×

zbMATH — the first resource for mathematics

On the influence of robustness measures on shape optimization with stochastic uncertainties. (English) Zbl 1364.74084
Summary: The unavoidable presence of uncertainties poses several difficulties to the numerical treatment of optimization tasks. In this paper, we discuss a general framework attacking the additional computational complexity of the treatment of uncertainties within optimization problems considering the specific application of optimal aerodynamic design. Appropriate measure of robustness and a proper treatment of constraints to reformulate the underlying deterministic problem are investigated. In order to solve the resulting robust optimization problems, we propose an efficient methodology based on a combination of adaptive uncertainty quantification methods and optimization techniques, in particular generalized one-shot ideas. Numerical results investigating the reliability and efficiency of the proposed method as well as the influence of different robustness measures on the resulting optimized shape will be presented.

MSC:
74P15 Topological methods for optimization problems in solid mechanics
90C90 Applications of mathematical programming
Software:
Spinterp; TAU
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ahmed S, Shapiro A (2008) Solving chance-constrained stochastic programs via sampling and integer programming. In: Chen ZL, Raghavan S (eds) TutORials in operations research. INFORMS, pp 261-269 · Zbl 1431.76015
[2] AIAA (1998) Guide for the verification and validation of computational fluid dynamics simulations. American Institute of Aeronautics & Astronautics · Zbl 0266.90046
[3] Babuska, IM; Nobile, F; Tempone, R, A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J Numer Anal, 45, 1005, (2007) · Zbl 1151.65008
[4] Ben-Tal A, Ghaoui LE, Nemirovski A (2009) Robust optimization. Princeton Series in Applied Mathematics. Princeton University Press, Princeton · Zbl 1221.90001
[5] Bertsimas, D; Sim, M, Tractable approximations to robust conic optimization problems, Math Program, 107, 5-36, (2006) · Zbl 1134.90026
[6] Beyer, H-G; Sendhoff, B, Robust optimization—a comprehensive survey, Comput Methods Appl Mech Eng, 196, 3190-3218, (2007) · Zbl 1173.74376
[7] Birge JR, Louveaux F (1997) Introduction to stochastic programming. Springer, Berlin · Zbl 0892.90142
[8] Bock, H; Egartner, W; Kappis, W; Schulz, V, Practical shape optimization for turbine and compressor blades by the use of PRSQP methods, Optim Eng, 3, 395-414, (2002) · Zbl 1079.90623
[9] Bock H, Kostina E, Schäfer A, Schlöder J, Schulz V (2007) Multiple set-point partially reduced SQP method for optimal control of PDE. In: Jäger RRW, Warnatz J (eds) Reactive flows, diffusion and transport. From experiments via mathematical modeling to numerical simulation and optimization final report of SFB, vol 359. Collaborative Research Center · Zbl 1205.35011
[10] Bungartz, H-J; Dirnstorfer, S, Multivariate quadrature on adaptive sparse grids, Computing, 71, 89-114, (2003) · Zbl 1031.65037
[11] Cameron, RH; Martin, WT, The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals, Ann Math, 48, 385-392, (1947) · Zbl 0029.14302
[12] Chkifa A, Cohen A, Schwab C (2013) High-dimensional adaptive sparse polynomial interpolation and applications to parametric PDEs. Found Comput Math 1-33 · Zbl 1298.65022
[13] Cohen, A; Devore, R; Schwab, C, Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDE’s, Anal Appl, 09, 11-47, (2011) · Zbl 1219.35379
