Non–Geometric Sensitivities Using the Adjoint Method

The adjoint method has been used as a successful tool to
obtain sensitivity gradients in aerodynamic design and optimisation
for many years. This work presents an alternative approach to the
continuous adjoint formulation that enables one to compute gradients
of a given measure of merit with respect to control parameters other
than those pertaining to geometry. The procedure is then applied to
the steady 2–D compressible Euler and incompressible Navier–Stokes
flow equations. Finally, the results are compared with sensitivities
obtained by finite differences and theoretical values for validation.

Authors:

• Marcelo Hayashi
• João Lima
• Bruno Chieregatti
• Ernani Volpe

Keywords:

• Adjoint method
• optimisation
• non–geometric sensitivities
• boundary conditions.

References:

[1] O. Pirroneau, “On optimal profiles in stokes flow,” Journal of Fluid
Mechanics, vol. 59, no. 1, pp. 117–128, 1973.
[2] A. Jameson, “Aerodynamic design via control theory,” in 12th IMACS
World Congress on Scientific Computation, ser. MAE Report 1824, Paris,
July 1988.
[3] H. Cabuk, C.-H. Sung, and V. Modi, “Adjoint operator approach to
shape design for incompressible flows,” in 3rd International Conference
on Inverse Design Concepts and Optimization in Engineering Sciences
(ICIDES), College Park, PA, 1991, pp. 391–404.
[4] S. Taasan, G. Kuruvila, and M. D. Salas, “Aerodynamic design and
optimization in one shot,” in 30th Aerospace Sciences Meeting and
Exhibit, Reno, NV, January 1992, aIAA 92–0025.
[5] G. Kuruvila, S. Taasan, and M. D. Salas, “Airfoil optimization by
the one–shot method, optimum design methods in aerodynamics,”
AGARD–FDP–VKI Special Course, 1994.
[6] L. C. C. Santos, “A study on aerodynamic design optimization using
an adjoint method,” Institut f¨ur Entwurfsaerodynamik, Braunschweig,
Tech. Rep. IB 129 - 95/12, July 1995.
[7] A. Jameson, L. Martinelli, and N. A. Pierce, “Optimum aerodynamic
design using the Navier-Stokes equations,” in 35th Aerospace Sciences
Meeting & Exhibit, American Institute of Aeronautics and Astronautics.
Reno, NV: AIAA, January 1997, pp. 1–20.
[8] S. Kim, “Design optimization of high–lift configurations using a viscous
adjoint–based method,” Ph.D. dissertation, Stanford University, 2001.
[9] B. Mohammadi and O. Pirroneau, Applied Shape Optimization for
Fluids, 1st ed. Oxford University Press, 2001.
[10] J. J. Alonso and I. M. Kroo, “Advanced algorithms for design
and optimization of quiet supersonic platforms,” in AIAA
Computational Fluid Dynamics Conference, Reno, NV, January
2002, aIAA–2002–0144.
[11] D. G. Cacuci, Weber, C. F., O. E. M., and J. H. Marable, “Sensitivity
theory for general systems of non–linear equations,” Nuclear Science
and Engineering, vol. 75, pp. 88–110, 1980.
[12] M. Hall and D. Cacuci, “Physical interpretation of the adjoint functions
for sensitivity analysis of atmospheric models,” Journal of Atmospheric
Sciences, vol. 40, pp. 2537–2546, October 1983.
[13] S. K. Nadarajah, “The discrete adjoint approach to aerodynamic shape
optimization,” Ph.D. dissertation, Stanford University, 2003.
[14] H. Kim and K. Nakahashi, “Unstructured adjoint method for
Navier-Stokes equations,” JSME International Journal, vol. 48, no. 2,
2005.
[15] S. Kim, J. J. Alonso, and A. Jameson, “Multi–element high–lift
configuration design optimization using viscous continuous adjoint
method,” Journal of Aircraft, vol. 41, no. 5, September–October 2004.
[16] H. J. Kim, D. Sasaki, S. Obayashi, and K. Nakahashi, “Aerodynamic
optimization of supersonic transport wing using unstructured adjoint
method,” AIAA Journal, vol. 39, no. 6, pp. 1011–1020, June 2001.
[17] J. P. Thomas, K. C. Hall, and E. H. Dowell, “Discrete adjoint approach
for modeling unsteady aerodynamic design sensitivities,” AIAA Journal,
