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51
CHARMS: A Simple Framework for Adaptive Simulation
 ACM Transactions on Graphics
, 2002
"... Finite element solvers are a basic component of simulation applications; they are common in computer graphics, engineering, and medical simulations. Although adaptive solvers can be of great value in reducing the often high computational cost of simulations they are not employed broadly. Indeed, bui ..."
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Cited by 148 (11 self)
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Finite element solvers are a basic component of simulation applications; they are common in computer graphics, engineering, and medical simulations. Although adaptive solvers can be of great value in reducing the often high computational cost of simulations they are not employed broadly. Indeed, building adaptive solvers can be a daunting task especially for 3D finite elements. In this paper we are introducing a new approach to produce conforming, hierarchical, adaptive refinement methods (CHARMS). The basic principle of our approach is to refine basis functions, not elements. This removes a number of implementation headaches associated with other approaches and is a general technique independent of domain dimension (here 2D and 3D), element type (e.g., triangle, quad, tetrahedron, hexahedron), and basis function order (piecewise linear, higher order Bsplines, Loop subdivision, etc.). The (un)refinement algorithms are simple and require little in terms of data structure support. We demonstrate the versatility of our new approach through 2D and 3D examples, including medical applications and thinshell animations.
Towards Algebraic Multigrid for Elliptic Problems of Second Order
, 1995
"... An algebraic multigrid method is developed which can be used as a preconditioner for the solution of linear systems of equations with positive definite matrices. The method is directed to equations which arise from the discretization of elliptic equations of second order, but only the matrix is the ..."
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Cited by 69 (0 self)
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An algebraic multigrid method is developed which can be used as a preconditioner for the solution of linear systems of equations with positive definite matrices. The method is directed to equations which arise from the discretization of elliptic equations of second order, but only the matrix is the source for the information used by the algorithm. One has only to know whether the matrix stems from a 2dimensional or 3dimensional problem and whether the elliptic equations are scalar equations or belong to a system.
The Auxiliary Space Method And Optimal Multigrid Preconditioning Techniques For Unstructured Grids
 Computing
, 1996
"... . An abstract framework of auxiliary space method is proposed and, as an application, an optimal multigrid technique is developed for general unstructured grids. The auxiliary space method is a (nonnested) two level preconditioning technique based on a simple relaxation scheme (smoother) and an auxi ..."
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Cited by 53 (5 self)
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. An abstract framework of auxiliary space method is proposed and, as an application, an optimal multigrid technique is developed for general unstructured grids. The auxiliary space method is a (nonnested) two level preconditioning technique based on a simple relaxation scheme (smoother) and an auxiliary space (that may be roughly understood as a nonnested coarser space). An optimal multigrid preconditioner is then obtained for a discretized partial differential operator defined on an unstructured grid by using an auxiliary space defined on a more structured grid in which a further nested multigrid method can be naturally applied. This new technique make it possible to apply multigrid methods to general unstructured grids without too much more programming effort than traditional solution methods. Some simple examples are also given to illustrate the abstract theory and for instance the Morley finite element space is used as an auxiliary space to construct a preconditioner for Argyris ...
WellSpaced Points for Numerical Methods
, 1997
"... mesh generation, mesh coarsening, multigrid Abstract A numerical method for the solution of a partial differential equation (PDE) requires the following steps: (1) discretizing the domain (mesh generation); (2) using an approximation method and the mesh to transform the problem into a linear system; ..."
