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  • 1
    Electronic Resource
    Electronic Resource
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 1-20 
    ISSN: 1070-5325
    Keywords: preconditioning saddle-point problems ; eigenvalue estimation ; mixed finite element method ; minimum residual method ; second-order elliptic problems ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We consider saddle-point problems that typically arise from the mixed finite element discretization of second-order elliptic problems. By proper equivalent algebraic operations the considered saddle-point problem is transformed to another saddle-point problem. The resulting problem can then be efficiently preconditioned by a block-diagonal matrix or by a factored block-matrix (the blocks correspond to the velocity and pressure, respectively). Both preconditioners have a block on the main diagonal that corresponds to the bilinear form(δ is a positive parameter) and a second block that is equal to a constant times the identity operator. We derive uniform bounds for the negative and positive eigenvalues of the preconditioned operator. Then any known preconditioner for the above bilinear form can be applied. We also show some numerical experiments that illustrate the convergence properties of the proposed technique.
    Additional Material: 2 Tab.
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  • 2
    Electronic Resource
    Electronic Resource
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996) 
    ISSN: 1070-5325
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
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  • 3
    Electronic Resource
    Electronic Resource
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 275-300 
    ISSN: 1070-5325
    Keywords: sparse QR factorization ; multifrontal method ; parallelism ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We describe the design and implementation of a parallel QR decomposition algorithm for a large sparse matrix A. The algorithm is based on the multifrontal approach and makes use of Householder transformations. The tasks are distributed among processors according to an assembly tree which is built from the symbolic factorization of the matrix ATA.We first address uniprocessor issues and then discuss the multiprocessor implementation of the method. We consider the parallelization of both the factorization phase and the solve phase. We use relaxation of the sparsity structure of both the original matrix and the frontal matrices to improve the performance. We show that, in this case, the use of Level 3 BLAS can lead to very significant gains in performance. We use the eight processor Alliant˜FX/80 at CERFACS to illustrate our discussion.
    Additional Material: 12 Ill.
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  • 4
    Electronic Resource
    Electronic Resource
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 113-124 
    ISSN: 1070-5325
    Keywords: iterative methods ; linear systems ; multisplittings ; overlap ; parallel algorithms ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Parallel two-stage multisplitting methods with overlap for the solution of linear systems of algebraic equations are studied. It is shown that, under certain hypotheses, the method with overlap is asymptotically faster than that without overlap. Experiments illustrating this phenomenon are presented.
    Additional Material: 2 Ill.
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  • 5
    Electronic Resource
    Electronic Resource
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 91-111 
    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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  • 6
    Electronic Resource
    Electronic Resource
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 173-177 
    ISSN: 1070-5325
    Keywords: diagonally dominant matrix ; determinant ; norm ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A property of strictly diagonally dominant matrices and a generalization of a Varga's bound for ∥ A-1∥∞ to the case ∥ A-1B∥∞ are given and the two-sided bounds for the determinants of strictly diagonally dominant matrices are derived
    Type of Medium: Electronic Resource
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  • 7
    Electronic Resource
    Electronic Resource
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 513-524 
    ISSN: 1070-5325
    Keywords: regularization ; I-norm ; discontinuous solutions ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In certain inverse problems it is useful to be able to compute solutions which are, in some sense, as simple as possible. For example,k one may wish to compute solutions which are piecewise constant and with as few discontinuities as possible. Such solutions are suited to describe models, e.g., geological layers, where the coarse structure is more important than the fine structure. A natural generalization of piecewise constant functions is piecewise polynomial solutions. In this paper we present a new algorithm which is capable of computing solutions that are piecewise polynomials, without having to specify a priori the positions of the break points between the polynomial pieces.
    Additional Material: 5 Ill.
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  • 8
    Electronic Resource
    Electronic Resource
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 491-512 
    ISSN: 1070-5325
    Keywords: unsymmetric ; Krylov subspace ; IOM(q) ; FOM ; truncated ; basis vector ; orthonormality ; convergence ; restarted ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The incomplete orthogonalization method (IOM(q)), a truncated version of the full orthogonalization method (FOM) proposed by Saad, has been used for solving large unsymmetric linear systems. However, no convergence analysis has been given. In this paper, IOM(q) is analysed in detail from a theoretical point of view. A number of important results are derived showing how the departure of the matrix A from symmetric affects the basis vectors generated by IOM(q), and some relationships between the residuals for IOM(q) and FOM are established. The results show that IOM(q) behaves much like FOM once the basis vectors generated by it are well conditioned. However, it is proved that IOM(q) may generate an ill-conditioned basis for a general unsymmetric matrix such that IOM(q) may fail to converge or at least cannot behave like FOM. Owing to the mathematical equivalence between IOM(q) and the truncated ORTHORES(q) developed by Young and Jea, insights are given into the convergence of the latter. A possible strategy is proposed for choosing the parameter q involved in IOM(q). Numerical experiments are reported to show convergence behaviour of IOM(q) and of its restarted version.
    Additional Material: 6 Tab.
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  • 9
    Electronic Resource
    Electronic Resource
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 413-426 
    ISSN: 1070-5325
    Keywords: iterative methods ; linear systems ; singular matrices ; block methods ; multisplitting ; two-stage ; non-stationary ; Markov chains ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The use of block two-stage methods for the iterative solution of consistent singular linear systems is studied. In these methods, suitable for parallel computations, different blocks, i.e., smaller linear systems, can be solved concurrently by different processors. Each of these smaller systems are solved by an (inner) iterative method. Hypotheses are provided for the convergence of non-stationary methods, i.e., when the number of inner iterations may vary from block to block and from one outer iteration to another. It is shown that the iteration matrix corresponding to one step of the block method is convergent, i.e., that its powers converge to a limit matrix. A theorem on the convergence of the infinite product of matrices with the same eigenspace corresponding to the eigenvalue 1 is proved, and later used as a tool in the convergence analysis of the block method. The methods studied can be used to solve any consistent singular system, including discretizations of certain differential equations. They can also be used to find stationary probability distribution of Markov chains. This last application is considered in detail.
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  • 10
    Electronic Resource
    Electronic Resource
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 349-350 
    ISSN: 1070-5325
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: No Abstract
    Type of Medium: Electronic Resource
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  • 11
    Electronic Resource
    Electronic Resource
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 427-453 
    ISSN: 1070-5325
    Keywords: mixed method ; nonconforming method ; multilevel preconditioner ; condition number ; second order elliptic problem ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A new approach for constructing algebraic multilevel preconditioners for mixed finite element methods for second order elliptic problems with tensor coefficients on general geometry is proposed. The linear system arising from the mixed methods is first algebraically condensed to a symmetric, positive definite system for Lagrange multipliers, which corresponds to a linear system generated by standard nonconforming finite element methods. Algebraic multilevel preconditioners for this system are then constructed based on a triangulation of the domain into tetrahedral substructures. Explicit estimates of condition numbers and simple computational schemes are established for the constructed preconditioners. Finally, numerical results for the mixed finite element methods are presented to illustrate the present theory.
