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2010 | 30 | 2 | 249-259
Tytuł artykułu

Graph centers used for stabilization of matrix factorizations

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Systems of consistent linear equations with symmetric positive semidefinite matrices arise naturally while solving many scientific and engineering problems. In case of a "floating" static structure, the boundary conditions are not sufficient to prevent its rigid body motions.
Traditional solvers based on Cholesky decomposition can be adapted to these systems by recognition of zero rows or columns and also by setting up a well conditioned regular submatrix of the problem that is used for implementation of a generalised inverse. Conditioning such a submatrix seems to be related with detection of so called fixing nodes such that the related boundary conditions make the structure as stiff as possible. We can consider the matrix of the problem as an unweighted non-oriented graph. Now we search for nodes that stabilize the solution well-fixing nodes (such nodes are sufficiently far away from each other and are not placed near any straight line). The set of such nodes corresponds to one type of graph center.
Słowa kluczowe
Wydawca
Rocznik
Tom
30
Numer
2
Strony
249-259
Opis fizyczny
Daty
wydano
2010
otrzymano
2008-12-31
poprawiono
2009-09-30
zaakceptowano
2009-11-09
Twórcy
  • Department of Applied Mathematics, FEI, VSB - Technical University of Ostrava, 17. listopadu 15, 708 33, Ostrava-Poruba, Czech Republic
Bibliografia
  • [1] O. Axelsson, Iterative Solution Methods (Cambridge University Press: Cambridge, 1994).
  • [2] T. Brzobohatý, Z. Dostál, T. Kozubek and A. Markopoulos, Combining Cholesky decomposition with SVD to stable evaluation of a generalised inverse of the stiffness matrix of a floating structure, submited, 2009.
  • [3] Z. Dostál, D. Horák, R. Kucera, V. Vondrák, J. Haslinger, J. Dobiás and S. Pták, FETI based algorithms for contact problems: scalability, large displacements and 3D Coulomb friction, Computer Methods in Applied Mechanics and Engineering 194 (2005) 395-409, doi: 10.1016/j.cma.2004.05.015.
  • [4] Z. Dostál, D. Horák and R. Kucera, Total FETI - an easier implementable variant of the FETI method for numerical solution of elliptic PDE, Communications in Numerical Methods in Engineering 22 (2006) 1155-1162, doi: 10.1002/cnm.881.
  • [5] C. Farhat and M. Géradin, On the general solution by a direct method of a large scale singular system of linear equations: application to the analysis of floating structures, Int. J. Numer. Meth. Engng 41 (1998) 675-696, doi: 10.1002/(SICI)1097-0207(19980228)41:4<675::AID-NME305>3.0.CO;2-8
  • [6] C. Farhat and F.-X. Roux, A method of finite element tearing and interconnecting and its parallel solution algorithm, Int. J. Numer. Meth. Engng 32 (1991) 1205-1227, doi: 10.1002/nme.1620320604.
  • [7] Ch. Godsil and G. Royle, Algebraic Graph Theory (Springer-Verlag, New York, ISBN 0-387-95241-1, 2001).
  • [8] G.H. Golub and C.F. Van Loan, Matrix Computations (2nd ed.) (John Hopkins University Press, Baltimore, 1989).
  • [9] P. Kabelíková, T. Kozubek, A. Markopoulos and T. Brzobohatý, Effective Algorithms for Implementation of FETI-Based Domain Decomposition Methods, submited, 2009.
  • [10] G. Karypis and V. Kumar, METIS manual Version 4.0, [online] http://glaros.dtc.umn.edu/gkhome/views/metis, University of Minnesota, 1998.
  • [11] C.T. Pan, On the existence and factorization of rank-revealing LU factorizations, Linear Algebra and Its Applications 316 (2000) 199-222, doi: 10.1016/S0024-3795(00)00120-8.
  • [12] M. Papadrakakis and Y. Fragakis, An integrated geometric-algebraic method for solving semi-definite problems in structural mechanics, Computer Methods in Applied Mechanics and Engineering 190 (2001) 6513-6532, doi: 10.1016/S0045-7825(01)00234-1.
  • [13] K.C. Park and C.A. Felipa, The construction of free-free flexibility matrices for multilevel structural analysis, Int. J. Numer. Meth. Engng 47 (2000) 395-418, doi: 10.1002/(SICI)1097-0207(20000110/30)47:1/3<395::AID-NME777>3.3.CO;2-0
  • [14] K.C. Park, C.A. Felipa and U.A. Gumaste, A localized version of the method of Lagrange multipliers, Computational Mechanics 24 (2000) 476-490, doi: 10.1007/s004660050007.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1498
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