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14
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 72 (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 quality: a function of geometry, error estimates or both
 Eng. Comput
, 1999
"... Abstract. The issue of mesh quality for unstructured triangular and tetrahedral meshes is considered. The theoretical background to finite element methods is used to understand the basis of presentday geometrical mesh quality indicators. A survey of more recent research in the development of finit ..."
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Cited by 24 (3 self)
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Abstract. The issue of mesh quality for unstructured triangular and tetrahedral meshes is considered. The theoretical background to finite element methods is used to understand the basis of presentday geometrical mesh quality indicators. A survey of more recent research in the development of finite element methods reveals work on anisotropic meshing algorithms and on providing good error estimates that reveal the relationship between the error and both the mesh and the solution gradients. The reality of solving complex three dimensional problems is that such indicators are presently not available for many problems of interest. A simple tetrahedral mesh quality measure using both geometrical and solution information is described. Some of the issues in mesh quality for unstructured tetrahedral meshes are illustrated by means of two simple examples.
Mesh ShapeQuality Optimization Using the Inverse MeanRatio Metric
 Preprint ANL/MCSP11360304, Argonne National Laboratory, Argonne
, 2004
"... Meshes containing elements with bad quality can result in poorly conditioned systems of equations that must be solved when using a discretization method, such as the finiteelement method, for solving a partial differential equation. Moreover, such meshes can lead to poor accuracy in the approximate ..."
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Cited by 20 (4 self)
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Meshes containing elements with bad quality can result in poorly conditioned systems of equations that must be solved when using a discretization method, such as the finiteelement method, for solving a partial differential equation. Moreover, such meshes can lead to poor accuracy in the approximate solution computed. In this paper, we present a nonlinear fractional program that relocates the vertices of a given mesh to optimize the average element shape quality as measured by the inverse meanratio metric. To solve the resulting largescale optimization problems, we apply an efficient implementation of an inexact Newton algorithm using the conjugate gradient method with a block Jacobi preconditioner to compute the direction. We show that the block Jacobi preconditioner is positive definite by proving a general theorem concerning the convexity of fractional functions, applying this result to components of the inverse meanratio metric, and showing that each block in the preconditioner is invertible. Numerical results obtained with this specialpurpose code on several test meshes are presented and used to quantify the impact on solution time and memory requirements of using a modeling language and generalpurpose algorithm to solve these problems. 1
A LogBarrier Method for Mesh Quality Improvement
"... Summary. The presence of a few poorquality mesh elements can negatively affect the stability and efficiency of a finite element solver and the accuracy of the associated partial differential equation solution. We propose a mesh quality improvement method that improves the quality of the worst eleme ..."
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Cited by 2 (1 self)
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Summary. The presence of a few poorquality mesh elements can negatively affect the stability and efficiency of a finite element solver and the accuracy of the associated partial differential equation solution. We propose a mesh quality improvement method that improves the quality of the worst elements. Mesh quality improvement of the worst elements can be formulated as a nonsmooth unconstrained optimization problem, which can be reformulated as a smooth constrained optimization problem. Our technique solves the latter problem using a logbarrier interior point method and uses the gradient of the objective function to efficiently converge to a stationary point. The technique can be used with convex or nonconvex quality metrics. The method uses a logarithmic barrier function and performs global mesh quality improvement. Our method usually yields better quality meshes than existing methods for improvement of the worst quality elements, such as the active set, pattern search, and multidirectional search mesh quality improvement methods.
The FeasNewt Benchmark
"... Abstract — We describe the FeasNewt meshquality optimization benchmark. The performance of the code is dominated by three phases—gradient evaluation, Hessian evaluation and assembly, and sparse matrixvector products—that have very different mixtures of floatingpoint operations and memory access pa ..."
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Cited by 2 (0 self)
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Abstract — We describe the FeasNewt meshquality optimization benchmark. The performance of the code is dominated by three phases—gradient evaluation, Hessian evaluation and assembly, and sparse matrixvector products—that have very different mixtures of floatingpoint operations and memory access patterns. The code includes an optional runtime data and iterationreordering phase, making it suitable for research on irregular memory access patterns. Meshquality optimization (or “mesh smoothing”) is an important ingredient in the solution of nonlinear partial differential equations (PDEs) as well as an excellent surrogate for finiteelement or finitevolume PDE solvers. I.
Geometric Error Estimation
 in "ADVCOMP 2010: The Fourth International Conference on Advanced Engineering Computing and Applications in Sciences, IARIA conference
"... Abstract—An essential prerequisite for the numerical finite element simulation of physical problems expressed in terms of PDEs is the construction of an adequate mesh of the domain. This first stage, which usually involves a fully automatic mesh generation method, is then followed by a computational ..."
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Abstract—An essential prerequisite for the numerical finite element simulation of physical problems expressed in terms of PDEs is the construction of an adequate mesh of the domain. This first stage, which usually involves a fully automatic mesh generation method, is then followed by a computational step. One can show that the quality of the solution strongly depends on the shape quality of the mesh of the domain. At the second stage, the numerical solution obtained with the initial mesh is generally analyzed using an appropriate a posteriori error estimator which, based on the quality of the solution, indicates whether or not the solution is accurate. The quality of the solution is closely related to how well the mesh corresponds to the underlying physical phenomenon, which can be quantified by the element sizes of the mesh. An a posteriori error estimation based on the interpolation error depending on the Hessian of the solution seems to be well adapted to the purpose of adaptive meshing. In this paper, we propose a new interpolation error estimation based on the local deformation of the Cartesian surface representing the solution. This methodology is generally used in the context of surface meshing. In our example, the proposed methodology is applied to minimize the interpolation error on an image whose grey level is considered as being the solution. Keywordsa posteriori error estimation; interpolation error; mesh adaptation; surface curvature. I.
Function approximation on triangular grids: some numerical results using adaptive techniques
, 1998
"... Applications of mesh adaption techniques could be found in the numerical solution of PDE’s or in the optimal triangulation of surfaces for shape representation or graphic display. The scope of this work is to verify through numerical experiments the effectiveness of some algorithms for the control ..."
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Cited by 1 (0 self)
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Applications of mesh adaption techniques could be found in the numerical solution of PDE’s or in the optimal triangulation of surfaces for shape representation or graphic display. The scope of this work is to verify through numerical experiments the effectiveness of some algorithms for the control of the L ∞ error norm for piece–wise linear approximation on 2D unstructured triangular meshes. The analysis could be extended to parametric surfaces and to the 3D case.
elastohydrodynamic
"... the linear finite element analysis of fullycoupled point contact ..."
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