Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

GCRO with dynamic deflated restarting for solving adjoint systems of equations for aerodynamic shape optimization

GCRO with dynamic deflated restarting for solving adjoint systems of equations for aerodynamic... This paper aims to present a dynamically adjusted deflated restarting procedure for the generalized conjugate residual method with an inner orthogonalization (GCRO) method.Design/methodology/approachThe proposed method uses a GCR solver for the outer iteration and the generalized minimal residual (GMRES) with deflated restarting in the inner iteration. Approximate eigenpairs are evaluated at the end of each inner GMRES restart cycle. The approach determines the number of vectors to be deflated from the spectrum based on the number of negative Ritz values, k∗.FindingsThe authors show that the approach restores convergence to cases where GMRES with restart failed and compare the approach against standard GMRES with restarts and deflated restarting. Efficiency is demonstrated for a 2D NACA 0012 airfoil and a 3D common research model wing. In addition, numerical experiments confirm the scalability of the solver.Originality/valueThis paper proposes an extension of dynamic deflated restarting into the traditional GCRO method to improve convergence performance with a significant reduction in the memory usage. The novel deflation strategy involves selecting the number of deflated vectors per restart cycle based on the number of negative harmonic Ritz eigenpairs and defaulting to standard restarted GMRES within the inner loop if none, and restricts the deflated vectors to the smallest eigenvalues present in the modified Hessenberg matrix. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png International Journal of Numerical Methods for Heat & Fluid Flow Emerald Publishing

GCRO with dynamic deflated restarting for solving adjoint systems of equations for aerodynamic shape optimization

Download PDF
27 pages

Loading next page...
 
/lp/emerald-publishing/gcro-with-dynamic-deflated-restarting-for-solving-adjoint-systems-of-JzJruPSdbX
Publisher
Emerald Publishing
Copyright
© Emerald Publishing Limited
ISSN
0961-5539
DOI
10.1108/hff-10-2018-0586
Publisher site
See Article on Publisher Site

Abstract

This paper aims to present a dynamically adjusted deflated restarting procedure for the generalized conjugate residual method with an inner orthogonalization (GCRO) method.Design/methodology/approachThe proposed method uses a GCR solver for the outer iteration and the generalized minimal residual (GMRES) with deflated restarting in the inner iteration. Approximate eigenpairs are evaluated at the end of each inner GMRES restart cycle. The approach determines the number of vectors to be deflated from the spectrum based on the number of negative Ritz values, k∗.FindingsThe authors show that the approach restores convergence to cases where GMRES with restart failed and compare the approach against standard GMRES with restarts and deflated restarting. Efficiency is demonstrated for a 2D NACA 0012 airfoil and a 3D common research model wing. In addition, numerical experiments confirm the scalability of the solver.Originality/valueThis paper proposes an extension of dynamic deflated restarting into the traditional GCRO method to improve convergence performance with a significant reduction in the memory usage. The novel deflation strategy involves selecting the number of deflated vectors per restart cycle based on the number of negative harmonic Ritz eigenpairs and defaulting to standard restarted GMRES within the inner loop if none, and restricts the deflated vectors to the smallest eigenvalues present in the modified Hessenberg matrix.

Journal

International Journal of Numerical Methods for Heat & Fluid Flow – Emerald Publishing

