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  • 1
    Electronic Resource
    Electronic Resource
    An explicit sparse unsymmetric finite element solver (1996)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 12 (1996), S. 21-29 
    Details
    ISSN: 1069-8299
    Keywords: finite element systems ; elliptic partial differential equations ; approximate LU factorization ; explicit matrix inversion ; preconditioning ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A new class of explicit generalized approximate inverse finite element matrix algorithmic methods, based on the concept of LU-sparse factorization procedures, without inverting the decomposition factors, has recently been introduced. The large sparse unsymmetric coefficient matrix of irregular structure is factorized approximately and, in conjunction with approximate inverse matrix techniques, yields explicit preconditioned methods for the finite element (FE) and finite difference (FD) method. The numerical implementation of these algorithms is presented and Fortran subroutines for the efficient solution of the sparse unsymmetric linear systems are given.
    Additional Material: 1 Tab.
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  • 2
    Electronic Resource
    Electronic Resource
    HIGH-PERFORMANCE PCG SOLVERS FOR FEM STRUCTURAL ANALYSIS (1996)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 39 (1996), S. 1313-1340 
    Details
    ISSN: 0029-5981
    Keywords: iterative methods for linear systems ; preconditioning ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The preconditioned conjugate gradient algorithm is a well-known and powerful method used to solve large sparse symmetric positive definite linear systems. Such systems are generated by the finite element discretization in structural analysis but users of finite elements in this context generally still rely on direct methods. It is our purpose in the present work to highlight the improvement brought forward by some new preconditioning techniques and show that the preconditioned conjugate gradient method performs better than efficient direct methods.
    Additional Material: 8 Ill.
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  • 3
    Electronic Resource
    Electronic Resource
    FE-BE Coupling Methods for Elastoplasticity (1997)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 13 (1997), S. 785-792 
    Details
    ISSN: 1069-8299
    Keywords: FEM-BEM coupling ; elastoplasticity ; relaxation ; preconditioning ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Non-linear physical systems of infinite extent are conveniently modelled using FE-BE coupling methods. By the combination of both methods, suitable use of the advantages of each one may be obtained. Several possibilities of FEM-BEM coupling and their performance in some practical cases are discussed in this paper. Parallelizable coupling algorithms based on domain decomposition are developed and compared with the most traditional coupling methods. © 1997 John Wiley & Sons, Ltd.
    Additional Material: 5 Ill.
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  • 4
    Electronic Resource
    Electronic Resource
    Upper bound of the constant in the strengthened C.B.S. inequality for FEM 2D elasticity equations (1994)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 1 (1994), S. 65-74 
    Details
    ISSN: 1070-5325
    Keywords: Elasticity ; Finite elements ; preconditioning ; Multilevel ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The basic theory of the strengthened Cauchy-Buniakowskii-Schwarz (C.B.S.) inequality is the main tool in the convergence analysis of the recently proposed algebraic multilevel iterative methods. An upper bound of the constant γ in the strengthened C.B.S. inequality for the case of the finite element solution of 2D elasticity problems is obtained. It is assumed that linear triangle finite elements are used, the initial mesh consisting of right isosceles triangles and the mesh refinement procedure being uniform. For the resulting linear algebraic systems we have proved that γ2〈0.75 uniformly on the mesh parameter and on Poisson's ratio ν ∊ (0, 1/2). Furthermore, the presented numerical tests show that the same relation holds for arbitrary initial right triangulations, even in the case of degeneracy of triangles.The theoretical results obtained are practically important for successful implementation of the finite element method to large-scale modeling of complicated structures. They allow us to construct optimal order algebraic multilevel iterative solvers for a wide class of real-life elasticity problems.
    Additional Material: 1 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nla.1680010107
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  • 5
    Electronic Resource
    Electronic Resource
    Multilevel preconditioners for discretizations of the biharmonic equation by rectangular finite elements (1995)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 2 (1995), S. 487-505 
    Details
    ISSN: 1070-5325
    Keywords: biharmonic equation ; rectangular finite elements ; preconditioning ; multilevel methods ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Recently, some new multilevel preconditioners for solving elliptic finite element discretizations by iterative methods have been proposed. They are based on appropriate splittings of the finite element spaces under consideration, and may be analyzed within the framework of additive Schwarz schemes. In this paper we discuss some multilevel methods for discretizations of the fourth-order biharmonic problem by rectangular elements and derive optimal estimates for the condition numbers of the preconditioned linear systems. For the Bogner-Fox-Schmit rectangle, the generalization of the Bramble-Pasciak-Xu method is discussed. As a byproduct, an optimal multilevel preconditioner for nonconforming discretizations by Adini elements is also derived.
