Matrix Method of Defect Analysis for Structures with Areas of Considerable Stiffness Differences

The paper presents an innovative matrix method for defect analysis in heterogeneous structures with significant differences in stiffness parameters. The proposed approach modifies the finite element method (FEM) by introducing a refined type of finite element capable of handling regions with varying material and geometrical properties. Unlike traditional FEM discretization, which increases computational time calculations due to mesh refinement in stiffness-variable areas, the proposed method maintains a consistent discretization while adjusting shape functions to reflect internal stiffness changes. An original functional for modifying linear shape function distributions is introduced, leading to piecewise broken-line shape functions. This enhancement significantly improves the accuracy of numerical calculations, as verified by computational calculations. The explicit formulation of the stiffness matrix enhances computational efficiency, making the approach particularly useful for periodic structures with inclusions, voids, or localized material defects. The results of numerical tests demonstrate that the proposed method provides solutions closely aligned with conventional FEM results while substantially reducing computational costs. The approach is particularly relevant for structural engineering applications where defect analysis and material heterogeneity play a critical role in design and safety evaluation.