B.F. Smith. Domain Decomposition Algorithms for the Partial Differential Equations of Linear Elasticity. PhD thesis, Courant Institute of Mathematical Sciences, September 1990. Tech. Rep. 517, Department of Computer Science, Courant Institute.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....from the eighties and nineties, have suggested different global coupling mechanisms and various combinations between them and the local preconditioners. In the framework of non overlapping domain decomposition techniques, we refer for instance to BPS (Bramble, Pasciak and Schatz) 3] Vertex Space [7, 17], and to some extent Balancing Neumann Neumann [13, 14, 15] as well as FETI [10, 16] for the presentation of major two level preconditioners. We refer to [6] and [18] for a more exhaustive overview of domain decomposition techniques. CERFACS France and COPPE UFRJ Brazil. This work was ....

B.F. Smith. Domain Decomposition Algorithms for the Partial Differential Equations of Linear Elasticity. PhD thesis, Courant Institute of Mathematical Sciences, September 1990. Tech. Rep. 517, Department of Computer Science, Courant Institute.

....belongs to the one level type preconditioners as, for instance, Dirichlet Neumann [BW86] and NeumannNeumann [RT91] Therefore, it does not implement any coarse grid component. Currently, most preconditioners include a coarse correction to propagate the error globally. We refer the reader to [BPS86, Smi90, Man93] for a detailed presentation of some of these preconditioners and to [SrG96, CM94] for complete surveys of domain decomposition methods. We should stress that the AAS approach, which first appeared in a few numerical experiments in [GT93] improves the convergence rate of the wellknown block ....

Smith B. F. (1990) Domain decomposition algorithms for the partial differential equations of linear elasticity. PhD thesis, Courant Institute of Mathematical Sciences, New York.

.... Subsequently, these techniques were extended to problems in three dimensions in [BPS89] and [Dry88] A critical ingredient in the three dimensional algorithms was a coarse grid problem involving the solution averages developed in [BPS87] Related work is contained in [CMW95] Nep91] Smi90] The papers [BPS86b] BPS86a] BPS87] BPS88] and [BPS89] developed domain decomposition preconditioners for the original discrete system. The alternative approach, to reduce to an iteration involving only the unknowns on the boundary, was taken in [BW86] BPX91] CMW95] and [Smi90] The ....

....[Smi90] The papers [BPS86b] BPS86a] BPS87] BPS88] and [BPS89] developed domain decomposition preconditioners for the original discrete system. The alternative approach, to reduce to an iteration involving only the unknowns on the boundary, was taken in [BW86] BPX91] CMW95] and [Smi90] The difference in the two techniques is important in that for the first, it is at least feasible to consider replacing the subproblem solves by preconditioners. Ninth International Conference on Domain Decomposition Methods Editor Petter E. Bjrstad, Magne S. Espedal and David E. Keyes c ....

Smith B. (1990) Domain Decomposition Algorithms for the Partial Differential Equations of Linear Elasticity. PhD thesis, Courant Institute of Mathematical Sciences, New York, NY.

.... Widlund [110] That work, in turn, was based on a result of Yserentant [117] We note that, in the conforming case, this was found to be an effective preconditioner with certain advantages over some similar iterative methods because of being relatively simple, and as effective as the others; cf. [107] for motivation and a comparative study. 109 Our algorithm is a preconditioned conjugate gradient method with a condition number bounded from above by C(1 ) 2 . Here is the maximum number of successive refinements of any individual subregion Omega i into elements, and C a constant which ....

Barry F. Smith. Domain Decomposition Algorithms for the Partial Differential Equations of Linear Elasticity. PhD thesis, Courant Institute of Mathematical Sciences, September 1990. Tech. Rep. 517, Department of Computer Science, Courant Institute.

