Stabilization and scalable block preconditioning for the Navier–Stokes equations

Cyr, Eric C; Shadid, John N.; Tuminaro, Raymond S.

doi:10.1016/j.jcp.2011.09.001

Cited by 35 publications

(47 citation statements)

References 26 publications

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“…and O U being the proper approximations to the matrices L, D, and U in the LDU block factorization of the coupled system matrix. Such a block factorization technique has been often utilized to construct (multilevel) preconditioners for the second-order elliptic problems (e.g., [20][21][22][23][24]) and the saddle point problems (e.g., [25][26][27][28][29][30]). In fact, the original Semi-Implicit Method for Pressure-Linked Equations method [31] and its variants for solving the fluid problem can be t s and t f \ t s D ;.…”

Section: Introductionmentioning

confidence: 99%

Robust and efficient monolithic fluid‐structure‐interaction solvers

Langer

Yang

2016

Numerical Meth Engineering

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Abstract. In this note, we consider preconditioned Krylov subspace methods for discrete fluid-structure interaction problems with a nonlinear hyperelastic material model and covering a large range of flows, e.g, water, blood, and air with highly varying density. Based on the complete LDU factorization of the coupled system matrix, the preconditioner is constructed in form ofLDÛ , whereL, D andÛ are proper approximations to L, D and U , respectively. The inverse of the corresponding Schur complement is approximated by applying one cycle of a special class of algebraic multigrid methods to the perturbed fluid sub-problem, that is obtained by modifying corresponding entries in the original fluid matrix with an explicitly constructed approximation of the exact perturbation coming from the sparse matrix-matrix multiplications. MotivationDuring the past years, robust and efficient monolithic fluid-structure interaction (FSI) solvers attract a lot of interests from many researchers; see, e.g., [13,29,18,16,4,9,6], that are mainly based on algebraic multigrid (AMG) [30] , geometry multigrid (GMG) [15], preconditioned Krylov subspace [32] and domain decomposition (DD) [28,35] methods. In our previous work [23], we implemented FSI monolithic AMG solvers with the W-cycle and with a variant of the W-cycle, i.e., a recursive Krylov-based multigrid cycle, somehow related to the algebraic multilevel method, see, e.g., [2,3,20,22,37,27,1] , as well as their corresponding preconditioners for the coupled FSI problem using the AMG preconditioners [21,14,38,24] for each sub-problem in the smoothing steps. In addition, we also considered the preconditioned GMRES method (see [33]), using a class of block-wise Gauss-Seidel type preconditioners (see the earlier work in [13]), that are based on the aforementioned AMG methods for the sub-problems. As well known, such block-wise Gauss-Seidel preconditioned Krylov subspace methods may lose the robustness with respect to the mesh size, i.e., the iteration numbers for solving the coupled FSI system nearly double, 1 arXiv:1412.6845v1 [math.NA]

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Section: Introductionmentioning

confidence: 99%

Robust and efficient monolithic fluid‐structure‐interaction solvers

Langer

Yang

2016

Numerical Meth Engineering

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“…As mentioned, preconditioners P t and P SIM P LE are expected to work well for a diagonally dominant Q. For the time-dependent incompressible Navier-Stokes equations on the transient simulations, it is reported in [9] that the performance of P SIM P LE in the context of stabilized FEM degrades with increasing the time-step sizes. This occurs because P SIM P LE assumes that the matrix Q can be well approximated by its diagonal.…”

Section: Block-triangular Preconditioner With Scaled Laplacian As Thementioning

confidence: 99%

Block-preconditioners for the incompressible Navier–Stokes equations discretized by a finite volume method

Vuik

Klaij

2017

Journal of Numerical Mathematics

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The modified augmented Lagrangian preconditioner has attracted much attention in solving nondimensional Navier–Stokes equations discretized by the finite element method. In industrial applications the governing equations are often in dimensional form and discretized using the finite volume method. This paper assesses the capability of this preconditioner for dimensional Navier–Stokes equations in the context of the finite volume method. Two main contributions are made. First, this paper introduces a new dimensionless parameter that is involved in the modified augmented Lagrangian preconditioner. Second, with a number of academic test problems this paper reveals that the convergence of both nonlinear and linear iterations depend on this dimensionless parameter. A way to choose the optimal value of the dimensionless parameter is suggested and it is found that the optimal value is dependent of the Reynolds number, instead of the fluid’s properties, e.g., density and dynamic viscosity. The outcomes of this paper yield a potential rule to choose the optimal dimensionless parameter in practice, namely, correspondingly increasing with enlarging the Reynolds number.

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“…Fig. 8 presents performance results for the Charon code [20] developed at Sandia National Laboratory being applied to solve the incompressible Navier-Stokes fluid equations. The specific application being analyzed is a Kelvin-Helmholtz shear layer.…”

Section: High Performance Computing Applicationsmentioning

confidence: 99%

“…8 shows for various Krylov preconditioners (see Ref. [20]) performance as measured regarding the number of Krylov iterations and Wall Clock Time per Newton step, with computational results for weak scaling on the Sandia Red Sky computer. A derivative of Charon, Drekar, has been used to obtain the turbulent pressure forces used in GTRF analysis.…”

Section: High Performance Computing Applicationsmentioning

confidence: 99%

Advances in Multi-Physics and High Performance Computing in Support of Nuclear Reactor Power Systems Modeling and Simulation

Turinsky¹

2012

Nuclear Engineering and Technology

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Stabilization and scalable block preconditioning for the Navier–Stokes equations

Cited by 35 publications

References 26 publications

Robust and efficient monolithic fluid‐structure‐interaction solvers

Robust and efficient monolithic fluid‐structure‐interaction solvers

Block-preconditioners for the incompressible Navier–Stokes equations discretized by a finite volume method

Advances in Multi-Physics and High Performance Computing in Support of Nuclear Reactor Power Systems Modeling and Simulation

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