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Article
Publication date: 2 October 2019

Makoto Kumagai, Shimpei Kakita and Yoshifumi Okamoto

This paper aims to present the affinity of BiCGStab and BiCGStab2 with successive over-relaxation (SOR) preconditioner supported by Eisenstat’s technique for a linear system…

Abstract

Purpose

This paper aims to present the affinity of BiCGStab and BiCGStab2 with successive over-relaxation (SOR) preconditioner supported by Eisenstat’s technique for a linear system derived from the time-periodic finite element method (TP-FEM). To solve the time domain electromagnetic field problem with long transient state, TP-FEM is very useful from the perspective of rapidly achieving a steady state. Because TP-FEM solves all of the state variables at once, the linear system derived from TP-FEM becomes the large scale and nonsymmetric, whereas the detailed performance of some preconditioned Krylov subspace method is not reported.

Design/methodology/approach

In this paper, BiCGStab and BiCGStab2 are used as the linear solver for a large-sparse nonsymmetric linear system derived from TP-FEM. In addition, incomplete LU (ILU) factorization is applied as a preconditioner to compare SOR supported by Eisenstat’s technique. As examples, the pot-type reactor and three-phase transformer is analyzed.

Findings

In the problem of the pot-type reactor, when SOR preconditioner supported by Eisenstat’s technique is applied to BiCGStab and BiCGStab2, the elapsed time can be reduced dramatically. However, in the problem of the three-phase transformer, the iterative process of the linear solvers with SOR preconditioner is not terminated, whereas the iterative process of linear solvers with ILU preconditioner is terminated. The preconditioner that can be supported by Eisenstat’s technique is not necessarily appropriate for the problem to derive from TP-FEM.

Originality/value

In this paper, the affinity of preconditioned linear solver supported by Eisenstat’s technique for the nonsymmetric linear system derived from TP-FEM is demonstrated.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering , vol. 38 no. 5
Type: Research Article
ISSN: 0332-1649

Keywords

Article
Publication date: 5 July 2021

Abhishek Kumar Singh and Krishna Mohan Singh

The work presents a novel implementation of the generalized minimum residual (GMRES) solver in conjunction with the interpolating meshless local Petrov–Galerkin (MLPG) method to…

Abstract

Purpose

The work presents a novel implementation of the generalized minimum residual (GMRES) solver in conjunction with the interpolating meshless local Petrov–Galerkin (MLPG) method to solve steady-state heat conduction in 2-D as well as in 3-D domains.

Design/methodology/approach

The restarted version of the GMRES solver (with and without preconditioner) is applied to solve an asymmetric system of equations, arising due to the interpolating MLPG formulation. Its performance is compared with the biconjugate gradient stabilized (BiCGSTAB) solver on the basis of computation time and convergence behaviour. Jacobi and successive over-relaxation (SOR) methods are used as the preconditioners in both the solvers.

Findings

The results show that the GMRES solver outperforms the BiCGSTAB solver in terms of smoothness of convergence behaviour, while performs slightly better than the BiCGSTAB method in terms of Central processing Unit (CPU) time.

Originality/value

MLPG formulation leads to a non-symmetric system of algebraic equations. Iterative methods such as GMRES and BiCGSTAB methods are required for its solution for large-scale problems. This work presents the use of GMRES solver with the MLPG method for the very first time.

Details

Engineering Computations, vol. 39 no. 2
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 13 November 2007

Yiqiang Yu and Andy McCowen

The purpose of this paper is to investigate and analyze the efficiency and stability of the implementation of the Crout version of ILU (ILUC) preconditioning on fast‐multipole…

Abstract

Purpose

The purpose of this paper is to investigate and analyze the efficiency and stability of the implementation of the Crout version of ILU (ILUC) preconditioning on fast‐multipole method (FMM) for solving large‐scale dense complex linear systems arising from electromagnetic open perfect electrical conductor (PEC).

Design/methodology/approach

The FMM is employed to reduce the computational complexity of the matrix‐vector product and the memory requirement of the impedance matrix. The numerical examples are initially solved by the quasi‐minimal residual (QMR) method with ILUC preconditioning. In order to fully investigate the performance of ILUC in connection with other iterative solvers, a case is also solved by bi‐conjugate gradient solver and conjugate gradient squared solver with ILUC preconditioning.

Findings

The solutions show that the ILUC preconditioner is stable and significantly improves the performance of the QMR solver on large ill‐conditioned open PEC problems compared to using ILU(0) and threshold‐based ILU (ILUT) preconditioners. It dramatically decreases the number of iterations required for convergence and consequently reduces the total CPU solving time with a reasonable overhead in memory.

Practical implications

The preconditioning scheme can be applied to large ill‐conditioned open PEC problems to effectively speed up the overall electromagnetic simulation progress while maintaining the computational complexity of FMM. More complex structures including wire‐PEC junctions and microstrip arrays may be addressed in future work.