....[60] We note that the additive Schwarz methods 1 can be regarded as an iterative method for an equivalent system with a better condition number. Iterative substructuring methods for second order problems were developed by Bramble, Pasciak and Schatz [11] 14] Dryja and Widlund [31] Smith [71], and Dryja, Smith and Widlund [28] It has been demonstrated that the Schwarz framework can be used to analyze many iterative substructuring methods. The best of these algorithms have condition numbers which are bounded independently of the number of subdomains and unknowns or grow like (1 ....

B. F. Smith, Domain Decomposition Algorithms for the Partial Differential Equations of Linear Elasticity, PhD thesis, Courant Institute of Mathematical Sciences, September 1990. Tech. Rep. 517, Department of Computer Science, Courant Institute.

.... Widlund [18] That work, in turn, was based on a result of Yserentant [20] We note that, in the conforming case, we had found this to be an effective preconditioner with certain advantages over some similar iterative methods because of being relatively simple, and as effective as the others; cf. [17] for motivation and a comparative study. Our algorithm is a preconditioned conjugate gradient method with a condition number bounded from above by C(1 ) 2 . Here is the maximum number of successive refinements of any individual subregion Omega i into elements, and C a constant which ....

B. F. Smith, Domain Decomposition Algorithms for the Partial Differential Equations of Linear Elasticity, PhD thesis, Courant Institute of Mathematical Sciences, September 1990. Tech. Rep. 517, Department of Computer Science, Courant Institute.

....of the number of elements and the jumps in the coefficients of the elliptic operator, and that it depends only weakly on the spectral degree. An alternative proof was later given by Casarin [9] This type of wire basket preconditioner was originally proposed for h version finite elements by Smith [32, 33, 34]; see also Bramble, Pasciak and Schatz [5] for earlier related work. Other iterative substructuring methods that have been successfully applied to elasticity problems and h version finite elements are the Neumann Neumann methods, see, e.g. 20] the balancing domain decomposition method of ....

B. F. Smith, Domain Decomposition Algorithms for the Partial Differential Equations of Linear Elasticity, PhD thesis, Dept. of Mathematics, Courant Institute of Mathematical Sciences, September 1990. Tech. Rep. 517, Dept. of Computer Science, Courant Institute.

....classifications. 65F10, 65N22, 65N30, 65N55, 73V05 1. Introduction. In recent years, modern iterative methods, e.g. domain decomposition and multigrid methods, have been applied to parameter dependent problems arising in solid mechanics; see Braess [6] Braess and Blomer [8] Jung [28] and Smith [38]. If the direct approach of (low order) conforming finite elements is used in a pure displacement setting, the phenomenon of locking leads to problems. Locking occurs when a parameter, e.g. the Poisson ratio of a material, approaches a limit. The convergence rate of the iterative method and that ....

Barry F. Smith. Domain decomposition algorithms for the partial differential equations of linear elasticity. PhD thesis, Courant Institute of the Mathematical Sciences, New York University, 1990.

....and a (low order) conforming finite element approach is used. In recent years, modern iterative methods, e.g. domain decomposition and multigrid methods, have been applied to parameter dependent problems arising in solid mechanics; see Braess [14] Braess and Blomer [16] Jung 29 [56] and Smith [78], p. 68 and Table 4.12. We note that one has to make a distinction between the convergence rate of the finite element model and the convergence rate of the iterative method. Both convergence rates can deteriorate severely when the limit of a certain parameter is approached, e.g. when the Poisson ....

Barry F. Smith. Domain decomposition algorithms for the partial differential equations of linear elasticity. PhD thesis, Courant Institute of the Mathematical Sciences, New York University, 1990.

.... Among them are two level, additive Schwarz methods first introduced in 1987; cf. Dryja and Widlund [28,25,29,30,55] For related work see also Bjrstad, Moe, and Skogen [1,2,3] Cai [10,11,12] Mathew [40,42,41] Matsokin and Nepomnyaschikh [43] Nepomnyaschikh [44] Skogen [47] Smith [48,52,49,50,51], and Zhang [59,60] As shown in Dryja and Widlund [30] a number of other domain decomposition methods, in particular those of Bramble, Pasciak, and Schatz [5,6] can also be derived and analyzed using the same framework. Recent efforts by Bramble, Pasciak, Wang, and Xu [7] and Xu [56] have ....

