A HDG formulation for nonlinear elasticity problems featuring finite deformations and frictionless contact constraints

In this work we introduce a Hybridizable Discontinuous Galerkin (HDG) formulation for Computational Contact Mechanics (CCM). In particular we extend the HDG formulation for nonlinear elasticity originally proposed by Soon, Cockburn, Stolarski in 2009 (https://doi.org/10.1002/nme.2646) and revisited by Kabaria, Lew and Cockburn in 2015 (https://doi.org/10.1016/j.cma.2014.08.012), introducing the possibility to enforce Dirichlet boundary conditions and non-penetration frictionless contact constraints using the Lagrange multi-pliers method. We consider hyperelastic materials and contact with rigid obstacles whose geometry admits an analytical description. The convergence of Newton's method towards the equilibrium configuration is globalized by means of an incremental load method coupled with an active set strategy, aimed at identifying the evolution of the contact interface. The HDG formulation is numerically validated by evaluating numerical convergence rates based on manufactured solutions. CCM computations of benchmark are performed based on higher-order accurate discretizations, in particular we tackle the interaction of a thin membrane with a spherical obstacle and we seek to improve the accuracy by increasing the polynomial degree. To conclude, we apply the CCM framework to perform a preliminary blow moulding computation involving inflation of a hyperelastic preform and contact with a cylindrical mould.