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16
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 95 (11 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.
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 64 (8 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.
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 51 (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.
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 51 (6 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...
An Efficient Parallel Algorithm for Mesh Smoothing
 INTERNATIONAL MESHING ROUNDTABLE
, 1995
"... Automatic mesh generation and adaptive refinement methods have proven to be very successful tools for the efficient solution of complex finite element applications. A problem with these methods is that they can produce poorly shaped elements; such elements are undesirable because they introduce ..."
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Cited by 50 (10 self)
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Automatic mesh generation and adaptive refinement methods have proven to be very successful tools for the efficient solution of complex finite element applications. A problem with these methods is that they can produce poorly shaped elements; such elements are undesirable because they introduce numerical difficulties in the solution process. However, the shape of the elements can be improved through the determination of new geometric locations for mesh vertices by using a mesh smoothing algorithm. In this paper we present a new parallel algorithm for mesh smoothing that has a fast parallel runtime both in theory and in practice. We present an efficient implementation of the algorithm that uses nonsmooth optimization techniques to find the new location of each vertex. Finally, we present experimental results obtained on the IBM SP system demonstrating the efficiency of this approach.
A Comparison of Tetrahedral Mesh Improvement Techniques
 Fifth International Meshing Roundtable
, 1996
"... . 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 le ..."
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Cited by 31 (6 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 faceswapping techniques that change local connectivity and optimizationbased mesh smoothing methods that adjust grid 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. The highest quality meshes are obtained by using a com...
A Cost/Benefit Analysis of Simplicial Mesh Improvement Techniques as Measured by Solution Efficiency
"... ..."
A Comparison of Optimization Software for Mesh ShapeQuality Improvement Problems
, 2002
"... Simplicial mesh shapequality can be improved by optimizing an objective function based on tetrahedral shape measures. If the objective function is formulated in terms of all elements in a given mesh rather than a local patch, one is confronted with a largescale, nonlinear, constrained numerical op ..."
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Cited by 12 (6 self)
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Simplicial mesh shapequality can be improved by optimizing an objective function based on tetrahedral shape measures. If the objective function is formulated in terms of all elements in a given mesh rather than a local patch, one is confronted with a largescale, nonlinear, constrained numerical optimization problem. We investigate the use of six generalpurpose stateoftheart solvers and two customdeveloped methods to solve the resulting largescale problem. The performance of each method is evaluated in terms of robustness, time to solution, convergence properties, and sealability on several two and threedimensional test cases.
Users Manual for OptMS: Local Methods for Simplicial Mesh Smoothing and Untangling
, 1999
"... Creating meshes containing goodquality elements is a challenging, yet critical, problem facing computational scientists today. Several researchers have shown that the size of the mesh, the shape of the elements within that mesh, and their relationship to the physical application of interest can p ..."
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Cited by 7 (1 self)
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Creating meshes containing goodquality elements is a challenging, yet critical, problem facing computational scientists today. Several researchers have shown that the size of the mesh, the shape of the elements within that mesh, and their relationship to the physical application of interest can profoundly affect the efficiency and accuracy of many numerical approximation techniques. If the application contains anisotropic physics, the mesh can be improved by considering both local characteristics of the approximate application solution and the geometry of the computational domain. If the application is isotropic, regularly shaped elements in the mesh reduce the discretization error, and the mesh can be improved a priori by considering geometric criteria only. The OptMS package provides several local node point smoothing techniques that improve elements in the mesh by adjusting grid point location using geometric criteria. The package is easy to use; only three subroutine ca...
Local OptimizationBased Untangling Algorithms for Quadrilateral Meshes
, 2001
"... The generation of a valid computational mesh is an essential step in the solution of many complex scientific and engineering applications. In this paper we present a new, robust algorithm, and several variants, for untangling invalid quadrilateral meshes. The primary computational aspect of the algo ..."
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Cited by 6 (0 self)
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The generation of a valid computational mesh is an essential step in the solution of many complex scientific and engineering applications. In this paper we present a new, robust algorithm, and several variants, for untangling invalid quadrilateral meshes. The primary computational aspect of the algorithm is the solution of a sequence of local linear programs, one for each interior mesh vertex. We show that the optimal solution to these local subproblems can be guaranteed and found efficiently. We present experimental results showing the effectiveness of this approach for problems where invalid, or negative area, elements can arise near highly concave domain boundaries.