[14] Conti, S; Held, H; Pach, M; Rumpf, M; Schultz, R, Risk averse shape optimization, SIAM J Control Optim, 49, 927-947, (2011) · Zbl 1230.49039
[15] El Ghaoui, L; Oks, M; Oustry, F, Worst-case value-at-risk and robust portfolio optimization: a conic programming approach, Oper Res, 51, 543-556, (2003) · Zbl 1165.91397
[16] Eldred M, Burkardt J (2009) Comparison of non-intrusive polynomial chaos and stochastic collocation methods for uncertainty quantification. In: AIAA 47th aerospace sciences meeting including the new horizons forum and aerospace exposition, AIAA
[17] Garcke J (2007) A dimension adaptive sparse grid combination technique for machine learning. In: Read W, Larson JW, Roberts AJ (eds). Proceedings of the 13th biennial computational techniques and applications conference, CTAC-2006. ANZIAM J 48:725-740 · Zbl 1334.68183
[18] Garcke, J; Griebel, M, Classification with sparse grids using simplicial basis functions, Intell Data Anal, 6, 483-502, (2002) · Zbl 1088.68550
[19] Garcke, J; Griebel, M; Thess, M, Data mining with sparse grids, Computing, 67, 225-253, (2001) · Zbl 0987.62045
[20] Gauger N (2003) Das Adjungiertenverfahren in der aerodynamischen Formoptimierung, PhD thesis, Technische Universitt Braunschweig · Zbl 1136.65308
[21] Gerstner, T; Griebel, M, Dimension-adaptive tensor-product quadrature, Computing, 71, 65-87, (2003) · Zbl 1030.65015
[22] Ghanem RG, Spanos PD (2003) Stochastic finite elements: a spectral approach. Dover Publications, New York
[23] Gherman I (2008) Approximate partially reduced SQP approaches for aerodynamic shape optimization problems. PhD thesis, University of Trier
[24] Giles MB, Pierce NA (1997) Adjoint equations in CFD: duality, boundary conditions and solution behaviour, AIAA, 97
[25] Giles, MB; Pierce, NA, An introduction to the adjoint approach to design, Flow Turbul Combust, 65, 393-415, (2000) · Zbl 0996.76023
[26] Glasserman P (2003) Monte Carlo methods in financial engineering (stochastic modelling and applied probability). Springer, Berlin
[27] Griewank, V, Projected hessians for preconditioning in one-step one-shot design optimization, Nonconv Optim Appl, 83, 151-172, (2006) · Zbl 1108.90040
[28] Gumbert CR, Newman PA, Hou GJ-W (2002) Effect of random geometric uncertainty on the computational design of a 3D flexible wing, in 20th AIAA applied aerodynamics conference · Zbl 0987.62045
[29] Gumbert, CR; Newman, PA; Hou, GJ-W, High-fidelity computational optimization for 3D flexible wings: part II—effect of random geometric uncertainty on design, Optim Eng, 6, 139-156, (2005) · Zbl 1145.76417
[30] Hammersley J, Handscomb D (1964) Monte Carlo methods. Methuen & Co, London · Zbl 0121.35503
[31] Heinrich R, Dwight R, Widhalm M, Raichle A (2005) Algorithmic developments in TAU. In: Kroll N, Fassbender J (eds). MEGAFLOW—numerical flow simulation for aircraft design. Springer, Berlin, pp 93-108 · Zbl 1273.76314
[32] Henrion R (2005) Structural properties of linear probabilistic constraints. Stochastic Programming E-Print Series · Zbl 1431.76015
[33] Hicks, RM; Henne, PA, Wing design by numerical optimization, J Aircraft, 15, 407412, (1978)
[34] Huyse, L; Lewis, R; Li, W; Padula, S, Probabilistic approach to free-form airfoil shape optimization under uncertainty, AIAA J, 40, 1764-1772, (2002)
[35] Jameson, A, Aerodynamic design via control theory, J Sci Comput, 3, 233-260, (1988) · Zbl 0676.76055
[36] Jameson, A, Computational aerodynamics for aircraft design, Science, 245, 361-371, (1989)
[37] Kall P, Wallace S (1994) Stochastic programming. Wiley, Chichester · Zbl 0812.90122