Originality/value

The performance of ILUC has been previously reported only on preconditioning sparse linear systems, in which the ILU preconditioner is constructed by the ILUC of the coefficient matrix (e.g. matrix arised from two‐dimensional finite element convection‐diffusion problem) and subsequently applied to the same sparse linear systems; so it is important to report its performance on the dense complex linear systems that arised from open PEC electromagnetic problems. In contrast, the preconditioner is constructed upon the near‐field matrix of the FMM and subsequently applied to the whole dense linear system. The comparison of its performance against the diagonal, ILU(0) and ILUT precoditioners is also presented.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol. 26 no. 5
Type: Research Article
ISSN: 0332-1649

Keywords

Article
Publication date: 1 February 2003

Orlando Soto, Rainald Löhner and Fernando Camelli

A parallel linelet preconditioner has been implemented to accelerate finite element (FE) solvers for incompressible flows when highly anisotropic meshes are used. The convergence…

Abstract

A parallel linelet preconditioner has been implemented to accelerate finite element (FE) solvers for incompressible flows when highly anisotropic meshes are used. The convergence of the standard preconditioned conjugate gradient (PCG) solver that is commonly used to solve the discrete pressure equations, greatly deteriorates due to the presence of highly distorted elements, which are of mandatory use for high Reynolds‐number flows. The linelet preconditioner notably accelerates the convergence rate of the PCG solver in such situations, saving an important amount of CPU time. Unlike other more sophisticated preconditioners, parallelization of the linelet preconditioner is almost straighforward. Numerical examples and some comparisons with other preconditioners are presented to demonstrate the performance of the proposed preconditioner.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 13 no. 1
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 11 July 2008

Cédric Doucet, Isabelle Charpentier, Jean‐Louis Coulomb, Christophe Guérin, Yann Le Floch and Gerard Meunier

The aim of this paper is to accelerate the convergence of iterative methods on ill‐conditioned linear systems of equations.

Abstract

Purpose

The aim of this paper is to accelerate the convergence of iterative methods on ill‐conditioned linear systems of equations.

Design/methodology/approach

First a brief numerical analysis is given of left preconditioners on ill‐conditioned linear systems of equations. From this result, it is deduced that a double preconditioning approach may be better. Then, a double preconditioner based on an iterative diagonal scaling method and an incomplete factorization method is proposed. The efficiency of this approach is illustrated on two finite element models produced by computational electromagnetism.

Findings

The double preconditioning approach is efficient for 2D and 3D finite element problems. The bi‐conjugate gradient algorithm always converges when it is double preconditioned. This is not the case when a simple incomplete factorization method is applied. Furthermore, when the two preconditioning techniques lead to the convergence of the iterative solving method, the double preconditioner significantly reduces the number of iterations in comparison with the simple preconditioner. On the proposed 2D problem, the speed‐up is between 6 and 32. On the proposed 3D problem, the speed‐up is between 13 and 20. Finally, the approach seems to reduce the growth of the condition number when higher‐order finite elements are used.

Research limitations/implications

The paper proposes a particular double preconditioning approach which can be applied to any invertible linear system of equations. A numerical evaluation on a singular linear system is also provided but no proof or analysis of stability is given for this case.

Originality/value

The paper presents a new preconditioning technique based on the combination of two very simple and elementary methods: a diagonal scaling method and an incomplete factorization process. Acceleration obtained from this approach is quite impressive.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol. 27 no. 4
Type: Research Article
ISSN: 0332-1649

Keywords

Article
Publication date: 1 February 1995

C.I. Goldstein

This paper is concerned with the numerical solution ofmulti‐dimensional convection dominated convection‐diffusionproblems. These problems are characterized by a large parameter…

Abstract

This paper is concerned with the numerical solution of multi‐dimensional convection dominated convection‐diffusion problems. These problems are characterized by a large parameter, K, multiplying the convection terms. The goal of this work is the development and analysis of effective preconditioners for iteratively solving the large system of linear equations arising from various finite element and finite difference discretizations with grid size h. When centered finite difference schemes and standard Galerkin finite element methods are used, h must be related to K by the stability constraint, Kh ≤ C0, where the constant C0 is sufficiently small. A class of preconditioners is developed that significantly reduces the condition number for large K and small h. Furthermore, these preconditioners are inexpensive to implement and well suited for parallel computation. It is shown that under suitable assumptions, the number of iterations remains bounded as h ↓0 with K fixed and, at worst, grows slowly as K ↓ ∞. Numerical results are presented illustrating the theory. It is also shown how to apply the theoretical results to more general convection‐diffusion problems and alternative discretizations (including streamline diffusion methods) that remain stable as Kh ↓ ∞.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 5 no. 2
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 21 March 2019

Zhenhan Yao, Xiaoping Zheng, Han Yuan and Jinlong Feng

Based on the error analysis, the authors proposed a new kind of high accuracy boundary element method (BEM) (HABEM), and for the large-scale problems, the fast algorithm, such as…

Abstract

Purpose

Based on the error analysis, the authors proposed a new kind of high accuracy boundary element method (BEM) (HABEM), and for the large-scale problems, the fast algorithm, such as adaptive cross approximation (ACA) with generalized minimal residual (GMRES) is introduced to develop the high performance BEM (HPBEM). It is found that for slender beams, the stress analysis using iterative solver GMRES will difficult to converge. For the analysis of slender beams and thin structures, to enhance the efficiency of GMRES solver becomes a key problem in the development of the HPBEM. The purpose of this paper is study on the preconditioning method to solve this convergence problem, and it is started from the 2D BE analysis of slender beams.

