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44
Algebraic mesh quality metrics
 SIAM J.Sci.Comput
"... Abstract. Quality metrics for structured and unstructured mesh generation are placed within an algebraic framework to form a mathematical theory of mesh quality metrics. The theory, based on the Jacobian and related matrices, provides a means of constructing, classifying, and evaluating mesh quality ..."
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Cited by 57 (3 self)
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Abstract. Quality metrics for structured and unstructured mesh generation are placed within an algebraic framework to form a mathematical theory of mesh quality metrics. The theory, based on the Jacobian and related matrices, provides a means of constructing, classifying, and evaluating mesh quality metrics. The Jacobian matrix is factored into geometrically meaningful parts. A nodally invariant Jacobian matrix can be defined for simplicial elements using a weight matrix derived from the Jacobian matrix of an ideal reference element. Scale and orientationinvariant algebraic mesh quality metrics are defined. The singular value decomposition is used to study relationships between metrics. Equivalence of the element condition number and mean ratio metrics is proved. The condition number is shown to measure the distance of an element to the set of degenerate elements. Algebraic measures for skew, length ratio, shape, volume, and orientation are defined abstractly, with specific examples given. Two combined metrics, shapevolume and shapevolume orientation, are algebraically defined and examples of such metrics are given. Algebraic mesh quality metrics are extended to nonsimplicial elements. A series of numerical tests verifies the theoretical properties of the metrics defined.
Reference Jacobian OptimizationBased Rezone Strategies for Arbitrary Lagrangian Eulerian Methods
 Journal of Computational Physics
"... The philosophy of the Arbitrary LagrangianEulerian (ALE) methodology for solving multidimensional uid ow problems is to move the computational grid, using the ow as a guide, to improve the accuracy and eciency of the simulation. A principal element of ALE is the rezone phase in which a "rezone ..."
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Cited by 42 (16 self)
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The philosophy of the Arbitrary LagrangianEulerian (ALE) methodology for solving multidimensional uid ow problems is to move the computational grid, using the ow as a guide, to improve the accuracy and eciency of the simulation. A principal element of ALE is the rezone phase in which a "rezoned" grid is created that is adapted to the uid motion. We will describe a general rezone strategy that ensures the continuing geometric quality of the computational grid, while keeping "rezoned" grid at each time step as close as possible to the Lagrangian grid. Although the methodology can be applied to more general grid types, here we restrict ourselves to logically rectangular grids in two dimensions. The rezoning phase consists of two components: a sequence of local optimizations followed by a single global optimization. The local optimization denes a "reference" Jacobians which incorporates our denition of mesh quality at each point of the grid. The set of reference Jacobians then is used in the global optimization. At each node we form a local patch from the adjacent cells of Lagrangian grid and construct a local realization of the Winslow smoothness functional on this patch. Minimization of this functional with respect to position of central node denes its 'virtual' location (the node is not actually moved at this stage). By connecting this virtuallymoved node to its (stationary) neighbors, we dene a reference Jacobian that represents the best locally achievable geometric grid quality. The "rezoned" grid results from a minimization (where the points are actually moved) of a global objective function that measures the distance (in a leastsquares sense) between the Jacobian of the rezoned grid and the reference Jacobian. This objective function includes a "barrier" ...
Tetrahedral Mesh Improvement Via Optimization of the Element Condition Number
, 2002
"... this paper for smoothing and untangling are local techniques; a globally optimal solution is not guaranteed although empirical evidence suggests Copyright c fl 2000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2000; 0:00 Prepared using nmeauth.cls ..."
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Cited by 28 (4 self)
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this paper for smoothing and untangling are local techniques; a globally optimal solution is not guaranteed although empirical evidence suggests Copyright c fl 2000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2000; 0:00 Prepared using nmeauth.cls
Matrix Norms & the Condition Number: A General Framework to Improve Mesh Quality via NodeMovement
 Eighth International Meshing Roundtable (Lake Tahoe, California
, 1999
"... Objective functions for unstructured hexahedral and tetrahedral mesh optimization are analyzed using matrices and matrix norms. Mesh untangling objective functions that create valid meshes are used to initialize the optimization process. Several new objective functions to achieve element invertibili ..."
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Cited by 21 (1 self)
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Objective functions for unstructured hexahedral and tetrahedral mesh optimization are analyzed using matrices and matrix norms. Mesh untangling objective functions that create valid meshes are used to initialize the optimization process. Several new objective functions to achieve element invertibility and quality are investigated, the most promising being the “condition number”. The condition number of the Jacobian matrix of an element forms the basis of a barrierbased objective function that measures the distance to the set of singular matrices and has the ideal matrix as a stationary point. The method was implemented in the Cubit code, with promising results.
Mesh Movement and Metamorphosis
 in Proceedings of the Tenth International Meshing Roundtable, Albuquerque, NM, 2001, Sandia National Laboratories
, 2001
"... Mesh coarsening and mesh enrichment are combined with an rre nement scheme to produce a exible approach for mesh adaptation of time evolving domains. The robustness of this method depends heavily on maintaining mesh quality during each adaptation cycle. This in turn is inuenced by the ability to i ..."