    Additional Material: 3 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nla.1680020603
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  • 6
    Electronic Resource
    Electronic Resource
    A Parallel Preconditioned Iterative Realization of the Panel Method in 3D (1996)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 65-80 
    Details
    ISSN: 1070-5325
    Keywords: boundary value problem ; boundary element method ; preconditioning ; iterative method ; fast Fourier transform ; parallel algorithm ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The parallel version of precondition iterative techniques is developed for matrices arising from the panel boundary element method for three-dimensional simple connected domains with Dirichlet boundary conditions. Results were obtained on an nCube-2 parallel computer showing that preconditioned iterative methods are very well suited also in three-dimensional cases for implementation on an MIMD computer and that they are much more efficient than usual direct solution techniques.
    Additional Material: 6 Ill.
    Type of Medium: Electronic Resource
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  • 7
    Electronic Resource
    Electronic Resource
    On Fast Computations with the Inverse to Harmonic Potential Operators via Domain Decomposition (1996)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 91-111 
    Details
    ISSN: 1070-5325
    Keywords: boundary integral operators ; domain decomposition ; interface operators ; fast elliptic problem solvers ; parallel algorithms ; preconditioning ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this paper a method for fast computations with the inverse to weakly singular, hypersingular and double layer potential boundary integral operators associated with the Laplacian on Lipschitz domains is proposed and analyzed. It is based on the representation formulae suggested for above-mentioned boundary operations in terms of the Poincare-Steklov interface mappings generated by the special decompositions of the interior and exterior domains. Computations with the discrete counterparts of these formulae can be efficiently performed by iterative substructuring algorithms provided some asymptotically optimal techniques for treatment of interface operators on subdomain boundaries. For both two- and three-dimensional cases the computation cost and memory needs are of the order O(N logp N) and O(N log2 N), respectively, with 1 ≤ p ≤ 3, where N is the number of degrees of freedom on the boundary under consideration (some kinds of polygons and polyhedra). The proposed algorithms are well suited for serial and parallel computations.
    Additional Material: 2 Ill.
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  • 8
    Electronic Resource
    Electronic Resource
    Stabilizing the Hierarchical Basis by Approximate Wavelets, I: Theory (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 4 (1997), S. 103-126 
    Details
    ISSN: 1070-5325
    Keywords: hierarchical basis ; multilevel methods ; preconditioning ; fine element elliptic equations ; approximate wavelets ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: This paper proposes a stabilization of the classical hierarchical basis (HB) method by modifying the HB functions using some computationally feasible approximate L2-projections onto finite element spaces of relatively coarse levels. The corresponding multilevel additive and multiplicative algorithms give spectrally equivalent preconditioners, and one action of such a preconditioner is of optimal order computationally. The results are regularity-free for the continuous problem (second order elliptic) and can be applied to problems with rough coefficients and local refinement. © 1997 by John Wiley & Sons, Ltd.
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  • 9
    Electronic Resource
    Electronic Resource
    A nearly optimal preconditioning based on recursive red-black orderings (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 4 (1997), S. 369-391 
    Details
    ISSN: 1070-5325
    Keywords: iterative methods for linear systems ; acceleration of convergence ; preconditioning ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Considering matrices obtained by the application of a five-point stencil on a 2D rectangular grid, we analyse a preconditioning method introduced by Axelsson and Eijkhout, and by Brand and Heinemann. In this method, one performs a (modified) incomplete factorization with respect to a so-called ‘repeated’ or ‘recursive’ red-black ordering of the unknowns while fill-in is accepted provided that the red unknowns in a same level remain uncoupled.Considering discrete second order elliptic PDEs with isotropic coefficients, we show that the condition number is bounded by O(n½ log2(√(5) -1)) where n is the total number of unknowns (½ log2(√(5) - 1) = 0.153), and thus, that the total arithmetic work for the solution is bounded by O(n1.077). Our condition number estimate, which turns out to be better than standard O(log2 n) estimates for any realistic problem size, is purely algebraic and holds in the presence of Neumann boundary conditions and/or discontinuities in the PDE coefficients.Numerical tests are reported, displaying the efficiency of the method and the relevance of our analysis. © 1997 John Wiley & Sons, Ltd.
    Additional Material: 2 Ill.
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  • 10
    Electronic Resource
    Electronic Resource
    A modern approach to the solution of problems of classic elastoplasticity on parallel computersDedicated to Professor Erwin Stein on the occasion of his 65th birthday. (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 4 (1997), S. 205-221 
    Details
    ISSN: 1070-5325
    Keywords: solid mechanics ; theory of elastoplasticity ; FEM ; scientific parallel computing ; preconditioning ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The numerical solution of non-linear problems in solid mechanics (e.g., elastic-plastic problems) may be very expensive in the case of the use of fine discretizations. There is a requirement for efficient numerical algorithms and very fast hardware. Modern parallel computers in combination with modern numerical techniques, such as the parallel preconditioned conjugate gradient method for solving large linear systems and the consequent use of a consistent linearization of the non-linear problem, may lead to a superlinear speed up in comparison with equivalent single processor machines. The authors have developed a parallel experimental computer code using both techniques. The code is used for scientific investigations in the area of the identification of mechanical material parameters of metals. © 1997 John Wiley & Sons, Ltd.
    Additional Material: 13 Ill.
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