Results 1  10
of
67
Quality Mesh Generation in Three Dimensions
, 1992
"... We show how to triangulate a three dimensional polyhedral region with holes. Our triangulation is optimal in the following two senses. First, our triangulation achieves the best possible aspect ratio up to a constant. Second, for any other triangulation of the same region into m triangles with b ..."
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Cited by 79 (4 self)
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We show how to triangulate a three dimensional polyhedral region with holes. Our triangulation is optimal in the following two senses. First, our triangulation achieves the best possible aspect ratio up to a constant. Second, for any other triangulation of the same region into m triangles with bounded aspect ratio, our triangulation has size n = O(m). Such a triangulation is desired as an initial mesh for a finite element mesh refinement algorithm. Previous three dimensional triangulation schemes either worked only on a restricted class of input, or did not guarantee wellshaped tetrahedra, or were not able to bound the output size. We build on some of the ideas presented in previous work by Bern, Eppstein, and Gilbert, who have shown how to triangulate a two dimensional polyhedral region with holes, with similar quality and optimality bounds. 1 Introduction Triangulation of polyhedral regions is a fundamental geometric problem for numerical analysis. In particular, if on...
Data Oscillation and Convergence of Adaptive FEM
 SIAM J. Numer. Anal
, 1999
"... Data oscillation is intrinsic information missed by the averaging process associated with finite element methods (FEM) regardless of quadrature. Ensuring a reduction rate of data oscillation, together with an error reduction based on a posteriori error estimators, we construct a simple and efficient ..."
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Cited by 76 (11 self)
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Data oscillation is intrinsic information missed by the averaging process associated with finite element methods (FEM) regardless of quadrature. Ensuring a reduction rate of data oscillation, together with an error reduction based on a posteriori error estimators, we construct a simple and efficient adaptive FEM for elliptic PDE with linear rate of convergence without any preliminary mesh adaptation nor explicit knowledge of constants. Any prescribed error tolerance is thus achieved in a finite number of steps. A number of numerical experiments in 2d and 3d yield quasioptimal meshes along with a competitive performance. Key words. A posteriori error estimators, data oscillation, adaptive mesh refinement, convergence, performance, quasioptimal meshes 1991 AMS subject classification. 65N12, 65N15, 65N30, 65N50, 65Y20 1 Introduction and Main Results Adaptive procedures for the numerical solution of partial differential equations (PDE) started in the late 70's and are now standard tools...
Mesh Smoothing Using A Posteriori Error Estimates
 SIAM JOURNAL ON NUMERICAL ANALYSIS
, 1997
"... We develop a simple mesh smoothing algorithm for adaptively improving finite element triangulations. The algorithm makes use of a posteriori error estimates which are now widely used in finite element calculations. In this paper, we derive the method, present some numerical illustrations, and give a ..."
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Cited by 58 (2 self)
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We develop a simple mesh smoothing algorithm for adaptively improving finite element triangulations. The algorithm makes use of a posteriori error estimates which are now widely used in finite element calculations. In this paper, we derive the method, present some numerical illustrations, and give a brief analysis of the issue of uniqueness.
Quality local refinement of tetrahedral meshes based on bisection
 SIAM J. Sci. Comput
, 1995
"... Abstract. Let T be a tetrahedral mesh. We present a 3D local refinement algorithm for T which is mainly based on an 8subtetrahedron subdivision procedure, and discuss the quality of refined meshes generated by the algorithm. It is proved that any tetrahedron T ∈T produces a finite number of classe ..."
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Cited by 57 (1 self)
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Abstract. Let T be a tetrahedral mesh. We present a 3D local refinement algorithm for T which is mainly based on an 8subtetrahedron subdivision procedure, and discuss the quality of refined meshes generated by the algorithm. It is proved that any tetrahedron T ∈T produces a finite number of classes of similar tetrahedra, independent of the number of refinement levels. Furthermore, η(Tn i) ≥ cη(T), where T ∈T,cis a positive constant independent of T and the number of refinement levels, Tn i is any refined tetrahedron of T, andηisa tetrahedron shape measure. It is also proved that local refinements on tetrahedra can be smoothly extended to their neighbors to maintain a conforming mesh. Experimental results show that the ratio of the number of tetrahedra actually refined to the number of tetrahedra chosen for refinement is bounded above by a small constant. 1.
An Algorithm for Coarsening Unstructured Meshes
 Numer. Math
, 1996
"... . We develop and analyze a procedure for creating a hierarchical basis of continuous piecewise linear polynomials on an arbitrary, unstructured, nonuniform triangular mesh. Using these hierarchical basis functions, we are able to define and analyze corresponding iterative methods for solving the lin ..."
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Cited by 47 (5 self)
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. We develop and analyze a procedure for creating a hierarchical basis of continuous piecewise linear polynomials on an arbitrary, unstructured, nonuniform triangular mesh. Using these hierarchical basis functions, we are able to define and analyze corresponding iterative methods for solving the linear systems arising from finite element discretizations of elliptic partial differential equations. We show that such iterative methods perform as well as those developed for the usual case of structured, locally refined meshes. In particular, we show that the generalized condition numbers for such iterative methods are of order J 2 , where J is the number of hierarchical basis levels. Key words. Finite element, hierarchical basis, multigrid, unstructured mesh. AMS subject classifications. 65F10, 65N20 1. Introduction. Iterative methods using the hierarchical basis decomposition have proved to be among the most robust for solving broad classes of elliptic partial differential equations, ...
