Using XFEM to Integrate the XIGA Method to Simulate the Stress Concentration Factor at the Circular Hole of an FGM Plate

In recent decades, numerical methods have become indispensable tools in solving complex problems in science and engineering, where the finite element method (FEM) is widely recognized as an effective computational approach. However, the traditional FEM often encounters significant limitations in modeling discontinuities such as cracks, holes, or material interfaces, especially when dealing with functionally graded materials (FGMs). To overcome these challenges, this study introduces an advanced framework that integrates the Extended Finite Element Method (XFEM) with the Isogeometric Analysis (XIGA) approach to simulate the stress concentration factors (SCFs) around circular holes in isotropic and FGM plates. The methodology employs the level-set method to represent discontinuous boundaries and incorporates enrichment functions into the displacement field, enabling accurate modeling of stress concentrations without remeshing. MATLAB codes were developed to implement this integration, offering a flexible computational platform for practical applications. The performance of the proposed method was evaluated through several benchmark problems, including isotropic plates with circular holes near material boundaries and FGMs subjected to uniaxial loading. The results obtained by means of XFEM–XIGA were compared against analytical solutions, standard FEM, and available experimental data. For isotropic plates, the XFEM–XIGA model achieved a stress concentration error of 1.71%. For FGM plates with cracks or circular holes, the error of XFEM–XIGA was 2.55% when compared with exact solutions. These findings highlight the robustness and accuracy of the integrated method in handling complex geometries and heterogeneous material properties. Overall, this study demonstrates that the combination of XFEM and XIGA provides an efficient and reliable tool for analyzing SCFs in FGM structures, paving the way for improved modeling of industrial components where stress concentrations at material boundaries are critical to structural performance.