Eigenvalues of Laplacian on parametric surface

[IgorMele]

Hello.

I would like to numerically determine eigenvalues of a rectangular membrane
which is twisted for $$\frac{\pi}{2}$$. Example picture:

I'm solving Helmholtz equation:
$$\nabla^2u+k^2u=0$$
where $$u=u(x,y)$$ and
$$\nabla^2 u=\frac{\partial^2u}{\partial x^2}+\frac{\partial^2v}{\partial y^2}$$
If we write in matrix form:
$$-Au=k^2u=\lambda u$$
$$A$$ is sparse matrix, which we get from second order finite difference approximation, $$\lambda$$ are eigenvalues. Analytical solution for rectangular membrane $$a\times b$$ is:
$$\lambda = h^2\pi^2\left(\frac{m^2}{a^2}+\frac{n^2}{b^2}\right)$$
$$h$$ is distance between two points in a grid.
Here is solution for second mode:

This was for normal membrane, now question is, what happens with eigenvalues when we twist membrane. Shape of membrane in parametric form:
$$x=u$$

$$y=v$$

$$z=uv$$

$$u,v\in [-1,1]$$

or perhaps:
$$x=u$$

$$y=v$$

$$z=(u-1)(v-1)$$

$$u,v\in [0,2]$$
In this way, we don't have negative $$u$$ and $$v$$. In this case parametrization applies only for quadratic membrane, in equations should be also $$a$$ and $$b$$, but it's not so relevant. I will add them later.

Here I'm stuck. I don't know, how to write Helmholtz equation for this parametrization.
Any tip or help would be appreciated.

 

[jvicens]

Hello,

Thank you for your question. To numerically determine the eigenvalues for a twisted rectangular membrane, we can use the same approach as for a normal rectangular membrane. However, we need to modify the Helmholtz equation to account for the twisting of the membrane.

One way to do this is to use a change of variables, where we define a new coordinate system (u,v) in terms of the original (x,y) coordinates. In this new coordinate system, the Helmholtz equation becomes:

\nabla^2u+k^2u=0

where
\nabla^2 u=\frac{\partial^2u}{\partial u^2}+\frac{\partial^2u}{\partial v^2}+\frac{\partial^2u}{\partial u \partial v}

We can then use a finite difference approximation to discretize this equation and solve for the eigenvalues. The exact form of the finite difference approximation will depend on the specific parametrization of the twisted membrane that you are using.

Another approach would be to use a spectral method, where we represent the solution as a sum of basis functions and use the Helmholtz equation to determine the coefficients of these basis functions. This can be more accurate and efficient for problems with smooth solutions, but may require more computational resources.

I hope this helps. Good luck with your research!