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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 86 (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.
Mesh Generation
 HANDBOOK OF COMPUTATIONAL GEOMETRY. ELSEVIER SCIENCE
, 2000
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Adaptive TriangularQuadrilateral Mesh Generation
 Intl. J. Numer. Methods Eng
, 1998
"... In this paper, we begin by recalling an adaptive mesh generation method governed by isotropic and anisotropic discrete metric maps, by means of the generation of a unit mesh with respect to a Riemannian structure. We propose then an automatic triangular to quadrilateral mesh conversion scheme, which ..."
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Cited by 47 (2 self)
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In this paper, we begin by recalling an adaptive mesh generation method governed by isotropic and anisotropic discrete metric maps, by means of the generation of a unit mesh with respect to a Riemannian structure. We propose then an automatic triangular to quadrilateral mesh conversion scheme, which generalizes the standard case to the anisotropic context. In addition, we introduce an optimal vertex smoothing procedure. Application test examples, in particular a CFD test, are given to demonstrate the e ciency of the proposed method.
Anisotropic Triangular Meshing of Parametric Surfaces via Close Packing of Ellipsoidal Bubbles
 In 6th International Meshing Roundtable
, 1996
"... This paper describes a new computational method of fully automated anisotropic triangulation of a trimmed parametric surface. Given as input: (1) a domain geometry and (2) a 3 x 3 tensor field that specifies a desired anisotropic nodespacing, this new approach first packs ellipsoids closely in the ..."
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Cited by 46 (5 self)
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This paper describes a new computational method of fully automated anisotropic triangulation of a trimmed parametric surface. Given as input: (1) a domain geometry and (2) a 3 x 3 tensor field that specifies a desired anisotropic nodespacing, this new approach first packs ellipsoids closely in the domain by defining proximitybased interacting forces among the ellipsoids and finding a forcebalancing configuration using dynamic simulation. The centers of the ellipsoids are then connected by anisotropic Delaunay triangulation for a complete mesh topology. Since a specified tensor field controls the directions and the lengths of the ellipsoids' principal axes, the method generates a high quality anisotropic mesh whose elements conform precisely to the given tensor field. Keywords: unstructured mesh, anisotropy, parametric surface, metric tensor, Delaunay triangulation 1 Introduction Although most automatic mesh generators try to create a regular isotropic mesh, in some FEM analysis an ...
Mesh Gradation Control
 Int. J. Numer. Methods Eng
, 1998
"... This paper gives a procedure to control the size variation in a mesh adaption scheme where the size speci cation (the socalled control space) is used to govern the mesh generation stage. The method consists in replacing the initial control space by a reduced one by means of size or metric. It allow ..."
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Cited by 36 (5 self)
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This paper gives a procedure to control the size variation in a mesh adaption scheme where the size speci cation (the socalled control space) is used to govern the mesh generation stage. The method consists in replacing the initial control space by a reduced one by means of size or metric. It allows to improve, a priori, the quality of the adapted mesh and to speed up the adaption procedure. Several numerical examples show the e ciency of the reduction scheme.? 1998 John Wiley & Sons, Ltd. KEY WORDS: mesh gradation; mesh adaption; surface mesh generation; anisotropic mesh 1.
Anisotropic Mesh Optimization And Its Application In Adaptivity
 in Adaptivity, International Journal for Numerical Methods in Engineering
, 1997
"... The construction of solutionadapted meshes is addressed within an optimization framework. An approximation of the second spatial derivative of the solution is used to get a suitable metric in the computational domain. A mesh quality is proposed and optimized under this metric, accounting for bot ..."
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Cited by 31 (2 self)
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The construction of solutionadapted meshes is addressed within an optimization framework. An approximation of the second spatial derivative of the solution is used to get a suitable metric in the computational domain. A mesh quality is proposed and optimized under this metric, accounting for both the shape and the size of the elements. For this purpose, a topological and geometrical mesh improvement method of high generality is introduced. It is shown that the adaptive algorithm that results recovers optimal convergence rates in singular problems, and that it captures boundary and internal layers in convectiondominated problems. Several important implementation issues are discussed. 1 Introduction Quite recently, we have developed mesh optimization algorithms suitable for both 2D and 3D unstructured meshes [1]. Our aim was to relieve the (frontal, Delaunay, etc.) meshing algorithm from the strict elementshape restrictions imposed by numerical solvers. It was shown that goo...