....measures the diameter of a subregion and ffi the overlap between neighboring subregions. We note that H=ffi is a measure of the aspect ratio of the subregion common to two overlapping neighboring subregions. We then turn our attention to a very interesting method, introduced in 1989 by Barry Smith [52,48]. It is known as the vertex space (or Copper Mountain) algorithm. Numerical experiments, for problems in the plane, have shown that this method converges quite rapidly even for problems, which were originally very ill conditioned, even if the overlap is very modest; cf. Smith [48] For additional ....

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Barry F. Smith. Domain Decomposition Algorithms for the Partial Differential Equations of Linear Elasticity. PhD thesis, Courant Institute of Mathematical Sciences, September 1990. Tech. Rep. 517, Department of Computer Science, Courant Institute.

....the systems of algebraic equations which correspond to the interfaces between the substructures. We also consider problems with discontinuous coefficients with a great variation in the values. The first method considered is an iterative substructuring algorithm recently developed by B. Smith [9], 10] The second is a domain decomposition method developed by R. Glowinski, P. Le Tallec, Y. H. De Roeck et al. see [1] 3] Finally, we consider a variant of a multigrid like method discovered by J. H. Bramble, J. E. Pasciak and J. Xu [2] The paper is organized as follows. In Section 2 a ....

....(ae(ffl) 1) a(u; u) u 2 V where ae(ffl) is the spectral radius of the matrix ffl = fffl ij g N i;j=1 . A proof of this theorem can be found in [8] 4. A wirebasket based method. In this section, we describe and analyze an iterative substructuring method, recently developed by B. Smith, see [9], 10] For the description and analysis we use the general framework of Section 3. Here V = V h ( Gamma) and (4:1) a(u; v) s(u; v) 4.1. A method without a vertex space. To describe the method, we need some notations. Let F ij , be the open faces and let W i be the wirebasket of the ....

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B. F. Smith, Domain Decomposition Algorithms for the Partial Differential Equations of Linear Elasticity, Ph.D. Thesis, Tech. Rep. 517, Dept. of Computer Science, Courant Institute, 1990.

....discontinuities have primarily been given for preconditioners that can be constructed from a direct sum of subspaces. Examples of such methods are given in Bramble, Pasciak, and Schatz [5] Mandel [28,27,26] and Smith [38] see also the discussion in Dryja and Widlund [18] or section 2. 5 of Smith [37]. For other recent work on Neumann Neumann preconditioners that incorporates a coarse solver, see Mandel [30,31] It has been known since 1958, cf. Hestenes [24] that the rate of convergence of a preconditioned conjugate gradient method can be estimated in terms of the condition number of a ....

....failing in the case of another, seemingly quite similar algorithm. It is known from numerical experiments, as well as theory, that the rate of convergence of many domain decomposition algorithms is adversely affected by high aspect ratios of the subregions; see e.g. Mandel and Lett [33] and Smith [37]. It is believed that the Neumann Neumann algorithms are less sensitive to extreme geometry than the iterative substructuring methods of Bramble, Pasciak, and Schatz [5] Dryja [13] and Smith [38] We know of no systematic experimental study comparing the performance of the Neumann Neumann and ....

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Barry F. Smith. Domain Decomposition Algorithms for the Partial Differential Equations of Linear Elasticity. PhD thesis, Courant Institute of Mathematical Sciences, September 1990. Tech. Rep. 517, Department of Computer Science, Courant Institute.

....solvers. From the point of view of the solver, we distinguish so called local methods from the global ones. The local DD methods involve the solution of local subproblems in subdomains. As an example, we consider the non overlapping DD methods, the main ideas of which are presented, e.g. in [3, 7, 8, 48, 49, 21, 22, 23, 39]. These methods have been applied successfully to the numerical solution of two dimensional linear and nonlinear b.v.p. s discretized by finite elements (FE) 19, 20, 30, 31] as well as to the a coupling of finite and boundary elements [33, 34, 14] The inherent parallel data distribution allows ....

