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Mesh Generation And Optimal Triangulation
, 1992
"... We survey the computational geometry relevant to finite element mesh generation. We especially focus on optimal triangulations of geometric domains in two and threedimensions. An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some cri ..."
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Cited by 213 (7 self)
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We survey the computational geometry relevant to finite element mesh generation. We especially focus on optimal triangulations of geometric domains in two and threedimensions. An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some criterion that measures the size, shape, or number of triangles. We discuss algorithms both for the optimization of triangulations on a fixed set of vertices and for the placement of new vertices (Steiner points). We briefly survey the heuristic algorithms used in some practical mesh generators.
FreeForm Shape Design Using Triangulated Surfaces
, 1994
"... We present an approach to modeling with truly mutable yet completely controllable freeform surfaces of arbitrary topology. Surfaces may be pinned down at points and along curves, cut up and smoothly welded back together, and faired and reshaped in the large. This style of control is formulated as a ..."
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Cited by 168 (0 self)
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We present an approach to modeling with truly mutable yet completely controllable freeform surfaces of arbitrary topology. Surfaces may be pinned down at points and along curves, cut up and smoothly welded back together, and faired and reshaped in the large. This style of control is formulated as a constrained shape optimization, with minimization of squared principal curvatures yielding graceful shapes that are free of the parameterization worries accompanying many patchbased approaches. Triangulated point sets are used to approximate these smooth variational surfaces, bridging the gap between patchbased and particlebased representations. Automatic refinement, mesh smoothing, and retriangulation maintain a good computational mesh as the surface shape evolves, and give sample points and surface features much of the freedom to slide around in the surface that oriented particles enjoy. The resulting surface triangulations are constructed and maintained in real time. 1 Introduction ...
Tetrahedral Mesh Improvement Using Swapping and Smoothing
 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
, 1997
"... Automatic mesh generation and adaptive refinement methods for complex threedimensional domains have proven to be very successful tools for the efficient solution of complex applications problems. These methods can, however, produce poorly shaped elements that cause the numerical solution to be less ..."
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Cited by 109 (12 self)
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Automatic mesh generation and adaptive refinement methods for complex threedimensional domains have proven to be very successful tools for the efficient solution of complex applications problems. These methods can, however, produce poorly shaped elements that cause the numerical solution to be less accurate and more difficult to compute. Fortunately, the shape of the elements can be improved through several mechanisms, including face and edgeswapping techniques, which change local connectivity, and optimizationbased mesh smoothing methods, which adjust mesh point location. We consider several criteria for each of these two methods and compare the quality of several meshes obtained by using different combinations of swapping and smoothing. Computational experiments show that swapping is critical to the improvement of general mesh quality and that optimizationbased smoothing is highly effective in eliminating very small and very large angles. Highquality meshes are obtained in a computationally efficient manner by using optimizationbased smoothing to improve only the worst elements and a smart variant of Laplacian smoothing on the remaining elements. Based on our experiments, we offer several recommendations for the improvement of tetrahedral meshes.
A Pliant Method for Anisotropic Mesh Generation
"... A new algorithm for the generation of anisotropic, unstructured triangular meshes in two dimensions is described. Inputs to the algorithm are the boundary geometry and a metric that specifies the desired element size and shape as a function of position. The algorithm is an example of what we call p ..."
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Cited by 91 (2 self)
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A new algorithm for the generation of anisotropic, unstructured triangular meshes in two dimensions is described. Inputs to the algorithm are the boundary geometry and a metric that specifies the desired element size and shape as a function of position. The algorithm is an example of what we call pliant mesh generation. It first constructs the constrained Delaunay triangulation of the domain, then iteratively smooths, refines, and retriangulates. On each iteration, a node is selected at random, it is repositioned according to attraction/repulsion with its neighbors, the neighborhood is retriangulated, and nodes are inserted or deleted as necessary. All operations are done relative to the metric tensor. This simple method generates high quality meshes whose elements conform well to the requested shape metric. The method appears particularly well suited to surface meshing and viscous flow simulations, where stretched triangles are desirable, and to timedependent remeshing problems.
A Simple Mesh Generator in MATLAB
 SIAM Review
, 2004
"... Abstract. Creating a mesh is the first step in a wide range of applications, including scientific computing and computer graphics. An unstructured simplex mesh requires a choice of meshpoints (vertex nodes) and a triangulation. We want to offer a short and simple MATLAB code, described in more detai ..."
