Topology optimization using the finite cell method

Abstract

Huge effort has been spent over the past decades to develop efficient numerical methods for topology optimization of mechanical structures. Most recent investigations have focused on increasing the efficiency and robustness, improving the optimization schemes and extending them to multidisciplinary objective functions. The vast majority of available methods is based on low order finite elements, assuming one element as the smallest entity which can be assigned material in the optimization process. Whereas the present paper uses only a very simple, heuristic optimization procedure, it investigates in detail the feasibility of high order elements for topology optimization. The Finite Cell Method, an extension of the p-version of FEM is used, which completely separates between the description of the geometry of a structure and cells, where the high order shape functions are defined. Whereas geometry is defined on a (very) fine mesh, the material grid, shape functions live on a much coarser grid of elements, the finite cells. The method takes advantage of the ability of high order elements to accurately approximate even strongly inhomogeneous material distribution within one element and thus boundaries between material and void which pass through the interior of the coarse cells. Very attractive properties of the proposed method can be observed: Due to the high order approach the stress field in the optimized structure is approximated very accurately, no checkerboarding is observed, the iteratively found boundary of the structure is very smooth and the observed number of iterations is in general very small.