B. F. Smith. Domain decomposition algorithms for the partial differential equations of linear elasticity. Technical Report 517, Department of Computer Science, Courant Institute, New York, 1990.

....thanks to the use of two level preconditioners that are composed by local and global terms acting either in an additive or in a multiplicative way. In the framework of non overlapping domain decomposition techniques, we refer for instance to BPS (Bramble, Pasciak and Schatz) 6] and Vertex Space [12, 19] for additive two level preconditioners, and to Balancing Neumann Neumann [15, 16] as well as FETI [13] for examples of multiplicative ones. We refer to [10] and [20] for a more exhaustive overview of domain decomposition techniques. We consider additive two level preconditioners similar to BPS ....

....two level preconditioners similar to BPS that can be written as the sum of a local and a global component. In Section 2, we describe a set of parallelizable local preconditioners that are the main focus of this paper and discuss the connections with well known preconditioners like Vertex Space [19] and Neumann Neumann [11] We also briefly describe the global coarse space component we have used for the numerical experiments reported in Section 3. These numerical experiments are conducted for two types of partial differential equations on two dimensional domains. For elliptic equations, we ....

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B.F. Smith. Domain Decomposition Algorithms for the Partial Differential Equations of Linear Elasticity. PhD thesis, Courant Institute of Mathematical Sciences, September 1990. Tech. Rep. 517, Department of Computer Science, Courant Institute.

....decomposition theory is to provide a good upper bound on the condition number of the preconditioned operator. Earlier work on iterative substructuring methods focused on the h version finite element methods; see e.g. Bramble, Pasciak, and Schatz [5] Dryja [10] Dryja and Widlund [12] and Smith [31,32,33] for work on three dimensional elliptic problems. A recent paper by Dryja, Smith, and Widlund [11] summarizes our knowledge of the h version case. The best of these results show that the condition number of the relevant preconditioned operator grows only linearly with the logarithm of the number ....

Barry F. Smith. Domain Decomposition Algorithms for the Partial Differential Equations of Linear Elasticity. PhD thesis, Courant Institute of Mathematical Sciences, September 1990. Tech. Rep. 517, Department of Computer Science, Courant Institute.

....more ill conditioned than for lower order methods. In this paper, we continue our recent work on spectral elements, described in detail in Pavarino and Widlund [30] and also announced in [29] 28] Just as our previous algorithm, the new methods are close relatives of a method developed by Smith [33], 34] for h Gammaversion finite elements. In this paper, we use tools and algorithmic ideas developed in our earlier paper to derive and analyze two new, closely related methods. The first provides a solver for the same Galerkin approximation considered in our previous work. Our second ....

B. F. Smith, Domain Decomposition Algorithms for the Partial Differential Equations of Linear Elasticity, PhD thesis, Courant Institute of Mathematical Sciences, September 1990. Tech. Rep. 517, Department of Computer Science, Courant Institute.

....PASCIAK AND A.T. VASSILEV Subsequently, these techniques were extended to problems in three dimensions in [8] and [15] A critical ingredient in the three dimensional algorithms was a coarse grid problem involving the solution averages developed in [6] Related work is contained in [13] 20] [21]. The papers [4] 5] 6] 7] and [8] developed domain decomposition preconditioners for the original discrete system. The alternative approach, to reduce to an iteration involving only the unknowns on the boundary, was taken in [1] 11] 13] and [21] The difference in the two techniques is ....

....Related work is contained in [13] 20] 21] The papers [4] 5] 6] 7] and [8] developed domain decomposition preconditioners for the original discrete system. The alternative approach, to reduce to an iteration involving only the unknowns on the boundary, was taken in [1] 11] 13] and [21]. The difference in the two techniques is important in that for the first, it is at least feasible to consider replacing the subproblem solves by preconditioners. The second approach for developing domain decomposition preconditioners involves the solution of subproblems on overlapping subdomains. ....