[38] Karhunen K (1946) Zur Spektraltheorie stochastischer Prozesse, Suomalaisen Tiedeakatemian toimituksia : Ser. A : 1, Mathematica, physica 34 · Zbl 1108.90040
[39] Klimke A (2006) Uncertainty modeling using fuzzy arithmetic and sparse grids, PhD thesis, Universität Stuttgart, Shaker Verlag, Aachen · Zbl 1087.65063
[40] Klimke, A; Wohlmuth, B, Algorithm 847: spinterp: piecewise multilinear hierarchical sparse grid interpolation in MATLAB, ACM Trans Math Softw, 31, 561-579, (2005) · Zbl 1136.65308
[41] Le Maître OP, Knio OM (2010) Spectral methods for uncertainty quantification: with applications to computational fluid dynamics. Scientific Computation, 1st edn · Zbl 1128.65009
[42] Lemieux C (2009) Monte Carlo and Quasi-Monte Carlo sampling, 1st edn. Springer Series in Statistics. Springer, Berlin · Zbl 1269.65001
[43] Li W, Huyse L, Padula S, Center LR (2001) Robust airfoil optimization to achieve consistent drag reduction over a mach range. National Aeronautics and Space Administration, Langley Research Center
[44] Li W, Padula SL (2003) Robust airfoil optimization in high resolution design space. ICASE NASA Langley Research Centre · Zbl 1047.76111
[45] Loève M (1977) Probability theory 1, 4th edn. Springer, Berlin
[46] Loève M (1978) Probability theory 2, 4th edn. Springer, Berlin · Zbl 0385.60001
[47] Loeven G (2010) Efficient uncertainty quantification in computational fluid dynamics, PhD thesis, Technische Universiteit Delft
[48] Loeven G, Bijl H (2008) Airfoil analysis with uncertain geometry using the probabilistic collocation method. In: 48th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference, AIAA-2008-2070 · Zbl 1232.76038
[49] Ma, X; Zabaras, N, An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations, J Comput Phys, 228, 3084-3113, (2009) · Zbl 1161.65006
[50] Mathelin, L; Hussaini, M; Zang, T, Stochastic approaches to uncertainty quantification in CFD simulations, Numer Algorithms, 38, 209-236, (2005) · Zbl 1130.76062
[51] Matthies H (2007) Quantifying uncertainty: modern computational representation of probability and applications. In: Ibrahimbegovic A, Kozar I (eds). Extreme man-made and natural hazards in dynamics of structures, vol 21 of NATO Security through Science Series. Springer, Dordrecht, pp 105-135 · Zbl 0676.76055
[52] Meyer M, Matthies H (2003) Efficient model reduction in nonlinear dynamics using the Karhunen-Loève expansion and dual-weighted-residual methods. Computat Mech 31:179-191 · Zbl 1038.74559
[53] Mishra, S; Schwab, C; Sukys, J, Multi-level Monte Carlo finite volume methods for nonlinear systems of conservation laws in multi-dimensions, J Comput Phys, 231, 3365-3388, (2012) · Zbl 1402.76083
[54] Najm, HN, Uncertainty quantification and polynomial chaos techniques in computational fluid dynamics, Ann Rev Fluid Mech, 41, 35-52, (2009) · Zbl 1168.76041
[55] Niederreiter H (1992) Random number generation and Quasi-Monte Carlo methods. Society for Industrial and Applied Mathematics, Philadelphia · Zbl 0761.65002
[56] Oberkampf, WL; DeLand, SM; Rutherford, BM; Diegert, KV; Alvin, KF, Error and uncertainty in modeling and simulation, Reliab Eng Syst Saf, 75, 333-357, (2002)
[57] Padula, S; Gumbert, C; Li, W, Aerospace applications of optimization under uncertainty, Optim Eng, 7, 317-328, (2006) · Zbl 1110.76049
[58] Pironneau, O, On optimum design in fluid mechanics, J Fluid Mech Digit Archive, 64, 97-110, (1974) · Zbl 0281.76020
[59] Prékopa A (1995) Stochastic programming. Kluwer Academic Publisher, Boston
[60] Putko M, Newman P, Taylor III A, Green L (2001) Approach for uncertainty propagation and robust design in CFD using sensitivity derivatives. In: AIAA 15-th computational fluid dynamics conference, AIAA, pp 2001-2528
