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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 64 (7 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...
Simulating needle insertion and radioactive seed implantation for prostate brachytherapy
 in Medicine Meets Virtual Reality 11
, 2003
"... Abstract. We are developing a simulation of needle insertion and radioactive seed implantation to facilitate surgeon training and planning for brachytherapy for treating prostate cancer. Inserting a needle into soft tissues causes the tissues to displace and deform: ignoring these effects during see ..."
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Cited by 45 (10 self)
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Abstract. We are developing a simulation of needle insertion and radioactive seed implantation to facilitate surgeon training and planning for brachytherapy for treating prostate cancer. Inserting a needle into soft tissues causes the tissues to displace and deform: ignoring these effects during seed implantation leads to imprecise seed placements. Surgeons should learn to compensate for these effects so seeds are implanted close to their preplanned locations. We describe a new 2D dynamic FEM model based on a 7phase insertion sequence where the mesh is updated to maintain element boundaries along the needle shaft. The locations of seed implants are predicted as the tissue deforms. The simulation, which achieves 24 frames per second using a 1250 triangular element mesh on a 750Mhz Pentium III PC, is available for surgeon testing by contacting ron@ieor.berkeley.edu. 1.
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
GuaranteedQuality Simplicial Mesh Generation with cell size and grading control
 Computers
, 2001
"... Unstructured mesh quality, as measured geometrically, has long been known to influence solution accuracy and efficiency for finiteelement and finitevolume simulations. ..."
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Cited by 11 (4 self)
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Unstructured mesh quality, as measured geometrically, has long been known to influence solution accuracy and efficiency for finiteelement and finitevolume simulations.
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 8 (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...
OptimizationBased Quadrilateral and Hexahedral Mesh Untangling and Smoothing Techniques
, 1999
"... The accuracy and efficiency of the solution to numerical systems depends on the quality of the mesh used. Automatic mesh generation and adaptive refinement methods can, however, produce poor quality elements, even invalid elements, for quadrilateral or hexahedral mesh. We present an optimization ..."
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Cited by 4 (1 self)
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The accuracy and efficiency of the solution to numerical systems depends on the quality of the mesh used. Automatic mesh generation and adaptive refinement methods can, however, produce poor quality elements, even invalid elements, for quadrilateral or hexahedral mesh. We present an optimizationbased approach for quadrilateral or hexahedral mesh untangling that maximizes the minimum area or volume of affected simplicial of an element in a local submesh. We formulate the problem as a linear program that is solved by using the computationally inexpensive simplex method. By proving that the function level sets are convex regardless of the position of the free vertex, we show that the local subproblem is guaranteed to converge. We combine the mesh untangling technique with optimizationbased mesh improvement techniques. Typical results showing the effectiveness of the combined untangling and smoothing techniques are given for both quadrilateral and hexahedral meshes.
Revisiting Delaunay Refinement Triangular Mesh Generation on Curvebounded Domains
"... An extension of Shewchuk’s Delaunay Refinement algorithm to planar domains bounded by curves is presented. A novel method is applied to construct constrained Delaunay triangulations from such geometries. The quality of these triangulations is then improved by inserting vertices at carefully chosen l ..."
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Cited by 1 (0 self)
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An extension of Shewchuk’s Delaunay Refinement algorithm to planar domains bounded by curves is presented. A novel method is applied to construct constrained Delaunay triangulations from such geometries. The quality of these triangulations is then improved by inserting vertices at carefully chosen locations. Amendments to Shewchuk’s insertion rules are necessary to handle cases resulting from the presence of curves. The algorithm accepts small input angles, although special provisions must be taken in their neighborhood. In practice, our algorithm generates graded triangular meshes where no angle is less than 30 ◦ , except near small input angles. Such meshes are suitable for use in numerical simulations. 1