Speaker
Description
The eigenfrequencies of a vibrating membrane generally depend on its shape. The associated inverse problem was made famous by the work of Kac and has been widely discussed in the literature. We consider this problem in the context of cracks. Is it possible to identify a crack in a membrane if its eigenfrequencies are known?
To answer this question, we first present an analytical approach based on the short-time asymptotic expansion of the heat trace, a well-known spectral invariant. Then, we introduce a data-based method that can be applied in practice. In particular, we train a neural network with simulated data to predict the shape of a crack from the corresponding eigenfrequencies of the domain. The data is computed using isogeometric analysis, a numerical method known for its excellent spectral approximation properties. The underlying eigenfunctions have a singularity at the crack tip, therefore the corresponding eigenvalues cannot be approximated well by standard mesh refinement. We remedy this with a local refinement scheme based on a singular isogeometric mapping and illustrate optimal convergence orders for the eigenfunctions and eigenvalues.
