A new solver for the elastic normal contact problem using conjugate gradients, deflation, and an FFT-based preconditioner

Authors:
E. A. H. Vollebregt
Affiliations:
DIAM, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands and VORtech, P.O. Box 260, 2600 AG Delft, The Netherlands
Venue:
Journal of Computational Physics
Year:
2014

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Abstract

This paper presents our new solver BCCG+FAI for solving elastic normal contact problems. This is a comprehensible approach that is based on the Conjugate Gradients (CG) algorithm and that uses FFTs. A first novel aspect is the definition of the ''FFT-based Approximate Inverse'' preconditioner. The underlying idea is that the inverse matrix can be approximated well using a Toeplitz or block-Toeplitz form, which can be computed using the FFT of the original matrix elements. This preconditioner makes the total number of CG iterations effectively constant in 2D and very slowly increasing in 3D problems. A second novelty is how we deal with a prescribed total force. This uses a deflation technique in such a way that CGs convergence and finite termination properties are maintained. Numerical results show that this solver is more effective than existing CG-based strategies, such that it can compete with Multi-Grid strategies over a much larger problem range. In our opinion it could be the new method of choice because of its simple structure and elegant theory, and because robust performance is achieved independently of any problem specific parameters.