New preprint on isogeometric analysis of eigenvalue problems

27 Februar 2024

In a new preprint, we investigate the solution of the Laplace eigenvalue equation on circular sectors using isogeometric analysis. The problem we address describes the vibrations of a membrane stretched over a circular sector. The corresponding eigenfunctions have characteristic corner singularities, causing standard simulation methods to produce suboptimal results.

We propose a numerical algorithm to resolve these singularities via a graded mesh refinement scheme for isogeometric analysis. Numerical tests show optimal convergence rates for the eigenvalues and eigenfunctions as well as a convincing approximation constant of the smooth splines. Furthermore, graded meshes prove to be particularly advantageous for a precise approximation of all low natural frequencies in only one simulation. One difficulty of the applied algorithm lies in the singularity of the isogeometric parameterization, but the approach proves to be robust. Finally, we present a hierarchical mesh structure that has some useful properties and leads to slightly improved approximation results.

Apel, T., Zilk, P.: Isogeometric analysis of the Laplace eigenvalue problem on circular sectors: Regularity properties, graded meshes & variational crimes, Preprint, submitted for publication, https://arxiv.org/abs/2402.16589