Partition of Unity-based four-node tetrahedral element for nonlinear structural analysis of nearly incompressible hyperelastic materials

This study introduces a refined four-node tetrahedral finite element employing the Partition of Unity method for nonlinear static and modal analysis of nearly incompressible hyperelastic materials. The proposed Partition of Unity-based element effectively reduces volumetric locking and improves solution accuracy without increasing the number of nodes. The Partition of Unity method enriches the displacement field by incorporating additional polynomial basis functions, enabling higher-order displacement approximation, and effectively alleviating volumetric locking. Mooney–Rivlin and Neo-Hookean material models are integrated with the penalty method, ensuring robust handling of nearly incompressible behavior. Large deformations are addressed using a total Lagrangian formulation. In addition, a displacement-based direct iterative nonlinear modal analysis procedure is employed to derive nonlinear natural frequencies and corresponding mode shapes. In nonlinear static analysis, the proposed element is validated through various numerical cases including blocks under compression, cylinders under large deformation, mesh distortion sensitivity analysis, and tires under compression. The present element effectively alleviates the volumetric locking phenomenon and provides excellent performance even when using a coarse mesh. Nonlinear modal analysis has been performed on cases such as free vibration of distorted plates, truncated cylindrical shells, and hyperelastic soft robots. The proposed elements effectively capture nonlinear natural frequencies and mode shapes even with distorted and coarse meshes.