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A simple yet effective modification to the standard finite element method is presented in this paper. The basic idea is an extension of a partial differential equation beyond the physical domain of computation up to the boundaries of an embedding domain, which can easier be meshed. If this extension is smooth, the extended solution can be well approximated by high order polynomials. This way, the finite element mesh can be replaced by structured or unstructured cells embedding the domain where classical h-or p-Ansatz functions are defined. An adequate scheme for numerical integration has to be used to differentiate between inside and outside the physical domain, very similar to strategies used in the level set method. In contrast to earlier works, e.g., the extended or the generalized finite element method, no special interpolation function is introduced for enrichment purposes. Nevertheless, when using p-extension, the method shows exponential rate of convergence for smooth problems and good accuracy even in the presence of singularities.

In many extended versions of the finite element method (FEM) the mesh does not conform to the physical domain. Therefore, discontinuity of variables is expected when some elements are cut by the boundary. Thus, the integrands are not continuous over the whole integration domain. Apparently, none of the well developed integration schemes such as Gauss quadrature can be used readily. This paper investigates several modifications of the Gauss quadrature to capture the discontinuity within an element and to perform a more precise integration. The extended method used here is the finite cell method (FCM), an extension of a high-order approximation space with the aim of simple meshing. Several examples are included to evaluate different modifications.

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 This work is the result of an institutional partnership of the three authors being supported by the Alexander von Humboldt Foundation. This support is gratefully acknowledged. J. Parvizian ( ) 58 J. Parvizian et al.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.

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