A New Methodology for Anisotropic Mesh Refinement Based Upon Error Gradients
, 2001
"... We introduce a new strategy for controlling the use of anisotropic mesh refinement based upon the gradients of an a posteriori approximation of the error in a computed finite element solution. The efficiency of this strategy is demonstrated using a simple anisotropic mesh adaption algorithm and the ..."
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Cited by 30 (7 self)
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We introduce a new strategy for controlling the use of anisotropic mesh refinement based upon the gradients of an a posteriori approximation of the error in a computed finite element solution. The efficiency of this strategy is demonstrated using a simple anisotropic mesh adaption algorithm and the quality of a number of potential a posteriori error estimates is considered.
Anisotropic Mesh Construction and Error Estimation in the Finite Element Method
, 2000
"... In an anisotropic adaptive finite element algorithm one usually needs an error estimator that yields the error size but also the stretching directions and stretching ratios of the elements of a (quasi) optimal anisotropic mesh. However the last two ingredients can not be extracted from any of the kn ..."
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Cited by 22 (2 self)
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In an anisotropic adaptive finite element algorithm one usually needs an error estimator that yields the error size but also the stretching directions and stretching ratios of the elements of a (quasi) optimal anisotropic mesh. However the last two ingredients can not be extracted from any of the known anisotropic a posteriori error estimators. Therefore a heuristic approach is pursued here, namely, the desired information is provided by the socalled Hessian strategy. This strategy produces favourable anisotropic meshes which result in a small discretization error. The focus of this paper is on error estimation on anisotropic meshes. It is known that such error estimation is reliable and efficient only if the anisotropic mesh is aligned with the anisotropic solution. The main result here is that the Hessian strategy produces anisotropic meshes that show the required alignment with the anisotropic solution. The corresponding inequalities are proven, and the underlying heuristic assum...
Anisotropic Triangulation of Parametric Surfaces via Close Packing of Ellipsoids
 International Journal on Computational Geometry and Applications
"... This paper describes a new computational method of fully automated anisotropic triangulation of a trimmed parametric surface. Given as input: (1) a domain geometry and (2) a 3 x 3 tensor field that specifies a desired anisotropic nodespacing, this new approach first packs ellipsoids closely in the ..."
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Cited by 18 (4 self)
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This paper describes a new computational method of fully automated anisotropic triangulation of a trimmed parametric surface. Given as input: (1) a domain geometry and (2) a 3 x 3 tensor field that specifies a desired anisotropic nodespacing, this new approach first packs ellipsoids closely in the domain by defining proximitybased interacting forces among the ellipsoids and finding a forcebalancing configuration using dynamic simulation. The centers of the ellipsoids are then connected by anisotropic Delaunay triangulation for a complete mesh topology. Since a specified tensor field controls the directions and the lengths of the ellipsoids ' principal axes, the method generates a graded anisotropic mesh whose elements conform precisely to the given tensor field.
Biting Ellipses to Generate Anisotropic Mesh
 In 8th International Meshing Roundtable
, 1999
"... . In numerical simulation where the underlying function is strongly directional, it is desirable to use a mesh that is adaptive both in size and in shape. In such simulation, a metric tensor is used to quantify the ideal size and direction locally at each point in the domain, which in turn defines t ..."
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Cited by 17 (4 self)
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. In numerical simulation where the underlying function is strongly directional, it is desirable to use a mesh that is adaptive both in size and in shape. In such simulation, a metric tensor is used to quantify the ideal size and direction locally at each point in the domain, which in turn defines the local stretching and size of the triangles or quadrilaterals of the mesh. Given a metric tensor, the anisotropic meshing problem is to construct a good quality mesh satisfying the metric tensor. We present a new anisotropic meshing method which is called the ellipse biting method. Our algorithm uses the framework of advancing front to generate a close to optimal packing of ellipses. We then use the aeDelaunay triangulation of the vertex set to generate the final mesh. Because it generates an ellipse packing that respects the underlying control spacing, this new method produce a high quality mesh whose element size and directionality conform well locally to the given input. As part of thi...