Eigenmodes of the biharmonic operator on the unit disk (vibrating plates)

[Kenny Pang]

Hi Chebfun users,

I am learning about the Chebyshev methods and have really enjoyed it so far. However, I am having some difficulties finding the eigenmodes of the biharmonic operator on the unit disk with clamped boundary condition at r=1. I have tried a few methods but the results are not satisfactory. I am hoping your expertise could help me with this.

I tried separation of variables and use the chebfun package to solve the radial equation, but the package fails to find a solution.

```

% biharmonic operator

k = 2; % wave number

A = chebop(0,1); % domain is [0,1]

A.op = @(r,u) r^4*diff(u,4) + 2*r^3*diff(u,3) - (2*k^2+1)*r^2*diff(u,2) + (2*k^2+1)*r*diff(u) + (k^4-4*k^2)*u;

A.lbc = @(u) [u; diff(u,1)]; A.rbc = [0; 0];

% right hand side

B = chebop(0,1); B.op = @(r,u) r^4*u;

% solve generalised eigenvalue problem

[V,D] = eigs(A,B);

plot(V, '-o')

```

I then try the same separation of variables but using the Chebyshev differentiation matricies following Trefethen (2000, Chapter 14) to construct the biharmonic operator. It is able to find the correct eigenvalues for wavenumber k=0,2, but got slightly wrong eigenvalues for other wavenumber k=1,3,4,5,... when compared to Lindsay et.al. I attached my code below.

Finally, I implemented a 2D solver following Trefethen (2000, Chapter 11) and got similar results where the eigenvalues are inaccurate for the same k=1,3,... Attached is a plot of the eigenmodes and corresponding eigenvalues. (I'm not sure why the first two modes with negative eigenvalues are there.) For example, u_{1,1} mode (subplot 4 and 5) has a frequency of 426.91 but Lindsay et.al. got 452.0. My codes for the 2D solver are on my GitHub.

Any feedback, insight, or discussion would be greatly appreciated.

- Lloyd N. Trefethen, Spectral Methods in MATLAB, SIAM, Philadelphia, 2000.

- Lindsay AE, Hao W, Sommese AJ. 2015 Vibrations of thin plates with small clamped patches. Proc. R. Soc. A 471: 20150474.

[Ian Ajzenszmidt]

The following url may show resources that may assist you in eigenmodes of the biharmonic operator on the unit disk (vibrating plates)  https://math.stackexchange.com/questions/1447956/the-biharmonic-eigenvalue-problem-on-a-rectangle-with-dirichlet-boundary-conditi/1470314#1470314

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[Kenny Pang]

(I am not sure why the system keeps deleting my reply)

Thank you for your reply Iajze. I am aware of the discussion surrounding the vibrating rectangular plates, but I still think it can be helpful to lay out my readings here.

In the StackExchange discussion, they mentioned that the solution to the biharmonic EV problem is not separable, but according to Everitt (2009), the solution is separable for k=2. This kind of explains why the 1D solver works for k=0,2 but not the rest. However, I would expect the 2D solver to be able to find the correct eigenvalues for the rest of the modes.

Another common method in the literature for solving the biharmonic EV problem is to factorise it into two Laplace EV problems. Unfortunately, this would not work for the problem I am trying to solve which is a thin-film equation https://en.wikipedia.org/wiki/Thin-film_equation.

So, to summarise my questions:

1. Is it possible to use the chebfun (or diskfun) package to solve the biharmonic EV problem, and more generally, any fourth-order linear eigenvalue problem?

2. If it is not possible to use an out-of-the-box solution, then I will have to implement my own. Am I missing something causing the 2D solver to not find the correct eigenvalues?

- W. N. Everitt, B. T. Johansson, L. L. Littlejohn, C. Markett, Quasi-separation of the biharmonic partial differential equation, IMA Journal of Applied Mathematics, Volume 74, Issue 5, October 2009, Pages 685–709.

[Sergio Manzetti]

Hello , I have a question regarding resolvent norm calculations by hand, and I have posted in on Math Stackexchange, since it is difficult to include symbols here.

The question is here:

https://math.stackexchange.com/questions/4652612/resolvent-norm-for-an-inhomogeneous-equation

Please let me know if you have some ideas regarding this!

Thanks

Sergio

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