A mixed polygonal finite element formulation for nearly-incompressible finite elasticity

In this work, we present a mixed formulation based on the scaled boundary parameterization for the analysis of nearly-incompressible hyperelastic materials at finite strains. The formulation leads to a polygonal element with an arbitrary number of edges. The element domain is parameterized in both the boundary and scaling directions and necessitates the definition of a scaling center. Interpolation functions are introduced in both directions, which allow for an application to physically and geometrically nonlinear problems. This is an extension to the original scaled boundary finite element method (SBFEM), in which an analytical solution in the scaling direction is employed. Whereas the SBFEM approach is restricted to star-convex element geometries, the present approach allows for non-star-convex polygonal element shapes. A mixed displacement-pressure formulation is proposed to alleviate volumetric locking, which typically occurs in case of nearly-incompressible material behavior. Several numerical examples demonstrate that the proposed approach is applicable to arbitrary polygonal meshes, such as Voronoi meshes. The results demonstrate an excellent behavior in terms of computational efficiency compared to standard quadrilateral elements and to the virtual element method (VEM). The ability of the present formulation to alleviate volumetric locking is shown and it is demonstrated that a higher accuracy can be reached on Voronoi meshes compared to quadrilateral discretizations.