On the stability of mixed polygonal finite element formulations in nonlinear analysis

This article discusses the accuracy and stability of the pressure field in nonlinear mixed displacement-pressure finite element formulations in solid mechanics. We focus on two-dimensional mixed polygonal finite element formulations with linear displacement and constant pressure approximations in particular. The inf-sup stability of these formulations is assessed and compared with classical mixed finite element formulations. An analytical proof is presented, which concludes that the occurrence of spurious pressure modes depends on the chosen meshing strategy. It is shown that these spurious modes are successfully suppressed on any Voronoi mesh in both linear elasticity and nonlinear hyperelasticity without the need for any kind of stabilization. Several linear and nonlinear nearly-incompressible examples with different discretization strategies and boundary conditions are considered to validate the analytical proof. A mixed polygonal finite element formulation based on the scaled boundary parameterization is used to approximate the field variables, however, the derivations presented herein hold for any lowest-order mixed polygonal finite element formulation. The nonexistence of checkerboard modes on linear elastic Voronoi discretizations is shown graphically. By evaluating the incremental pressure in each Newton-Raphson iteration, the stabilization effect of the Voronoi discretization is demonstrated for nonlinear problems. In addition, the analytical proof is validated by the numerical (generalized) inf-sup test.