Examples of Eigenvalue Problems
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Examples of Eigenvalue Problems
This section describes the solution of some eigenvalue PDE problems. The problems are solved using the graphical user interface and the command-line functions of the PDE Toolbox.
Eigenvalues and Eigenfunctions for the L-Shaped
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Examples2-28
Finally, solve the L-shaped membrane problem by pressing the = button. The solution displayed is the first eigenfunction. The value of the first (smallest) eigenvalue is also displayed. You find the number of eigenvalues on the information line at the bottom of the GUI. You can open the Plot Selection dialog box and choose which eigenfunction to plot by selecting from a pop-up menu of the corresponding eigenvalues.
Using Command-Line Functions
The geometry of the L-shaped membrane is described in the file lshapeg.m and the boundary conditions in the file lshapeb.m.
First, initialize the mesh and refine it twice using the command line functions at the MATLAB prompt:
» [p,e,t]=initmesh('lshapeg');
» [p,e,t]=refinemesh('lshapeg',p,e,t);
» [p,e,t]=refinemesh('lshapeg',p,e,t);
Recall the general eigenvalue PDE problem description:
This means that in this case you have c = 1, a = 0, and d = 1. The syntax of pdeeig, the eigenvalue solver in the PDE Toolbox, is:
[v,l]=pdeeig(b,p,e,t,c,a,d,r)
The input argument r is a two-element vector indicating the interval on the real axis where pdeeig searches for eigenvalues. Here you are looking for eigenvalues < 100, so the interval you use is [0 100].
Now you can call pdeeig and see how many eigenvalues you find:
» [v,l]=pdeeig('lshapeb',p,e,t,1,0,1,[0 100]);
There are 19 eigenvalues smaller than 100. Plot the first eigenmode and compare it to MATLAB’s membrane function:
» pdesurf(p,t,v(:,1))
» figure
» membrane(1,20,9,9)
∇
– ⋅(c∇u)+au = λdu
Examples of Eigenvalue Problems
2-29 membrane can produce the first 12 eigenfunctions for the L-shaped membrane.
Compare also the 12th eigenmodes:
» figure
» pdesurf(p,t,v(:,12))
» figure
» membrane(12,20,9,9)
Looking at the following eigenmodes, you can see how the number of oscillations increases. The eigenfunctions are symmetric or antisymmetric around the diagonal from (0,0) to (1,-1), which divides the L-shaped membrane into two mirror images. In a practical computation, you could take advantage of such symmetries in the PDE problem, and solve over a region half the size.
The eigenvalues of the full L-shaped membrane are the union of those of the half with Dirichlet boundary condition along the diagonal (eigenvalues 2, 4, 7, 11, 13, 16, and 17) and those with Neumann boundary condition (eigenvalues 1, 3, 5, 6, 10, 12, 14, and 15).
The eigenvalues λ8 and λ9 make up a double eigenvalue for the PDE at around 49.64. Also, the eigenvalues λ18 and λ19 make up another double eigenvalue at around 99.87. You may have gotten two different but close values. The default triangulation made by initmesh is not symmetric around the diagonal, but a symmetric grid gives a matrix with a true double eigenvalue. Each of the eigenfunctions u8 and u9 consists of three copies of eigenfunctions over the unit square, corresponding to its double second eigenvalue. You may not have obtained the zero values along a diagonal of the square — any line through the center of the square may have been computed. This shows a general fact about multiple eigenvalues for symmetric matrices; namely that any vector in the invariant subspace is equally valid as an eigenvector. The two eigenfunctions u8 and u9 are orthogonal to each other if the dividing lines make right angles.
Check your solutions for that.
Actually, the eigenvalues of the square can be computed exactly. They are (m2+n2)π2, e.g., the double eigenvalue λ18 and λ19 is 10π2, which is pretty close to 100. If you compute the FEM approximation with only one refinement, you would only find 16 eigenvalues, and you obtain the wrong solution to the original problem. You can of course check for this situation by computing the eigenvalues over a slightly larger range than the original problem.
