Title: An adaptive preconditioned conjugate gradient approach for extremely large optimisation problems arising in computational geomechanics
Authors: Omid Kardani; Navid Kardani
Addresses: Center for Geotechnical Science and Engineering, School of Engineering, University of Newcastle, University Drive, Callaghan, NSW, 2308, Australia ' Institute of Petroleum Engineering, Faculty of Engineering, University of Tehran, North Kargar, Tehran, Iran
Abstract: The feasibility of preconditioned conjugate gradient (PCG) for solving extremely large sparse linear systems arising in three-dimensional finite element limit analysis of geotechnical stability problems is studied. With problem size increasing beyond few millions of variables, the use of direct solvers becomes prohibitive (both in storage and computational times) even on modern computers. This motivates the study on possible employment of iterative methods to solve systems of equations resulting from interior point method-based limit analysis formulations. In this paper, an adaptive preconditioning technique based on incomplete Cholesky (IC) preconditioner, as well as a novel implementation scheme for preconditioned conjugate gradient method, is proposed to effectively deal with highly ill-conditioned systems, such as those which occur at last iterations of IPM methods. Numerical tests on some ultra-large scale sample geotechnical problems, along with comparison with other potential solution methods, verify the robustness and efficiency of the proposed approach.
Keywords: computational geomechanics; conjugate gradient method; interior point method; IPM; incomplete Cholesky factorisation; finite element limit analysis; FELA; geotechnical stability; adaptive preconditioning; extremely large sparse linear systems; optimisation.
International Journal of Mathematical Modelling and Numerical Optimisation, 2015 Vol.6 No.4, pp.299 - 320
Received: 06 Jul 2015
Accepted: 29 Sep 2015
Published online: 28 Dec 2015 *