    Additional Material: 4 Ill.
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  • 12
    Electronic Resource
    Electronic Resource
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 455-457 
    ISSN: 1070-5325
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
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  • 13
    Electronic Resource
    Electronic Resource
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 473-489 
    ISSN: 1070-5325
    Keywords: conjugate gradients ; biconjugate gradients ; conjugate directions ; non-symmetric linear systems ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this paper three basic conjugate direction methods for solving non-symmetric linear systems are described and compared. All three have entirely different basic structures and, consequently, completely different characteristics. The motivation for this work is the comparative assessment of the methods in order to decide which of the three is the most suitable for further development.
    Additional Material: 8 Ill.
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  • 14
    Electronic Resource
    Electronic Resource
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996) 
    ISSN: 1070-5325
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
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  • 15
    Electronic Resource
    Electronic Resource
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 543-545 
    ISSN: 1070-5325
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
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  • 16
    Electronic Resource
    Electronic Resource
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 21-44 
    ISSN: 1070-5325
    Keywords: first-order PDEs ; iterative methods ; preconditioners ; Toeplitz ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Preconditioners to nonsymmetric, nondiagonally dominant systems of equations are constructed and examined numerically. The preconditioners are based on a Toeplitz approach with a certain symmetry that we define. The inversion of the preconditioners is defined through a Fast Modified Sine Transform. As a model problem we study the systems of equations arising from a implicit time-discretization with a large time-step of a scalar hyperbolic PDE.
    Additional Material: 17 Ill.
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  • 17
    Electronic Resource
    Electronic Resource
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 45-64 
    ISSN: 1070-5325
    Keywords: Toeplitz least squares problems ; circulant preconditioned conjugate gradient method ; deconvolution ; image restoration ; atmospheric imaging ; medical imaging ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this paper, we propose a method to generalize Strang's circulant preconditioner for arbitrary n-by-n matrices An. The th column of our circulant preconditioner Sn is equal to the th column of the given matrix An. Thus if An is a square Toeplitz matrix, then Sn is just the Strang circulant preconditioner. When Sn is not Hermitian, our circulant preconditioner can be defined as . This construction is similar to the forward-backward projection method used in constructing preconditioners for tomographic inversion problems in medical imaging. We show that if the matrix An has decaying coefficients away from the main diagonal, then is a good preconditioner for An. Comparisons of our preconditioner with other circulant-based preconditioners are carried out for some 1-D Toeplitz least squares problems: min ∥ b - Ax∥2. Preliminary numerical results show that our preconditioner performs quite well, in comparison to other circulant preconditioners. Promising test results are also reported for a 2-D deconvolution problem arising in ground-based atmospheric imaging.
    Additional Material: 5 Ill.
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  • 18
    Electronic Resource
    Electronic Resource
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 65-80 
    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.
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  • 19
    Electronic Resource
    Electronic Resource
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 91-93 
    ISSN: 1070-5325
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
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  • 20
    Electronic Resource
    Electronic Resource
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996) 
    ISSN: 1070-5325
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
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  • 21
    Electronic Resource
    Electronic Resource
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 345-348 
    ISSN: 1070-5325
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
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  • 22
    Electronic Resource
    Electronic Resource
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 391-411 
    ISSN: 1070-5325
    Keywords: finite difference discretization ; fast adaptive composite grid method ; convergence rate ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The fast adaptive composite grid (FAC) method is an iterative method for solving discrete boundary value problems on composite grids. McCormick introduced the method in [8] and considered the convergence behaviour for discrete problems resulting from finite volume element discretization on composite grids. In this paper we consider discrete problems resulting from finite difference discretization on composite grids. We distinguish between two obvious discretization approaches at the grid points on the interfaces between fine and coarse subgrids. The FAC method for solving such discrete problems is described. In the FAC method several intergrid transfer operators appear. We study how the convergence behaviour depends on these intergrid transfer operators. Based on theoretical insights, (quasi-)optimal intergrid transfer operators are derived. Numerical results illustrate the fast convergence of the FAC method using these intergrid transfer operators.
    Additional Material: 3 Ill.
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  • 23
    Electronic Resource
    Electronic Resource
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996) 
    ISSN: 1070-5325
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
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  • 24
    Electronic Resource
    Electronic Resource
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 459-471 
    ISSN: 1070-5325
    Keywords: alternating projection method ; Dykstra's algorithm ; patterned matrix ; constrained least-squares ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We apply Dykstra's alternating projection algorithm to the constrained least-squares matrix problem that arises naturally in statistics and mathematical economics. In particular, we are concerned with the problem of finding the closest symmetric positive definite bounded and patterned matrix, in the Frobenius norm, to a given matrix. In this work, we state the problem as the minimization of a convex function over the intersection of a finite collection of closed and convex sets in the vector space of square matrices.We present iterative schemes that exploit the geometry of the problem, and for which we establish convergence to the unique solution. Finally, we present preliminary numberical results to illustrate the performance of the proposed iterative methods.
    Additional Material: 4 Tab.
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  • 25
    Electronic Resource
    Electronic Resource
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 125-145 
    ISSN: 1070-5325
    Keywords: orthogonalization ; QR factorization ; polar decomposition ; subspace tracking ; error analysis ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We consider the problem of computing Uk = QkUk-1(where U0 is given) in finite precision (εM = machine precision) where U0 and theQi are known to be unitary. The problem is that Ûk, the computed product may not be unitary, so one applies an O(n2) orthogonalizing step after each multiplication to(a) prevent Ûk from drifing too far from the set of untary matrices(b) prevent Ûk from drifting too far from Uk the true product.Our main results are1. Scaling the rows to have unit length after each multiplication (the cheaptest of the algorithms considered) is usually as good as any other method with respect to either of the criteria (a) or (b).2. A new orthogonalization algorithm that guarantees the distance of Ûk (k = 1, 2, …) to the set of unitary matrices is bounded by n3.5εM for any choice of Qi.
    Additional Material: 4 Ill.
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  • 26
    Electronic Resource
    Electronic Resource
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996) 
    ISSN: 1070-5325
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
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  • 27
    Electronic Resource
    Electronic Resource
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 147-159 
    ISSN: 1070-5325
    Keywords: elasticity ; finite-element ; multilevel ; a posteriori error estimation ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The constant γ in the strengthened Cauchy-Buniakowski-Schwarz (C.B.S.) inequality plays a crucial role in the convergence rate of multilevel iterative methods as well as in the efficiency of a posteriori error estimators, that is in the framework of finite element approximations of SPD problems.We consider the approximation of the 2D elasticity problem by the Courant element. Concerning multilevel convergence rate, that is the γ corresponding to nested general triangular meshes of size h and 2h, we have proved that γ2≤ 3/4$ uniformly on the mesh and the Poisson ratio. Concerning error estimator, that is the γ corresponding to quadratic and linear approximations on the same mesh, numerical computations have shown that the exact γ for a reference element deteriorates that is goes to one, when the Poisson ratio tends to 1/2
    Additional Material: 1 Ill.
