Developments in structural analysis by direct energy minimization.

Fox, Richard L.; Stanton, E. L.

doi:10.2514/3.4670

Cited by 74 publications

(12 citation statements)

References 7 publications

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“…References [24][25][26][27]) for solving symmetric positive definite linear systems. Coupled with element-by-element (EBE) iterative strategies that obviate the assembly of the global coefficient matrix [28,29], the PCG method should allow large 3-D problems to be solved on relatively modest computing platforms, e.g. PCs, owing to its lower memory demand (e.g.…”

Section: Pcg and Sj Preconditionermentioning

confidence: 99%

Fast iterative solution of large undrained soil‐structure interaction problems

Phoon

Chan

Toh

et al. 2003

Num Anal Meth Geomechanics

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SUMMARYIn view of rapid developments in iterative solvers, it is timely to re-examine the merits of using mixed formulation for incompressible problems. This paper presents extensive numerical studies to compare the accuracy of undrained solutions resulting from the standard displacement formulation with a penalty term and the two-field mixed formulation. The standard displacement and two-field mixed formulations are solved using both direct and iterative approaches to assess if it is cost-effective to achieve more accurate solutions. Numerical studies of a simple footing problem show that the mixed formulation is able to solve the incompressible problem 'exactly', does not create pressure and stress instabilities, and obviate the need for an ad hoc penalty number. In addition, for large-scale problems where it is not possible to perform direct solutions entirely within available random access memory, it turns out that the larger system of equations from mixed formulation also can be solved much more efficiently than the smaller system of equations arising from standard formulation by using the symmetric quasi-minimal residual (SQMR) method with the generalized Jacobi (GJ) preconditioner. Iterative solution by SQMR with GJ preconditioning also is more elegant, faster, and more accurate than the popular Uzawa method.

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Section: Pcg and Sj Preconditionermentioning

confidence: 99%

Fast iterative solution of large undrained soil‐structure interaction problems

Phoon

Chan

Toh

et al. 2003

Num Anal Meth Geomechanics

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“…It has been pointed out in [3] and [4] that it is not essential to assemble the stiffness matrix A from the sum of many simple matrices. The code which produces A from its usual definition (2) A=SeaeSt can be modified to produce, for any given column vector u,…”

mentioning

confidence: 99%

“…The motivation in [3] and [4] was to avoid cancellation of digits during the assembling of A. Ours is to reduce storage demands at the cost of increasing the arithmetic effort.…”

mentioning

confidence: 99%

Element Preconditioning Using Splitting Techniques

Nour-Omid¹,

Parlett²

1985

SIAM J. Sci. and Stat. Comput.

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The finite element method is a good way to turn elliptic boundary value problems into large symmetric systems of equations. These large matrices, A, are usually assembled from small ones. It is simple to omit the assembly process and use the code to accumulate the product Av for any v. Consequently the conjugate gradient algorithm (CG) can be used to solve Ax b without ever forming A.It is well known, however, that CG performs best when applied to preconditioned systems. In this paper we show one way to precondition without forming any large matrices.The trade-off between time and storage is examined for the I-D model problem and for the analysis of several realistic structures.

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“…One of the most conventional way to improve the condition number is a scaling transformation with the diagonal scaling matrix< 5 l. In recent years, Meijerink & Van der V orst<•l developed incomplete Choleski (IC) factorizations and combined them with the conjugate gradient method. This method is so called ICCG method.…”

Section: Introductionmentioning

confidence: 99%

Conjugate Gradient Like Methods and Their Application to Fixed Source Neutron Diffusion Problems

Suetomi

Sekimoto

1989

Journal of Nuclear Science and Technology

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This paper presents a number of fast iterative methods for solving systems of linear equations appearing in fixed source problems for neutron diffusion. We employed the conjugate gradient and conjugate residual methods. In order to accelerate the conjugate residual method, we proposed the conjugate residual squared method by transforming the residual polynomial of the conjugate residual method. Since the convergence of these methods depends on the spectrum of coefficient matrix, we employed the incomplete Choleski (IC) factorization and the modified IC (MIC) factorization as preconditioners. These methods were applied to some neutron diffusion problems and compared with the successive overrelaxation (SOR) method. The results of these numerical experiments showed superior convergence characteristics of the conjugate gradient like method with MIC factorization to the SOR method, especially for a problem involving void region. The CPU time of the MICCG, MICCR and MICCRS methods showed no great difference. In order to vectorize the conjugate gradient like methods based on (M) IC factorization, the hyperplane method was used and implemented on the vector computers, the HIT AC S-820/80 and ET AlO-E (one processor mode). Significant decrease of the CPU times was observed on the S-820/80. Since the scaled conjugate gradient (SCG) method can be vectorized with no manipulation, it was also compared with the above methods. It turned out the SCG method was the fastest with respect to the CPU times on the ET AlO-E. These results suggest that one should implement suitable algorithm for different vector computers.

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Developments in structural analysis by direct energy minimization.

Cited by 74 publications

References 7 publications

Fast iterative solution of large undrained soil‐structure interaction problems

Fast iterative solution of large undrained soil‐structure interaction problems

Element Preconditioning Using Splitting Techniques

Conjugate Gradient Like Methods and Their Application to Fixed Source Neutron Diffusion Problems

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