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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 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.

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.

The aim of this paper is to present two independent ways in which a simple approximation to a Green's function for a differential equation can be used to improve the performance of well‐known iterative methods for linear equations.

The indefinite nature of the mixed finite element formulation of the Navier‐Stokes equations is treated by segregation of the variables. The segregation algorithm assembles the coefficients which correspond to the velocity variables in the upper part of the equation matrix and the coefficients which corresponds to the pressure variables in the lower part of the equation matrix. During the incomplete; elimination of the velocity matrix, fill‐in will occur in the pressure matrix, hence, divisions with zero are avoided. The fill‐in rule applied here is related to the location of the node in the finite element mesh, rather than the magnitude of the fill‐in or the magnitude of the coefficient at the location of the fill‐in. Different orders of fill‐in are explored for ILU preconditioning of the mixed finite element formulation of the Navier‐Stokes equations in two dimensions.

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 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.

The paper aims to focus on: implementation of the fast‐multipole method (FMM) to open perfect electric conductors (PEC) problems involving triangular type wire‐to‐surface junctions; investigation and analysis of the effect of wire‐to‐surface junction configuration on the conditioning of the linear systems; application of the preconditioning technique to improve the efficiency of the FMM scheme on such problems.

A complete set of formulations is proposed to evaluate the far‐field terms of the impedance matrix that represent the couplings between the wire‐to‐surface junction and standard wire and PEC surfaces. The formulations are derived in a convenient form suitable for the application of the FMM. An iterative scheme is adopted to estimate the condition number of the linear systems arising from open‐PEC problems with wire‐to‐surface junctions and to investigate the effect of wire‐to‐surface junction configuration on the conditioning of the linear systems. The Crout version of ILU (ILUC) preconditioning strategy is applied to improve the convergence rate of the iterative solver on such problems.

The solutions show that the proposed formulations have accurately evaluated the far‐field terms that represent the couplings between the wire‐to‐surface junction and standard wire and PEC surfaces. The investigation of the conditioning of open‐PEC problems with junctions shows that the effect of the wire‐to‐surface junction configuration induced to the conditioning of the linear systems is negligible. The convergence records of several open‐PEC problems involving wire‐to‐surface junctions show that the ILUC preconditioning strategy is suitable to apply to such problems, as it significantly improves the performance of the iterative solver.

The proposed FMM strategy can be applied to many practical large‐scale open‐PEC problems that involve wire‐to‐surface junctions, such as antenna arrays and electromagnetic compatibility problems, to effectively speed up the overall electromagnetic simulation progress and overcome the bottleneck associated with the dense impedance matrix of the method‐of‐moments.

The application of the FMM to open‐PEC problems that involve wire‐to‐surface junctions has yet to be reported, which has been addressed in this work. This work also investigates the conditioning of such problems and analyzes the effect of wire‐to‐surface junction configuration on the conditioning of the linear systems. In addition, the performance of the ILUC preconditioner on such problems has not been reported, which has also been included in this report.

This paper develops C++ and Fortran-90 solvers to establish parallel solution procedures in a finite element or meshless analysis program using shared memory computers. The paper aims to discuss these issues.

The stiffness matrix can be symmetrical or unsymmetrical, and the solution schemes include sky-line Cholesky and parallel preconditioned conjugate gradient-like methods.

By using the features of C++ or Fortran-90, the stiffness matrix and its auxiliary arrays can be encapsulated into a class or module as private arrays. This class or module will handle how to allocate, renumber, assemble, parallelize and solve these complicated arrays automatically.

The source codes can be obtained online at http//myweb.ncku.edu.tw/∼juju. The major advantage of the scheme is that it is simple and systematic, so an efficient parallel finite element or meshless program can be established easily.

With the minimum requirement of computer memory, an object-oriented C++ class and a Fortran-90 module were established to allocate, renumber, assemble, parallel, and solve the global stiffness matrix, so that the programmer does not need to handle them directly.

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

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∗.

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.

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.