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Investigates the efficiency of hybrid solution methods when incorporated into large‐scale topology and shape optimization problems and to demonstrate their influence on the overall performance of the optimization algorithms. Implements three innovative solution methods based on the preconditioned conjugate gradient (PCG) and Lanczos algorithms. The first method is a PCG algorithm with a preconditioner resulted from a complete or an incomplete Cholesky factorization, the second is a PCG algorithm in which a truncated Neumann series expansion is used as preconditioner, and the third is a preconditioned Lanczos algorithm properly modified to treat multiple right‐hand sides. The numerical tests presented demonstrate the computational advantages of the proposed methods which become more pronounced in large‐scale and/or computationally intensive optimization problems.

The advantages of a direct superposition of the Ritz vector in dynamic response analysis (developed by Wilson, Yuan, and Dickens in 1982 and termed the WYD method) are that: no iteration is involved; the method is at least four times faster than the subspace iteration method; and fewer Ritz vectors are necessary for the mode superposition of dynamic response analysis than exact eigenvectors are used. The major purpose of this paper is to illustrate that the WYD method can also be used as a general approximate algorithm to calculate eigenvalues and eigenvectors. The WYD and Lanczos algorithms are very similar and a formula that relates the two is given in this paper. Although the exact algebraic value of only a single eigenvector of a multi‐eigenvalue can be calculated using either the WYD or Lanczos methods, an artificial round‐off is presented that can be used to solve the eigenvalue problem. A method of estimating the error introduced by the WYD method is also developed. A dynamic substructuring technique, based on the WYD method, and which assumes that the connectivities on the interfaces among the substructures need not be considered is also presented.

Finite element discretizations of low‐frequency, time‐harmonic magnetic problems lead to sparse, complex symmetric systems of linear equations. The question arises which Krylov subspace methods are appropriate to solve such systems. The quasi minimal residual method combines a constant amount of work and storage per iteration step with a smooth convergence history. These advantages are obtained by building a quasi minimal residual approach on top of a Lanczos process to construct the search space. Solving the complex systems by transforming them to equivalent real ones of double dimension has to be avoided as such real systems have spectra that are less favourable for the convergence of Krylov‐based methods. Numerical experiments are performed on electromagnetic engineering problems to compare the quasi minimal residual method to the bi‐conjugate gradient method and the generalized minimal residual method.

The purpose of this paper is to develop a new finite difference method for solving the seismic wave propagation in fluid-solid media, which can be described by the acoustic and viscoelastic wave equations for the fluid and solid parts, respectively.

In this paper, the authors introduced a coordinate transformation method for seismic wave simulation method. In the new method, the irregular fluid–solid interface is transformed into a horizontal interface. Then, a multi-block coordinate transformation method is proposed to mesh every layer to curved grids and transforms every interface to horizontal interface. Meanwhile, a variable grid size is used in different regions according to the shape and the velocity within each region. Finally, a Lebedev-standard staggered coupled grid scheme for curved grids is applied in the multi-block coordinate transformation method to reduce the computational cost.

The instability in the auxiliary coordinate system caused by the standard staggered grid scheme is resolved using a curved grid viscoelastic wave field separation strategy. Several numerical examples are solved using this new method. It has been shown that the new method is stable, efficient and highly accurate in solving the seismic wave equation defined on domain with irregular fluid–solid interface.

First, the irregular fluid–solid interface is transformed into a horizontal interface by using the coordinate transformation method. The conversion between pressures and stresses is easy to implement and adaptive to different irregular fluid–solid interface models, because the normal stress and shear stress vanish when the normal angle is 90° in the interface. Moreover, in the new method, the strong false artificial boundary reflection and instability caused by ladder-shaped grid discretion are resolved as well.

Presents a review on implementing finite element methods on supercomputers, workstations and PCs and gives main trends in hardware and software developments. An appendix included at the end of the paper presents a bibliography on the subjects retrospectively to 1985 and approximately 1,100 references are listed.

The purpose of this paper is to find the best solver for parallelizing particle methods based on solving Pressure Poisson Equation (PPE) by taking Moving Particle Semi-Implicit (MPS) method as an example because the solution for PPE is usually the most time-consuming part difficult to parallelize.

To find the best solver, the authors compare six Krylov solvers, namely, Conjugate Gradient method (CG), Scaled Conjugate Gradient method (SCG), Bi-Conjugate Gradient Stabilized (BiCGStab) method, Conjugate Gradient Squared (CGS) method with Symmetric Lanczos Algorithm (SLA) method and Incomplete Cholesky Conjugate Gradient method (ICCG) in terms of convergence, time consumption, parallel efficiency and memory consumption for the semi-implicit particle method. The MPS method is parallelized by the hybrid Open Multi-Processing (OpenMP)/Message Passing Interface (MPI) model. The dam-break flow and channel flow simulations are used to evaluate the performance of different solvers.

It is found that CG converges stably, runs fastest in the serial way, uses the least memory and has highest OpenMP parallel efficiency, but its MPI parallel efficiency is lower than SLA because SLA requires less synchronization than CG.

With all these criteria considered and weighed, the recommended parallel solver for the MPS method is CG.

The application of the preconditioned Lanczos method is proposed for the solution of the linearized equations resulting from a non‐linear solution routine based on Newton methods. A path‐following solution algorithm with an arc length method is employed for tracing all types of post‐critical branches of a load‐displacement curve. The proposed methodology retains all characteristics of an iterative method by avoiding the complete factorization of the current stiffness matrix. The necessary eigenvalue information is retained in the tridiagonal matrix of the Lanczos approach.

The evaluation of linear dynamic response analysis of large structures by vector superposition requires, in its traditional formulation, the solution of a large and expensive eigenvalue problem. A method of solution based on a Ritz transformation to a reduced system of generalized coordinates using load dependent vectors generated from the spatial distribution of the dynamic loads is shown to maintain the high expected accuracy of modern computer analysis and significantly reduces the execution time over eigensolution procedures. New computational variants to generate load dependent vectors are presented and error norms are developed to control the convergence characteristics of load dependent Ritz solutions. Numerical applications on simple structural systems are used to show the relative efficiency of the proposed solution procedures.

The Lanczos method with the Golub and Kahan bidiagonalization algorithm is used to solve non‐symmetric diagonal dominant simultaneous equations. The method is very suitable for sparse matrix and vector computer. The problem with loss of orthogonal property is dealt with by restating the iteration. Numerical examples together with FORTRAN 77 routine are given to illustrate the algorithm. Both scalar and vector CPU times are given for comparison.

The computer implementation of a method for calculating coupled vibration eigenmodes and eigenfrequencies and the transient response of an elastic enclosure filled with a compressible fluid is described. The formalism is based on a simultaneous discretization of the fluid pressure and the fluid displacement potential. Attention is focused on the resulting form of the dynamic equilibrium equations and on their solution both in free vibration and transient analyses. In the eigenvalue analysis, an adapted Lanczos iteration scheme is proposed which takes account of the specific form of the equations. Applications are presented which demonstrate the effectiveness of the method.