ABSTRACT
This chapter summarizes a new approach, using polygonal and polyhedral elements in nonlinear elasticity problems involving extremely large and heterogeneous deformations. It presents both displacement-based and two-field mixed variational principles for finite elasticity, together with the corresponding lower- and higher-order polygonal and polyhedral finite element approximations. The chapter utilizes a gradient correction scheme that adds minimal perturbations to the gradient field at the element level in order to restore polynomial consistency and recover (expected) optimal convergence rates when the weak form integrals are evaluated using quadrature rules. It applies the gradient correction scheme to polygonal and polyhedral finite elements for finite elasticity problems and demonstrates that the gradient correction scheme can effectively and efficiently render optimally convergent polygonal and polyhedral finite element methods. The chapter recalls the equations of elastostatics applied to hyperelastic materials. Two variational formulations are presented: the standard displacement-based formulation, and a general mixed formulation involving the displacement field and a pressure-like scalar field as trial fields.
