - \\boldsymbol{r}_j |$, where $\\boldsymbol{r}_i$ is the position of cell $i$ at the time $t_{r,i}$ of reorientation, and temporal separations $\\Delta t_{ij} = |t_{r,i} - t_{r,j}|$ between pairs of reorientation events for (\\textbf{a}) $\\ell_0 = \\SI{1}{\\micrometer}$ and (\\textbf{b}) $\\ell_0 = \\SI{2}{\\micrometer}$. (\\textbf{c-d}) Joint radial distribution $P(\\Delta r_{ij} , \\Delta t_{r,ij} ) \/ \\Delta r_{ij}$ of spatial and temporal separations among pairs of reorientation events in the null model, which consists of randomizing the angular positions of cells within the biofilm, for (\\textbf{c}) $\\ell_0 = \\SI{1}{\\micrometer}$ and (\\textbf{d}) $\\ell_0 = \\SI{2}{\\micrometer}$.\n}\n\\end{figure}\n\nTo rule out that the peak structure in the joint radial distribution of verticalization events was caused by the finite size and growth rate of the annular region, we compared our results to a null model that accounts for this effect by randomizing the angular positions of cells within the biofilm. Specifically, for the null model, we compute the spatiotemporal separations between verticalization events from a given biofilm to those in ten copies of the same biofilm that have been randomly rotated around its center. This model respects the radial symmetry of the biofilm and also allows for comparison between biofilms of different average cell lengths. Specifically, if correlations within a given biofilm exceed those obtained for the corresponding null model, then any excess correlation implies a nontrivial source of correlations. We found that the probability at the peak was significantly increased compared to the null model, which demonstrates that verticalization events are cooperative. This effect is similar to the phenomenon of dynamical facilitation observed in glassy systems\\cite{S4}. A possible explanation for the nontrivial correlations comes from the inverse domino effect, which consists of a cell standing up and applying an out-of-plane torque that triggers one or more neighboring cells to stand up. We expect this effect to occur for long cells because long cells are more likely to become vertical due to the peeling instability, which is triggered by torques from neighboring vertical cells.\n\nDoes the inverse domino effect explain the spatiotemporal extent of the peak? The inverse domino effect can occur when a vertical cell comes into contact with a horizontal cell that is susceptible to becoming vertical. The requirement of cell-cell contact for this effect to occur is consistent with the observed spatial separation of the peak, which is approximately equal to the average distance between the centers of horizontal cells (Supplementary Fig. 8). Furthermore, the inverse domino effect also suggests a limit on the temporal extent of the peak, because the reduction of cell footprint upon verticalization opens up space for local rearrangements that rapidly alleviate the surface pressure as the cell configuration relaxes (Supplementary Fig. 5). Indeed, the time it takes for the surface pressure to relax is roughly a few minutes, consistent with the temporal extent of the peak beyond its maximum (Supplementary Figs. 5, 8). Thus, taken together, the requirement for spatial proximity, along with the decrease in surface pressure associated with verticalization, can explain the rapid spatiotemporal decay of $P(\\Delta r_{ij} , \\Delta t_{r,ij} )$.\n\nOur observations of the behavior of $P(\\Delta r_{ij} , \\Delta t_{{r},ij} )$ provide a natural definition for the extent of cooperativity in cell verticalization. That is, since facilitation occurs on short spatiotemporal scales, we can capture the extent of cooperative effects by computing the number of cells involved in a series of verticalization events that are proximal in space and time. Specifically, we define proximity in space as $\\Delta r_{ij} < \\ell_{{f}}$, where $\\ell_{{f}}$ is the cell division length, and define proximity in time as $\\Delta t_{{r},ij} < t_{{f}}$, where $t_{{f}}$ is the facilitation time scale, defined as the time scale of the decay of the spatiotemporal separation probability $P(\\Delta r_{ij} , \\Delta t_{r,ij} )\/ \\Delta r_{ij}$ after the peak. For the growth of a given biofilm cluster, connecting reorientation events that are spatiotemporally proximal results in a graph. We refer to the connected components of this graph as ``cell avalanches'', following the literature on glasses\\cite{S4}.\n\nInterestingly, the distribution of avalanche sizes decays roughly exponentially for all values of cell length we studied (Supplementary Fig. 9), with only a modest number of cells involved in typical avalanches (Fig. 2d). Moreover, the distribution of avalanche sizes does not change substantially as a function of time, unlike the overall number of horizontal cells, which grows proportionally to the biofilm radius (Fig. 1). Thus, as time goes on, a vanishing fraction of the overall number of horizontal cells are involved in a typical avalanche, which demonstrates that avalanches are localized. The dependence of the mean avalanche size on cell length demonstrates that the scale of localization is determined by the geometrical and mechanical properties of individual cells (Fig. 2d).\n\nWhat limits the size of cell avalanches? In order to be susceptible to becoming vertical due to the inverse domino effect, horizontal cells must be poised near the threshold torque for verticalization. Thus, a natural explanation for the size limit comes from the reduction of cell footprint upon reorientation from horizontal to vertical, which rapidly alleviates the local surface pressure and thereby lowers the susceptibility of nearby horizontal cells to becoming vertical (Supplementary Fig. 5). This effect combines with the inherent disorder in the configuration of cells, which generically results in extremely heterogeneous contact geometries and forces throughout the biofilm (Supplementary Video 3, Supplementary Fig. 10). These heterogeneous local conditions segregate horizontal cells poised near the threshold torque for verticalization into small groups. Although within such groups, the verticalization cooperativity is transiently increased by the inverse domino effect, verticalization rapidly exhausts the local supply of horizontal cells. Thus, the rapid timescale of verticalization and the disorder in the cell configuration limit the propagation of cell avalanches.\n\n\\newpage\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{fig_s9.jpg}\n\\caption{\\label{fig:FG7}\nRepresentative examples of avalanche size distributions. (\\textbf{a-c}) Distributions of avalanche sizes $N$ (red data points), defined as the number of cells in a group of verticalization events that are proximal in space (i.e. separated by a distance $\\Delta r_{ij} < \\ell_{{f}}$, where $\\ell_{{f}}$ is the cell division length) and time (i.e. with temporal separation $\\Delta t_{{r},ij} < t_{{f}}$) on a logarithmic scale for (\\textbf{a}) experimental biofilm, (\\textbf{b}) agent-based model with $\\ell_0 = \\SI{1}{\\micrometer}$, and (\\textbf{c}) agent-based model with $\\ell_0 = \\SI{2}{\\micrometer}$. The black data points indicate the corresponding distribution of avalanche sizes for a null model. For reference, gray straight dashed lines correspond to exponential decay over a scale (\\textbf{a}) $N=1.8$ cells, (\\textbf{b}) $N=1.2$ cells, and (\\textbf{c}) $N=2.4$ cells.\n}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{fig_s10.jpg}\n\\caption{\\label{fig:FG9005}\nMechanical heterogeneity in the agent-based model. (\\textbf{a}) Joint distribution of cell surface pressures $p$, defined as the sum of all horizontal forces acting on a cell divided by its perimeter, and radial coordinates $r$ for cells in modeled biofilm with $\\ell_0 = \\SI{1.2}{\\micrometer}$, showing cell fraction in color. Dashed white curve shows the average cell surface pressure $\\langle p \\rangle$ versus $r$. (\\textbf{b}) Visualization of surface layer of a modeled biofilm with $\\ell_0 = \\SI{2}{\\micrometer}$, showing horizontal (blue) and vertical (red) cells as spherocylinders, the surface (brown), and cell-to-cell contact forces (yellow lines connecting the centers of cells, with thickness proportional to the force). Cells with $n_z < 0.5$ ($>0.5$) are considered horizontal (vertical), where $\\boldsymbol{\\hat{n}}$ is the orientation vector. The length of the scale bar is $\\SI{3}{\\micrometer}$, and its thickness corresponds to $\\SI{300}{\\pico\\newton}$.\n}\n\\end{figure}\n\n\n\\newpage\n\n\n\n\n\n\\section*{\\textbf{Supplementary Note: Continuum models for verticalizing biofilms}}\n\nIn this Supplementary Note, we present minimal continuum models that provide insight into the verticalization transition. We first present a simplified model for verticalizing biofilms in the incompressible limit. We go on to explore the compressible biofilm model that was discussed in the main text.\n\n\\subsection*{\\textbf{Origin of vertical ordering}}\n\nHow does cell growth drive biofilm expansion and verticalization? To gain qualitative insight into this question, we started by considering a simple continuum model in the limit of approximately incompressible cells. We first assumed that cells in 2D grow exponentially at rate $\\alpha$. For an isotropic 2D biofilm, this growth implies that the total radius $R_{{B}}$ increases as:\n\n\\begin{equation}\nR_{{B}} \\sim e^{\\alpha t \/ 2}.\n\\end{equation}\n\n\\noindent Similarly, the local radial velocity must be $v = \\alpha r \/ 2$, where $r$ is the radial coordinate. This velocity must arise from the cell surface pressure $p$, associated with the compression of cells. The local gradient of this surface pressure $dp\/dr$ required to drive cells with velocity $v$ is\n\n\\begin{equation}\n\\frac{dp}{dr} = -\\eta v = - \\frac{\\alpha \\eta  r }{2},\n\\end{equation}\n\n\\noindent where $\\eta$ is the surface drag coefficient of the medium. Spatially integrating this equation gives the pressure field:\n\n\\begin{equation}\np = \\frac{\\alpha \\eta}{4} (R_{{B}}^2 - r^2),\n\\end{equation}\n\n\\noindent which is quadratic and peaked at the center of the biofilm.\n\nNow we assume that as soon as the local surface pressure exceeds some verticalization threshold $p_{t}$, the cells start becoming vertical. These transitions occur first at the center of the biofilm, resulting in an inner region of vertical cells surrounded by an annular periphery of horizontal cells. Since vertical cells do not contribute to growth along the surface, the surface pressure remains constant throughout the region of vertical cells. Furthermore, to satisfy the boundary condition $p=p_{t}$ at the interface between horizontal and vertical cells, the width of the annular periphery of horizontal cells must remain constant. This results in a biofilm front and a region of vertical cells that both expand outward at a fixed rate $c^* \\sim \\sqrt{\\alpha  p_{t}\/ \\eta }$. Thus, this simple continuum model of incompressible cells provides a qualitative explanation of the verticalization transition.\n\nAlthough this model roughly captures the spreading of vertical ordering, it cannot capture the crossover between the radial density profiles of the horizontal and vertical cells (Fig. 1c,d), or the saturation of the expansion speed as a function of increasing verticalization threshold (Results, Equation 2).\n\n\\subsection*{\\textbf{Compressible, two-fluid model for verticalizing biofilms}}\n\nTo better quantify the growth of verticalizing biofilms, we developed a continuum model that treats horizontal and vertical cells, respectively, as densities $\\rho_h$ and $\\rho_v$. These densities specify the number of cells per unit of surface area. In what follows, we define the total cell density as $\\tilde{\\rho}_{\\mathrm{tot}} = \\rho_h + \\xi \\rho_v$, where $\\xi$ is the ratio of vertical to horizontal cell footprints. The cell densities evolve according to the following hydrodynamic equations:\n\n\\begin{equation}\n\\dot{\\rho_h} + \\nabla \\cdot ( \\rho_h \\boldsymbol{v}) = \\left[ \\alpha - \\beta \\Theta(p - p_t) \\right] \\rho_h,\n\\end{equation}\n\n\\begin{equation}\n\\dot{\\rho_v} =  \\beta \\Theta(p - p_t) \\rho_h,\n\\end{equation}\n\n\\begin{equation}\n- \\eta \\tilde{\\rho}_{\\mathrm{tot}} \\boldsymbol{v} = \\nabla p.\n\\end{equation}\n\n\\noindent Here, $\\boldsymbol{v}$ is the cell velocity, $\\alpha$ is the growth rate, $\\beta$ is the verticalization rate, $\\eta$ is a viscous drag coefficient, $p$ is the surface pressure, $p_t$ is the threshold surface pressure for verticalization, $\\tilde{\\rho}_0$ is the close-packing density, and $\\Theta$ is the Heaviside step function. The first two equations describe the conservation of cell number, and the third equation describes the balance between growth forces and surface drag. In the first equation, we have assumed that the change in local horizontal cell density is determined by the effect of cell transport, i.e. the change in cell density due to the motion of cells, as well as in-plane cell growth and cell verticalization. In the second equation, we have neglected the transport of vertical cells, and so the vertical cell density only changes due to cell verticalization. Below, we will revisit this approximation and show that it does not change the results for rapid enough verticalization rates $\\beta > \\alpha(1-\\xi)$ consistent with the behavior of the experimental biofilms and the agent-based modeled biofilms (Fig. 4b).\n\nWe take the surface pressure to be given by the Young's modulus $\\lambda$ of the biofilm times the areal strain, which becomes nonzero when cells are close-packed but uncompressed:\n\n\\begin{equation} \np=\\begin{cases}\n    \\lambda ( \\tilde{\\rho}_{\\mathrm{tot}} - \\tilde{\\rho}_0) & \\text{if}\\  \\tilde{\\rho}_{\\mathrm{tot}} > \\tilde{\\rho}_0,\\\\\n    0              & \\text{otherwise}.\n\\end{cases}\n\\end{equation}\n\n\\noindent Therefore, in our model, the threshold surface pressure for verticalization corresponds to a threshold surface density $\\tilde{\\rho}_t$ for verticalization:\n\n\\begin{equation}\np_t = \\lambda (\\tilde{\\rho}_t - \\tilde{\\rho}_0).\n\\end{equation}\n\nUpon substituting this relation between the pressure and cell density into the hydrodynamic equations, we obtain the following equations of motion for the cell densities:\n\n\\begin{equation}\n\\dot{\\rho_h} =  \\frac{\\lambda}{\\eta} \\nabla \\cdot \\left(\\Theta(\\tilde{\\rho}_{\\mathrm{tot}} - \\tilde{\\rho}_0)  \\nabla \\tilde{\\rho}_{\\mathrm{tot}}\\right)  +  \\left[ \\alpha - \\beta \\Theta(p - p_t) \\right] \\rho_h,\n\\end{equation}\n\n\\begin{equation}\n\\dot{\\rho_v}  =  \\beta \\Theta(p - p_t) \\rho_h.\n\\end{equation}\n\nIn the following sections, we solve for the dynamics of the cell densities. For simplicity, we will first solve the case of spreading in one spatial dimension, which will allow us to characterize the different dynamical regimes of the model. In the last few sections, we will discuss how the results are modified for the cases of non-stationary vertical cells, two dimensional growth, and surface curvature.\n\n\\subsection*{\\textbf{Existence of a linearly-expanding front}}\n\nTo understand whether our continuum model can give rise to stable, linearly-expanding fronts, we first consider the growth of the total number of horizontal cells throughout the entire biofilm. The change in the number of horizontal cells is given by:\n\n\\begin{equation}\n\\dot{\\rho}_h = \\int_{\\mathcal{R}_1} \\alpha \\rho_h d\\boldsymbol{r} + \\int_{\\mathcal{R}_2} (\\alpha - \\beta) \\rho_h d\\boldsymbol{r},\n\\end{equation}\n\n\\noindent where $\\mathcal{R}_1$ corresponds to regions with $p<p_t$ and $\\mathcal{R}_2$ corresponds to regions with $p>p_t$. For $\\alpha > \\beta$, both terms are positive, and the total number of horizontal cells must grow exponentially with rate $\\alpha - \\beta$ at long times. At long times, the regions $\\mathcal{R}_2$ will dominate the growth. Assuming a uniform growth rate $\\alpha - \\beta$, the radius $R_B$ of the biofilm is given by:\n\n\\begin{equation}\nR_B \\sim \\sqrt{(\\alpha - \\beta) \\gamma ( t^2 - t \\log t )},\n\\end{equation}\n\n\\noindent at long times\\cite{S5}, which yields an expansion speed $c^*$ given by:\n\n\\begin{equation}\nc^* \\sim \\sqrt{(\\alpha-\\beta) \\gamma} \\left(1 - \\frac{1}{t}\\right).\n\\end{equation}\n\nThus, the speed of the edge of the biofilm cluster increases over time. However, for $\\beta>\\alpha$, the contribution from region $\\mathcal{R}_2$ is negative and thus could potentially compensate the contribution from region $\\mathcal{R}_1$ to limit the total growth of horizontal cells and thereby allow for a linearly-expanding front.\n\n\\subsection*{\\textbf{Solving for steady-state motion}}\n\nTo search for linearly-expanding solutions to the equations that govern the dynamics of the cell densities, we now assume that the biofilm expands linearly with a speed $c^*$ and we seek consistent solutions to the equations of motion. For now, we treat $c^*$ as an undetermined constant. To solve the equations of motion, we start by shifting to a reference frame that moves at speed $c^*$:\n\n\\begin{equation}\n0 =  \\frac{\\lambda}{\\eta} \\Theta(\\tilde{\\rho}_{\\mathrm{tot}} - \\tilde{\\rho}_0) \\nabla^2\\tilde{\\rho}_{\\mathrm{tot}}  + c^* \\nabla \\rho_h + \\left[ \\alpha - \\beta \\Theta(p - p_t) \\right] \\rho_h,\n\\end{equation}\n\n\\begin{equation}\n0  =  c^* \\nabla \\rho_v + \\beta \\Theta(p - p_t) \\rho_h.\n\\end{equation}\n\n\\noindent We can eliminate the density of vertical cells from the first equation by substituting in the second equation. Doing this yields the following equation:\n\n\\begin{equation}\n0 =  \\frac{\\lambda}{\\eta}  \\Theta(\\rho_h + \\xi \\rho_v - \\tilde{\\rho}_0) \\nabla^2 \\rho_h  + \\left( c^* - \\frac{\\lambda \\beta \\xi}{\\eta c^*}\\Theta(p - p_t) \\right) \\nabla \\rho_h + \\left[ \\alpha - \\beta \\Theta(p - p_t) \\right] \\rho_h,\n\\end{equation}\n\nTo be consistent with the biofilm morphology observed in experiment, we assume that the leading edge of the biofilm consists of a periphery of horizontal cells trailed by an interior region containing a mixture of horizontal and vertical cells with $p>p_t$. For the continuum model, we verified, using numerical simulations, that this pattern generically arises from initial conditions that consist of a small, localized region of horizontal cells (Methods, Supplementary Video 4). Below, we will determine the conditions under which these preliminary solutions are valid. These assumptions lead to the following equations for the horizontal periphery (``$P$'') and the mixed interior (``$I$''):\n\n\\begin{equation}\n0 =   \\frac{\\lambda}{\\eta} \\nabla^2 \\rho_h^{(P)}   + c^* \\nabla \\rho_h^{(P)} + \\alpha  \\rho_h^{(P)},\n\\end{equation}\n\n\\begin{equation}\n0 =   \\frac{\\lambda}{\\eta} \\nabla^2 \\rho_h^{(I)}  + \\left( c^* - \\frac{\\lambda \\beta \\xi}{\\eta c^*}  \\right) \\nabla \\rho_h^{(I)}  +(\\alpha - \\beta) \\rho_h^{(I)}.\n\\end{equation}\n\n\\subsection*{\\textbf{Boundary conditions}}\n\nAt the leading edge of the horizontal periphery, we must have:\n\n\\begin{equation}\n\\rho_h^{(P)} = \\tilde{\\rho}_0,\n\\end{equation}\n\n\\noindent since the pressure that drives cell motion drops to zero when the cell density declines below the packing density $\\tilde{\\rho}_0$. Furthermore, the leading edge of the horizontal periphery must be moving at speed $c^*$:\n\n\\begin{equation}\n- \\rho_h^{(P)}c^* = \\frac{\\lambda}{\\eta} \\nabla \\rho_h^{(P)}.\n\\end{equation}\n\n\\noindent The interface between the horizontal periphery and the mixed interior marks the onset of verticalization, which implies:\n\n\\begin{equation}\n\\rho_h^{(I)} = \\rho_h^{(P)} = \\tilde{\\rho}_t.\n\\end{equation}\n\nFinally, the surface pressure gradient must be continuous at the interface:\n\n\\begin{equation}\n\\nabla \\rho_h^{(I)} + \\xi \\nabla \\rho_v^{(I)} = \\nabla \\rho_h^{(P)},\n\\end{equation}\n\n\\subsection*{\\textbf{The density of horizontal cells in the mixed interior must be given by a single, exponentially-decaying term}}\n\nFrom the above equation of motion for the mixed interior, we find:\n\n\\begin{equation}\n\\rho_h^{(I)} = q_1 e^{\\gamma_+ \\tilde{x}} + q_2 e^{\\gamma_- \\tilde{x}},\n\\end{equation}\n\n\\noindent where $q_1$ and $q_2$ are constants to be determined by the boundary conditions,\n\n\\begin{equation}\n\\gamma_{+,-} = \\left( \\frac{\\beta \\xi}{2c^*} - \\frac{\\eta c^*}{2 \\lambda}  \\right) \\pm \\sqrt{ \\left(\\frac{\\beta \\xi}{2c^*} - \\frac{\\eta c^*}{2 \\lambda} \\right)^2 - \\frac{\\eta}{\\lambda}(\\alpha - \\beta)},\n\\end{equation}\n\n\\noindent and we have chosen the spatial coordinate $\\tilde{x}$ such that $\\tilde{x} = 0$ is the location of the interface between the two regions. We can further simplify this by inserting the boundary condition for the cell density at the interface to eliminate one of the undetermined constants. We find:\n\n\\begin{equation}\n\\rho_h^{(I)} = q_1 e^{\\gamma_+ \\tilde{x}} + (\\rho_t - q_1)  e^{\\gamma_- \\tilde{x}}.\n\\end{equation}\n\nFor $\\alpha < \\beta$, $\\gamma_+$ and $\\gamma_-$ must both be purely real. Furthermore, since $  | \\frac{\\beta \\xi}{2c^*} - \\frac{\\eta c^*}{2 \\lambda} | <  \\sqrt{ \\left( \\frac{\\beta \\xi}{2c^*} - \\frac{\\eta c^*}{2 \\lambda} \\right)^2 -  \\frac{\\eta}{\\lambda}(\\alpha - \\beta) }$, the constants must have opposite signs. That is, $\\gamma_+$ is positive and $\\gamma_-$ is negative.\n\nWe now show that if the density of horizontal cells is 0 at the inner boundary, the density of horizontal cells must be given by a decaying exponential. Consider a density of horizontal cells that is $\\rho_t$ at $\\tilde{x}=0$ and approaches zero at some finite negative value $\\tilde{x}=\\tilde{x}_t$. Since the density of horizontal cells at $\\tilde{x}=\\tilde{x}_t$ is zero, the cell flux must also be zero (since $-\\eta \\rho_h \\boldsymbol{v}_h = \\lambda \\nabla \\rho_h$). Therefore, at this inner boundary, we have:\n\n\\begin{equation}\nq_1 e^{\\gamma_+ \\tilde{x}_t } + (\\tilde{\\rho}_t - q_1) e^{\\gamma_- \\tilde{x}_t } = 0,\n\\end{equation}\n\n\\begin{equation}\n\\gamma_+ q_1 e^{\\gamma_+ \\tilde{x}_t } - \\gamma_- (\\tilde{\\rho}_t - q_1) e^{\\gamma_- \\tilde{x}_t } = 0.\n\\end{equation}\n\n\\noindent These equations only have a non-trivial solution for $b_1 = 0$, which means that the density of horizontal cells must be given by an exponential function with a decay constant $\\gamma_+$:\n\n\\begin{equation} \\label{eq:hord}\n\\rho_h^{(I)} =  \\tilde{\\rho}_t e^{\\gamma_+ \\tilde{x} }.\n\\end{equation}\n\n\\subsection*{\\textbf{Solving for the expansion speed}}\n\nWe now determine the expansion speed $c^*$ by solving for the steady-state density of horizontal cells at the leading edge of the biofilm. We choose coordinates $x$ such that the front of this leading edge is at $x=0$. Upon insertion of the boundary conditions at the front of the leading edge, we find that:\n\n\\begin{equation}\n\\rho_h^{(P)} = \\tilde{\\rho}_0 e^{-\\frac{\\eta c^* x}{2 \\lambda}} \\left[ \\cosh \\left(\\frac{\\eta c^* x}{2\\lambda}  \\sqrt{ 1 - \\frac{4 \\alpha \\lambda}{\\eta c^{*2}} }\\right)-\\frac{\\sinh \\left(\\frac{\\eta c^* x}{2\\lambda}  \\sqrt{ 1 - \\frac{4 \\alpha \\lambda}{\\eta c^{*2}} } \\right)}{  \\sqrt{ 1 - \\frac{4 \\alpha \\lambda}{\\eta c^{*2}}  }}\\right].\n\\end{equation}\n\nThis profile extends to negative values of $x$ until the density of horizontal cells reaches $\\rho_h = \\tilde{\\rho}_t$ at the interface $x=x_t$. The value of $x_t$ may be obtained from the following non-dimensionalized form of the above equation:\n\n\\begin{equation}\n\\frac{\\tilde{\\rho}_t}{\\tilde{\\rho}_0} = e^{-\\kappa q} \\left(\\cosh(\\kappa q)- \\frac{\\sinh(\\kappa q)}{\\kappa} \\right),\n\\end{equation}\n\n\\noindent where $q = \\frac{\\eta c^*}{2 \\lambda} x_t$ and $\\kappa = \\sqrt{1 - \\frac{4 \\alpha \\lambda } {\\eta c^{*2}} }$. Finally, at the interface, we insert the boundary condition for the balance of cell flux, which states:\n\n\\begin{equation}\n\\tilde{\\rho}_t \\left( \\gamma_+  - \\frac{\\xi \\beta}{c^*} \\right) = \\nabla \\rho_h^{(P)}(x=x_t).\n\\end{equation}\n\nThis equation has the following non-dimensional form:\n\n\\begin{equation}\n\\frac{\\tilde{\\rho}_t}{\\tilde{\\rho}_0} = \\frac{ e^{-w} \\sinh ({w^2 - \\delta^2}) }{\\sqrt{w^2 - \\delta^2}} \\left( \\frac{\\xi \\chi \\delta}{4w} - \\frac{w}{\\delta} - \\sqrt{ \\left(\\frac{w}{\\delta} + \\frac{\\xi \\chi \\delta}{4w} \\right)^2 - \\delta \\left[ 1 + \\chi (\\xi - 1) \\right] } \\right)^{-1},\n\\end{equation}\n\n\n\\noindent where $\\delta^2= \\alpha \\eta x_t^2 \/ \\lambda$, $w = \\eta x_t v \/ 2 \\lambda$, and $\\chi = \\beta \/ \\alpha$. This equation determines the speed of the expanding front, provided our assumption holds that the surface pressure exceeds the threshold surface pressure for reorientation in the mixed interior.\n\n\\subsection*{\\textbf{The assumption of simple verticalization in the mixed interior can break down}}\n\nFor the above solution to be consistent, the surface pressure in the mixed interior region must always exceed the threshold surface density $\\tilde{\\rho}_{t}$ for reorientation:\n\n\\begin{equation}\n\\rho_h + \\xi \\rho_v > \\tilde{\\rho}_{t}.\n\\end{equation}\n\n\\noindent The density $\\rho_v$ of vertical cells is obtained by integrating the density of horizontal cells:\n\n\\begin{equation}\n\\rho_v = -\\frac{\\beta}{c^*} \\int_0^x \\rho_h dx.\n\\end{equation}\n\n\\noindent Inserting the solution for $\\rho_h$ from above (see Supplementary Eq. \\ref{eq:hord}), we find that $\\rho_v$ is given by:\n\n\\begin{equation}\n\\rho_v = \\frac{\\beta}{c^* \\gamma_+} (1 - e^{\\gamma_+ x}). \n\\end{equation}\n\n\\noindent The condition for the surface density to exceed the threshold surface density is:\n\n\\begin{equation}\n\\tilde{\\rho}_t \\left( \\frac{\\xi \\beta}{c^* \\gamma_+} - \\frac{\\xi \\beta}{c^* \\gamma_+} e^{\\gamma_+ x} + e^{\\gamma_+ x} \\right) > \\tilde{\\rho}_t.\n\\end{equation}\n\n\\noindent The slope of the surface density is given by $(\\gamma_+ - \\xi \\beta \/ c^* )e^{\\gamma_+ x}$. The exponential part is always positive, and its prefactor is:\n\n\\begin{equation}\n(\\gamma_+ - \\xi \\beta \/ c^* ) = \\frac{-(\\xi \\beta + c^{*2}) + \\sqrt{(\\xi \\beta + c^{*2})^2 - 4 \\frac{\\lambda}{\\eta} c^{*2} (\\alpha + \\xi \\beta - \\beta)}}{2c^*(\\lambda\/\\eta)}.\n\\end{equation}\n\n\\noindent For large enough ratio of growth rate to verticalization rate, i.e. $\\alpha \/ \\beta > 1 - \\xi$, this quantity is always negative since $\\xi \\beta + c^{*2} > \\sqrt{(\\xi \\beta + c^{*2})^2 - 4 \\frac{\\lambda}{\\eta} c^{*2} (\\alpha + \\xi \\beta - \\beta) }$. In this case, the surface density increases monotonically as $x\\rightarrow -\\infty$, and the surface density always exceeds the threshold surface density for verticalization in the mixed interior.\n\n\\subsection*{\\textbf{For high verticalization rates, dynamical isobaricity determines the cell density profiles}}\n\nIn the previous section, we found that our candidate solution for the steady-state cell density in the mixed interior could yield a surface density profile that was too small to sustain the assumed verticalization. Specifically, the solution fails when $\\alpha \/ \\beta < 1 - \\xi$. Intuitively, this occurs because the horizontal cells become vertical faster than the maximum rate at which the combined effects of cell growth and cell transport can replenish the threshold surface density of cells needed for further verticalization to occur. Thus, when $\\alpha \/ \\beta < 1-\\xi$, the surface density must constantly fluctuate between verticalizing and non-verticalizing values. This fluctuation stabilizes the surface density throughout the mixed interior at the verticalization threshold. The stability of the uniform surface density is apparent from examining the equation for the surface pressure $p \\sim \\rho_h + \\xi \\rho_v - \\tilde{\\rho}_0$:\n\n\\begin{equation}\n\\dot{p}= \\frac{\\lambda}{\\eta} p'' + c^* p' + \\alpha p + \\left( \\xi \\beta - \\beta \\right) \\Theta(p - p_t) p.\n\\end{equation}\n\n\\noindent Consider a distribution of cells throughout the mixed interior such that the surface pressure is everywhere equal to the threshold surface pressure for verticalization. Here, a verticalization event may bring the total surface pressure below the threshold surface pressure at some particular location. At such a location, the total surface pressure is at a local minimum, which means that $p' = 0$ and $p'' > 0$. Therefore the rate of change of surface pressure is positive:\n\n\\begin{equation}\n\\dot{p} =  \\frac{\\lambda}{\\eta} p'' + \\alpha p > 0.\n\\end{equation}\n\n\\noindent Conversely, if the total surface pressure ever exceeds the threshold surface pressure at some location, we have $p' = 0$ and $p'' < 0$, and the rate of change in the surface pressure must be negative:\n\n\\begin{equation}\n\\dot{p}=  \\frac{\\lambda}{\\eta} p'' +\\alpha p + \\left( \\xi \\beta - \\beta \\right) \\Theta(p - p_t) p.