Stabilization of mixed displacement-pressure finite elements at finite strains using polyhedral formulations and Voronoi meshing

Abstract

The hexahedral mixed displacement-pressure finite element of the lowest order (H1/P0) has shown to be simple and effective during both linear and nonlinear analysis of incompressible solids. While the discrete displacement field is generally considered to be sufficiently accurate, the discrete pressure field can sometimes be heavily polluted by spurious pressure modes. This results from the fact that the element does not fulfill the inf-sup condition. While postprocessing techniques, such as pressure filtering or smoothing, exist to remove the spurious pressure modes from the solution, this contribution aims on the exclusion of spurious pressure modes from the solution a priori due to the element geometry. By employing polyhedral finite element formulations on Voronoi tessellations in three dimensions, we show that the discrete kernel of the linearized mixed bilinear form only consists of the hydrostatic pressure mode. A spurious pressure mode is automatically suppressed due to the vertex-to-volume ratio in the finite element mesh. These considerations hold for any arbitrary physically admissible displacement state that can occur within a Newton-Raphson framework. A nonlinear numerical example shows that spurious pressure modes are indeed suppressed if the type of tessellation is changed from hexahedral to Voronoi.

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