Partial Differential Equation Toolbox |

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.

First Eigenmode for an L-Shaped Membrane with a Rounded Corner

| Eigenvalue Problems |  | Eigenvalues and Eigenmodes of a Square |