Generalized eigenvalue stabilization for immersed explicit dynamics

2025-09-09

AI Summary

In immersed finite element methods, poorly cut elements severely reduce the critical time step for explicit time integration, drastically impairing dynamic simulation efficiency. To address this, we propose a generalized eigenvalue stabilization technique: (i) the mass matrix of cut elements is reconstructed to ensure positive definiteness; (ii) a lumped, diagonal mass matrix is constructed using $C^0$ spectral basis functions on Gauss–Lobatto–Legendre (GLL) nodes combined with GLL quadrature; and (iii) the finite cell method (FCM) is integrated for robust geometric cutting treatment, while boundary conditions are enforced via Nitsche’s method or penalty formulation. This approach elevates the critical time step to levels comparable to body-fitted meshes, restores optimal high-order convergence rates, and maintains high accuracy and stability even under complex cutting configurations. The method is broadly applicable across diverse immersed boundary frameworks.

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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.

Problem

• Stabilizing cut elements in immersed explicit dynamics
• Improving critical time step size via eigenvalue stabilization
• Ensuring system matrix definiteness with spectral basis functions

Innovation

• Generalized eigenvalue stabilization for cut elements
• Spectral basis functions with GLL quadrature
• Combined with finite cell method for definiteness