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Cited by 21 (0 self)
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Mesh coarsening and mesh enrichment are combined with an rre nement scheme to produce a exible approach for mesh adaptation of time evolving domains. The robustness of this method depends heavily on maintaining mesh quality during each adaptation cycle. This in turn is inuenced by the ability to identify and remove badly shaped elements after the rre nement stage. Measures of both element quality and element deformation can be de ned in terms of unitarily invariant matrix norms. The construction of these element deformation and quality measures is described, and details are provided of the three stages of the adaptation cycle.
Simultaneous untangling and smoothing of tetrahedral meshes
 Comput. Meth. in
, 2003
"... The quality improvement in mesh optimisation techniques that preserve its connectivity are obtained by an iterative process in which each node of the mesh is moved to a new position that minimises a certain objective function. The objective function is derived from some quality measure of the local ..."
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Cited by 20 (2 self)
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The quality improvement in mesh optimisation techniques that preserve its connectivity are obtained by an iterative process in which each node of the mesh is moved to a new position that minimises a certain objective function. The objective function is derived from some quality measure of the local submesh, that is, the set of tetrahedra connected to the adjustable or free node. Although these objective functions are suitable to improve the quality of a mesh in which there are noninverted elements, they are not when the mesh is tangled. This is due to the fact that usual objective functions are not defined on all R 3 and they present several discontinuities and local minima that prevent the use of conventional optimisation procedures. Otherwise, when the mesh is tangled, there are local submeshes for which the free node is out of the feasible region, or this does not exist. In this paper we propose the substitution of objective functions having barriers by modified versions that are defined and regular on all R 3. With these modifications, the optimisation process is also directly applicable to meshes with inverted elements, making a previous untangling procedure unnecessary. This simultaneous procedure allows to reduce the number of iterations for reaching a prescribed quality. To illustrate the effectiveness of our approach, we present several applications where it can be seen that our results clearly improve those obtained by other authors.
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 9 (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.
A Mesh Warping Algorithm Based on Weighted Laplacian Smoothing
, 2004
"... We present a new mesh warping algorithm for tetrahedral meshes based upon weighted laplacian smoothing. We start with a 3D domain which is bounded by a triangulated surface mesh and has a tetrahedral volume mesh as its interior. We then suppose that a movement of the surface mesh is prescribed and u ..."
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Cited by 8 (1 self)
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We present a new mesh warping algorithm for tetrahedral meshes based upon weighted laplacian smoothing. We start with a 3D domain which is bounded by a triangulated surface mesh and has a tetrahedral volume mesh as its interior. We then suppose that a movement of the surface mesh is prescribed and use our mesh warping algorithm to update the nodes of the volume mesh. Our method determines a set of local weights for each interior node which describe the relative distances of the node to its neighbors. After a boundary transformation is applied, the method solves a system of linear equations based upon the weights to determine the final position of the interior nodes. We study mesh invertibility and prove a theorem which gives su#cient conditions for a mesh to resist inversion by a transformation. We prove that our algorithm yields exact results for a#ne mappings and state a conjecture for more general mappings. In addition, we prove that our algorithm converges to the same point as both the local weighted laplacian smoothing algorithm and the GaussSeidel algorithm for linear systems. We test our algorithm's robustness and present some numerical results. Finally, we use our algorithm to study the movement of the canine heart.
ForceDirected Methods For Smoothing Unstructured Triangular And Tetrahedral Meshes
 In Proceedings of the 9th International Meshing Roundtable
, 2000
"... We develop and implement new algorithms for smoothing triangular and tetrahedral unstructured meshes. Our approach is based on a variation of the forcedirected method used in graph drawing. This method assumes that on each vertex a certain force is applied that moves the vertex relative to its neig ..."
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Cited by 8 (0 self)
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We develop and implement new algorithms for smoothing triangular and tetrahedral unstructured meshes. Our approach is based on a variation of the forcedirected method used in graph drawing. This method assumes that on each vertex a certain force is applied that moves the vertex relative to its neighbors so that the shapes of its incident elements are improved. The final stable configuration often corresponds to a graph with good global properties. In this paper we show that this method can be successfully applied to mesh smoothing and describe some details of our implementation and test results.
Hexahedral Mesh Generation for the Simulation of the Human Mandible
 In Proceedings of the 9th International Meshing Roundtable, Sandia National Laboratories
, 2000
"... A combinatorial approach for the generation of hexahedral meshes by means of successive dual cycle elimination has been proposed by the second author in previous work. We provide a case study for the applicability of our hexahedral mesh generation approach to the simulation of physiological stress s ..."
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Cited by 6 (0 self)
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A combinatorial approach for the generation of hexahedral meshes by means of successive dual cycle elimination has been proposed by the second author in previous work. We provide a case study for the applicability of our hexahedral mesh generation approach to the simulation of physiological stress scenarios of the human mandible. Due to its complex and very detailed freeform geometry, the mandible model is very demanding. This test case is used as a running example to report on the progress and recent advances of the cycle elimination scheme. The given input data, a surface triangulation, requires a substantial mesh reduction and a suitable conversion into a quadrilateral surface mesh as a first step, for which we use mesh clustering and bmatching techniques. Several strategies for improved cycle elimination orders are proposed. They lead to a significant reduction in the mesh size and a better structural quality. Based on the resulting combinatorial meshes, gradientbased optimize...