Design/methodology/approach

The conventional sparse approximate inverse (SAI) based on adjacent nodes is modified to that based on adjacent nodes along the boundary line. In addition, the authors proposed a dual node variable merging (DNVM) preprocessing for slender thin-plate beams. As benchmark problems, the pure bending of thin-plate beam and the local stress analysis (LSA) of real thin-plate cantilever beam are applied to verify the effect of these two preconditioning method.

Findings

For the LSA of real thin-plate cantilever beams, as GMRES (m) without preconditioning applied, it is difficult to converge provided the length to height ratio greater than 50. Even with the preconditioner SAI or DNVM, it is also difficult to obtain the converged results. For the slender real beams, the iteration of GMRES (m) with SAI or DNVM stopped at wrong deformation state, and the computation failed. By changing zero initial solution to the analytical displacement solution of conventional beam theory, GMRES (m) with SAI or DNVM will not be stopped at wrong deformation state, but the stress error is still difficult to converge. However, by GMRES (m) combined with both SAI and DNVM preconditioning, the computation efficiency enhanced significantly.

Originality/value

This paper presents two preconditioners: DNVM and a modified SAI based on adjacent nodes along the boundary line of slender thin-plate beam. In the LSA, by using GMRES (m) combined with both DNVM and SAI, the computation efficiency enhanced significantly. It provides a reference for the further development of the 3D HPBEM in the LSA of real beam, plate and shell structures.

Details

Engineering Computations, vol. 36 no. 8
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 1 July 2006

Oleg Iliev, Joachim Linn, Mathias Moog, Dariusz Niedziela and Vadimas Starikovicius

This study proposes to develop and investigate different iterative solvers for non‐Newtonian flow equations.

Abstract

Purpose

This study proposes to develop and investigate different iterative solvers for non‐Newtonian flow equations.

Design/methodology/approach

Existing approaches for the time discretization of the flow equation and for an iterative solution of the discrete systems are discussed. Ideas for further development of existing preconditioners are proposed, implemented and investigated numerically.

Findings

A two‐level preconditioning, consisting of a transformation of the original system in the first step and subsequent preconditioning of the transformed system is suggested. The GMRES iterative method, which usually performs well when applied to academic problems, showed dissatisfactory performance for the type of industrial flow simulations investigated in this work. It was found that the BiCGStab method performed best in the tests presented here.

Research limitations/implications

The iterative solvers considered here were investigated only for a certain class of polymer flows. More detailed studies for other non‐Newtonian flows should be carried out.

Originality/value

The work presented in this paper fills a gap related to the usage of efficient iterative methods for non‐Newtonian flow simulations.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 16 no. 5
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 30 September 2014

Zixiang Hu, Zhenmin Wang, Shi Zhang, Yun Zhang and Huamin Zhou

The purpose of this paper is to propose a combined reordering scheme with a wide range of application, called Reversed Cuthill-McKee-approximate minimum degree (RCM-AMD), to…

191

Abstract

Purpose

The purpose of this paper is to propose a combined reordering scheme with a wide range of application, called Reversed Cuthill-McKee-approximate minimum degree (RCM-AMD), to improve a preconditioned general minimal residual method for solving equations using Lagrange multiplier method, and facilitates the choice of the reordering for the iterative method.

Design/methodology/approach

To reordering the coefficient matrix before a preconditioned iterative method will greatly impact its convergence behavior, but the effect is very problem-dependent, even performs very differently when different preconditionings applied for an identical problem or the scale of the problem varies. The proposed reordering scheme is designed based on the features of two popular ordering schemes, RCM and AMD, and benefits from each of them.

Findings

Via numerical experiments for the cases of various scales and difficulties, the effects of RCM-AMD on the preconditioner and the convergence are investigated and the comparisons of RCM, AMD and RCM-AMD are presented. The results show that the proposed reordering scheme RCM-AMD is appropriate for large-scale and difficult problems and can be used more generally and conveniently. The reason of the reordering effects is further analyzed as well.

Originality/value

The proposed RCM-AMD reordering scheme preferable for solving equations using Lagrange multiplier method, especially considering that the large-scale and difficult problems are very common in practical application. This combined reordering scheme is more wide-ranging and facilitates the choice of the reordering for the iterative method, and the proposed iterative method has good performance for practical cases in in-house and commercial codes on PC.

Details

Engineering Computations, vol. 31 no. 7
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 30 September 2014

Zixiang Hu, Shi Zhang, Yun Zhang, Huamin Zhou and Dequn Li

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…

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.

Details

Engineering Computations, vol. 31 no. 7
Type: Research Article
ISSN: 0264-4401

Keywords

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