A finite element formulation in boundary representation for the analysis of nonlinear problems in solid mechanics

Abstract

AbstractThe contribution is concerned with a numerical element formulation for the analysis of the nonlinear behavior in solid mechanics. It is based on the so‐called scaled boundary finite element method (SB‐FEM). The SB‐FEM is as a semi‐analytical formulation to analyze problems of linear elasticity. The basic idea is to scale the boundary with respect to a scaling center. Hence, the boundary denoted as circumferential direction and the scaling direction span the parameter space. The original SB‐FEM approach is based on an analytical ansatz for the displacement response preserving equilibrium in scaling direction. In the present approach, an interpolation in scaling direction is introduced to account for nonlinear problems. Lagrange interpolations with different polynomial order are therefore discussed. The interpolation in circumferential direction is independent of the scaling direction and is defined by independent Lagrange functions. The displacement degrees of freedom are located at the nodes on the boundary and in the interior element domain. The degrees of freedom located in the interior of the domain are eliminated by static condensation, which leads to a polygonal finite element formulation with an arbitrary number of nodes on the boundary. The element formulation allows a priori for Voronoi meshes and quadtree mesh generation. Numerical examples give rise to the performance of the present approach in comparison to other element formulations, like mixed formulations.