A CFL condition for the finite cell method

Abstract

Immersed boundary finite element methods allow the user to bypass the potentially troublesome task of boundary-conforming mesh generation. When combined with explicit time integration, poorly cut elements with little support in the physical domain lead to a severely reduced critical time step size, posing a major challenge for immersed wave propagation simulations. The finite cell method stabilizes cut elements by defining the weak form of the problem also in the fictitious domain, but scaled by a small value $α$. This paper investigates the effect of the finite cell method on the critical time step size for explicit time integration. Starting with an analytical one-degree-of-freedom model, we systematically study the influence of $α$-stabilization on the maximum eigenvalue, and thus on the critical time step size, for corner and sliver cuts. The analysis is complemented by a numerical study of an example with one element and increasing polynomial degree, confirming that the critical time step size does not decrease below a certain limit, even as the cut fraction tends to zero. This lower bound is controlled by the choice of $α$. In higher dimensions, sliver cuts are found to be more detrimental than corner cuts, thus determining the minimum critical time step size. Increasing the polynomial degree has only little effect on this degradation. Based on these observations, we derive an estimate of the minimum critical time step size as a function of $α$, which we use to propose a modified CFL condition for the finite cell method. The validity of this condition is demonstrated on a two-dimensional perforated plate example.