B.F. Smith, Domain Decomposition Algorithms for the Partial Differential Equations of Linear Elasticity, Ph.D. Thesis, Courant Institute of Mathematical Sciences, Dept. of Computer Science Tech. Rep. 517, New York, 1990.

....theory in general, is to provide good upper bounds on the condition number of the preconditioned operator. Early work on iterative substructuring methods focused on the h version finite element methods; see, e.g. Bramble, Pasciak, and Schatz [10] Dryja [19] Dryja and Widlund [21] and Smith [49, 50, 51] for work on three dimensional elliptic problems. A recent paper by Dryja, Smith, and Widlund [20] summarizes our knowledge of the h version case. The best of these results show that the condition number of the relevant preconditioned operator grows only linearly with the logarithm of the number ....

Barry F. Smith. Domain Decomposition Algorithms for the Partial Differential Equations of Linear Elasticity. PhD thesis, Courant Institute of Mathematical Sciences, September 1990. Tech. Rep. 517, Department of Computer Science, Courant Institute.

....with a set e I i defining the AAS(3) preconditioner. Basically, this approach attempts to recover information around the cross points or vertices. In a different framework, linear elasticity problems in three dimensions, the idea of capturing the information around the cross points appears in [15]. In the cited work, the author proposes the Vertex Space preconditioner that combines coarse and local parts. In the additive version of the Vertex preconditioner, the local part consists of three terms: one related to the edges, the second related to the vertices, and the last to the faces of ....

B.F. Smith. Domain Decomposition Algorithms for the Partial Differential Equations of Linear Elasticity. PhD thesis, Courant Institute of Mathematical Sciences, September 1990. Tech. Rep. 517, Department of Computer Science, Courant Institute.

....itself. Hence other Schur complement preconditioners have to be employed for the spaces V edge k and V face l , such as the mentioned preconditioner of Dryja. The resulting convergence rates can be found in Table 1. An extension of this construction principle leads to the vertex space method [146]. It is related to the fictitious domain methods in [111, 119,120] and [1] In addition to the splitting in equation (25) for each vertex, a space is constructed with consists of functions on the separator Gamma in the vicinity of the vertex xm , i.e. Omega vertex m = B fiH (x m ) Gamma ; ....

....this respect, these are even hard test problems. We have to mention that in the area of elasticity, domain decomposition methods are also very popular and a number of industrial parallel application are based on these algorithms. We will not cover this aspect and refer instead to the literature [68, 102, 146] and other articles of this issue. The following numerical experiments are based on an adaptive parallel multigrid solver which uses a Hilbert space filling curve for decomposition and a hash table for addressing, see [81] A finite difference discretization is used, where the degrees of freedom ....

B. F. Smith, Domain decomposition algorithms for the partial differential equations of linear elasticity, PhD thesis, Dept. Computer Science, Courant Institute, New York, 1990.

....from the eighties and nineties, have suggested different global coupling mechanisms and various combinations between them and the local preconditioners. In the framework of non overlapping domain decomposition techniques, we refer for instance to BPS (Bramble, Pasciak and Schatz) 3] Vertex Space [7, 17], and to some extent Balancing Neumann Neumann [13, 14, 15] as well as FETI [10, 16] for the presentation of major two level preconditioners. We refer to [6] and [18] for a more exhaustive overview of domain decomposition techniques. CERFACS France and COPPE UFRJ Brazil. This work was ....

B.F. Smith. Domain Decomposition Algorithms for the Partial Differential Equations of Linear Elasticity. PhD thesis, Courant Institute of Mathematical Sciences, September 1990. Tech. Rep. 517, Department of Computer Science, Courant Institute.

