Taming Slivers: A Robust TFEM Framework for Reliable Computations on Degenerate Tetrahedral Meshes

Abstract

Sliver elements are an intrinsic difficulty of three-dimensional tetrahedral mesh generation and remain costly, and sometimes impractical, to eliminate completely. Although isolated degenerate elements do not necessarily prevent finite element convergence, connected clusters or sheets of slivers may impose artificial constraints on the discrete solution, leading to locking and severe loss of accuracy. In this work, we revisit the effect of slivers from the viewpoint of the finite element solution and propose a robust solver-side treatment based on the Tempered Finite Element Method (TFEM). The method limits the singular contribution of degenerate elements by introducing a lower bound on the Jacobian determinant, which can be interpreted as a vanishing added-volume correction. The resulting formulation prevents the effective element volume from falling below a threshold while preserving the relevant physical modes of the solution. We analyze the stiffness matrices of degenerate tetrahedra, identify the mechanisms responsible for locking in sliver bands, and assess the method on a range of representative physical problems, including incompressible flow, Cahn--Hilliard phase-field dynamics, transient wave propagation, and vibro-acoustic fluid--structure interaction. The numerical results show that TFEM consistently recovers accurate and physically meaningful solutions on meshes for which standard FEM exhibits locking or loss of convergence, providing a simple and broadly applicable alternative to exhaustive geometric sliver removal.