A CFL Condition for the Finite Cell Method
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Immersed boundary finite element methods simplify the modeling process by circumventing boundary-conforming mesh generation. However, they face challenges due to cut elements, i.e. elements that intersect the domain’s boundaries. In wave simulation, they are typically combined with explicit time integration, where poorly cut elements drastically restrict the critical time step size. Various stabilization techniques and other remedies have been proposed to mitigate these difficulties. In this presentation, we discuss the impact of material stabilization, as introduced by the finite cell method, on explicit time integration. Using the analytical solution of an example with one degree of freedom and arbitrary dimension, we systematically investigate how material stabilization affects the maximum eigenvalue and, consequently, the critical time step size. The analysis is augmented by a numerical investigation of an example with a single element with increasing polymial degree. Our results show that the finite cell method enforces a lower bound on the critical time step size, that the severity of the critical time step size decreases in higher dimensions, and that increasing the polynomial degree has minimal effect on the minimum critical time step size. Finally, we provide a modified CFL condition to accurately estimate the critical time step size for systems discretized with the finite cell method and demonstrate its validaty on a perforated plate example.