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  • 28
    Electronic Resource
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    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 161-172 
    ISSN: 1070-5325
    Keywords: smoothing property ; multigrid method ; semi-iterative smoother ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this paper we present an extension of Reusken's Lemma about the smoothing property of a multigrid method for solving non-symmetric linear systems of equations. One of the consequences of this extended lemma is the verification of the smoothing property for all damping factors οε(0, 1). Additionally, a semi-iterative smoother is constructed which gives, in some sense, optimal smoothing rate estimates.
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  • 29
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    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 179-183 
    ISSN: 1070-5325
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
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  • 30
    Electronic Resource
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    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 185-203 
    ISSN: 1070-5325
    Keywords: orthotropic partial differential equation ; preconditioned conjugate gradient method ; parallel algorithm ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Finite element meshes and node-numberings suitable for parallel solution with equally loaded processors are presented for linear orthotropic elliptic partial differential equations. These problems are of great importance, for instance in the oil and airfoil industries. The linear systems of equations are solved by the conjugate gradient method preconditioned by modified incomplete factorization, MIC. The basic method presented, is based on fronts of uncoupled nodes and unlike earlier methods it has the advantage of no requirement of a specific orientation of the mesh. This method is however, in general, restricted to small degree of anisotropy in the differential equation. Another method, which does not suffer from this limitation, uses rotation of the differential equation and spectral equivalence. The rotation is made in such a way that in the new co-ordinate system, the basic method is applicable. The spectral equivalence property is used for estimation of the condition number of the preconditioned system. Both methods are suitable for implementation on parallel computers. The computer architecture could be single instruction multiple data (SIMD) as well as multiple instruction multiple data (MIMD) with shared or distributed memory. Implementation of the basic method on a shared memory parallel computer shows a significant improvement by use of the MIC method compared with the diagonal scaling preconditioning method.
    Additional Material: 13 Ill.
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  • 31
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    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 221-237 
    ISSN: 1070-5325
    Keywords: sparse matrix ; iterative methods ; preconditioning ; graph partitioning ; domain decomposition ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Domain decomposition methods for finite element problems using a partition based on the underlying finite element mesh have been extensively studied. In this paper, we discuss algebraic extensions of the class of overlapping domain decomposition algorithms for general sparse matrices. The subproblems are created with an overlapping partition of the graph corresponding to the sparsity structure of the matrix. These algebraic domain decomposition methods are especially useful for unstructured mesh problems. We also discuss some difficulties encountered in the algebraic extension, particularly the issues related to the coarse solver.
    Additional Material: 4 Ill.
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  • 32
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    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996), S. 205-220 
    ISSN: 1070-5325
    Keywords: elliptic equations ; multilevel methods ; finite element ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: New uniform estimates for multigrid algorithms are established for certain non-symmetric indefinite problems. In particular, we are concerned with the simple additive algorithm and multigrid (V(1,0)-cycle) algorithms given in [5]. We prove, without full elliptic regularity assumption, that these algorithms have uniform reduction per iteration, independent of the finest mesh size and number of refinement levels, provided that the coarsest mesh size is sufficiently small.
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  • 33
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    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 3 (1996) 
    ISSN: 1070-5325
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
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  • 34
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    Numerical Linear Algebra with Applications 3 (1996), S. 239-249 
    ISSN: 1070-5325
    Keywords: non-linear equations ; global convergence ; Lagrangians ; non-linear programming ; Lagrangian globalization ; detour potentials ; homotopy methods ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The Lagrangian globalization (LG) method for non-linear equation-solving proposed in [10] is developed through theoretical analysis, the formulation of a particular LG algorithm, and a numerical illustration. New merit functions (termed detour potentials) for non-linear equation-solving, which broaden the LG concept, are also defined.
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  • 35
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    Numerical Linear Algebra with Applications 3 (1996), S. 251-254 
    ISSN: 1070-5325
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
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  • 36
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    Numerical Linear Algebra with Applications 3 (1996), S. 255-274 
    ISSN: 1070-5325
    Keywords: conjugate gradient method ; preconditioning ; two-level method ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The presented paper contains a suggestion and an analysis of a two-level preconditioner appropriate for unstructured meshes. The convergence analysis is done for H1-equivalent forms in the two-dimensional case; ways for generalization are discussed, too. The efficiency is demonstrated by numerical experiments.
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  • 37
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    Numerical Linear Algebra with Applications 3 (1996), S. 329-343 
    ISSN: 1070-5325
    Keywords: iterative methods ; GMRES ; Krylov methods ; incomplete orthogonalization ; quasi-minimization ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We describe a Krylov subspace technique, based on incomplete orthogonalization of the Krylov vectors, which can be considered as a truncated version of GMRES. Unlike GMRES(m), the restarted version of GMRES, the new method does not require restarting. Like GMRES, it does not break down. Numerical experiments show that DQGMRES(k) often performs as well as the restarted GMRES using a subspace of dimension m=2k. In addition, the algorithm is flexible to variable preconditioning, i.e., it can accommodate variations in the preconditioner at every step. In particular, this feature allows the use of any iterative solver as a right-preconditioner for DQGMRES(k). This inner-outer iterative combination often results in a robust approach for solving indefinite non-Hermitian linear systems.
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  • 38
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    Numerical Linear Algebra with Applications 3 (1996), S. 81-90 
    ISSN: 1070-5325
    Keywords: linear systems ; regularization ; divided differences ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: A regularization procedure for linear systems of the type fi(zj)xi = g(zj), (j = 1, 2, …, n) is presented, which is particularly useful in the case when z1, z2, …, zn are close to each other. The associated numerical algorithm was tested on several examples for which analytic solutions do exist and was found to yield highly accurate results.
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  • 39
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    Numerical Linear Algebra with Applications 3 (1996), S. 301-327 
    ISSN: 1070-5325
    Keywords: information ; latent semantic indexing ; low-rank ; orthogonal ; matrices ; metrieval ; singular value decomposition ; sparse ; ULV and URV decompositions ; updating ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Current methods to index and retrieve documents from databases usually depend on a lexical match between query terms and keywords extracted from documents in a database. These methods can produce incomplete or irrelevant results due to the use of synonyms and polysemus words. The association of terms with documents (or implicit semantic structure) can be derived using large sparse {\it term-by-document} matrices. In fact, both terms and documents can be matched with user queries using representations in k-space (where 100 ≤ k ≤ 200) derived from k of the largest approximate singular vectors of these term-by-document matrices. This completely automated approach called latent semantic indexing or LSI, uses subspaces spanned by the approximate singular vectors to encode important associative relationships between terms and documents in k-space. Using LSI, two or more documents may be closeto each other in k-space (and hence meaning) yet share no common terms. The focus of this work is to demonstrate the computational advantages of exploiting low-rank orthogonal decompositions such as the ULV (or URV) as opposed to the truncated singular value decomposition (SVD) for the construction of initial and updated rank-k subspaces arising from LSI applications.
