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

Examples of Eigenvalue Problems

Eigenvalues and Eigenmodes of a Square