Linear Elasticity and Hyperelasticity

In this chapter, we show how to discretize using HHO methods linear elasticity and nonlinear hyperelasticity problems. In particular, we pay particular attention to the robustness of the discretization in the quasi-incompressible limit. For linear elasticity, we reconstruct the strain tensor in the space composed of symmetric gradients of vector-valued polynomials. For nonlinear hyperelasticity, we reconstruct the deformation gradient in a full tensor-valued polynomial space, and not just in a space composed of polynomial gradients. We also consider a second gradient reconstruction in an even larger space built using Raviart–Thomas polynomials, for which no additional stabilization is necessary. Finally, we present some numerical examples.