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  • 40
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    Numerical Linear Algebra with Applications 3 (1996), S. 369-390 
    ISSN: 1070-5325
    Keywords: multigrid ; incomplete Gaussian elimination ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this paper we introduce and analyse a new Schur complement approximation based on incomplete Gaussian elimination. The approximate Schur complement is used to develop a multigrid method. This multigrid method has an algorithmic structure that is very similar to the algorithmic structure of classical multigrid methods. The resulting method is almost purely algebraic and has interesting properties with respect to variation in problem parameters.
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  • 41
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    Numerical Linear Algebra with Applications 3 (1996), S. 351-367 
    ISSN: 1070-5325
    Keywords: multilevel preconditioning ; Krylov subspace methods ; GMRES, nonsymmetric elliptic problems ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The subject of this paper is to study the performance of multilevel preconditioning for nonsymmetric elliptic boundary value problems. In particular, a minimal residual method with respect to an appropriately scaled norm, measuring the size of the residual projections on all levels, is studied. This norm, induced by the multilevel splitting, is also the basis for a proper stopping criterion. Our analysis shows that the convergence rate of this minimal residual method using the multilevel preconditioner by Bramble, pasciak and Xu is bounded independently of the mesh-size. However, the convergence rate deteriorates with increasing size of the skew-symmetric part. Our numerical results show that by incorporating this into a multilevel cycle starting on the coarsest level, one can save fine-level-iterations and, therefore, computational work.
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  • 42
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    Communications in Numerical Methods in Engineering 12 (1996), S. 43-50 
    ISSN: 1069-8299
    Keywords: view factor ; radiation ; parallel computation ; heat transfer ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Two implementations for parallel computation of radiation heat transfer view factors are formulated and tested for a model problem. Using a sufficiently large number of processors and a suitable communications paradigm, the solution time for the problem considered here scales linearly with the number of surface elements in the parallel implementation rather than the quadratic scaling obtained using a serial approach. For larger problems, significantly shorter solution times are obtained using the massively parallel Connection Machine 5 than those obtained on a single processor of the Cray C-90, a traditional vector supercomputer. Parallel performance was degraded for implementations using too few processors or data layouts leading to inefficient processor communication.
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  • 43
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    Communications in Numerical Methods in Engineering 12 (1996), S. 51-62 
    ISSN: 1069-8299
    Keywords: boundary element method ; hypersingular integrals ; singular integration ; elasticity ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: An algorithm based on the Taylor series expansion is extended to deal with the problem of near-hypersingular integrals occurring in a 3D electrostatic BEM formulation. The integral is evaluated by subtracting out the leading terms in the near-hypersingular part of the integrand and adding it back. The accuracy of the proposed method is demonstrated for a curved element. It is shown that accurate values can be obtained with low orders of Gaussian quadrature.
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  • 44
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    Communications in Numerical Methods in Engineering 12 (1996) 
    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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  • 45
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    Communications in Numerical Methods in Engineering 12 (1996), S. 147-148 
    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
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  • 46
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    Communications in Numerical Methods in Engineering 12 (1996), S. 149-150 
    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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  • 47
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    Communications in Numerical Methods in Engineering 12 (1996), S. 63-73 
    ISSN: 1069-8299
    Keywords: asymptotic solution ; natural frequencies ; membrane vibrations ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: In the paper an asymptotic formula has been developed to correct the discretization error for the finite element predicted natural frequencies of membrane transverse vibration problems. The general idea behind deriving this asymptotic formula is that, when the finite element size approaches zero, a discretized finite element system approaches a continuous system and the predicted natural frequencies of the system from the finite element analysis therefore approach the exact solutions of the system. Without losing generality, several different finite element mesh patterns have been considered and the same asymptotic formula for correcting the finite element predicted natural frequency has been obtained for all the different mesh patterns because of the uniqueness of the exact solution to the natural frequency of a real structure. The usefulness, effectiveness and efficiency of the present asymptotic formula have been assessed by a simple but critical problem, for which the exact solution is available for comparison. In order to investigate the applicability of the asymptotic formula to practical engineering problems, two challenging membrane vibration problems of irregular shapes, an L-shape and a tapered shape with a circular hole in the centre, have also been analysed. The related numerical results have demonstrated that the asymptotic formula provides a very useful post-processing error corrector for the finite element predicted natural frequencies of membrane transverse vibration problems, even though the problem domains are of irregular shape. The greatest advantage in using the present asymptotic formula is that it yields a solution of higher accuracy, by simply using the formula to correct the rough solution obtained from a much coarser finite element mesh with fewer degrees of freedom, without any further finite element calculation.
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  • 48
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    Communications in Numerical Methods in Engineering 12 (1996), S. 77-83 
    ISSN: 1069-8299
    Keywords: eigenvalue extraction ; conjugate gradient method ; finite element method ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A new conjugate gradient algorithm is presented for extracting eigenvalues from large systems of equations encountered in finite element analysis. The new algorithm involves applying the conjugate gradient method (CGM) to a static problem to generate an equivalent tridiagonal matrix used for eigenvalue computation. The eigenvalues of the tridiagonal matrix are then extracted using a QR factorization. The similarity of the new CGM with the Lanczos method is discussed regarding the need for matrix inversion and reorthogonalization. Several examples using the new method are presented to illustrate its performance.
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  • 49
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    Communications in Numerical Methods in Engineering 12 (1996), S. 185-196 
    ISSN: 1069-8299
    Keywords: homogenization ; boundary conditions ; periodicity ; finite element ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The numerical solution of homogenization equations by the finite element (FE) method is explained briefly. The issue of extracting boundary conditions from the periodicity assumption is addressed and a direct method utilizing symmetry is presented. Using this method, the computation of the elements of the constitutive matrix of a composite material is reduced to a very conventional boundary value problem with known forces and boundary conditions which can be carried out with any FE code. Two examples are presented.
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    Communications in Numerical Methods in Engineering 12 (1996) 
    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Communications in Numerical Methods in Engineering 12 (1996), S. 229-234 
    ISSN: 1069-8299
    Keywords: higher-order beam element ; best-fit stress prediction ; a priori error analysis ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: It is known that finite elements try to capture stresses within each discretized local region in a ‘best-fit’ sense. In the paper we examine the performance of a beam element based on a higher-order shear deformation theory and show that the best-fit paradigm accounts for the manner in which through-the-thickness displacement and stresses are modelled. An a priori prediction derived from the paradigm is confirmed by a carefully chosen numerical experiment. This provides a measure of the quality of approximation as well as another ‘falsification’ of the best-fit paradigm.
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    Communications in Numerical Methods in Engineering 12 (1996), S. 223-228 
    ISSN: 1069-8299
    Keywords: Discretization ; heat conduction ; simulation ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A different approach to discretization is described with which complicated three-dimensional heat transfer problems can be solved with a finite volume approach on a general curvilinear grid. It represents an improvement on the existing methods in that it can easily be expanded to three-dimensional problems. A concise explanation of the transformation process is given, together with a discussion of the discretization procedure. The method is evaluated by solving two simple test problems and comparing the results with those of existing methods and the analytical solution. In conclusion it is found that this method yields equally or more accurate results than the existing methods, with the additional advantage of being easily expandable to three-dimensional problems.