\n\\end{equation}\n\n\\noindent Therefore, in the regime of rapid verticalization, any deviation of the total surface pressure away from the threshold surface pressure will decay in time. This argument suggests that at any specific location in the mixed interior, the surface pressure will be close to the threshold surface pressure, and the horizontal cells will constantly fluctuate between verticalizable and non-verticalizable conditions. To predict the cell density profile in the mixed interior, we assume that the surface pressure is maintained at the threshold surface pressure by cells that spend a fraction of time $\\kappa$ in the verticalizable state. This results in the following equations of ``steady state'':\n\n\\begin{equation}\n0 = c^* \\rho_h' + \\left[ (\\alpha - \\beta)\\kappa + \\alpha(1 - \\kappa) \\right] \\rho_h,\n\\end{equation}\n\n\\begin{equation}\n0 = c^* \\rho_v' + \\beta \\kappa \\rho_v.\n\\end{equation}\n\nSince the surface pressure is constant, we have $\\tilde{\\rho}_t = \\rho_h + \\xi \\rho_v$, which implies:\n\n\\begin{equation}\n\\rho_h' = -\\xi \\rho_v',\n\\end{equation}\n\n\\noindent which, together with the above equations, allows us to solve for the fraction of time $\\kappa$ spent in the verticalizable state:\n\n\\begin{equation}\n\\kappa = \\frac{\\alpha}{\\beta (1 - \\xi)}.\n\\end{equation}\n\nInserting this into the equations of steady state yields horizontal and vertical cell densities that decay and grow exponentially, respectively, at a rate $\\mu$ given by:\n\n\\begin{equation} \\label{eq:dec}\n\\mu = \\frac{\\alpha \\xi }{c^*(1-\\xi)},\n\\end{equation}\n\n\\noindent which, interestingly, does not depend on the verticalization rate $\\beta$. The resulting horizontal and vertical cell surface density profiles determine the boundary conditions at the interface between the mixed interior and the horizontal cell periphery, which thereby determines the overall expansion speed of the biofilm.\n\n\\subsection*{\\textbf{Phase diagram for verticalizing biofilms}}\n\nWe summarize our results from the continuum modeling in the phase diagram in Supplementary Fig. 11\\textbf{a}. The dynamics of the cell densities fall into three different regimes, depending on the values of the growth rate $\\alpha$, the verticalization rate $\\beta$, and the ratio $\\xi$ of vertical to horizontal cell footprints. For $\\alpha>\\beta$, the overall number of horizontal cells increases exponentially, which precludes the existence of a stable, linearly-propagating front. For $\\alpha < \\beta$, the biofilm develops into a mixed interior of vertical and horizontal cells surrounded by a periphery of horizontal cells, which both spread outwards linearly in time. For $\\beta (1-\\xi) < \\alpha < \\beta$, the surface pressure and density continue to build up inside the mixed interior and ultimately saturate at values above the threshold values for verticalization. However, when $\\beta (1-\\xi) < \\alpha$, verticalization can deplete the cell density in the mixed interior more rapidly than cell density can be replenished by cell growth and cell transport due to gradients in surface pressure. Thus, in this regime, the surface pressure and density rapidly fluctuate around the threshold values for verticalization, which effectively tunes the verticalization rate to $\\beta (1-\\xi) = \\alpha$.\n\n\\subsection*{\\textbf{The effect of vertical cell transport}}\n\nIncorporating an in-plane velocity for vertical cells into the equations for the change in cell densities yields:\n\n\\begin{equation}\n\\dot{\\rho_h} + \\nabla \\cdot ( \\rho_h \\boldsymbol{v}) = \\left[ \\alpha - \\beta \\Theta(p - p_t) \\right] \\rho_h,\n\\end{equation}\n\n\\begin{equation}\n\\dot{\\rho_v} + \\nabla \\cdot ( \\rho_v \\boldsymbol{v})=  \\beta \\Theta(p - p_t) \\rho_h,\n\\end{equation}\n\n\\begin{equation}\n- \\eta \\tilde{\\rho}_{\\mathrm{tot}} \\boldsymbol{v} = \\nabla p.\n\\end{equation}\n\nThe presence of vertical cell transport could potentially influence the dynamics in regions with a finite vertical cell density, i.e. the mixed interior. For the isobaric regime $\\alpha < \\beta(1-\\xi)$, the vertical cell transport has no effect because there the cell velocity is zero in the mixed interior due to the uniformity of the surface pressure. Furthermore, for $\\alpha < \\beta$, the same argument given above (see \\textbf{Existence of a linearly-expanding front}) implies that there is no stable, linearly-expanding steady state. To examine how vertical cell transport impacts the non-isobaric regime, we incorporated this effect into the simulations. We found that the presence of vertical cell transport does not qualitatively change the cell density profiles. Furthermore, incorporating vertical cell transport only results in small changes to the cell densities of around a few percent (inset, Supplementary Fig. 11b). Thus, although the presence of vertical cell transport can slightly affect the quantitative details of the cell density profiles in the non-isobaric regime, vertical cell transport does not influence the qualitative behaviors of the different phases of our model.\n\n\\subsection*{\\textbf{Growth in two spatial dimensions}}\n\nIn two spatial dimensions, the equation for the total cell density (assuming no vertical cell transport) becomes:\n\n\\begin{equation}\n\\dot{\\tilde{\\rho}}_{\\mathrm{tot}} = \\gamma \\left(\\frac{1}{r}\\frac{d}{dr}\\tilde{\\rho}_{\\mathrm{tot}} + \\frac{d^2}{dr^2}\\tilde{\\rho}_{\\mathrm{tot}} \\right) + \\alpha  \\rho_h - (1 - \\xi) \\beta \\Theta(\\tilde{\\rho}_{\\mathrm{tot}} - \\tilde{\\rho}_t) \\rho_h.\n\\end{equation}\n\nThe effect of the additional spatial dimension is to add an advection-like term $\\frac{1}{r}\\frac{d}{dr}\\tilde{\\rho}_{\\mathrm{tot}}$ to the change in cell density, which, as above, does not alter the qualitative features of the model's phase behavior. Furthermore, this term becomes less important for larger values of the radial coordinate $r$. To estimate the importance of this term, we compare $r$ to the maximum value of $\\frac{d}{dr}\\tilde{\\rho}_{\\mathrm{tot}}$, which is given by $\\frac{d}{dr}\\tilde{\\rho}_{\\mathrm{tot}}=\\rho_0 c^* \/ \\gamma$ at the edge of the biofilm. This estimate suggests that for large values of the radial coordinate $r \\gg \\rho_0 c^* \/ \\gamma$, the dynamics become effectively one-dimensional.\n\n\\subsection*{\\textbf{The effect of surface curvature}}\n\nHow does surface curvature influence the global build-up of surface pressure? Here, we answer this question by extending our incompressible model (see \\textbf{Origin of vertical ordering}) to the case of a curved surface. Specifically, we consider an incompressible biofilm growing exponentially at a rate $\\alpha$ along a sphere of radius $R$. We assume that growth begins from a single point and that the biofilm remains azimuthally symmetric. Thus, the extent of the biofilm along the surface, defined as the geodesic distance along the surface between the origin $\\mathcal{O}$ of growth and the edge of the biofilm, is described by a single coordinate $R_B$. For a given value of $R_B$, the surface area covered by the biofilm is given by:\n\n\\begin{equation}\nA = 2 \\pi R^2 \\left( 1 - \\cos \\left( \\frac{R_B}{R} \\right) \\right).\n\\end{equation}\n\nThe exponential growth of $A$ implies that the velocity $v = dR_B \/ dt$ of the biofilm extent is given by:\n\n\\begin{equation}\nv = \\frac{\\alpha A}{2\\pi \\sqrt{1 - R \\left( 1 - \\frac{A}{2\\pi R^2} \\right)^2}}.\n\\end{equation}\n\nSimilarly, the local velocity of a point at a distance $r$ along the surface from $\\mathcal{O}$ is given by the same expression, but with $A$ instead giving the area of the spherical cap that contains all points nearer to the origin of growth than $r$. As before for growth on a flat surface, we assume that this velocity is driven by the local gradient $dp\/dr$ of surface pressure:\n\n\\begin{equation}\n\\frac{dp}{dr} = -\\eta v,\n\\end{equation}\n\n\\noindent where $\\eta$ is the surface drag coefficient of the cell medium. Spatially integrating this equation gives the pressure field:\n\n\\begin{equation}\np = 2 \\eta \\alpha R^2 \\left[ \\log \\cos \\left( \\frac{r}{2R} \\right)  - \\log \\cos \\left( \\frac{R_B}{2R} \\right)   \\right],\n\\end{equation}\n\n\\noindent which is peaked at the origin of growth.\n\nThis equation indicates that, for a given extent $R_B$, the effect of surface curvature is to increase the surface pressure throughout the biofilm. Intuitively, this increase arises because, for a fixed biofilm extent $R_B$, the biofilm footprint is larger for a flat surface than for a spherical surface. This difference in footprint implies that, assuming equal growth rates, cells must spread out more rapidly on the spherical surface to accommodate the increase in total biofilm surface area. In turn, the increase in biofilm expansion speed implies a larger gradient in surface pressure. This argument suggests that negatively-curved surfaces, e.g. saddle-like surfaces, would have the opposite effect; they would decrease the rate at which surface pressure builds up.\n\nFor values of $r \\ll R$, the surface of the sphere is effectively flat on the scale of the biofilm, and the equation for the surface pressure on a sphere reduces to the expression for a flat surface. However, when the biofilm extent becomes comparable to the radius of curvature of the surface, the increase in surface pressure becomes substantial (Supplementary Fig. 14). This increase in surface pressure can trigger verticalization at much smaller values of $R_B$ for a spherical surface than for a flat surface. Thus, surface curvature provides an additional geometrical mechanism that regulates the transition of biofilms to three-dimensional growth.\n\n\\newpage\n\n\\section*{\\textbf{Supplementary Figure 11: Phase diagram for verticalizing biofilms}}\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{fig_s13.jpg}\n\\caption{\\label{fig:FG9019}\nPhases of front propagation of verticalizing biofilms. (\\textbf{a}) Phase diagram of continuum model for verticalizing biofilms. For $\\alpha < \\beta(1-\\xi)$, the mixed interior is isobaric, i.e. the surface pressure is uniform. For $\\beta  >\\alpha > \\beta(1-\\xi)$, the surface pressure of the mixed interior decreases monotonically with the radial distance from the biofilm center. For $\\alpha > \\beta$, there is no stable, steady-state linearly-propagating front. (\\textbf{b}) Numerical simulations of the continuum model assuming no vertical cell transport, showing radial densities of horizontal cells ($\\rho_h$, blue), vertical cells ($\\rho_v$, red), and total density ($\\tilde{\\rho}_{\\mathrm{tot}}$, black), versus shifted radial coordinate $\\tilde{r}$ for isobaric regime with $\\tilde{\\rho}_0 = \\SI{1}{\\meter^{-2}}$, $\\tilde{\\rho}_t = \\SI{1.5}{\\meter^{-2}}$, $\\beta = 2.5 \\alpha$, and $\\xi=0.5$ (left) and non-isobaric regime with $\\tilde{\\rho}_0 = \\SI{1}{\\meter^{-2}}$, $\\tilde{\\rho}_t = \\SI{1.5}{\\meter^{-2}}$, $\\beta = 1.25 \\alpha$, and $\\xi=0.5$ (right). Dashed gray line shows $\\tilde{\\rho}_t$. Inset of right panel shows the change $\\Delta \\tilde{\\rho}_{\\mathrm{tot}} \/ \\tilde{\\rho}_{\\mathrm{tot,vel}}$, where $\\Delta \\tilde{\\rho}_{\\mathrm{tot}} = \\tilde{\\rho}_{\\mathrm{tot,vel}} - \\tilde{\\rho}_{\\mathrm{tot}}$ and $\\tilde{\\rho}_{\\mathrm{tot,vel}}$ is the total cell density for the variation of the continuum model that assumes vertical cell transport, versus shifted radial coordinate $\\tilde{r}$.}\n\\end{figure}\n\n\\newpage\n\n\\section*{Supplementary Figures 12-13: Fitting the continuum model to the agent-based model}\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{fig_s14.jpg}\n\\caption{\\label{fig:FG9004}\nDefining ``pressure'' in the agent-based model. (\\textbf{a}) Average of normalized total force $f$, where we define $f$ as the total force on a cell divided by the projected area of the cell onto the surface (i.e., the cell footprint), and the average is taken over horizontal cells (blue data points) and vertical cells (red data point) in the mixed interior with projected cell cylinder length around $\\ell$, versus projected cell cylinder length $\\ell$ for ten simulated biofilms with initial cell cylinder length $\\ell_0 = \\SI{1.2}{\\micrometer}$. Black dashed line shows average of $f$ over both horizontal and vertical cells. (\\textbf{b}) Average surface pressure $p$, where we define $p$ as the total force on a cell divided by the perimeter of the cell footprint, and the average is taken over horizontal cells (blue data points) and vertical cells (red data point) in the mixed interior with projected cell cylinder length around $\\ell$, versus projected cell cylinder length $\\ell$ for ten simulated biofilms with initial cell cylinder length $\\ell_0 = \\SI{1.2}{\\micrometer}$. Black dashed line shows the average of horizontal and vertical surface pressures.\n}\n\\end{figure}\n\n\n\\textbf{Mapping the surface pressure in the continuum model to forces in the agent-based model} What is the relationship between the surface pressure in our continuum model and the microscopic, cell-cell contact forces? On the cell scale, the disorder of the cell configuration yields forces on cells that are extremely heterogeneous in space and time, even for a fixed radial distance along the moving front (Supplementary Fig. 10). A further contribution to this disorder comes from polydispersity in the cell lengths. That is, since larger cells can have more cell-cell contacts, we expect the forces acting on a cell to increase with cell size, on average. Therefore, to understand how the cell-cell contact forces relate to the surface pressure in the continuum model, we considered the forces acting on cells as a function of cell length.\n\nFor convenience, we focused on the mixed interior of the biofilm, since there our continuum theory predicts a uniform value of the macroscopic surface pressure. We found that the sum of the magnitudes of the in-plane forces on such cells scales with the perimeter of the cell footprint, but not with other quantities such as the cell footprint area (Supplementary Fig. 12), which is consistent with the behavior of an object embedded inside a two-dimensional, homogeneous fluid in mechanical equilibrium. Therefore, we quantified the surface pressure acting on a cell within the agent-based model as the sum of the magnitudes of the in-plane forces acting on a cell divided by the perimeter of its footprint. \\\\\n\n\\newpage\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{fig_s11.jpg}\n\\caption{\\label{fig:FG9006}\nObtaining parameters for the continuum model from the agent-based model. (\\textbf{a}) Average surface pressure $p_{\\mathrm{int}}$ acting on cells in the mixed interior versus average interior density $\\tilde{\\rho}_{\\mathrm{int}}$, where $\\tilde{\\rho}_{\\mathrm{int}}$ is defined as the total footprint of cells in the mixed interior divided by the surface area of the mixed interior. Each data point is extracted using a different value of the initial cell length $\\ell_0$, and averaged over ten simulated biofilms. (\\textbf{b}) Interpolated expansion speed $c^*$ versus threshold surface density $\\tilde{\\rho}_{t}$, defined as the cell density in an annular window of $\\SI{2}{\\micrometer}$ centered at the radius of the maximum horizontal cell density, for agent-based models simulated over a range of different initial cell lengths. Inset shows the interpolated threshold surface density versus initial cell cylinder length. For reference, the dashed green lines in the main panel and the inset indicate the same five values of threshold surface density.\n}\n\\end{figure}\n\n\\textbf{Choice of parameters for the continuum model} We fitted the parameters of our continuum model from results of the agent-based model as follows:\n\n\\begin{itemize}\n\\item { Cell stiffness $\\lambda$: we fitted this parameter by measuring the average surface pressure and density in the central region of the biofilm for a range of initial cell lengths (Supplementary Fig. 13). A linear fit was then performed to extract $\\tilde{\\rho}_0$ and $\\lambda$. Data were averaged over ten simulated biofilms. }\n\\item {  Threshold surface density $\\tilde{\\rho}_{t}$: the threshold surface density was calculated by averaging the pressure acting on all horizontal cells in a small radial window around the peak horizontal cell radial density (inset, Supplementary Fig. 13b). }\n\\item { The verticalization rate $\\beta$ was obtained by fitting our continuum model for the horizontal cell density profile to the mixed interior of the biofilm in agent-based simulations, which has a decay constant given by Supplementary Eq. \\ref{eq:dec}. For $\\ell_0 = 1$, comparable to those in our experiments, we find $\\beta=\\SI{2.5}{hour^{-1}}$.}\n\\item { The ratio of vertical to horizontal cell footprints $\\xi$: we computed $\\xi$ as the average ratio of footprints in the mixed interior, where we defined footprint as the projected area of the cell onto the surface. }\n\\item { The cell growth rate $\\alpha$ was determined from Supplementary Eq. \\ref{eq:td} as $\\alpha = \\log 2 \/ t_d$, which yields $\\alpha = \\SI{1.4}{hour^{-1}}$.}\n\\item { The cell stiffness $\\lambda$: we chose the cell stiffness equal to the Young's modulus times the cell radius, with the Young's modulus $Y=\\SI{450}{\\pascal}$ measured from bulk rheology, which yields $\\lambda=\\SI{360}{\\pico\\newton\/\\micrometer}$.}\n\\item { Surface viscosity coefficient $\\eta$: we determined $\\eta$ by fitting the expansion speeds in the continuum model to those of the agent-based model (Fig. 4a), which yields $\\eta \\simeq 10^5\\ \\SI{}{\\pascal\\second}$. }\n\\end{itemize}\n\n\\newpage\n\n\\section*{\\textbf{Supplementary Figure 14: The effect of surface curvature}}\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.4\\columnwidth]{fig_s12.jpg}\n\\caption{\\label{fig:FG9007}\nSurface pressure $p$ at the origin $\\mathcal{O}$ of growth versus biofilm extent $R_B$ for expansion along the surface of a sphere (solid curve) and a flat surface (dashed curve). Inset shows schematic illustration of biofilm expansion along the surface of a sphere.}\n\\end{figure}\n\n\\newpage\n\n\\section*{Supplementary Figure 15: Analysis of biofilms with cell-to-cell adhesion}\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{fig_s15.jpg}\n\\caption{\\label{fig:FG9001}\nDevelopment of experimental biofilms with cell-to-cell adhesion. (\\textbf{a}) Top-down and perspective visualizations of the surface layer of a living biofilm with cells producing the cell-cell adhesion protein RbmA, showing positions and orientations of horizontal (blue) and vertical (red) surface-adhered cells as spherocylinders of radius $R=\\SI{0.8}{\\micrometer}$, with the surface shown at height $z=\\SI{0}{\\micrometer}$ (brown). Cells with $n_z < 0.5$ ($>0.5$) are considered horizontal (vertical), where $\\boldsymbol{\\hat{n}}$ is the orientation vector. Scale bar: $\\SI{5}{\\micrometer}$. (\\textbf{b}) 2D growth of a biofilm surface layer containing cells that produce RbmA. The color of each spatiotemporal bin indicates the fraction of vertical cells at a given radius from the biofilm center, averaged over the angular coordinate of the biofilm (gray regions contain no cells). The dashed pink line shows the onset of verticalization. The black dashed line shows the edge of the biofilm. The insets shows the distribution of cell orientations at time $t=300$ minutes, with color highlighting horizontal and vertical orientations.\n}\n\\end{figure}\n\nFor the majority of the data presented, we used a \\emph{V. cholerae} strain in which the gene (\\emph{rbmA}) encoding the cell-to-cell adhesion protein RbmA, was deleted. We also performed experiments with a \\emph{V. cholerae} strain in which the \\emph{rbmA} gene was present and so RbmA protein was produced at wild type levels (Supplementary Fig. 15). We found that the transition to verticalization still occurred at roughly the same time, albeit with somewhat reduced vertical ordering. Interestingly, the horizontal expansion speed of the RbmA$^+$ biofilm was roughly $20\\%$ more rapid than the biofilm we considered in the main text. This increase in expansion speed is consistent with the reduction in vertical ordering that we observed for the RbmA$^+$ biofilm.\n\n\\newpage\n\n\\section*{Captions for Supplementary Videos}\n\n\\textbf{Supplementary Video 1} Growth of a \\emph{V. cholerae} biofilm cluster, showing cross-sectional images of the bottom cell layer. The strain constitutively expresses \\emph{mKO}. The viewing window is 45 by 45 $\\SI{}{\\micrometer^2}$ and the total duration is 8 hours with 10 min time steps.\n\n\\textbf{Supplementary Video 2} Visualization of the surface layer of a modeled biofilm with initial cell cylinder length $\\ell_0 = \\SI{1}{\\micrometer}$, showing positions and orientations of horizontal (blue) and vertical (red) surface-adhered cells as spherocylinders of radius $R=\\SI{0.8}{\\micrometer}$, with the surface shown at height $z=\\SI{0}{\\micrometer}$ (brown). Cells with $n_z < 0.5$ ($>0.5$) are considered horizontal (vertical), where $\\boldsymbol{\\hat{n}}$ is the orientation vector. Scale bar: $\\SI{3}{\\micrometer}$. The total duration is 10 hours.\n\n\\textbf{Supplementary Video 3} Visualization of the surface layer of a modeled biofilm with initial cell cylinder length $\\ell_0 = \\SI{2}{\\micrometer}$, showing horizontal (blue) and vertical (red) cells as spherocylinders, the surface (brown), and cell-to-cell contact forces (yellow lines connecting the centers of cells, with thicknesses proportional to the force). Cells with $n_z < 0.5$ ($>0.5$) are considered horizontal (vertical), where $\\boldsymbol{\\hat{n}}$ is the orientation vector. The length of the scale bar is $\\SI{3}{\\micrometer}$, and its thickness corresponds to $\\SI{300}{\\pico\\newton}$.\n\n\\textbf{Supplementary Video 4} Numerical simulation of the continuum model assuming no vertical cell transport, showing radial densities of horizontal cells ($\\rho_h$, blue), vertical cells ($\\rho_v$, red), and total density ($\\tilde{\\rho}_{\\mathrm{tot}}=\\rho_h + \\xi \\rho_v$, black), versus radial coordinate $r$ for isobaric regime with $\\tilde{\\rho}_0 = \\SI{1}{\\meter^{-2}}$, $\\tilde{\\rho}_t = \\SI{1.5}{\\meter^{-2}}$, $\\beta = 2.5 \\alpha$, and $\\xi=0.5$. Dashed gray line shows $\\tilde{\\rho}_t$.\n\n\\textbf{Supplementary Video 5} Expansion of \\emph{V. cholerae} biofilm clusters grown with the drug A22 at a concentration of $\\SI{0.4}{\\microgram\/\\milli\\liter}$ (left) and the drug Cefalexin at a concentration of $\\SI{4}{\\microgram\/\\milli\\liter}$ (right). Cross-sectional images of the bottom cell layers are shown. The strain constitutively expresses \\emph{mKO}. Scale bar: $\\SI{30}{\\micrometer}$. The total duration is 10 hours with 30 min time steps.\n\n\\newpage\n\n\\section*{\\textbf{Supplementary Discussion}}\n\n\\subsection*{\\textbf{The effects of cell-scale geometrical and mechanical properties}} We have focused the current analysis on mutant \\emph{V. cholerae} biofilms that have been engineered to have simpler interactions than wild type cells in biofilms (see Methods). For example, wild type cells are slightly curved\\cite{S7}, and produce a cell-to-cell adhesion protein called RbmA\\cite{S3}. The presence of cell curvature endows the cell with an additional rotational degree of freedom, which provides an additional direction along which a mechanical instability can proceed. In curved cells, due to the reduced extent of the cell in the transverse direction, we expect such an instability to occur with a lower reorientation threshold. This effect could allow verticalization to occur at lower values of surface pressure by proceeding in two stages: first by rotating away from the surface along the transverse direction, and then by being peeled off the surface at the remaining point(s) of contact. To understand how cell-to-cell adhesion could influence verticalization, we preliminarily analyzed biofilms that produce the cell-to-cell adhesion protein and found that the horizontal to vertical transition still occurs, albeit with somewhat reduced vertical ordering (Supplementary Fig. 15). Understanding the modifying effects of cell curvature and cell-cell adhesion will be important directions for future studies.\n\nOur agent-based model does not explicitly incorporate the VPS matrix secreted by cells. A more detailed treatment of the VPS matrix could potentially explain small differences we observed between the orientational patterning and spreading dynamics of the agent-based model biofilms and those of the experimental biofilms. Furthermore, previous studies have shown that matrix production can allow the bacterial cells of the biofilm to establish an osmotic pressure difference between the biofilm and the external environment\\cite{S9}, which could potentially impact the mechanics of the verticalization transition. Thus, understanding the interplay between cell and matrix mechanics will be an important direction for future studies.\n\nBy varying the parameters in our agent-based model that reflect cell-scale features, we observed a wide range of biofilm architectures of varying size, shape, orientational ordering, and dimensionality. Importantly, with regard to the features we analyzed, we found that the verticalization transition relies primarily on the presence of cell-to-surface adhesion, and so we expect our results to apply to a wide range of bacterial biofilms. In particular, our findings on mechanical instabilities are general enough to describe analogous transitions for other cell shapes, including spherical cells \\cite{S10, S11, S12}, for which compression will induce vertical center of mass displacements. There are other types of biofilm architectures that we did not observe in our simulations. For example, \\emph{Bacillus subtilis} have been observed to create planar Y-shaped formations, which appear to have an extended bending mode\\cite{S13}. In addition, \\emph{Escherichia coli} colonies that are compressed against the surface undergo a variant of the 2D-3D transition\\cite{S14}, but with the cells reportedly remaining horizontal in a layered, wedding-cake type formation\\cite{S15}. \\emph{Pseudomonas aeruginosa} biofilms can form 3D streamers under the influence of flow\\cite{S16}. These examples of known architectures already suggest a grand challenge in the study of biofilms: we must develop a systematic method to account for the diversity of architectures that can be produced by local mechanical interactions.\n\n\\subsection*{\\textbf{The effects of surface curvature}}  For simplicity, we have considered expansion of biofilms along a flat surface. However, many surfaces in nature are curved, which would locally change the cell-to-cell and cell-to-surface contact geometries, as well as globally influencing the build-up of pressure throughout a biofilm. We expect the resulting changes in cell-cell contact geometry to decrease the threshold surface pressure for reorientation by facilitating the application of verticalizing torque. On the other hand, changes in the cell-to-surface geometry can both increase and decrease the threshold surface pressure for verticalization depending on the sign of the curvature. Thus, cells on a concave surface might undergo more spreading in two dimensions, while those on a convex surface might undergo more rapid three-dimensional expansion. Finally, surface curvature can increase or decrease the rate at which pressure builds up throughout a biofilm, since spreading out to a given distance has the consequence of covering a smaller or larger footprint depending on whether the surface is ball-like or saddle-like, respectively. This effect becomes significant when the extent of the biofilm along the surface becomes comparable to the radius of surface curvature (Supplementary Note), and thus could serve as an additional mechanism to modulate the onset of vertical growth.\n\n\\subsection*{\\textbf{Evolution and adaptation of global biofilm morphology}} Our results suggest that bacteria have harnessed the physics of mechanical instabilities to generate complex architectures. What impact does a biofilm's morphology have on its growth and survival? \\emph{V. cholerae} biofilm clusters have been observed to form as monocultures that exclude competitors\\cite{S17}. When two clusters impinge upon one another, for example during resource competition, the structural properties of a biofilm become crucial determinants of its success in edging out competitors\\cite{S17}. The morphology of a biofilm could also be important in driving how biofilm cells access nutrients. Nutrients can be delivered from surfaces, e.g. when the biofilm forms on a solid food source such as chitin or marine snow\\cite{S18}, as well as by the surrounding fluid, e.g. via the diffusion of oxygen and other chemicals. Therefore, since both two-dimensional and three-dimensional growth can be beneficial, we expect a balance between horizontal and vertical growth to be most advantageous. We therefore speculate that individual cell features have evolved in response to selective pressures on the global morphologies of biofilm communities\\cite{S10, S11}. Since optimal morphology may be condition dependent, cells may also have evolved adaptive strategies for biofilm formation, which could be investigated experimentally by screening environmental influences on cell size, shape, and surface adhesion. Intriguingly, data exists which suggests that, in nature, \\emph{V. cholerae} undergoes morphological changes during starvation, including developing into small cocci and long filaments\\cite{S12}.