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  • 1
    Electronic Resource
    Electronic Resource
    Multigrid solution procedures for structural dynamics eigenvalue problems (1992)
    Springer
    Computational mechanics 10 (1992), S. 247-262 
    Details
    ISSN: 1432-0924
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: Abstract Three multigrid methods are described for solving the generalized symmetric eigenvalue problem encountered in structural dynamics. Two implicit algorithms are discussed that use a multigrid method to solve the linear matrix equations encountered in each iteration of the standard subspace and block Lanczos methods. An explicit method is also outlined which explicitly applies the basic multigrid philosophy of fine mesh relaxation and coarse mesh correction to the eigenvalue problem. All of these algorithms are capable of extracting the lower modes of the system, provided each required eigenvector can be represented on each coarse mesh. The behavior of the methods is studied by examining the selection of convergence tolerances and the solution of some ill-conditioned problems. A well-conditioned plate problem is solved to demonstrate the computational resources required by the algorithms. The explicit method is observed to be the most efficient method (in terms of storage and CPU time), whereas the implicit Lanczos method requires the most computational effort. A comparison between the multigrid algorithms and a commercially available implementation of the subspace iteration method is also presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00370092
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    The multigrid method in solid mechanics: Part I - Algorithm description and behaviour (1990)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 29 (1990), S. 719-737 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A multigrid algorithm is described that can be used to obtain the finite element solution of linear elastic solid mechanics problems. The method is applied to some two dimensional problems to evaluate its strengths and weaknesses. Extensive studies are made to determine the convergence behaviour of the method. In general, this depends on many factors: the number of degrees-of-freedom in the discretization, characteristics of the algorithm, Poisson's ratio when it is close to 0·5, the amount of bending deformation in the problem under consideration, and the degree of non-uniformity in the mesh. Only certain values of the multigrid parameters allow a converged solution to be obtained with a computational effort proportional to the number of degrees-of-freedom. These values include the optimum ones, i.e. those that lead to convergence with the least computational effort. The constant of proportionality is only independent of the number of degrees-of-freedom and still depends on the other factors listed above.
    Additional Material: 11 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620290404
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    The multigrid method in solid mechanics: Part II - Practical applications (1990)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 29 (1990), S. 739-753 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The performance of the multigrid algorithm is investigated by solving some large, practical, three dimensional solid mechanics problems. The convergence of the method is sensitive to factors such as the amount of bending present and the degree of mesh non-uniformity, as was also observed in Part I for two dimensional problems. However, in contrast to Part I, no proportionality is observed between the total number of operations to convergence and the problem size. Despite such behaviour, the multigrid algorithm proves to be an effective matrix equation solver for solid mechanics poblems. It is orders of magnitude faster than a direct factorization method, and yields converged solutions several times faster than the Jacobi preconditioned conjugate gradient method.
    Additional Material: 14 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620290405
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    A multigrid method for the generalized symmetric eigenvalue problem: Part II - performance evaluation (1992)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 35 (1992), S. 1677-1696 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The behaviour of the multigrid method is studied by solving some simple test problems. Optimum choices for some of the parameters are discussed, together with effective techniques for solving the coarse mesh correction equation. The effects of ill-conditioning on the performance of the algorithm are examined. In particular, thin shells and non-uniform meshes are observed to slow convergence. The solution of practical, large scale problems demonstrates the utility and speed of the proposed multigrid method. For example, the first 10 eigensolutions of a stiffened plate problem with 193 536 degrees-of-freedom were computed in 1.6 CPU hours using 42 Mbytes of memory on a Convex C240.
    Additional Material: 17 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620350808
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    Paper (German National Licenses)
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  • 5
    Electronic Resource
    Electronic Resource
    A multigrid method for the generalized symmetric eigenvalue problem: Part I - algorithm and implementation (1992)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 35 (1992), S. 1663-1676 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A multigrid method is described that can solve the generalized eigenvalue problem encountered in structural dynamics. The algorithm combines relaxation on a fine mesh with the solution of a singular equation on a coarse mesh. A sequence of coarser meshes may be used to quickly solve this singular equation using another multigrid method. The hierarchy of increasingly finer meshes can be further exploited using a nested iteration scheme, whereby initial approximations to the fine mesh eigenvectors are computed using interpolated coarse mesh eigenvectors. The solution of some simple plate problems on a Convex C240 demonstrates the efficiency of a vectorized version of the multigrid algorithm.
    Additional Material: 6 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620350807
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    Paper (German National Licenses)
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