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  • 53
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    Communications in Numerical Methods in Engineering 12 (1996), S. 243-248 
    ISSN: 1069-8299
    Keywords: finite elements ; automatic generation ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Finite element generation methods, such as Delaunay, allow the meshing of shapes from a set of nodes. These nodes must previously exist. The authors present a simple, fast and reliable method to generate them in an unstructured way in N-dimensional space. Its main characteristic is to easily allow variation of the node density by introducing a density function, either analytical or discretized. This function can also be anisotropic. Some nodes can have a pre-fixed position. Inner lines and holes can also be specified in the domain to be meshed. Generated nodes are to be connected according to triangles in 2D and tetrahedrals in 3D.
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  • 54
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    Communications in Numerical Methods in Engineering 12 (1996), S. 235-242 
    ISSN: 1069-8299
    Keywords: finite member element ; modified theory ; geometrical non-linearity ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: In the paper a modified non-linear member element is derived which can lead to satisfactory computed results even for large loading and displacement increments from pre-instability to post-instability of structures. The high accuracy of this element is proved by numerical examples.
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    Communications in Numerical Methods in Engineering 12 (1996) 
    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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  • 56
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    Communications in Numerical Methods in Engineering 12 (1996), S. 249-256 
    ISSN: 1069-8299
    Keywords: obstacle problems ; quadratic programming ; finite element ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The numerical solution of problems involving frictionless contact between an elastic body and a rigid obstacle is considered. The elastic body may undergo small or large deformation. Finite element discretization and repetitive linearization lead to a sequence of quadratic programming (QP) problems for incremental displacement. The performances of several QP algorithms, including two new versions of a modified steepest descent algorithm, are compared in this context. Numerical examples include a string, a membrane and an Euler-Bernoulli beam, in contact with flat and non-flat rigid obstacles.
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    Communications in Numerical Methods in Engineering 12 (1996), S. 317-319 
    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
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    Communications in Numerical Methods in Engineering 12 (1996), S. 271-280 
    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The authors study the shape optimization of a complex cracked shell under complex criteria. The shell is one of various cases of a turboshaft, and optimization criteria are associated to the cost, the technology, and above all the working conditions for the turboshaft. The optimization criteria involved are of course the weight of the structure, but also the plastic instability and critical stress intensity factor. All computations have been made with the Ansys finite element program in which an optimization module exists.
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    Communications in Numerical Methods in Engineering 12 (1996), S. 257-267 
    ISSN: 1069-8299
    Keywords: mode-matching ; Helmholtz' equation ; DtN ; finite element ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Finite element (FE) mode-matching procedures for the solution of Helmholtz' equation on an unbounded domain are reviewed and a symmetric general formulation is presented. This is a formal restatement of procedures applied previously to computations involving scattering of shallow water waves, acoustic transmission in non-uniform ducts and acoustic radiation from prismatic sheet metal ducts. An essential feature of the method is the use of a Galerkin procedure, rather than collocation, to match a finite computational model to a truncated modal expansion with the desired radiation characteristics. The method produces a symmetric set of linear equations which can be solved to give the unknown nodal values of the dependent variable and the modal coefficients of an outer expansion. Either of these sets of variables can be eliminated prior to solution to yield a reduced set of equations in the remaining parameters. The reduced equations obtained by eliminating the modal coefficients are shown to be identical to those obtained by applying a truncated Dirichlet-to-Neumann (DtN) boundary condition. If applied in this form, mode-matching can therefore be regarded as an alternative to the DtN method for generating this common set of discrete equations while permitting simultaneous solution for the modal coefficients in the outer region.
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    Communications in Numerical Methods in Engineering 12 (1996), S. 281-294 
    ISSN: 1069-8299
    Keywords: wavelets ; measures ; stability ; domains ; differential equations ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The paper investigates the accuracy and numerical stability of a class of wavelet Galerkin formulations on irregular domains. The method of numerical boundary measures is based upon a domain embedding strategy in which the irregular domain of interest is embedded in a larger domain having regular geometry. One advantage of the domain embedding method is that the boundary conditions on the larger, regular domain can be enforced in a straightforward manner, and the solution procedure can exploit the highly structured form of the resulting governing equations. The defining characteristic of this method is that the calculation of integrals along the irregular boundary are carried out using recently derived numerical boundary measures. In addition, the coercive bilinear forms characterizing the boundary value problem of interest must be calculated when restricted to the actual domain. In the case of wavelet Galerkin formulations, this calculation is accomplished with the three term connection coefficients that characterize the numerical boundary measure. The numerical stability and accuracy of the domain embedding procedure is compared to a newly developed wavelet-based finite element formulation.
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    Communications in Numerical Methods in Engineering 12 (1996), S. 371-372 
    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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    Communications in Numerical Methods in Engineering 12 (1996), S. 295-302 
    ISSN: 1069-8299
    Keywords: mesh generation ; adaptivity ; iterated fractal systems ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: In the paper the author presents a novel point of view for the refinement and derefinement algorithms of triangular nested meshes using fractal concepts and iterated function systems (IFS). The fractal behaviour can be understood in the sense that these meshes feature a remarkable amplifying invariance under changes of magnification. Here we compare the meshes obtained by the combination of these algorithms with those presented by Bova and Carey (1992). Although both of the meshes are very similar, the current algorithms automatically build and manage sequences of nested irregular discretizations of the domain. The author illustrates here how the application of IFS families is equivalent to the use of an adaptive strategy that combines the refinement procedure with the derefinement one.
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    Communications in Numerical Methods in Engineering 12 (1996), S. 303-316 
    ISSN: 1069-8299
    Keywords: stretching functions ; mesh refinement ; finite differences ; truncation error ; composite grids ; regularity ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: In this work the truncation-error criteria of Thompson and Mastin (1985) are combined with conditions of vanishing second and higher derivatives at both endpoints for the purpose of deriving new classes of one-dimensional stretching functions for mesh refinement in finite-difference numerics. With these elementary stretching functions, matching of the slopes between adjacent grid patches then automatically confers Cn regularity upon the composite stretching function. Formulated with reference to two conceptions of truncation order (fixed relative distribution against fixed number of nodes) the resulting mappings are shown to provide particularly advantageous node distributions at both ends simultaneously (with concomitantly higher truncation error in between). Viewed overall, the truncation-error functions compare favourably with those for sinh, tanh and erf - mappings whose utility for mesh refinement was established by Thompson and Mastin. The numerical labour of implementing the new stretching functions is only slightly greater than that required for the error function. An illustrative derivation involving Cn patching leads to two-sided stretching functions, which allow the slopes at both ends to be prescribed arbitrarily. This formulation differs from a previous approach described by Vinokur (1983).
