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首页> 美国政府科技报告 >Iterative Solvers and Preconditioners for Fully-coupled Finite Element Formulations of Incompressible Fluid Mechanics and Related Transport Problems.
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Iterative Solvers and Preconditioners for Fully-coupled Finite Element Formulations of Incompressible Fluid Mechanics and Related Transport Problems.

机译：不可压缩流体力学及相关运输问题全耦合有限元形式的迭代求解器和预处理器。

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Finite element discretization of fully-coupled, incompressible flow problems with the classic mixed velocity-pressure interpolation pro duces matrix systems that render the best and most robust iterative solvers and preconditioners ineffective. The indefinite nature of the discretized continuity equation is the root cause and is one reason for the advancement of pressure penalty formulations, least-squares pres sure stabilization techniques, and pressure projection methods. These alternatives have served as admirable expedients and have enabled routine use of iterative matrix solution techniques; but all remain plagued by exceedingly slow convergence in the corresponding non- linear problem, lack of robustness, or limited range of accuracy. The purpose of this paper is to revisit matrix systems produced by this old mixed velocity-pressure formulation with two approaches: (1) deploy ing well-established tools consisting of matrix system reordering, GM- RES, and ILU preconditioning on modern architectures with substantial distributed or shared memory, and (2) tuning the preconditioner by managing the condition number using knowledge of the physical causes leading to the large condition number. Results obtained thus far using these simple techniques are very encouraging when measured against the reliability (not efficiency) of a direct matrix solver. Here we demonstrate routine solution for an incompressible flow problem using the Galerkin finite element method, Newton-Raphson iteration, and the robust and accurate LBB element. We also critique via an historical survey the limitations of pressure-stabilization strategies and all other commonly used alternatives to the mixed formulation for acceleration of iterative solver convergence. The performance of the new iterative solver approaches on other classes of problems, including fluid-structural interaction, multi-mode viscoelasticity, and free surface flow is also demonstrated. Iterative solvers, matrix preconditioners, ILU, pressure stabilization, finite element method

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