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58
Solving elliptic finite element systems in nearlinear time with support preconditioners
 Manuscript, Sandia National
"... Abstract. We show in this note how support preconditioners can be applied to a class of linear systems arising from use of the finite element method to solve linear elliptic problems. Our technique reduces the problem, which is symmetric and positive definite, to a symmetric positive definite diagon ..."
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Cited by 32 (1 self)
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Abstract. We show in this note how support preconditioners can be applied to a class of linear systems arising from use of the finite element method to solve linear elliptic problems. Our technique reduces the problem, which is symmetric and positive definite, to a symmetric positive definite diagonally dominant problem. Significant theory has already been developed for preconditioners in the diagonally dominant case. We show that the degradation in the quality of the preconditioner using our technique is only a small constant factor. 1. Introduction. Finite
Adaptivity with moving grids
, 2009
"... In this article we look at the modern theory of moving meshes as part of an radaptive strategy for solving partial differential equations with evolving internal structure. We firstly examine the possible geometries of a moving mesh in both one and higher dimensions, and the discretization of partia ..."
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Cited by 28 (5 self)
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In this article we look at the modern theory of moving meshes as part of an radaptive strategy for solving partial differential equations with evolving internal structure. We firstly examine the possible geometries of a moving mesh in both one and higher dimensions, and the discretization of partial differential equation on such meshes. In particular, we consider such issues as mesh regularity, equidistribution, variational methods, and the error in interpolating a function or truncation error on such a mesh. We show that, guided by these, we can design effective moving mesh strategies. We then look in more detail as to how these strategies are implemented. Firstly we look at positionbased methods and the use of moving mesh partial differential equation (MMPDE), variational and optimal transport methods. This is followed by an analysis of velocitybased methods such as the geometric conservation law (GCL) methods. Finally we look at a number of examples where the use of a moving mesh method is effective in applications. These include scaleinvariant problems, blowup problems, problems with moving fronts and problems in meteorology. We conclude that, whilst radaptive methods are still in a relatively new stage of development, with many outstanding questions remaining, they have enormous potential for development, and for many problems they represent an optimal form of adaptivity.
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
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.
Benchmarking optimization software with cops 3.0
 MATHEMATICS AND COMPUTER SCIENCE DIVISION, ARGONNE NATIONAL LABORATORY
, 2004
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A twodimensional unstructured cellcentered multimaterial ALE scheme using VOF interface reconstruction
, 2010
"... We present a new cellcentered multimaterial Arbitrary Lagrangian Eulerian (ALE) scheme to solve the compressible gaz dynamics equations on twodimensional unstructured grid. Our ALE method is of the explicit timemarching Lagrange plus remap type. Namely, it involves the following three phases: a ..."
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Cited by 13 (2 self)
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We present a new cellcentered multimaterial Arbitrary Lagrangian Eulerian (ALE) scheme to solve the compressible gaz dynamics equations on twodimensional unstructured grid. Our ALE method is of the explicit timemarching Lagrange plus remap type. Namely, it involves the following three phases: a Lagrangian phase wherein the flow is advanced using a cellcentered scheme; a rezone phase in which the nodes of the computational grid are moved to more optimal positions; a cellcentered remap phase which consists in interpolating conservatively the Lagrangian solution onto the rezoned grid. The multimaterial modeling utilizes either concentration equations for miscible fluids or the Volume Of Fluid (VOF) capability with interface reconstruction for immiscible fluids. The main original feature of this ALE scheme lies in the introduction of a new mesh relaxation procedure which keeps the rezoned grid as close as possible to the Lagrangian one. In this formalism, the rezoned grid is defined as a convex combination between the Lagrangian grid and the grid resulting from condition number smoothing. This convex combination is constructed through the use of a scalar parameter which is a scalar function of the invariants of the CauchyGreen tensor over the Lagrangian phase. Regarding the cellcentered remap phase, we employ two classical methods based on a partition of the rezoned cell in terms of its overlap with the Lagrangian cells. The first one is a simplified swept facebased method whereas the second one is a cellintersectionbased method. Our multimaterial ALE methodology is assessed through several demanding twodimensional tests. The corresponding numerical results provide a clear evidence of the robustness and the accuracy of this new scheme.
Uniform random Voronoi meshes
 In 20th International Meshing Roundtable
, 2011
"... Summary. We generate Voronoi meshes over three dimensional domains with prescribed boundaries. Voronoi cells are clipped at onesided domain boundaries. The seeds of Voronoi cells are generated by maximal Poissondisk sampling. In contrast to centroidal Voronoi tessellations, our seed locations are ..."
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Cited by 12 (6 self)
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Summary. We generate Voronoi meshes over three dimensional domains with prescribed boundaries. Voronoi cells are clipped at onesided domain boundaries. The seeds of Voronoi cells are generated by maximal Poissondisk sampling. In contrast to centroidal Voronoi tessellations, our seed locations are unbiased. The exception is some bias near concave features of the boundary to ensure wellshaped cells. The method is extensible to generating Voronoi cells that agree on both sides of twosided internal boundaries. Maximal uniform sampling leads naturally to bounds on the aspect ratio and dihedral angles of the cells. Small cell edges are removed by collapsing them; some facets become slightly nonplanar. Voronoi meshes are preferred to tetrahedral or hexahedral meshes for some Lagrangian fracture simulations. We may generate an ensemble of random Voronoi meshes. Point location variability models some of the material strength variability observed in physical experiments. The ensemble of simulation results defines a spectrum of possible experimental results.
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.
A Comparison of Two Optimization Methods for Mesh Quality Improvement
"... We compare inexact Newton and coordinate descent optimization methods for improving the quality of a mesh by repositioning the vertices, where the overall quality is measured by the harmonic mean of the meanratio metric. The e#ects of problem size, element size heterogeneity, and various vertex dis ..."
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Cited by 7 (2 self)
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We compare inexact Newton and coordinate descent optimization methods for improving the quality of a mesh by repositioning the vertices, where the overall quality is measured by the harmonic mean of the meanratio metric. The e#ects of problem size, element size heterogeneity, and various vertex displacement schemes on the performance of these algorithms are assessed for a series of tetrahedral meshes.