# Geometric PDEs

## Bilayer Plates

The bending of polymer bilayers is a mechanism which allows for large deformations via small externally induced lattice mismatches of the underlying materials. This technology recently regained popularity due to new techniques allowing for the fabrication of these materials at the micro scale. Typical applications are thermostats, nanotubes, micro-robots, micro-switches, micro-grippers, micro-scanners, micro-probes, micro-capsules... The proposed reduced model consists of a nonlinear fourth order problem with a pointwise isometry constraint. The bilayer material is modeled as a single plate endowed with an energy proportional to $$J(y)= \left\lbrace\begin{array}{ll}\frac12 \int_\Omega | II-Z|^2 \ \ \ & \textrm{when } y \textrm{ is an isometry}, \\ +\infty & \textrm{otherwise.} \end{array}\right.$$ Here $II$ denotes the second fundamental form of the plate parametrized by $y:\Omega \rightarrow \mathbb R^3$ and $Z$ is a spontaneous curvature representing the material mismatch.

### Effect of the Aspect Ratio and Spontaneous Curvature

Equilibrium shapes of bilayer plates for several aspect-ratios $\rho$ (from left to right $\rho=\frac52,\frac32,1,\frac12$) and spontaneous curvatures $Z=-rI_2$ (from top to bottom $r=5,3,1$). Decreasing the aspect ratio restores the ability for the plate to fold into a cylindrical shape for larger spontaneous curvatures. For instance, this is the case for plates with parameters $r=3$ and $\rho=\frac32$ or $r=5$ and $\rho=\frac12$. Notice, however, that small regions around the free corners have not completely relaxed to equilibrium. This effect is due to the violation of the isometry constraint and reduced upon decreasing the discretization parameters as well as the stopping criteria. The numbers below each stationary configuration are the corresponding approximate energies.

### Box

In this simulation, the hinges of a box are made of two different polymers reacting differently to heat. The sides are made of material allowing for heat propagation so that closing and opening the box can be controlled by the temperature set at the base (non-moving side).
Actual laboratory experiment of a self assembling box, see E. Smela's page at the univeristy of Maryland for complete information of the fabrication process. Experimental video reproduced here courtesy of E. Smela.

### Different Clamped Configuration

Different snapshots of the deformed corner-clamped plate with spontaneous curvature $Z=-I_2$. The equilibrium shape has energy 11.616 and is not a cylinder. For comparison, the one side fully clamped edge simulation predicts a smaller equilibrium energy (9.81).

### Humidity Control

Bilayer devices can be used to control the temperature or moisture inside a room. The HygroSkin project by Achim Menges exposed at FRAC Centre Orlean takes advantage of this technology by designing humidity responsive apertures to a pavilion. Heat and moisture are thus dynamically controlled without any high-tech equipment.

### Anisotropic Curvature

(LEFT) Deformation of a plate with anisotropic curvature given by $$Z=\begin{pmatrix}1 & 0 \\ 0 & 5 \end{pmatrix}.$$ The spontaneous curvature value is 1 in the clamped direction, its effect being barely noticeable, whereas it is 5 in the orthogonal direction. The equilibrium shape is a cylinder (absolute minimizer) with an energy of 42.09.

(RIGHT) Deformation of a plate with anisotropic curvature $$Z=\begin{pmatrix}-3 & 2 \\ 2 & -3 \end{pmatrix}.$$ The principal curvatures are 5 and 1 but the principal directions form an angle $\frac{\pi}4$ with the coordinate axes. The plate adopts a corkscrew shape before self-intersecting.

### Hyperbolic Spontaneous Curvature

Deformation of a plate with anisotropic curvature given by $$Z=\begin{pmatrix}5 & 0 \\ 0 & -5 \end{pmatrix}.$$ The spontaneous curvatures are $−5$ in the clamped direction and 5 in the perpendicular direction, which eventually dominates the former and leads to a cylindical shape after three full rotations.

### Anisotropic Curvature

(LEFT) Deformation of a plate with anisotropic curvature given by $$Z=\begin{pmatrix}1 & 0 \\ 0 & 5 \end{pmatrix}.$$ The spontaneous curvature value is 1 in the clamped direction, its effect being barely noticeable, whereas it is 5 in the orthogonal direction. The equilibrium shape is a cylinder (absolute minimizer) with an energy of 42.09.

(RIGHT) Deformation of a plate with anisotropic curvature $$Z=\begin{pmatrix}-3 & 2 \\ 2 & -3 \end{pmatrix}.$$ The principal curvatures are 5 and 1 but the principal directions form an angle $\frac{\pi}4$ with the coordinate axes. The plate adopts a corkscrew shape before self-intersecting.

