MetaFEM: A generic FEM solver by meta-expressions

Jiaxi Xie, Kornel Ehmann, Jian Cao*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


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.

Original languageEnglish (US)
Article number114907
JournalComputer Methods in Applied Mechanics and Engineering
StatePublished - May 1 2022


  • Continuum mechanics
  • Finite element
  • MetaFEM
  • Rewriting system

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • Physics and Astronomy(all)
  • Computer Science Applications


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