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Plot 2-D Solutions and Their Gradients

Plot Solutions Without Explicit Interpolation

To quickly visualize a 2-D scalar PDE solution, use the pdeplot function. This function lets you plot the solution without explicitly interpolating the solution. For example, solve the scalar elliptic problem -u=1 on the L-shaped membrane with zero Dirichlet boundary conditions and plot the solution.

Create the PDE model, 2-D geometry, and mesh. Specify boundary conditions and coefficients. Solve the PDE problem.

model = createpde;
geometryFromEdges(model,@lshapeg);
applyBoundaryCondition(model,'dirichlet','edge',1:model.Geometry.NumEdges,'u',0);
c = 1;
a = 0;
f = 1;
specifyCoefficients(model,'m',0,'d',0,'c',c,'a',a,'f',f);
generateMesh(model);

results = solvepde(model);

Use pdeplot to plot the solution.

u = results.NodalSolution;
pdeplot(model,'XYData',u,'ZData',u,'Mesh','on')
xlabel('x')
ylabel('y')

To get a smoother solution surface, specify the maximum size of the mesh triangles by using the Hmax argument. Then solve the PDE problem using this new mesh, and plot the solution again.

generateMesh(model,'Hmax',0.05);
results = solvepde(model);
u = results.NodalSolution;

pdeplot(model,'XYData',u,'ZData',u,'Mesh','on')
xlabel('x')
ylabel('y')

Interpolate and Plot Solutions and Gradients

Alternatively, you can interpolate the solution and, if needed, its gradient in separate steps, and then plot the results by using MATLAB™ functions, such as surf, mesh, quiver, and so on. For example, solve the same scalar elliptic problem -Δu=1 on the L-shaped membrane with zero Dirichlet boundary conditions. Interpolate the solution and its gradient, and then plot the results.

Create the PDE model, 2-D geometry, and mesh. Specify boundary conditions and coefficients. Solve the PDE problem.

model = createpde;
geometryFromEdges(model,@lshapeg);
applyBoundaryCondition(model,'dirichlet','edge',1:model.Geometry.NumEdges,'u',0);
c = 1;
a = 0;
f = 1;
specifyCoefficients(model,'m',0,'d',0,'c',c,'a',a,'f',f);
generateMesh(model,'Hmax',0.05);
results = solvepde(model);

Interpolate the solution and its gradients to a dense grid from -1 to 1 in each direction.

v = linspace(-1,1,101);
[X,Y] = meshgrid(v);
querypoints = [X(:),Y(:)]';
uintrp = interpolateSolution(results,querypoints);

Plot the resulting solution on a mesh.

uintrp = reshape(uintrp,size(X));
mesh(X,Y,uintrp)
xlabel('x')
ylabel('y')

Interpolate gradients of the solution to the grid from -1 to 1 in each direction. Plot the result using quiver.

[gradx,grady] = evaluateGradient(results,querypoints);
figure
quiver(X(:),Y(:),gradx,grady)
xlabel('x')
ylabel('y')

Zoom in to see more details. For example, restrict the range to [-0.2,0.2] in each direction.

axis([-0.2 0.2 -0.2 0.2])

Plot the solution and the gradients on the same range.

figure
h1 = meshc(X,Y,uintrp);
set(h1,'FaceColor','g','EdgeColor','b')
xlabel('x')
ylabel('y')
alpha(0.5)
hold on

Z = -0.05*ones(size(X));
gradz = zeros(size(gradx));

h2 = quiver3(X(:),Y(:),Z(:),gradx,grady,gradz);
set(h2,'Color','r')
axis([-0.2,0.2,-0.2,0.2])

Slice of the solution plot along the line x = y.

figure
mesh(X,Y,uintrp)
xlabel('x')
ylabel('y')
alpha(0.25)
hold on

z = linspace(0,0.15,101);
Z = meshgrid(z);
surf(X,X,Z')

view([-20 -45 15])
colormap winter

Plot the interpolated solution along the line.

figure
xq = v;
yq = v;
uintrp = interpolateSolution(results,xq,yq);

plot3(xq,yq,uintrp)
grid on
xlabel('x')
ylabel('y')

Interpolate gradients of the solution along the same line and add them to the solution plot.

[gradx,grady] = evaluateGradient(results,xq,yq);

gradx = reshape(gradx,size(xq));
grady = reshape(grady,size(yq));

hold on
quiver(xq,yq,gradx,grady)
view([-20 -45 75])