\n\n\\subsection*{\\textbf{Dynamical isobaricity}} Our study of a two-fluid model for verticalizing biofilms led us to discover a novel type of front propagation in which mechanical feedback stabilizes a linearly-expanding density profile. Remarkably, this density profile is precisely uniform in the biofilm interior starting at some finite distance from the front, whereas previous models of front propagation saturate continuously toward uniformity \\cite{S22, S23, S24}. The spatial uniformity is a hallmark of an isolated fluid in mechanical equilibrium. However, in our system, the internal state and volume of the biofilm surface layer are constantly changing due to cell growth. Verticalizing biofilms thus provide a striking example of how equilibrium-like features can emerge naturally in a system that is driven far from equilibrium. Indeed, the self-organized nature of this process yields a universal behavior for the expansion speed that is independent of details of the mechanical feedback, including the verticalization rate $\\beta$ (which sets the rate of feedback) and the ratio of vertical to horizontal footprints $\\xi$ (which sets the strength of feedback).\n\n\\subsection*{\\textbf{Fluctuations in biofilm shape and pressure}} Our continuum model, along with our agent-based model, both address the case of growth in nutrient-rich conditions, which pertains to our experiments. These models capture the observations of compact, circular morphologies. However, bacterial range expansions have also been studied under nutrient-poor conditions, which are known to cause branching morphologies\\cite{S25}. Thus, it would be interesting to investigate the connections between mechanical and chemical feedback on biofilm growth.\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\IEEEPARstart{A}{ssistive} robots have been applied in gait rehabilitation for Parkinson's Disease (PD), stroke, spinal cord injury, and others \\cite{lunenburger2007biofeedback, louie2016powered}. The robots can take the form of exoskeletons (e.g. Lokomat and ReWalk \\cite{louie2016powered}), capable of monitoring the patients' joint motion in real time and providing assistive torques to guide the movement of patients \\textcolor{blue}{who require weight bearing assistance} \\cite{lunenburger2007biofeedback}. \\textcolor{blue}{On the other hand, patients capable of weight bearing can use wearable devices that provide simple feedback such as visual, auditory, and tactile cues to help their gait in both rehabilitation and everyday settings} \\cite{lunenburger2007biofeedback, sweeney2019technological}. Visual cues such as laser projections provide spatial information on where to step, whereas auditory and tactile cues provide temporal information on when to step using metronome beats or vibrations \\cite{sweeney2019technological}. \n\n\nExisting research on cues focuses on providing visual cues at a fixed distance or auditory\/tactile cues at a fixed pace calibrated to each patient \\cite{delval2008effect, schaefer2014auditory, Bachlin2010, Mikos2019}. \\textcolor{blue}{These cue provision paradigms have several limitations over long-term use and for diseases with progressive symptoms such as PD}. For instance, the cueing mechanisms are often used in conjunction with medications for PD treatment, and the same patient can respond differently to the cues depending on the medication state \\cite{sweeney2019technological}. In addition, long-term use of the cues can result in habituation, where the cues become less salient and lose their effectiveness. Patients \\textcolor{blue}{might also become reliant on the cues} even without gait abnormalities \\cite{ginis2018cueing}. Current cueing mechanisms do not address symptom fluctuation, habituation or cue-dependency and thus, there is a need to develop a cue adaptation strategy to address issues with static cues. \n\nWe propose an adaptive cue-provision framework that can simultaneously monitor the person's gait performance and provide personalized cues to change the person's gait to a \\textcolor{blue}{target} state. Personalized cues are provided by continuously learning a model of the individual's response to the provided cues, and utilizing the model to optimise cue selection. The performance of the adaptive cueing strategy is compared to two alternative approaches, the fixed cue and the proportional cue. The fixed approach implements the typical cueing approach in the literature. The proportional approach is a semi-adaptive strategy that generates cues based on individual user performance, but using a fixed control strategy. The results show that adaptive cueing outperforms the other two cueing methods in changing the participant's gait once the personalized response model has been learned. \n\n\\section{Related Work}\\label{sec:related_work}\n\\textcolor{blue}{A review of the} two primary features in an assistive feedback system, patient monitoring and providing personalized feedback, is presented in this section.\n\n\\subsection{Monitoring Gait Parameters}\nGait performance can be quantified using parameters such as stride length, cadence, velocity, and double support time \\cite{conklyn2010home}. Methods for calculating gait parameters for cue-provision can be categorized into approaches that are used in clinical or everyday settings. \n\n\\textcolor{blue}{In clinical settings, commercially available marker (e.g. VICON Motion System) or pressure-mat (e.g. GAITRite System) measuring devices can be used \\cite{conklyn2010home, wu2020novel}. Marker-based systems use cameras to track markers attached on the limbs, from which angles are derived to compute gait parameters. Pressure-mat based systems estimate the parameters using the position and timing of the foot landing on a pressure-sensitive mat. While these systems are considered the gold standard, they require specialized equipment and hence are typically employed in validation studies, rather than being used as a feedback signal to a cueing system.} \n\nSensors that are portable, unintrusive, and easy to set up, such as inertial measurement units (IMU) or encoders, have been embedded onto wearable devices to measure gait metrics outside of the clinical setting and provide information for \\textcolor{blue}{cue adjustment}. For instance, in \\cite{wu2020novel}, the authors developed a laser projection system that can be mounted onto a walking frame. The system adjusts the location of the visual cues based on the movement of the person measured through encoders embedded on the walking aid to ensure that the projection is always a fixed distance ahead of the person. Stride length has also been measured through sensor fusion algorithms using the gyroscope and accelerometer signals from IMUs to adjust for the location of the visual cues \\cite{ahn2017smart}. \n\n\n\n\\subsection{Providing Personalized Assistance}\n\\textcolor{blue}{Personalization of the cues usually involves changing cue modality, location, and form factors. For instance, visual cue personalization can include adjusting the projection according to the user's step length, or changing the location of the projection device on the user} (e.g. foot-strapped wearables like Path Finder LaserShoes (Walk with Path, Essex, England), walking aid based system like U-Step Walker (U-Step Mobility Products, Inc; Illinois, USA), or Augmented-reality glasses \\cite{ahn2017smart}).\n\nAuditory and tactile cue adjustments share common tuning parameters, such as the duration of the cues (i.e. continuous or on-demand), frequency of the cues (i.e. set speed or patient-specific speed), and timing of the cues (e.g. reactive or proactive, synchronization to the gait cycle events). \\textcolor{blue}{A characteristic unique to auditory cues is the provision of music melody, human voice, or metronome beats \\cite{Ghai2018}}. The parameters unique to tactile cues are the amplitude and the pattern of the cue. Various forms of vibration have been tested (e.g. constant \\cite{Punin2018}, or variable \\cite{yasuda2020development}) and can be provided via electrical stimulation \\cite{rosenthal2018sensory} or vibration motor \\cite{Punin2018}. The effect of the tactile cue location has also been examined \\cite{ sweeney2019technological, rosenthal2018sensory, pereira2016freezing}. \\textcolor{blue}{A recent study developed a wearable system that provides tactile cues to induce a new walking speed in healthy participants \\cite{zhang2020wearable}. The system, which consists of pressure resistive sensors, ERM motors, and an IMU sensor, monitors walking speed and gait events and adjusts feedback provided by the motors using a PI controller. While the closed-loop system was better at inducing the desired speed changes compared to an open-loop algorithm that provided feedback at a constant pace, it is unclear how the controller gains were selected. As well, the same gains were used across all participants.} \n\nOnline feedback adaption, including human-in-the-loop (HIL) optimization, has been investigated for robotic exoskeletons. In the HIL framework, real-time adjustment of the assistance is implemented based on the current performance of the user. Specifically, the HIL framework has been applied in optimizing the assistive force provided by exoskeletons to reduce metabolic cost during walking \\cite{kim2017human, zhang2017human, felt2015body}. A fundamental requirement for HIL is building a model that relates the input assistive force to the output performance metrics. Previous studies have used a set of pre-defined assistive forces to uniformly explore the parameter space for the model \\cite{zhang2017human}. Others have also investigated more sample-efficient methods without the initial parameter exploration by using gradient descent \\cite{kim2017human, felt2015body} or Bayesian optimization \\cite{kim2017human}. \\textcolor{blue}{Overall, the current cue adaptation strategy in gait rehabilitation is limited to the one-time adjustment to calibrate the cue for the user's height, preferred cadence, or preferred location. While adaptive cue-provision is under investigation, there is a lack of personalization to account for the user's immediate motor capability and response}. \n\n\\section{Proposed Approach}\n\\label{sec:propApp}\nThe proposed adaptive cue-provision framework, shown in Figure~\\ref{sysID}, can continuously monitor the person's gait performance and periodically adjust the assistance based on the person's response to the feedback. \n\n\\begin{figure*}[htb]\n\\begin{center}\n\\includegraphics[width=\\textwidth]{images\/block-diagram-a4.pdf}\n\\caption{\\textcolor{blue}{The proposed system is a feedback loop that consists of the human, gait measurement and estimation, cue response learning and cue provision. The human gait model computes metrics to monitor gait performance using the data from the IMU. The Gaussian process (GP) regression model then uses the gait performance metrics, along with the history of the provided cues, to model the gait performance as a function of the provided cues. Finally, the optimization algorithm utilizes the GP model to provide personalized cues that would prompt the participant's current state (the blue circle in the optimization block) to move towards the target state (the red circle)}.}\n\\label{sysID}\n\\end{center}\n\\end{figure*}\n\n\\subsection{Online Gait Parameter Estimation}\nThe canonical dynamical system (CDS) proposed in \\cite{petrivc2011line} \\textcolor{blue}{is used in the study. The system models periodic signals using Fourier series and} has been previously applied in online learning and modelling of an individual's gait \\cite{waugh2019online}. The gait is captured by a single inertial measurement unit (IMU) fixed above the individual's knee of the dominant leg. The sensor is oriented such that the y-axis aligns with the normal of the sagittal plane. Once the gait model is learned, the associated model coefficients allows metrics to be derived for continuous monitoring. The CDS is defined as:\n\n\\begin{equation}\n    \\hat{y_t} = \\sum_{m=0}^{M}\\hat\\alpha_{m,t} \\sin(m\\hat\\phi_t) + \\hat\\beta_{m,t} \\cos(m\\hat\\phi_t) \n    \\label{eq:cds}\n\\end{equation}\n\n\\noindent where $\\hat{y_t}$ is the estimated signal, $t$ is the current timestep, \\textit{M} is the total number of harmonics, $\\hat\\alpha_{m,t}$ and $\\hat\\beta_{m,t}$ are the Fourier series coefficients associated with the $m^{th}$ harmonic, and $\\hat\\phi_t$ is the phase of the signal. The coefficients are updated iteratively through the equations below:\n\\begin{align}\n&e_t=y_t-\\hat y_t \\notag \\\\\n&\\hat\\phi_{t+1}= mod(\\hat\\phi_t+T(\\hat\\omega_t-\\mu e_t sin(\\hat\\phi_t)),2\\pi) \\notag \\\\\n&\\hat\\omega_{t+1}= \\mid \\hat\\omega_{t}-T\\mu e_t sin(\\hat\\phi_t)\\mid \\notag \\\\\n&\\hat\\alpha_{m,t+1}=\\hat\\alpha_{m,t}+T\\eta e_t cos(m\\hat\\phi_t)\\notag \\\\\n&\\hat\\beta_{m,t+1}=\\hat\\beta_{m,t}+T\\eta e_t sin(m\\hat\\phi_t) \\notag \n\\end{align}\n\n\\noindent where $y$ is the input signal to be learned (i.e. the gyroscope signal in the y-axis), $\\hat\\omega$ is the estimated frequency, $T$ is the sampling period in seconds, and $\\mu$ and $\\eta$ are the learning rates associated with the estimated frequency and Fourier series coefficients, respectively. $\\hat\\omega$ is \\textcolor{blue}{used to estimate the person's} cadence in the experiment described in Section \\ref{sec:exp}. \n\n\n\n\\subsection{Learning of the Cue Response Model}\\label{subsec:gp}\nIn order to provide personalized assistance that accounts for the individual's response to the feedback, a solution is formulated based on the HIL framework. A Gaussian process (GP) is used to model the person's response to a given auditory cue while walking at a given cadence. Specifically, \n\\begin{align}\n&\\hat\\omega_k = \\hat\\omega_t \\label{incr}\\\\\n&Y=f(X)+H\\beta, \\\\\n&\\mbox{where } f(X)\\sptilde GP(m(X), k(X,X')) \\notag\\\\\n&Y = \\hat\\omega_k, X = (\\hat\\omega_{k-1},c_{k-1}) \\notag\n\\end{align}\n\n\\noindent where $\\hat\\omega_k$ is the estimated cadence at time \\textit{t} from the CDS model; the index, \\textit{k}, increments every four strides; \\textcolor{blue}{when $k$ is incremented at time $t$, $\\hat\\omega_t$ is} directly written to $\\hat\\omega_k$ as shown in Eq~\\ref{incr}; $c_k$ denotes the auditory cue frequency provided at increment $k$ and is zero when no cue is provided. Both $\\hat\\omega$ and \\textit{c} are in Hertz (Hz). The GP prior, \\textit{f(X)}, is computed over the available data up to index \\textit{k}, where $Y$ is a list of the cadences, and $X$ is a list of the preceding cadences and cue frequencies. New data gets appended to \\textit{X} and \\textit{Y} with each $k$. \\textit{m(X)} is the mean function and \\textit{k(X,X')} is the square exponential kernel of the GP. An explicit, constant basis function, $H$, is specified, where $H$ is a k-by-one vector of ones and $\\beta$ is a scalar basis coefficient estimated from the data.\n\n\\textcolor{blue}{The approach is similar to the Bayesian approach described in \\cite{kim2017human}. However, an initial exploration with a pre-defined set of parameters was not performed. Instead, initialization was done through random exploration until sufficient data is collected to compute the gradient, since the action space is small and the GP only requires a small number of samples.}\n\n\\subsection{Cue Provision and Optimization}\\label{subsec:opt}\n\nDuring the GP update, we also check whether the participant's current cadence is within a threshold of the target cadence ($w_{target}$). If $\\mid\\hat\\omega_t-w_{target}\\mid > threshold$, the expected value of the predictive posterior distribution is computed using the GP model given the current cadence and the available range of cue frequencies ($\\pm35$\\% of the baseline cadence, $\\omega_{baseline}$) to minimize the difference between the mean and the target cadence, as follows: \n\\begin{align}\\label{eq:minimize}\n&\\hat{\\overline{\\omega}}_{k+1} = k((\\hat\\omega_{k},c_{k}), X)(k(X,X)+\\sigma^2 I_n)^{-1}(Y-H\\beta) \\notag \\\\\n&\\mathbf{J}(\\hat\\omega_{k},c_{k}) = (\\omega_{target}-\\hat{\\overline{\\omega}}_{k+1})^2 \\notag \\\\\n&c_{k} = \\argmin_{c_{k}} \\mathbf{J}(\\hat\\omega_{k},c_{k}), \\mbox{ subject to } \\notag \\\\\n&\\omega_{baseline}\\times0.65 \\leq c_{k} \\leq \\omega_{baseline}\\times1.35\n\\end{align}\n\\noindent where $\\hat{\\overline{\\omega}}_{k+1}$ is the next cadence estimated from the GP model, $I_n$ is a square identity matrix. The minimization algorithm is initialized with a randomly generated number. This random initialization is used for response space exploration; when a random starting point far away from the kernel is selected, the optimizer will exit immediately as the size of the gradient is less than the optimality tolerance. The random selection behaviour will stop once the gradient can be computed. Based on this property, the model can be interpreted as having two phases: the exploration (exp) phase (i.e. random sampling) versus the converged (cvg) phase (i.e. when there is a valid gradient). The two phases are discussed in Section \\ref{sec:discussion}.\n\nThe personalized cue provision algorithm described above is summarized in Figure \\ref{fig:Code}. The algorithm is implemented in MATLAB, using the GP models (Statistics and Machine Learning Toolbox) and nonlinear least-squares optimization using trust region methods (Optimization Toolbox) \\cite{MATLAB:2019}.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[scale=0.7]{images\/sudo-code.pdf}\n\\caption{With every new CDS estimate, \\textcolor{blue}{the algorithm utilizes the phase wraparound to track the number of strides. The algorithm checks whether the person's cadence is within the threshold every 4 strides. The system computes and outputs the optimal cue if the threshold is exceeded} and updates the response model.}\n\\label{fig:Code}\n\\end{center}\n\\end{figure}\n\n\\section{Experiments}\\label{sec:exp}\nWe examined the effect of different auditory cue-provision strategies on cadence in the experiment, as auditory cues have been shown to have a strong influence on cadence \\cite{sweeney2019technological}.\n\n\\subsection{Participants} \nA convenience sample of 25 participants (5 female\/20 male; age 26.08$\\pm$3.58 years; height 174.52$\\pm$8.65 cm; mass 70.88$\\pm$12.84 kg; mean$\\pm$standard deviation) enrolled in the study. All participants provided consent \\textcolor{blue}{at} the start of the experiment. The study (Project ID 22556) was approved by the Monash University Human Research Ethics Committee.\n\n\\subsection{\\textcolor{blue}{Equipment and Parameter Initialization}}\\label{sec:materials}\nThe motion data was recorded using a single IMU sensor with the WaveTrack Inertial System (Cometa Systems, Milan, IT).  \\textcolor{blue}{The equipment is shown in Figure~\\ref{fig:exp-diagram}B.} The data was sampled at 285 Hz and streamed wirelessly into a custom program in C\\#. The C\\# program ran on a laptop (Windows 10, i7 core with no GPU), which controlled the timing of the auditory cues played from the computer and interfaced with MATLAB. The coefficients of the gait parameter estimation algorithm, CDS, were initialized as follows: M = 7, $\\mu$ = 0.1, $\\eta$ = 1, $\\phi_0$ = 0, $\\omega_0$ = $2\\pi\\cdot\\frac{4}{5}$, $\\alpha_{m,0}$ = 0 for all $\\alpha_m$, and $\\beta_{m,0}$ = 0 for all $\\beta_m$. \\textcolor{blue}{The initial values for $\\phi_0$, $\\alpha_{m}$, and $\\beta_{m}$ were set to 0 as there is no strong prior, whereas $\\omega_0$ was based on the typical walking speed for the healthy population \\cite{waugh2019online}. The parameters were the same as \\cite{waugh2019online}, except for M, $\\mu$, and $\\eta$. Specifically, M was reduced as gyroscope data contains less high frequency content. $\\mu$, $\\eta$ were manually tuned until the frequency converged within four strides while minimizing oscillations around the settled value. }\n\n\n\\subsection{Experimental Conditions}\nThere was 1 control and 6 levels in the study. In the control condition, the participants walked at their natural cadence with no cueing. The baseline cadence of each participant ($\\omega_{baseline}$) was measured during control and was used to calculate the two target cadences ($\\pm20\\% \\omega_{baseline}$). \n\nFollowing the control condition, each of the three cueing approaches was implemented for each target cadence: fixed, proportional, and adaptive. In the fixed cue approach, beats were provided directly at the target cadence, emulating the baseline cueing mechanisms in the literature \\cite{Bachlin2010, Mikos2019}. In the proportional cue approach, the pace of the cue was proportional to the error between the participant's current cadence and the target cadence. The proportional approach serves as an intermediate comparison between the fixed and adaptive approach\\textcolor{blue}{, similar to the approach in \\cite{zhang2020wearable}. The proportional approach} accounts for the person's current cadence but the error gain requires manual tuning and the gain remains the same throughout the experiment. The proportional gain (i.e. $p_{gain}$) was chosen to be 0.5, which was set empirically during pilot tests. Since the gain was small, the pace of the provided cues was close to the person's current cadence. Finally, the adaptive approach was the algorithm that incorporated the participant's individualized cue response model and optimization, described in Section~\\ref{sec:propApp}. \\textcolor{blue}{The experiment conditions are summarized in Figure~\\ref{fig:exp-diagram}A.}\n\nAll three approaches provided cues only when the participant's cadence was out of the acceptable boundary, set to $\\pm$ 1\\% of the target cadence, as described in Figure \\ref{fig:Code}. Eight metronome beats were provided if the acceptable condition was not met, one for each step. The number of beats was selected empirically as observed in the pilot study, where participants were able to change their gait within eight beats and the CDS model was able to converge to the new pace. Each experimental condition took 7 minutes to complete. The duration was as an extension of the Six-Minute-Walk clinical test. During each 7-minute session, cues were provided based on the corresponding condition in the first 6 minutes, and no cue was provided in the last minute. \n\n\\subsection{Experimental Protocol}\\label{sec:exp_protocol}\nThe participant watched an introduction video and placed the IMU sensor above the knee of the dominant leg during preparation. A short training session ($<$1 minute) was provided to allow the participant to become familiar with the act of syncing one's gait to the metronome beats, where a metronome beat at 1.3 Hz was played continuously for the participant to follow. After training, the participant completed the control condition where they walked in a big circle for 7 minutes without cues. They were told to walk naturally and forget about the practice metronome beats. The participant completed a demographic survey and proceeded to the experimental conditions. The order of the conditions was randomly generated for each participant \\textcolor{blue}{and the participants were blinded to the conditions. Each condition was followed by a NASA Task Load Index (TLX) survey.} Finally, the experiment was concluded after a debriefing session.\n\n\\begin{figure*}[htb]\n\\begin{center}\n\\includegraphics[width=\\textwidth]{images\/experiment-diagram.pdf}\n\\caption{\\textcolor{blue}{Experimental Procedure:  Panel A illustrates an example experiment sequence and the cueing approaches described in Section~\\ref{sec:exp_protocol}. Panel B, with the IMU, illustrates the experiment equipment. Panel C shows the walking space and route. The background image was taken during the experiment.}}\n\\label{fig:exp-diagram}\n\\end{center}\n\\end{figure*}\n\n\n\\subsection{Analysis}\nThe convergence of the GP was first assessed to validate its ability to model the person's response to cues. The relationship between the experimental conditions and the resultant gait changes was then analyzed using linear mixed-effect models (LME) in R \\cite{citeR}. Square root transformation was applied to all data except the task load index score for the following analysis. The transformation helped with the normality and homoscedasticity assumptions during visual inspection of the residual plots. In general, the model satisfies these assumptions but contains outliers towards the tails. Data shown in the box plots in Section \\ref{sec:results} are the un-transformed data for an easier interpretation. The fixed effects of the LME model are the different cue-providing approaches and the random effects are the intercepts for the individual participants. P-values were calculated using the likelihood ratio tests between the model without the fixed effect and the model with the fixed and random effects. \n\nThe performance of the cueing approaches (proportional and adaptive) was benchmarked against the baseline fixed cue approach. The analysis was grouped into the speeding up (UP) and slowing down (DOWN) conditions. \\textcolor{blue}{The adaptive approach was further divided} based on the two phases of the GP model: the initial exploration (exp) phase during the first 70 seconds of the experiment and converged (cvg) phase for the remaining time in the experiment. \\textcolor{blue}{Figure~\\ref{present_data} illustrates the exploration and converged phases.  The first three cues in the Adaptive-Down panel are exploratory, as the goal is to slow down but cues that are faster than the target were provided. Afterwards, the cues were consistently around the target cadence, indicating the start of the converged phase. Based on visual inspection of all the data, 70 seconds was chosen as the upper bound for the exploration phase.} \n\n\\section{Results} \\label{sec:results}\n\\noindent\\textbf{Adaptive Framework: Response Model Convergence}\nThe GP modelling error and convergence, averaged over all participants and UP\/DOWN conditions, are shown in Figure \\ref{gp_performance}. \\textcolor{blue}{Both the variance and prediction error are high} during exploration. The variance quickly drops off within 5 iterations as the algorithm learns the response model. However, the error variance\\textcolor{blue}{, which indicates the model's confidence,} does not decrease further until later in the experiment due to the fact that similar cues are often provided after the initial exploration. The result shows that GP \\textcolor{blue}{can} capture the participant's behaviour around the target cadence.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3.2in]{images\/GP_convergence.png}\n\\caption{Evolution of the GP prediction error and the model error variance averaged over all participants for the adaptive cueing trials. The x-axis is the index value which increments every 4 cycles (i.e. when the acceptable condition is checked). The exploration v.s. the convergence phase is displayed using the average index from all participants.}\n\\label{gp_performance}\n\\end{figure}\n\\noindent\\textbf{Sample Experimental Data} A sample dataset from a participant is shown in Figure \\ref{present_data}. The following metrics were used to quantify the performance of the cueing approaches: \n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3.4in]{images\/present_data.png}\n\\caption{Sample data from a participant who had a strong tendency to return to the baseline. Both the fixed and proportional approach attempt to change the cadence by providing more cues, whereas the adaptive approach provides cues at a faster\/slower frequency than the target frequency. \\textcolor{blue}{The adaptive approach was able to shift the participant's cadence to the desired boundary more effectively compared to other approaches.}}\n\\label{present_data}\n\\end{figure}\n\n\\noindent\\textbf{Target mean absolute error (MAE)}\nThe target MAE \\textcolor{blue}{(Figure \\ref{fig:target_mae})} is calculated as the mean absolute error between the participant's estimated cadence and the target cadence. A low target MAE means the participant is able to change their original cadence to match the new target cadence. The LME model for UP shows that the effect of cueing method is significant (likelihood-ratio test statistic ($\\lambda_{LR}$) = 25.8837, p $<<$ 0.05, standard deviation of the random effect (StdDev R.N.) = 0.0164). On average, the proportional approach has a higher target MAE compared to the fixed approach (Value = 0.0159, 95\\% Confidence Interval (CI) = [-0.0147, 0.0338], Standard Error (SE) = 0.016). The adaptive approach during exploration also has a higher target MAE than the fixed approach (Value = 0.0671, CI = [0.0266, 0.0756], SE = 0.0162). The \\textcolor{blue}{converged} adaptive approach has a lower target MAE than the fixed approach (Value = -0.0216, CI = [-0.038, 0.0137], SE = 0.017). \n\nFor the DOWN conditions, the effect of the cueing method is also significant ($\\lambda_{LR}$ = 56.3132, p $<<$ 0.05, StdDev R.N. = 0.0276). The target MAE is higher in the proportional approach than the fixed approach (Value = 0.0383, CI = [0.0135, 0.0631], SE = 0.0127) and it is also higher for the adaptive approach during exploration compared to the fixed approach (Value = 0.0975, CI = [0.0724, 0.1227], SE = 0.0129). The \\textcolor{blue}{converged }adaptive approach has a lower target MAE compared to the fixed approach (Value = -0.0076, CI = [-0.0338, 0.0185], SE = 0.0134).