A fully-hybrid finite element formulation for general compressible-incompressible elasticity problems using de Rham compatible spaces

Resumo

This work proposes a novel fully-hybrid finite element formulation for general elasticity problems, combining $H(\text{div},\Omega)$ conforming vector functions for displacement and $L^2(\Omega)$ discontinuous scalar functions for pressure. This pair is De Rham compatible, which means that within the incompressible regime, the divergence-free constrain will hold strongly at element level. As the $H(\text{div})$ spaces only present continuity of the normal displacements across elements boundaries, the tangential displacement continuity can be weakly imposed performing a hybridization of the shear stresses using $H^1(\partial\Omega_e)$ functions. From past researches conducted at LabMeC, this approach has demonstrated to pose some numerical difficulties as it leads to a saddle point problem with two constraint variables - the pressure and the shear-stresses. In this work, a second hybridization is done by approximating the tangential component of the primal displacement variable using $H^1(\partial\Omega_e)$ space. Two benchmarks are used to verify the developed numerical scheme - the classical Cook's membrane and a tridimensional cantilever beam subjected to an end shear load. Optimum convergence rate is achieved under compressible, quasi-incompressible and full incompressible scenarios.