An EigenValue Stabilization Technique for Immersed Boundary Finite Element Methods in Explicit Dynamics

TL;DR

This paper tackles the small cut-element problem in immersed boundary methods for explicit dynamics by introducing an eigenvalue stabilization (EVS) technique that operates at the element level. The method stabilizes near-zero eigenmodes of the cut-element mass and, optionally, stiffness matrices, with a scaling approach to ensure unit-independence and robust performance. In explicit dynamics, stabilizing only the mass matrix (with HRZ-lumped mass) significantly increases the critical time step $Δt_ ext{cr}$ while preserving accuracy, and avoiding unnecessary stabilization of the stiffness matrix. Across wave-propagation benchmarks, EVS yields substantial time-step gains, reduces ill-conditioning, and maintains comparable accuracy to reference SEM solutions, establishing a practical pathway to efficient high-frequency immersed boundary simulations. The work also discusses variant choices, parameter recommendations ($ε_S\approx10^{-3}$, $ε_λ\approx10^{-3}$), and future directions such as force-correction techniques for dynamic problems.

Abstract

The application of immersed boundary methods in static analyses is often impeded by poorly cut elements (small cut elements problem), leading to ill-conditioned linear systems of equations and stability problems. While these concerns may not be paramount in explicit dynamics, a substantial reduction in the critical time step size based on the smallest volume fraction $χ$ of a cut element is observed. This reduction can be so drastic that it renders explicit time integration schemes impractical. To tackle this challenge, we propose the use of a dedicated eigenvalue stabilization (EVS) technique. The EVS-technique serves a dual purpose. Beyond merely improving the condition number of system matrices, it plays a pivotal role in extending the critical time increment, effectively broadening the stability region in explicit dynamics. As a result, our approach enables robust and efficient analyses of high-frequency transient problems using immersed boundary methods. A key advantage of the stabilization method lies in the fact that only element-level operations are required. This is accomplished by computing all eigenvalues of the element matrices and subsequently introducing a stabilization term that mitigates the adverse effects of cutting. Notably, the stabilization of the mass matrix $\mathbf{M}_\mathrm{c}$ of cut elements -- especially for high polynomial orders $p$ of the shape functions -- leads to a significant raise in the critical time step size $Δt_\mathrm{cr}$. To demonstrate the efficacy of our technique, we present two specifically selected dynamic benchmark examples related to wave propagation analysis, where an explicit time integration scheme must be employed to leverage the increase in the critical time step size.

Figures (23)

• Figure 1: Fundamental idea of immersed boundary methods (Cartesian mesh) in comparison to a typical finite element discretization (body-fitted mesh).
• Figure 2: Construction of the element-level integration grid based on the quadtree decomposition of a cut element. Three different refinement levels $k = 1,3,5$ are depicted. Dark gray subcells indicate that the integration domain belongs to the physical domain, while a light grey color refers to the fictitious domain, and cut integration subcells are marked in yellow.
• Figure 3: Model of a rectangular plate intersected by a circular domain boundary (Discretization: $1\times1$ finite elements, Subdivision depth: $k\,{=}\,8$; Red solid line: physical boundary of the circular hole). Color coding for the numerical integration -- Dark gray: integration domains located in the physical domain; Light gray: integration domains located in the fictitious domain; Yellow: cut integration domains (leaf cells of the tree data structure).
• Figure 4: Normalized critical time step size $\Delta t^\mathrm{norm}_\mathrm{cr}$ for variant 2b considering different values of $\epsilon_\mathrm{S}$ and $\epsilon_\lambda$.
• Figure 5: Model of a rectangular plate with a circular hole (Discretization: $2\times2$ finite element, Subdivision depth: $k\,{=}\,4$; Red solid line: physical boundary of the circular hole). Color coding for the numerical integration -- White: conventional finite elements; Dark gray: integration domains located in the physical domain; Light gray: integration domains located in the fictitious domain; Yellow: cut integration domains (leaf cells).