SUMMARYA high-order generalized finite element method (GFEM) for non-planar three-dimensional crack surfaces is presented. Discontinuous p-hierarchical enrichment functions are applied to strongly graded tetrahedral meshes automatically created around crack fronts. The GFEM is able to model a crack arbitrarily located within a finite element (FE) mesh and thus the proposed method allows fully automated fracture analysis using an existing FE discretization without cracks. We also propose a crack surface representation that is independent of the underlying GFEM discretization and controlled only by the physics of the problem. The representation preserves continuity of the crack surface while being able to represent non-planar, non-smooth, crack surfaces inside of elements of any size. The proposed representation also provides support for the implementation of accurate, robust, and computationally efficient numerical integration of the weak form over elements cut by the crack surface. Numerical simulations using the proposed GFEM show high convergence rates of extracted stress intensity factors along non-planar curved crack fronts and the robustness of the method.

We present an algorithm to construct meshes suitable for space-time discontinuous Galerkin finite-element methods. Our method generalizes and improves the 'Tent Pitcher' algorithm ofÜngör and Sheffer. Given an arbitrary simplicially meshed domain X of any dimension and a time interval [0, T ], our algorithm builds a simplicial mesh of the space-time domain X × [0, T ], in constant time per element. Our algorithm avoids the limitations of previous methods by carefully adapting the durations of space-time elements to the local quality and feature size of the underlying space mesh.

Meshes composed of well-centered simplices have nice orthogonal dual meshes (the dual Voronoi diagram). This is useful for certain numerical algorithms that prefer such primal-dual mesh pairs. We prove that well-centered meshes also have optimality properties and relationships to Delaunay and minmax angle triangulations. We present an iterative algorithm that seeks to transform a given triangulation in two or three dimensions into a well-centered one by minimizing a cost function and moving the interior vertices while keeping the mesh connectivity and boundary vertices fixed. The cost function is a direct result of a new characterization of well-centeredness in arbitrary dimensions that we present. Ours is the first optimization-based heuristic for well-centeredness, and the first one that applies in both two and three dimensions. We show the results of applying our algorithm to small and large two-dimensional meshes, some with a complex boundary, and obtain a well-centered tetrahedralization of the cube. We also show numerical evidence that our algorithm preserves gradation and that it improves the maximum and minimum angles of acute triangulations created by the best known previous method.Comment: Content has been added to experimental results section. Significant edits in introduction and in summary of current and previous results. Minor edits elsewher

We propose a new algorithm for constructing finite-element meshes suitable for spacetime discontinuous Galerkin solutions of linear hyperbolic PDEs. Given a triangular mesh of some planar domain Ω and a target time value T , our method constructs a tetrahedral mesh of the spacetime domain Ω × [0, T ] in constant running time per tetrahedron in IR 3 using an advancing front method. Elements are added to the evolving mesh in small patches by moving a vertex of the front forward in time. Spacetime discontinuous Galerkin methods allow the numerical solution within each patch to be computed as soon as the patch is created. Our algorithm employs new mechanisms for adaptively coarsening and refining the front in response to a posteriori error estimates returned by the numerical code. A change in the front induces a corresponding refinement or coarsening of future elements in the spacetime mesh. Our algorithm adapts the duration of each element to the local quality, feature size, and degree of refinement of the underlying space mesh. We directly exploit the ability of discontinuous Galerkin methods to accommodate discontinuities in the solution fields across element boundaries.

This paper presents a study of generalized enrichment functions for 3D curved crack fronts. Two coordinate systems used in the definition of singular curved crack front enrichment functions are analyzed. In the first one, a set of Cartesian coordinate systems defined along the crack front is used. In the second case, the geometry of the crack front is approximated by a set of curvilinear coordinate systems. A description of the computation of derivatives of enrichment functions and curvilinear base vectors is presented. The coordinate systems are automatically defined using geometrical information provided by an explicit representation of the crack surface. A detailed procedure to accurately evaluate the surface normal, conormal and tangent vectors along curvilinear crack fronts in explicit crack surface representations is also presented. An accurate and robust definition of orthonormal vectors along crack fronts is crucial for the proper definition of enrichment functions. Numerical experiments illustrate the accuracy and robustness of the proposed approaches.