Block Krylov Methods to Solve Adjoint Problems in Aerodynamic Design Optimization

A new resolution method for solving linear systems resulting from the discrete adjoint approach in aerodynamic shape optimization with a single objective and multiple constraint functions is presented. The steady flow is governed by the three-dimensional compressible Favre-averaged Navier–Stokes equations combined with a two-equation turbulence model. Block flexible variants of the generalized minimal residual method are Krylov methods designed for the solution of linear systems with multiple right-hand sides allowing variable preconditioning. Then, they can be applied to the discrete adjoint equations of Favre-averaged Navier–Stokes equations for all objective and constraint functions as a right-hand side. The paper focuses on the development of a new algorithm combining the three following features: flexibility, multiple right-hand-side concurrent processing, and spectral deflation. This latter property enables to recycle approximate spectral information from one restart to the next one in order to avoi...

[1]  D K Smith,et al.  Numerical Optimization , 2001, J. Oper. Res. Soc..

[2]  Elizabeth R. Jessup,et al.  On Improving Linear Solver Performance: A Block Variant of GMRES , 2005, SIAM J. Sci. Comput..

[3]  Jason E. Hicken,et al.  Aerodynamic Optimization Algorithm with Integrated Geometry Parameterization and Mesh Movement , 2010 .

[4]  Katya Scheinberg,et al.  Introduction to derivative-free optimization , 2010, Math. Comput..

[5]  J. Meijerink,et al.  An iterative solution method for linear systems of which the coefficient matrix is a symmetric -matrix , 1977 .

[6]  G. D. van Albada,et al.  A comparative study of computational methods in cosmic gas dynamics , 1982 .

[7]  John H. Holland,et al.  Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence , 1992 .

[8]  J. N. Lyness,et al.  Numerical Differentiation of Analytic Functions , 1967 .

[9]  Valeria Simoncini,et al.  Flexible Inner-Outer Krylov Subspace Methods , 2002, SIAM J. Numer. Anal..

[10]  D. O’Leary,et al.  Efficient iterative algorithms for the stochastic finite element method with application to acoustic scattering , 2005 .

[11]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[12]  Hans-Paul Schwefel,et al.  Evolution strategies – A comprehensive introduction , 2002, Natural Computing.

[13]  J. Eric,et al.  Aerodynamic Design Optimization on Unstructured Meshes Using the Navier-Stokes Equations , 1998 .

[14]  W. K. Anderson,et al.  Recent improvements in aerodynamic design optimization on unstructured meshes , 2001 .

[15]  R. Freund,et al.  A block QMR algorithm for non-Hermitian linear systems with multiple right-hand sides , 1997 .

[16]  Ronald B. Morgan,et al.  GMRES with Deflated Restarting , 2002, SIAM J. Sci. Comput..

[17]  Jason E. Hicken,et al.  A Simplified and Flexible Variant of GCROT for Solving Nonsymmetric Linear Systems , 2010, SIAM J. Sci. Comput..

[18]  H. Sadok,et al.  A block version of BiCGSTAB for linear systems with multiple right-hand sides. , 2003 .

[19]  G. R. Shubin,et al.  A comparison of optimization-based approaches for a model computational aerodynamics design problem , 1992 .

[20]  Jean-François Boussuge,et al.  Development of a new hybrid compressible solver inside the CFD elsA software , 2011 .

[21]  Antony Jameson,et al.  Steady-State Solution of the Euler Equations for Transonic Flow , 1982 .

[22]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[23]  A. Jameson,et al.  Optimum Aerodynamic Design Using the Navier–Stokes Equations , 1997 .

[24]  Luc Giraud,et al.  Flexible GMRES with Deflated Restarting , 2010, SIAM J. Sci. Comput..

[25]  D. Keyes,et al.  Jacobian-free Newton-Krylov methods: a survey of approaches and applications , 2004 .

[26]  T. Barth,et al.  An unstructured mesh Newton solver for compressible fluid flow and its parallel implementation , 1995 .

[27]  E. Sturler,et al.  Truncation Strategies for Optimal Krylov Subspace Methods , 1999 .

[28]  D. Wilcox.  Reassessment of the scale-determining equation for advanced turbulence models , 1988 .

[29]  W. K. Anderson,et al.  Sensitivity Analysis for Navier-Stokes Equations on Unstructured Meshes Using Complex Variables , 2001 .

[30]  O. Pironneau.  On optimum design in fluid mechanics , 1974 .

[31]  Axel Ruhe.  Implementation aspects of band Lanczos algorithms for computation of eigenvalues of large sparse sym , 1979 .

[32]  D. Darmofal,et al.  An implicit, exact dual adjoint solution method for turbulent flows on unstructured grids , 2004 .

[33]  Joaquim R. R. A. Martins,et al.  The complex-step derivative approximation , 2003, TOMS.

[34]  D. Zingg,et al.  Newton-Krylov Algorithm for Aerodynamic Design Using the Navier-Stokes Equations , 2002 .

[35]  Antony Jameson,et al.  Aerodynamic design via control theory , 1988, J. Sci. Comput..

[36]  R. Morgan.  Restarted block-GMRES with deflation of eigenvalues , 2005 .

[37]  Oktay Baysal,et al.  Three-dimensional aerodynamic shape optimization using discrete sensitivity analysis , 1996 .

[38]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .

[39]  C. L. Hilmes,et al.  Nonlinear Elimination in Aerodynamic Analysis and Design Optimization , 2003 .

[40]  Dirk Roose,et al.  Ritz and harmonic Ritz values and the convergence of FOM and GMRES , 1999 .

[41]  M. Sadkane,et al.  Exact and inexact breakdowns in the block GMRES method , 2006 .

[42]  M. Gutknecht,et al.  The block grade of a block Krylov space , 2009 .

[43]  O. Baysal,et al.  Aerodynamic Sensitivity Analysis Methods for the Compressible Euler Equations , 1991 .

[44]  A. Jameson,et al.  Lower-upper Symmetric-Gauss-Seidel method for the Euler and Navier-Stokes equations , 1988 .

[45]  Oktay Baysal,et al.  Aerodynamic design optimization using sensitivity analysis and computational fluid dynamics , 1991 .

[46]  John H. Holland,et al.  Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence , 1992 .

[47]  Yousef Saad,et al.  A Flexible Inner-Outer Preconditioned GMRES Algorithm , 1993, SIAM J. Sci. Comput..

[48]  R. M. Hicks,et al.  Wing Design by Numerical Optimization , 1977 .

[49]  Henri Calandra,et al.  Flexible Variants of Block Restarted GMRES Methods with Application to Geophysics , 2012, SIAM J. Sci. Comput..

[50]  J. N. Lyness.  Numerical algorithms based on the theory of complex variable , 1967, ACM National Conference.