Eigenvalues and Eigenmodes of L-Shaped Membrane
This example shows how to calculate eigenvalues and eigenvectors. The eigenvalue problem is . This example computes all eigenmodes with eigenvalues smaller than 100.
Create a model and include this geometry. The geometry of the L-shaped membrane is described in the file lshapeg.
model = createpde(); geometryFromEdges(model,@lshapeg);
Set zero Dirichlet boundary conditions on all edges.
applyBoundaryCondition(model,'dirichlet','Edge',1:model.Geometry.NumEdges,'u',0);
Specify the coefficients for the problem: d = 1 and c = 1. All other coefficients are equal to zero.
specifyCoefficients(model,'m',0,'d',1,'c',1,'a',0,'f',0);
Set the interval [0 100] as the region for the eigenvalues in the solution.
r = [0 100];
Create a mesh and solve the problem.
generateMesh(model,'Hmax',0.05);
results = solvepdeeig(model,r); Basis= 10, Time= 0.44, New conv eig= 0
Basis= 11, Time= 0.50, New conv eig= 0
Basis= 12, Time= 0.56, New conv eig= 0
Basis= 13, Time= 0.60, New conv eig= 0
Basis= 14, Time= 0.61, New conv eig= 0
Basis= 15, Time= 0.64, New conv eig= 0
Basis= 16, Time= 0.66, New conv eig= 0
Basis= 17, Time= 0.68, New conv eig= 0
Basis= 18, Time= 0.71, New conv eig= 1
Basis= 19, Time= 0.73, New conv eig= 1
Basis= 20, Time= 0.78, New conv eig= 1
Basis= 21, Time= 0.83, New conv eig= 1
Basis= 22, Time= 0.89, New conv eig= 3
Basis= 23, Time= 1.06, New conv eig= 3
Basis= 24, Time= 1.15, New conv eig= 4
Basis= 25, Time= 1.26, New conv eig= 5
Basis= 26, Time= 1.30, New conv eig= 6
Basis= 27, Time= 1.37, New conv eig= 6
Basis= 28, Time= 1.44, New conv eig= 6
Basis= 29, Time= 1.53, New conv eig= 7
Basis= 30, Time= 1.75, New conv eig= 7
Basis= 31, Time= 1.88, New conv eig= 10
Basis= 32, Time= 1.98, New conv eig= 10
Basis= 33, Time= 2.07, New conv eig= 11
Basis= 34, Time= 2.14, New conv eig= 11
Basis= 35, Time= 2.22, New conv eig= 14
Basis= 36, Time= 2.33, New conv eig= 14
Basis= 37, Time= 2.53, New conv eig= 14
Basis= 38, Time= 2.66, New conv eig= 14
Basis= 39, Time= 2.78, New conv eig= 14
Basis= 40, Time= 2.85, New conv eig= 14
Basis= 41, Time= 2.91, New conv eig= 15
Basis= 42, Time= 2.97, New conv eig= 15
Basis= 43, Time= 3.06, New conv eig= 15
Basis= 44, Time= 3.12, New conv eig= 16
Basis= 45, Time= 3.17, New conv eig= 16
Basis= 46, Time= 3.23, New conv eig= 16
Basis= 47, Time= 3.29, New conv eig= 16
Basis= 48, Time= 3.35, New conv eig= 17
Basis= 49, Time= 3.47, New conv eig= 18
Basis= 50, Time= 3.63, New conv eig= 18
Basis= 51, Time= 3.77, New conv eig= 18
Basis= 52, Time= 3.89, New conv eig= 18
Basis= 53, Time= 4.02, New conv eig= 18
Basis= 54, Time= 4.25, New conv eig= 21
End of sweep: Basis= 54, Time= 4.26, New conv eig= 21
Basis= 31, Time= 4.85, New conv eig= 0
Basis= 32, Time= 4.95, New conv eig= 0
Basis= 33, Time= 5.05, New conv eig= 0
End of sweep: Basis= 33, Time= 5.05, New conv eig= 0
There are 19 eigenvalues smaller than 100.
length(results.Eigenvalues)
ans = 19
Plot the first eigenmode and compare it to the MATLAB's membrane function.
u = results.Eigenvectors; pdeplot(model,'XYData',u(:,1),'ZData',u(:,1));
figure membrane(1,20,9,9)
Eigenvectors can be multiplied by any scalar and remain eigenvectors. This explains the difference in scale that you see.
membrane can produce the first 12 eigenfunctions for the L-shaped membrane. Compare the 12th eigenmodes.
figure pdeplot(model,'XYData',u(:,12),'ZData',u(:,12));
figure membrane(12,20,9,9)
