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  • 1
    Electronic Resource
    Electronic Resource
    Phase-space representation of iterative design processes (1991)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 32 (1991), S. 327-346 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A large variety of design optimization methods have been proposed in recent years. Comparison of the relative performance of each method is a difficult task, and attempts to do so are often based on a limited number of numerical experiments. Recently, a ‘basins of attraction’ construction has been proposed as a graphical tool for investigating global performance of iterative design optimization methods, and as a basis for comparison of different methods. The phase-space representation presented in this paper is a companion to the basins of attraction construction. Basins of attraction reveal the relationship between starting design and final outcome of the solution process; the phase-space construction reveals that nature of the paths connecting the starting design and final outcome. The two constructions complement one another in summarizing the performance of design optimization processes. Both constructions are demonstrated in this paper, applied to the optimal design of an elastic grillage structure using Newton's method and the stress ratio method.
    Additional Material: 21 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620320207
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    The monomial method and asymptotic properties of algebraic systems (1994)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 37 (1994), S. 3939-3955 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The monomial method is a numerical method for solving systems of algebraic equations. It is related to Newton's method, but is based on an approximation that is monomial, rather than linear, in form. It has been shown previously that the monomial method has a number of properties not shared by Newton's method that are responsible for enhanced performance. This paper demonstrates that many of the performance characteristics of the monomial method can be explained in terms of asymptotic properties of algebraic systems, and the ability of the monomial method to exploit these properties. The monomial method recasts the algebraic system to have monomial asymptotes in all directions, so that the monomial approximation becomes asymptotically exact. This explains the very rapid movement toward a solution in the first iteration from distant starting points that has been observed with the monomial method. Problem types with ‘good asymptotics’ are found to be solved very effectively by the monomial method. Several specific engineering applications with good asymptotics are identified, and suggestions are made regarding other types of engineering applications that are likely to be handled effectively by the monomial method.
    Additional Material: 9 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620372211
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    The monomial method: Extensions, variations, and performance issues (1994)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 37 (1994), S. 2093-2107 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The monomial method solves systems of non-linear algebraic equations by constructing a sequence of approximating monomial (single-term polynomial) systems, much as Newton's method generates a sequence of linear systems to do this. Since the monomial system becomes linear through a logarithmic transformation of variables, the monomial method can be considered to be an alternative linearization scheme. Although the monomial method is closely related to Newton's method, it exhibits many special invariance properties not shared by Newton's method that enhance performance. This paper first briefly reviews the monomial method and its special properties. Two new versions of the algorithm are presented, both of which, are simplified and computationally more efficient to implement in comparison to the original algorithm. The monomial method is also extended to apply to certain non-algebraic systems. Since the monomial method can be interpreted as Newton's method applied to a three-part reformulation of the algebraic system, graphical experiments are presented which investigate the role that each part of the reformulation plays in contributing to the enchanced performance. Finally, instances in which difficulties have arisen using the monomial method are discussed.
    Additional Material: 3 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620371206
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  • 4
    Electronic Resource
    Electronic Resource
    A move coordination method for alternative loads in structural optimization (1989)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 28 (1989), S. 1041-1060 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A decomposition technique for alternative loading conditions in the integrated optimal structural design is developed. The method, called the move coordination, consists of partitioning the large structural optimization problem into a set of smaller coupled subproblems. In each subproblem only one loading condition is considered and the subproblems are solved in a parallel cyclic way. The coupling among the subproblems is accomplished through the introduction of coordinating constraints between each subproblem. These constraints ensure that the final design is the same and feasible for all subproblems. The method developed is illustrated by two examples of member sizing of truss structures using the integrated optimal design formulation and geometric programming. The method presents the advantage of reducing the size of the optimization problem as well as the computer processing time. The method is also suitable for implementation on computers using parallel processing.
    Additional Material: 9 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620280505
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    Paper (German National Licenses)
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  • 5
    Electronic Resource
    Electronic Resource
    Generalized geometric programming with many equality constraints (1987)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 24 (1987), S. 725-741 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A monomial treatment for solving mixed equality/inequality constrained generalized geometric programs is presented. The method is shown to be equivalent to a Newton-Raphson procedure carried out in log space on the equality constraints at the same time that a cutting plane procedure is performed on a convex region constrained within the feasible region of the inequality constraints. Known limitations of this monomial treatment are outlined and a technique for improving the efficiency of the GGP method is suggested. A discussion of how other methods of solving mixed equality/inequality constrained GGP problems compare to the monomial treatment is presented and two example problems are solved.
    Additional Material: 3 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620240406
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    Paper (German National Licenses)
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  • 6
    Electronic Resource
    Electronic Resource
    A finite element algorithm for elliptical equations over unstructured domains on a data parallel computerComputational resources were provided by Sandia National Laboratories, Albuquerque, New Mexico and Los Alamos National Laboratories, Los Alamos, New Mexico. (1994)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 37 (1994), S. 3153-3167 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: In the past, most finite element algorithms on data-parallel computers have been limited to structured problem domains. Here, we consider methods to generalize the finite element method on data-parallel architectures to unstructured domains. A nodal assembly algorithm is described which effectively allows for both the generation of the sparse interaction matrix (coefficient matrix) and its solution via a preconditioned conjugate-gradient type routine utilizing several polynomial preconditioners on the Connection Machine 200 (CM-200). Jacobi preconditioning along with Neumann series and least-squares polynomial preconditioners are presented and implemented are presented and implemented. Only the Jacobi preconditioner produces an improvement in the convergence time for the problems examined. Several irregular interprocessor communication protocols available on the CM-200 are investigated in the solution portion of the algorithm, yielding differing performance characteristics. For one such protocol, sustained performance of over two-hundred MFlops/s, is demonstrated for a test problem on a 512 processing element CM-200 in slicewise mode. The results are discussed and conclusions are drawn concerning this finite element algorithm.
    Additional Material: 6 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620371809
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    Paper (German National Licenses)
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