Optimality of a standard adaptive finite element method
"... In this paper, an adaptive ¯nite element method is constructed
for solving elliptic equations that has optimal computational complexity.
Whenever for some s > 0, the solution can be approximated to accuracy
O(n¡s) in energy norm by a continuous piecewise linear function on some
partition with n tria ..."
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Cited by 40 (3 self)
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In this paper, an adaptive ¯nite element method is constructed
for solving elliptic equations that has optimal computational complexity.
Whenever for some s > 0, the solution can be approximated to accuracy
O(n¡s) in energy norm by a continuous piecewise linear function on some
partition with n triangles, and one knows how to approximate the righthand
side in the dual norm with the same rate with piecewise constants, then
the adaptive method produces approximations that converge with this rate,
taking a number of operations that is of the order of the number of triangles
in the output partition. The method is similar in spirit to that from [SINUM,
38 (2000), pp.466{488] by Morin, Nochetto, and Siebert, and so in particular
it does not rely on a recurrent coarsening of the partitions. Although the
Poisson equation in two dimensions with piecewise linear approximation is
considered, it can be expected that the results generalize in several respects.
Parallel Multigrid in an Adaptive PDE Solver Based on Hashing and SpaceFilling Curves
, 1997
"... this paper is organized as follows: In section 2 we discuss data structures for adaptive PDE solvers. Here, we suggest to use hash tables instead of the usually employed tree type data structures. Then, in section 3 we discuss the main features of the sequential adaptive multilevel solver. Section 4 ..."
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Cited by 39 (3 self)
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this paper is organized as follows: In section 2 we discuss data structures for adaptive PDE solvers. Here, we suggest to use hash tables instead of the usually employed tree type data structures. Then, in section 3 we discuss the main features of the sequential adaptive multilevel solver. Section 4 deals with the partitioning and distribution of adaptive grids with spacefilling curves and section 5 gives the main features of our new parallelized adaptive multilevel solver. In section 6 we present the results of numerical experiments on a parallel cluster computer with up to 64 processors. It is shown that our approach works nicely also for problems with severe singularities which result in locally refined meshes. Here, the work overhead for load balancing and data distribution remains only a small fraction of the overall work load. 2. DATA STRUCTURES FOR ADAPTIVE PDE SOLVERS 2.1. Adaptive Cycle
Hierarchical and Adaptive Visualization on Nested Grids
 Computing
, 1997
"... Zusammenfassung Hierarchical and Adaptive Visualization on Nested Grids. Modern numerical methods are capable to resolve fine structures in solutions of partial differential equations. Thereby they produce large amounts of data. The user wants to explore them interactively by applying visualization ..."
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Cited by 35 (5 self)
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Zusammenfassung Hierarchical and Adaptive Visualization on Nested Grids. Modern numerical methods are capable to resolve fine structures in solutions of partial differential equations. Thereby they produce large amounts of data. The user wants to explore them interactively by applying visualization tools in order to understand the simulated physical process. Here we present a multiresolution approach for a large class of hierarchical and nested grids. It is based on a hierarchical traversal of mesh elements combined with an adaptive selection of the hierarchical depth. The adaptation depends on an error indicator which is closely related to the visual impression of the smoothness of isosurfaces or isolines, which are typically used to visualize data. Significant examples illustrate the applicability and efficiency on different types of meshes. Keywords : multiresolutional visualization, adaptive methods, error indicator, isosurfaces. Hierarchische und adaptive Visualisierung auf ges...
Quasioptimal convergence rate for an adaptive finite element method
, 2007
"... We analyze the simplest and most standard adaptive finite element method (AFEM), with any polynomial degree, for general second order linear, symmetric elliptic operators. As it is customary in practice, AFEM marks exclusively according to the error estimator and performs a minimal element refineme ..."
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Cited by 33 (8 self)
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We analyze the simplest and most standard adaptive finite element method (AFEM), with any polynomial degree, for general second order linear, symmetric elliptic operators. As it is customary in practice, AFEM marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property. We prove that AFEM is a contraction for the sum of energy error and scaled error estimator, between two consecutive adaptive loops. This geometric decay is instrumental to derive optimal cardinality of AFEM. We show that AFEM yields a decay rate of energy error plus oscillation in terms of number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity.
Computational Results for Parallel Unstructured Mesh Computations
, 1994
"... : The majority of finite element models in structural engineering are composed of unstructured meshes. These unstructured meshes are often very large and require significant computational resources; hence they are excellent candidates for massively parallel computation. Parallel solution of the spar ..."
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Cited by 30 (5 self)
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: The majority of finite element models in structural engineering are composed of unstructured meshes. These unstructured meshes are often very large and require significant computational resources; hence they are excellent candidates for massively parallel computation. Parallel solution of the sparse matrices that arise from such meshes has been studied heavily, and many good algorithms have been developed. Unfortunately, many of the other aspects of parallel unstructured mesh computation have gone largely ignored. We present a set of algorithms that allow the entire unstructured mesh computation process to execute in parallelincluding adaptive mesh refinement, equation reordering, mesh partitioning, and sparse linear system solution. We briefly describe these algorithms and state results regarding their runningtime and performance. We then give results from the 512processor Intel DELTA for a largescale structural analysis problem. These results demonstrate that the new algorith...