Continuous and discrete adjoint approaches for aerodynamic shape optimization with low Mach number preconditioning

Discrete and continuous adjoint approaches for use in aerodynamic shape optimization problems at all flow speeds are developed and assessed. They are based on the Navier–Stokes equations with low Mach number preconditioning. By alleviating the large disparity between acoustic waves and fluid speeds, the preconditioned flow and adjoint equations are numerically solved with affordable CPU cost, even at the so‐called incompressible flow conditions. Either by employing the adjoint to the preconditioned flow equations or by preconditioning the adjoint to the ‘standard’ flow equations (under certain conditions the two formulations become equivalent, as proved in this paper), efficient optimization methods with reasonable cost per optimization cycle, even at very low Mach numbers, are derived. During the mathematical development, a couple of assumptions are made which are proved to be harmless to the accuracy in the computed gradients and the effectiveness of the optimization method. The proposed approaches are validated in inviscid and viscous flows in external aerodynamics and turbomachinery flows at various Mach numbers. Copyright © 2007 John Wiley & Sons, Ltd.

[1]  O. Pironneau.  Optimal Shape Design for Elliptic Systems , 1983 .

[2]  Earl H. Dowell,et al.  A Discrete Adjoint Approach for Modeling Unsteady Aerodynamic Design Sensitivities , 2003 .

[3]  C. L. Merkle,et al.  The application of preconditioning in viscous flows , 1993 .

[4]  Jeffrey P. Thomas,et al.  Discrete Adjoint Approach for Modeling Unsteady Aerodynamic Design Sensitivities , 2005 .

[5]  Antony Jameson,et al.  NON-LINEAR FREQUENCY DOMAIN BASED OPTIMUM SHAPE DESIGN FOR UNSTEADY THREE-DIMENSIONAL FLOWS , 2006 .

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

[7]  Michael B. Giles,et al.  Discrete Adjoint Approximations with Shocks , 2003 .

[8]  Antony Jameson,et al.  Aerodynamic Shape Optimization of Complete Aircraft Configurations using Unstructured Grids , 2004 .

[9]  P. Spalart.  A One-Equation Turbulence Model for Aerodynamic Flows , 1992 .

[10]  A. Jameson.  Optimum aerodynamic design using CFD and control theory , 1995 .

[11]  Kyriakos C. Giannakoglou,et al.  A continuous adjoint method with objective function derivatives based on boundary integrals, for inviscid and viscous flows , 2007 .

[12]  Philip L. Roe,et al.  Characteristic time-stepping or local preconditioning of the Euler equations , 1991 .

[13]  L. Eriksson.  A preconditioned Navier-Stokes solver for low Mach number flows , 1996 .

[14]  S. Obayashi,et al.  Aerodynamic Optimization of Supersonic Transport Wing Using Unstructured Adjoint Method , 2001 .

[15]  E. Turkel,et al.  Preconditioned methods for solving the incompressible low speed compressible equations , 1987 .

[16]  K. Giannakoglou,et al.  COMPRESSOR BLADE OPTIMIZATION USING A CONTINUOUS ADJOINT FORMULATION , 2006 .

[17]  W. K. Anderson,et al.  Airfoil Design on Unstructured Grids for Turbulent Flows , 1999 .

[18]  M. D. Salas,et al.  Airfoil Design and Optimization by the One-Shot Method , 1995 .

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

[20]  Antony Jameson,et al.  Exploring the Limits of Shock-free Transonic Airfoil Design , 2005 .

[21]  Joel Brezillon,et al.  Aerodynamic shape optimization using simultaneous pseudo-timestepping , 2005 .

[22]  Rainald Löhner,et al.  On the Boundary Computation of Flow Sensitivities , 2004 .

[23]  Bernard Grossman,et al.  Fast convergence of inviscid fluid dynamic design problems , 2003 .

[24]  A. Jameson,et al.  A COMPARISON OF THE CONTINUOUS AND DISCRETE ADJOINT APPROACH TO AUTOMATIC AERODYNAMIC OPTIMIZATION , 2000 .

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

[26]  O. Pironneau,et al.  SHAPE OPTIMIZATION IN FLUID MECHANICS , 2004 .

[27]  Eric J. Nielsen,et al.  Sensitivity analysis for the Navier-Stokes equations on unstructured meshes using complex variables , 1999 .

[28]  A. Robin Forrest,et al.  Interactive interpolation and approximation by Bezier polynomials , 1972, Comput. J..

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

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