Mixed node's residual descent method for hyperelastic problem analysis

Abstract

Geometric nonlinearities, material nonlinearities, and volume locking are the notable challenges faced in hyperelastic analysis. Traditional methods in this regard are complex and laborious for implementation as they require linearization and formulation of global matrix equations while simultaneously addressing volumetric locking. A mixed node's residual descent method (NRDM) proposed herein can effectively address the numerical challenges associated with geometric nonlinearities, material nonlinearities, force nonlinearities, and incompressibility. First, the implementation of the NRDM is considerably simplified as unlike traditional incremental–iterative methods, it's a matrix-free iterative method that does not require incremental linear equations. Second, the NRDM addresses the geometric and material nonlinearities with relative ease as it flexibly describes the deformation with the initial configuration or assumed deformed configuration as the reference frame. Third, the NRDM prevents the occurrence of volumetric locking by employing hydrostatic pressure as an independent variable. Furthermore, the NRDM can easily treat the force nonlinearities and boundary nonlinearities by controlling the relation between load and deformation during iteration. Moreover, a notable accuracy of the NRDM is confirmed through numerical verifications, and several critical matters are discussed, including the scheme for adjusting the basic independent variables, treatment of displacement boundaries, scheme for imposing loads, and computational parameter setting.