....3.3 Additive Schwarz methods Domain decomposition preconditioners based on ideas from Schwarz [26] has received much attention, see [16] 17] and the references in these papers. Until recently, most of this work related to model problems and scalar elliptic equations. In his thesis Barry Smith [27] developed and tested a method for two and three dimensional linear elasticity. He proved the optimality of the method and carried out extensive tests using the SESAM structural analysis program to provide the underlying stiffness matrices. This method appears very robust and quite promising, it ....

....level substructures combined with a very compute intensive structure level matrix. This is a very attractive special problem class where the combination of direct elimination of the first level matrices (in parallel) combined with an optimal iterative solver like the one proposed by Barry Smith [27], may challenge the traditional use of direct algorithms. ....

B. F. Smith, Domain decomposition algorithms for the partial differential equations of linear elasticity, PhD thesis, Courant Institute of Mathematical Sciences, New York, September 1990.

.... which such a framework is developed and used, see Bramble, Pasciak, Wang, and Xu [8] Cai [10] Cai and Widlund [12] 13] Dryja and Widlund [23] 24] 26] 27] 28] 29] Lions [35] Mathew [42] 43] Nepomnyaschikh [45] Pavarino [47] 48] Pavarino and Widlund [49] Sarkis [51] Smith [54], 55] 56] 57] Widlund [61] Xu [63] and Zhang [64] 66] In Section 2, we will demonstrate that rapid convergence of the iterative methods occurs if and only if all u 2 V can be decomposed into components in V i ; i.e. u = P i u i ; u i 2 V i ; in such a way that P b i (u i ; u i ) can ....

....of continuous, piecewise linear functions using the substructures as elements. This approach is successful in the case of two dimensions but for the three dimensional problems considered in this paper quite unsatisfactory algorithms can result; cf. Bramble, Pasciak, and Schatz [6] 7] Smith [54], 55] and Section 6 for a discussion. In certain cases when the decomposition of the functions into subspaces is unique, i.e. when V is a direct sum of the subspaces V i ; we necessarily obtain a poor bound on b 0 (u 0 ; u 0 ) and, as a consequence, a poor convergence rate. However, by ....

[Article contains additional citation context not shown here]

B. F. Smith, Domain Decomposition Algorithms for the Partial Differential Equations of Linear Elasticity, Ph. D. thesis, Courant Institute of Mathematical Sciences, September 1990. Tech. Rep. 517, Department of Computer Science, Courant Institute.

.... such a framework is developed and used, see Bramble, Pasciak, Wang, and Xu [8] Cai [10] Cai and Widlund [12] 13] Dryja and Widlund [24] 25] 27] 28] 29] 30] Lions [36] Mathew [43] 44] Nepomnyaschikh [46] Pavarino [48] 49] Pavarino and Widlund [50] 51] Sarkis [53] Smith [56], 57] 58] 59] Widlund [63] Xu [65] and Zhang [66] 68] In Section 2, we will demonstrate that rapid convergence of the iterative methods occurs if and only if all u 2 V can be decomposed into components in V i ; i.e. u = P i u i ; u i 2 V i ; in such a way that P b i (u i ; u i ) ....

....of continuous, piecewise linear functions using the substructures as elements. This approach is successful in the case of two dimensions but for the three dimensional problems considered in this paper quite unsatisfactory algorithms can result; see Bramble, Pasciak, and Schatz [6] 7] Smith [56], 57] and Section 6 for a discussion. In certain cases when the decomposition of the functions into subspaces is unique, i.e. when V is a direct sum of the subspaces V i ) we necessarily obtain a poor bound on b 0 (u 0 ; u 0 ) and, as a consequence, a poor convergence rate. However, by ....

[Article contains additional citation context not shown here]

B. F. Smith, Domain Decomposition Algorithms for the Partial Differential Equations of Linear Elasticity, Ph. D. thesis, Courant Institute of Mathematical Sciences, September 1990. Tech. Rep. 517, Department of Computer Science, Courant Institute.

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Smith B. F. (1991) Domain Decomposition Algorithms for the Partial Differential Equations of Linear Elasticity. PhD thesis, New York University.

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