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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

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: 16 July 2019

Chih-Hao Chen and Siva Nadarajah

This paper aims to present a dynamically adjusted deflated restarting procedure for the generalized conjugate residual method with an inner orthogonalization (GCRO) method.

Abstract

Purpose

This paper aims to present a dynamically adjusted deflated restarting procedure for the generalized conjugate residual method with an inner orthogonalization (GCRO) method.

Design/methodology/approach

The proposed method uses a GCR solver for the outer iteration and the generalized minimal residual (GMRES) with deflated restarting in the inner iteration. Approximate eigenpairs are evaluated at the end of each inner GMRES restart cycle. The approach determines the number of vectors to be deflated from the spectrum based on the number of negative Ritz values, k∗.

Findings

The authors show that the approach restores convergence to cases where GMRES with restart failed and compare the approach against standard GMRES with restarts and deflated restarting. Efficiency is demonstrated for a 2D NACA 0012 airfoil and a 3D common research model wing. In addition, numerical experiments confirm the scalability of the solver.

Originality/value

This paper proposes an extension of dynamic deflated restarting into the traditional GCRO method to improve convergence performance with a significant reduction in the memory usage. The novel deflation strategy involves selecting the number of deflated vectors per restart cycle based on the number of negative harmonic Ritz eigenpairs and defaulting to standard restarted GMRES within the inner loop if none, and restricts the deflated vectors to the smallest eigenvalues present in the modified Hessenberg matrix.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 29 no. 7
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 March 1995

Gh. Juncu and I. Iliuta

The paper presents the numerical performance of the preconditionedgeneralized conjugate gradient (PGCG) methods in solvingnon‐linear convection — diffusion equations…

Abstract

The paper presents the numerical performance of the preconditioned generalized conjugate gradient (PGCG) methods in solving non‐linear convection — diffusion equations. Three non‐linear systems which describe a non‐isothermal chemical reactor, the chemically driven convection in a porous medium and the incompressible steady flow past a sphere are the test problems. The standard second order accurate centred finite difference scheme is used to discretize the models equations. The discrete approximations are solved with a double iterative process using the Newton method as outer iteration and the PGCG algorithm as inner iteration. Three PGCG techniques, which emerge to be the best performing, are tested. Laplace‐type operators are employed for preconditioning. The results show that the convergence of the PGCG methods depends strongly on the convection—diffusion ratio. The most robust algorithm is GMRES. But even with GMRES non‐convergence occurs when the convection—diffusion ratio exceeds a limit value. This value seems to be influenced by the non‐linearity type.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 5 no. 3
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: 1 June 2005

B. Auchmann, S. Kurz, O. Rain and S. Russenschuck

To introduce a Whitney‐element based coupling of the Finite Element Method (FEM) and the Boundary Element Method (BEM); to discuss the algebraic properties of the resulting system…

1409

Abstract

Purpose

To introduce a Whitney‐element based coupling of the Finite Element Method (FEM) and the Boundary Element Method (BEM); to discuss the algebraic properties of the resulting system and propose solver strategies.

Design/methodology/approach

The FEM is interpreted in the framework of the theory of discrete electromagnetism (DEM). The BEM formulation is given in a DEM‐compatible notation. This allows for a physical interpretation of the algebraic properties of the resulting BEM‐FEM system matrix. To these ends we give a concise introduction to the mathematical concepts of DEM.

Findings

Although the BEM‐FEM system matrix is not symmetric, its kernel is equivalent to the kernel of its transpose. This surprising finding allows for the use of two solution techniques: regularization or an adapted GMRES solver.

Research limitations/implications

The programming of the proposed techniques is a work in progress. The numerical results to support the presented theory are limited to a small number of test cases.

Practical implications

The paper will help to improve the understanding of the topological and geometrical implications in the algebraic structure of the BEM‐FEM coupling.

