Effect of Small-Scale Output Unsteadiness on Adjoint-Based Sensitivity

This paper investigates the impact of small-scale unsteadiness on adjoint-based output sensitivity analysis. In particular, when iterative methods for nonlinear flows fail to converge to a steady state, it is demonstrated that the resulting sensitivity analysis can be highly inaccurate, even when the unsteadiness in the outputs is small. The specific example considered is the viscous subsonic flow around an airfoil over a range of angles of attack. If a strengthened solver is used to solve the adjoint equation (even though the flow equations did not fully converge), it is demonstrated that the sensitivity of the lift with respect to angle of attack can vary significantly, due to linearizing about different solution iterates of the steady flow solver. Further, the unsteady iterates from the time-inaccurate steady-state solver to the time-accurate solution are compared. The unsteadiness of the time-accurate solution is markedly different from the iterate solutions of the steady-state solver. If a strengthened solver is applied to the nonlinear flow equations, steady solutions can be achieved whose lift is significantly different from the time-averaged lift of the time-accurate simulations. Time-accurate unsteady adjoint analysis is then shown to provide accurate sensitivities for the time-averaged lift.

[1]  Michael B. Giles,et al.  Stabilization of linear flow solver for turbomachinery aeroelasticity using Recursive Projection method , 2004 .

[2]  J. Derber.  A Variational Continuous Assimilation Technique , 1989 .

[3]  D. Mavriplis.  Solution of the Unsteady Discrete Adjoint for Three-Dimensional Problems on Dynamically Deforming Unstructured Meshes , 2008 .

[4]  David L. Darmofal,et al.  An unsteady adaptation algorithm for discontinuous Galerkin discretizations of the RANS equations , 2007 .

[5]  Dimitri J. Mavriplis,et al.  Implicit Solution of the Unsteady Euler Equations for High-Order Accurate Discontinuous Galerkin Discretizations , 2006 .

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

[7]  Andreas Griewank,et al.  Algorithm 799: revolve: an implementation of checkpointing for the reverse or adjoint mode of computational differentiation , 2000, TOMS.

[8]  P. Courtier,et al.  Variational Assimilation of Meteorological Observations With the Adjoint Vorticity Equation. I: Theory , 2007 .

[9]  Ralf Hartmann,et al.  Error Estimation and Adaptive Mesh Refinement for Aerodynamic Flows , 2010 .

[10]  Niles A. Pierce,et al.  An Introduction to the Adjoint Approach to Design , 2000 .

[11]  Michael B. Giles,et al.  Adjoint Recovery of Superconvergent Functionals from PDE Approximations , 2000, SIAM Rev..

[12]  Rolf Rannacher,et al.  An optimal control approach to a posteriori error estimation in finite element methods , 2001, Acta Numerica.

[13]  Dimitri J. Mavriplis,et al.  An Unsteady Discrete Adjoint Formulation for Two-Dimensional Flow Problems with Deforming Meshes , 2007 .

[14]  Gautam M. Shroff,et al.  Stabilization of unstable procedures: the recursive projection method , 1993 .

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

[16]  Mats G. Larson,et al.  A posteriori eror estimation for higher order Godunov finite volume methods on unstructured meshes , 2002 .

[17]  Antony Jameson,et al.  OPTIMAL CONTROL OF UNSTEADY FLOWS USING A TIME ACCURATE METHOD , 2002 .

[18]  David L. Darmofal,et al.  Output Error Estimation and Adaptation in Computational Fluid Dynamics: Overview and Recent Results , 2009 .

[19]  M. Giles,et al.  Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality , 2002, Acta Numerica.

[20]  David W. Zingg,et al.  Unsteady Optimization Using a Discrete Adjoint Approach Applied to Aeroacoustic Shape Design , 2008 .

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

[22]  Arthur E. Bryson,et al.  Applied Optimal Control , 1969 .

[23]  Mathias Wintzer,et al.  Adjoint-Based Adaptive Mesh Refinement for Complex Geometries , 2008 .

[24]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[25]  Laslo T. Diosady,et al.  Preconditioning methods for discontinuous Galerkin solutions of the Navier-Stokes equations , 2009, J. Comput. Phys..

[26]  S. Rebay,et al.  GMRES Discontinuous Galerkin Solution of the Compressible Navier-Stokes Equations , 2000 .

[27]  M. C. Hall,et al.  Application of adjoint sensitivity theory to an atmospheric general circulation model , 1986 .

[28]  M. Giles.  On the iterative solution of adjoint equations , 2000 .

[29]  Endre Süli,et al.  Acta Numerica 2002: Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality , 2002 .