\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3.4in]{images\/images_simplified_legends\/target_mae.pdf}\n\\caption{The target MAE grouped by the UP (left) and DOWN (right) trials. The metric is further divided for the adaptive approach into two stages based on the GP model convergence: exploration (exp) and converged model (cvg). The \\textcolor{blue}{converged} adaptive approach has the lowest target MAE.}\n\\label{fig:target_mae}\n\\end{figure}\n\n\\noindent\\textbf{Intermediate MAE and Decay Rate}\nIntermediate MAE and decay rate indicate how well the participant is able to maintain the target cadence in the absence of the cue. Intermediate MAE measures the mean absolute error between the participant's current cadence and the target cadence in the periods of silence during the first 6 minutes of the experiment. Decay rate is the rate at which the participant returns to a new steady state cadence after the final cue is provided. We calculate decay rate by fitting an exponential function to the cadence estimate. An example of the fitted decay rate data can be seen in in Figure \\ref{present_data}. A low intermediate MAE and a low decay rate would indicate a better maintenance of the new cadence. \n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3.4in]{images\/images_simplified_legends\/intermediate_rmse.pdf}\n\\caption{The intermediate MAE grouped by the UP (left) and DOWN (right) trials. The intermediate MAE is the highest for the proportional approach in both UP and DOWN trials. However, the effect of cueing conditions is only significant for the DOWN condition.}\n\\label{fig:intermediate}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3.4in]{images\/images_simplified_legends\/decay.pdf}\n\\caption{The decay rate grouped by the UP (left) and DOWN (right) trials. The effect of the cueing condition is not significant for both trials.}\n\\label{fig:decay}\n\\end{figure}\n\nThe effect of the cueing method is not significant \\textcolor{blue}{for the UP conditions} for the intermediate MAE outcome ($\\lambda_{LR}$ = 5.9734, p = 0.1129 $>$ 0.05, StdDev R.N. = 0.0115). For the DOWN conditions, the effect of the cueing approach is significant ($\\lambda_{LR}$ = 35.9315, p $<<$ 0.05, StdDev R.N. = 0.0235). The proportional approach has a higher intermediate MAE compared to the fixed approach (Value = 0.0512, CI = [0.0302, 0.0721], SE = 0.0107); the adaptive approach during exploration is also higher than the fixed approach (Value = 0.0434, CI = [0.0217, 0.0653], SE = 0.0111). The converged adaptive approach has an intermediate MAE lower than the fixed cue condition (Value = -0.007, CI = [-0.0285, 0.0144], SE = 0.011). The results are shown in Figure \\ref{fig:intermediate}.\n\nCueing approaches do not significantly affect the decay rate in the UP conditions ($\\lambda_{LR}$ = 5.3566, p = 0.0687 $>$ 0.05, StdDev R.N. = 0.1505). Similarly, the cueing approaches also do not significantly affect the decay rate for the DOWN conditions ($\\lambda_{LR}$ = 3.8148, p = 0.1485 $>$ 0.05, StdDev R.N. = 0.0756). The results are shown in Figure \\ref{fig:decay}.\n\n\\noindent\\textbf{Percent On}\nIn terms of minimizing the cue duration to reduce habituation, we quantified the cueing strategy performance using the percent on metric. Percent on represents the amount of time a strategy is providing beats expressed as a percentage of the first 6 minutes of the experiment. \n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3.4in]{images\/images_simplified_legends\/percent_on.pdf}\n\\caption{The percent on time grouped by the UP (left) and DOWN (right) trials. The percent on time is the highest during the exploration phase of the adaptive approach. \\textcolor{blue}{The percent on on time of the converged adaptive approach is close to the fixed approach.}}\n\\label{fig:percent_on}\n\\end{figure}\n\nFor the UP conditions, the effect of the cueing method is significant. ($\\lambda_{LR}$ = 43.2037, p $<<$ 0.05, StdDev R.N. = 0.141). The percent on time is higher for the proportional approach (Value = 0.1625, CI = [0.0679, 0.257], SE = 0.0484); the adaptive approach during exploration also has a higher percent on time than the fixed approach (Value = 0.2507, CI = [0.1562, 0.3453], SE = 0.0484). Once the adaptive approach has converged, the percent on time is lower than the fixed approach (Value = -0.0731, CI = [-0.1676, 0.0214], SE = 0.0484).\n\nFor the DOWN conditions, the effect of the cueing approach is also significant ($\\lambda_{LR}$ = 44.7463, p $<<$ 0.05, StdDev R.N. = 0.1644). The proportional approach playing cues more than the fixed approach (Value = 0.1517, CI = [0.0596, 0.2436], SE = 0.0471). The adaptive approach during exploration has the highest percent on time compared to the fixed approach (Value = 0.3164, CI = [0.2244, 0.4085], SE = 0.0471). The adaptive approach when converged is also higher than the fixed approach (Value = 0.014, CI = [-0.0779, 0.1061], SE = 0.0471). \\textcolor{blue}{The results are displayed in Figure~\\ref{fig:percent_on}}.\n\n\\noindent\\textbf{Participant Perception: NASA-Task Load Index (TLX)}\nThe sum of the raw TLX scores (\\textcolor{blue}{Figure \\ref{fig:tlx_raw}}) represents the participant's cognitive workload in each condition. For the UP conditions, the TLX score is not significantly influenced by the cueing approaches ($\\lambda_{LR}$ = 1.2837, p = 0.5263 $>$ 0.05, StdDev R.N. = 4.7480). Similarly, the TLX score is not significantly affected by the cueing approaches for the DOWN conditions ($\\lambda_{LR}$ = 5.0589, p = 0.0797 $>$ 0.05, StdDev R.N. = 3.4140). \n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3.4in]{images\/images_simplified_legends\/tlx_raw.pdf}\n\\caption{The sum of TLX scores across conditions. Overall, the UP conditions are more demanding than the DOWN conditions.}\n\\label{fig:tlx_raw}\n\\end{figure}\n\n\n\n\n\\section{Discussion}\\label{sec:discussion}\n\n\\textcolor{blue}{Comparing the different cueing approaches, the adaptive approach is the most effective in achieving the new target cadence. Specifically, the adaptive approach in the converged phase has the lowest target MAE for both target speeds.} The converged adaptive approach reduces the target MAE by providing cues at a very different pace compared to the participant's current state, prompting the participants to be more proactive, as seen in Figure \\ref{cue_vs_curr_state}. \\textcolor{blue}{Participants also perceived the proactive cueing,  reflected in the higher TLX score. Another factor for the high TLX score is caused by the participants having to follow a random set of beats during the exploration phase. Once GP has converged, the adaptive approach is able to achieve a lower target MAE while having a comparable percent on time to the baseline fixed approach. The adaptive approach might be able to outperform the baseline approach in the future by penalizing cue-playing in the cost function to reduce habituation.} \n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3.4in]{images\/images_simplified_legends\/cue_vs_curr_state.pdf}\n\\caption{The absolute difference gives insights into how the cues are changing the person's cadence. In the UP conditions, the adaptive approach once converged is providing cues at a much higher pace from the participant's current cadence. The proportional approach on the other hand provides participants with cues that are close to their current cadence.}\n\\label{cue_vs_curr_state}\n\\end{figure}\n\n\\textcolor{blue}{Of the three approaches, the proportional approach is the least effective in changing the natural cadence as seen in the high target MAE and intermediate MAE.} This might be because the proportional cue is not prompting the person to change much from the original cadence. In Figure \\ref{cue_vs_curr_state}, it can be seen that the difference between the cue and the current gait cadence is always the smallest for the proportional approach. The phenomenon is due to the choice of the controller gain, which is designed to provide gradual changes in the pace of the cue. The gradual change allows for a lower cognitive workload (as seen in the TLX scores), but is less effective in altering the cadence. While the proportional cue might have been more effective with a higher gain or personalized gain tuning, the cueing method highlights the difficulty in manual gain selection. \n\n\\textcolor{blue}{The three approaches are not significantly different in terms of intermediate MAE and decay rate. This might be due to the decay rate and the intermediate MAE being} influenced by the participant's memory (i.e. forgetting the pace of the cue over time) and their ability or willingness to adjust their cadence. In the post-study interviews, participants reflected that it was difficult to recall the pace of the cue in the long periods of silence. We also observed patterns in the experiment similar to the participant in Figure \\ref{present_data}, where some participants immediately deviate from the target cadence when the cues are off, causing the drastic fluctuations and the on-off cueing pattern.\n\n\\textcolor{blue}{Our study has several limitations. Only healthy participants were tested in a single trial. We also assumed that the participant could attain the target, which might not be possible with the patient population. Finally, only auditory cueing was used, which may not be effective with all users and in all environments.}\n\n\\textcolor{blue}{The proposed approach could be adopted to provide assistance using exoskeletons. The current HIL approaches (e.g. \\cite{kim2017human, zhang2017human, felt2015body}) utilize respiration measurements in the optimization step, which can be difficult to obtain in an everyday setting. With the proposed adaptive framework, kinematics that could replace the respiration measurements, improving usability.}\n\n\\section{Conclusions and Future Work}\nWe proposed an adaptive cueing framework that can simultaneously monitor gait performance of a person and adjust the auditory cues based on the person's response. In the framework, a Gaussian Process \\textcolor{blue}{was used to model} the person's gait as a function of the provided cues and past gait performance. Using \\textcolor{blue}{GP}, personalized assistance can be provided \\textcolor{blue}{through optimization} to improve gait performance. We investigated the effectiveness of the adaptive cueing strategy with healthy participants in a gait study, where \\textcolor{blue}{the aim was} to change the participant's cadence with the cues. The adaptive cue method \\textcolor{blue}{was compared} to the fixed and the proportional methods. The results show that the proportional cues perform the worst among the three cueing approaches, highlighting the need for individualization and adaptation. The adaptive strategy outperforms the both comparison strategies when the GP model has converged. \n\n\n\\textcolor{blue}{Future work involves developing a wearable device that can provide multi-modal cues for patients with gait impairments. To relax the assumption that participants will be able to achieve to a fixed target gait state, we will adapt the target gait state in real time, and include additional objectives within the cost function to support the provision of multi-modal cues. An advantage of the proposed approach is that the system continuously learns the user's gait profile and response model, and can therefore be used when the patient's response profile is changing (e.g. due to medication or fatigue). We plan to recruit patients to examine the effectiveness of the adaptive cueing strategy.} \n\n\n\n\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nTo describe the stability of a stellar compact object, in usual\nway, it is necessary to consider the Tolman-Oppenheimer-Volkoff\nequations \\cite{1} and the equation of state of the star.\nStability criteria of relativistic spherical symmetric compact\nobjects with isotropic pressure in the framework of general\nrelativity include boundary conditions, non-singularity, electric\ncharge, surface redshift, energy conditions, the speed of sound in\ncausal conditions and relativistic adiabatic index. In a stable\nmodel, the energy and pressure densities are finite at the center\nof compact object and decrease uniformly toward the boundary. The\nmetric potentials are regular and the electric field intensity is\nzero at the center and increases towards the surface. In addition,\nthe gravitational redshift follows $Z_s <2$  and four energy\nconditions are satisfied, the speed of sound is less than the\nspeed of light and decreases uniformly toward the surface. In\naddition, the adiabatic index is strongly higher than\n$\\frac{4}{3}$ \\cite{2}. Relativistic Compact object with gravity\nand strong internal density have two different pressures, radial\nand tangential \\cite{3}. The stability of a stellar model can be\nincreased by an anisotropic repulsive force that $\\Delta =\np_t-p_r> 0$. This property leads to more compact stable\nconfigurations compared to the states of isotropic \\cite{4}.\nHydrostatic equilibrium of solutions of anisotropic relativistic\nstars in scale-dependent gravity, where Newton's constant is\nallowed to vary with radial coordinates across the star, shows\nthat a decrease in Newton's constant across objects leads to\nslightly more massive and compact stars  \\cite{5}. A stability\nanalysis for Einstein-Kline-Gordon  model with static real scalar\nfield interaction express that the initial value of the field at\nthe origin is a function of the energy density of the matter at\nthe origin and in the far regions the field behaves Yukawa-like\npotential. Such a model for compact stellar object is stable if\nthe gradient of the total mass versus energy density is positive\nand the weak energy condition is satisfied (positive total\ndensity) \\cite{6}. The stability of the star can be investigated\nin the presence of both electric and magnetic fields. Solving the\nEinstein-Maxwell field equations for compact objects with the\ncharged anisotropic fluid model gives more stable solutions than\nfor neutral stars. The presence of charges creates a repulsive\nforce against the gravitational force, and this factor causes\ndenser stable stars, higher maximum mass and larger redshift\n\\cite{7}. Charged quarks can create more stable quark stars than\nneutron nuclei. Also, for a white dwarf  with a charged perfect\nfluid, there is a direct correlation between the increase in\nelectric charge and its size. Near the surface of the star, the\nradial pressure is close to zero and the electric charge density\nis non-zero, leading to a stable star with more mass \\cite{8}. The\nmass-radius relation of some kinds of Neutron stars, which can\ncontain a core of quark matter, have a large frequency range of\nradial fluctuations near the transition point in their core versus\nmass. These induce nonlinear general relativistic effects which\ncause to be the stars unstable dynamically. The core of the\nNeutron stars becomes several times larger, making the Neutron\nstars highly unstable \\cite{9}. While for the charged\nboson-fermion stars with a charged fluid related to fermion and a\ncomplex scalar field related to boson, the charge increase can\nreduce the stellar radius and create a denser and more massive\nstar. In the whole parameter space, the critical curve can show\nstable and unstable regions \\cite{10}. If the number of baryons in\ncompact pulsar-like stars exceeds the critical value $10^9$, the\nstrangeon star model is proposed. In fact the strangeon star\natmosphere  model describes the radiation from interstellar medium\naccreted plasma atmosphere on a strangeon star surface and its\nspectrum. This object could simply be regarded as the upper layer\nof a normal neutron star because the radiation from strangeon\nmatter can be neglected \\cite{11}. The atmosphere is in radiative,\nthermal equilibrium and two-temperature. The strangeon star\nspectrum is based on bremsstrahlung from an extremely thin\nhydrogen plasma. More details of this model are described in\n\\cite{12}. Since the extra strange flavor provides more degrees of\nfreedom to lower the Fermi energy in the free quark approximation,\nmacroscopic bulk strong matter with 3-flavor symmetry (up, down,\nand strange quarks) is more stable than up quark matter. The\ndifference in the strangeness level between a strange star and a\ntypical neutron star can have a profound effect on the\nmagnetospheres activity associated with the coherent radio\nemission of the Compact stars. Content of this paper is as\nfollows:\\\\\nIn section 2 we describe shortly  generalized Einstein Maxwell\ngravity. Then we obtain Lagrangian form of the model for a general\nspherically symmetric  static metric and corresponding\nEuler-Lagrange equations. These  are nonlinear second order\ndifferential equations and so we use dynamical systems approach to\nsolve them in the section 3. In the latter section we obtain\nalternative first order differential equations in phase space and\ncalculate Jacobi matrix of the equations. We determine eigenvalues\nof the Jacobi matrix by solving the secular equation. At last we\ndetermine sign of the eigenvalues for which the system has  stable\nnatures for some suitable numeric values of the parameter of the\ncoupling constant of the model. They are collected in a table.\nSection 4 dedicated to concluding remarks and outlook of the work.\n\\section{The gravity model }\nAs a generalization of the Einstein-Maxwell gravity theory we\nconsider non-minimally coupling between the Maxwell vector\npotential $A_{\\mu}$ and the Recci scalar and the Ricci tensor such\nthat \\cite{MS}\n\\begin{equation}\\label{action}I=-\\int\ndx^4\\sqrt{g}\\bigg[\\frac{1}{4}F_{\\mu\\nu}F^{\\mu\\nu}+\\frac{\\alpha}{2}A^2R+\\frac{\\beta}{2}R_{\\mu\\nu}A^\\mu\nA^\\nu\\bigg],\n\\end{equation}where $g$ is absolute value of determinant of the metric field and anti symmetric electromagnetic tensor\nfield $F_{\\mu\\nu}$ is defined versus the partial derivatives of\nthe four vector electromagnetic potential $A_{\\mu}$ as follows.\n\\begin{equation} \\label{fmunu}\nF_{\\mu\\nu}=\\nabla_\\mu A_\\nu-\\nabla_\\nu A_\\mu=\\partial_\\mu\nA_\\nu-\\partial_\\nu A_\\mu\n\\end{equation} with $A^2=g_{\\mu\\nu}A^\\mu A^\\nu$ and $R_{\\mu\\nu}$ is Ricci\ntensor. It is easy to check that this model has not gauge\ninvariance symmetry same as \\cite{GHN} in which the action\nfunctional remain unchanged by transforming $A_{\\mu}\\to\nA_{\\mu}+\\partial_\\mu \\xi$. In this transformation $\\xi$ is gauge\nfield for which $F_{\\mu\\nu}\\to F_{\\mu\\nu}.$ In this work we like\nto investigate  effects of the electromagnetic fields on metric\nfield solutions of a spherically symmetric stellar object. For\nspherically symmetric time-independent static metric\n\\begin{equation}\\label{line}ds^2=-X(r)dt^2+Y(r)dr^2+r^2(d\\theta^2+\\sin^2\\theta d\\varphi^2)\\end{equation}\nwe know that non-vanishing components of the vector potential is\njust $A_t(r)=\\phi(r)$ means electric potential for a\nelectrostatics spherical stellar object which by substituting into\nthe action function (\\ref{action}) we obtain exact form for the\nLagrangian density of the fields such that\n\\begin{equation}L=\\frac{2\\pi}{\\sqrt{XY}}\\bigg\\{\\dot{\\phi}^2+\n\\beta\\phi^2\\bigg[\\frac{\\dot{Y}}{Y}+Y-1\\bigg]\\bigg\\}\\end{equation}\nwhere we defined $\\dot{~}$ as logarithmic derivative of the radial\ncoordinate as\n\\begin{equation}\\dot{~}=\\frac{d}{d\\tau}=r\\frac{d}{dr}=\\frac{d}{d\\ln(r\/D)}\\end{equation}\nin which $D$ is a suitable length parameter. Also we use the\nansatz \\begin{equation}\\alpha=-\\frac{\\beta}{2}\\end{equation} to\nremove second order derivatives of the $X$ field coming from the\nRicci scalar and Ricci tensor. By defining an new field $\\psi(r)$\nsuch that\n\\begin{equation}\\dot{\\phi}=\\psi\\phi\\end{equation} one can show\nthat the Euler-Lagrange equations of the fields\n$\\phi(r),X(r),Y(r)$ are obtained after some simple mathematical\ncalculations respectively as follows.\n\\begin{equation}\\label{psid}\\dot{\\psi}=-\\frac{\\psi^2}{\\beta}(4\\beta Y+\\beta\\psi+3\\psi^2-3\\beta)\\end{equation}\n\\begin{equation}\\label{xd}\\dot{X}=-\\frac{X}{\\beta}(7\\beta Y-4\\beta\\psi+5\\psi^2-5\\beta)\\end{equation}\nand\n\\begin{equation}\\label{yd}\\dot{Y}=-\\frac{Y}{\\beta}(\\beta Y+\\psi^2-\\beta).\\end{equation}\nThe above equations are non-linear first order differential\nequations not having exact analytic solutions and so we can solve\nvia dynamical systems approach. This is done in the next section.\n\\section{Metric solutions}\nGeneral strategy in the dynamical system approach to obtain \nanalytic solutions for the fields equations $\\dot{X}_i=F(X_i,t)$\nhave 4 different steps as follows: (a) By solving the equations\n$\\dot{X}_i=0; i=1,2,3,\\cdots n$ for $n$ dimensional phase space of\nthe system one determine the critical points. (b) For each of\nthese critical points he\/she should calculate Jacobi matrix\n$J_{ij}=\\frac{\\partial \\dot{X}_i}{\\partial X_j}$ and (c) then\nsolve the corresponding secular equation\n$\\det(J_{ij}-E\\delta_{ij})=0$ to obtain eigenvalues for each of\ncritical points. (d) By choosing some eigenvalues which have\nnegative numeric values (if are real) or real part of them are\nnegative (if they are complex numbers) he\/she should solve\nalternative equations $\\dot{X}_i=\\Sigma_{j=1}^nJ_{ij}X_j$ instead\nof the original equations $\\dot{X}_i=F(X_i,t).$ In fact the\nobtained\nsolutions are valid near the stable critical points (see \\cite{15} or \\cite{HH} for more discussion).\\\\\nWe now continue to use these 4 steps for the equations given in\nthe previous section. In this case $\\dot{X}=\\dot{Y}=\\dot{\\psi}=0$\ngive us two different critical points as\n\\begin{equation}\\label{210}(1):~~~\\{\\psi_c=0,~~~X_c=0,~~~Y_c=1\\}\\end{equation}\n$$(2):~~~\\{\\psi_c=\\frac{1}{2},~~~X_c=0,~~Y_c=1-\\frac{1}{4\\beta}\\}.$$\nAt these critical points one can calculate the Jacobi matrix as\nfollows.\n\\begin{equation} J_{ij}^{(1)}=\\frac{\\partial{\\dot{X}_i}}{\\partial X_j}=\\left\n\\begin{array}{ccc}\n0 & 0 & 0 \\\\\n0 & -2 & 0 \\\\\n0 & 0 & -1 \\\\\n\\end{array\n\\right)\n\\end{equation}\nand\n\\begin{equation}J_{ij}^{(2)}=\\left\n\\begin{array}{ccc}\n-\\frac{(7\\beta+2)}{4\\beta} & 0 & -1 \\\\\n0 & \\frac{1}{2\\beta} & 0 \\\\\n\\frac{(1-4\\beta)}{4\\beta^2} & 0 & \\frac{(1-4\\beta)}{4\\beta} \\\\\n\\end{array\n\\right).\n\\end{equation}\nTo obtain eigenvalues $E$ of the above Jacobi matrixes  we should\nsolve the secular equation $\\det(J^{(1,2)}_{ij}-E\\delta_{ij})=0$\nwhich reads\n\\begin{equation}\\label{E}E^{(1)}_1=0,~~~E^{(1)}_2=-2,~~~E_3^{(3)}=-1\\end{equation}\nfor the first critical point (1)  and\n\\begin{equation}\\label{E2}E_1^{(2)}=\\frac{1}{4\\beta},~~~E_2^{(2)}=\\frac{-(1+11\\beta)+3\\sqrt{(\\beta-\\beta_{+})(\\beta-\\beta_-)}}{8\\beta}\\end{equation}\n$$E_3^{(2)}=\\frac{-(1+11\\beta)-3\\sqrt{(\\beta-\\beta_{+})(\\beta-\\beta_-)}}{8\\beta}$$\nwhere we defined\n\\begin{equation}\\beta_{\\pm}=\\frac{-41\\pm4\\sqrt{109}}{9}.\\end{equation}\nBy looking at these critical points one can infer that the first\ncritical point (1) is quasi stable because all three eigenvalues\nfor it are not negative numbers. The eigenvalues for the second\ncritical point (2) are parametric. For $\\beta<0$ the first\neigenvalue is negative number $E_1^{(2)}<0$ and for $\\beta>0$ the\nfirst part in the second and third eigenvalues are negative umbers\n$E_{2,3}^{(2)}<0$ and for $\\beta_-<\\beta<\\beta_+$ in which\n$\\beta_-=-9.1956$ and $\\beta_+=0.0844$ the square roots in these\neigenvalues become imaginary and so nature of the system is spiral\nstable. We give out numeric values for the eigenvalues (\\ref{E2})\nwith describing whose stability nature in the table 1. To obtain\nmetric solutions near the critical point (2) one should solve\nalternative equations instead of the original equations\n(\\ref{psid}),(\\ref{xd}) and (\\ref{yd}) such that\n\\begin{equation}\\frac{d}{d\\tau}\\left\n\\begin{array}{c}\n\\psi \\\\\nX \\\\\nY \\\\\n\\end{array\n\\right)=\\left\n\\begin{array}{ccc}\n-\\frac{(7\\beta+2)}{4\\beta} & 0 & -1 \\\\\n0 & \\frac{1}{2\\beta} & 0 \\\\\n\\frac{(1-4\\beta)}{4\\beta^2} & 0 & \\frac{(1-4\\beta)}{4\\beta} \\\\\n\\end{array\n\\right)\\left\n\\begin{array}{c}\n\\psi \\\\\nX \\\\\nY \\\\\n\\end{array\n\\right)\\end{equation} which have solutions with the following\nforms.\n\\begin{equation}\\label{psi1}\\psi(\\tau)=\\psi_+e^{E_2^{(2)}\\tau}+\\psi_-e^{E_3^{(2)}\\tau}\n\\end{equation}\n\\begin{equation}\\label{x1}X(\\tau)=X_0 e^{(\\tau\/2\\beta)}\\end{equation}\nand\n\\begin{equation}\\label{y1}Y(\\tau)=Y_+e^{E_2^{(2)}\\tau}+Y_-e^{E_3^{(2)}\\tau}\\end{equation}\nin which  $\\psi_{\\pm},Y_{\\pm}$ and $X(0)=X_c$ are constants and\nshould be fix by initial conditions of the system. We collected\nnumeric values for the eigenvalues together with stability nature\nin the table 1.   This metric solution is not for a black hole\nbecause the horizon equations $X(\\tau)=0$ and $Y^{-1}(\\tau)=0$\ngive not a finite radius for position of horizon. In other words\nthis metric solution describes line element of internal region of\nan electrostatic star.\n\\section{Concluding remarks}\nIn this work we added a nonminimal directionally interaction\nLagrangian between geometry and the electromagnetic vector\npotential for Einstein-Maxwell gravity and investigate this\nadditional contribution on a spherically symmetric static space\ntime of a stellar compact object. After to solve the\nEuler-Lagrange equations of the fields via dynamical systems\napproach, we were determine stabilization conditions of the\nobtained solutions near parametric critical points in phase space.\nWe obtained that for negative values of the coupling constant of\nthe interaction part of the Lagrangian density the solutions take\non stable nature but not for positive coupling constant. As\nextension of this work we like investigate magnetic fields effects\nof the stellar matter on stabilization of the stellar compact\nobject. In our previous work \\cite{GHN} we investigated effects of\nmagnetic monopoles on stability of a  spherically symmetric\nstellar compact object for a modified gauge invariance\nEinstein-Maxwell gravity. Same as the one, we checked this effect\nagain for the present work but we obtained a problem regretfully\nwhere the interaction parameter must be complex (imaginary) which\nis not physical. This comes from integral of the action functional\nfor the polar coordinate $\\theta$ which must be used the Cauchy`s\nintegral residue. In fact the integrand is singular on the poles\n$\\theta=0,\\pi.$  Hence we like to investigate effects of magnetic\nmultipoles on stabilization of cylindrically symmetric stellar\ncompact objects for the present gravity model in our future work.\nAlso we will seek that did stable the stellar systems with these\nmagnetic multipole sources?\n\\newpage\n\\begin{center}\nTable 1: Numeric values of Jacobi matrix eigenvalues vs $\\beta.$\nHere, `S` means `stable`, `S.S`, means `spiral stable`, `Indet.`\nmeans `indeterminate` and `Q.S`, means `quasi stable`\n\\end{center}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n$\\beta$ & $E_{1}$ & $E_{2}$ & $E_{3}$ & $nature$ \\\\\n\\hline\n-15.& -0.0166667& -360.134& -254.866& S\\\\\n-14.& -0.0178571&-310.936& -224.564&S\\\\\n-13.& -0.0192308& -265.145& -196.355&S\\\\\n-12.& -0.0208333& -222.696& -170.304&S\\\\\n-11.& -0.0227273&-183.448& -146.552&S\\\\\n-10.& -0.025& -146.93& -125.57&S\\\\\n-9.& -0.0277778& -110.25 -4.5 I& -110.25 + 4.5 I&S.S \\\\\n-8.& -0.03125& -87. - 9.32738 I& -87. +   9.32738 I& S.S\\\\\n-7.& -0.0357143& -66.5 - 10.3531 I& -66.5 +   10.3531 I&S.S\\\\\n-6.& -0.0416667& -48.75 - 9.92157 I& -48.75 +   9.92157 I&S.S\\\\\n-5.& -0.05& -33.75 - 8.66025 I& -33.75 +   8.66025 I&S.S\\\\\n-4.& -0.0625& -21.5 - 6.91014 I& -21.5 +   6.91014 I&S.S\\\\\n-3.& -0.0833333& -12. - 4.91808 I& -12. +   4.91808 I&S.S\\\\\n-2.& -0.125& -5.25 - 2.90474 I& -5.25 +   2.90474 I&S.S\\\\\n-1.& -0.25& -1.25 - 1.11803 I& -1.25 +   1.11803 I&S.S\\\\\n0.& Complex~Infinity& 0.& 0.&Indet. \\\\\n1.&  0.25& -0.354356& -2.64564&Q.S\\\\\n2.& 0.125& -2.27689& -9.22311&Q.S\\\\\n3.&  0.0833333& -6.0418& -19.4582&Q.S\\\\\n4.&  0.0625& -11.7181& -33.2819& Q.S\\\\\n5.& 0.05& -19.3375& -50.6625&Q.S\\\\\n6.&  0.0416667& -28.9178& -71.5822&Q.S\\\\\n7.&  0.0357143& -40.4696& -96.0304&Q.S\\\\\n8.& 0.03125& -54.& -124.&Q.S\\\\\n9.&  0.0277778& -69.5138& -155.486&Q.S\\\\\n10.&  0.025& -87.0145& -190.486&Q.S\\\\\n11.&  0.0227273& -106.505& -228.995&Q.S\\\\\n12.&  0.0208333& -127.986& -271.014&Q.S\\\\\n13.&  0.0192308& -151.46& -316.54&Q.S\\\\\n14.&  0.0178571& -176.928& -365.572&Q.S\\\\\n15.&  0.0166667& -204.391& -418.109& Q.S\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIt is a classical result that,\nfor any bounded convex domain $\\Omega\\subseteq\\mathbb R^d$, \nthe Hessian of any function $v\\in H^1_0(\\Omega)\\cap H^2(\\Omega)$\ncan be bounded, in the $L^2$ norm, by the Laplacian:\n\\begin{equation}\n \\|D^2 v\\|_0\\leq \\|\\triangle v\\|_0\n \\quad\\text{for all } v\\in H^1_0(\\Omega)\\cap H^2(\\Omega).