Advanced Modeling of Crack Propagation Using Extended Finite Element Method (XFEM): Module Theory and Computational Approaches

Fracture and crack propagation are critical challenges in computational mechanics, as they directly affect the safety and durability of structural components in aerospace, automotive, and civil engineering. The extended finite element method (XFEM) has become a powerful tool for modeling discontinuities without remeshing. However, enrichment functions in XFEM are often introduced in an ad hoc manner, leading to issues of computational cost, numerical instability, and difficulty in extending the framework to optimization and nonlinear dynamic problems. This study develops a systematic enrichment strategy for XFEM by integrating module theory into the enrichment framework. The approach decomposes enrichment functions into modular components, enabling consistent management of crack-tip singularities and discontinuities while ensuring stability and scalability. A weak form of equilibrium is formulated with enrichment functions satisfying the partition of unity (POU). A MATLAB implementation is developed for two-dimensional linear elastic fracture problems, with validation performed against analytical linear elastic fracture mechanics (LEFM) benchmarks. The method is also extended to topology optimization using the solid isotropic material with penalization (SIMP) scheme, where material distribution evolves under fracture-driven constraints. The modular enrichment strategy accurately reproduces stress intensity factors and crack paths under various loading conditions, with results closely matching LEFM solutions. Figures demonstrate the evolution of displacement fields, crack propagation trajectories, and stress distributions, confirming the robustness of the enrichment functions. Compared with conventional XFEM, the proposed method reduces computational overhead and improves numerical stability in crack growth simulations. Integration with SIMP topology optimization further enables automatic crack-aware material redistribution, illustrating the framework’s potential for dynamic structural design applications. The integration of module theory with XFEM provides a mathematically consistent and computationally efficient approach for fracture analysis. The framework addresses key limitations of traditional enrichment methods, including stability and scalability, while enabling natural extension to topology optimization. This work establishes a foundation for future studies on nonlinear fracture, three-dimensional crack propagation, and lightweight design optimization under dynamic loading conditions.