Minimal Surface Problem

This example shows how to solve the minimal surface equation

on the unit disk $\Omega =\left\{\left(\mathit{x},\mathit{y}\right)|{\mathit{x}}^2+{\mathit{y}}^2\le 1\right\}$, with $\mathit{u}\left(\mathit{x},\mathit{y}\right)={\mathit{x}}^2$ on the boundary $\partial \Omega$. An elliptic equation in the toolbox form is

$-\nabla \cdot \left(\mathit{c}\nabla \mathit{u}\right)+\mathit{au}=\mathit{f}$.

Therefore, for the minimal surface problem, the coefficients are:

$\mathit{c}=\frac{1}{\sqrt{1+{\left|\nabla \mathit{u}\right|}^2}},\mathit{a}=0,\mathit{f}=0$.

Because the coefficient c is a function of the solution u, the minimal surface problem is a nonlinear elliptic problem.

To solve the minimal surface problem using the programmatic workflow, first create a PDE model with a single dependent variable.

model = createpde;

Create the geometry and include it in the model. The circleg function represents this geometry.

geometryFromEdges(model,@circleg);

Plot the geometry with the edge labels.

pdegplot(model,EdgeLabels="on")
axis equal
title("Geometry with Edge Labels")

Specify the coefficients.

a = 0;
f = 0;
cCoef = @(region,state) 1./sqrt(1+state.ux.^2 + state.uy.^2);
specifyCoefficients(model,m=0,d=0,c=cCoef,a=a,f=f);

Specify the boundary conditions using the function $\mathit{u}\left(\mathit{x},\mathit{y}\right)={\mathit{x}}^2$.

bcMatrix = @(region,~)region.x.^2;
applyBoundaryCondition(model,"dirichlet", ...
                       Edge=1:model.Geometry.NumEdges, ...
                       u=bcMatrix);

Generate and plot a mesh.

generateMesh(model,Hmax=0.1);
figure; 
pdemesh(model); 
axis equal

Clear figure for future plots.

clf

Solve the problem by using the solvepde function.

result = solvepde(model);
u = result.NodalSolution;

Plot the solution by using the Visualize PDE Results Live Editor task. First, create a new live script by clicking the New Live Script button in the File section on the Home tab.

On the Live Editor tab, select Task > Visualize PDE Results. This action inserts the task into your script.

To plot the solution, follow these steps.

You also can plot the solution at the MATLAB® command line by using the pdeplot function. For example, plot the solution as a 3-D plot, using the solution values for plot heights.

figure; 
pdeplot(model,XYData=u,ZData=u);
xlabel x
ylabel y
zlabel u(x,y)
title("Minimal Surface")
colormap jet