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    Communications in Numerical Methods in Engineering 12 (1996), S. 383-394 
    ISSN: 1069-8299
    Keywords: symmetrically laminated plates ; anisotropy ; fundamental solutions ; boundary integral equations ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: In the paper the bending problem of moderately thick symmetrically laminated anisotropic plates is considered, based on the first-order transverse shear deformation plate theory. Using the method of plane wave decomposition and Hörmander's operator method, the fundamental solution of the plates is presented. The boundary integral equation of the plates is formulated by taking the fundamental solution presented as the weighted function and using the method of weighted residuals. The numerical calculation of the boundary integral equation presented is discussed in detail. Some examples are presented and compared with the exact solutions and the numerical solutions available in the literature. The numerical results show that the present method has a satisfactory rate of convergence and acceptable accuracy with a reasonable boundary mesh.
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  • 65
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    Communications in Numerical Methods in Engineering 12 (1996), S. 373-381 
    ISSN: 1069-8299
    Keywords: finite element method ; mesh refinement ; adaptivity ; singularities near edges ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: This paper is concerned with several refinement techniques of finite element meshes for treating elliptic boundary value problems in domains with re-entrant edges and corners. A priori mesh grading is explained, and it is combined with the well-known adaptive finite element method. For two representative examples the numerically determined error norms are recorded, and the different strategies are compared.
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  • 66
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    Communications in Numerical Methods in Engineering 12 (1996), S. 395-411 
    ISSN: 1069-8299
    Keywords: finite volume ; high-order discretization ; non-oscillatory schemes ; pressure interpolation ; non-staggered grid ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The paper addresses the problem of convection discretization by extension and application of numerical schemes used in compressible flows: SONIC-A, SONIC-B, UNO2, MUSCL and MINMOD to predict steady incompressible recirculating convection dominated flows. A new scheme, SONIC-Q, is proposed together with a third-order non-oscillatory practice for pressure interpolation in non-staggered grids. Finite-volume calculations of the Navier-Stokes equations of a standard 2D driven square cavity standard test case and the laminar flow over a fence using primitive variables and non-staggered grid systems have shown that the schemes are alternatives to the conventional ones used in general algorithms for incompressible recirculating flows. In general these composite high-order schemes have proved to be good candidates to overcome the problems of false-diffusion and unboundedness encountered in non-composite high-order upwind schemes used in incompressible flows.
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    Communications in Numerical Methods in Engineering 12 (1996), S. 425-431 
    ISSN: 1069-8299
    Keywords: time integration ; finite element method ; least squares formulation ; multistep methods ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The paper presents a formulation and analysis of three and four step least squares algorithms for first order IVPs. The three step algorithm is derived using cubic Lagrangian interpolation, and is found to be third order accurate but only conditionally stable. Fourth order Lagrangian interpolation is used to obtain a four step least squares scheme which is A0-stable but inconsistent.
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  • 68
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    Communications in Numerical Methods in Engineering 12 (1996), S. 413-424 
    ISSN: 1069-8299
    Keywords: boundary integral equation ; necessary and sufficient condition ; plane elasticity problem ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: With respect to a given boundary value problem, the corresponding conventional boundary integral equation is shown to yield non-equivalent solutions, which are dependent upon Poisson's ratio and geometry. In the paper a systematic method for establishing a necessary and sufficient boundary integral formulation has been proposed for two-dimensional elastostatic problems. Numerical analyses show that the conventional boundary integral equation yields incorrect results when the scale in the fundamental solution approaches a degenerate scale value. However, the results of the necessary and sufficient boundary integral equation are in good agreement with analytical solutions of the boundary value problem.
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  • 69
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    Communications in Numerical Methods in Engineering 12 (1996), S. 519-520 
    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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  • 70
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    Communications in Numerical Methods in Engineering 12 (1996) 
    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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  • 71
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    Communications in Numerical Methods in Engineering 12 (1996), S. 433-444 
    ISSN: 1069-8299
    Keywords: two-phase flow ; solute transport ; interphase exchange ; porous media ; finite element ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The development of a numerical method for modelling two-phase flows and solute transport, particularly with interphase exchange in porous media, is presented. The governing equations are derived to describe two immiscible and compressible fluids flows such as water-air and two-phase solute transport with interphase exchange. Technically, the standard finite element method and a strongly implicit procedure are employed to solve the fully coupled governing equations. Pressures of two-phase fluids and solute concentrations in two-phase fluids are taken as the primary unknown variables, and the discretized equations are solved by a direct type of solver. Application examples are shown to confirm the applicability of the numerical method.
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  • 72
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    Communications in Numerical Methods in Engineering 12 (1996), S. 447-454 
    ISSN: 1069-8299
    Keywords: inviscid drops ; coalescence ; boundary element method ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A numerical method is used to simulate the motion of inviscid drops colliding and coalescing at a solid surface. The equations of motion are solved by a boundary element method in which the free surface of the drop is represented by a moving grid. The numerical results include the configuration of the drop during coalescence and the kinetic and potential energies. A numerical example is used to demonstrate the way in which coalescence affects the configuration of the free surface.
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  • 73
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    Communications in Numerical Methods in Engineering 12 (1996), S. 455-459 
    ISSN: 1069-8299
    Keywords: harmonic differential quadrature method ; computational complexity ; differential quadrature ; numerical method ; centrosymmetric matrix ; computational mechanics ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The structure of weighting coefficient matrices of harmonic differential quadrature (HDQ) is found to be either centrosymmetric or skew centrosymmetric, depending on the order of the corresponding derivatives. The properties of both matrices are briefly discussed in the paper. It is noted that the computational effort of the harmonic quadrature for some problems can be further reduced by up to 75 per cent by using the properties of the above-mentioned matrices.
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  • 74
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    Communications in Numerical Methods in Engineering 12 (1996), S. 471-482 
    ISSN: 1069-8299
    Keywords: numerical methods ; constitutive equations ; composite ; mixing formulation ; anisotropic elastoplastic model ; mapping stress tensor space ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A general constitutive model adequate for analysis of the thermomechanical response of composite materials is presented. The model is based on the mixture of the basic substances of the composite and allows the evaluation of the interdependence between the constitutive behaviour of different compounding materials. The behaviour of the each compound is modelled by a general anisotropic thermo-elasto-plastic model, termed the ‘base model’. The different base models for each compound are combined using mixing theory to simulate the behaviour of the multiphase material.
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  • 75
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    Communications in Numerical Methods in Engineering 12 (1996), S. 581-594 
    ISSN: 1069-8299
    Keywords: acoustics ; finite element method ; error estimation ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A posteriori error estimation has become very popular, mainly in linear elasticity. A robust implementation of the superconvergent patch recovery technique of O. C. Zienkiewicz and J. Z. Zhu is presented for acoustic finite element analyses: the original concepts are extended to complex variables, and both local and global behaviours of the recovery procedure and the error estimation are studied. The numerical tests confirm the improvement of the rates of convergence for the recovered solution and also show the reliability of the error estimator except at frequencies corresponding either to the analytical or to the finite element eigenfrequencies.
