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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.

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.

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.

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.

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.

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.

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.

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.

In this paper we investigate the performance of CGS, BCGSTAB and GMRES with ILU preconditioner for solving convection‐diffusion problems. Numerical experiments indicate that BCGSTAB appears to be an efficient and stable method. CGS sometimes suffers from severe numerical instability. GMRES shows a higher suitability and stability but the overall convergence rate may be lower.

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).

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.

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.

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.

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.

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.