### Free Boundary Conditions

When no boundary conditions are applied to (part of) the boundary, the algorithm automatically filter out the translations. We present two experiments in this case.    (LEFT) Actual experiment taken from [Alben, Balakrisnan and Smela. Nano letters (2011)] and approximate equilibrium shape with spontaneous curvature $$Z=\begin{pmatrix}5 & 0 \\ 0 & 0 \end{pmatrix}.$$

(RIGHT) Actual experiment taken from [Simpson, Nunnery, Tannenbaum, and Kalaitzidou. Journal of Materials Chemistry (2010).] and approximate equilibrium shape with anisotropic curvature $$Z=\begin{pmatrix}-3 & 2 \\ 2 & -3 \end{pmatrix}.$$

### Collaborators and Relevant Literature

• BONITO, A. AND NOCHETTO, R.H. AND NTOGKAS, D.; DG Approach to Large Bending Plate Deformations with Isometry Constraint. M3AS, 31(1), 133-175, 2021.
• BONITO, A. AND NOCHETTO, R.H. AND NTOGKAS, D.; Discontinuous Galerkin Approach to Large Bending Deformation of a Bilayer Plate with Isometry Constraint. J. Comput. Phys., 423 (2020).
• BARTELS, S. AND BONITO, A. AND MULIANA, A. AND NOCHETTO, R.H.; Modeling and simulation of thermally actuated bilayer plates. J. Comput. Phys, 354, 512--528, 2018.
• BARTELS, S. AND BONITO, A. AND NOCHETTO, R.H.; Bilayer Plates: Model Reduction, $\Gamma$-Convergent Finite Element Approximation and Discrete Gradient Flow, Comm. Pure Appl. Math., 70(3), 547--589, 2017.

## Prestrained Plates

Prestrained thin materials develop internal stresses, deform out of plane, and exhibit nontrivial 3D shapes even without external forces. Chief examples exhibiting such phenomena are nematic glasses, natural growth of soft tissues or manufactured polymer gels. After reduction from 3D to 2D [Battacharya, Lewicka and Shaffner (2016)], denoting by $\Omega\subset\mathbb{R}^2$ the midplane of the plate, the model consists in minimizing the bending energy $$E(\mathbf{y}) = \frac{\mu}{12}\int_{\Omega}\big|g^{-\frac{1}{2}} \, {\rm II}(\mathbf{y}) \, g^{-\frac{1}{2}}\big|^2+\frac{\lambda}{2\mu+\lambda}{\rm tr}\big(g^{-\frac{1}{2}}\, {\rm II}(\mathbf{y}) \, g^{-\frac{1}{2}} \big)^2$$ under the constraint that $$\nabla\mathbf{y}^T\nabla\mathbf{y}=g \quad \mbox{a.e. in } \Omega.$$ Here, ${\rm II}(\mathbf{y})$ denote the second fundamental form of the deformed plate $\mathbf{y}(\Omega)$, $\mu$ and $\lambda$ are the Lamé coefficients, and $g$ is the given (symmetric positive definite) target metric. The second fundamental form can actually be replaced by the Hessian of the deformation without affecting the minimizers. In the proposed numerical method, an LDG approach is used for the discretization, where the Hessian is replaced by a so-called discrete Hessian consisting of the broken Hessian and the liftings of jump terms, and an $H^2$ gradient flow is used for the energy minimization.

### Manufactured gel discs

A laboratory experiment, where prestrained discs made of NIPA gel with various monomer concentrations are manufactured, is presented in [Sharon and Efrati (2010)]. Considering a similar metric, the proposed numerical method produces deformations that are comparable to the experimental ones.    Laboratory manufactured disc gels using an elliptic prestrained metric $g$ (taken from [Sharon and Efrati (2010)]) along with the approximate equilibrium shape.

Laboratory manufactured disc gels using a hyperbolic prestrained metric $g$ (taken from [Sharon and Efrati 2010]) along with the approximate equilibrium shape.

### Metric with positive Gaussian curvature

In this example, we consider the bending of the unit disc when a metric with positive Gaussian curvature is prescribed, namely $$g = \nabla\mathbf{y}^T\nabla\mathbf{y} \qquad \mbox{with} \qquad \mathbf{y}(x_1,x_2)=\left(x_1,x_2,\sqrt{0.2}\sin\Big(\frac{\pi}{2}\big(1-\sqrt{x_1^2+x_2^2}\big)\Big)\right)^T,$$ where $\mathbf{y}$ corresponds to the parametrization of a bubble. We do not impose any boundary conditions (free boundary case). Before running the gradient flow algorithm, a pre-processing step is used to get an initial deformation with a prestrain defect below $0.1$.

