Nonconforming Finite Elements Methods for Nonlinear Elasticity Problems Featuring Finite Deformations and Frictionless Contact Constraints

In this work, motivated by the ambitious goal of simulating e Blow Moulding manufacturing processes, we devise, implement and test a discontinuous Galerkin (dG) framework and a Hybridizable Discontinuous Galerkin (HDG) framework for nonlinear elasticity problems featuring large deformations. The dG framework encompasses compressible and incompressible hyperelastic materials. In order to achieve stability, we combine higher-order lifting operators for the BR2 stabilization term with an adaptive strategy which relies on the BR2 Laplace operator stabilization and a penalty parameter based on the spectrum of the fourth-order elasticity tensor. Dirichlet boundary conditions for the displacement can be imposed by means of Lagrange multipliers and Nitsche method. Efficiency of the solution strategy is achieved by means of state-of-the-art agglomeration based h-multigrid. Several benchmark test cases are proposed to investigate some relevant computational aspects, namely the performance of the h-multigrid iterative solver varying the stabilization parameters and the influence of Dirichlet boundary conditions on Newton's method globalisation strategy. The HDG framework is developed to solve nonlinear elasticity problems for compressible materials. Lagrange multipliers are introduced to enforce both Dirichlet boundary conditions and frictionless contact conditions: in particular, we consider the geometrical constraint of non-penetration which ensures that the region of space occupied by a rigid fixed obstacle is precluded to the deforming body. Once the analytical descriptions of obstacles surface boundaries is given, allowing to evaluate normal gap functions, normal vectors and Hessian matrices, an incremental method equipped with an active set strategy provides the ability to capture and follow the contact surface all along the deformation history. Two and three dimensional simulations featuring complex contact surfaces are performed to validate the numerical strategy.