MetaFEM: A generic FEM solver by meta-expressions

• Published in: Computer methods in applied mechanics and engineering Vol. 394; p. 114907
• Main Authors: Xie, Jiaxi, Ehmann, Kornel, Cao, Jian
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.05.2022; Elsevier BV
• Subjects: Algorithms; Complex systems; Complexity; Computer algebra; Continuum mechanics; Finite element; Finite element method; Fluid flow; Incompressible flow; MetaFEM; Partial differential equations; Rewriting system; Solvers; MetaFEM; Continuum mechanics; Rewriting system; Finite element
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2022.114907

Current multi-physics Finite Element Method (FEM) solvers are complex systems in terms of both their mathematical complexity and lines of code. This paper proposes a skeleton generic FEM solver, named MetaFEM, in total about 6,000 lines of Julia code, which translates generic input Partial Differential Equation (PDE) weak forms into corresponding GPU-accelerated simulations with a grammar similar to FEniCS or FreeFEM. Two novel approaches differentiate MetaFEM from the common solvers: (1) the FEM kernel is based on an original theory/algorithm which explicitly processes meta-expressions, as the name suggests, and (2) the symbolic engine is a rule-based Computer Algebra System (CAS), i.e., the equations are rewritten/derived according to a set of rewriting rules instead of going through completely fixed routines, supporting easy customization by developers. Example cases in thermal conduction, linear elasticity and incompressible flow are presented to demonstrate utility. •A novel formulation/architecture for a generic FEM solver, significantly reducing system complexity.•An applied symbolic rewriting system for representing continuum mechanics.•Generating GPU-accelerated FEM simulation for generic/various physics on generic geometry.