SUMMARYThis paper investigates two approaches for the handling of hanging nodes in the framework of extended finite element methods (XFEM). Allowing for hanging nodes, locally refined meshes may be easily generated to improve the resolution of general, i.e. model-independent, steep gradients in the problem under consideration. Hence, a combination of these meshes with XFEM facilitates an appropriate modeling of jumps and kinks within elements that interact with steep gradients. Examples for such an interaction are, e.g. found in stress fields near crack fronts or in boundary layers near internal interfaces between two fluids. The two approaches for XFEM based on locally refined meshes with hanging nodes basically differ in whether (enriched) degrees of freedom are associated with the hanging nodes. Both approaches are applied to problems in linear elasticity and incompressible flows.

SUMMARYThis paper discusses higher-order extended finite element methods (XFEMs) obtained from the combination of the standard XFEM with higher-order FEMs. Here, the focus is on the embedding of the latter into the partition of unity method, which is the basis of the XFEM. A priori error estimates are discussed, and numerical verification is given for three benchmark problems. Moreover, methodological aspects, which are necessary for hp-adaptivity in XFEM and allow for exponential convergence rates, are summarized. In particular, the handling of hanging nodes via constrained approximation and an hp-adaptive strategy are presented.

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This paper presents an algorithm for the refinement of two-or threedimensional meshes with respect to an implicitly given domain, so that its surface is approximated by facets of the resulting polytopes. Using a Cartesian grid, the proposed algorithm may be used as a mesh generator. Initial meshes may consist of polytopes such as quadrilaterals and triangles, as well as hexahedrons, pyramids, and tetrahedrons. Given the ability to compute edge intersections with the surface of an implicitly given domain, the proposed marching volume polytopes algorithm uses predefined refinement patterns applied to individual polytopes depending on the intersection pattern of their edges. The refinement patterns take advantage of rotational symmetry. Since these patterns are applied independently to individual polytopes, the resulting mesh may encompass the so-called orientation problem, where two adjacent polytopes are rotated against one another. To allow for a repeated application of the marching volume polytopes algorithm, the proposed data structures and algorithms account for this ambiguity. A simple example illustrates the advantage of the repeated application of the proposed algorithm to approximate domains with sharp corners. Furthermore, finite element simulations for two challenging real-world problems, which require highly accurate approximations of the considered domains, demonstrate its applicability. For these simulations, a variant of the fictitious domain method is used. KEYWORDSfictitious domain method, finite element method, marching cubes, marching tetrahedrons, mesh generation, volume mesh Delaunay, quadtree/octree, and advancing-front methods for mesh generation. Using quadtree or octree techniques, 2-6 Cartesian grids covering a given domain are repeatedly refined via isotropic bisections of quadrilaterals into four similar sub-quadrilaterals or of hexahedrons into eight similar sub-hexahedrons for

Summary This paper presents a (higher‐order) finite element approach for the simulation of heat diffusion and thermoelastic deformations in NC‐milling processes. The inherent continuous material removal in the process of the simulation is taken into account via continuous removal‐dependent refinements of a paraxial hexahedron base‐mesh covering a given workpiece. These refinements rely on isotropic bisections of these hexahedrons along with subdivisions of the latter into tetrahedrons and pyramids in correspondence to a milling surface triangulation obtained from the application of the marching cubes algorithm. The resulting mesh is used for an element‐wise defined characteristic function for the milling‐dependent workpiece within that paraxial hexahedron base‐mesh. Using this characteristic function, a (higher‐order) fictitious domain method is used to compute the heat diffusion and thermoelastic deformations, where the corresponding ansatz spaces are defined for some hexahedron‐based refinement of the base‐mesh. Numerical experiments compared to real physical experiments exhibit the applicability of the proposed approach to predict deviations of the milled workpiece from its designed shape because of thermoelastic deformations in the process.