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L-Shaped Membrane with a 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 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'); 
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 Partial Differential Equation Toolbox™ software 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 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