Discrete Adjoint-Based Approach for Optimization Problems on Three-Dimensional Unstructured Meshes

A comprehensive strategy for developing and implementing discrete adjoint methods for aerodynamic shape optimization problems is presented. By linearizing each procedure in the entire optimization problem, transposing each linearization, and reversing the sequential order of operations, the adjoint of the complete optimization problem, including flow equations and mesh motion equations is constructed in a modular and verifiable fashion. This construction is also shown to produce minimal memory overheads, and retain the same convergence characteristics of the original analysis problem in the sensitivity analysis. These techniques are implemented in a three-dimensional unstructured multigrid NavierStokes solver, and demonstrated on a transonic drag reduction problem for a wing body configuration.

[1]  C. Hirsch,et al.  Numerical Computation of Internal and External Flows. By C. HIRSCH. Wiley. Vol. 1, Fundamentals of Numerical Discretization. 1988. 515 pp. £60. Vol. 2, Computational Methods for Inviscid and Viscous Flows. 1990, 691 pp. £65. , 1991, Journal of Fluid Mechanics.

[2]  J. Batina Unsteady Euler airfoil solutions using unstructured dynamic meshes , 1989 .

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

[4]  Shahyar Pirzadeh,et al.  Three-dimensional unstructured viscous grids by the advancing-layers method , 1996 .

[5]  V. Venkatakrishnan,et al.  IMPLICIT METHOD FOR THE COMPUTATION OF UNSTEADY FLOWS ON UNSTRUCTURED GRIDS , 1995 .

[6]  J. Peraire,et al.  Practical Three-Dimensional Aerodynamic Design and Optimization Using Unstructured Meshes , 1997 .

[7]  Christian Bischof,et al.  The ADIFOR 2.0 system for the automatic differentiation of Fortran 77 programs , 1997 .

[8]  Dimitri J. Mavriplis,et al.  Directional Agglomeration Multigrid Techniques for High-Reynolds Number Viscous Flows , 1998 .

[9]  A. Jameson,et al.  Aerodynamic shape optimization techniques based on control theory , 1998 .

[10]  D. Mavriplis Multigrid Strategies for Viscous Flow Solvers on Anisotropic Unstructured Meshes , 1997 .

[11]  Dimitri J. Mavriplis,et al.  Directional agglomeration multigrid techniques for high Reynolds number viscous flow solvers , 1998 .

[12]  S. Pirzadeh,et al.  Large-Scale Parallel Unstructured Mesh Computations for Three-Dimensional High-Lift Analysis , 1999 .

[13]  Dimitri J. Mavriplis,et al.  AIAA 99 – 0537 LARGE-SCALE PARALLEL UNSTRUCTURED MESH COMPUTATIONS FOR 3 D HIGH-LIFT ANALYSIS , 1999 .

[14]  M. Giles,et al.  Adjoint Code Developments Using the Exact Discrete Approach , 2001 .

[15]  D. Mavriplis An assessment of linear versus non-linear multigrid methods for unstructured mesh solvers , 2001 .

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

[17]  Charbel Farhat,et al.  A three-dimensional torsional spring analogy method for unstructured dynamic meshes , 2002 .

[18]  Timothy J. Baker,et al.  Mesh Movement and Metamorphosis , 2002, Engineering with Computers.

[19]  M. Giles,et al.  Algorithm Developments for Discrete Adjoint Methods , 2003 .

[20]  Rainald Löhner,et al.  An Adjoint-Based Design Methodology for CFD Optimization Problems , 2003 .

[21]  Antony Jameson,et al.  Aerodynamic Shape Optimization Using the Adjoint Method , 2003 .

[22]  David L. Darmofal,et al.  An Exact Dual Adjoint Solution Method for Turbulent Flows on Unstructured Grids , 2003 .

[23]  Edward N. Tinoco,et al.  Data Summary from Second AIAA Computational Fluid Dynamics Drag Prediction Workshop , 2003 .

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

[25]  Edward N. Tinoco,et al.  Summary of Data from the Second AIAA CFD Drag Prediction Workshop (Invited) , 2004 .

[26]  Shengtai Li,et al.  Adjoint sensitivity analysis for time-dependent partial differential equations with adaptive mesh refinement , 2004 .

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

[28]  Zhi Yang,et al.  Unstructured Dynamic Meshes with Higher-order Time Integration Schemes for the Unsteady Navier-Stokes Equations , 2005 .

[29]  E. Nielsen,et al.  Using an Adjoint Approach to Eliminate Mesh Sensitivities in Computational Design , 2005 .

[30]  D. Mavriplis Multigrid Solution of the Discrete Adjoint for Optimization Problems on Unstructured Meshes , 2006 .

[31]  E. Nielsen,et al.  Efficient Construction of Discrete Adjoint Operators on Unstructured Grids Using Complex Variables , 2005 .

[32]  Luis Santos,et al.  Aerodynamic shape optimization using the adjoint method , 2007 .

[33]  Michel van Tooren,et al.  Development of the Discrete Adjoint for a Three-Dimensional Unstructured Euler Solver , 2008 .

[34]  Byung Joon Lee,et al.  Aerodynamic Redesign Using Discrete Adjoint Approach on Overset Mesh System , 2008 .

[35]  Dimitri J. Mavriplis,et al.  Unsteady Discrete Adjoint Formulation for Two-Dimensional Flow Problems with Deforming Meshes , 2008 .