\n \\label{MT}\n\\end{equation}\nHere and throughout this article, the spaces $H^k(\\Omega)$\nwith $k\\geq 0$ denote the usual $L^2$-based Sobolev spaces,\n(the subindex $0$ refers to a homogeneous Dirichlet boundary\ncondition in the sense of traces)\nand $\\|\\cdot\\|_0$ is the $L^2$ norm over $\\Omega$.\nEstimate \\eqref{MT} is known as the\n\\emph{Miranda--Talenti inequality},\nnamed after the authors of \\cite{Miranda} and \\cite{Talenti},\nand proofs thereof can be found in \\cite{bihar-regular,Suli2013}.\nIf $\\Omega$ is polytopal (not necessarily convex),\nestimate \\eqref{MT} becomes even an identity.\nThe Miranda--Talenti estimate is critical for the well-posedness\nof second-order partial differential equations (PDEs) in\nnondivergence form\n\\begin{equation}\nA:D^2u \\equiv \\sum_{i,j=1}^d A_{ij}\\partial^2_{ij} u = f\n\\quad \\text{in } \\Omega,\\quad u=0 \\text{ on }\\partial\\Omega\n\\label{nondiv}\n\\end{equation}\nwhere $f\\in L^2(\\Omega)$ is a given right-hand side and the \ncoefficient $A:\\Omega\\to\\mathbb R^{d\\times d}$ is essentially\nbounded, uniformly elliptic, and satisfies the Cordes condition\n(details follow in Section~\\ref{s:nondiv}),\nsee \\cite{MT,Suli2013}.\nIn this setting,\nGalerkin methods in subspaces of $H^1_0(\\Omega)\\cap H^2(\\Omega)$\nare automatically well-posed because the discrete functions involved\nenjoy \\eqref{MT}. On the other hand, such subspaces are generally\ncumbersome already in the planar case $d=2$ while they are considered\ncompletely impractical for $d=3$.\nA classical alternative are nonconforming finite element spaces,\nwhich are not subspaces of $H^2(\\Omega)$ but rather involve weaker\ncontinuity constraints on the discrete functions.\nIn this paper, we are interested in nonconforming methods\nsatisfying the following discrete version of \\eqref{MT}.\n\n\\begin{myDef}[strong discrete Miranda--Talenti property]\\label{def:dmt}\nLet $\\mathcal T$ be a simplicial triangulation of $\\Omega$\nand let $V_h$ be a space of piecewise polynomial functions\nwith respect to $\\mathcal T$.\nWe say that $V_h$ satisfies the \nstrong discrete Miranda--Talenti property, if \n\t\\begin{equation}\n\t \\|D_h^2v_h\\|_0\\leq \\|\\triangle_h v_h\\|_0\n            \\quad \\text{for all }v_h\\in V_h.\n\t\t\\label{dMT}\n\t\\end{equation}\nHere $D_h^2$ and $\\triangle_h$ denote the piecewise action of\nthe Hessian and the Laplacian, respectively.\n\\end{myDef}\n\nFor equations of the biharmonic type, such as Kirchhoff plates,\nnonconforming methods are a well established tool \\cite{Ciarlet2002},\nbut they are usually unstable when applied to problems in\nnondivergence form; the reason being that their\nderivatives (piecewise with respect to some simplicial\ntriangulation of the domain)\ndo not satisfy \\eqref{dMT}.\nA simple example for this phenomenon is the Morley finite element,\nsome basis functions of which are nonzero,\npiecewise quadratic, and have normal derivatives with vanishing \njump integrals across all $(d-1)$-dimensional hyperfaces.\nThe divergence theorem therefore implies that the\npiecewise Laplacian (which is piecewise constant)\nof such a function must vanish, making a discrete version of\n\\eqref{MT} impossible.\n\nThere exist several discretizations of \\eqref{nondiv} that do\nnot require \\eqref{dMT} as a discrete strong version\nof \\eqref{MT} and instead\ninvolve stabilization terms:\nthe discontinuous Galerkin finite element method\nof \\cite{Suli2013},\nmixed methods \\cite{Mixednondiv},\nmethods based on\nthe discrete Miranda--Talenti inequality from \\cite{MTlag},\nand the stabilized Hermite element \nof \\cite{Wuhermite}.\nIn this paper we are concerned with constructing\nnonconforming methods that satisfy \\eqref{dMT}\nand are thus suited for approximating \\eqref{nondiv}\nwithout any stabilization term in the method.\nWe note that, in two dimensions, there exists a nonconforming\nscheme satisfying a strong discrete Miranda--Talenti inequality\n\\cite{MTzhang}, which ---similar to the Fortin--Soulie element\n\\cite{FortinSoulie}--- is not a finite element in the sense of\nlocal degrees of freedom.\nIn three space dimensions, we are not aware of any nonconforming\nmethod satisfying \\eqref{dMT}.\nThis article focus on constructing nonconforming elements satisfying \ndiscrete Miranda--Talenti inequality.\nWe contribute results for two and three space dimensions.\nThe construction is motivated by the $H^2$-nonconforming elements\nwhich are $C^0$-elements with $C^1$ continuity on $(d-2)$-dimensional\nfaces. The latter turns out to be an important ingredient in the\nconstruction whilst it is not sufficient,\nas can be seen, e.g., from\nthe Adini element \\cite{finiteBrenner,Ciarlet2002,finitewang} \nand the first order Specht element \\cite{Specht1ord}.\nIn two dimensions, we first prove that the \nsecond order Specht element \\cite{specht2,specht3}\nsatisfies \\eqref{dMT} (see Lemma~\\ref{Specht}).\nThis element belongs is $C^0$ continuous\nand has $C^1$ continuity in the element vertices.\nWe then propose a high order family of \n$H^2$-nonconforming finite elements for $d=2$,\nwhich is defined for any odd approximation order.\nIn the case $d=3$ we propose a family of nonconforming\nfinite elements satisfying \\eqref{dMT}.\nIt is worth mentioning that, in the three-dimensional case,\nour new element seems to be the first such element\nin the literature.\nThe approximation of the lowest-order version\nin the $H^2$ norm is of fourth order.\nSince in three dimensions the $C^1$-continuity across edges \nrequires polynomial shape functions to be of at least fifth order,\nour element appears to be in some sense minimal,\nwhence we call it a low-order method.\nIt also has less degrees of freedom compared to other\nknown $H^2$-nonconforming elements of similar order\n\\cite{Guzm2012,HZT}.\n\n\nThe organisation of this paper as follows. \nIn Section~\\ref{s:prelim}, we introduce some notation\n and the strong discrete Miranda--Talenti inequality. \nThe proof of Specht element satisfying discrete Miranda--Talenti inequality and \ndesigning 2D odd order elements are presented in \nSection~\\ref{s:2d}.\nIn Section~\\ref{s:3d}, we construct the 3D satisfying \ndiscrete Miranda--Talenti property $H^2$-nonconforming elements. \nSection~\\ref{s:bih} and Section~\\ref{s:nondiv} present\nthe application the the biharmonic equation and \nelliptic problems in nondivergence form.\nIn the Section~\\ref{s:num} we present numerical experiments.\n\n\n\n\n\\section{Preliminaries}\\label{s:prelim}\n\n\\subsection{General notation}\nThrougout this paper, $\\Omega\\subset \\mathbb{R}^d$, $d=2,3$ is an open and bounded\ndomain with polytopal Lipschitz boundary $\\partial\\Omega$.\nThe standard Sobolev spaces are denoted by $H^m(\\Omega)$, $m\\geq 0$, with\n$H^0(\\Omega)=L^2(\\Omega)$.\nThe space of tensor-valued functions with all components in $H^m(\\Omega)$\nis denoted by $H^m(\\Omega;\\mathbb{R}^{d\\times d})$.\nThe space $H_0^1(\\Omega)$ is the subspace of $H^1(\\Omega)$ of functions\nwith vanishing trace on $\\partial\\Omega$.\nFor any measurable subdomain $G\\subseteq \\Omega$,\nthe $L^2$ inner product is denoted by\n$(\\cdot,\\cdot)_G$.\nThe symbols $\\nabla$ and $D^2$ denote the gradient and\nthe Hessian, respectively.\nThe space of polynomial functions over $G\\subseteq \\Omega$ of degree not greater than $\\ell$\nis denoted by $P_{\\ell}(G)$.\nGiven a unit vector $\\xi\\in\\mathbb R^d$, the directional derivative\nwith respect to $\\xi$\nis denoted by $\\partial_\\xi u= \\nabla u \\cdot \\xi$.\n\n\n\n\\subsection{Meshes and discrete functions}\nLet $\\mathcal{T}$ be a shape-regular simplicial triangulation of \n$\\Omega$, and $h_T$ be the diameter of the element $T\\in\\mathcal T$.\nThe global mesh size reads $h=\\max_{T\\in\\mathcal{T}}\\{h_T\\}$.\nLet $H^m(\\mathcal{T})$ be the spaces of functions that belong to\n$H^m(\\operatorname{int} T)$ when restricted to the interior \n$\\operatorname{int} T$ of any simplex $T\\in\\mathcal T$.\nLet $\\mathcal F$ denote the set of all $(d-1)$-dimensional hyperfaces\nof the triangulation $\\mathcal T$.\nTo each face $F\\in\\mathcal F$ we assign a fixed unit normal vector\n$n_F$ with the convention that $n_F$ coincides with the outward-pointing\nunit vector to $\\Omega$ if $F\\subset\\partial\\Omega$ is a boundary face.\nGiven $F\\in\\mathcal F$ is shared by two elements\n$T_+,T_-$, the normal derivative jump of a function $v$ across $F$ is defined as\n$$\n[\\partial_n v]|_F:=\\partial_{n_F}|_{F\\cap T_+} - \\partial_{n_F}|_{F\\cap T_-}. \n$$\nIf $F\\subset \\partial \\Omega$ is a boundary face,\n$[\\partial_nu]|_F$ denotes the trace of the normal derivative,\nand we will denote \nthe integration of the normal derivative of $v$ on \n$(d-1)$-dimensional sides by $\\int_F\\partial_nv\\mathrm{d}S$ .\nThe piecewise versions of the Laplacian and the Hessian\nare defined as\n$\\triangle_h v :=\\sum_{T \\in \\mathcal{T}} \\triangle v|_T$,\n$D^2_h v :=\\sum_{T \\in \\mathcal{T}} D^2 v|_T$\nwhere the convention is used that $\\triangle v|_T$\nand $D^2 v|_T$ are extended to the whole domain $\\Omega$\nby zero.\n\n\n\\subsection{An identity from integration by parts}\n\nOur construction of nonconforming elements which satisfy the strong discrete \nMiranda-Talenti property \\eqref{dMT}\nwill be based on $C^0$-continuous elements with $C^1$ continuity across \n$(d-2)$-dimensional hyperfaces. A proof of \nthe following identity based on integration\nby parts can be found in \\cite{Wuhermite}.\n\\begin{mylem}\nLet $\\mathcal{T}$ be a regular triangulation of \n$\\Omega\\subset \\mathbb{R}^d,d=2,3$ and $V_h\\subseteq H^2(\\mathcal T_h)$\nbe a $C^0$ finite element space has $C^1$ continuity on $(d-2)$-dimensional faces.\nThen any $v_h\\in V_h$ satisfies\n\\begin{equation*}\n\\|\\triangle_hv_h\\|_0^2 \n = \\|D_h^2v_h\\|_0^2+2\\sum_{i=1}^{d-1}\\sum_{F\\in \\mathcal{F}}([\\partial_n v_h],\\partial^2_{t_i}v_h)_{F}.\n\\end{equation*}\nHere $(t_i)_{i=1}^{d-1}$ is any orthonormal set of tangential vectors to \n$F\\in\\mathcal F$.\n\\label{DMT1}\n\\end{mylem}\n\\begin{comment} \n \\begin{proof}\n On each elements, by integration by parts\n \\begin{align*}\n (\\triangle v_h,\\triangle v_h)_T = \\sum_{F\\subset \\partial T}\\int_{F}\\partial_n v_h\\triangle v_h\\mathrm{d}S-\\sum_{F\\subset \\partial T}\\int_{F}\\nabla v_h\\cdot(D^2v_h\\pmb n)\\mathrm{d}S+(D^2v_h,D^2v_h)_T.  \n \\end{align*}\t\nSince the $\\triangle$ operator is coordinate invariant,\n\\begin{align*}\n&\\sum_{F\\subset \\partial T}\\int_{F}\\partial_n v_h\\triangle v_h\\mathrm{d}S-\\sum_{F\\subset \\partial T}\\int_{F}\\nabla v_h\\cdot(D^2v_h\\pmb n)\\mathrm{d}S\\\\\n&= \\sum_{F\\subset \\partial T}\\int_{F}\\partial_n v_h(\\partial^2_nv_h+\\sum_{i=1}^{d-1}\\partial^2_{t_i}v_h)\\mathrm{d}S-\\sum_{F\\subset \\partial T}\\int_{F}(\\partial_nv_h\\pmb n+\\sum_{i=1}^{d-1}\\partial_{t_i}v_h\\pmb t_i )\\cdot(D^2v_h\\pmb n)\\mathrm{d}S\\\\\n&=\\sum_{F\\subset \\partial T}\\int_{F}\\partial_n v_h\\sum_{i=1}^{d-1}\\partial^2_{t_i}v_h\\mathrm{d}S-\\sum_{F\\subset \\partial T}\\int_{F}\\sum_{i=1}^{d-1}\\partial_{t_i}v_h\\pmb t_i \\cdot(D^2v_h\\pmb n)\\mathrm{d}S\n\\end{align*}\nSince $v_h$ is $C^1$ continuous on $(d-2)$-dimensional sides and $\\partial^2_{t_i}v_h,i=1,\\cdots,d-1$ are continuous across the $(d-1)$-dimensional sides, summing all the elements and integration by part the second term on the $(d-1)$-dimensional sides, finish the proof.\n\\end{proof}\n\\end{comment}\n\n\n\n\\section{Design of the finite element in two dimensions}\\label{s:2d}\n \nWe will work with shape function spaces $\\mathcal{P}_{T}$ \nwith respect to an element $T\\in\\mathcal T$ that\nconsist of the full polynomial space of degree $\\ell$\nplus a space $B_{T,\\ell}$ of some particular bubble functions, i.e.,\n\\begin{equation} \n\\mathcal{P}_{T} = P_{\\ell}(T)+B_{T,\\ell}. \n\\label{SFspace}\n\\end{equation}\nThe careful choice of the space \n$B_{T,\\ell}\\subseteq H^1_0(\\operatorname{int}T)$\nturns out crucial for the desired property \\eqref{dMT}.\n\n\\subsection{Notation}\nWe use the following convention on enumeration of vertices and \nedges of a triangle.\nThe three vertices of a triangle are denoted by\n$p_i,i=1,2,3$ \nand $F_i,i=1,2,3$ are the three edges.\nWe assume a counter-clockwise enumeration as in Figure~\\ref{f:triangle}.\nIn particular, $F_i$ is the convex combination of \n$p_{i-1}$ and $p_{i+1}$.\nHere and throughout this section,\nwe always omit $\\mathrm{mod}3$ in the subscript of $p_{i-1}$ and $p_{i+1}$. \nThe functions\n$\\lambda_i,i=1,2,3$ are the three barycentric coordinates on the element, i.e., $\\lambda_i$ is an\naffine function on $T$ satisfying $\\lambda_i(a_j)=\\delta_{i,j},~ 1\\leq i,j\\leq 3$.\nWe define the non-negative cubic volume bubble function as\n$b_T=\\lambda_1\\lambda_2\\lambda_3$ \nand the quadratic edge bubble functions by $b_{F_i}=\\lambda_{i-1}\\lambda_{i+1}$.\n\n\t\\begin{figure} \\setlength\\unitlength{1pt}\n\t\\begin{center} \n\t\\begin{tikzpicture}\n\\draw (0,0) -- (2,0);\t\n\\draw (0,0) -- (1,1.732);\n\\draw (2,0) -- (1,1.732);\n\\node [below] at (1,0) {$F_1$};\n\\node [above] at (1,1.732) {$p_1$};\n\\node [right] at (2,0) {$p_3$};\n\\node [left] at (0,0) {$p_2$};\t\n\\node [left] at (0.5,1.732\/2) {$F_3$};\n\\node [right] at (1.5,1.732\/2) {$F_2$};\n\\draw[fill] (1,1.732) circle [radius=0.05];\n\\draw[fill] (2,0) circle [radius=0.05];\n\\draw[fill] (0,0) circle [radius=0.05];\n\t\\end{tikzpicture}\n\\end{center}\n\\caption{Enumeration of vertices and edges in a triangle.\n         \\label{f:triangle}}\n\\end{figure}\n\n\\subsection{Review of the second-order Specht element}\nReferences\n\\cite{specht,specht2,specht3} introduce the second-order Specht element, \nwhich has the convergence order $2$ under optimal regularity\nassumptions.\nThe local shape functions on an element $T\\in\\mathcal T$ are as follows\n$$\n \\mathcal{P}_T=P_3(T)+\\mathrm{span}\\{\\phi_i:i=1,2,3\\},\n$$\nwhere the three functions $\\phi_i$ are defined as\n$$\n\\phi_i = 2b_T\\big(5(\\lambda_i-\\lambda_i^2-2\\lambda_{i-1}\\lambda_{i+1})-1\\big),\n~1\\leq i\\leq 3\n.\n$$\nThe space of functions over $\\Omega$ that belong to $\\mathcal P_T$\nwhen restricted to any $T\\in\\mathcal T$ is denoted by $\\mathcal P_{\\mathcal T}$.\nThe global finite element space $V_h$ reads\n$$\n\\begin{aligned}\nV_h := \nH^1_0(\\Omega)\\cap \n\\left\\{\n   v \\in \\mathcal P_{\\mathcal T } :\n   \\begin{aligned}\n   &v\\text{ and }\\nabla v \\text{ are continuous at all vertices}\n   \\\\\n   &\\text{and }\n         \\textstyle\\int_F [\\partial_n v]\\,ds = 0 \\text{ for any interior edge } F\n\\end{aligned}\n\\right\\}\n.\n\\end{aligned}\n$$\nIt is direct to verify that the functions of $V_h$ are continuous\nwith $C^1$ continuity at the vertices.\nThe next result establishes the strong discrete Miranda--Talenti\nproperty \\eqref{dMT} for\nthe second order Specht element.\n\n\n\\begin{mylem}\nAny $v_h\\in V_h$ satisfies\n$$\n\\|\\triangle_hv_h\\|_0^2 = \\|D_h^2v_h\\|_0^2.\n$$ \n\\label{Specht}\n\\end{mylem}\n\n\n\\begin{proof}\n\tAccording to Lemma \\ref{DMT1}, the continuity of $v_h$ over interior edges\n\tand the continuity of $\\nabla v_h$ at the vertices imply\n\t \\begin{align*}\n\\|\\triangle_hv_h\\|_0^2 = \\|D_h^2v_h\\|_0^2+2\\sum_{F\\in \\mathcal{F}}([\\partial_n v_h],\\partial^2_{t}v_h)_{F}.\n\t\\end{align*}\n Let $F_i\\in\\mathcal F$ be an interior edge\n  with endpoints\n $p_{i-1}$ and $p_{i+1}$ and normal vector $n=n_{F_i}$.\n By the defining degrees of freedom,\n the integral over $F_i$ of the normal derivative jump\n of $v_h$ against constants vanishes.\n Therefore the second-order tangential derivative\n and its integral mean $m$ over $F_i$ satisfy\n$$\n \\int_{F_i}\\partial^2_{t}v_h[\\partial_nv_h]\\mathrm{d}s =\n  \\int_{F_i} (\\partial^2_{t}v_h - m ) [\\partial_nv_h] \\mathrm{d}s\n  .\n$$ \n We note that\n $\\partial^2_{t}v_h|_{F}$ is an affine function along $F_i$,\n whence $\\partial^2_{t}v_h - m$ vanishes\n in the midpoint of $F_i$ and thus is a multiple of $\\lambda_{i+1}-1\/2$.\n Therefore there exists a constant $c$ such that\n\\begin{equation}\n \\int_{F_i}\\partial^2_{t}v_h[\\partial_nv_h]\\mathrm{d}s =\n  c\\int_{F_i} (\\lambda_{i+1}-1\/2) [\\partial_nv_h] \\mathrm{d}s\n  .\n  \\label{e:simpson_prep}\n\\end{equation}\nWe next claim the identity\n\\begin{equation}\n\\int_{F_i}\\lambda_{i+1} [\\partial_{n}v_h]\\mathrm{d}s=\n\\frac{1}{12}([\\partial_{n}v_h](p_{i+1})-[\\partial_{n}v_h](p_{i-1}))\n   +\\frac{1}{2}\\int_{F_i}[\\partial_{n}v_h]\\mathrm{d}s\n\\label{simpson}\n\\end{equation}\nwhere $[\\cdot]$ always denotes the jump across $F_i$.\nThe proof of the identity utilizes the fact that $\\lambda_{i+1}-1\/2$\nvanishes in the midpoint of $F_i$ and takes the value $\\pm1$ at the \nvertices so that the result for the piecewise cubic part of $v_h$\nfollows from an application of the Simpson\nquadrature rule, which is exact for polynomials of degree $3$.\nSince the normal derivatives of the bubble functions $\\phi_i$\nhave vanishing integrals against purely linear functions \n(multiples of $\\lambda_i -1\/2$)\nover the edges and vanish in\nthe vertices, the result holds for any $v_h\\in V_h$.\nThe combination of \\eqref{e:simpson_prep} and \\eqref{simpson}\nshows\n$$\n \\int_{F_i}\\partial^2_{t}v_h[\\partial_nv_h]\\mathrm{d}s = 0\n$$\nbecause the normal derivative jumps vanish in the vertices\ndue to the defining constraints of $V_h$.\n\\end{proof}\n\\subsection{Extension of the second order Specht element to any odd order}\\label{section2D}\nIn this section, we will extend the Specht element to any odd order in 2 dimensions.\nFor any triangle $T\\in\\mathcal T$ and $i=1,2,3$\nwe define degrees of freedom (linear functionals) \nmapping any $v\\in C^\\infty(\\mathbb R^2)$ to\n\\begin{subequations}\n\\begin{align}\n    \\label{e:dof2d_nodal_edge}\n\tv(a_i) \\text{ and }\n\t\\nabla v(a_i)\n    \\text{ and }\n\t\\fint_{F_i}v q_{\\ell-4}\\mathrm{d}s \\text{ for any }q_{\\ell-4}\\in P_{\\ell-4}(F_i),\\\\\n\t\t  \t\\label{e:dof2d_normalA}\n\t\\fint_{F_i}\\partial_nv q_{\\ell-4}\\mathrm{d}s \n\t\\text{ for any }q_{\\ell-4}\\in P_{\\ell-4}(F_i),\\\\\n\t\\label{e:dof2d_normalB}\n\t\\fint_{F_i}\\partial_nv \\,\\lambda_{i+1}^{\\ell-3}\\mathrm{d}s,\n\t \\\\\n    \\label{e:dof2d_volume}\n\t\\fint_T v q_{\\ell-6} \\mathrm{d}x , \n\t\\text{ for any }q_{\\ell-6} \\in P_{\\ell-6}(T).\n\\end{align}\n\t \\end{subequations}\nIf $\\ell<6$, we interpret \\eqref{e:dof2d_volume} as being void.\nThe dimension of the space\n$\\Sigma_{T,\\ell}$ spanned by the linear functionals \n\\eqref{e:dof2d_nodal_edge}--\\eqref{e:dof2d_volume}\nat most $\\mathrm{dim}P_{\\ell}(T)+3$. \n\n\\begin{mylem}\nLet $\\ell\\geq 4$ be even.\nThen,\nthe degrees of freedom \\eqref{e:dof2d_nodal_edge}, \n\\eqref{e:dof2d_normalA}, \\eqref{e:dof2d_volume}\nare linear independent as functionals over\n$P_{\\ell}(T)$.\n\\label{uni2D}\n\\end{mylem}\n\n\\begin{proof}\n\tLet $v_h\\in P_{\\ell}(T)$ such that all degrees of freedom\n\tfrom \\eqref{e:dof2d_nodal_edge}, \\eqref{e:dof2d_normalA}, \\eqref{e:dof2d_volume}\n\tvanish on $v_h$.\n\tWe will prove $v_h\\equiv 0$.\n\tSince the functionals \\eqref{e:dof2d_nodal_edge} vanish on $v_h$,\n\tthe values of $v_h$ and $\\nabla v_h$ vanish at all vertices\n\t$a_i$ and all edge moments $\\int_{F_i}v q_{\\ell-4}\\mathrm{d}s$,\n\tagainst polynomials $q_{\\ell-4}$ of degree at most $\\ell-4$\n\tare zero.\n\tThus $v_h$ vanishes on the boundary of $T$ and there exists some \n\t$g_{\\ell-3}\\in P_{\\ell-3}(T)$ such that\n\t$$\n\tv_h = b_T g_{\\ell-3}.\n\t$$\n\tSince the functionals\n\t\\eqref{e:dof2d_normalA} vanish,\n\twe have for all $q_{\\ell-4}\\in P_{\\ell-4}(F_i)$ that\n\t$$\n\t0=\\int_{F_i}\\partial_nv_hq_{\\ell-4}\\mathrm{d}s\n\t =\\int_{F_i}\\partial_n (b_Tg_{\\ell-3})q_{\\ell-4}\\mathrm{d}s\n\t =\\partial_n\\lambda_j\\int_{F_i} b_{F_i}g_{\\ell-3}q_{\\ell-4}\\mathrm{d}s\n\t$$\n\tfor $i=1,2,3$.\n\tTherefore,  $g_{\\ell-3}|_{F_i}$ is a multiple of the $(\\ell-3)$rd orthogonal\n\tpolynomial $L_{\\ell-3}^i$ over $F_i$ with respect to the even weight function\n\t$b_{F_i}$. That is, there is a coefficient $c_i$ such that\n\t$g_{\\ell-3}|_{F_i} = c_i L_{\\ell-3}^i$.\n\tSince $\\ell-3$ is odd, it is konwn \\cite{Spectral} that $L_{\\ell-3}^i$\n\tis an odd function with\n\t$L_{\\ell-3}^i(p_{i-1})= -L_{\\ell-3}^i(p_{i+1})\\neq 0$. \n    Since $g_{\\ell-3}$ is continuous, evaluation of at the three vertices $p_1,p_2,p_3$\n    results in\n    $$\n    c_1=-c_2, \\quad c_2=-c_3, \\quad c_3=-c_1.\n    $$\n    Therefore $c_1=c_2=c_3=0$ and so $g_{\\ell-3}$ vanishes on the boundary\n    $\\partial T$.\n    In consequence,\n    $\\partial_nv_h|_{\\partial T}=0$.\n    This proves the assertion in the case case $\\ell=4$.\n\tIf $\\ell\\geq 6$, we see that there exists a polynomial\n\t$g_{\\ell-6}$ such that\n\t$$\n\tv_h = b_T^2 g_{\\ell-6}.\n\t$$\n    Since, by the vanishing degrees of freedom \\eqref{e:dof2d_volume},\n    the integral\n    $\\int_T v_h g_{\\ell-6}\\mathrm dx$ equals zero,\n    we conclude that $g_{\\ell-6}=0$ whence $v_h$ vanishes identically.\n\\end{proof}\n\nLemma \\ref{uni2D} states that the functionals \\eqref{e:dof2d_nodal_edge},\n\\eqref{e:dof2d_normalA}, \\eqref{e:dof2d_volume} are linear independent\nover $P_\\ell(T)$.\nIn order to include the remaining three degrees of freedom\n\\eqref{e:dof2d_normalB}, we enlarge the space \nby three additional basis functions.\nFor any $i=1,2,3$ we set\n$$\n \\psi_i:=\\lambda_{i+1}^{\\ell-3}+\\sum_{k=0}^{\\ell-4}c_k\\lambda_{i+1}^k \\in P_{\\ell-3}(T)\n$$ \nwhere the coefficients $c_k$ are chosen such that such that $\\psi_i$ satisfies\n\\begin{equation} \n  \\int_{F_i}b^2_{F_i}\\psi_iq_{\\ell-4}\\mathrm{d}s=0,\n  \\text{ for all } q_{\\ell-4}\\in P_{\\ell-4}(F_i).\n  \\label{2Dbubble'}\n\\end{equation}\nWe then define bubble functions $\\psi^B_i$, $i=1,2,3$ by\n\\begin{equation} \n\\psi_i^B := b_Tb_{F_{i}}\\psi_i\n\\label{2Dbubble}\n\\end{equation}\nand define the shape function space as\n\\begin{equation}\n\t\\mathcal{P}_{T,\\ell} = P_{\\ell}(T) + \\mathrm{span}\\{\\psi_i^B : i=1,2,3\\}.\n\\end{equation}  \n\n\\begin{mylem}\n If, $\\ell\\geq4$ is even,\n  the degrees of freedom spanned by the functionals\n  \\eqref{e:dof2d_nodal_edge}--\\eqref{e:dof2d_volume} are unisolvent for the space\n  $\\mathcal{P}_{T,\\ell}$.\n  \\label{uni2Dgen}\n\\end{mylem}\n\\begin{proof}\nLet $v_h\\in \\mathcal{P}_{T,\\ell}$ vanish on degrees of freedom  \\eqref{e:dof2d_nodal_edge}--\\eqref{e:dof2d_volume}. We will show that $v_h\\equiv 0.$\nBy definition of $\\mathcal{P}_{T,\\ell}$,\nthe function $v_h$ can be decomposed as\n$$\n v_h= q_\\ell +  \\sum_{k=1}^{3}a_k\\psi^B_k\n$$\nwith some $q_\\ell\\in P_\\ell(T)$ and real coefficients $a_1,a_2,a_3$.\nSince the functions $\\psi_k^B$ are zero on degrees of freedom \\eqref{e:dof2d_nodal_edge},\n\t\\eqref{e:dof2d_normalA},\n\tso must be the function $q_\\ell$.\nBy Lemma~\\ref{uni2D}, such function $q_\\ell$ is uniquely determined by\nthe degrees of freedom \\eqref{e:dof2d_volume}\nand thus there exists a polynomial $q_{\\ell-6}\\in P_{\\ell-6}(T)$\n(zero if $\\ell<6$) such that $q_\\ell=b_T^2 q_{\\ell-6}$.\n\nSince the normal derivative of $q_\\ell$ is zero over $\\partial T$,\nthe vanishing of degrees of freedom \\eqref{e:dof2d_normalB}\nand elementary computations imply that\n$$\n  0 = \\fint_{F_i} \\lambda_{i+1}^{\\ell-3}\n       \\partial_n \\left(\\sum_{k=1}^{3}a_k\\psi^B_k\\right)\n      \\mathrm{d}s\n    =\n    a_i (\\partial_n\\lambda_{i}) \\fint_{F_i}\\lambda_{i+1}^{\\ell-3}b_{F_i}^2\\psi_i\\mathrm{d}s\n    \\quad\\text{for }i=1,2,3.\n$$\nIt follows from the definition of $\\psi_i$ that the integral on the right-hand\nside is nonzero, whence $a_i=0$ for $i=1,2,3$.\nTherefore we have $v_h=q_\\ell=b_T^2 q_{\\ell-6}$.\nThe fact that $v_h$ vanishes on degrees of freedom \\eqref{e:dof2d_volume}\nfinally implies that $q_{\\ell-6}=0$ and thus $v_h=0$.\n\\end{proof}\n\nWe define the space of functions over $\\Omega$ whose restriction to\nany $T\\in\\mathcal T$ belongs to $\\mathcal{P}_{T,\\ell}$\nby $\\mathcal{P}_{\\mathcal T,\\ell}$\nThe global finite element space $V^{\\ell}_h$ defined as \n$$\n\\begin{aligned}\n\tV^{(\\ell)}_h := \n\tH^1_0(\\Omega)\\cap \n\t\\{&\n\tv\\in \\mathcal{P}_{\\mathcal T,\\ell}: \\nabla v \\text{ is continuous at all vertices and}\n\t\\\\\n\t&\n\t\\int_F [\\partial_n v]q_{\\ell-3}\\,ds = 0\n\t\\text{ for all interior faces } F \\text{ and all } q_{k}\\in P_{k}(F)\n\t\\}\n\t.\n\\end{aligned}\n$$\n\nThe crucial property of functions $v_h\\in V^{(\\ell)}_h$ is\nthat the normal derivative of $v_h$ has some extra continuity in terms\nof moments of degree one higher than required by the definition.\nThis implies that the finite element \n$(\\Sigma_{T,\\ell},\\mathcal{P}_{T,\\ell},\\mathcal{T}_h)$\nsatisfies the strong discrete Miranda--Talenti property \\eqref{dMT}.\n\n\\begin{mylem}\n\tLet $v_h\\in V^{(\\ell)}_h$ with $\\ell\\geq 4$ even.\n\tFor any interior edge $F$, $v_h$ satisfies\n\t\\begin{equation*}\n\t\t\\int_F [\\partial_nv_h]q_{\\ell-2}\\mathrm{d}s = 0\n\t\t\\quad\\text{for all } q_{\\ell-2}\\in P_{\\ell-2}(F).\n\t\\end{equation*}\n\t\\label{2Dsuper}  \n\\end{mylem}\n\\begin{proof} \n    Let $F$ be an interior edge shared by elements $T_+$, $T_-$.\n    In view of the degrees of freedom,\n    it suffices to prove that\n    $$\n\t\t\\int_F [\\partial_nv_h]\\varphi_F\\mathrm{d}s = 0\n    $$\n    for some $\\varphi_F\\in P_{\\ell-2}(F)$ that is linear independent\n    of all elements from $P_{\\ell-3}(F)$.\n    To this end, we\n    choose $\\varphi_F\\in P_{\\ell-2}(F)$ to be some nonzero function satisfying\n\t\\begin{equation}\n    \\int_{F}b_F\\varphi_Fq_{\\ell-3}\\mathrm{d}s=0\n    \\quad\\text{for all } q_{\\ell-3}\\in P_{\\ell-3}(F)\n    .\n    \\label{temp1}\t\n    \\end{equation}\n\tAny function $v_h\\in V_h$ can be expressed as \n\t$v_h|_{T_\\pm} = p_{\\ell,\\pm}+\\psi_{\\pm}^B$,\n\twhere $p_{\\ell,\\pm}\\in P_{\\ell}(T_\\pm)$ \n\tand \n\t$\\psi_{\\pm}^B\\in \\mathrm{span}\\{\\psi^B_i(T_\\pm),i=1,2,3\\}. $ \n\t Then\n\t\\begin{align}\\label{e:super_split}\n\t\t\\int_{F}[\\partial_n v_h]\\varphi_F\\mathrm{d}s\n\t\t=\\int_{F}(\\partial_np_{\\ell,+}-\\partial_np_{\\ell,-})\\varphi_F\\mathrm{d}s+\\int_{F}(\\partial_n\\psi^B_{\\ell,+}-\\partial_n\\psi^B_{\\ell,-})\\varphi_F\\mathrm{d}s.\n\t\\end{align}\n\tWe consider the first term on the right-hand side.\n\tSince $\\nabla v_h$ continuous on all vertices and $\\nabla\\psi^B_{\\pm}$ vanishes\n\tthere,\n\tthere exists some $q_{\\ell-3}\\in P_{\\ell-3}(F)$ such that\n\t$(\\partial_np_{\\ell,+}-\\partial_np_{\\ell,-})$\n\tequals $b_Fq_{\\ell-3}$. Relation \\eqref{temp1} thus implies\n\t\\begin{align*}\n\t\t\\int_{F}(\\partial_np_{\\ell,+}-\\partial_np_{\\ell,-})\\varphi_F\\mathrm{d}s =\n\t\t\\int_{F}b_Fq_{\\ell-3}\\varphi_F\\mathrm{d}s = 0.\n\t\\end{align*}\n\tFor the second term of \\eqref{e:super_split},\n\tby the definition of $\\psi_i^B$, there exists a function $\\psi_F\\in P_{\\ell-3}(F)$, such that\n\t$$\n\t\\int_{F}(\\partial_n\\psi^B_{+}-\\partial_n\\psi^B_{-})\\varphi_F\\mathrm{d}s\n\t=\\int_F b_F^2\\psi_F\\varphi_F\\mathrm{d}s.\n\t$$\n\tAccording to \\eqref{2Dbubble'}, ${\\psi}_F$ satisfies\n\t$$\n\t\\int_F b_F^2 \\psi_F q_{\\ell-4}\\mathrm{d}s=0\n\t\\quad\\text{for all } q_{\\ell-4}\\in P_{\\ell-4}(F).\n\t$$\n\tThis shows that $\\psi_F$ is an odd function\n\t(after an affine change of coordinates where $F$ is\n\tmapped to the real interval $[-1,1]$).\n\tOn the other hand $\\varphi_F$ satisfies \n    \\eqref{temp1} and therefore $\\varphi_F$ is an even function.\n    Thus,\n\t$$\n\t\\int_F b_F^2 \\psi_F \\varphi_F\\mathrm{d}s \n\t=\n\t0.\n\t$$\n\tHence, the right-hand side of \\eqref{e:super_split} is zero\n\tand the assertion is proven.\n\\end{proof}\n\n\\begin{corollary}\n The spaces $V_h^{(\\ell)}$, with $\\ell\\geq 4$ even, satisfy the discrete\n Miranda--Talenti property from Definition~\\ref{def:dmt}.\n\\end{corollary}\n\\begin{proof}\n The proof is consequence of Lemma~\\ref{DMT1}\n and Lemma~\\ref{2Dsuper}.\n\\end{proof}\n\n\n\n\n\n\\subsection{Example for $\\ell=4$}\nAs an illustration, we give details on the \nthe new finite element from Section~\\ref{section2D}\nfor the case $\\ell=4$.\nOn a triangle $T$, the shape function space reads\n$$\n\\mathcal{P}_{T,4} = P_4(T)+\\mathrm{span}\\{b_Tb_{F_i}(\\lambda_i-1\/2):i=1,2,3\\}.\n$$\nThe degrees of freedom are the evaluation of the \nfunction and its gradient at the three vertices, the averages of the \nfunction over the three edges, and the zeroth and first order moments\nof the normal derivative over the three edges.\nAn illustration is displayed in Figure~\\ref{fig:2d_illustration}.\nThe global finite element space consists of all globally continuous\nfunctions with homogeneous Dirichlet boundary values\nsuch that the gradient is continuous in all vertices and the \nnormal jumps over all interior edges have vanishing zeroth and \nfirst-order moments.\nLemma~\\ref{2Dsuper} states that, additionally, the normal jumps\nof such functions across interior edges automatically vanish\nwhen integrated against quadratic polynomials.\nOne implication of this property is, as mentioned earlier, the validity\nof the discrete Miranda--Talenti property.