Adjoint Formulation for an Embedded-Boundary Cartesian Method

A discrete-adjoint formulation is presented for the three-dimensional Euler equations discretized on a Cartesian mesh with embedded boundaries. The solution algorithm for the adjoint and flow-sensitivity equations leverages the Runge‐Kutta time-marching scheme in conjunction with the parallel multigrid method of the flow solver. The matrix-vector products associated with the linearization of the flow equations are computed on-the-fly, thereby minimizing the memory requirements of the algorithm at a computational cost roughly equivalent to a flow solution. Three-dimensional test cases, including a wing-body geometry at transonic flow conditions and an entry vehicle at supersonic flow conditions, are presented. These cases verify the accuracy of the linearization and demonstrate the eciency and robustness of the adjoint algorithm for complex-geometry problems.

[1]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

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

[3]  D. P. Young,et al.  A locally refined rectangular grid finite element method: application to computational fluid dynamics and computational physics , 1990 .

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

[5]  Michael B. Bieterman,et al.  Practical Design and Optimization in Computational Fluid Dynamics , 1993 .

[6]  P. A. Newman,et al.  An Approximately Factored Incremental Strategy for Calculating Consistent Discrete Aerodynamic Sensitivity Derivatives , 1992 .

[7]  David J. Thuente,et al.  Line search algorithms with guaranteed sufficient decrease , 1994, TOMS.

[8]  David E. Keyes,et al.  Application of Newton-Krylov methodology to a three-dimensional unstructured Euler code , 1995 .

[9]  Timothy J. Barth,et al.  Parallel CFD Algorithms for Aerodynamical Flow Solvers on Unstructured Meshes. Parts 1 and 2 , 1995 .

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

[11]  W. K. Anderson,et al.  Aerodynamic design optimization on unstructured grids with a continuous adjoint formulation , 1997 .

[12]  B. Leer,et al.  Flux-vector splitting for the Euler equations , 1997 .

[13]  M. Aftosmis.  Solution adaptive cartesian grid methods for aerodynamic flows with complex geometries , 1997 .

[14]  Michael J. Aftosmis,et al.  Robust and efficient Cartesian mesh generation for component-based geometry , 1997 .

[15]  J. Peraire,et al.  Constrained, multipoint shape optimisation for complex 3D configurations , 1998, The Aeronautical Journal (1968).

[16]  Jonathan Kindred Elliott.  Aerodynamic optimization based on the Euller and Navier-Stokes equations using unstructured grids , 1998 .

[17]  Thomas Kaminski,et al.  Recipes for adjoint code construction , 1998, TOMS.

[18]  M. Berger,et al.  Robust and efficient Cartesian mesh generation for component-based geometry , 1998 .

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

[20]  Michael B. Bieterman,et al.  Recent progress in aerodynamic design optimization , 1999 .

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

[22]  Scott M. Murman,et al.  Applications of Space-Filling-Curves to Cartesian Methods for CFD , 2004 .

[23]  Michael B. Giles,et al.  On the use of Runge-Kutta time-marching and multigrid for the solution of steady adjoint equations , 2000 .

[24]  Gediminas Adomavicius,et al.  A Parallel Multilevel Method for Adaptively Refined Cartesian Grids with Embedded Boundaries , 2000 .

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

[26]  Jonathan Elliott,et al.  A Chimera approach to aerodynamic shape optimization for the compressible, high-Re Navier-Stokes equations , 2000 .

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

[28]  O. Pironneau,et al.  Applied Shape Optimization for Fluids , 2001 .

[29]  Michael B. Giles,et al.  Analytic adjoint solutions for the quasi-one-dimensional Euler equations , 2001, Journal of Fluid Mechanics.

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

[31]  Oh-Hyun Rho,et al.  Sensitivity Analysis for the Navier-Stokes Equations with Two-Equation Turbulence Models , 2001 .

[32]  Perry A. Newman,et al.  Some Advanced Concepts in Discrete Aerodynamic Sensitivity Analysis , 2001 .

[33]  David W. Bogdanoff,et al.  Free Flight Testing in Support of the Mars Smart Lander Aerodynamics Database , 2002 .

[34]  Michael J. Aftosmis,et al.  Multilevel Error Estimation and Adaptive h-Refinement for Cartesian Meshes with Embedded Boundaries , 2002 .

[35]  Michael Andrew Park,et al.  Adjoint-Based, Three-Dimensional Error Prediction and Grid Adaptation , 2002 .

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

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

[38]  D. Venditti,et al.  Anisotropic grid adaptation for functional outputs: application to two-dimensional viscous flows , 2003 .

[39]  Marian Nemec,et al.  Optimal Shape Design Of Aerodynamic Configurations: A Newton-Krylov Approach , 2003 .

[40]  A. Dadone,et al.  Efficient Fluid Dynamic Design Optimization Using Cartesian Grids , 2003 .

[41]  Michael B. Giles,et al.  Adjoint and defect error bounding and correction for functional estimates , 2003 .

[42]  A. Dadone,et al.  Ghost-Cell Method for Inviscid Two-Dimensional Flows on Cartesian Grids , 2004 .

[43]  P. Cusdin.  GENERATING EFFICIENT CODE WITH AUTOMATIC DIFFERENTIATION , 2004 .

[44]  Thomas H. Pulliam,et al.  CAD-Based Aerodynamic Design of Complex Configurations using a Cartesian Method , 2004 .

[45]  Michael Andrew Park,et al.  Adjoint-Based, Three-Dimensional Error Prediction and Grid Adaptation , 2004 .

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

[47]  M. Berger,et al.  Analysis of Slope Limiters on Irregular Grids , 2005 .

[48]  D. Mavriplis.  Formulation and Multigrid Solution of the Discrete Adjoint Problem on Unstructured Meshes , 2006 .