Iterative method for constrained and mixed approximation. An inexpensive improvement of F.E.M. performance

Zienkiewicz, O. C.; Vilotte, J. P.; Toyoshima, Shiro; Nakazawa, S.

doi:10.1016/0045-7825(85)90025-8

Cited by 64 publications

(23 citation statements)

References 18 publications

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“…Although the examples shown are restricted to undrained problems in the context of geomechanics, its general applicability to other examples such as incompressible flow and metal forming problems studied by Zienkiewicz et al [9] is clear.…”

Section: Resultsmentioning

confidence: 96%

“…The Uzawa method [39] is a popular iterative method for solving the 2 Â 2 block system by recursively solving two decoupled systems [4,9,25]. The standard implementation of the Uzawa's method for solving a system of the form shown by Equation (13) …”

Section: Uzawa Methodsmentioning

confidence: 99%

“…Zienkiewicz et al [9] demonstrated that the advantages of mixed formulation can be retained at a cost comparable to that of the irreducible form by using a global iterative procedure. The proposed Uzawa procedure involves iterating two decoupled systems of equations, one of which is complex and requires another iterative procedure for solution.…”

Section: Introductionmentioning

confidence: 99%

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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: Resultsmentioning

confidence: 96%

Section: Uzawa Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Fast iterative solution of large undrained soil‐structure interaction problems

Phoon

Chan

Toh

et al. 2003

Num Anal Meth Geomechanics

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“…Hence, Assumptions A2 and A3 are commonly encountered restrictions in practice. We note in passing that a wide class of constrained problems involving mixed finite element formulations will produce the block equation structure A 0 [1]. Hence, the theoretical and numerical results presented below have potential applications beyond geomechanics.…”

Section: Introductionmentioning

confidence: 97%

Block preconditioners for symmetric indefinite linear systems

Toh

Phoon

Chan

2004

Numerical Meth Engineering

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SUMMARYThis paper presents a systematic theoretical and numerical evaluation of three common block preconditioners in a Krylov subspace method for solving symmetric indefinite linear systems. The focus is on large-scale real world problems where block approximations are a practical necessity. The main illustration is the performance of the block diagonal, constrained, and lower triangular preconditioners over a range of block approximations for the symmetric indefinite system arising from large-scale finite element discretization of Biot's consolidation equations. This system of equations is of fundamental importance to geomechanics. Numerical studies show that simple diagonal approximations to the (1,1) block K and inexpensive approximations to the Schur complement matrix S may not always produce the most spectacular time savings when K is explicitly available, but is able to deliver reasonably good results on a consistent basis. In addition, the block diagonal preconditioner with a negative (2,2) block appears to be reasonably competitive when compared to the more complicated ones. These observation are expected to remain valid for coefficient matrices whereby the (1,1) block is sparse, diagonally significant (a notion weaker than diagonal dominance), moderately well-conditioned, and has a much larger block size than the (2,2) block.

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“…There are many ways to solve the Stokes problem, such as Uzawa-like [4±10, 16,25], SIM-PLE-like [14] and multigrid-like methods [5,23]. In this work we only consider the Uzawa-like methods.…”

Section: Introductionmentioning

confidence: 98%

An new improved Uzawa method for finite element solution of Stokes problem

Liu

2001

Computational Mechanics

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Tchebychev iteration may be used for acceleration convergence of an iterative algorithm to solve a general linear system equation. Associating it with the Uzawa method, we suggest a new iterative solution method for the Stokes problems. The new algorithm retains the simplicity and robustness of the Uzawa method. So it requires almost no additional cost of computation, in terms of storage or CPU time, yet it provides the property of speed up convergence. Numerical tests showed that the algorithm of this type have much faster convergence rates than both the original Uzawa iterative algorithm and the augmented Lagrangian method.

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Iterative method for constrained and mixed approximation. An inexpensive improvement of F.E.M. performance

Cited by 64 publications

References 18 publications

Fast iterative solution of large undrained soil‐structure interaction problems

Fast iterative solution of large undrained soil‐structure interaction problems

Block preconditioners for symmetric indefinite linear systems

An new improved Uzawa method for finite element solution of Stokes problem

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