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Tetrahedral element shape optimization via the jacobian determinant and condition number
 IN PROCEEDINGS OF THE 8TH INTERNATIONAL MESHING ROUNDTABLE
, 1999
"... We present a new shape measure for tetrahedral elements that is optimal in the sense that it gives the distance of a tetrahedron from the set of inverted elements. This measure is constructed from the condition number of the linear transformation between a unit equilateral tetrahedron and any tetra ..."
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Cited by 45 (6 self)
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We present a new shape measure for tetrahedral elements that is optimal in the sense that it gives the distance of a tetrahedron from the set of inverted elements. This measure is constructed from the condition number of the linear transformation between a unit equilateral tetrahedron and any tetrahedron with positive volume. We use this shape measure to formulate two optimization objective functions that are differentiated by their goal: the first seeks to improve the average quality of the tetrahedral mesh; the second aims to improve the worstquality element in the mesh. Because the element condition number is not defined for tetrahedral with negative volume, these objective functions can be used only when the initial mesh is valid. Therefore, we formulate a third objective function using the determinant of the element Jacobian that is suitable for mesh untangling. We review the optimization techniques used with each objective function and present experimental results that demonstrate the effectiveness of the mesh improvement and untangling methods. We show that a combbed optimization approach that uses both condition number objective functions obtains the bestquality meshes.
Volumetric Parameterization and Trivariate Bspline Fitting using Harmonic Functions
"... We present a methodology based on discrete volumetric harmonic functions to parameterize a volumetric model in a way that it can be used to fit a single trivariate Bspline to data so that simulation attributes can also be modeled. The resulting model representation is suitable for isogeometric anal ..."
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Cited by 43 (3 self)
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We present a methodology based on discrete volumetric harmonic functions to parameterize a volumetric model in a way that it can be used to fit a single trivariate Bspline to data so that simulation attributes can also be modeled. The resulting model representation is suitable for isogeometric analysis [Hughes T.J. 2005]. Input data consists of both a closed triangle mesh representing the exterior geometric shape of the object and interior triangle meshes that can represent material attributes or other interior features. The trivariate Bspline geometric and attribute representations are generated from the resulting parameterization, creating trivariate Bspline material property representations over the same parameterization in a way that is related to [Martin and Cohen 2001] but is suitable for application to a much larger family of shapes and attributes. The technique constructs a Bspline representation with guaranteed quality of approximation to the original data. Then we focus attention on a model of simulation interest, a femur, consisting of hard outer cortical bone and inner trabecular bone. The femur is a reasonably complex object to model with a single trivariate Bspline since the shape overhangs make it impossible to model by sweeping planar slices. The representation is used in an elastostatic isogeometric analysis, demonstrating its ability to suitably represent objects for isogeometric analysis.
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
Surface smoothing and quality improvement of quadrilateral/hexahedral meshes with geometric flow
 Communications in Numerical Methods in Engineering
, 2009
"... Abstract: This paper describes an approach to smooth the surface and improve the quality of quadrilateral/hexahedral meshes with feature preserved using geometric flow. For quadrilateral surface meshes, the surface diffusion flow is selected to remove noise by relocating vertices in the normal direc ..."
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Cited by 26 (8 self)
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Abstract: This paper describes an approach to smooth the surface and improve the quality of quadrilateral/hexahedral meshes with feature preserved using geometric flow. For quadrilateral surface meshes, the surface diffusion flow is selected to remove noise by relocating vertices in the normal direction, and the aspect ratio is improved with feature preserved by adjusting vertex positions in the tangent direction. For hexahedral meshes, besides the surface vertex movement in the normal and tangent directions, interior vertices are relocated to improve the aspect ratio. Our method has the properties of noise removal, feature preservation and quality improvement of quadrilateral/hexahedral meshes, and it is especially suitable for biomolecular meshes because the surface diffusion flow preserves sphere accurately if the initial surface is close to a sphere. Several demonstration examples are provided from a wide variety of application domains. Some extracted meshes have been extensively used in finite element simulations. Key words: quadrilateral/hexahedral mesh, surface smoothing, feature preservation, quality improvement, geometric flow. 1
Untangling of 2D meshes in ALE simulations
 J. Comput. Phys
, 2004
"... Department of Energy under contract W7405ENG36. By acceptance of this article, the publisher recognizes that the U.S. Government retains a nonexclusive, royaltyfree license to publish or reproduce the published form of this contribution, or to allow others to do so, for U.S. Government purposes. ..."
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Cited by 23 (5 self)
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Department of Energy under contract W7405ENG36. By acceptance of this article, the publisher recognizes that the U.S. Government retains a nonexclusive, royaltyfree license to publish or reproduce the published form of this contribution, or to allow others to do so, for U.S. Government purposes. Los Alamos National Laboratory requests that the publisher identify this article as work performed under the
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.
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.
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 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 15 (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.