Generalized eigenvalue stabilization for immersed explicit dynamics

Abstract

Explicit time integration for immersed finite element discretizations severely suffers from the influence of poorly cut elements. In this contribution, we propose a generalized eigenvalue stabilization (GEVS) strategy for the element mass matrices of cut elements to cure their adverse impact on the critical time step size of the global system. We use spectral basis functions, specifically $C^0$ continuous Lagrangian interpolation polynomials defined on Gauss-Lobatto-Legendre (GLL) points, which, in combination with its associated GLL quadrature rule, yield high-order convergent diagonal mass matrices for uncut elements. Moreover, considering cut elements, we combine the proposed GEVS approach with the finite cell method (FCM) to guarantee definiteness of the system matrices. However, the proposed GEVS stabilization can directly be applied to other immersed boundary finite element methods. Numerical experiments demonstrate that the stabilization strategy achieves optimal convergence rates and recovers critical time step sizes of equivalent boundary-conforming discretizations. This also holds in the presence of weakly enforced Dirichlet boundary conditions using either Nitsche's method or penalty formulations.