Eigenvalues and Eigenmodes of L-Shaped Membrane: PDE Modeler App

This example shows how to compute all eigenmodes with eigenvalues smaller than 100 for the eigenmode PDE problem

–Δu = λu

on the geometry of the L-shaped membrane. The boundary condition is the Dirichlet condition u = 0. This example uses the PDE Modeler app. For a programmatic workflow, see Eigenvalues and Eigenmodes of L-Shaped Membrane.

To solve this problem in the PDE Modeler app, follow these steps:

1. Draw a polygon with the corners (0,0), (–1,0), (–1,–1), (1,–1), (1,1), and (0,1) by using the pdepoly function.

   pdepoly([0,-1,-1,1,1,0],[0,0,-1,-1,1,1])

2. Check that the application mode is set to Generic Scalar by selecting Options > Application.

3. Use the default Dirichlet boundary condition u = 0 for all boundaries. To verify it, switch to boundary mode by selecting Boundary > Boundary Mode. Use Edit > Select all to select all boundaries. Select Boundary > Specify Boundary Conditions and verify that the boundary condition is the Dirichlet condition with h = 1, r = 0.

4. Specify the coefficients by selecting PDE > PDE Specification or clicking the button on the toolbar. This is an eigenvalue problem, so select the Eigenmodes type of PDE. The general eigenvalue PDE is described by $$-\nabla \cdot \left(c\nabla u\right)+au=\lambda du$$. Thus, for this problem, use the default coefficients c = 1, a = 0, and d = 1.

5. Specify the maximum edge size for the mesh by selecting Mesh > Parameters. Set the maximum edge size value to 0.05.

6. Initialize the mesh by selecting Mesh > Initialize Mesh.

7. Specify the eigenvalue range by selecting Solve > Parameters. In the resulting dialog box, use the default eigenvalue range [0 100].

8. Solve the PDE by selecting Solve > Solve PDE or clicking the button on the toolbar. By default, the app plots the first eigenfunction.

   

9. Plot other eigenfunctions by selecting Plot > Parameters and then selecting the corresponding eigenvalue from the drop-down list at the bottom of the dialog box. For example, plot the 12th eigenfunction in the specified range.