Multiscale analysis of heterogeneous materials in boundary representation

Abstract

AbstractThis paper deals with a numerical method for modeling heterogeneous materials with nonlinear behavior. Many novel high‐performance materials consist of different components. In the frame of nonlinear behavior the resulting macroscopic material properties cannot be determined without great experimental effort. In order to simulate the nonlinear material behavior, a nested finite element method (FE2) is used. An accompanying homogenization is performed for each load step. The material behavior is determined from the represented volume element (RVE) on the heterogeneous micro structure. The present approach is dealing with a discretization method of the heterogeneous structure. It based on the geometrical data of the interface boundary of the different constituents. A quad‐tree algorithm, in combination with a trimming algorithm is used for the mesh generation. Based on the geometrical data it yields an initial mesh for the analysis. Within the algorithm a staircase approximation of voids or inclusions is circumvented. However, the resulting mesh yields polygonal elements with an arbitrary number of nodes on the element boundary. To this end a novel polygonal element formulation is provided. It is based on the scaled boundary finite element method (SBFEM). The basic idea is to scale the boundary representation in relation to a scaling center. In contrast to SBFEM the present method makes us of an approximation of the displacement response in scaling direction. This enables the analysis of geometrical and material‐related nonlinear problems in solid mechanics. The interpolation at the boundary in circumferential direction is independent of the interpolation in scaling direction. This allows for novel refinement strategies, where e.g. p‐refinement is only applied for the interior element domain. The internal degrees of freedom are eliminated by static condensation. It leads to an element formulation with an arbitrary number of nodes at the element boundary. Thus, the present element is perfectly suited for a quad‐tree meshing algorithm of the RVE. Its hierarchical, tree‐like data structure makes it very useful for adaptive mesh refinements of complex geometries and regions with localized gradients. The classical hanging‐nodes problem for standard elements is omitted by the present element formulation. Some numerical examples show the capability of the formulation with respect to the numerical homogenization of heterogeneous materials.