Overview
This research introduces a novel approach termed the Tempered Finite Element Method (TFEM). This method aims to expand the applicability of the Finite Element Method (FEM) to mesh classes that incorporate zero-measure or nearly degenerate elements, conditions under which traditional FEM approaches typically do not achieve convergence. The paper details a modification to classical FEM, specifically for elliptic problems, verifying its theoretical and numerical soundness.
Research Context
The standard FEM is foundational in computational mechanics and engineering for solving partial differential equations. A known limitation of classical FEM relates to mesh quality, particularly the presence of highly distorted or degenerate elements. The 'maximum angle condition' is often cited in discussions regarding FEM convergence. This study reviews the necessity of the maximum angle condition for FEM convergence, identifying the actual limitations imposed by mesh characteristics. Specifically, it addresses scenarios where meshes contain “bands of caps” which are known to induce locking in standard FEM formulations.
Approach
The core of the TFEM involves a simple modification to the classical FEM framework for elliptic problems. The researchers designed this modification to achieve provable convergence across a broader spectrum of meshes, including those with zero-measure or nearly degenerate elements. The method’s implementation is reported as trivial within an existing FEM code infrastructure. For the specific case of exactly zero-measure elements, the TFEM is mathematically proven to correspond to mortaring. The researchers validate the proposed method through both numerical simulations and theoretical analyses.
Findings
- The maximum angle condition is not a universal necessity for FEM convergence.
- The actual limitations on FEM convergence stem from mesh characteristics that include zero-measure or nearly degenerate elements.
- The Tempered Finite Element Method (TFEM) is a proposed modification to classical FEM that enables convergence for a wider class of meshes.
- TFEM specifically allows for convergence in meshes containing zero-measure or nearly degenerate elements.
- The method is compatible with existing FEM codebases due to its simple implementation.
- Theoretical analysis demonstrates the provable convergence of TFEM.
- In the instance of exactly zero-measure elements, TFEM is equivalent to mortaring.
- Numerical and theoretical validations support the functionality and soundness of TFEM.
- The TFEM framework has extensions to linear elasticity, mortaring of non-conforming meshes, high-order elements, and advection problems.
Why This Matters
This research addresses a long-standing challenge in finite element analysis by providing a method that ensures convergence even with traditionally problematic mesh geometries. This could potentially extend the reliability and applicability of FEM in scenarios where mesh quality is compromised, such as in complex engineering designs or simulations involving extreme deformations.