Polytopal mixed finite elements for nearly-incompressible finite elasticity : discretization, approximation, stabilization

This dissertation addresses the development and stability of polytopal finite element formulations in structural analysis. It focuses particularly on nearly-incompressible problems and contains finite strain applications. The first part of the thesis is concerned with the construction of two novel finite element formulations based on the scaled boundary parameterization. A low-order polygonal mixed displacement-pressure formulation is presented, which mitigates volumetric locking on arbitrary polygonal discretizations, such as Voronoi meshes. Additionally, a novel isoparametric framework for polygons is presented, facilitating the construction of linear complete interpolation functions on arbitrary polygonal parametric domains. The interpolants avoid the need for additional interior degrees-of-freedom and improve the accuracy of the solution in the radial direction. The second part of this thesis investigates the well-posedness of polytopal mixed displacement-pressure formulations, focusing on the inf-sup stability and consequently, the possible appearance of spurious pressure oscillations in the solution. The stability analysis in 2D reveals that the properties of non-degenerate Voronoi meshes can be employed to suppress spurious pressure modes. An extension of this mesh topology-based stabilization technique to 3D finite elasticity is presented. Throughout the thesis, it is shown that Voronoi meshes can contribute to an increase in accuracy and to the stabilization of mixed finite element formulations for nearly-incompressible problems in nonlinear structural analysis.