Comparative study of inner–outer Krylov solvers for linear systems in structured and high-order unstructured CFD problems

Jadoui, Mehdi; Blondeau, Christophe; Martin, E. Dale; Renac, Florent; Roux, François‐Xavier

doi:10.1016/j.compfluid.2022.105575

Cited by 7 publications

(6 citation statements)

References 29 publications

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“…However, due to the loss of precision, Krylov-subspace-based methods must often be accompanied by sophisticated subalgorithms implementing restart and orthogonality checking [43]. Variants include the generalized minimal residual algorithm (or Saad-Schultz [30]) and restarted versions [44][45][46][47].…”

Section: Eigenvaluementioning

confidence: 99%

Eigenproblem Basics and Algorithms

Jäntschi

2023

Symmetry

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Some might say that the eigenproblem is one of the examples people discovered by looking at the sky and wondering. Even though it was formulated to explain the movement of the planets, today it has become the ansatz of solving many linear and nonlinear problems. Formulation in the terms of the eigenproblem is one of the key tools to solve complex problems, especially in the area of molecular geometry. However, the basic concept is difficult without proper preparation. A review paper covering basic concepts and algorithms is very useful. This review covers the basics of the topic. Definitions are provided for defective, Hermitian, Hessenberg, modal, singular, spectral, symmetric, skew-symmetric, skew-Hermitian, triangular, and Wishart matrices. Then, concepts of characteristic polynomial, eigendecomposition, eigenpair, eigenproblem, eigenspace, eigenvalue, and eigenvector are subsequently introduced. Faddeev–LeVerrier, von Mises, Gauss–Jordan, Pohlhausen, Lanczos–Arnoldi, Rayleigh–Ritz, Jacobi–Davidson, and Gauss–Seidel fundamental algorithms are given, while others (Francis–Kublanovskaya, Gram–Schmidt, Householder, Givens, Broyden–Fletcher–Goldfarb–Shanno, Davidon–Fletcher–Powell, and Saad–Schultz) are merely discussed. The eigenproblem has thus found its use in many topics. The applications discussed include solving Bessel’s, Helmholtz’s, Laplace’s, Legendre’s, Poisson’s, and Schrödinger’s equations. The algorithm extracting the first principal component is also provided.

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Section: Eigenvaluementioning

confidence: 99%

Eigenproblem Basics and Algorithms

Jäntschi

2023

Symmetry

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“…The elsA adjoint and coupled-adjoint solvers have also been used for the computation of rigid and flexible derivatives. The latest improvements to the Krylov solvers for the solution of the adjoint linear system are described in [8]. A multi-block structured mesh featuring a C-H topology is used (Fig.…”

Section: Onera-m6 Wing Aeroelastic Analysismentioning

confidence: 99%

“…The number of relaxations is fixed at 6 and the CFL is varied. In [8] we point out that a proper tuning of the CFL coefficient is crucial for this type of preconditioners and how new block ILU type preconditioners accelerate the rate of convergence. Also, after each fluid-structure coupling, a cold restart is performed and the spectral information from the previous fluid-structure cycle is discarded.…”

Section: Introductionmentioning

confidence: 99%

“…For deflation by augmentation, adding eigenvectors to the Krylov subspace can effectively deflate corresponding eigenvalues from the spectrum because when these directions are included in the solution approximation, the convergence of the Krylov solver continues according to the modified spectrum. The deflation by augmentation led to the well-known FGMRES-DR solver [19] and its extension to inner-outer Krylov solver [20,8].…”

Section: Introductionmentioning

confidence: 99%

“…More specifically, we compare GMRES-DR and FGMRES-DR to GCRO-DR and FGCRO-DR with and without subspace recycling. This work benefits from the recent achievements to improve efficiency of the fluid adjoint solution by applying nested Krylov subspace methods [8]. The numerical experiments are performed on an aeroelastic configuration of the ONERA M6 fixed wing in transonic viscous flow.…”

Section: Introductionmentioning

confidence: 99%

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Recycling Krylov Subspaces for Efficient Partitioned Solution of Aerostructural Coupled Adjoint Systems

Jadoui

Blondeau

Roux

2022

AIAA AVIATION 2022 Forum

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Robust and efficient solvers for coupled-adjoint linear systems are crucial to successful aerostructural optimization. Monolithic and partitioned strategies can be applied. The monolithic approach is expected to offer better robustness and efficiency for strong fluid-structure interactions. However, it requires a high implementation cost and convergence may depend on appropriate scaling and initialization strategies. On the other hand, the modularity of the partitioned method enables a straightforward implementation while its convergence may require relaxation. In addition, a partitioned solver leads to a higher number of iterations to get the same level of convergence as the monolithic one. The objective of this paper is to accelerate the partitioned solver by considering techniques borrowed from Krylov subspace recycling strategies adapted to sequences of linear systems with varying right-hand sides. Indeed, in a partitioned framework, the structural source term attached to the fluid block of equations affects the right-hand side with the nice property of quickly converging to a constant value. We will also consider error approximation and approximate eigenvectors deflation in conjunction with advanced inner-outer Krylov solvers for the fluid block equations. We demonstrate the benefit of these techniques by computing the coupled derivatives for an aeroelastic configuration of the ONERA-M6 fixed wing in transonic flow. For this exercise the fluid grid was coupled to a structural model specifically designed to exhibit a high flexibility. All computations are performed using RANS flow modeling and a fully linearized one-equation Spalart-Allmaras turbulence model.

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