\nA further consequence is that the finite element is a third order element \nA conforming finite element of the same order is the Bell element \\cite{Bell1969A},\nwhich also has 18 degrees of freedom and has third order convergence.\nA practical advantage of our nonconforming method is that the degrees of freedom\nonly involve first-order derivatives as degrees of freedom at the vertices,\nwhich simplifies the implementation of boundary conditions.\n\n\n\\begin{figure}\n \\begin{tikzpicture}[scale=2.2]\n \\draw (0,0)--(1,0)--(.5,.8)--cycle;\n \\foreach \\x\/\\y  in {0\/0,1\/0,.5\/.8}\n      {  \\fill (\\x,\\y) circle (1pt);\n         \\draw (\\x,\\y) circle (2pt);}\n \\foreach \\a\/\\b\/\\c\/\\d  in {.35\/0\/.35\/-.2,\n                           .65\/0\/.65\/-.2,  \n                     .825\/.28\/.975\/.38,\n                     .675\/.52\/.825\/.62,\n                     .175\/.28\/.025\/.38,\n                     .325\/.52\/.175\/.62}\n      {\\draw[->,thick] (\\a,\\b)--(\\c,\\d);}\n \\draw (.17,.05)--(.83,.05)\n       (.9,.06)--(.55,.62)\n       (.1,.06)--(.45,.62);\n\\end{tikzpicture}\n\\qquad\\qquad\n\\begin{tikzpicture}[scale=2.2]\n \\draw (0,0)--(1,0)--(.5,.8)--cycle;\n \\foreach \\x\/\\y  in {0\/0,1\/0,.5\/.8}\n      {  \\fill (\\x,\\y) circle (1pt);\n         \\draw (\\x,\\y) circle (2pt);}\n \\foreach \\a\/\\b\/\\c\/\\d  in {.5\/0\/.5\/-.2,\n                     .75\/.4\/.9\/.5,\n                      .25\/.4\/.1\/.5}\n      {\\draw[->,thick] (\\a,\\b)--(\\c,\\d);}\n \\fill (.5,.2667) circle (1pt);\n\\end{tikzpicture}\n\\caption{Mnemonic diagram of the two-dimensional finite element for $\\ell=4$\n         (left) and $\\ell=3$ (right).\n         \\label{fig:2d_illustration}}\n\\end{figure}\n\n\n\n\\subsection{Example for $\\ell=3$}\n\nThe finite element from Section~\\ref{section2D} is defined for even values\nof $\\ell\\geq 4$.\nIn the low-order case, we can define an analogous element for the odd value\n$\\ell=3$.\nIt can be viewed as an alternative to the second-order Specht element.\nOur element offers a simpler shape function space,\nat the expense of one additional interior degree of freedom.\nThe construction is analogous to that of Section~\\ref{section2D}.\nThe shape function space reads\n\\begin{equation*}\n\t\\mathcal{P}_{T,3} = P_{3}(T)+\\mathrm{span}\\{b_Tb_{F_i}:i=1,2,3\\}.\n\\end{equation*}\nThe degrees of freedom\n(illustrated in Figure~\\ref{fig:2d_illustration}) are the evaluation of the \nfunction and its gradient at the three vertices, the averages of the \nnormal derivative over the three edges, and the volume average.\nWe denote this set of linear functionals as by $\\Sigma_{T,3}$.\n\n\\begin{mylem}\n\tThe shape function space $\\mathcal{P}_{T,3}$ is unisolvent \n    by the degrees of freedom  $\t\\Sigma_{T,3}$.\n\\end{mylem}\n\\begin{proof}\n   The functions\n\tSince $b_Tb_{F_i}$ and their gradients vanish at the vertices of \n    the triangle $T$.\n    An argument along the lines of Lemma~\\ref{uni2Dgen} therefore shows \n    that any function $v_h$ from $\\mathcal{P}_{T,3}$ vanishing at all degrees \n    of freedom is the sum of a multiple of the volume bubble $b_T$\n    and a linear combination of the functions $b_Tb_{F_i}$.\n    Therefore there are real coefficients $c,a_1,a_2,a_3$ such that\n    $$\n     v_h = c\\, b_T + \\sum_{i=1}^3 a_i b_Tb_{F_i} .\n    $$\n    Since the degrees of freedom related to the normal derivative \n    vanish, an elementary computation reveals that\n    $a_1=a_2=a_3$ and $a_i=-5c$.\n    On the other hand, the volume average of $b_T$ equals $1\/60$\n    and the volume average of $b_Tb_{F_i}$ equals $1\/630$.\n    The constraint $\\fint_T v_h\\mathrm{d}x=0$ therefore implies\n    $c=0$ and thus $v_h=0$.    \n\\end{proof}\n\nThe global finite element space for the the case $\\ell=3$ is defined as\n\\begin{align*}\n\tV^{(3)}_{h}=\n\t H^1_0(\\Omega)\\cap \n\t \\{ v \\in \\mathcal{P}_{\\mathcal T,3} :\n\t & \\nabla v \\text{ is continuous \n\t at all vertices and} &\\\\\n\t&\\int_F [\\partial_{n} v] \\mathrm{d}s=0\\text{ for any interior edge } F\n\t\\}.\n\\end{align*}\nHere, as in prior sections, the space $\\mathcal{P}_{\\mathcal T,3}$ consists of\nfunctions that piecewise belong to $\\mathcal{P}_{T,3}$.\nA proof analogous to that of Lemma~\\ref{2Dsuper}\nshows that the finite element spaces enjoys a higher-order normal\ncontinuity.\n\\begin{mylem}\n\t\tAny function $v_h\\in V^{(3)}_h$ satisfies for all interior edges $F$\n\t\tthat\n\t\\begin{equation*}\n\t\t\\int_F [\\partial_nv_h]q_1\\mathrm{d}s = 0\n\t\t\\quad\\text{for all } q_1\\in P_1(F).\n\t\\end{equation*}  \n\n\\end{mylem}\nBy Lemma~\\ref{DMT1}, the space $V^{(3)}_{h}$ then satisfies the discrete\nMiranda--Talenti property.\n\n\n\\section{Design of the finite element on three dimensions}\\label{s:3d}\nThe construction of piecewise polynomial finite element functions \nin three dimensions\nwith $C^1$ continuity across edges requires the use of polynomials\nof degree at least 5.\nAnalogous to the two-dimensional case, our our shape functions\nconsists of two parts,\nnamely functions from the space $P_\\ell(T)$ for polynomials of degree\n$\\ell\\geq 5$ with respect to a simplex $T$,\nplus carefully chosen\nparticular $H_0^1(\\operatorname{int}T)$ bubble functions.\nIn view of the $C^1$ continuity at the edges,\nthe degrees of freedom related to the normal derivatives on faces\nare the moments of order up to $\\ell-4$, which is less than in\nthe two-dimensional case.\n\n\\subsection{Notation}\nWe begin by introducing some notation.\nThroughout this section, $\\mathcal T$ is a regular simplicial partition\nof the open, bounded, connected \npolytopal Lipschitz domain $\\Omega\\subseteq\\mathbb R^3$.\nWe denote by $\\mathcal{N}, \\mathcal{E},\\mathcal{F}$ the sets\nof vertices, edges, faces, respectively. \nGiven a simplex $T\\in\\mathcal T$, the sets\n$\\mathcal{N}(T), \\mathcal{E}(T),\\mathcal{F}(T)$ are sets of the vertices,\nedges, face of $T$.\nWe assign every edge $e\\in\\mathcal E$ with two fixed \nlinear independent unit normals $n_1,n_2$.\nOn a simplex $T$ with vertices $a_1,a_2,a_3,a_4$,\nthe four barycentric coordinates are denoted by\n$\\lambda_1,\\lambda_2,\\lambda_3,\\lambda_4$.\nGiven an edge $e$ with endpoints $a_i$, $a_j$,\nthe quadratic edge bubble function $b_e$ is defined as\n$b_e:=\\lambda_i\\lambda_j$.\nGiven a face $F$ with vertices $a_i$, $a_j$, $a_k$,\nthe cubic face bubble function $b_F$ is defined as\n$b_F:=\\lambda_i\\lambda_j\\lambda_k$.\nThe quartic volume bubble is defined as \n$b_T:=\\lambda_1\\lambda_2\\lambda_3\\lambda_4$.\n\n\n \n\\subsection{Degrees of freedom}\nGiven any simplex $T\\in\\mathcal T$,\nwe define degrees of freedom (linear functionals) \nmapping any $v\\in C^\\infty(\\mathbb R^3)$ to\n\\begin{subequations}\n\t\\begin{align}\n\t\t\\label{e:dof3d_nodal}\n\t\tv(p),~\\nabla v(p), \n\t\tD^2 v(p)&\\quad \\text{ for any }p\\in \\mathcal{N}(T)\\\\\n\t\t\t\\label{e:dof3d_edge}\n\t\t\\fint_{e}v q_{\\ell-6}\\mathrm{d}s~\n\t\t&\\quad\\text{ for any }q_{\\ell-6}\\in P_{\\ell-6}(e),~e\\in\\mathcal{E}(T)\\\\\n\t\t\\label{e:dof3d_edge_normal}\n\t\t\t\\fint_{e}\\partial_{n_j}v q_{\\ell-5}\\mathrm{d}s\n\t\t\t&\\quad\\text{ for any } q_{\\ell-5}\\in P_{\\ell-5}(e),~e\\in\\mathcal{E}(T),~ j =1,2\\\\\n\t\t\\label{F:dof3d_normalA}\n\t\t\\fint_{F}\\partial_nv q_{\\ell-4}\\mathrm{d}S \n        &\\quad\n\t\t\\text{ for any }q_{\\ell-4}\\in P_{\\ell-4}(F),~F\\in \\mathcal{F}(T)\\\\\n\t\t\\label{F:dof3d_normalB}\n\t\t\\fint_{F}v q_{\\ell-6}\\mathrm{d}S \n\t\t&\\quad\n\t\t\\text{ for any }q_{\\ell-6}\\in P_{\\ell-6}(F),~F\\in \\mathcal{F}(T)\n\t\t\\\\\n\t\t\\label{e:dof3d_volume}\n\t\t\\fint_T v q_{\\ell-4} \\mathrm{d}x , \n\t\t&\\quad \n\t\t\\text{ for any }q_{\\ell-4}\\in P_{\\ell-4}(T)\n\t\t.\n\t\\end{align}\n\\end{subequations}\nWe use the convention that \\eqref{e:dof3d_edge} and \\eqref{F:dof3d_normalB}\nare void if $\\ell=5$.\n\n\\subsection{Shape functions}\n\n\n\tFrom the definition of degrees of freedom \\eqref{e:dof3d_nodal}--\\eqref{e:dof3d_volume}, \n\tthe jump of the normal derivative has $(\\ell-4)$th order moment continuity. \n\tOur selection for of $H_0^1(T)$ bubble functions are functions\n\torthogonal to particular $(\\ell-2)$nd order polynomials \n\t$\\varphi_1,\\dots,\\varphi_{2\\ell-3}$ on each face. \n\tThe latter polynomials are defined as follows.\nLet $\\ell\\geq 5$ be a fixed integer and let $F\\in\\mathcal F(T)$ be a face\nof the simplex $T\\in\\mathcal T$.\nWe abbreviate\n$$\nN_{\\ell-4}:=\\dim P_{\\ell-4}(F).\n$$\nLet \n$(\\phi_1,\\dots,\\phi_{N_{\\ell-4}})$\nbe a given basis of $P_{\\ell-4}(F)$, which\nwe extend to a basis of $P_{\\ell-2}(F)$ \nby adding suitable polynomials $\\varphi_1,\\dots,\\varphi_{2\\ell-3}$\nof degree $\\ell-2$ satisfying the orthogonality relation\n\\begin{equation}\n\\int_{F}b_{F} q_{\\ell-4} \\varphi_{i}\\mathrm{d}S=0\n\\quad\\text{for all }q_{\\ell-4}\\in P_{\\ell-4}(F)\n\\text{ and all } i=1,\\dots,2\\ell-3.\n\\label{express1}\n\\end{equation}\nNext we will define $N_{\\ell-4}$ bubble functions that\nvanish on $\\partial T$.\nLet $X_F$ denote the subspace of $P_{\\ell-2}(T)$ spanned\nby the Lagrange basis functions that do not vanish identically\non $F$. The dimension of $X_F$ equals that of $P_{\\ell-2}(F)$.\nLet $\\tilde\\phi_1,\\dots,\\tilde\\phi_{N_{\\ell-4}} \\in X_F$ satisfy\n\\begin{equation}\\label{e:tildephi_orth}\n\\fint_{F}b_{F}^2\\tilde{\\phi}_i\\phi_{j}\\mathrm{d}S = \\delta_{i,j} \n\\text{ and }~\\fint_{F}b_{F}^2\\tilde{\\phi}_i\\varphi_{k}\\mathrm{d}S = 0\n\\quad\\text{for all }\n\\begin{cases}\n i,j=1,\\dots,N_{\\ell-4}\\\\k=1,\\dots,2\\ell-3.\n\\end{cases}\n\\end{equation}\nNote that we have chosen the space $X_F$ for the purpose of uniqueness\nof the functions $\\tilde\\phi_i$.\nFor the functions $\\varphi_1,\\dots,\\varphi_{2\\ell-3}$ from \\eqref{express1}\nwe have\n\\begin{equation}\n\\begin{aligned}\n\t\\int_{F}\\partial_n b_Tb_{F}\\tilde{\\phi}_i\\varphi_k\\mathrm{d}S \n\t= \n\t(\\partial_n\\lambda_F)\\int_{F}b^2_{F} \\tilde{\\phi}_i\n\\varphi_k \n\t\\mathrm{d}S\n=\t\n0\n\\quad\\text{ for all }\n\\begin{cases}\n i=1,\\dots,N_{\\ell-4}\\\\k=1,\\dots,2\\ell-3.\n\\end{cases}\n\t\\label{express2}\n\\end{aligned}\n\\end{equation}\nTherefore, the normal derivative  of $\tb_Tb_F\\tilde{\\phi}_i$\n$i=1,\\cdots,N_{\\ell-4}$, is orthogonal to $\\varphi_k,k=1,\\cdots,2\\ell-3$. \nGiven any $F\\in \\mathcal{F}(T)$, define the $N_{\\ell-4}$ bubble functions as\n\\begin{equation}\n\t\\xi_i^F :=  \n\tb_Tb_F\\tilde{\\phi}_i\n\t   -\\tilde q_i,\n\t   \\quad i=1,\\cdots, N_{\\ell-4}.\n\\label{express3}\n\\end{equation}\nHere $\\tilde q_1,\\dots,\\tilde q_{N_{\\ell-4}}\\in b^2_T P_{\\ell-4}(T)$  satisfy\n$$\n\\int_T\\tilde q_i q_{\\ell-4}\\mathrm{d}x=\n\\int_T\\tilde b_Tb_F\\tilde{\\phi}_i\nq_{\\ell-4}\\mathrm{d}x\n\\quad\\text{for all } q_{\\ell-4}\\in P_{\\ell-4}(T).\n$$\n\n\\begin{remark}\\label{rem:xidof}\nAccording to \\eqref{express2} and \\eqref{express3},\n$\\xi_i^F$ span the nodal basis functions corresponding to degrees of freedom \n\\eqref{F:dof3d_normalA}.\n\\end{remark}\n\nWe define the shape function space of our finite element as \n\\begin{equation*}\n\\mathcal{P}_{T,\\ell} = P_{\\ell}(T) + \\Phi_{T,\\ell}^B\n\\end{equation*}\nwhere\n$\\Phi_{T,\\ell}^B \n= \\mathrm{span}\\{\\xi_i^F: F\\in\\mathcal F(T),~\n          i=1,\\cdots,N_{\\ell-4}\\}.$\n\n\\begin{mylem}\n    Let $\\ell\\geq 5$.\n\tThe degrees of freedom defined by the functionals\n\tfrom \\eqref{e:dof3d_nodal} -- \\eqref{e:dof3d_volume} are \n\tunisolvent for the shape function space\n\t$\\mathcal{P}_{T,\\ell}$.\n\\end{mylem}\n\\begin{proof}\nThe dimension of \\eqref{e:dof3d_nodal} -- \\eqref{e:dof3d_volume} at most \n$\\mathrm{dim}P_{\\ell}(T)+4N_{\\ell-4}=\\mathrm{dim}\\mathcal{P}_{T,\\ell}$.\nLet $v_h\\in \\mathcal{P}_{\\ell}(T)$ vanish on the degrees of freedom\n\\eqref{e:dof3d_nodal} -- \\eqref{e:dof3d_volume}.\nWe will show that $v_h \\equiv 0$.\nSince $v_h|_F\\in P_\\ell(F)$\nfor any face $F\\in\\mathcal{F}(T)$ and $v_h$ vanishes on\n\\eqref{e:dof3d_nodal},\n\\eqref{e:dof3d_edge},\n\\eqref{e:dof3d_edge_normal},\n\\eqref{F:dof3d_normalB},\nthe function $v_h$ can be written as   \t\n$$\n  v_h = b_Tq_{\\ell-4} + \\sum_{i=1}^{N_{\\ell-4}}\\sum_{F\\in \\mathcal{F}(T)}c^F_i\\xi_i^F\n$$\nfor some $q_{\\ell-4}\\in P_{\\ell-4}(T).$ \nBy their definition \\eqref{express3}, \nthe functions $\\xi_i^F$ vanish on \\eqref{e:dof3d_volume}. Therefore, $v_h$ can be expressed as\n$$\nv_h = \\sum_{i=1}^{N_{\\ell-4}}\\sum_{F\\in \\mathcal{F}(T)}c^F_i\\xi_i^F.\n$$\nAs $v_h$ vanishes on degrees of freedom \\eqref{F:dof3d_normalA}\nand, by Remark~\\ref{rem:xidof}, the $\\xi_i^F$ span the corresponding\nnodal basis functions, we deduce\n$c_i^F=0$ for all $i=1,\\cdots,N_{\\ell-4}$ and all $F\\in \\mathcal{F}(T)$.\nThis shows $v_h=0$ and concludes the proof.\n\\end{proof}\n\n\nWe denote by $\\mathcal{P}_{\\mathcal T,\\ell}$ the space of functions whose restriction\nto any $T\\in\\mathcal T$ belongs to $\\mathcal{P}_{T,\\ell}$.\nThe global finite element space reads\n\\begin{align*}\n\tV^{(\\ell)}_{h}=\n\tH^1_0(\\Omega) \\cap\n\t\\left\\{\n\tv\\in \\mathcal{P}_{\\mathcal T,\\ell}:\n\t\\begin{aligned}\n\t & v,~\\nabla v, D^2 v~\\text{are continuous at }\\mathcal N,\\\\\n\t&v, \\nabla v~\\text{ are continuous at }\\mathcal E,\n\t\\text{ and }\\int_F[\\partial_nv]q \\mathrm{d}S=0 \\\\&\\text{ for all interior faces }F\n\t\\text{ and }q\\in P_{\\ell-4}(F)\n\t\\end{aligned}\n\t\\right\\}.\n\\end{align*}    \n\nThe subsequent lemma establishes improved normal continuity\nfor the elements of $V_h^{(\\ell)}$.\n\\begin{mylem}\n Any $v_h\\in V^{(\\ell)}_{h}$, $\\ell\\geq 5$,\n satisfies for all interior faces\n $F$ that\n\\begin{equation*}\n\t\\int_F [\\partial_nv_h]p_{\\ell-2}\\mathrm{d}S = 0\n\t\\quad\\text{for all } p_{\\ell-2}\\in P_{\\ell-2}(F).\n\\end{equation*}\n\\label{3Dsuper}\n\\end{mylem}\n\\begin{proof}\nLet $F\\in\\mathcal F$ be an interior face shared by elements\n$T_+$, $T_-$.\nAccording to the definition \\eqref{F:dof3d_normalA}, \nthe asserted relation is satisfied for all test polynomials\nof degree $\\ell-4$ so that we only need to verify it\nfor polynomials $p_{\\ell-2}\\in P_{\\ell-2}(F)$ that do not belong to \n$P_{\\ell-4}(F)$.\nWe will take the linear independent\npolynomials from \\eqref{express1} as such test functions.\n Similar as in the 2D case, by the definition of\n $\\mathcal P_{T,\\ell}$,\n the function $v_h|_{T_\\pm}$ can be rewritten \n as $v_h|_{T_\\pm}=p_{\\ell,\\pm} + \\xi^B_\\pm$, \n where $p_{\\ell,\\pm}\\in P_{\\ell}(T_\\pm)$ and\n $\\xi^B_\\pm\\in \\Phi^B_{T_\\pm,\\ell}.$\n Then we test with functions $\\varphi_i$ defined in \\eqref{express1}\n \\begin{align}\\label{e:3dproof_split} \n \\int_{F}[\\partial_nv_h]\\varphi_i\\mathrm{d}S = \\int_{F}(\\partial_n p_{\\ell,+}-\\partial_n p_{\\ell,-})\\varphi_i\\mathrm{d}S + \\int_{F}(\\partial_n\\xi^B_+-\\partial_n\\xi^B_-)\\varphi_i\\mathrm{d}S.\n \\end{align}\n The functions $\\xi^B_\\pm$ and $\\nabla\\xi^B_\\pm$ vanish on all vertices and \n edges.\n  From the continuity properties of $ V^{(\\ell)}_h$,\n we see that the piecewise polynomial function $p_{\\ell,\\pm}$\n and its piecewise gradient are continuous at the vertices and edges.\n Therefore, the normal derivative jump on face $F$ vanishes on $\\partial F$\n and thus there\n exists a polynomial $q_{\\ell-4}\\in P_{\\ell-4}(F)$ such that\n $(\\partial_np_{\\ell,+}-\\partial_np_{\\ell,-})|_F = b_Fq_{\\ell-4}$.\n The orthogonality relation \\eqref{express1} shows that the first integral\n on the right-hand side of \\eqref{e:3dproof_split} equals zero.\n  In the remaining part of the proof we will show that the\n integrals\n $$\n \\int_F\\partial_n\\xi^B_\\pm\\varphi_i\\mathrm dS\n $$\n equal zero, which implies that the left-hand side of\n \\eqref{e:3dproof_split} vanishes.\n Consider $\\partial_n\\xi^F_k$,\n $k=1,\\cdots,N_{\\ell-4}$.\n Since the volume bubble $\\tilde{q}_j\\in H_0^2(T)$ does not contribute\n to boundary integrals, we have from \\eqref{e:tildephi_orth} that\n\\begin{align*}\n\t\\int_{F\\cap T_+} \\partial_n\\xi^F_j\\varphi_i \\mathrm{d}S\n\t& = (\\partial_n\\lambda_F)\n\t     \\int_{F\\cap T_+}b^2_F\\tilde{\\phi}_j\n\t       \\varphi_i\n\t        \\mathrm{d}S=0.\n\\end{align*}\nThis finishes the proof.\n\\end{proof}\n\nAs a consequence of Lemma~\\ref{3Dsuper}\nwe note the following.\n\\begin{corollary}\n The spaces $V_h^{(\\ell)}$, with $\\ell\\geq 5$ even, satisfy the discrete\n Miranda--Talenti property from Definition~\\ref{def:dmt}.\n\\end{corollary}\n\\begin{proof}\n The proof is consequence of Lemma~\\ref{DMT1}\n and Lemma~\\ref{3Dsuper}.\n\\end{proof}\n\n\\subsection{Example for $\\ell=5$}\nWe give details for the lowest-order version of our three-dimensional\nfinite element.\nOn a simplex $T$, the shape function space reads\n$$\n\\mathcal{P}_{T,5} = P_5(T)+\\mathrm{span}\\{\\xi_i^F:F\\in\\mathcal F(T),i=1,2,3\\}.\n$$\nThe degrees of freedom are the point evaluation of the function,\nthe gradient, and the Hessian at the vertices;\nthe averages of the two normal derivatives along each edge;\nthe zeroth and first moments of the normal derivative over each face;\nand the zeroth and first moments of the function over the volume.\nThe degrees of freedom are visualized in Figure~\\ref{fig:3d_illustration}.\nThe local number of degrees of freedom therefore amounts to $68$.\n\n\n\\begin{figure}\n\\begin{tabular}{c|c|c|c}\n4 vertices &\n6 edges & 4 faces & volume\n\\\\\n\\hline\n\\begin{tikzpicture}[scale=2.2]\n\\fill (0,0) circle (1pt);\n\\draw (0,0) circle (2.5pt);\n\\draw (0,0) circle (4.4pt);\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=1.8]\n\\draw[very thick] (0,0) -- (1,.6);\n\\draw[->,thick] (.5,.3)--(.7,.2);\n\\draw[->,thick] (.5,.3)--(.66,.5);\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=2.2]\n\\draw[fill=gray] (.6,.3) -- (1,.5) -- (.5,1)--cycle;\n\\draw[->,thick](.675,.525)--(.845,.695);\n\\draw[->,thick] (.775,.575)--(.975,.775);\n\\draw[->,thick] (.65,.7)--(.8,.85);\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=2.2]\n\\draw[] (.6,.3) -- (1,.5) -- (.5,1) -- cycle;\n\\draw[] (.6,.3) -- (.3,.5)--(.5,1);\n\\draw[dashed] (.3,.5)--(1,.5);\n\\fill (.5,.64) circle (1pt);\n\\fill (.58,.57) circle (1pt);\n\\fill (.5,.55) circle (1pt);\n\\fill (.59,.66) circle (1pt);\n\\end{tikzpicture}\n\\end{tabular}\n\n\\caption{Visualization of the degrees of freedom \n         of the three-dimensional finite element for $\\ell=5$.\n         \\label{fig:3d_illustration}}\n\\end{figure}\n\n\n\\section{Application I: Non-divergence form equations}\\label{s:nondiv}\n\nAs the first application of the finite elements satisfying a strong discrete \nMiranda--Talenti property \\eqref{dMT} we present\nequations in non-divergence form \\eqref{nondiv}. \n\n\\subsection{Cordes condition and strong solutions}\nThis section will list some basic results on strong solutions of \nproblem \\eqref{nondiv}. \nWe assume that $A\\in L^{\\infty}(\\Omega;\\mathbb{R}^{d\\times d})$, $d=2,3$,\nis uniformly elliptic, i.e., there exist positive constants $\\lambda$, $\\Lambda$\nsuch that\n\\begin{equation}\n\t\\lambda \\leq \\eta^TA\\eta\\leq \\Lambda\\quad \\text{a.e. in }\\Omega\n\t\\quad \\text{for all }\\eta\\in \\mathbb{R}^d \\text{ with } \\eta^T\\eta = 1.\n\t\\label{uniell}\n\\end{equation}\nFor the well-posedness of \\eqref{nondiv} we impose an additional\ncondition known as Cordes condition \\cite{MT,Suli2013}.\n\\begin{myDef}[Cordes condition]\n\tThere exist an $\\varepsilon\\in (0,1]$ such that \n\t\\begin{equation}\n\t\t\\frac{|A|^2}{(\\mathrm{tr}A)^2}\\leq \\frac{1}{d-1+\\varepsilon}.\n\t\t\\label{cordes}\n\t\\end{equation}\n\\end{myDef}\nIn the case $d=2$ the Cordes condition is a consequence of the \nuniform ellipticity \\eqref{uniell} while it is an essential condition\nif $d=3$.\nFollowing \\cite{Suli2013} we introduce the scaling function\n$\\gamma := |A|^{-2}\\operatorname{tr}A$. Then the variational formulation of \\eqref{nondiv} is given as  \n\\begin{equation}\n\tA(u,v) : = \\int_{\\Omega}\\gamma (A:D^2u)\\triangle v\\mathrm{d}x = \\int_{\\Omega}\\gamma f\\triangle v\\mathrm{d}x,\\quad \\text{for all } v\\in V. \n\t\\label{nondivvar}\n\\end{equation} \n\nThe Cordes condition guarantees adequate control on\nthe distance between $A:D^2 v$ and $\\triangle v$.\nThe combination with the Miranda--Talenti inequality \\eqref{MT} \nresults in the following well-posedness result for \\eqref{nondiv}.\n\\begin{mylem}[\\cite{Suli2013}, Theorem 3]\n\tLet $A$ satisfy \\eqref{uniell} and \\eqref{cordes}.\n     Then, for any open set $U\\subset \\Omega$ and $v\\in H^2(U)$, we have\n\t\\begin{equation}\n\t\t|\\gamma A:D^2 v- \\triangle v|\\leq \\sqrt{1-\\varepsilon}|D^2 v|,\\quad a.e. \\text{ in } U,\n\t\t\\label{cordes2}\n\t\\end{equation}\n\twhere the $\\varepsilon$ is defined in \\eqref{cordes}. \n\tMoreover, for any $f\\in L^2(\\Omega)$ problem \\eqref{nondivvar}\n\thas a unique solution $u\\in H^2(\\Omega)\\cap H_0^1(\\Omega)$ and satisfies\n\t\\begin{equation}\n\t\t\\|D^2 u\\|_0 \\leq\n\t\t\\|\\triangle u\\|_0 \\leq\n\t\t\\frac{\\|\\gamma\\|_{\\infty}}{1-\\sqrt{1-\\varepsilon}}\\|f\\|_0.\n\t\\end{equation}\n\\end{mylem}\n\n\n\n\\subsection{Finite element discretization}\nThe discrete problem corresponding to \\eqref{nondivvar}\nseeks \n$u_h\\in V_h\\subseteq H^2(\\mathcal{T})\\cap H_0^1(\\Omega)$, such that\n\\begin{equation}\n\tA_h(u_h,v_h):=\\int_\\Omega (\\gamma A:D^2_h u_h)\\triangle_h v_h\\mathrm{d}x \n\t= \n\t\\int_\\Omega \\gamma f \\triangle_h v_h \\mathrm{d}x\n\t \\quad \\text{for all } v_h\\in V_h.\n\t\\label{dnondiv}\t\n\\end{equation} \n\nSince the exact solution $u$ satisfies $A:D^2 u=f$ pointwise\nalmost everywhere in $\\Omega$, the following analogue of what\nis called Galerkin orthogonality in conforming methods is valid\n\\begin{equation}\\label{e:galerkinorthogonality}\nA_h(u-u_h,v_h)=0 \\quad\\text{for all }v_h\\in V_h. \n\\end{equation}\n\nThe basic error estimate is as follows.\n\n\\begin{myTheo}\n\tLet $\\Omega$ be a bounded convex polytopal domain with simplicial triangulation \n\t$\\mathcal{T}$ and let $A\\in L^{\\infty}(\\Omega;\\mathbb R^{d\\times d})$\n\tsatisfy uniform ellipticity \\eqref{uniell} and the Cordes condition\n\t\\eqref{cordes}.\n\tLet $u$ be the solution of \\eqref{nondivvar}.\n\tIf $V_h\\subseteq H^2(\\mathcal{T}_h)\\cap H_0^1(\\Omega) $ \n\tsatisfies the strong discrete Miranda--Talenti property, \n\tthen \\eqref{dnondiv} has a unique solution $u_h\\in V_h$,\n\twhich satisfies\n    \\begin{equation*}\n\t\t\\|D^2_h(u-u_h)\\|_0\n    \\leq \n    \\left(1+\\frac{\\Lambda}{1-\\sqrt{1-\\varepsilon}}\\right)\n    \\inf_{w_h\\in V_h}\\|D_h^2(u-w_h)\\|_0 . \n    \\end{equation*}\n\t\\label{duni}\n\\end{myTheo}\n\n\\begin{proof}\nWe abbreviate $\\delta:=1-\\sqrt{1-\\varepsilon}$.\n\tSince $V_h$ satisfies \\eqref{dMT}, \n\t$\\|\\triangle_h\\cdot\\|_0$ defines a norm on $V_h$.\n\tCombining \\eqref{cordes2} and \\eqref{dMT}, we have\n\tfor any $v_h\\in V_h$ that\n\t\\begin{align}\\label{e:d_coerc}\n\t\\begin{aligned}\n\t\tA_h(v_h,v_h) &= \\|\\triangle_h v_h\\|^2_0 \n\t\t- \\sum_{T \\in \\mathcal{T}_h}\\int_T(\\gamma A:D^2v_h-\\triangle v_h)\\triangle v_h\\mathrm{d}x\n\t\t\\geq \\delta\\|\\triangle_h v_h\\|^2_{0}.\n    \\end{aligned}\n\t\\end{align}\n\tSince, by the discrete Miranda--Talenti inequality,\n\t$A_h(\\cdot,\\cdot)$ is also bounded on $V_h$,\n\tthe Lax--Milgram Theorem implies that \\eqref{dnondiv} has unique \n\tsolution $u_h\\in V_h$. \n    Then for any $w_h\\in V_h$, the triangle inequality implies\n\\begin{equation}\\label{e:errest_triang}\n\\|D^2_h(u-u_h)\\|_0\\leq \\|D_h^2(u-w_h)\\|_0 + \\|D^2_h(u_h-w_h)\\|_0.\n\\end{equation}\nFor the analysis of the the second term on the right-hand side,\nwe abbreviate $e_h:=u_h-w_h$.\nThe discrete coercivity\n\\eqref{e:d_coerc} implies\n$$\n\\|\\triangle_h e_h\\|_0^2\n\\leq \\delta^{-1}A_h(e_h,e_h)\n= \\delta^{-1} A_h(u_h-w_h,e_h).\n$$\nThe orthogonality \\eqref{e:galerkinorthogonality} and the Cauchy\ninequality show that\n$$\nA_h(u_h-w_h,e_h) \n= A_h(u-w_h,e_h)\n\\leq \\Lambda\\|D_h^2(u-w_h)\\|_0 \\|\\triangle_he_h\\|_0.\n$$\nWe use the discrete Miranda--Talenti estimate \\eqref{dMT} and\nthe combination of the two foregoing displayed estimates to\nconclude\n$$\n\\|D^2_h e_h\\|_0\n\\leq \\|\\triangle_h e_h\\|_0\n\\leq \\delta^{-1}\\Lambda\\|D_h^2(u-w_h)\\|_0\n.\n$$\nSince $w_h\\in V_h$ was arbitrary, the combination with\n\\eqref{e:errest_triang} proves the assertion.\n\\end{proof}\n\nWe apply Theorem~\\ref{duni} to our families of finite elements\nfrom the previous sections and obtain the following \nasymptotic error estimates.\n\\begin{myTheo}\n\tLet $\\Omega\\subseteq \\mathbb R^d$ for $d=2,3$ be a bounded convex polygonal domain \n\ttriangulated by $\\mathcal{T}$ and \n\tlet $A\\in L^{\\infty}(\\Omega)$  satisfy conditions\n\t\\eqref{uniell} and \\eqref{cordes}.\n\tLet $u_h\\in V^{(\\ell)}_h$ denote the discrete solution to\n\t\\eqref{dnondiv}, where\n\t$$\n\t\\begin{cases}\n\t \\ell=3 \\text{ or } \\ell \\text{ is even with }\\ell\\geq 4 &\\text{if } d=2,\\\\\n\t \\ell\\geq 5 &\\text{if }d=3 .\n\t \\end{cases}\n\t $$\n\tIf the solution $u$ to \\eqref{nondivvar}\n\tsatisfies $u\\in H^s(\\Omega)\\cap H_0^1(\\Omega)$ for some $s>2$, and $u_h$ is the solution of \n\t \\eqref{dnondiv}, then\n\t\\begin{equation*}\n\t\t\\|D_h^2 (u-u_h)\\|_{0}\\leq Ch^{s-2}|u|_s\n\t\\end{equation*}\n\twhere $|\\cdot|_s$ denotes the usual $H^s(\\Omega)$ seminorm.\n\\end{myTheo}\n\n\\begin{proof}\nFor $d=2$ and for $d=3$ with $s\\geq 7\/2$, the result follows from\nstandard interpolation error estimates \\cite{finiteBrenner}.\nFor the remaining case $d=3$ and $2<s<7\/2$, a suitable quasi-interpolation\noperator can be designed by averaging as outlined in \\cite{finitewang}.\nWe skip the details, which are based on standard techniques.\n\\end{proof}\n\n\\begin{remark}\nLet\t$V_h$ be a finite element space and satisfies \\eqref{dMT}\nand let $u_h\\in V_h$ be finite element solution of \\eqref{nondivvar}.\nThen the a~posteriori error estimator as\n\\begin{equation}\n\t\\eta = \\sqrt{  \\sum_{T\\in\\mathcal T} \\|f-A:D^2 u_h\\|_{L^2(T)}^2 }.\n\\end{equation}  \nis reliable and locally efficient with respect to the \nerror $\\|D_h^2 (u-u_h)\\|_0$. \n\\end{remark}\n\n\n\n\n\n\\section{Application II: Biharmonic equation}\\label{s:bih}\nThe second application of our new finite element families is the biharmonic equation.\nLet $\\Omega\\subset \\mathbb{R}^d$, $d=2,3$ be a bounded,\nopen, polytopal Lipschitz domain.\nThe weak form of the biharmonic equation $\\triangle^2 u =f$ for some\nright-hand side $f\\in L^2(\\Omega)$ and clamped boundary conditions\n$u=\\partial_n u=0$ over $\\partial\\Omega$\nseeks $u\\in H^2_0(\\Omega)$ such that\n\\begin{equation}\n\\int_\\Omega D^2u:D^2v\\mathrm{d}x=\\int_\\Omega fv\\mathrm{d}x\n\\quad\\text{for all }v\\in H^2_0(\\Omega).\n\\label{bihar}\n\\end{equation}\nGiven a space $V_h$ of piecewise polynomial functions with respect to\na triangulation $\\mathcal T$, the discrete problem seeks\n$u_h\\in V_h$ such that\n\\begin{equation}\n\\int_\\Omega D_h^2u_h:D_h^2v_h\\mathrm{d}x=\\int_\\Omega fv_h\\mathrm{d}x\n\\quad\\text{for all }v_h\\in V_h.\n\\label{dbihar}\n\\end{equation}\nGiven a face $F\\in\\mathcal F$,\nlet the face patch $\\omega_F$ be the union of the simplices sharing the face $F$.\nGiven a measurable subdomain $G\\subseteq\\overline\\Omega$,\nlet $\\mathrm{P}^{m}_G$ denote the $L^2$-orthogonal projection operator to the \npolynomials of degree not larger than $m$ over $G$.\n\n\\begin{remark}\\label{rem:clampedBC}\n In order to apply subspaces of the spaces\n $V^{(\\ell)}_h$ from Sections~\\ref{s:2d}--\\ref{s:3d},\n to the problem with clamped boundary conditions, we additionally need to\n enforce a discrete analogue of the condition $\\partial_n u=0$\n on $\\partial\\Omega$.\n To this end, all degrees of freedom related to the normal derivative\n on boundary faces and all edge degrees of freedom for\n boundary edges are set to zero.\n For $d=2$ all degrees of freedom related to the boundary vertices\n are set to zero. For $d=3$ the vertex degrees of freedom\n $\\partial_{nn}^2 u(z)$ for any boundary vertex $z\\in\\mathcal N$\n is left free unless $z$ is a boundary point where $\\partial\\Omega$\n is not differentiable.\n\\end{remark}\n\nBy applying a result of \\cite{Hu&Zhang2019}, we directly obtain an error \nestimate for our finite element families adapted to the biharmonic\nproblem.\n\n\\begin{myTheo}\nLet $u\\in H^2_0(\\Omega)$ be the solution to \\eqref{bihar}.\nLet the integer $\\ell$ satisfy\n\t$$\n\t\\begin{cases}\n\t \\ell=3 \\text{ or } \\ell \\text{ is even with }\\ell\\geq 4 &\\text{if } d=2,\\\\\n\t \\ell\\geq 5 &\\text{if }d=3 \n\t \\end{cases}\n\t $$\nand let $V_h$ denote the subspace of the space\n$V^{(\\ell)}_h$ (from Sections~\\ref{s:2d}--\\ref{s:3d})\ndescribed in Remark~\\ref{rem:clampedBC}.\nThen problem \\eqref{dbihar} has a unique solution $u_h\\in V_h$,\nwhich satisfies\n\\begin{align*}\n\\|D_h^2(u-u_h)\\|_0\\leq C&\\bigg(\\inf_{v_h\\in V_h}\\|D_h^2(u-v_h)\\|_0 \n + \\Big(\\sum_{F\\in\\mathcal F}\\|(1-\\mathrm{P}^{\\ell-2}_{\\omega_F})D^2u\\|^2_{0,\\omega_F}\\Big)^{1\/2}\\\\\n \\quad\n& +  \\Big(\\sum_{T \\in \\mathcal{T}}h_T^4\\|(1-\\mathrm{P}^{\\ell-3}_{T})f\\|^2_{0,T}\\Big)^{1\/2}   \n\\bigg).\n\\end{align*} \n\\end{myTheo}\n\\begin{proof}\nThe definition of the spaces $V_h$ ensures that\n$\\|D_h^2\\cdot\\|_0$ defines a norm over $V_h$\nso that existence and uniqueness of discrete solutions follow from\nstandard arguments \\cite{finiteBrenner}.