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Cited by 50 (2 self)
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mesh generation, mesh coarsening, multigrid Abstract A numerical method for the solution of a partial differential equation (PDE) requires the following steps: (1) discretizing the domain (mesh generation); (2) using an approximation method and the mesh to transform the problem into a linear system; (3) solving the linear system. The approximation error and convergence of the numerical method depend on the geometric quality of the mesh, which in turn depends on the size and shape of its elements. For example, the shape quality of a triangular mesh is measured by its element's aspect ratio. In this work, we shift the focus to the geometric properties of the nodes, rather than the elements, of well shaped meshes. We introduce the concept of wellspaced points and their spacing functions, and show that these enable the development of simple and efficient algorithms for the different stages of the numerical solution of PDEs. We first apply wellspaced point sets and their accompanying technology to mesh coarsening, a crucial step in the multigrid solution of a PDE. A good aspectratio coarsening sequence of an unstructured mesh M0 is a sequence of good aspectratio meshes M1; : : : ; Mk such that Mi is an approximation of Mi\Gamma 1 containing fewer nodes and elements. We present a new approach to coarsening that guarantees the sequence is also of optimal size and width up to a constant factor the first coarsening method that provides these guarantees. We also present experimental results, based on an implementation of our approach, that substantiate the theoretical claims.
Multilevel Solvers For Unstructured Surface Meshes
 SIAM J. Sci. Comput
"... Parameterization of unstructured surface meshes is of fundamental importance in many applications of Digital Geometry Processing. Such parameterization approaches give rise to large and exceedingly illconditioned systems which are difficult or impossible to solve without the use of sophisticated mu ..."
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Cited by 39 (3 self)
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Parameterization of unstructured surface meshes is of fundamental importance in many applications of Digital Geometry Processing. Such parameterization approaches give rise to large and exceedingly illconditioned systems which are difficult or impossible to solve without the use of sophisticated multilevel preconditioning strategies. Since the underlying meshes are very fine to begin with, such multilevel preconditioners require mesh coarsening to build an appropriate hierarchy. In this paper we consider several strategies for the construction of hierarchies using ideas from mesh simplification algorithms used in the computer graphics literature. We introduce two novel hierarchy construction schemes and demonstrate their superior performance when used in conjunction with a multigrid preconditioner.
Natural Hierarchical Refinement for Finite Element Methods
 International Journal for Numerical Methods in Engineering
, 2001
"... Introduction Adaptive finite element computations rely on adjustments of th spatial resolution of th domain discretization to deliverh igh er accuracywh ere it is neededWh enth e domain is discretized into a finite element mesh a possible option, albeit somewh at expensive and in some cases complex ..."
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Cited by 27 (8 self)
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Introduction Adaptive finite element computations rely on adjustments of th spatial resolution of th domain discretization to deliverh igh er accuracywh ere it is neededWh enth e domain is discretized into a finite element mesh a possible option, albeit somewh at expensive and in some cases complex, is to create a new mesh with th desired resolution, i e remeshing Anoth0 alternative is to adjustthdensityofth mesh by performing local refinement (or coarsening) of th existing mesh soth t in some regions finite elements are split to decreaseth eir "size", inoth er regionsth ey are merged to reduce th e resolution Both ch oices, remesh ing and refinement,h ave th eir advantages and disadvantages We are not going to argue for one or th oth r option Rath r, we assume th t refinement h d been adopted asth meth d of ch ice Wh t are th desirable properties of a mesh refinement algorith0 It sh uld certainly be e#cient in th at itsh ould not becom
Local refinement of simplicial grids based on the skeleton
 APPLIED NUMERICAL MATHEMATICS 32 (2000) 195–218
, 2000
"... In this paper we present a novel approach to the development of a class of local simplicial refinement strategies. The algorithm in two dimensions first subdivides certain edges. Then each triangle, if refined, is subdivided in two, three or four subelements depending on the previous division of its ..."
