2024/11/26 by Antoine Quiriny, Quiriny, Antoine, Václav Kučera +7 · 1 citation
Engineering · #FOS: Mathematics #Metal Forming Simulation Techniques #Numerical Analysis (math.NA) #Numerical methods in engineering #Soil, Finite Element Methods
paper · pdf · doi:10.48550/arxiv.2411.17564
openalex publication_date 2024/11/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we propose a new approach -- the Tempered Finite Element Method (TFEM) -- that extends the Finite Element Method (FEM) to classes of meshes that include zero-measure or nearly degenerate elements for which standard FEM approaches do not allow convergence. First, we review why the maximum angle condition [2] is not necessary for FEM convergence and what are the real limitations in terms of meshes. Next, we propose a simple modification of the classical FEM for elliptic problems that provably allows convergence for a wider class of meshes including bands of caps that cause locking of the solution in standard FEM formulations. The proposed method is trivial to implement in an existing FEM code and can be theoretically analyzed. We prove that in the case of exactly zero-measure elements it corresponds to mortaring. We show numerically and theoretically that what we propose is functional and sound. The remainder of the paper is devoted to extensions of the TFEM method to linear elasticity, mortaring of non-conforming meshes, high-order elements, and advection.