Published: Aug 30, 2019

Keywords: GMRES; Adjoint method; Deflated restarting; GCRO-DR; Krylov solvers

References

  • Recycling Krylov subspaces for CFD applications and a new hybrid recycling solver
    Amritkar, A.; de Sturler, E.; Świrydowicz, K.; Tafti, D.; Ahuja, K.
  • Aerodynamic optimization for cruise and high-lift configurations
    Brezillon, J.; Dwight, R.P.; Widhalm, M.
  • 2D and 3D aerodynamic shape optimisation using the adjoint approach
    Brezillon, J.; Gauger, N.
  • MIRACLE – a joint DLR/ONERA effort on harmonization and development of industrial and research aerodynamic computational environment
    Cambier, L.; Kroll, N.
  • A dynamically deflated GMRES adjoint solver for aerodynamic shape optimization
    Chen, C.-H.; Nadarajah, S.; Castonguay, P.
  • Nested Krylov methods based on GCR
    de Sturler, E.
  • Truncation strategies for optimal Krylov subspace methods
    de Sturler, E.
  • Efficiency improvements of RANS-based analysis and optimization using implicit and adjoint methods on unstructured grids
    Dwight, R.
  • Restarted GMRES preconditioned by deflation
    Erhel, J.; Burrage, K.; Pohl, B.
  • Flexible GMRES with deflated restarting
    Giraud, L.; Gratton, S.; Pinel, X.; Vasseur, X.
  • An aerodynamic design optimization framework using a discrete adjoint approach with OpenFOAM
    He, P.; Mader, C.A.; Martins, J.R.; Maki, K.J.
  • Shape optimization towards stability in constrained hydrodynamic systems
    Heuveline, V.; Strauß, F.
  • A simplified and flexible variant of GCROT for solving nonsymmetric linear systems
    Hicken, J.E.; Zingg, D.W.
  • Induced-drag minimization of nonplanar geometries based on the euler equations
    Hicken, J.E.; Zingg, D.W.
  • Multi-element high-lift configuration design optimization using viscous continuous adjoint method
    Kim, S.; Alonso, J.J.; Jameson, A.
  • Flow simulation and shape optimization for aircraft design
    Kroll, N.; Gauger, N.R.; Brezillon, J.; Dwight, R.; Fazzolari, A.; Vollmer, D.; Becker, K.; Barnewitz, H.; Schulz, V.; Hazra, S.U.
  • Aerodynamic shape optimization investigations of the common research model wing benchmark
    Lyu, Z.; Kenway, G.K.; Martins, J.R.
  • A restarted GMRES method augmented with eigenvectors
    Morgan, R.B.
  • Implicitly restarted GMRES and Arnoldi methods for nonsymmetric systems of equations
    Morgan, R.B.
  • GMRES with deflated restarting
    Morgan, R.B.
  • Restarted block-GMRES with deflation of eigenvalues
    Morgan, R.B.
  • A harmonic restarted Arnoldi algorithm for calculating eigenvalues and determining multiplicity
    Morgan, R.B.; Zeng, M.
  • Optimum shape design for unsteady three-dimensional viscous flows using a nonlinear frequency-domain method
    Nadarajah, S.; Jameson, A.
  • Aerodynamic shape optimization for unsteady three-dimensional flows
    Nadarajah, S.K.; McMullen, M.S.; Jameson, A.
  • Approximate solutions and eigenvalue bounds from Krylov subspaces
    Paige, C.C.; Parlett, B.N.; Van der Vorst, H.A.
  • A continuous adjoint method with objective function derivatives based on boundary integrals, for inviscid and viscous flows
    Papadimitriou, D.I.; Giannakoglou, K.C.
  • Recycling Krylov subspaces for sequences of linear systems
    Parks, M.L.; de Sturler, E.; Mackey, G.; Johnson, D.D.; Maiti, S.
  • Simultaneous solution of large-scale linear systems and eigenvalue problems with a parallel GMRES method
    Pashos, G.; Kavousanakis, M.E.; Spyropoulos, A.N.; Palyvos, J.A.; Boudouvis, A.G.
  • Block krylov methods to solve adjoint problems in aerodynamic design optimization
    Pinel, X.; Montagnac, M.
  • A one-equation turbulence model for aerodynamic flows
    Spalart, P.; Allmaras, S.
  • Aerodynamic shape optimization of hovering rotor blades using a non-linear frequency domain approach
    Tatossian, C.A.; Nadarajah, S.K.; Castonguay, P.
  • The superlinear convergence behavior of GMRES
    Van der Vorst, H.A.; Vuik, C.
  • Constrained adjoint-based aerodynamic shape optimization of a single-stage transonic compressor
    Walther, B.; Nadarajah, S.
  • Robust and efficient adjoint solver for complex flow conditions
    Xu, S.; Timme, S.
  • Deflation strategies to improve the convergence of communication-avoiding GMRES
    Yamazaki, I.; Tomov, S.; Dongarra, J.
  • On the influence of optimization algorithm and initial design on wing aerodynamic shape optimization
    Yu, Y.; Lyu, Z.; Xu, Z.; Martins, J.R.