A VDQ-transformed approach to the 3D compressible and incompressible finite hyperelasticity

Based on the ideas of variational differential quadrature (VDQ) method and position transformation, an efficient numerical variational strategy is proposed in this paper to analyze the large deformations of hyperelastic structures in the context of three-dimensional (3D) compressible and incompressible nonlinear elasticity theories. Based on the minimum total potential energy principle together with the Neo-Hookean model, the governing equations are derived. The relations of paper are presented in novel vector–matrix format. Replacing the tensor form of formulations with matricized ones is a novelty of present work since the matricized formulations can be readily employed for the programming in numerical approaches. Discretizing is also carried out via VDQ operators. For applying the VDQ technique, the irregular domain of elements is transformed into a regular one by the method of mapping of position field based on the finite element shape functions. This feature enables the proposed VDQ-transformed approach to solve problems with irregular domains. Moreover, the developed formulation is simple, compact and easy to implement. Considering structures with various shapes, several illustrative convergence and comparative investigations are given to assess the performance of the approach in both compressible and incompressible regimes. Good accuracy and computational efficiency can be reported as the features of developed VDQ-based approach.