Finite Element Theory for PHIMATS

This document summarizes the main concepts of the finite element (FE) theory and constitutive relations as implemented in the open-source code phase-field multiphysics materials simulator PHIMATS https://github.com/ahcomat/PHIMATS. PHIMATS is written in C++ and uses Python for pre- and post-processing. It provides tools for discretizing the weak form of partial differential equations (PDE), interfacing with PETSc data structures (Vec/Mat) and solvers (KSP/SNES). The framework supports both single-physics and coupled multiphysics problems primarily using staggered coupling schemes. Hands-on examples can be found in the CaseStudies directory on GitHub repository. Rather than detailing the derivations of specific models, this document focuses on the key mathematical formulations and numerical strategies used within the implementation. For in-depth theoretical discussions, the reader is encouraged to consult the references. For citing this document, please use: [Abdelrahman Hussein. Finite Element Theory for PHIMATS. 2025. doi: 10.48550/ARXIV.2502.16283].