vol. 43, no. 9, pp. 1931–1936, September 2005.
[18] S. Nadarajah and A. Jameson, “Optimum shape design for unsteady
three-dimensional viscous flows using a non-linear frequency domain
method,” in 24th Applied Aerodynamics Conference, American Institute
of Aeronautics and Astronautics. San Francisco, CA: AIAA, June 2006.
[19] S. K. Nadarajah and A. Jameson, “Optimum shape design for unsteady
flows with time–accurate continuous and discrete adjoint methods,”
AIAA Journal, vol. 45, no. 7, pp. 1478–1491, July 2007.
[20] K. Mani and D. Mavriplis, “Unsteady discrete adjoint formulation for
two–dimensional flow problems with deforming meshes,” AIAA Journal,
vol. 46, no. 6, pp. 1351–1364, June 2008.
[21] M. Giles, N. Pierce, and E. S¨uli, “Progress in adjoint error correction
for integral functionals,” Comput Visual Sci, vol. 6, pp. 113–121, 2004.
[22] M. Giles and E. S¨uli, “Adjoint methods for pdes: a posteriori error
analysis and postprocessing by duality,” Acta Numerica, vol. 11, pp.
145–236, 2002.
[23] D. A. Venditti and D. L. Darmofal, “A multilevel error estimation and
grid adaptive strategy for improving the accuracy of integral outputs,”
AIAA Paper 99–3292, 1999.
[24] D. A. Venditti and D. L. Darmofal, “Adjoint error estimation and grid
adaptation for functional outputs: Application to quasi–one–dimensional
flow,” Journal of Computational Physics, vol. 164, pp. 204–227, 2000.
[25] D. A. Venditti and D. L. Darmofal, “Anisotropic grid adaptation
for functional outputs: application to two-dimensional viscous flows,”
Journal of Computational Physics, vol. 187, pp. 22–46, 2003.
[26] M. B. Giles and N. A. Pierce, “Superconvergent
lift estimates through adjoint error analysis,” 1998,
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.27.152.
[27] M. B. Giles and N. A. Pierce, “Adjoint recovery of superconvergent
functionals from approximate solutions of partial differential equations,”
Oxford University Computing Laboratory, Oxford, Report 98/18, August
1999.
[28] M. B. Giles and N. A. Pierce, “Improved lift and drag estimates using
adjoint Euler equations,” AIAA Paper 99–3293, 1999.
[29] N. Pierce and M. Giles, “Adjoint and defect error bounding and
correction for functional estimates,” Journal of Computational Physics,
vol. 200, pp. 769–794, 2004.
[30] A. Jameson, L. Martinelli, and N. A. Pierce, “Optimum aerodynamic
design using the Navier-Stokes equations,” Theoretical and
Computational Fluid Dynamics, vol. 1, no. 10, pp. 213–237, 1998.
[31] A. Jameson, A. Sriram, and L. Martinelli, “A continuous adjoint
method for unstructured grids,” in AIAA Computational Fluid Dynamics
Conference, Orlando, FL, June 2003, aIAA 2003–3955.
[32] L. A. Lusternick and V. J. Sobolev, Elements of Functional Analysis,
1st ed., ser. Russian Monographs and Texts on Advanced Mathematics
and Physics. Delhi: Hindustan Pub. Co., 1961, vol. V. [33] P. M. Morse and H. Feshbach, Methods of Theoretical Physics, 1st ed.
NY: MacGraw–Hill, 1953, vol. 1.
[34] M. Hayashi, M. Ceze, and E. Volpe, “Characteristics-based boundary
conditions for the Euler adjoint problem,” Int. J. Numer. Meth. Fluids,
vol. 71, no. 10, pp. 1297–1321, April 2013.
[35] A. Jameson and S. Kim, “Reduction of the adjoint gradient formula
for aerodynamic shape optimization problems,” AIAA Journal, vol. 41,
no. 11, pp. 2114–2129, November 2003.
[36] E. V. Volpe, “Continuous formulation of the adjoint problem for
unsteady incompressible Navier–Stokes flows,” ANP–NDF–EPUSP, SP,
Technical Report 5, June 2013.
[37] A. Jameson, W. Schimidt, and E. Turkel, “Numerical solution of
the euler equations by finite volume methods using runge–kutta
time–stepping schemes,” AIAA Journal, 1981.
[38] D. Barkley, H. M. Blackburn, and S. J. Sherwin, “Direct optimal
growth for timesteppers,” International Journal for Numerical Methods
in Fluids, vol. 1, no. 231, pp. 1–21, September 2002.