A framework for residual-based stabilization of incompressible finite elasticity: Stabilized formulations and F methods for linear triangles and tetrahedra

A new Variational Multiscale framework for finite strain incompressible elasticity is presented. Significant contributions in this work are: (i) a systematic derivation of multiscale formulations that include the classical F method as a particular subclass, (ii) an error estimation procedure for nonlinear elasticity that emanates naturally from within the present multiscale framework, and (iii) robust performance of linear triangular and tetrahedral elements for modeling nearly incompressible materials in the finite strain range. When viewed from the Variational Multiscale perspective, the classical F method is shown to include only a volumetric or diagonal fine-scale deformation gradient while the proposed formulation includes the full spectrum of inter-scale coupling effects. Also, the error estimation procedure readily carries over from developments conducted for small strain linear problems. The formulation is presented first in the context of displacement-based methods and then extended to more general mixed methods accommodating arbitrary combinations of displacement-pressure interpolations. An extensive set of benchmark problems is investigated to show the performance of the method for a variety of hyperelastic materials exhibiting incompressible response.