Originality/value

Several original concepts are presented: a new interpretation of the FEM boundary term leads to an intuitive understanding of the coupling of BEM and FEM. The adapted GMRES solver allows for an accurate solution of a singular, unsymetric system with a right‐hand side that is not in the image of the matrix. The issue of a grid‐transfer matrix is briefly mentioned.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol. 24 no. 2
Type: Research Article
ISSN: 0332-1649

Keywords

Article
Publication date: 1 February 1998

S. Alfonzetti, G. Borzì and N. Salerno

The Robin iteration procedure is a technique for the FEM computation of electromagnetic scattering fields in unbounded domains. It is based on the iterative improvement of the…

173

Abstract

The Robin iteration procedure is a technique for the FEM computation of electromagnetic scattering fields in unbounded domains. It is based on the iterative improvement of the known term of a non‐homogeneous Robin condition on a fictitious boundary enclosing the scatterer. In this paper it is shown that the procedure is equivalent to the application of the Richardson method to a reduced system and that the use of GMRES significantly reduces the computational effort.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol. 17 no. 1
Type: Research Article
ISSN: 0332-1649

Keywords

Article
Publication date: 1 March 2002

Johan Driesen and Kay Hameyer

A general framework for the application of the Newton methods in non‐linear coupled electromagnetic‐thermal problems solved with the FEM on independent subproblem meshes is…

Abstract

A general framework for the application of the Newton methods in non‐linear coupled electromagnetic‐thermal problems solved with the FEM on independent subproblem meshes is presented. The explicit derivation of the Jacobian matrix is outlined and discussed. A matrix‐free quasi‐Newton method, to be used along with linear system solvers built around Jacobian‐vector products is presented. This method does not require explicit derivatives and can be parallelised. The numerical aspects of these methods are discussed. The different Newton methods are demonstrated using a steady‐state conductive heating example problem.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol. 21 no. 1
Type: Research Article
ISSN: 0332-1649

Keywords

Article
Publication date: 1 May 2001

M.F. Carfora

A semi‐implicit semi‐Lagrangian mixed finite‐difference finite‐volume model for the shallow water equations on a rotating sphere is considered. The main features of the model are…

Abstract

A semi‐implicit semi‐Lagrangian mixed finite‐difference finite‐volume model for the shallow water equations on a rotating sphere is considered. The main features of the model are the finite‐volume approach for the continuity equation and the vectorial treatment of the momentum equation. Pressure and Coriolis terms in the momentum equation and velocity in the continuity equation are treated semi‐implicitly. Discretization of this model led to the introducion, in a previous paper, of a splitting technique which highly reduces the computational effort for the numerical solution. In this paper we solve the full set of equations, without splitting, introducing an ad hoc algorithm. A von Neumann stability analysis of this scheme is performed to establish the unconditional stability of the new proposed method. Finally, we compare the efficiency of the two approaches by numerical experiments on a standard test problem. Results show that, due to the devised algorithm, the solution of the full system of equations is much more accurate while slightly increasing the computational cost.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 11 no. 3
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 2 May 2017

Boštjan Mavrič and Božidar Šarler

In this study, the authors aim to upgrade their previous developments of the local radial basis function collocation method (LRBFCM) for heat transfer, fluid flow, electromagnetic…

Abstract

Purpose

In this study, the authors aim to upgrade their previous developments of the local radial basis function collocation method (LRBFCM) for heat transfer, fluid flow, electromagnetic problems and linear thermoelasticity to dynamic-coupled thermoelasticity problems.

Design/methodology/approach

The authors solve a thermoelastic benchmark by considering a linear thermoelastic plate under thermal and pressure shock. Spatial discretization is performed by a local collocation with multi-quadrics augmented by monomials. The implicit Euler formula is used to perform the time stepping. The system of equations obtained from the formula is solved using a Newton–Raphson algorithm with GMRES to iteratively obtain the solution. The LRBFCM solution is compared with the reference finite-element method (FEM) solution and, in one case, with a solution obtained using the meshless local Petrov–Galerkin method.

Findings

The performance of the LRBFCM is found to be comparable to the FEM, with some differences near the tip of the shock front. The LRBFCM appears to converge to the mesh-converged solution more smoothly than the FEM. Also, the LRBFCM seems to perform better than the MLPG in the studied case.

Research limitations/implications

The performance of the LRBFCM near the tip of the shock front appears to be suboptimal because it does not capture the shock front as well as the FEM. With the exception of a solution obtained using the meshless local Petrov–Galerkin method, there is no other high-quality reference solution for the considered problem in the literature yet. In most cases, therefore, the authors are able to compare only two mesh-converged solutions obtained by the authors using two different discretization methods. The shock-capturing capabilities of the method should be studied in more detail.

Originality/value

For the first time, the LRBFCM has been applied to problems of coupled thermoelasticity.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 27 no. 5
Type: Research Article
ISSN: 0961-5539

Keywords

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