A non‐uniform rational B‐splines enhanced finite element formulation based on the scaled boundary parameterization for the analysis of heterogeneous solids

Abstract

AbstractThe contribution is concerned with a finite element formulation for the nonlinear analysis of heterogeneous solids in boundary representation. It results in an element with an arbitrary number of curved boundary edges. The curved edges can be parametrized by, for example, non‐uniform rational B‐splines (NURBS). The presented element formulation is based on the scaling concept, which is adopted from the so‐called scaled boundary finite element method (SBFEM). In contrast to SBFEM, the proposed method uses a numerical approximation for the displacement response in scaling direction. This enables the analysis of geometrically and physically nonlinear problems in solid mechanics. The interpolation at the boundary in circumferential direction is independent of interpolation in scaling direction. Thus, different basis functions can be used for each direction, for example, NURBS basis functions in circumferential and Lagrange basis functions in radial direction. It allows the construction of polygonal elements with an arbitrary number of curved sides, which are described by either whole NURBS curves or NURBS curves' segments. The advantage of the presented element formulation is the flexibility in mesh generation. For example, using Quadtree algorithms, a fast and reliable mesh generation can be achieved. Furthermore, in connection with trimming algorithms, the element formulation allows a precise representation of the geometry even with coarse meshes. Some benchmark tests are presented to evaluate the accuracy of the proposed numerical method against analytical solutions, and a comparison to standard element formulations is given as well.