[61] Reuther J, Jameson A, Farmer J, Martinelli L, Saunders D (1996) Aerodynamic shape optimization of complex aircraft configurations via an adjoint formulation. Research Institute for Advanced Computer Science, NASA Ames Research Center · Zbl 1047.76111
[62] Ruszczyński A, Shapiro A (2003) Stochastic programming. Handbook in operations research and management science, vol 10. Elsevier · Zbl 1173.74376
[63] Ruszczyński A, Shapiro A (2004) Optimization of risk measures, risk and insurance. EconWPA · JFM 64.0887.02
[64] Sachs, EW; Volkwein, S, POD-Galerkin approximations in PDE-constrained optimization, GAMM-Mitteilungen, 33, 194-208, (2010) · Zbl 1205.35011
[65] Schillings C (2011) Optimal aerodynamic design under uncertainties, PhD thesis, Universität Trier · Zbl 0996.76023
[66] Schillings, C; Schmidt, S; Schulz, V, Efficient shape optimization for certain and uncertain aerodynamic design, Comput Fluids, 46, 78-87, (2011) · Zbl 1431.76015
[67] Schillings, C; Schwab, C, Sparse, adaptive Smolyak quadratures for Bayesian inverse problems, Inverse Probl, 29, 065011, (2013) · Zbl 1278.65008
[68] Schmidt S, Ilic C, Gauger N, Schulz V (2008) Shape gradients and their smoothness for practical aerodynamic design optimization. Tech. Report SPP1253-10-03, DFG-SPP 1253, submitted (OPTE)
[69] Schmidt, S; Schulz, V, Impulse response approximations of discrete shape hessians with application in CFD, SIAM J Control Optim, 48, 2562-2580, (2009) · Zbl 1387.49064
[70] Schulz V, Gherman I (2008) One-shot methods for aerodynamic shape optimization, In: Kroll N, Schwamborn D, Becker K, Rieger H, Thiele F (eds). MEGADESIGN and MegaOpt—aerodynamic simulation and optimization in aircraft design. Notes on numerical fluid mechanics and multidisciplinary design. Springer · Zbl 0996.76023
[71] Schulz, V; Schillings, C, On the nature and treatment of uncertainties in aerodynamic design, AIAA J, 47, 646-654, (2009)
[72] Schwab, C; Todor, RA, Karhunen-Loève approximation of random fields by generalized fast multipole methods, J Comput Phys, 217, 100-122, (2006) · Zbl 1104.65008
[73] Schwamborn D, Gerhold T, Heinrich R (2006) The DLR TAU-code: recent applications in research and industry. In: European conference on computational fluid dynamics, ECCOMAS CFD
[74] Smolyak, SA, Quadrature and interpolation formulas for tensor products of certain classes of functions, Sov Math Doklady, 4, 240-243, (1963)
[75] Soyster, AL, Convex programming with set-inclusive constraints and applications to inexact linear programming, Oper Res, 21, 1154-1157, (1973) · Zbl 0266.90046
[76] Ta’asan S, Kuruvila G, Salas M (1992) Aerodynamic design and optimization in one shot. In: 30th Aerospace Sciences Meeting, Reno, NV, AIAA Paper 92-0025
[77] Wan, X; Karniadakis, GE, Multi-element generalized polynomial chaos for arbitrary probability measures, SIAM J Sci Comput, 28, 901-928, (2006) · Zbl 1128.65009
[78] Wasilkowski, GW; Wozniakowski, H, Explicit cost bounds of algorithms for multivariate tensor product problems, J Complex, 11, 1-56, (1994) · Zbl 0819.65082
[79] Wiener, N, The homogeneous chaos, Am J Math, 60, 897-936, (1938) · JFM 64.0887.02
[80] Xiu, D; Karniadakis, GE, Modeling uncertainty in flow simulations via generalized polynomial chaos, J Comput Phys, 187, 137-167, (2003) · Zbl 1047.76111
[81] Zingg DW, Elias S (2006) On aerodynamic optimization under a range of operating conditions. In: 44th AIAA aerospace sciences meeting, Reno
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.