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  • 76
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    Communications in Numerical Methods in Engineering 12 (1996), S. 595-595 
    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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  • 77
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    Communications in Numerical Methods in Engineering 12 (1996) 
    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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  • 78
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    Communications in Numerical Methods in Engineering 12 (1996), S. 597-597 
    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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  • 79
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    Communications in Numerical Methods in Engineering 12 (1996), S. 521-529 
    ISSN: 1069-8299
    Keywords: bellows ; shell of revolution ; integral equation method ; compressed angle ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: In the paper a U-shaped bellows is treated as a flexible shell of revolution which consists of circular ring shells and truncated shallow conical shells; then the non-linear problem of U-shaped bellows under the action of axial compression force and internal pressure is solved by means of the non-linear theory of shells and the integral equation method. Numerical solutions obtained are compared with previous theoretical and experimental results. The present theory is more appropriate to the analysis of bellows in the light of real profile shape, and shows that the influence of compressed angle on the characteristic relation and peak stresses is noticeable.
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  • 80
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    Communications in Numerical Methods in Engineering 12 (1996), S. 599-608 
    ISSN: 1069-8299
    Keywords: finite elements ; moving grids ; moving bodies ; mesh velocity ; ALE ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A Laplacian smoothing of the mesh velocities with variable diffusivity based on the distance from moving bodies is introduced. This variable diffusivity enforces a more uniform mesh velocity in the region close to the moving bodies. Given that in most applications these are regions where small elements are located, the new procedure decreases element distortion considerably, reducing the need for local or global remeshing, and in some cases avoiding it altogether.
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  • 81
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    Communications in Numerical Methods in Engineering 12 (1996), S. 531-542 
    ISSN: 1069-8299
    Keywords: vibration ; inverse eigenvalue problem ; modal analysis ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A method for determining mass and stiffness modifications to achieve desired natural frequencies is presented. The given data are modal testing results, which consist of a truncated set of natural frequencies and mode shapes. The difficulty arising from the incompleteness of data is overcome by solving an optimization problem rather than seeking an exact solution. The obtained modifications are optimal in a Rayleigh-Ritz sense. The case where the mass and stiffness matrices are interrelated is also considered. Numerical examples demonstrating the various results and the sensitivity of the problem to perturbations are presented.
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  • 82
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    Communications in Numerical Methods in Engineering 12 (1996), S. 609-616 
    ISSN: 1069-8299
    Keywords: hybrid-Trefftz element ; plate ; transient analysis ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The paper presents a hybrid Trefftz (HT) element approach for the numerical solution of transient plate bending problems. In the proposed method, the dynamic plate equation is first discretized with respect to time and then the resulting set of elliptic equations is solved by the corresponding time independent hybrid Trefftz element approach. Two examples are considered to assess the effectiveness of the numerical method.
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  • 83
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    Communications in Numerical Methods in Engineering 12 (1996), S. 617-630 
    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Three different solution methods for the finite-volume discretized incompressible Navier-Stokes equations have been tested: segregated approaches, built around coupling methods such as SIMPLE, SIMPLER and PISO plus a line Gauss-Seidel linear solver; coupled methods, incorporating a penalty formulation to eliminate zero diagonal elements in the coefficient matrices, plus preconditioned GMRES as a linear solver;- and a FAS-full multigrid algorithm accelerating a classical segregated method based upon SIMPLE and the line Gauss-Seidel solver. Results demonstrate that the coupled method compares favourably to the segregated technique at small grid sizes but becomes too expensive for large problems. The FAS-full multigrid algorithm outperforms the other two methods when large numbers of nodes are employed in the simulation.
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  • 84
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    Communications in Numerical Methods in Engineering 12 (1996), S. 631-641 
    ISSN: 1069-8299
    Keywords: numerical analysis ; fast direct method ; elliptic problems ; finite difference methods ; eigenvalue problems ; harmonic ; biharmonic ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A high accuracy finite difference scheme has been developed for solving some elliptic problems which appear in engineering and applied sciences. These include Laplace, Poisson, Helmholtz and related equations. The second- and fourth-order problems dealing with vibration of membranes and plates have also been examined. Numerically, the problem reduces to a block tridiagonal system which can be solved by suitably modifying the fast direct method developed by Hockney. Comparison has been made with results obtained from some alternative numerical methods or analytical methods whenever available
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  • 85
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    Communications in Numerical Methods in Engineering 12 (1996), S. 643-655 
    ISSN: 1069-8299
    Keywords: viscoelastic flows ; sphere ; annular flow ; adaptive meshing ; finite elements ; error analysis ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: This work describes the implementation of an adaptive procedure for viscoelastic flows. Finite element simulations are conducted using a Taylor-Galerkin/pressure correction scheme. The constitutive equations considered are those for an Oldroyd-B and a Phan-Thien/Tanner model. The adaptive meshing technique is h-refined by grid remeshing, based on a Delaunay procedure. Results are presented for two benchmark problems, namely flow past a sphere in a cylindrical tube and flow in an annular converging tube. It is concluded from the results that the use of an adaptive procedure with mesh regeneration enables an optimized mesh to be devised and a predicted accuracy to be attained.
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  • 86
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    Communications in Numerical Methods in Engineering 12 (1996), S. 657-671 
    ISSN: 1069-8299
    Keywords: continuous casting ; free boundary problem ; co-ordinate transformation ; solidification front ; non-orthogonal control volume ; temperature profile ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: An efficient computational simulation scheme based on non-orthogonal control volume discretization and co-ordinate transformation techniques has been developed for solving the thermal transport phenomena, which involves tracking of the interface between solid and liquid phases (solidification front) and evaluation of the temperature profile during continuous casting operation. Conservation equations are reformulated in differential-integral form in terms of the transformed co-ordinates. All the terms arising from the non-orthogonality of the control volume have been retained in the numerical solution methodology, and a front tracking procedure has been formulated based on an iterative solution scheme. The formulation has been applied to solve the thermal transport phenomena in solidification processing of an A1-Mg alloy cylindrical ingot during continuous casting, which also includes axial conduction of heat. Theoretical evaluation of the solidification front and temperature distribution in the ingot are in good agreement with the experimentally measured data.
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  • 87
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    Communications in Numerical Methods in Engineering 12 (1996) 
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    Keywords: Engineering ; Engineering General
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  • 88
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    Communications in Numerical Methods in Engineering 12 (1996), S. 703-703 
    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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  • 89
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    Communications in Numerical Methods in Engineering 12 (1996), S. 705-705 
    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
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  • 90
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    Communications in Numerical Methods in Engineering 12 (1996), S. 707-717 
    ISSN: 1069-8299
    Keywords: hybrid element ; shear locking ; penalty-equilibrating approach ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The paper presents a penalty-equilibrating approach which is used to enforce the equilibrium constraints to individual hybrid elements so as to optimize the element performance. The approach has been applied successfully to improve the 4-noded hybrid stress membrane element (P-S) by Pian and Sumihara. It turns out that some defects of the P-S element, e.g. the shear locking phenomenon and the inability to pass the MacNeal's bending test, can be overcome and the element performance is significantly improved, without having to add any extra variables to the P-S element or to modify its stress fields.