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Cited by 86 (4 self)
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Abstract. Creating a mesh is the first step in a wide range of applications, including scientific computing and computer graphics. An unstructured simplex mesh requires a choice of meshpoints (vertex nodes) and a triangulation. We want to offer a short and simple MATLAB code, described in more detail than usual, so the reader can experiment (and add to the code) knowing the underlying principles. We find the node locations by solving for equilibrium in a truss structure (using piecewise linear forcedisplacement relations) and we reset the topology by the Delaunay algorithm. The geometry is described implicitly by its distance function. In addition to being much shorter and simpler than other meshing techniques, our algorithm typically produces meshes of very high quality. We discuss ways to improve the robustness and the performance, but our aim here is simplicity. Readers can download (and edit) the codes from
On Combining Laplacian And OptimizationBased Mesh Smoothing Techniques
 TRENDS IN UNSTRUCTURED MESH GENERATION
, 1997
"... Local mesh smoothing algorithms have been shown to be effective in repairing distorted elements in automatically generated meshes. The simplest such algorithm is Laplacian smoothing, which moves grid points to the geometric center of incident vertices. Unfortunately, this method operates heuristical ..."
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Cited by 77 (9 self)
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Local mesh smoothing algorithms have been shown to be effective in repairing distorted elements in automatically generated meshes. The simplest such algorithm is Laplacian smoothing, which moves grid points to the geometric center of incident vertices. Unfortunately, this method operates heuristically and can create invalid meshes or elements of worse quality than those contained in the original mesh. In contrast, optimizationbased methods are designed to maximize some measure of mesh quality and are very effective at eliminating extremal angles in the mesh. These improvements come at a higher computational cost, however. In this article we propose four smoothing techniques that combine a smart variant of Laplacian smoothing with an optimizationbased approach. Several numerical experiments are performed that compare the mesh quality and computational cost for each of the methods in two and three dimensions. We find that the combined approaches are very cost effective and yield highquality meshes.
An Approach to Combined Laplacian and OptimizationBased Smoothing for Triangular, Quadrilateral, and QuadDominant Meshes
 INTERNATIONAL MESHING ROUNDTABLE
, 1998
"... Automatic finite element mesh generation techniques have become commonly used tools for the analysis of complex, realworld models. All of these methods can, however, create distorted and even unusable elements. Fortunately, several techniques exist which can take an existing mesh and improve its qu ..."
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Cited by 70 (4 self)
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Automatic finite element mesh generation techniques have become commonly used tools for the analysis of complex, realworld models. All of these methods can, however, create distorted and even unusable elements. Fortunately, several techniques exist which can take an existing mesh and improve its quality. Smoothing (also referred to as mesh relaxation) is one such method, which repositions nodal locations, so as to minimize element distortion. In this paper, an overall mesh smoothing scheme is presented for meshes consisting of triangular, quadrilateral, or mixed triangular and quadrilateral elements. This paper describes an efficient and robust combination of constrained Laplacian smoothing together with an optimizationbased smoothing algorithm. The smoothing algorithms have been implemented in ANSYS and performance times are presented along with several example models.
Mesh Smoothing Using A Posteriori Error Estimates
 SIAM JOURNAL ON NUMERICAL ANALYSIS
, 1997
"... We develop a simple mesh smoothing algorithm for adaptively improving finite element triangulations. The algorithm makes use of a posteriori error estimates which are now widely used in finite element calculations. In this paper, we derive the method, present some numerical illustrations, and give a ..."
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Cited by 67 (2 self)
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We develop a simple mesh smoothing algorithm for adaptively improving finite element triangulations. The algorithm makes use of a posteriori error estimates which are now widely used in finite element calculations. In this paper, we derive the method, present some numerical illustrations, and give a brief analysis of the issue of uniqueness.
Local OptimizationBased Simplicial Mesh Untangling And Improvement
 International Journal of Numerical Methods in Engineering
"... . We present an optimizationbased approach for mesh untangling that maximizes the minimum area or volume of simplicial elements in a local submesh. These functions are linear with respect to the free vertex position; thus the problem can be formulated as a linear program that is solved by using the ..."
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Cited by 63 (7 self)
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. We present an optimizationbased approach for mesh untangling that maximizes the minimum area or volume of simplicial elements in a local submesh. These functions are linear with respect to the free vertex position; thus the problem can be formulated as a linear program that is solved by using the computationally inexpensive simplex method. We prove that the function level sets are convex regardless of the position of the free vertex, and hence the local subproblem is guaranteed to converge. Maximizing the minimum area or volume of mesh elements, although wellsuited for mesh untangling, is not ideal for mesh improvement, and its use often results in poor quality meshes. We therefore combine the mesh untangling technique with optimizationbased mesh improvement techniques and expand previous results to show that a commonly used twodimensional mesh quality criterion can be guaranteed to converge when starting with a valid mesh. Typical results showing the effectiveness of the combine...
Mesh Generation
 Handbook of Computational Geometry. Elsevier Science
, 2000
"... this article, we emphasize practical issues; an earlier survey by Bern and Eppstein [24] emphasized theoretical results. Although there is inevitably some overlap between these two surveys, we intend them to be complementary. ..."
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Cited by 57 (8 self)
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this article, we emphasize practical issues; an earlier survey by Bern and Eppstein [24] emphasized theoretical results. Although there is inevitably some overlap between these two surveys, we intend them to be complementary.