Порівняння скінченно-елементних сіток у разі чисельного моделювання колінного суглоба та аналітична валідація результатів

Abstract

This study investigates the influence of finite element mesh parameters on the accuracy and computational efficiency of numerical modeling of the human knee joint under static axial compression corresponding to quiet standing. The knee joint is one of the most highly loaded and injury-prone components of the musculoskeletal system, and reliable prediction of its stress–strain state is essential for biomechanics, orthopedics, and rehabilitation engineering. Although the finite element method is widely used for analyzing such complex biological structures, the obtained results strongly depend on mesh density, element type, and mesh quality, which requires a dedicated quantitative assessment. A three-dimensional geometric model of the knee joint was developed in SolidWorks environment based on computed tomography data and included the femur, tibia, meniscus, and ligamentous structures. Finite element simulations were performed in ANSYS (Static Structural) for an axial compressive load of 750 N applied along the vertical axis. Three tetrahedral mesh configurations were analyzed, consisting of 30,001, 501,090, and 1,006,936 elements. Mesh quality was evaluated using Skewness and Orthogonal Quality metrics, with local refinement applied in regions prone to stress concentration. For each mesh configuration, axial displacements, von Mises equivalent stresses in the meniscus and ligaments, and total computation time were evaluated using the same hardware setup. To independently verify the numerical results, an analytical validation was conducted by representing the meniscus–ligament system as an equivalent parallel spring model. The overall stiffness and compressive deformation were determined using Hooke’s law and compared with the finite element predictions. The results demonstrate that the mesh containing approximately 5.0×10⁵ elements provides the best balance between accuracy and computational cost. The deviation between finite element displacements and analytical predictions is approximately 4.5 %, while further mesh refinement beyond 10⁶ elements does not lead to a noticeable improvement in accuracy but significantly increases computational time. These findings highlight the importance of controlled mesh design in knee joint biomechanics and show that well-quality-controlled meshes of moderate density with local refinement can ensure reliable and reproducible results at reasonable computational expense.