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Cited by 23 (7 self)
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In this paper we present a novel approach to the development of a class of local simplicial refinement strategies. The algorithm in two dimensions first subdivides certain edges. Then each triangle, if refined, is subdivided in two, three or four subelements depending on the previous division of its edges. Similarly, in three dimensions the algorithm begins by subdividing the twodimensional triangulation composed by the faces of the tetrahedra (the skeleton) and then subdividing each tetrahedron in a compatible manner with the division of the faces. The complexity of the algorithm is linear in the number of added nodes. The algorithm is fully automatic and has been implemented to achieve global as well as local refinements. The numerical results obtained appear to confirm that the measure of degeneracy of subtetrahedra is bounded, and converges asymptotically to a fixed value when the
An Agglomeration Multigrid Method For Unstructured Grids
 in Tenth international conference on Domain Decomposition
"... . A new agglomeration multigrid method is proposed in this paper for general unstructured grids. By a proper local agglomeration of finite elements, a nested sequence of finite dimensional subspaces are obtained by taking appropriate linear combinations of the basis functions from previous level of ..."
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Cited by 16 (3 self)
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. A new agglomeration multigrid method is proposed in this paper for general unstructured grids. By a proper local agglomeration of finite elements, a nested sequence of finite dimensional subspaces are obtained by taking appropriate linear combinations of the basis functions from previous level of space. Our algorithm seems to be able to solve, for example, the Poisson equation discretized on any shaperegular finite element grids with nearly optimal complexity. 1. Introduction In this paper, we discuss a multilevel method applied to problems on general unstructured grids. We will describe an approach for designing a multilevel method for the solution of large systems of linear algebraic equations, arising from finite element discretizations on unstructured grids. Our interest will be focused on the performance of an agglomeration multigrid method for unstructured grids. One approach of constructing coarse spaces is based on generating nodenested coarse grids, which are created by s...
Boundary Treatments For Multilevel Methods On Unstructured Meshes
, 1996
"... . In applying multilevel iterative methods on unstructured meshes, the grid hierarchy can allow general coarse grids whose boundaries may be nonmatching to the boundary of the fine grid. In this case, the standard coarsetofine grid transfer operators with linear interpolants are not accurate enoug ..."
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Cited by 15 (8 self)
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. In applying multilevel iterative methods on unstructured meshes, the grid hierarchy can allow general coarse grids whose boundaries may be nonmatching to the boundary of the fine grid. In this case, the standard coarsetofine grid transfer operators with linear interpolants are not accurate enough at Neumann boundaries so special care is needed to correctly handle di#erent types of boundary conditions. We propose two e#ective ways to adapt the standard coarsetofine interpolations to correctly implement boundary conditions for twodimensional polygonal domains, and we provide some numerical examples of multilevel Schwarz methods (and multigrid methods) which show that these methods are as e#cient as in the structured case. In addition, we prove that the proposed interpolants possess the local optimal L 2 approximation and H 1 stability, which are essential in the convergence analysis of the multilevel Schwarz methods. Using these results, we give a condition number bound for ...
Optimal Coarsening of Unstructured Meshes
 J. Algorithms
, 1997
"... A bounded aspectratio coarsening sequence of an unstructured mesh M 0 is a sequence of meshes M 1 ; : : : ; M k such that: ffl M i is a bounded aspectratio mesh, and ffl M i is an approximation of M i\Gamma1 that has fewer elements, where a mesh is called a bounded aspectratio mesh if all it ..."
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Cited by 13 (2 self)
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A bounded aspectratio coarsening sequence of an unstructured mesh M 0 is a sequence of meshes M 1 ; : : : ; M k such that: ffl M i is a bounded aspectratio mesh, and ffl M i is an approximation of M i\Gamma1 that has fewer elements, where a mesh is called a bounded aspectratio mesh if all its elements are of bounded aspectratio. The sequence is nodenested if the set of the nodes of M i is a subset of that of M i\Gamma1 . The problem of constructing good quality coarsening sequences is a key step for hierarchical and multilevel numerical calculations. In this paper, we give an algorithm for finding a bounded aspectratio, nodenested, coarsening sequence that is of optimal size: that is, the number of meshes in the sequence, as well as the number of elements in each mesh, are within a constant factor of the smallest possible. 1 Introduction Numerical methods such as the finite element, finite difference, and finite volume methods apply the following basic steps to sol...