Extension of fixed point PDE solvers for optimal design by one-shot method
With first applications to aerodynamic shape optimization
DOI:
https://doi.org/10.13052/REMN.17.87-102Keywords:
automated optimal design, one-shotAbstract
This paper concerns mathematical methods, algorithmic techniques and software tools for the transition from simulation to optimization. We focus in particular on applications in aerodynamics. The methodology is applicable to all areas of scientific computing, where large scale governing equations involving discretized PDEs are treated by custom made fixed point solvers. To exploit the domain specific experience and expertise invested in these simulation tools we propose to extend them in a semi-automated fashion. First they are augmented with an adjoint solvers to obtain (reduced) derivatives and then this sensitivity information is immediately used to determine optimization corrections.
Downloads
References
Biros G., Ghattas O., “ Inexactness Issues in the Lagrange-Newton-Krylov-Schur Method for
PDE-Constrained optimization”, Springer’s Lecture Notes in Computational Science and
Engineering, 2002.
Bischof C. H., Bücker H. M., Lang B., Rasch A., “ Automatic Differentiation of the FLUENT
CFD program: procedure and first results”, Working Note of the Institute for Scientific
Computing RWTH-CS-SC-01-22, Aachen University of Technology, Aachen, 2001.
Das I., Dennis J., “ Normal boundary intersection: a new method for generating Pareto optimal
points in multicriteria optimization problems”, SIAM J. on Optimization, vol. 8, n° 3, p. 631-
, 1998.
Giering R., Kaminski T., Slawig T., “ Generating efficient derivative code with TAF: Adjoint
and tangent linear Euler flow around an airfoil”, Future Generation Computer Systems, vol.
, n° 8, p. 1345-1355, 2005.
Griewank A., “ Projected Hessians for Preconditioning in One-Step One-Shot Design Optimization”,
Large Scale Nonlinear Optimization, vol. 83, p. 151-171, 2006.
Griewank A., Faure C., “ Reduced Functions, Gradients and Hessians from Fixed Point Iteration
for State Equations”, Numerical Algorithms, vol. 30, n° 2, p. 113-139, 2002.
Griewank A., Kressner D., “ Time-lag in Derivative Convergence for Fixed Point Iterations”,
Revue ARIMA, vol. Numéro spécial CARI’04, p. 87-102, 2005.
Hascoët L., Vázquez M., Dervieux A., “ Automatic Differentiation for Optimum Design, Applied
to Sonic Boom Reduction”, in V.Kumar et al. (ed.), Proceedings of the International
Conference on Computational Science and its Applications, ICCSA’03, Montreal, Canada,
LNCS 2668, Springer, p. 85-94, 2003.
Hazra S., Schulz V., “ Simultaneous pseudo-timestepping for state constrained optimization
problems in aerodynamics”, in L. Biegler, O. Ghattas, M. Heinkenschloss, D. Keyes, B. van
Bloemen Waanders (eds), Large-scale PDE-Constrained Optimization: Towards Real-time
and Online PDE-Constrained Optimization, SIAM series on Computational Science and
Engineering, 2005.
Heinkenschloss M., Vicente L. N., “ Analysis of Inexact Trust-Region Interior-Point Algorithms”,
SIAM J. Optim.,p. 282-302, 2001.
Heinrich R., “ Implementation and usage of structured algorithms within an unstructured CFDcode”,
Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 92, 2006.
Hicks R. M., Henne P. A., “ Wing design by numerical optimization”, Journal of Aircraft, vol.
, p. 407-412, 1978.
Hinrichsen D., Pritchard A. J., Mathematical Systems Theory I, vol. 48 of Texts in Applied
Mathematics, Springer-Verlag, 2005.
Jameson A., “ Multigrid algorithms for compressible flow calculations”, in W. Hackbusch,
U. Trottenberg (eds), Multigrid Methods II, vol. 1228 of Lecture Notes in Mathematics,
Springer, p. 166-201, 1986.
Jameson A., Reuther J., “ Control theory based airfoil design using the Euler equations”, AIAA
Proceedings 94-4272-CP, 1994.
Jameson A., Schmidt W., Turkel E., “ Numerical Solutions of the Euler Equation by Finite
Volume Methods Using Runge-Kutta Time-Stepping Schemes”, AIAA 81-1259, 1981.
Kroll N., Eisfeld B., Bleecke H.M., “ FLOWer”, Notes on Numerical Fluid Mechanics, vol. 71,
p. 58-68, 1999.
Nelder J. A., Mead R. A., “ A simplex method for function minimisation”, Comput. J., vol. 7,
p. 308-313, 1964.
Pillo G. D., “ Exact penalty methods”, in E. Spedicato (ed.), Algorithms for continuous optimization:
the state of the art, Kluwer Academic Publishers, p. 209-253, 1994.
Schlenkrich S.,Walther A., Gauger N., Heinrich R., “ Differentiating fixed point iterations with
ADOL-C: Gradient calculation for fluid dynamics”, Proceedings of HPSC, Hanoi, March
-10, 2006.
Swanson R. C., Turkel E.,Multistage Scheme withMultigrid for Euler and Navier-Stokes Equations
(Components and Analysis), Technical Report n° 3631, NASA, 1997.
