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An efficient preconditioned Krylov subspace method for large-scale finite element equations with MPC using Lagrange multiplier method

Zixiang Hu (State Key Laboratory of Material Processing and Die & Mold Technology, Huazhong University of Science and Technology, Wuhan, P. R. China and Research Institute of Huazhong University of Science and Technology in Shenzhen, Shenzhen, P. R. China)
Shi Zhang (State Key Laboratory of Material Processing and Die & Mold Technology, Huazhong University of Science and Technology, Wuhan, P. R. China)
Yun Zhang (State Key Laboratory of Material Processing and Die & Mold Technology, Huazhong University of Science and Technology, Wuhan, P. R. China)
Huamin Zhou (State Key Laboratory of Material Processing and Die & Mold Technology, Huazhong University of Science and Technology, Wuhan, P. R. China and Research Institute of Huazhong University of Science and Technology in Shenzhen, Shenzhen, P. R. China)
Dequn Li (State Key Laboratory of Material Processing and Die & Mold Technology, Huazhong University of Science and Technology, Wuhan, P. R. China)

Engineering Computations

ISSN: 0264-4401

Article publication date: 30 September 2014

215

Abstract

Purpose

The purpose of this paper is to propose an efficient iterative method for large-scale finite element equations of bad numerical stability arising from deformation analysis with multi-point constraint using Lagrange multiplier method.

Design/methodology/approach

In this paper, taking warpage analysis of polymer injection molding based on surface model as an example, the performance of several popular Krylov subspace methods, including conjugate gradient, BiCGSTAB and generalized minimal residual (GMRES), with diffident Incomplete LU (ILU)-type preconditions is investigated and compared. For controlling memory usage, GMRES(m) is also considered. And the ordering technique, commonly used in the direct method, is introduced into the presented iterative method to improve the preconditioner.

Findings

It is found that the proposed preconditioned GMRES method is robust and effective for solving problems considered in this paper, and approximate minimum degree (AMD) ordering is most beneficial for the reduction of fill-ins in the ILU preconditioner and acceleration of the convergence, especially for relatively accurate ILU-type preconditioning. And because of concerns about memory usage, GMRES(m) is a good choice if necessary.

Originality/value

In this paper, for overcoming difficulties of bad numerical stability resulting from Lagrange multiplier method, together with increasing scale of problems in engineering applications and limited hardware conditions of computer, a stable and efficient preconditioned iterative method is proposed for practical purpose. Before the preconditioning, AMD reordering, commonly used in the direct method, is introduced to improve the preconditioner. The numerical experiments show the good performance of the proposed iterative method for practical cases, which is implemented in in-house and commercial codes on PC.

Keywords

Acknowledgements

The authors would like to acknowledge the financial support from the National Natural Science Foundation Council of China (Grant No. 51125021, 51105152), the Major State Basic Research Project of China (Grant No. 2012CB025900) and the Shenzhen Basic Research Fund (Grant No. JC201005280644A, JC201105160599A).

Citation

Hu, Z., Zhang, S., Zhang, Y., Zhou, H. and Li, D. (2014), "An efficient preconditioned Krylov subspace method for large-scale finite element equations with MPC using Lagrange multiplier method", Engineering Computations, Vol. 31 No. 7, pp. 1169-1197. https://doi.org/10.1108/EC-03-2013-0077

Publisher

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Emerald Group Publishing Limited

Copyright © 2014, Emerald Group Publishing Limited

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