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.
lshapec file is an extension of an ordinary model file
created using the PDE Modeler app. 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
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 Partial Differential Equation Toolbox™ software is a part of
the open MATLAB environment.
lshapec you can create L-shaped rounded
geometries with different degrees of roundness. If you use
an input argument, a default radius of 0.5 is used. Otherwise, use
a is the radius of the circle.
Experimenting using different values for the radius
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 following figure, 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