Mechanical analysis of functionally graded materials with complex singularities by singularity-informed deep finite element method

Abstract

Functionally graded materials (FGMs) containing interacting cracks and holes exhibit strong local singularities and steep displacement gradients, which pose significant challenges to conventional numerical analysis. This study proposes a Singularity-Informed Deep Finite Element Method (SIDFEM) for the elastic analysis of FGM plates with a central elliptical hole and asymmetric collinear cracks. Built on the Deep Finite Element Method (DFEM), the proposed method reconstructs the approximation space by embedding crack-tip distance information, near-tip square-root asymptotics, crack-surface topology, and hole-boundary enrichment, so that the neural approximation is better aligned with the local singular structure of the mechanical field. The displacement solution is obtained through discrete potential-energy minimization with FEM-style quadrature and a two-stage Adam/Limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) optimization strategy. Compared with Abaqus reference solutions, the benchmark case yields a relative \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_2$$\end{document} error of 0.026 for the displacement vector and a maximum absolute error of 0.015. In addition, compared with standard DFEM, SIDFEM reduces the relative \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_2$$\end{document} errors of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_x$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_y$$\end{document}, and the displacement vector by more than \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$92\%$$\end{document}. Further studies on ablation, geometric degeneration, defect rotation, load reversal, and generalized J-integral evaluation demonstrate the robustness and mechanical applicability of the proposed framework for fracture analysis of nonhomogeneous materials.