\nAn adaptation of Lemma~\\ref{2Dsuper} and Lemma~\\ref{3Dsuper} \nshows for all faces $F$ that\n$$\n\\int_F[\\nabla v_h]q_{\\ell-2}\\mathrm{d}S = 0\\quad \n\\text{for all } q_{\\ell-2}\\in P_{\\ell-2}(F).\n$$ \nAn application of Theorem~2.1 of \\cite{Hu&Zhang2019} thus concludes the proof. \n\\end{proof}\n\n\n\n\n\\section{Numerical Experiments}\\label{s:num}\nIn this section, we present numerical results for the \nelliptic problem in nondivergence form and the biharmonic equation\n(with clamped boundary conditions) in two and three space dimensions\non uniformly refined meshes.\nWe always display relative errors in the $L^2$ norm and the \ndiscrete $H^2$ seminorm,\nabbreviated by $e_{L^2}^{\\mathrm{rel}}$\nand $e_{H^2}^{\\mathrm{rel}}$, respectively.\n\n\n\\subsection{Nondivergence-form operator in two dimensions}\\label{ss:nondiv2d_num}\nWe consider the elliptic operator in nondivergence form given through\nthe coefficient\n$$\nA = \\left(\\begin{matrix}\n\t2&xy\/|xy|\\\\\n\txy\/|xy|&2\n\\end{matrix}\\right)\n$$\non the square $\\Omega=(-1,1)^2$.\nThen $\\gamma =2\/5,\\varepsilon = 3\/5.$\nThe exact solution is given by\n$u(x,y) =xy(1-e^{(1-|x|)})(1-e^{1-|y|})$.\nWe compare three numerical methods:\nthe quadratic Specht triangle and our finite element space\n$V_h^{(\\ell)}$ from Section~\\ref{s:2d} with $\\ell=3$ and $\\ell=4$.\nThe results are displayed in Tables~\\ref{tab:2dnondiv_a}--\\ref{tab:2dnondiv_b}.\nThe quadratic Specht element and our element with $\\ell=3$\nconverge with order close to 2 in the discrete $H^2$ seminorm\nand with an order better than 3 in the $L^2$ norm.\nFor $\\ell=4$, we observe third-order convergence in the discrete $H^2$ seminorm\nand a convergence order better than 4 in $L^2$.\n\n\\begin{table}\n\t\\begin{center} \n\t\t\\begin{tabular}{c|cc|cc|cc|cc}\n\t\t&\t\\multicolumn{4}{c|}{quadratic Specht}& \\multicolumn{4}{|c}{$V_h^{(3)}$}\\\\\n \\hline\n\t$h$             &  $e_{L^2}^{\\mathrm{rel}}$&  rate & $e_{H^2}^{\\mathrm{rel}}$\n\t               & rate & $e_{L^2}^{\\mathrm{rel}}$ & rate &$e_{H^2}^{\\mathrm{rel}}$ &  rate \\\\\n\t\t\t\\hline\n\t1\/16\t       & 3.85e-03&    &2.73e-02&   & 3.72e-03 &  & 2.93e-02  &  \\\\\n\t1\/32\t      &4.13e-04&  3.22   &7.60e-03& 1.85   & 4.01e-04 & 3.21 & 8.13e-03  &  1.85\\\\\n\t1\/64\t       &4.16e-05& 3.31   &2.00e-03&1.92   & 4.03e-05 & 3.32 & 2.16e-03  & 1.94 \\\\\n\t1\/128\t       & 4.04e-06& 3.36   &5.17e-04& 1.92  & 2.27e-06 & 4.15 & 5.54e-04  &1.92       \n\t\t\\end{tabular}\n\t\\end{center} \n\t\\caption{Convergence history for the nondivergence-form operator in two dimensions:\n    quadratic Specht element and $V_h^{(\\ell)}$ with $\\ell=3$.\n    \\label{tab:2dnondiv_a}}\n\\end{table}\n\n\\begin{table}\n\t\\begin{center} \n\t\t\t\\begin{tabular}{c|cc|cc|cc|cc}\n     $h$                  &1\/8    & &1\/16    &     &1\/32   &      &  1\/64 &\\\\\n     \\hline\n    $e_{L^2}^{\\mathrm{rel}}$, rate          &  5.17e-04& &2.73e-05&4.24 &1.33e-06&4.36 & 4.69e-08& 4.82  \\\\\n    $e_{H^2}^{\\mathrm{rel}}$, rate &  6.12e-02& &8.20e-04&2.88 &1.06e-04&2.95 & 1.35e-05&2.98\n\t\t\\end{tabular}\n\\end{center} \n\\caption{Convergence history for the nondivergence-form operator in two dimensions:\n    $V_h^{(\\ell)}$ with $\\ell=4$.\n    \\label{tab:2dnondiv_b}}\n\\end{table}\n\n\n\\newpage\n\\subsection{Biharmonic operator in two dimensions}\nWe consider the biharmonic equation with clamped boundary conditions on\nthe square $\\Omega=(0,1)^2$.\nThe exact solution is chosen as\n$u(x,y)=(\\mathrm{sin}(\\pi x)\\mathrm{sin}(\\pi y))^2$.\nWe compare the three methods based \nthe quadratic Specht triangle and our finite element space\n$V_h^{(\\ell)}$ from Section~\\ref{s:2d} with $\\ell=3$ and $\\ell=4$.\nThe results are displayed in Tables~\\ref{tab:2dbih_a}--\\ref{tab:2dbih_b}.\nThe quadratic Specht element and our element with $\\ell=3$\nconverge with order close to 2 in the discrete $H^2$ seminorm\nand with order close to 4 in the $L^2$ norm.\nFor $\\ell=4$, we observe third-order convergence in the discrete $H^2$ seminorm\nand a convergence order better than 5 in $L^2$.\n\n\n\n\n\\begin{table}\n\t\\begin{center} \n\t\t\\begin{tabular}{c|cc|cc|cc|cc}\n\t\t\t&\t\\multicolumn{4}{|c|}{quadratic Specht}& \\multicolumn{4}{|c}{$V_h^{(3)}$}\\\\\n\t\t\t\\hline\n\t\t\t$h$             &  $e_{L^2}^{\\mathrm{rel}}$&  rate & $e_{H^2}^{\\mathrm{rel}}$ & rate & $e_{L^2}^{\\mathrm{rel}}$ & rate &$e_{H^2}^{\\mathrm{rel}}$ &  rate \\\\\n\t\t\t\\hline\n\t1\/16\t       & 6.53e-03&       &7.78e-02&       & 6.40e-03 &      & 5.21e-02  &  \\\\\n\t1\/32\t       &6.61e-04&  3.32  &2.52e-02& 1.67  & 6.27e-04 & 3.34 & 1.63e-02  &  1.68\\\\\t\n\t1\/64\t       &5.04e-05& 3.71   &6.71e-03& 1.87  & 4.72e-05 & 3.73 & 4.41e-03  & 1.88 \\\\\n\t1\/128\t       &3.35e-06& 3.91  & 1.72e-03& 1.96  & 3.14e-06 & 3.91 & 1.13e-04  &1.96       \n\t\t\\end{tabular}\n\t\\end{center} \n\t\\caption{Convergence history for the biharmonic equation in two dimensions:\n    quadratic Specht element and $V_h^{(\\ell)}$ with $\\ell=3$.\n    \\label{tab:2dbih_a}}\n\\end{table}\n\n\\begin{table}\n\t\\begin{center} \n\t\t\\begin{tabular}{c|cc|cc|cc|cc}\n\t\t\t$h$                  &1\/8    & &1\/16    &     &1\/32   &      &  1\/64 &\\\\\n\t\t\t\\hline\n\t\t\t$e_{L^2}^{\\mathrm{rel}}$           &  4.26e-03& &1.09e-05&5.27 &2.35e-06&5.54 & 5.79e-08& 5.34  \\\\\n\t\t\t$e_{H^2}^{\\mathrm{rel}}$ &  5.99e-02& &9.06e-03&2.73 &1.20e-03&2.91 & 1.53e-04&2.97\n\t\t\\end{tabular}\n\t\\end{center} \n     \\caption{Convergence history for the biharmonic equation in two dimensions:\n    $V_h^{(\\ell)}$ with $\\ell=4$.\n    \\label{tab:2dbih_b}}\n\\end{table}\n\n\n\\subsection{Nondivergence-form operator in three dimensions}\nWe consider the elliptic operator in nondivergence form given through\nthe coefficient\n$$\nA = \\left(\\begin{matrix}\n\t3&xyz\/|xyz|&xyz\/|xyz|\\\\\n\txyz\/|xyz|&3&xyz\/|xyz|\\\\\n\txyz\/|xyz|&xyz\/|xyz|&3\n\\end{matrix}\\right).\n$$\nThe domain is $\\Omega=(-1,1)^3$\nand we compute \n$\\gamma = 11\/27,~\\varepsilon = 5\/11.$ \nWe choose $u(x,y,z) = \\mathrm{sin}(\\pi x) \\mathrm{sin}(\\pi y) \\mathrm{sin}(\\pi z)$\nas the exact solution.\nWe approximate the problem with our finite element space\n$V_h^{(\\ell)}$ from Section~\\ref{s:3d} with $\\ell=5$.\nThe results are displayed in Table~\\ref{tab:3dnondiv}.\nWe observe convergence of order 4 in the discrete $H^2$ seminorm\nand convergence of order 6 in the $L^2$ norm.\n\n\n\\begin{table}\n\t\\begin{center} \n\t\t\\begin{tabular}{c|cc|cc|cc|cc}\n                 \t$h$  &1\/2    & &1\/4    & rate    &1\/8   &     rate &  1\/16 &rate\\\\\n\t\\hline $e_{L^2}^{\\mathrm{rel}}$   &  4.96e-01& &1.51e-02&5.04 &2.81e-04&5.75 & 2.54e-06&6.79  \\\\\n\t\t\t$e_{H^2}^{\\mathrm{rel}}$  &  5.41e-01& &8.68e-02&2.64 &8.97e-03&3.28 & 5.20e-04&4.11\n\t\t\\end{tabular}\n\t\\end{center} \n\t\\caption{Convergence history for the nondivergence-form operator in three dimensions\n\t         with the lowest-order method $(\\ell=5)$.\n\t      \\label{tab:3dnondiv}}\n\\end{table}\n\n\n\\subsection{Biharmonic operator in three dimensions}\nWe consider the biharmonic equation with clamped boundary conditions on\nthe cube $\\Omega=(0,1)^3$.\nThe exact solution is given by\n$u(x,y,z) = (\\mathrm{sin}(\\pi x)\\mathrm{sin}(\\pi y)\\mathrm{sin}(\\pi z) )^2.$\nWe approximate the problem with our finite element space\n$V_h^{(\\ell)}$ from Section~\\ref{s:3d} with $\\ell=5$.\nThe results are displayed in Table~\\ref{tab:3dbih}.\nWe observe convergence of order 4 in the discrete $H^2$ seminorm\nand convergence better than of order 6 in the $L^2$ norm.\n\n\\begin{table}\n\t\\begin{center} \n\t\t\\begin{tabular}{c|cc|cc|cc|cc}\n\t\t\t$h$    &   1\/2& &1\/4& &1\/8& &1\/16& \\\\\n\t\t\t\\hline\n\t\t\t$e_{L^2}^{\\mathrm{rel}}$ &6.14e-02&  &4.79e-03 & 3.68&5.69e-05&6.40 &4.59e-07 &6.95\\\\\n\t   \t$e_{H^2}^{\\mathrm{rel}}$  &1.71e-01 &  &4.07e-02 &2.07 &3.35e-03&3.60 &1.78e-04 &4.23\\\\\n\t\t\\end{tabular}\n\t\\end{center} \n\t\\caption{Convergence history for the biharmonic equation in three dimensions\n\t         with the lowest-order method $(\\ell=5)$.\n\t\\label{tab:3dbih}}\n\\end{table}\n\n\\subsection{Conclusions from the compuations}\nIn all experiments, we observe the optimal convergence rates in the \ndiscrete $H^2$ seminorm. \nMorever, we observe higher convergence orders for the error in the \n$L^2$ norm.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAs the penetration of renewable energy resources and electric vehicles on modern power grids continues to increase, so does the amount of uncertainty in the network's power balance.  Energy storage presents a natural solution to mitigating these uncertainties by smoothing short-term fluctuations in load and generation.  However, modeling energy storage devices presents three fundamental challenges: (1) a variety storage technologies exists, such as pumped hydro, compressed air, flywheels, molten salt, and batteries, each with unique characteristics; (2) many storage devices can provide valuable ancillary services, \\textit{e.g.}, frequency control and reactive power support, which are important to capture when making operational or investment decisions; (3) storage devices are available in an assortment of sizes, from massive bulk electric system facilities to small household batteries.  The combination of these features presents a notable challenge to modeling storage devices in the context of network operations and investment.\n\nInspired by the success of the $\\Pi$-equivalent circuit for modeling branches in transmissions systems, this work proposes a generic and flexible grid-connected storage model that strives to strike a balance between model simplicity and flexibility.\n\nIn the most general case, the proposed storage model captures a three-phase steady-state AC grid connection with a variety of losses, suitable for detailed operational models in power distribution systems.  Consistent single-phase equivalent and active-power-only approximations of this model, which are suitable for applications requiring less detail, such as transmission network investment planning have been developed with care.  These mathematical variants provide a natural trade-off between model accuracy and scalability.  Furthermore, convex relaxations of the storage model are proposed to provide lower bounds on AC Optimal Power Flow (OPF) studies with storage devices.\n\nThis paper begins with a review and analysis of storage simulation and optimization models for power system operation in Section \\S \\ref{sec_lit_review}.\nA research-grade, easy-to-use, simplified storage model is then developed in Section \\S \\ref{sec_model}.\nRelaxations and approximations of the nonlinear model are proposed in Section \\S \\ref{sec_model_relax}.\nSection \\S \\ref{sec_study} illustrates the use of the proposed models in the context of reducing generation fuel costs in power grid operations, and Section \\S \\ref{sec_conclusions} concludes the paper.\n\n\\section{Literature Review}\\label{sec_lit_review}\nBoth \\cite{Das2018} and \\cite{Diaz-Gonzalez2012} provide detailed discussions of the storage technology options and have focus on how to best support renewable integration in power systems.\nIn contrast, this section provides a high level overview of storage models with a focus on optimal operation.\n\n\\paragraph*{Energy Storage}\nStorage technologies can be broadly categorized by the nature of their energy conversion processes:  batteries (\\textit{e.g.}, in electric vehicles or stationary storage) rely on a reversible electrochemical process,  pumped hydro on a gravitational field,  flywheels on a kinetic energy system, and pressure is leveraged by compressed air storage.  Interestingly, grid-level energy storage need-not be bi-directional.  One such case is thermal storage, where electric power is converted to heat preemptively to be utilized later (\\textit{e.g.}, in residential hot water heaters).  Another common case is demand response protocols, which can modulate power consumption temporarily to time-shift power demands.\nIn this work, these uni-directionally interfaced systems are treated as a special case of a bi-directional model.\n\nConsuming power from the grid generally means the energy buffer increases, which is referred to as storing energy.  Injecting power into the grid results in reductions in the energy buffer.\nIn the context of battery storage, we use the terminology of charging (energy content increases) and discharging  (energy content decreases)  the batteries; for pumped hydro it is often called pumping and turbining.\nIn both cases, the underlying principle for storing energy is through a buffer $E$,\n\\begin{IEEEeqnarray}{C}\nP(t) = \\frac{dE(t)}{dt}\n\\end{IEEEeqnarray}\nwhere a nonzero power $P(t)$ will lead to a change in energy $E(t)$ over time $t$.\n\n\nRegardless of storage approach, the key feature that distinguish storage devices from other grid connected devices is that their energy content can be controlled independently of the electrical interface, and can therefore be optimized to meet objectives like cost reduction costs or quality-of-service increases.  However, it is important to note that energy losses occur during charging, discharging, and while idle (\\textit{e.g.}, energy dissipation).  Such losses are important contributors to the overall costs of operating and optimizing such systems.\n\n\n\\paragraph*{Storage System Simulation}\nFor simulation it is common to develop highly fidelity storage device models.  For instance, \\cite{Tremblay2007} developed a detailed electrical battery storage model for validation of electric vehicle simulation, which incorporates the dependence of battery voltage on the state of charge, \\textit{i.e.},\n\\begin{IEEEeqnarray}{C}\nU(t) = f(Q(t))\n\\end{IEEEeqnarray}\nand\n models a charge buffer $Q$ instead of an energy buffer, \\textit{i.e.},\n\\begin{IEEEeqnarray}{C}\nI(t) = \\frac{dQ(t)}{dt}\n\\end{IEEEeqnarray}\nIt is noted that the charge-dependency of the terminal voltage is very limited with technologies such as Li-ion over a broad range of state of charge, therefore these effects are commonly ignored.\nFurthermore, when the battery voltage is assumed to be constant, \\textit{e.g.}, $U^{\\text{nom}}$, in the normal operation range, one can represent the charge buffer $Q$ as an energy buffer $E$,\n\\begin{IEEEeqnarray}{C}\nP(t) \\approx U^{\\text{nom}} \\cdot I(t) = U^{\\text{nom}} \\cdot \\frac{dQ(t)}{dt} =  \\frac{dE(t)}{dt},\n\\end{IEEEeqnarray}\nwhere a nonzero power $P(t)$ will lead to a change in energy $E(t)$ over time $t$.\nSuch an energy buffer expression can easily represent other storage processes.\nFor simulation, these equations are solved using numerical integration methods.\n\nHigh fidelity models for simulation of storage systems are available in such tools as OpenDSS \\cite{Dugan2011,Dugan2013a} and GridLAB-D \\cite{gridlabd}.  However, these simulators rely on local control heuristics or user-provided schedules for storage device operation decisions.  A natural solution would be embed these models into mathematical {\\em optimization} tool chains, where it is possible to define an objective, and allow an algorithm to determine the ideal operation of the storage system.  Unfortunately, these detailed storage models are often too complex for state-of-the-art optimization tools and hence there is a need to balance the storage model's accuracy for the abilities of modern optimization tools.\n\nTo that end, a number of papers have proposed optimization-compatible storage models, including \\cite{Tant12,Eyisi2019,Alnaser2015,hari2018hierarchical} and similar models have been incorporated into software tools such as {\\sc Matpower}'s MOST \\cite{Zimmerman2011,MOST19} and pandapower \\cite{Meinecke2018}.\nWe note that Heussen et al.  \\cite{Heussen2011} developed a approach for representing different storage technologies through an optimization-compatible parameterized mathematical abstraction called the `Power Nodes Framework'.\nThese models have varying levels of sophistication, however we find that their scope is relatively narrow in terms of technologies \\emph{and} support for different power flow representations.  Specifically, they exclusively focus on single-phase models with an active-power emphasis; it is not immediately clear how these models could be generalized to consider all of the modeling contexts targeted by this work.\n\n\\paragraph*{Unit Commitment and Multi-Period OPF}\nPumped hydro has historically been the most popular electricity storage technology (bidirectionally interfaced). Therefore, substantial work has occurred in the context of economic dispatch and unit commitment, including pumped hydro units \\cite{Almassalkhi2016,Bruninx2016a}. Typically, such approaches have very limited representations of the grid's electrical physics; for instance, the `DC', network flow, and copper plate approximations are commonly used.\n\nNevertheless, driven by the interest in storage in distribution grids, extending optimal power flow with storage models has been a topic of interest \\cite{Jabr2015}. Commonly, optimal power flow is considered as a snap-shot problem, \\textit{i.e.}, all time steps are solved independently. Extending OPF to include multiple time steps simultaneously is commonly referred to as multi-period OPF \\cite{Electric2013,Gemine2014b}. In multi-period OPF frameworks, one can incorporate battery storage models, including representations of the energy storage dynamics and\/or other intertemporal constraints, such as ramp rates. Kouronis et al. \\cite{Kourounis2018} demonstrate that large-scale multi-period AC OPF with storage (no complementarity constraint) can be done using customized interior-point NLP codes. Stai et al. propose a successive convex approximation scheme to improve scalability \\cite{Stai2018}.\nOne can further extend such frameworks with higher fidelity physical models to represent phase unbalance \\cite{Connell2014, Stai2018}.\n\n\\paragraph*{Balanced and Unbalanced Optimal Power Flow}\nOptimal power flow has historically focused on modelling transmission networks, where assuming that the grid impedance and the loading allow for balanced voltage and current phasors is typically considered adequate.\nIn distribution systems, and especially in low-voltage grids, these assumptions are difficult to justify.\nIn this context, unbalanced OPF has been proposed as a generalization of OPF where phase unbalance effects are explicitly modeled.\n\nStorage devices are particularly interesting in unbalanced OPF, as one can choose separate power set-points for each phase of the converter, assuming the technology allows it. In the most general case, both active and reactive power set-points can be chosen independently for each phase, subject to the active power being balanced with the energy storage subsystem while charging or discharging. Nevertheless, even without a storage subsystem, converters can circulate active power between phases. Such functionality can be delivered by inverter-based systems \\cite{Stuyts2018}. The differences in control freedom have been explored \\cite{Tant2015a}, but software implementations for modeling such features remained unavailable.\n\n\n\\paragraph*{Convex Relaxation}\nRecently, for both balanced and unbalanced OPF, a significant amount of work has been published on reformulating and relaxing the mathematical models to obtain better numerical performance and stronger optimality guarantees \\cite{Molzahn2017a}.\nOne of the key works describing relaxations of unbalanced OPF is \\cite{Gan2014}.\nTo the best of our knowledge, few works have considered the use of storage optimization models in combination with convex relaxation of the power flow physics.  Papers that do derive such storage models include \\cite{Marley2016,Geth2016}.\n\nCareful consideration must be taken when combining published optimization formulations for storage systems with OPF formulations.\nIntersecting two preexisting feasible sets with known complexity may require the definition of new constraints to serve as bridges between the variable spaces, potentially increasing the overall complexity.\nFor instance, a current-voltage variable space with a linear formulation of the branch physics, when combined with a linear power-voltage model for a generator, requires the addition of the definition of complex power, $S = U(I)^{*}$, as a bridge; this equation is nonlinear, which makes the integrated problem nonlinear (and non-convex).\nTo mitigate this, this paper defines a number of storage models with different complexities, and in a number of variable spaces that are compatible with recently developed formulations for balanced and unbalanced OPF.\n\n\n\\paragraph*{Discussion}\nIn summary, we observe that there is a need to develop a research-grade mathematical model for representing a broad range of storage technologies, which is suitable for different levels of grid modeling fidelity, \\textit{e.g.}, AC NLP vs linearized `DC' and single-phase vs multi-phase, and is easy to implement in various grid optimization applications.  Developing such a model is the objective of this work, and a detailed derivation is provided in the subsequent sections.\nNext to the features that it offers, we also want it to be convenient template to implement extensions for, e.g., for storage sizing, of technology-specific detailed models such as for battery storage systems.\n\n\\section{The Proposed Storage Model} \\label{sec_model}\nIn this section we develop a foundational mathematical model for a simple energy storage system composed of two core parts: (1) a flexible energy storage subcomponent, which supports adaptive time discretization; and (2) a grid interface that can represent both inverters and synchronous machines.  The derivation begins with a general grid interface, which is a nonlinear multi-conductor model.  It then integrates that interface into the energy buffer.\n\n\\subsection{Time Discretization}\nUsing differential equations directly in optimization models is often prohibitive.\nConsequently the first step in developing this model is to discretize time.\nNatural time is defined by $t$. Time is sampled at instances $t \\in \\mathcal{T}$, and there exists a bijection with respect to indices $k  \\in \\mathcal{K}$ (integers). The time step length $\\textcolor{\\paramcolor}{\\symtime_{\\indextime}}$ is the amount of time between time $t_k$ and $t_{k+1}$. Note that we do not assume this to be uniform.\nTherefore, continuous time symbols map to discrete time as follows:\n\\begin{IEEEeqnarray}{C}\nP(t) \\rightarrow P_k,\nE(t) \\rightarrow E_k\n\\end{IEEEeqnarray}\nNext, we integrate the ODE w.r.t. the sample times, to obtain approximate algebraic expressions.\n\\begin{IEEEeqnarray}{C}\n\\int_{t_k}^{t_k+\\textcolor{\\paramcolor}{\\symtime_{\\indextime}}} P(t)dt = \\int_{t_k}^{t_k+\\textcolor{\\paramcolor}{\\symtime_{\\indextime}}} dE(t) = E_{k+1} -  E_{k} \\label{eq_int_e}\n\\end{IEEEeqnarray}\nIntegrating the $P(t)dt$ term approximately via the trapezoidal rule yields,\n\\begin{IEEEeqnarray}{C}\n\\int_{t_k}^{t_k+\\textcolor{\\paramcolor}{\\symtime_{\\indextime}}} P(t)dt \\approx \\left(\\frac{P_{k+1} +  P_{k}}{2}\\right) \\textcolor{\\paramcolor}{\\symtime_{\\indextime}} \\label{eq_trap}\n\\end{IEEEeqnarray}\nWith the following special cases for in the initial and final points respectively,\n\\begin{IEEEeqnarray}{C}\n\\int_{t_k}^{t_k+\\textcolor{\\paramcolor}{\\symtime_{\\indextime}}} P(t)dt \\approx P_{k} \\textcolor{\\paramcolor}{\\symtime_{\\indextime}} \\label{eq_init} \\\\\n\\int_{t_k}^{t_k+\\textcolor{\\paramcolor}{\\symtime_{\\indextime}}} P(t)dt \\approx P_{k+1} \\textcolor{\\paramcolor}{\\symtime_{\\indextime}}  \\label{eq_final}\n\\end{IEEEeqnarray}\nNote that \\eqref{eq_trap} is exact when $P(t)$ is piece-wise linear with breakpoints at $t_k$ and $t_k+\\textcolor{\\paramcolor}{\\symtime_{\\indextime}}$, and \\eqref{eq_init},\\eqref{eq_final} are exact when $P(t)$ is a step function (piece-wise constant) with break points at $t_k$ and $t_k+\\textcolor{\\paramcolor}{\\symtime_{\\indextime}}$.\nCombining \\eqref{eq_int_e} together with any of \\eqref{eq_trap},\\eqref{eq_init},\\eqref{eq_final} results in an algebraic expression, which can be generated for any sampled time $t_k$. This results in a set of \\emph{sparse} difference equations that link the power and energy variables.\n\nNote that in all cases, $t$ and $\\textcolor{\\paramcolor}{\\symtime_{\\indextime}}$ are known, but we want to decide on the values of $P_{k}\\,\\,\\forall k$, which then determines $E_{k}$. Both $E_{k}$ and $P_{k}$ must nevertheless remain within user-specified ranges. So, it is not a valid decision discharge when the energy content is at its lower bound (empty), or charge when the energy content is at the upper bound (full). The fact that a decision now, \\textit{e.g.}, to charge at a certain power setting, impacts the future decision making freedom, suggests that an optimal decision may not be obvious.\n\nLeveraging this derivation, the proposed optimization model is then defined in descretized time with time steps $k \\in \\mathcal{K} = \\{1,2,\\dots, n \\}$ and $\\textcolor{\\paramcolor}{\\symtime_{\\indextime}}$ being the duration of time step $k$.  Note that time step durations need not be uniform.\n\n\\subsection{Nonlinear Multi-Conductor Converter Model}\nThe complex power flow from the grid into the converter $c$, through conductor $p$, at time $k$, is given by,\n\\begin{IEEEeqnarray}{C}\n\\symcomplexpower_{\\indexconverter, \\indexconductor, \\indextime} = \\symactivepower_{\\indexconverter, \\indexconductor, \\indextime} + j \\symreactivepower_{\\indexconverter, \\indexconductor, \\indextime}. \\label{eq_power}\n\\end{IEEEeqnarray}\nWe define the set of conductors (phases), $p \\in \\mathcal{P}$.\nThis power flow is subject to a per-conductor apparent power flow limit $\\textcolor{\\paramcolor}{\\symcomplexpower_{\\indexconverter, \\indexconductor}^{\\text{rating}}}$ and a time-dependent operational status 0\/1 parameter $\\textcolor{\\paramcolor}{\\symstatus_{\\indexconverter, \\indextime}}$,\n\\begin{IEEEeqnarray}{C}\n  |\\symcomplexpower_{\\indexconverter, \\indexconductor, \\indextime}|^2 =   (\\symactivepower_{\\indexconverter, \\indexconductor, \\indextime})^2 + (\\symreactivepower_{\\indexconverter, \\indexconductor, \\indextime})^2 \\leq (\\textcolor{\\paramcolor}{\\symstatus_{\\indexconverter, \\indextime}} \\cdot \\textcolor{\\paramcolor}{\\symcomplexpower_{\\indexconverter, \\indexconductor}^{\\text{rating}}})^2\n    \\end{IEEEeqnarray}\nCurrent limits are defined per conductor,\n\\begin{IEEEeqnarray}{C}\n  |\\symcurrent_{\\indexconverter,  \\indexconductor,\\indextime}|  \\leq \\textcolor{\\paramcolor}{\\symstatus_{\\indexconverter, \\indextime}} \\cdot \\textcolor{\\paramcolor}{\\symcurrent_{\\indexconverter, \\indexconductor}^{\\text{rating}}}\n      \\end{IEEEeqnarray}\n  The relationship between power, current and voltage magnitudes is given by,\n\\begin{IEEEeqnarray}{C}\n     |\\symcomplexpower_{\\indexconverter, \\indexconductor, \\indextime}|^2 = |\\symvoltage_{\\indexnode, \\indexconductor, \\indextime}|^2 |\\symcurrent_{\\indexconverter,  \\indexconductor,\\indextime}|^2. \\label{eq_complex_power_def}\n     \\end{IEEEeqnarray}\nCombining these properties, the converter's power balance and losses are defined as,\n\\begin{IEEEeqnarray}{C}\n \\!\\!\\!\\!\\!\\!\\underbrace{ \\sum_{p \\in \\mathcal{P}}  \\symcomplexpower_{\\indexconverter, \\indexconductor, \\indextime} }_{\\text{grid side power}} \\!+ \\!\\!\\!\\!\\!\\!\\underbrace{\\symactivepower_{\\indexconverter, \\indextime}^{\\text{stor}}}_{\\text{energy buffer}} \\!\\!\\! =  \\!\\!\\! \\underbrace{j \\textcolor{\\paramcolor}{\\symreactivepower_{\\indexconverter, \\indextime}^{\\text{int}}}}_{\\text{reactive source}} \\!\\!\\! + \\!\\! \\underbrace{\\textcolor{\\paramcolor}{\\symcomplexpower_{\\indexconverter, \\indextime}^{\\text{ext}}}}_{\\text{other flow}} \\!\\!\\!+\\!\\!  \\underbrace{\\sum_{p \\in \\mathcal{P}} \\textcolor{\\paramcolor}{\\symimpedance_{\\indexconverter, \\indexconductor}} |\\symcurrent_{\\indexconverter,  \\indexconductor,\\indextime}|^2}_{\\text{copper loss}} \\label{eq_balance_loss}\n\\end{IEEEeqnarray}\nThis equation is composed of five components (left to right):\n\\begin{itemize}\n    \\item the complex power flow supplied through the all of the converter's conductors $\\symcomplexpower_{\\indexconverter, \\indexconductor, \\indextime}$;\n    \\item the storage subsystem active power $\\symactivepower_{\\indexconverter, \\indextime}^{\\text{stor}}$;\n    \\item a variable $\\textcolor{\\paramcolor}{\\symreactivepower_{\\indexconverter, \\indextime}^{\\text{int}}}$ that represents the converter's ability to control the generation and\/or absorption of reactive power;\n    \\item the user-defined complex power $\\textcolor{\\paramcolor}{\\symcomplexpower_{\\indexconverter, \\indextime}^{\\text{ext}}}$;\n    \\item copper losses proportional to internal\\footnote{effectively modelling the converter as a voltage source with a series impedance} impedance $\\textcolor{\\paramcolor}{\\symimpedance_{\\indexconverter, \\indexconductor}}$ and current magnitude $\\symcurrent_{\\indexconverter,  \\indexconductor,\\indextime}$ squared.