On the use of rigid body modes in the deflated preconditioned conjugate gradient method

On the use of rigid body modes in the deflated preconditioned conjugate gradient method

Jönsthövel, T.B.
Van Gijzen, M.B.
Vuik, C.
Scarpas, A.

Electrical Engineering, Mathematics and Computer Science

2011-12-31

Large discontinuities in material properties, such as encountered in composite materials, lead to ill-conditioned systems of linear equations. These discontinuities give rise to small eigenvalues that may negatively affect the convergence of iterative solution methods such as the Preconditioned Conjugate Gradient (PCG) method. This paper considers the Deflated Preconditioned Conjugate Gradient (DPCG) method for solving such systems. Our deflation technique uses as the deflation space the rigid body modes of sets of elements with homogeneous material properties. We show that in the deflated spectrum the small eigenvalues are mapped to zero and no longer negatively affect the convergence. We justify our approach through mathematical analysis and we show with numerical experiments on both academic and realistic test problems that the convergence of our DPCG method is independent of discontinuities in the material properties.

deflation, preconditioners, conjugate gradients, rigid body modes, CT scan, structural mechanics

http://resolver.tudelft.nl/uuid:246be1b7-d899-4052-8b27-e38cf985ad3b

Delft University of Technology, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft Institute of Applied Mathematics

1389-6520

Reports of the Department of Applied Mathematical Analysis, 11-04

Institutional Repository

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(c)2011 Jönsthövel, T.B., Van Gijzen, M.B., Vuik, C., Scarpas, A.