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1.
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. 相似文献
2.
A multi-lithology diffusive stratigraphic model is considered, which simulates at large scales in space and time the infill
of sedimentary basins governed by the interaction between tectonics displacements, eustatic variations, sediment supply, and
sediment transport laws. The model accounts for the mass conservation of each sediment lithology resulting in a mixed parabolic,
hyperbolic system of partial differential equations (PDEs) for the lithology concentrations and the sediment thickness. It
also takes into account a limit on the rock alteration velocity modeled as a unilaterality constraint. To obtain a robust,
fast, and accurate simulation, fully and semi-implicit finite volume discre tization schemes are derived for which the existence
of stable solutions is proved. Then, the set of nonlinear equations is solved using a Newton algorithm adapted to the unilaterality
constraint, and preconditioning strategies are defined for the solution of the linear system at each Newton iteration. They
are based on an algebraic approximate decoupling of the sediment thickness and the concentration variables as well as on a
proper preconditioning of each variable. These algorithms are studied and compared in terms of robustness, scalability, and
efficiency on two real basin test cases. 相似文献
3.
The finite element (FE) simulation of large‐scale soil–structure interaction problems (e.g. piled‐raft, tunnelling, and excavation) typically involves structural and geomaterials with significant differences in stiffness and permeability. The symmetric quasi‐minimal residual solver coupled with recently developed generalized Jacobi, modified symmetric successive over‐relaxation (MSSOR), or standard incomplete LU factorization (ILU) preconditioners can be ineffective for this class of problems. Inexact block diagonal preconditioners that are inexpensive approximations of the theoretical form are systematically evaluated for mitigating the coupled adverse effects because of such heterogeneous material properties (stiffness and permeability) and because of the percentage of the structural component in the system in piled‐raft foundations. Such mitigation led the proposed preconditioners to offer a significant saving in runtime (up to more than 10 times faster) in comparison with generalized Jacobi, modified symmetric successive over‐relaxation, and ILU preconditioners in simulating piled‐raft foundations. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献
4.
In this paper we present a study of the performance of sparse iterative solvers regarding the resolution of three-dimensional and non-linear problems encountered in soil/structure interaction. It is composed of two parts. In the first one, we present briefly iterative methods and preconditioners used in this study, then we analyse their performance on three soil/structure interaction problems: a shallow foundation under a vertical loading, a single pile subjected to a lateral loading and the construction of a lined tunnel in a soft soil. Tests are performed assuming an elastic–perfectly plastic constitutive law for the soil material with a non-associated Mohr–Coulomb flow rule. Copyright © 1999 John Wiley & Sons, Ltd. 相似文献
5.
本文研究了大型方程组的解算方法，给出了改进的共轭梯度算法和预条件共轭梯度算法，推导了相应的计算公式。对各种算法的计算效率进行了比较，结果表明，改进的共轭梯度法和预条件共轭梯度法能有效提高大型方程组的解算效率，其缺点是不能精确给出参数估值的验后协方差矩阵。 相似文献
6.
Finite element discretization of Biot's consolidation equations can produce a symmetric indefinite system (commonly used in geomechanics) or a non‐symmetric system. While this difference appears to be minor, however, it will require the selection of entirely different Krylov subspace solvers with potentially significant impact on solution efficiency. The former is solved using the symmetric quasi‐minimal residual whereas the latter is solved using the popular bi‐conjugate gradient stabilized. This paper presents an extensive comparison of the symmetric and non‐symmetric forms by varying the time step, size of the spatial domain, choice of physical units, and left versus left–right preconditioning. The generalized Jacobi (GJ) preconditioner is able to handle the non‐symmetric version of Biot's finite element method equation, although there are no practical incentives to do so. The convergence behaviour of GJ‐preconditioned systems and its relation to the spectral condition number or the complete spectrum are studied to clarify the concept of ill‐conditioning within the context of iteration solvers. Copyright © 2007 John Wiley & Sons, Ltd. 相似文献
7.
The problem of finite element simulation of incompressible fluid flow in porous medium is considered. The porous medium is characterized by the X‐ray microtomography technique in three dimensions. The finite calculus‐based stabilization technique is reviewed to implement the equal order finite element interpolation functions for both velocity and pressure. A noble preconditioner, the nodal block diagonal preconditioner, is considered whose performance is thoroughly investigated. Combining this preconditioner with a standard iterative solver during the computational homogenization procedure, it is possible to carry out the large‐scale fluid flow simulation for estimating permeability of the porous medium with reasonable accuracy and reliability. Copyright © 2013 John Wiley & Sons, Ltd. 相似文献
8.
The boundary-volume integral equation numerical technique can be a powerful tool for piecewise heterogeneous media, but it is limited to small problems or low frequencies because of great computational cost. Therefore, a restarted GMRES method is applied to solve large-scale boundary-volume scattering problems in this paper to overcome the computational barrier. The iterative method is firstly applied to responses of dimensionless frequency to a semicircular alluvial valley filled with sediments, compared with the standard Gaussian elimination method. Then the method is tested by a heterogeneous multilayered model to show its applicability. Numerical experiments indicate that the preconditioned GMRES method can significantly improve computational efficiency especially for large Earth models and high frequencies, but with a faster convergence for the left diagonal preconditioning. 相似文献
9.
Soil–structure interaction problems are commonly encountered in geotechnical practice and remarkably characterized with significant material stiffness contrast. When solving the soil–structure interaction problems, the employed Krylov subspace iterative method may converge slowly or even fail, indicating that the adopted preconditioning method may not suit for such problems. The inexact block diagonal preconditioners proposed recently have been shown effective for the soil–structure interaction problems; however, they haven't been exploited to full capabilities. By using the same partition strategy according to the structure elements and soil elements, the partitioned block symmetric successive over‐relaxation preconditioners or partitioned block constraint preconditioners are proposed. Based on two pile‐group foundation problems and a tunnel problem, the proposed preconditioners are evaluated and compared with the available preconditioners for the consolidation analysis and the drained analysis, respectively. In spite of one additional solve associated with the structure block and multiplications with off‐diagonal blocks in the preconditioning step, numerical results reveal that the proposed preconditioners obviously possess better performance than the recently developed inexact block preconditioners. Copyright © 2013 John Wiley & Sons, Ltd. 相似文献
10.