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  • 91
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    Communications in Numerical Methods in Engineering 12 (1996), S. 683-702 
    ISSN: 1069-8299
    Keywords: unstructured grid generation ; finite elements ; advancing front ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: We describe extensions and improvements to the advancing front grid generation technique that have proven useful over the years. The following areas are treated in detail: situations with thin or crossing surfaces, meshing of surfaces defined by triangulations, and ease of user input to define the desired element size in space. The first extension is important if one considers the generation of volumetric grids around shells, membranes, fabrics, or CAD-data that exhibit cusps. Traditional advancing front generators are likely to fail in these situations. We propose the introduction of a crossing environment variable attached to faces and points in order to filter out undesired or incorrect information during the grid generation process. The second extension is required for situations where the surfaces to be gridded are not defined analytically, but via a triangulation. Typical cases where such triangulations are used to define the domain are geophysical problems, climate modelling and medical problems. The third topic deals with the reduction of manual labour to specify element size in space. Sources, element size attached directly to CAD-data, and adaptive background grids are discussed. Adaptive background grids, in combination with surface deviation tolerances, are used to obtain surface triangulations that represent the geometry faithfully, and at the same time enable a smooth transition to volumetric meshes.
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  • 92
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    Communications in Numerical Methods in Engineering 12 (1996), S. 731-743 
    ISSN: 1069-8299
    Keywords: explicit integration ; finite elements ; stiffness matrix ; symbolic manipulation ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The paper deals with the symbolic integration of a 4-noded isoparametric finite element for plane elasticity. An efficient approach to generate explicit formulas for computing the elementary stiffness matrix is discussed. The procedure is based on the use of the Derive symbolic manipulation code as well as in a posteriori manipulation of the expressions obtained. The accuracy of the results is tested in extremely distorted and geometrically ill-conditioned elements. Three practical engineering models are presented and the accuracy of the results is discussed. A computer time comparison between both numerical and symbolic integration approaches is also included, showing that relevant CPU savings are obtained when applying symbolic integration.
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  • 93
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    Communications in Numerical Methods in Engineering 12 (1996), S. 719-729 
    ISSN: 1069-8299
    Keywords: air humidification process ; experiments ; predictions with finite-difference technique ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The paper deals with a numerical and an experimental analysis of the humidification process of air in a nearly horizontal tube. The numerical analysis is based on a finite-difference solution of the three-dimensional momentum, heat and mass transfer differential equations within the tube for the evaporation process of a water stream of circular meniscus cross-section counterflowing to an air stream. In order to provide reliable, high-accuracy data to assist in the development of the theoretical model, tests have been conducted using specially constructed apparatus to measure temperature and velocity profiles within the tube. The proposed computer algorithm was validated by evaluating its ability to predict the measurements of temperatures and velocities along the stream's flow. The theoretical predictions agree well with the experimental data. The model may be used to give performance characteristics for a wide range of humidification processes with different working fluids and state conditions.
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  • 94
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    Communications in Numerical Methods in Engineering 12 (1996), S. 745-753 
    ISSN: 1069-8299
    Keywords: boundary element method ; foundations ; soil-structure interaction ; contact problem ; optimal solutions ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The problem of providing settlement uniformity of a rigid foundation on a non-uniformly compressible base has been reduced to the contact problem from elasticity theory, with controlling parameters which allow one to select either co-ordinates of the point of application of the vertical load resultant or contours in plan at which the foundation slope will be equal to zero. The solution of the problem is illustrated by numerical examples.
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  • 95
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    Communications in Numerical Methods in Engineering 12 (1996), S. 755-766 
    ISSN: 1069-8299
    Keywords: finite elements ; isoparametric ; Jacobian ; transformation ; valid ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Isoparametric elements are only valid if the Jacobian determinant of the transformation between a given element and a master element does not change sign within or on the element boundary. Some algorithms are known which analyse Jacobians for various element types. Some necessary conditions are presented for determining the validity of an element.
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  • 96
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    Communications in Numerical Methods in Engineering 12 (1996), S. 767-773 
    ISSN: 1069-8299
    Keywords: curved beam element ; membrane locking ; shear locking ; strain based element ; transformation matrix ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Two curved beam elements with two nodes and three nodes are designed based on strain fields. At the element level, curvature and membrane strain fields are approximated independently and shear strain fields are incorporated into the formulation by the equilibrium equations. The displacement fields are obtained by integrating the assumed strain fields. Two examples are given to verify the formulations and demonstrate the numerical performance of the two curved beam elements. Analysis results obtained reveal that the elements describe the curved beam behaviour correctly and show exceptional accuracy throughout a wide slenderness range.
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  • 97
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    Communications in Numerical Methods in Engineering 12 (1996), S. 775-785 
    ISSN: 1069-8299
    Keywords: membrane ; wrinkle ; non-linear ; elasticity ; finite-element method ; sail ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: This is a presentation of a quadrilateral finite element for wrinkled curved elastic membranes. A modified form of the deformation gradient enables us to avoid the spurious compressive stresses generated by a classical model. It results in non-linear relations for the eigencomponents of the Cauchy stress tensor, which are solved by means of a secant method. The application of the element to sail design is presented in this paper.
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  • 98
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    Communications in Numerical Methods in Engineering 12 (1996), S. 897-903 
    ISSN: 1069-8299
    Keywords: finite element analysis ; reinforced concrete element ; embedded reinforcement ; stress analysis ; brick element ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: In the paper a mathematical formulation for embedding reinforcements in 3D brick concrete elements is presented. The computational aspects and the implementation process of the model have been discussed. The proposed algorithm may be implemented in 3D finite element programs for stress analysis of reinforced concrete structures.
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  • 99
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    Communications in Numerical Methods in Engineering 12 (1996), S. 107-114 
    ISSN: 1069-8299
    Keywords: beam ; vibration ; trial functions ; Ritz method ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Free vibration of beams with intermediate point supports is studied by the classical Ritz method within the context of Euler beam theory. For the Ritz method, the displacement of a beam is approximated by a set of admissible trial functions which must satisfy the kinematic conditions at the ends and intermediate supports of the beam. To this end, a polynomial is superimposed on the conventional single-span beam vibration functions to form continuous-span or modified beam vibration functions. These modified beam functions are taken as the admissible trial functions for subsequent formulation. Stiffness and mass matrices are formulated using the conventional procedure and the resulting linear eigen-equation can be solved easily. A number of numerical examples are given to demonstrate the accuracy and efficiency of the present method.
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  • 100
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    Communications in Numerical Methods in Engineering 12 (1996), S. 135-140 
    ISSN: 1069-8299
    Keywords: finite elements ; QUAD4 elements ; RBF corrections ; shear force predictions ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The residual bending flexibility correction is an extra-variational trick used to enhance the convergence rate of shear flexible beam and plate elements. In the paper we show how this can affect the shear deflection and shear force estimates in the QUAD4 element.
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