Abstract.The extended and generalized finite element methods are reviewed with an emphasis on their applications to problems in material science: (1) fracture (2) dislocations (3) grain boundaries and (4) phases interfaces. These methods facilitate the modeling of complicated geometries and the evolution of such geometries, particularly when combined with level set methods, as for example in the simulation growing cracks or moving phase interfaces. The state of the art for these problems is described along with the history of developments.

SUMMARYThe extended finite element method (XFEM) enables local enrichments of approximation spaces. Standard finite elements are used in the major part of the domain and enriched elements are employed where special solution properties such as discontinuities and singularities shall be captured. In elements that blend the enriched areas with the rest of the domain problems arise in general. These blending elements often require a special treatment in order to avoid a decrease in the overall convergence rate. A modification of the XFEM approximation is proposed in this work. The enrichment functions are modified such that they are zero in the standard elements, unchanged in the elements with all their nodes being enriched, and varying continuously in the blending elements. All nodes in the blending elements are enriched. The modified enrichment function can be reproduced exactly everywhere in the domain and no problems arise in the blending elements. The corrected XFEM is applied to problems in linear elasticity and optimal convergence rates are achieved.

The extended finite element method (XFEM) enables the accurate approximation of solutions with jumps or kinks within elements. Optimal convergence rates have frequently been achieved for linear elements and piecewise planar interfaces. Higher-order convergence for arbitrary curved interfaces relies on two major issues: (i) an accurate quadrature of the Galerkin weak form for the cut elements and (ii) a careful formulation of the enrichment, which should preclude any problems in the blending elements. For (i), we employ a strategy of subdividing the elements into subcells with only one curved side. Reference elements that are higher-order on only one side are then used to map the integration points to the real element. For (ii), we find that enrichments for strong discontinuities are easily extended to higher-order accuracy. In contrast, problems in blending elements may hinder optimal convergence for weak discontinuities. Different formulations are investigated, including the corrected XFEM. Numerical results for several test cases involving strong or weak curved discontinuities are presented. Quadratic and cubic approximations are investigated. Optimal convergence rates are achieved using the standard XFEM for the case of a strong discontinuity. Close-to-optimal convergence rates for the case of a weak discontinuity are achieved using the corrected XFEM.HIGHER-ORDER XFEM FOR CURVED DISCONTINUITIES 565 characteristics, the construction of an appropriate mesh is crucial for the success of the finite element (FE) approximation (i.e. element edges have to align with a discontinuity and a mesh refinement is required where the solution is expected to possess singularities or large gradients). Furthermore, if the discontinuity evolves with time, the nodes and elements must be moved or remeshed continuously [1][2][3][4]. For multiple discontinuities and three-dimensional problems, this becomes rapidly intractable. Therefore, a method for modelling arbitrary discontinuities in FEs on a fixed mesh without remeshing is desirable.An extension of the standard FEM, the extended finite element method (XFEM) [5, 6] has been found to yield accurate results and yet does not require the mesh to conform to discontinuities in the approximating function or its derivatives. The XFEM also avoids remeshing for moving discontinuities. This is accomplished by extending the piecewise polynomial approximation space of the FEM to include discontinuous function spaces in local regions of the computational domain where the solution exhibits jumps or kinks. This local enrichment of the approximation space is realized by means of the partition-of-unity concept [7,8]. As a result, optimal convergence rates are achieved for linear elements and (piecewise) planar interfaces. It is worthwhile to mention the close similarity between the XFEM and other partition-of-unity-based methods such as the partition-of-unity method (PUM) [7,8], the generalized finite element method (GFEM) [9-13] and the hp-cloud method [14][15][16].Formally, the XFEM approximat...

SUMMARYA unified strategy for the higher-order accurate integration of implicitly defined geometries is proposed. The geometry is represented by a higher-order level-set function. The task is to integrate either on the zero-level set or in the sub-domains defined by the sign of the level-set function. In three dimensions, this is either an integration on a surface or inside a volume. A starting point is the identification and meshing of the zero-level set by means of higher-order interface elements. For the volume integration, special sub-elements are proposed where the element faces coincide with the identified interface elements on the zero-level set. Standard Gauss points are mapped onto the interface elements or into the volumetric sub-elements. The resulting integration points may, for example, be used in fictitious domain methods and extended finite element methods. For the case of hexahedral meshes, parts of the approach may also be seen as a higher-order marching cubes algorithm.

SUMMARYA new method for treating arbitrary discontinuities in a finite element (FE) context is presented. Unlike the standard extended FE method (XFEM), no additional unknowns are introduced at the nodes whose supports are crossed by discontinuities. The method constructs an approximation space consisting of meshbased, enriched moving least-squares (MLS) functions near discontinuities and standard FE shape functions elsewhere. There is only one shape function per node, and these functions are able to represent known characteristics of the solution such as discontinuities, singularities, etc. The MLS method constructs shape functions based on an intrinsic basis by minimizing a weighted error functional. Thereby, weight functions are involved, and special mesh-based weight functions are proposed in this work. The enrichment is achieved through the intrinsic basis. The method is illustrated for linear elastic examples involving strong and weak discontinuities, and matches optimal rates of convergence even for crack-tip applications.

Summary Stationary and instationary Stokes and Navier‐Stokes flows are considered on two‐dimensional manifolds, ie, on curved surfaces in three dimensions. The higher‐order surface FEM is used for the approximation of the geometry, velocities, pressure, and Lagrange multiplier to enforce tangential velocities. Individual element orders are employed for these various fields. Streamline‐upwind stabilization is employed for flows at high Reynolds numbers. Applications are presented, which extend classical benchmark test cases from flat domains to general manifolds. Highly accurate solutions are obtained, and higher‐order convergence rates are confirmed.

An accurate implicit description of geometries is enabled by the level-set method.Level-set data is given at the nodes of a higher-order background mesh and the interpolated zero-level sets imply boundaries of the domain or interfaces within. The higher-order accurate integration of elements cut by the zero-level sets is described.The proposed strategy relies on an automatic meshing of the cut elements. Firstly, the zero-level sets are identified and meshed by higher-order interface elements. Secondly, the cut elements are decomposed into conforming sub-elements on the two sides of the zero-level sets. Any quadrature rule may then be employed within the sub-elements.The approach is described in two and three dimensions without any requirements on the background meshes. Special attention is given to the consideration of corners and edges of the implicit geometries.

An isogeometric boundary element method for problems in elasticity is presented, which is based on an independent approximation for the geometry, traction and displacement field. This enables a flexible choice of refinement strategies, permits an efficient evaluation of geometry related information, a mixed collocation scheme which deals with discontinuous tractions along non-smooth boundaries and a significant reduction of the right hand side of the system of equations for common boundary conditions. All these benefits are achieved without any loss of accuracy compared to conventional isogeometric formulations. The system matrices are approximated by means of hierarchical matrices to reduce the computational complexity for large scale analysis. For the required geometrical bisection of the domain, a strategy for the evaluation of bounding boxes containing the supports of NURBS basis functions is presented. The versatility and accuracy of the proposed methodology is demonstrated by convergence studies showing optimal rates and real world examples in two and three dimensions.