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66
Tetrahedral Mesh Improvement Using Swapping and Smoothing
 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
, 1997
"... Automatic mesh generation and adaptive refinement methods for complex threedimensional domains have proven to be very successful tools for the efficient solution of complex applications problems. These methods can, however, produce poorly shaped elements that cause the numerical solution to be less ..."
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Cited by 109 (12 self)
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Automatic mesh generation and adaptive refinement methods for complex threedimensional domains have proven to be very successful tools for the efficient solution of complex applications problems. These methods can, however, produce poorly shaped elements that cause the numerical solution to be less accurate and more difficult to compute. Fortunately, the shape of the elements can be improved through several mechanisms, including face and edgeswapping techniques, which change local connectivity, and optimizationbased mesh smoothing methods, which adjust mesh point location. We consider several criteria for each of these two methods and compare the quality of several meshes obtained by using different combinations of swapping and smoothing. Computational experiments show that swapping is critical to the improvement of general mesh quality and that optimizationbased smoothing is highly effective in eliminating very small and very large angles. Highquality meshes are obtained in a computationally efficient manner by using optimizationbased smoothing to improve only the worst elements and a smart variant of Laplacian smoothing on the remaining elements. Based on our experiments, we offer several recommendations for the improvement of tetrahedral meshes.
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.
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 63 (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...
Mesh Generation
 HANDBOOK OF COMPUTATIONAL GEOMETRY. ELSEVIER SCIENCE
, 2000
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Adaptive numerical treatment of elliptic systems on manifolds
 Advances in Computational Mathematics, 15(1):139
, 2001
"... ABSTRACT. Adaptive multilevel finite element methods are developed and analyzed for certain elliptic systems arising in geometric analysis and general relativity. This class of nonlinear elliptic systems of tensor equations on manifolds is first reviewed, and then adaptive multilevel finite element ..."
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Cited by 56 (25 self)
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ABSTRACT. Adaptive multilevel finite element methods are developed and analyzed for certain elliptic systems arising in geometric analysis and general relativity. This class of nonlinear elliptic systems of tensor equations on manifolds is first reviewed, and then adaptive multilevel finite element methods for approximating solutions to this class of problems are considered in some detail. Two a posteriori error indicators are derived, based on local residuals and on global linearized adjoint or dual problems. The design of Manifold Code (MC) is then discussed; MC is an adaptive multilevel finite element software package for 2 and 3manifolds developed over several years at Caltech and UC San Diego. It employs a posteriori error estimation, adaptive simplex subdivision, unstructured algebraic multilevel methods, global inexact Newton methods, and numerical continuation methods for the numerical solution of nonlinear covariant elliptic systems on 2 and 3manifolds. Some of the more interesting features of MC are described in detail, including some new ideas for topology and geometry representation in simplex meshes, and an unusual partition of unitybased method for exploiting parallel computers. A short example is then given which involves the Hamiltonian and momentum constraints in the Einstein equations, a representative nonlinear 4component covariant elliptic system on a Riemannian 3manifold which arises in general relativity. A number of operator properties and solvability results recently established are first summarized, making possible two quasioptimal a priori error estimates for Galerkin approximations which are then derived. These two results complete the theoretical framework for effective use of adaptive multilevel finite element methods. A sample calculation using the MC software is then presented.
A New Paradigm for Parallel Adaptive Meshing Algorithms
 SIAM J. Sci. Comput
, 2003
"... We present a new approach to the use of parallel computers with adaptive finite element methods. This approach addresses the load balancing problem in a new way, requiring far less communication than current approaches. It also allows existing sequential adaptive PDE codes such as PLTMG and MC to ru ..."