\n\\end{itemize}\nNote that the inclusion of a copper loss term, due to its quadratic nature, incentivizes charging and discharging slowly over time, as well as avoiding unnecessary reactive power injection or consumption.  This has an added benefit of avoiding degeneracy issues related to multiple reactive power sources and sinks on a single node.  If reactive power is absorbed or generated in the inverter, the reactive power in \\eqref{eq_balance_loss} remains balanced through variable $\\textcolor{\\paramcolor}{\\symreactivepower_{\\indexconverter, \\indextime}^{\\text{int}}}$.\n\nThe $\\textcolor{\\paramcolor}{\\symcomplexpower_{\\indexconverter, \\indextime}^{\\text{ext}}}$ parameter often represents unavoidable losses that are incurred while the system is idle, but it can also be used to capture a wide variety of exogenous energy sinks or sources. Examples of exogenous flows are, self-discharge in a battery,  heat losses from a heat  buffer to the environment, water flow into the upper reservoir of a pumped hydro plant.\n\n\\subsection{Nonlinear Energy Storage Model}\nBecause reactive power cannot be stored, $\\symcomplexpower_{\\indexconverter, \\indextime}^{\\text{stor}} = \\symactivepower_{\\indexconverter, \\indextime}^{\\text{stor}}+ j 0$, and the amount of power stored is separated into charge $\\symactivepower_{\\indexconverter, \\indextime}^{\\text{c}}$ and discharge $\\symactivepower_{\\indexconverter, \\indextime}^{\\text{d}}$ power values, both of which are positive,\n\\begin{IEEEeqnarray}{C}\n\\symactivepower_{\\indexconverter, \\indextime}^{\\text{stor}} = \\symactivepower_{\\indexconverter, \\indextime}^{\\text{d}} - \\symactivepower_{\\indexconverter, \\indextime}^{\\text{c}}  \\label{eq:ac-6}\n     \\end{IEEEeqnarray}\n Charging and discharging are typically complementary,\n\\begin{IEEEeqnarray}{C}\n\\symactivepower_{\\indexconverter, \\indextime}^{\\text{d}} \\cdot \\symactivepower_{\\indexconverter, \\indextime}^{\\text{c}} = 0 \\label{eq:ac-7} \\label{eq_complementarity}\n     \\end{IEEEeqnarray}\nThe energy content in the buffer changes when charging and discharging, which also incurs losses which are captured by $\\textcolor{\\paramcolor}{\\symefficiency_{\\indexconverter}^{\\text{c}}}$ and $\\textcolor{\\paramcolor}{\\symefficiency_{\\indexconverter}^{\\text{d}}}$ as follows,\n\\begin{IEEEeqnarray}{C}\n\\forall k \\in \\mathcal{K} \\setminus 1: \\symenergy_{\\indexconverter, \\indextime} - \\symenergy_{\\indexconverter, \\indextime-1} = \\textcolor{\\paramcolor}{\\symtime_{\\indextime}} \\left(\\textcolor{\\paramcolor}{\\symefficiency_{\\indexconverter}^{\\text{c}}} \\symactivepower_{\\indexconverter, \\indextime}^{\\text{c}} - \\frac{\\symactivepower_{\\indexconverter, \\indextime}^{\\text{d}}}{\\textcolor{\\paramcolor}{\\symefficiency_{\\indexconverter}^{\\text{d}}}} \\right) \\label{eq:ac-8}\n     \\end{IEEEeqnarray}\nwhen using the end-point time discretization. Special care is taken to initialize the energy buffer in a certain state at the first time step,\n\\begin{IEEEeqnarray}{C}\nk=1: \\symenergy_{\\indexconverter, \\indextime} - \\textcolor{\\paramcolor}{\\symenergy_{\\indexconverter}^{\\text{init}}} = \\textcolor{\\paramcolor}{\\symtime_{\\indextime}} \\left(\\textcolor{\\paramcolor}{\\symefficiency_{\\indexconverter}^{\\text{c}}} \\symactivepower_{\\indexconverter, \\indextime}^{\\text{c}} - \\frac{\\symactivepower_{\\indexconverter, \\indextime}^{\\text{d}}}{\\textcolor{\\paramcolor}{\\symefficiency_{\\indexconverter}^{\\text{d}}}} \\right)\n\\end{IEEEeqnarray}\nIt is possible to avoid defining an initial energy content by adding the a constraint that the final energy content must be higher than the initial one. For instance,\n\\begin{IEEEeqnarray}{C}\n\\symenergy_{\\indexconverter, \\indextime=n} \\geq \\symenergy_{\\indexconverter, \\indextime=1}.\n\\end{IEEEeqnarray}\nIf so included, one can  enforce a specific end-time value instead.\n\\subsection{Variable Bounds }\n  We define  the total power rating of the converter  as the sum of the per-phase values.\n    \\begin{IEEEeqnarray}{C}\n\\textcolor{\\paramcolor}{\\symcomplexpower_{\\indexconverter}^{\\text{rating}}} = \\sum_p \\textcolor{\\paramcolor}{\\symcomplexpower_{\\indexconverter, \\indexconductor}^{\\text{rating}}}.\n  \\end{IEEEeqnarray}\n  If per-phase values are unknown, we can derive them from the total through $\\textcolor{\\paramcolor}{\\symcomplexpower_{\\indexconverter, \\indexconductor}^{\\text{rating}}} = \\textcolor{\\paramcolor}{\\symcomplexpower_{\\indexconverter}^{\\text{rating}}} \/ |\\mathcal{P}|$.\nThe magnitude bounds on the complex variables are,\n\\begin{IEEEeqnarray}{C}\n (\\symactivepower_{\\indexconverter, \\indexconductor, \\indextime})^2 + (\\symreactivepower_{\\indexconverter, \\indexconductor, \\indextime})^2 \\leq  |\\symvoltage_{\\indexnode, \\indexconductor, \\indextime}|^2 (\\textcolor{\\paramcolor}{\\symstatus_{\\indexconverter, \\indextime}} \\cdot \\textcolor{\\paramcolor}{\\symcurrent_{\\indexconverter, \\indexconductor}^{\\text{rating}}})^2, \\label{eq_current_bounds}\\\\\n (\\symactivepower_{\\indexconverter, \\indexconductor, \\indextime})^2 + (\\symreactivepower_{\\indexconverter, \\indexconductor, \\indextime})^2 \\leq (\\textcolor{\\paramcolor}{\\symstatus_{\\indexconverter, \\indextime}} \\cdot \\textcolor{\\paramcolor}{\\symcomplexpower_{\\indexconverter, \\indexconductor}^{\\text{rating}}})^2. \\label{eq_power_bounds}\n\\end{IEEEeqnarray}\nIt is not obvious whether just the current rating \\eqref{eq_current_bounds} or power ratings \\eqref{eq_power_bounds} are to be considered, so we include both. Note that both ratings are defined per-phase, thereby enabling two\/single-phase connected systems as edge cases through conductor-wise values set to 0.\nIn rectangular variables the converter bounds are,\n\\begin{IEEEeqnarray}{C}\n-\\textcolor{\\paramcolor}{\\symstatus_{\\indexconverter, \\indextime}} \\cdot\\textcolor{\\paramcolor}{\\symcomplexpower_{\\indexconverter, \\indexconductor}^{\\text{rating}}} \\leq \\symactivepower_{\\indexconverter, \\indexconductor, \\indextime} \\leq \\textcolor{\\paramcolor}{\\symstatus_{\\indexconverter, \\indextime}} \\cdot\\textcolor{\\paramcolor}{\\symcomplexpower_{\\indexconverter, \\indexconductor}^{\\text{rating}}},\\\\\n-\\textcolor{\\paramcolor}{\\symstatus_{\\indexconverter, \\indextime}} \\cdot\\textcolor{\\paramcolor}{\\symcomplexpower_{\\indexconverter, \\indexconductor}^{\\text{rating}}} \\leq \\symreactivepower_{\\indexconverter, \\indexconductor, \\indextime} \\leq \\textcolor{\\paramcolor}{\\symstatus_{\\indexconverter, \\indextime}} \\cdot\\textcolor{\\paramcolor}{\\symcomplexpower_{\\indexconverter, \\indexconductor}^{\\text{rating}}},\\\\\n- \\textcolor{\\paramcolor}{\\symstatus_{\\indexconverter, \\indextime}}  \\cdot \\textcolor{\\paramcolor}{\\symcomplexpower_{\\indexconverter}^{\\text{rating}}} \\leq \\textcolor{\\paramcolor}{\\symreactivepower_{\\indexconverter, \\indextime}^{\\text{int}}} \\leq \\textcolor{\\paramcolor}{\\symstatus_{\\indexconverter, \\indextime}}  \\cdot \\textcolor{\\paramcolor}{\\symcomplexpower_{\\indexconverter}^{\\text{rating}}},\\\\\n -\\textcolor{\\paramcolor}{\\symcomplexpower_{\\indexconverter}^{\\text{rating}}} \\leq \\symactivepower_{\\indexconverter, \\indextime}^{\\text{stor}} \\leq \\textcolor{\\paramcolor}{\\symcomplexpower_{\\indexconverter}^{\\text{rating}}}. \\label{eq:ac-12}\n\\end{IEEEeqnarray}\nThe storage subsystem bounds are,\n\\begin{IEEEeqnarray}{C}\n0 \\leq \\symactivepower_{\\indexconverter, \\indextime}^{\\text{c}} \\leq \\textcolor{\\paramcolor}{\\symactivepower_{\\indexconverter}^{\\text{c,max}}}, \\\\\n 0 \\leq \\symactivepower_{\\indexconverter, \\indextime}^{\\text{d}} \\leq \\textcolor{\\paramcolor}{\\symactivepower_{\\indexconverter}^{\\text{d,max}}}, \\\\\n0 \\leq \\symenergy_{\\indexconverter, \\indextime} \\leq \\textcolor{\\paramcolor}{\\symenergy_{\\indexconverter}^{\\text{max}}}, \\\\\n0 \\leq \\textcolor{\\paramcolor}{\\symvoltage_{\\indexnode, \\indexconductor}^{\\text{min}}} \\leq |\\symvoltage_{\\indexnode, \\indexconductor, \\indextime}| \\leq \\textcolor{\\paramcolor}{\\symvoltage_{\\indexnode, \\indexconductor}^{\\text{max}}}.  \\label{eq:ac-18}\n\\end{IEEEeqnarray}\n\nTogether, this system of equations and inequalities defines the most general form of the storage model proposed by this work. Note that model the only nonconvex constraint is \\eqref{eq_complementarity}.\n\n\n\\subsection{Data Model}\n\n\n\\begin{table}[tb]\n\\setlength{\\tabcolsep}{3pt}\n  \\centering\n   \\caption{Parameters}\\label{tab_params}\n    \\begin{tabular}{l l l l l  }\n\\hline\nSymbol & SI& data model  & condition & meaning \\\\\n\\hline\n$\\textcolor{\\paramcolor}{\\symtime_{\\indextime}}$ &  s&  h & $>0$  &  time step length   \\\\\n\\hline\n$\\textcolor{\\paramcolor}{\\symvoltage_{\\indexnode, \\indexconductor}^{\\text{min}}}$ &  V&  pu & $\\geq0$  & voltage lower bound  \\\\\n$\\textcolor{\\paramcolor}{\\symvoltage_{\\indexnode, \\indexconductor}^{\\text{max}}}$ &  V&  pu & $\\geq \\textcolor{\\paramcolor}{\\symvoltage_{\\indexnode, \\indexconductor}^{\\text{min}}}$  &  voltage upper bound \\\\\n\\hline\n$\\textcolor{\\paramcolor}{\\symstatus_{\\indexconverter, \\indextime}} $&  - &  - & $\\in \\{0,1\\}$  & converter status  \\\\\n$\\textcolor{\\paramcolor}{\\symcomplexpower_{\\indexconverter, \\indextime}^{\\text{ext}}} $& VA& MVA & $-$  & external flow  \\\\\n\\hline\n$\\textcolor{\\paramcolor}{\\symcomplexpower_{\\indexconverter}^{\\text{rating}}}, \\textcolor{\\paramcolor}{\\symcomplexpower_{\\indexconverter, \\indexconductor}^{\\text{rating}}}$ &  VA&  MVA & $\\geq0$  &  apparent power rating \\\\\n$\\textcolor{\\paramcolor}{\\symcurrent_{\\indexconverter}^{\\text{rating}}}, \\textcolor{\\paramcolor}{\\symcurrent_{\\indexconverter, \\indexconductor}^{\\text{rating}}} $&  A &  MA & $\\geq0$  & current rating  \\\\\n$\\textcolor{\\paramcolor}{\\symefficiency_{\\indexconverter}^{\\text{d}}} $&  -&  - & $>0, \\leq1$  &  discharge efficiency \\\\\n$\\textcolor{\\paramcolor}{\\symefficiency_{\\indexconverter}^{\\text{c}}}$ &  -&  - & $\\geq0, \\leq1$  & charge efficiency  \\\\\n$\\textcolor{\\paramcolor}{\\symenergy_{\\indexconverter}^{\\text{init}}} $&  J&  MWh & $\\geq0$  & initial energy content  \\\\\n$\\textcolor{\\paramcolor}{\\symenergy_{\\indexconverter}^{\\text{max}}} $&  J&  MWh & $\\geq0$  & energy rating  \\\\\n$\\textcolor{\\paramcolor}{\\symactivepower_{\\indexconverter}^{\\text{c,max}}} $&  W&  MW & $\\geq0$  & charge power rating  \\\\\n$\\textcolor{\\paramcolor}{\\symactivepower_{\\indexconverter}^{\\text{d,max}}}$ &  W&  MW & $\\geq0$  & discharge power rating  \\\\\n$\\textcolor{\\paramcolor}{\\symimpedance_{\\indexconverter, \\indexconductor}}$ &  $\\Omega$ & pu & $- $  & converter impedance \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nTable \\ref{tab_params} summarizes the constant parameters defined in the mathematical models.\nWe note that while all equations in this report are given in SI units, the user-facing data model utilizes the indicated engineering units.\nTable \\ref{tab_examples} illustrates how the data model can be parameterized to capture common storage technologies such batteries and pumped hydro can be achieved.  Time step length is uniform at $\\textcolor{\\paramcolor}{\\symtime_{\\indextime}}$ = 0.25h and number of time steps $|\\mathcal{K}|= 96$.\n\n\n\\begin{table}[tbh]\n  \\centering\n   \\caption{Illustrative Parameterization for Specific Technologies}\\label{tab_examples}\n    \\begin{tabular}{l l l l l  }\n\\hline\nParameter & Unit  & BESS & PHS & Flywheel \\\\\n\\hline\n$\\textcolor{\\paramcolor}{\\symstatus_{\\indexconverter, \\indextime}} $&   - & 1  & 1  & 1 \\\\\n$\\textcolor{\\paramcolor}{\\symcomplexpower_{\\indexconverter, \\indextime}^{\\text{ext}}} $&  MVA & standby  & flow into & standby \\\\\n&   &  power &  upper reservoir & power \\\\\n&  MVA & 0.00005 & time series & 0.003 \\\\\n\\hline\n$\\textcolor{\\paramcolor}{\\symcomplexpower_{\\indexconverter}^{\\text{rating}}}$ &  MVA & 0.005   & 1000  & 0.1 \\\\\n$\\textcolor{\\paramcolor}{\\symefficiency_{\\indexconverter}^{\\text{d}}} $&  - & 0.95  & 0.90 & 0.92 \\\\\n$\\textcolor{\\paramcolor}{\\symefficiency_{\\indexconverter}^{\\text{c}}}$ &  - & 0.95 & 0.90 & 0.92 \\\\\n$\\textcolor{\\paramcolor}{\\symenergy_{\\indexconverter}^{\\text{init}}} $&  MWh & 0.010  & 1000 & 0.015 \\\\\n$\\textcolor{\\paramcolor}{\\symenergy_{\\indexconverter}^{\\text{max}}} $&  MWh & 0.010  & 2000 & 0.030\\\\\n$\\textcolor{\\paramcolor}{\\symactivepower_{\\indexconverter}^{\\text{c,max}}} $&  MW & 0.005  & 1.0  &  0.1\\\\\n$\\textcolor{\\paramcolor}{\\symactivepower_{\\indexconverter}^{\\text{d,max}}}$ &  MW & 0.005 & 0.72 &  0.1\\\\\n$\\textcolor{\\paramcolor}{\\symresistance_{\\indexconverter, \\indexconductor}}$ &pu  &0.1   & 0 & 0 \\\\\n$\\textcolor{\\paramcolor}{\\symreactance_{\\indexconverter, \\indexconductor}}$ &pu  & 0   & 0.1 & 0\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\section{Reformulation, Relaxation and Approximation}\n\\label{sec_model_relax}\nThe key step to integrate the storage models into OPF, is to aggregate the storage systems for a specific bus, and then add the aggregated variables $\\symactivepower_{\\indexconverter, \\indexconductor, \\indextime}, \\symreactivepower_{\\indexconverter, \\indexconductor, \\indextime}$ to the nodal power balance equations for each conductor.\n\nNevertheless, this approach  is not immediately applicable to power system optimization tasks that do not typically consider multi-conductor network models, such as balanced OPF and production cost models.\nFurthermore, such a  nonlinear model can present significant computational challenges in large-scale optimization contexts.\nTo make this model more widely applicable, this section develops a series of simplifications for using the proposed model in convex and approximate power flow models.\n\n\\paragraph*{Balanced Power Flow Approximation}\nThe most natural approximation of the proposed model is a single-phase balanced power flow approximation.  This is easily accomplished\nby considering the single-phase case of \\eqref{eq_balance_loss} in the general model, namely,\n\\begin{IEEEeqnarray}{C}\n    \\symcomplexpower_{\\indexconverter, \\indextime} + \\symactivepower_{\\indexconverter, \\indextime}^{\\text{stor}} =  j \\textcolor{\\paramcolor}{\\symreactivepower_{\\indexconverter, \\indextime}^{\\text{int}}} + \\textcolor{\\paramcolor}{\\symcomplexpower_{\\indexconverter, \\indextime}^{\\text{ext}}}+  \\textcolor{\\paramcolor}{\\symimpedance_{\\indexconverter}} |\\symcurrent_{\\indexconverter, \\indextime}|^2  \\label{eq:ac-5}\n\\end{IEEEeqnarray}\nA similar transformation can be applied to any of the model variants presented in this section.\n\n\n\\paragraph*{Discrete Complementarity Formulation}\nThe charge-discharge complementarity constraint \\eqref{eq:ac-7} is a non-convex nonlinear constraint that has undesirable numerical properties.  Consequently, it is common to reformulate this constraint by introducing a binary variable,\n\\begin{IEEEeqnarray}{C}\n \\symbinary_{\\indexconverter, \\indextime}^{\\text{c}} \\in \\{0,1\\} \\label{eq_integrality}\n \\end{IEEEeqnarray}\nthat indicates which of the charge\/discharge variables is active at a given point in time by replacing \\eqref{eq_complementarity} with,\n\\begin{IEEEeqnarray}{C}\n 0 \\leq \\symactivepower_{\\indexconverter, \\indextime}^{\\text{c}} \\leq \\textcolor{\\paramcolor}{\\symactivepower_{\\indexconverter}^{\\text{c,max}}} \\cdot  \\symbinary_{\\indexconverter, \\indextime}^{\\text{c}} \\label{eq_complementarity_mi_1}\\\\\n 0 \\leq \\symactivepower_{\\indexconverter, \\indextime}^{\\text{d}} \\leq \\textcolor{\\paramcolor}{\\symactivepower_{\\indexconverter}^{\\text{d,max}}} \\cdot(1-  \\symbinary_{\\indexconverter, \\indextime}^{\\text{c}})  \\label{eq_complementarity_mi_2}\n\\end{IEEEeqnarray}\nThe discrete nature of this formulation does not typically present a overwhelming burden in commercial optimization software.  However, if such a problem is encountered, the integrality of this indicator variable can be relaxed as follows, providing a fast continuous relaxation of the complementarity requirement,\n\\begin{IEEEeqnarray}{C}\n0 \\leq \\symbinary_{\\indexconverter, \\indextime}^{\\text{c}} \\leq 1.\n\\end{IEEEeqnarray}\n\n\n\\paragraph*{Convex-Quadratic Relaxation}\nConvex quadratic relaxations of the power flow equations have gained interest recently and we observe that the proposed storage model has a natural relaxation in terms of the squared magnitudes of voltage and current.  First, the voltage and current squared expressions are lifted to new variables as follows:\n\\begin{IEEEeqnarray}{C}\n  |\\symvoltage_{\\indexnode, \\indexconductor, \\indextime}|^2    \\rightarrow \\symvoltagelifted_{\\indexnode, \\indexconductor, \\indextime}, \\\\\n    |\\symcurrent_{\\indexconverter,  \\indexconductor,\\indextime}|^2 \\rightarrow \\symcurrentlifted_{\\indexconverter, \\indexconductor, \\indextime} .\n\\end{IEEEeqnarray}\nUsing these lifted variables one can rewrite \\eqref{eq_complex_power_def} and \\eqref{eq_balance_loss} as convex constraints,\n\\begin{IEEEeqnarray}{C}\n     (\\symactivepower_{\\indexconverter, \\indexconductor, \\indextime})^2 + (\\symreactivepower_{\\indexconverter, \\indexconductor, \\indextime})^2 \\leq \\symvoltagelifted_{\\indexnode, \\indexconductor, \\indextime} \\symcurrentlifted_{\\indexconverter, \\indexconductor, \\indextime}, \\label{eq_lifted_power_relax}\\\\\n       \\sum_{p \\in \\mathcal{P}}  \\symcomplexpower_{\\indexconverter, \\indexconductor, \\indextime} + \\symactivepower_{\\indexconverter, \\indextime}^{\\text{stor}} = j \\textcolor{\\paramcolor}{\\symreactivepower_{\\indexconverter, \\indextime}^{\\text{int}}} + \\textcolor{\\paramcolor}{\\symcomplexpower_{\\indexconverter, \\indextime}^{\\text{ext}}} + \\sum_{p \\in \\mathcal{P}} \\textcolor{\\paramcolor}{\\symimpedance_{\\indexconverter, \\indexconductor}} \\symcurrentlifted_{\\indexconverter, \\indexconductor, \\indextime}. \\label{eq_lifted_storage_relax}\n\\end{IEEEeqnarray}\nNote that \\eqref{eq_lifted_power_relax} is a rotated second order cone, and therefore convex.\nIt is advantageous to combine this relaxation with the discrete complementary formulation resulting in a mixed-integer second-order conic optimization problem, which can be solved with highly optimized commercial software packages.\n\n\\paragraph*{Linearized Approximation}\nGiven the high performance of the DC power flow approximation, a linear active-power-only approximation of the storage model is proposed by replacing the converter balance \\eqref{eq_balance_loss} with,\n\\begin{IEEEeqnarray}{C}\n  \\sum_{p \\in \\mathcal{P}}  \\symactivepower_{\\indexconverter, \\indexconductor, \\indextime} + \\symactivepower_{\\indexconverter, \\indextime}^{\\text{stor}} = \\textcolor{\\paramcolor}{\\symactivepower_{\\indexconverter, \\indextime}^{\\text{ext}}} \\label{eq_storage_approx}.\n\\end{IEEEeqnarray}\nSimilar to the DC power flow approximation, this formulation does not capture the copper losses incurred from operating the converter, but it does benefit from a significant reduction in model size because reactive power is ignored.\n\n\\section{Computational Validation} \\label{sec_study}\n\nThe generic storage model proposed in the previous sections is implemented in the open-source power network optimization software PowerModels v0.17 \\cite{Coffrin2017}.  This section presents two proof-of-concept validation studies demonstrating the soundness of the model and how the dispatch of a storage device can vary across different formulations of the power flow equations.  Throughout this section the following solvers were used: Ipopt v12.4 \\cite{Wachter2006} for the NLP version of AC model; Juniper v0.6 \\cite{Kroger} using Ipopt as an NLP sub-solver for the MINLP version of AC model; Gurobi v9.0 \\cite{gurobi} for the MIP and MISOCP models presented by the DC and SOC models.\n\n\\subsection{Test Case}\nA multi-period AC-OPF test case is required to study the storage model proposed in this work.  In the interest of utilizing open-access data, a test case was constructed by combining the 14-bus network available in the PGLib-OPF v19.05 \\cite{Babaeinejadsarookolaee2019}  with the hourly load profile scalars provided in the RTS 96 test case for a typical summer weekday.  Linear interpolation was used to increase the load profile's resolution from hourly to 15 minute increments (\\textit{i.e.}, $\\textcolor{\\paramcolor}{\\symtime_{\\indextime}}$ = 0.25h, $|\\mathcal{K}|= 96$).  A moderately sized storage device was added to Bus 13 in the test case with the following parameters, $\\textcolor{\\paramcolor}{\\symenergy_{\\indexconverter}^{\\text{init}}} = 1, \\textcolor{\\paramcolor}{\\symenergy_{\\indexconverter}^{\\text{max}}} = 200, \\textcolor{\\paramcolor}{\\symactivepower_{\\indexconverter}^{\\text{c,max}}} = 100, \\textcolor{\\paramcolor}{\\symactivepower_{\\indexconverter}^{\\text{d,max}}} = 75, \\textcolor{\\paramcolor}{\\symefficiency_{\\indexconverter}^{\\text{c}}} = 0.85, \\textcolor{\\paramcolor}{\\symefficiency_{\\indexconverter}^{\\text{d}}} = 0.90,  Z = 0.1 + j 0.01, \\textcolor{\\paramcolor}{\\symcomplexpower_{\\indexconverter}^{\\text{rating}}} = 1000$; rather than focus on a specific storage technology, this invented device was designed to exercise the key mathematical features of the proposed model.  Finally, a quadratic cost term of $0.2$ \\$\/MWh$^2$ was added to the generators to provide a cost incentive to charge the storage during off-peak times.\n\nThe proposed extended version of the 14-bus network is suitable for single-phase multi-period AC-OPF studies.  To test the multi-phase variant of the proposed storage model a three-phase extension of this network was developed by creating three replicates of network (one for each phase) and splitting the load by 36\\%, 33\\%, and 31\\% on phases A, B, and C respectively.  This proposed network is a highly stylized version of a multi-phase network model, though it is sufficient to highlight some of the non-trivial interactions that can occur between multi-phase storage devices and generator cost functions.\n\n\\subsection{AC Optimal Power Flow}\nThe first and simplest experiment considers solving the multi-period AC-OPF with storage; note that both models are only guaranteed to converge to a locally optimal solution. Both the NLP and MINLP variants of the AC-OPF with storage model were considered (see Table \\ref{tbl:obj_rt}).  It was observed that both formulations found similar quality solutions.  While the NLP variant was significantly faster to converge it also occasionally suffered from numerical issues, suggesting the MINLP formulation is preferable for robustness.\n\nThe operation profiles presented in Figure \\ref{fig:ac_storage} are qualitatively what is expected from the introduction of a storage device.  A time shift of the load allows the total generation cost to be reduced from 882,439 to 871,971.  This is accomplished by storing energy in the evening that is expended during the day, as illustrated by the peak-shaving behavior in Figure \\ref{fig:ac_storage}.  However encouraging these results are, it is important to verify the quality of the proposed charge\/discharge schedule due to the non-convex nature of these models.  The SOC relaxation and DC approximations are natural choices for verification because their convexity ensures globally optimal schedules.\n\n\\begin{figure}[t]\n    \\begin{center}\n    \\includegraphics[width=8.8cm]{ac-power.pdf}\n    \\end{center}\n    \\vspace{-0.6cm}\n    \\caption{A comparison of storage impacts to generator dispatch in single-phase AC power flow. Positive\/negative storage dispatch indicates charging\/discharging, respectively.}\n    \\vspace{-0.2cm}\n    \\label{fig:ac_storage}\n\\end{figure}\n\n\\subsection{Power Flow Model Comparison}\nThe second experiment compares the charge\/discharge schedule proposed by three power flow models: the non-convex AC power flow, the convex nonlinear SOC power flow and the linear `DC' power flow.  The results of this experiment are highlighted in Table \\ref{tbl:obj_rt} and Figure \\ref{fig:ac_storage2}.  Focusing on Figure \\ref{fig:ac_storage2}, the first observation is that AC and SOC dispatch schedules are remarkably similar, which suggests that the AC solution is of high quality.  Furthermore, because the SOC model is a convex relaxation and provides a lower bound on the optimal solution, we can compute an optimally gap of less than $0.17\\%$ on the AC solution, based on the values reported in Table \\ref{tbl:obj_rt}.  The second observation in Figure \\ref{fig:ac_storage2} is that the DC power flow generally tracks the behavior of the other two models but is notably more aggressive in the charge\/discharge schedule.  A preliminary sensitivity investigation suggested that this discrepancy can be explained in large part by the lack of converter loss modeling in this linear approximation.\n\n\\begin{figure}[t]\n    \\begin{center}\n    \\includegraphics[width=8.8cm]{storage-comp.pdf}\n    \\end{center}\n    \\vspace{-0.6cm}\n    \\caption{A comparison of power flow models on storage dispatch. Positive\/negative storage dispatch indicates charging\/discharging, respectively.}\n    \\label{fig:ac_storage2}\n\\end{figure}\n\n\\begin{table}[tbh]\n  \\centering\n   \\caption{Storage Model Cost and Runtime Comparison}\\label{tbl:obj_rt}\n    \\begin{tabular}{r r r r r  }\n\\hline\nModel & Storage Equations & Objective & Runtime (s)  \\\\\n\\hline\nAC Base & none & 882\\,439 & 5.06 \\\\\nAC-NL & \\eqref{eq:ac-5},\\eqref{eq_complementarity} & 871\\,971 & 82.26 \\\\\nAC-MI & \\eqref{eq:ac-5},\\eqref{eq_complementarity_mi_1},\\eqref{eq_complementarity_mi_2} & 871\\,971 & 686.48 \\\\\nSOC-MI & \\eqref{eq_lifted_power_relax},\\eqref{eq_lifted_storage_relax},\\eqref{eq_complementarity_mi_1},\\eqref{eq_complementarity_mi_2} & 870\\,519 & 5.43 \\\\\nDC-MI & \\eqref{eq_storage_approx},\\eqref{eq_complementarity_mi_1},\\eqref{eq_complementarity_mi_2} & 807\\,625 & 0.12 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\subsection{Single-Phase vs Multi-Phase}\nThis third experiment highlights the value of the proposed model in a simplified multi-phase setting.  Figure \\ref{fig:ac_3p_storage} provides a comparison of the storage dispatch and state in both the AC single-phase and AC three-phase settings.  The first observation is that the total utilization of the storage device is reduced significantly.  This is driven primarily by the added value of the converter to exchange power across phases, which reduces the relative value of storing energy over time.\nThis effect is highlighted by the storage dispatch in the last few hours of the day where energy is consumed on Phases B\/C and delivered to Phase A to reduce the cost of generating power on the more heavily loaded phase.  This stylized example highlights the importance of considering converter models and phase unbalance information when evaluating the sizing, operation and value of storage devices.\n\n\\begin{figure}[t]\n    \\begin{center}\n    \\includegraphics[width=8.8cm]{ac-3p-storage-charge-comp.pdf}\n    \\end{center}\n    \\vspace{-0.6cm}\n    \\caption{A comparison of storage dispatch (top) and energy state (bottom) in single-phase and multi-phase power flow models.}\n    \\vspace{-0.5cm}\n    \\label{fig:ac_3p_storage}\n\\end{figure}\n\n\\section{Conclusions}\n\\label{sec_conclusions}\nMotivated by the increasing role of storage in future energy systems, this work proposed a generic and flexible storage model that can be leveraged in a variety of power system optimization studies, ranging from operation scheduling to production cost modeling.  Based on a computational validation of the proposed model, we conclude that the details of both the storage device and its converter can have a non-trivial impact on the utilization and value of energy storage.  These results highlight the value of the proposed model in switching between different levels of modeling fidelity.\n\nThe proposed model has been implemented in PowerModels since v0.9 \\cite{Coffrin2017} and PowerModelsDistribution since v0.2 to provide easy access by the research community.  A core feature of the PowerModels framework is that it allows the user to easily switch between different types of power flow formulations (\\textit{e.g.}, non-convex, convex and linear), which provided a clear path to implement the storage model variants proposed in this work.  In future work we will continue testing the proposed model in more complex optimization contexts, such as OPF with unit commitment and optimal storage sizing, to better understand how storage model fidelity impacts optimal decisions.  We also plan to continue validation efforts with direct comparisons to the detailed storage models provided by power system simulators.\n\n\\section{Acknowledgements}\nThis work was supported by funding from the U.S. Department of Energy's (DOE) Office of Electricity (OE) as part of the CleanStart-DERMS project of the Grid Modernization Laboratory Consortium, and by the U.S. Department of Energy through the Los Alamos National Laboratory LDRD Program and the Center for Nonlinear Studies.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}