Eigenvalues and Eigenmodes of L-Shaped Membrane

This example shows how to calculate eigenvalues and eigenvectors. The eigenvalue problem is $$-\Delta u=\lambda u$$. 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=   0.13,  New conv eig=  0
              Basis= 11,  Time=   0.15,  New conv eig=  0
              Basis= 12,  Time=   0.16,  New conv eig=  0
              Basis= 13,  Time=   0.16,  New conv eig=  0
              Basis= 14,  Time=   0.16,  New conv eig=  0
              Basis= 15,  Time=   0.16,  New conv eig=  0
              Basis= 16,  Time=   0.16,  New conv eig=  0
              Basis= 17,  Time=   0.16,  New conv eig=  0
              Basis= 18,  Time=   0.17,  New conv eig=  1
              Basis= 19,  Time=   0.17,  New conv eig=  1
              Basis= 20,  Time=   0.17,  New conv eig=  1
              Basis= 21,  Time=   0.17,  New conv eig=  1
              Basis= 22,  Time=   0.17,  New conv eig=  2
              Basis= 23,  Time=   0.17,  New conv eig=  3
              Basis= 24,  Time=   0.18,  New conv eig=  3
              Basis= 25,  Time=   0.18,  New conv eig=  3
              Basis= 26,  Time=   0.18,  New conv eig=  3
              Basis= 27,  Time=   0.18,  New conv eig=  3
              Basis= 28,  Time=   0.18,  New conv eig=  4
              Basis= 29,  Time=   0.18,  New conv eig=  4
              Basis= 30,  Time=   0.19,  New conv eig=  5
              Basis= 31,  Time=   0.19,  New conv eig=  5
              Basis= 32,  Time=   0.19,  New conv eig=  5
              Basis= 33,  Time=   0.19,  New conv eig=  5
              Basis= 34,  Time=   0.19,  New conv eig=  5
              Basis= 35,  Time=   0.19,  New conv eig=  5
              Basis= 36,  Time=   0.19,  New conv eig=  5
              Basis= 37,  Time=   0.19,  New conv eig=  7
              Basis= 38,  Time=   0.20,  New conv eig=  7
              Basis= 39,  Time=   0.21,  New conv eig=  7
              Basis= 40,  Time=   0.21,  New conv eig=  7
              Basis= 41,  Time=   0.21,  New conv eig=  7
              Basis= 42,  Time=   0.21,  New conv eig=  8
              Basis= 43,  Time=   0.21,  New conv eig=  8
              Basis= 44,  Time=   0.21,  New conv eig=  8
              Basis= 45,  Time=   0.22,  New conv eig=  8
              Basis= 46,  Time=   0.22,  New conv eig= 11
              Basis= 47,  Time=   0.22,  New conv eig= 11
              Basis= 48,  Time=   0.22,  New conv eig= 11
              Basis= 49,  Time=   0.23,  New conv eig= 11
              Basis= 50,  Time=   0.23,  New conv eig= 12
              Basis= 51,  Time=   0.23,  New conv eig= 13
              Basis= 52,  Time=   0.23,  New conv eig= 14
              Basis= 53,  Time=   0.23,  New conv eig= 14
              Basis= 54,  Time=   0.23,  New conv eig= 14
              Basis= 55,  Time=   0.24,  New conv eig= 14
              Basis= 56,  Time=   0.25,  New conv eig= 14
              Basis= 57,  Time=   0.25,  New conv eig= 14
              Basis= 58,  Time=   0.26,  New conv eig= 14
              Basis= 59,  Time=   0.26,  New conv eig= 15
              Basis= 60,  Time=   0.26,  New conv eig= 15
              Basis= 61,  Time=   0.27,  New conv eig= 15
              Basis= 62,  Time=   0.27,  New conv eig= 16
              Basis= 63,  Time=   0.28,  New conv eig= 16
              Basis= 64,  Time=   0.28,  New conv eig= 16
              Basis= 65,  Time=   0.28,  New conv eig= 16
              Basis= 66,  Time=   0.29,  New conv eig= 18
              Basis= 67,  Time=   0.29,  New conv eig= 18
              Basis= 68,  Time=   0.29,  New conv eig= 18
              Basis= 69,  Time=   0.89,  New conv eig= 18
              Basis= 70,  Time=   0.92,  New conv eig= 19
              Basis= 71,  Time=   0.94,  New conv eig= 20
              Basis= 72,  Time=   0.96,  New conv eig= 21
              Basis= 73,  Time=   0.99,  New conv eig= 21
              Basis= 74,  Time=   1.03,  New conv eig= 22
End of sweep: Basis= 74,  Time=   1.04,  New conv eig= 22
              Basis= 32,  Time=   1.09,  New conv eig=  0
              Basis= 33,  Time=   1.23,  New conv eig=  0
              Basis= 34,  Time=   1.26,  New conv eig=  0
              Basis= 35,  Time=   1.27,  New conv eig=  0
              Basis= 36,  Time=   1.29,  New conv eig=  0
              Basis= 37,  Time=   1.30,  New conv eig=  0
              Basis= 38,  Time=   1.32,  New conv eig=  0
              Basis= 39,  Time=   1.33,  New conv eig=  0
              Basis= 40,  Time=   1.34,  New conv eig=  0
              Basis= 41,  Time=   1.36,  New conv eig=  0
              Basis= 42,  Time=   1.37,  New conv eig=  0
              Basis= 43,  Time=   1.38,  New conv eig=  0
              Basis= 44,  Time=   1.40,  New conv eig=  0
End of sweep: Basis= 44,  Time=   1.41,  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)

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