Keywords: singular matrix, regularization, fixing nodes, factorization, TFETI.

Due to the rounding errors, effective elimination of the displacements of "floating'' subdomains is a nontrivial aspect of the implementation of the FETI domain decomposition methods, since it can be difficult to recognize the positions of zero pivots when the nonsingular diagonal block of A is ill-conditioned. Moreover, even if the zero pivots are recognized properly, it turns out that the ill-conditioning of the nonsingular submatrix defined by the nonzero pivots can have a devastating effect on the precision of the solution.

Most of the results are related to the first problem, i.e., to identify reliably the zero pivots. Thus [1] proposed to combine the Cholesky decomposition with the singular value decomposition (SVD) of the related Schur complement S in order to guarantee a proper rank of the generalized inverse. The dimension of S is typically very small but not for ill-conditioned problems.

Here we review the results of [2], where we proposed a regularization technique enabling us to define a non-singular matrix whose inverse is the generalized inverse to A. It avoids the necessity to identify zero pivots. The favourable feature of our regularization is that an extra fill-in effect in the pattern of the matrix may be negligible.

The regularization is based on the fixing nodes strategies (uniform, kernel, and geometrical) described in [3,4,2] and demonstrated on both academic and engineering problems of mechanics therein. For the numerical solution we used the total FETI, a variant [5] of the FETI domain decomposition method that implements both prescribed displacements and interface conditions by the Lagrange multipliers. Thus the kernels of the stiffness matrices of the subdomains, i.e., their rigid body motions, are known a priori and we can use them in combination with fixing nodes for the effective regularization.

1

C.  Farhat, 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." Internat. J. Numer. Methods Engrg., 41(4):675-696, 1998.

2

R. Kucera, T. Kozubek, A. Markopoulos. "On large-scale generalized inverses in solving two-by-two block linear systems."Linear Algebra Appl., accepted, 2012.

3

T. Brzobohatý, Z. Dostál, T. Kozubek, P. Kovár, A. Markopoulos. "Cholesky decomposition with fixing nodes to stable computation of a generalized inverse of the stiffness matrix of a floating structure." Internat. J. Numer. Methods Engrg., (5):493-509 (electronic), 2011.

4

Z. Dostál, T. Kozubek, A. Markopoulos, M. Menšík. "Cholesky decomposition of a positive semidefinite matrix with known kernel" Appl. Math. Comput., 217(13):6067-6077, 2011.

5

Z. Dostál, T. Kozubek, A. Markopoulos, T. Brzobohatý, V. Vondrák, and P. Horyl. "A theoretically supported scalable TFETI algorithm for the solution of multibody 3d contact problems with friction." Comput Method Appl M, 205-208:110-120, 2012.

purchase the full-text of this paper (price £20)

go to the next paper
return to the table of contents
return to the book description
purchase this book (price £40 +P&P)