Evolution of the deformation throughout the gradient flow for the unit disc with a positive Gaussian curvature metric.

### Metric with negative Gaussian curvature

We use the same setting as in the previous example but with the metric $$g(x_1,x_2) = \begin{bmatrix} 1+x_2^2 & x_1x_2 \\ x_1x_2 & 1+x_1^2 \end{bmatrix}$$ which has a negative Gaussian curvature and is achieved by the deformation $\mathbf{y}(x_1,x_2)=(x_1,x_2,x_1x_2)^T$ corresponding to a hyperbolic paraboloid.

Evolution of the deformation throughout the gradient flow for the unit disc with a negative Gaussian curvature metric.

### Metric corresponding to a catenoid/helicoid

For this simulation, $\Omega\subset\mathbb{R}^2$ is a rectangular domain and we prescribe the metric $$g(x_1,x_2) = \begin{bmatrix} \cosh(x_2)^2 & 0 \\ 0 & \cosh(x_2)^2 \end{bmatrix}.$$ The deformations $$\mathbf{y}^{\alpha}(x_1,x_2)=\cos(\alpha) \begin{bmatrix} \sinh(x_2)\sin(x_1) \\ -\sinh(x_2)\cos(x_1) \\ x_1 \end{bmatrix} + \sin(\alpha) \begin{bmatrix} \cosh(x_2)\cos(x_1) \\ \cosh(x_2)\sin(x_1) \\ x_2 \end{bmatrix}$$ are all compatible with the metric and all give the same energy. The case $\alpha=0$ corresponds to an helicoid while $\alpha=\pi/2$ represents a catenoid.   Final configurations for the free boundary case with different initial deformations: from left to right, the prestrain defect of the initial deformation is about 2.6, 1.4 and 0.9. The smaller the defect, the closer the final shape is to be a closed catenoid.

Evolution of the deformation throughout the gradient flow. Left: free boundary case; right: view from the side and from the top when boundary conditions for $\nabla\mathbf{y}$ and $\mathbf{y}$ are prescribed on the bottom side of the rectangle.

### Collaborators and Relevant Literature

• BONITO, A. AND GUIGNARD, D. AND NOCHETTO, R.H. AND YANG, SH.; LdG Approximation of Large Deformations of Prestrained Plates. ArXiv 2011.01086.

## Thin Structures Foldings

The folding of thin elastic sheets along a prepared curved arc is considered. This naturally leads to bending effects and has recently attracted considerable attention in applied sciences. Particular applications arise in the design of cardboxes and bistable switching devices that make use of corresponding flapping mechanisms. To describe our approach we let $S \subset \mathbb R^2$ be a bounded Lipschitz domain that represents the sheet without thickness and $\Sigma \subset \overline{S}$ be a curve that models the crease, i.e., the arc along which the sheet is folded. The corresponding three-dimensional model involves a thickness parameter $h>0$ and the thin domain $\Omega_h = S \times (-h/2,h/2)$. The material is weakened (or damaged) in a neighbourhood of width $r>0$ around the arc $\Sigma$ and we consider for given functions $W$ and $f_{\epsilon,r}$ the hyperelastic energy functional for a deformation $z:\Omega_h \to \mathbb R^3$ $$E^h(z) = \int_{\Omega_h} f_{\epsilon,r}(x) |(\nabla z)^T \nabla z - I|^2.$$ Here $f_{\epsilon,r}(x) \in (0,1]$ is small with value $\epsilon>0$ close to the arc $\Sigma$ and approximately $1$ away from the arc. Hence, the factor $f_{\epsilon,r}$ models a reduced elastic response of the material close to $\Sigma$. By appropriately relating the thickness $h$, the intactness fraction $\epsilon$, and width $r$ of the prepared region, we obtain for $(h,\epsilon,r) \to 0$ a meaningful dimensionally reduced model which seeks a minimizing deformation $y:S\to \mathbb R^3$ for the functional $$E_K(y) = \frac{1}{24} \int_{S \setminus \Sigma} | D^2 y|^2,$$ where $y$ is required to satisfy the pointwise isometry condition $(\nabla y)^T (\nabla y) = I.$ Compare with the prestrained or bilayer models described above.