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Examples2-30
You get some information from the printout in the MATLAB command window that is printed during the computation. For this problem, the algorithm computed a new set of eigenvalue approximations and tested for convergence every third step. In the output, you get the step number, the time in seconds since the start of the eigenvalue computation, and the number of converged eigenvalues with eigenvalues both inside and outside the interval counted.
Here is what MATLAB wrote:
Basis= 10, Time= 2.70, New conv eig= 0 Basis= 13, Time= 3.50, New conv eig= 0 Basis= 16, Time= 4.36, New conv eig= 0 Basis= 19, Time= 5.34, New conv eig= 1 Basis= 22, Time= 6.46, New conv eig= 2 Basis= 25, Time= 7.61, New conv eig= 3 Basis= 28, Time= 8.86, New conv eig= 3 Basis= 31, Time= 10.23, New conv eig= 5 Basis= 34, Time= 11.69, New conv eig= 5 Basis= 37, Time= 13.28, New conv eig= 7 Basis= 40, Time= 14.97, New conv eig= 8 Basis= 43, Time= 16.77, New conv eig= 9 Basis= 46, Time= 18.70, New conv eig= 11 Basis= 49, Time= 20.73, New conv eig= 11 Basis= 52, Time= 22.90, New conv eig= 13 Basis= 55, Time= 25.13, New conv eig= 14 Basis= 58, Time= 27.58, New conv eig= 14 Basis= 61, Time= 30.13, New conv eig= 15 Basis= 64, Time= 32.83, New conv eig= 16 Basis= 67, Time= 35.64, New conv eig= 18 Basis= 70, Time= 38.62, New conv eig= 22 End of sweep: Basis= 70, Time= 38.62, New conv eig= 22
Basis= 32, Time= 43.29, New conv eig= 0 Basis= 35, Time= 44.70, New conv eig= 0 Basis= 38, Time= 46.22, New conv eig= 0 Basis= 41, Time= 47.81, New conv eig= 0 Basis= 44, Time= 49.52, New conv eig= 0 Basis= 47, Time= 51.35, New conv eig= 0 Basis= 50, Time= 53.27, New conv eig= 0 Basis= 53, Time= 55.30, New conv eig= 0 End of sweep: Basis= 53, Time= 55.30, New conv eig= 0
Examples of Eigenvalue Problems
2-31 You can see that two Arnoldi runs were made. In the first, 22 eigenvalues
converged after a basis of size 70 was computed; in the second, where the vectors were orthogonalized against all the 22 converged vectors, the smallest eigenvalue stabilized at a value outside of the interval [0, 100], so the algorithm signaled convergence. Of the 22 converged eigenvalues, 19 were inside the search interval.
L-Shaped Membrane with Rounded Corner
An extension of this problem is to compute the eigenvalues for an L-shaped membrane where the inner corner at the “knee” is rounded. The roundness is created by adding a circle so that the circle’s arc is a part of the L-shaped membrane’s boundary. By varying the circle’s radius, the degree of roundness can be controlled. The M-file lshapec is an extension of an ordinary model file created using pdetool. It contains the lines
pdepoly([–1, 1, 1, 0, 0, –1],...
[–1, –1, 1, 1, 0, 0],'P1');
pdecirc(–a,a,a,'C1');
pderect([–a 0 a 0],'SQ1');
The extra circle and rectangle that are added using pdecirc and pderect to create the rounded corner are affected by the added input argument a through a couple of extra lines of MATLAB code. This is possible since the PDE Toolbox is a part of the open MATLAB environment.
With lshapec you can create L-shaped rounded geometries with different degrees of roundness. If you use lshapec without an input argument, a default radius of 0.5 is used. Otherwise, use lshapec(a), where a is the radius of the circle.
Experimenting using different values for the radius a, shows you that the eigenvalues and the frequencies of the corresponding eigenmodes decrease as the radius increases, and the shape of the L-shaped membrane becomes more rounded. In the figure below, the first eigenmode of an L-shaped membrane with a rounded corner is plotted.
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Examples2-32