Generalized Eigenvalue stabilization for immersed explicit dynamics

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 C0 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 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.

Subjects

Explicit dynamics

Finite cell method

Generalized eigenvalue stabilization

Immersed boundary method

Spectral cell method

Spectral element method

Wave equation

DDC Class

518: Numerical Analysis

531: Classical Mechanics

620: Engineering

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https://creativecommons.org/licenses/by/4.0/

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publishedVersion

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1-s2.0-S0045782526000010-main-1.pdf

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Main Article

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3.04 MB

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Generalized Eigenvalue stabilization for immersed explicit dynamics