### Flapping Mechanisms

We consider a rectangular plate with a quadratic folding arc. The plate is compressed at two corners to generate bistable a flapping mechanism. The experimental setting, the laboratory experiment and the actual simulation are provided below. ### Flowers

Our second experiment simulates a typical origami folding construction with curved arcs which is also known as kirigami folding which includes cutting and bending a piece of paper. In our example a square domain with a square hole is prepared using four arcs that connect midpoints of the outer boundary with the corners of the inner boundary. The experimental setting, the laboratory experiment and the actual simulation are provided below. The next illustration relates to the NASA starshade experiment. To facilite direct imaging of planets around other stars, the starlights of the stars must be suppressed. One possibility is to emply occulters that are external to the telescope. These are called starshade. They are compactly packed in the shuttle and deployed when in space. ### Mixing it up

Bilayer materials can also leverage folding mechanisms. The next example is inspired from Tahouni et al. (2020) and consists of 8 folding curves on a bilayer material. ### Collaborators and Relevant Literature

• BARTELS, S. AND BONITO, A. AND HORNUNG, P.; Modeling and simulation of thin sheet folding. Interfaces and Free Boundaries, 2022. ArXiv 2108.00937.

## Biomembranes When lipid molecules are immersed in aqueous environment they aggregate spontaneously into a bilayers or membrane that forms an encapsulating bag called vesicle. This happens because lipids consist of a hydrophilic head group and a hydrophobic tail, which isolate themselves in the interior of the membrane. This phenomenon is of interest in biophysics because lipid membranes are ubiquitous in biological systems, and an understanding of vesicles provides an important element to understand real cells.

The elastic behavior under large deformations and the dynamics of such deformations are poorly understood. Helfrich in 1973 was one of the first to introduce a model for the equilibrium shape of vesicles where the bending or curvature energy had to be minimized. Jenkins in 1976 arrived to the same model starting from continuum mechanical principles.        An ellipsoid is deformed to a shape that resembles a red blood cell. The second row shows the same frame with a middle cut.

### Bend and Twist

A 3D simulation with a bended initial shape is considered. As expected, the flow decreases the bending and smoothes the sharp tails.
Two different time scales are observed: (left) tails are rounded - rapid initial time scale and (right) unbending - slow time scale.

### Inertial Effect

The effect of the bulk fluid inertia is emphasized here.
Simulations not accounting (left) or accounting (right) for the bulk fluids are compared. The colors indicate the magnitude of the pression (red = high, blue = low).
A 3D simulation with a bended initial shape and taking into account the bulk fluid.

### Red Blood Cells and Dumbell Shapes

The equilibrium shape corresponding to ellipsoidal shape depends on the principal axis aspect ratios.
8x1x1 initial ellipsoid leads to a dumbell like equilibrium shape.
10x10x2 initial ellipsoid leads to a bi-concave equilibrium shape.
The fluid inside an 5x1x1 ellipsoid is accounted for.

### Collaborators and Relevant Literature

• BONITO, A. AND NOCHETTO, R.H. AND PAULETTI, M.S.; Dynamics of Biomembranes: Effect of the Bulk Fluid, Math. Model. Nat. Phenom., 6(5), 25--43, 2011.
• BONITO, A. AND NOCHETTO, R.H. AND PAULETTI, M.S.; Parametric FEM for Geometric Biomembranes, J. Comput. Phys., 229, 3171--3188, 2010.

## Electrowetting on Dielectric

The ability to manipulate fluids at the micro-scale is an important tool in the area of bio-medical applications. Micro-fluidic devices often exploit surface tension forces to actuate or control liquids by taking advantage of the large surface-to-volume ratios found at the micro-scale. In particular, some MEMS (micro-electro-mechanical system) devices use electro-wetting which effectively modifies surface tension effects through the use of electric fields, which allows for the manipulation of fluid droplets.     Time evolution of splitting of a droplet. Experiment (blurry line, courtesy Kim's Lab (UCLA)) vs numerical simulation (dashed line).

For a given time evolution of the electric potential, we are able to predict the dynamic of the droplet and simulate splitting.
Predicted droplet motion for given electric potential.
Controlled splitting of droplet using electrowetting.

### Collaborators and Relevant Literature

• WALKER, S.W. AND BONITO, A. AND NOCHETTO, R.H.; Mixed Finite Element Method for Electrowetting On Dielectric with Contact Line Pinning, Interfaces Free Bound., 12(1), 85--119, 2010.

## Support

This material is based upon work partially supported by the National Science Foundation. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.