MetaFEM: A Generic FEM Solver By Meta-expressions

TL;DR

MetaFEM introduces a compact, GPU-accelerated FEM solver in Julia that uses meta-expressions and a rule-based symbolic engine, simplifying customization and broadening applicability for multi-physics simulations.

Contribution

It presents a novel FEM solver framework based on meta-expressions and a rule-based symbolic engine, enabling flexible, GPU-accelerated PDE simulations with minimal code.

Findings

Successfully implemented for thermal conduction, elasticity, and flow.

Achieves GPU acceleration with about 5,000 lines of Julia code.

Supports easy customization through rewriting rules.

Abstract

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 5,000 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.