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Cited by 46 (9 self)
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We present a new approach to the use of parallel computers with adaptive finite element methods. This approach addresses the load balancing problem in a new way, requiring far less communication than current approaches. It also allows existing sequential adaptive PDE codes such as PLTMG and MC to run in a parallel environment without a large investment in recoding. In this new approach, the load balancing problem is reduced to the numerical solution of a small elliptic problem on a single processor, using a sequential adaptive solver, without requiring any modifications to the sequential solver. The small elliptic problem is used to produce a posteriori error estimates to predict future element densities in the mesh, which are then used in a weighted recursive spectral bisection of the initial mesh. The bulk of the calculation then takes place independently on each processor, with no communication, using possibly the same sequential adaptive solver. Each processor adapts its region of the mesh independently, and a nearly loadbalanced mesh distribution is usually obtained as a result of the initial weighted spectral bisection. Only the initial fanout of the mesh decomposition to the processors requires communication. Two additional steps requiring boundary exchange communication may be employed after the individual processors reach an adapted solution, namely, the construction of a global conforming mesh from the independent subproblems, followed by a final smoothing phase using the subdomain solutions as an initial guess. We present a series of convincing numerical experiments that illustrate the e#ectiveness of this approach. The justification of the initial refinement prediction step, as well as the justification of skipping the two communicationintensive steps, ...
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.
Local and parallel finite element algorithms based on twogrid discretizations
 Math. Comput
"... Abstract. A number of new local and parallel discretization and adaptive finite element algorithms are proposed and analyzed in this paper for elliptic boundary value problems. These algorithms are motivated by the observation that, for a solution to some elliptic problems, low frequency components ..."
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Cited by 37 (13 self)
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Abstract. A number of new local and parallel discretization and adaptive finite element algorithms are proposed and analyzed in this paper for elliptic boundary value problems. These algorithms are motivated by the observation that, for a solution to some elliptic problems, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel procedure. The theoretical tools for analyzing these methods are some local a priori and a posteriori estimates that are also obtained in this paper for finite element solutions on general shaperegular grids. Some numerical experiments are also presented to support the theory. 1.
Variational mesh adaptation II: error estimates and monitor functions
, 2003
"... The ke tothe succev of a variationalmer adaptationmeapt is tode99 aprope monitor function which controlsmet adaptation. In this papewe studythe choice ofthe monitor function forthe variationaladaptive mep mept deptive inthe pre)]]E work [J. Comput. Phys. 174 (2001) 924]. Twotype of monitor functions ..."
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Cited by 37 (12 self)
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The ke tothe succev of a variationalmer adaptationmeapt is tode99 aprope monitor function which controlsmet adaptation. In this papewe studythe choice ofthe monitor function forthe variationaladaptive mep mept deptive inthe pre)]]E work [J. Comput. Phys. 174 (2001) 924]. Twotype of monitor functions, scalar matrix and nonscalar matrixonei are deix base on asymptoticesymptot ofinteq9vC)E)x eee obtaine usingthe intev]q)x4vC thee offinite eite4 mete4A The choice ofthe adaptationintetati parameti is alsodiscusse for eor ofthe( monitor functions. Asymptotic bounds oninte)9vC)A4 eee are obtaine foradaptive mepti that satisfythe resfyvA and ev(]]49vC)xExv conditions.Twodimes.v)xnumedime reedi are give tove]x9 the thex9E(vC) findings.
A Parallel Algorithm for Mesh Smoothing
"... Maintaining good mesh quality during the generation and refinement of unstructured meshes in finiteelement applications is an important aspect in obtaining accurate discretizations and wellconditioned linear systems. In this article, we present a meshsmoothing algorithm based on nonsmooth optimiz ..."
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Cited by 29 (7 self)
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Maintaining good mesh quality during the generation and refinement of unstructured meshes in finiteelement applications is an important aspect in obtaining accurate discretizations and wellconditioned linear systems. In this article, we present a meshsmoothing algorithm based on nonsmooth optimization techniques and a scalable implementation of this algorithm. We report mesh improvement results for twodimensional simplicial meshes that demonstrate the effectiveness of this approach for a number of different test cases. We also show the scalability of the parallel algorithm on the IBM SP supercomputer and an ATMconnected network of SPARC Ultras. 1 Introduction Unstructured meshes have proven to be an essential tool in the numerical solution of largescale scientific and engineering applications on complex computational domains. A problem with such meshes is that the shape of the elements in the mesh can vary significantly, and this variation can affect the accuracy of the numerical ...