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Incompressible Problem (incompressible + problem)

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Engineering100%

Selected Abstracts

Fast iterative solution of large undrained soil-structure interaction problems

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 3 2003
Kok-Kwang Phoon
Abstract In 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. Copyright © 2003 John Wiley & Sons, Ltd. [source]

Performance of Jacobi preconditioning in Krylov subspace solution of finite element equations

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 4 2002
F.-H. Lee
Abstract This paper examines the performance of the Jacobi preconditioner when used with two Krylov subspace iterative methods. The number of iterations needed for convergence was shown to be different for drained, undrained and consolidation problems, even for similar condition number. The differences were due to differences in the eigenvalue distribution, which cannot be completely described by the condition number alone. For drained problems involving large stiffness ratios between different material zones, ill-conditioning is caused by these large stiffness ratios. Since Jacobi preconditioning operates on degrees-of-freedom, it effectively homogenizes the different spatial sub-domains. The undrained problem, modelled as a nearly incompressible problem, is much more resistant to Jacobi preconditioning, because its ill-conditioning arises from the large stiffness ratios between volumetric and distortional deformational modes, many of which involve the similar spatial domains or sub-domains. The consolidation problem has two sets of degrees-of-freedom, namely displacement and pore pressure. Some of the eigenvalues are displacement dominated whereas others are excess pore pressure dominated. Jacobi preconditioning compresses the displacement-dominated eigenvalues in a similar manner as the drained problem, but pore-pressure-dominated eigenvalues are often over-scaled. Convergence can be accelerated if this over-scaling is recognized and corrected for. Copyright © 2002 John Wiley & Sons, Ltd. [source]

Fast iterative solution of large undrained soil-structure interaction problems

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 3 2003
Kok-Kwang Phoon
Abstract In 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. Copyright © 2003 John Wiley & Sons, Ltd. [source]

A promising boundary element formulation for three-dimensional viscous flow

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1 2005
Xiao-Wei Gao
Abstract In this paper, a new set of boundary-domain integral equations is derived from the continuity and momentum equations for three-dimensional viscous flows. The primary variables involved in these integral equations are velocity, traction, and pressure. The final system of equations entering the iteration procedure only involves velocities and tractions as unknowns. In the use of the continuity equation, a complex-variable technique is used to compute the divergence of velocity for internal points, while the traction-recovery method is adopted for boundary points. Although the derived equations are valid for steady, unsteady, compressible, and incompressible problems, the numerical implementation is only focused on steady incompressible flows. Two commonly cited numerical examples and one practical pipe flow problem are presented to validate the derived equations. Copyright © 2004 John Wiley & Sons, Ltd. [source]