Reliability-based design optimization of aeroelastic structures

  • M. Allen
  • K. MauteEmail author
Research paper


Aeroelastic phenomena are most often either ignored or roughly approximated when uncertainties are considered in the design optimization process of structures subject to aerodynamic loading, affecting the quality of the optimization results. Therefore, a design methodology is proposed that combines reliability-based design optimization and high-fidelity aeroelastic simulations for the analysis and design of aeroelastic structures. To account for uncertainties in design and operating conditions, a first-order reliability method (FORM) is employed to approximate the system reliability. To limit model uncertainties while accounting for the effects of given uncertainties, a high-fidelity nonlinear aeroelastic simulation method is used. The structure is modelled by a finite element method, and the aerodynamic loads are predicted by a finite volume discretization of a nonlinear Euler flow. The usefulness of the employed reliability analysis in both describing the effects of uncertainties on a particular design and as a design tool in the optimization process is illustrated. Though computationally more expensive than a deterministic optimum, due to the necessity of solving additional optimization problems for reliability analysis within each step of the broader design optimization procedure, a reliability-based optimum is shown to be an improved design. Conventional deterministic aeroelastic tailoring, which exploits the aeroelastic nature of the structure to enhance performance, is shown to often produce designs that are sensitive to variations in system or operational parameters.


design optimization first-order reliability method nonlinear aeroelasticity stochastic analysis  


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  1. 1.
    Allen, M.; Maute, K. 2003: Shape Optimization of Aeroelastic Structures under Uncertainties. 16th AIAA Computational Fluid Dynamics Conference, June 23–26, Orlando, FL Google Scholar
  2. 2.
    Box, G. 1988: Signal-to-Noise Rations, Performance Criteria, and Transformations. Technometrics 30(1), pp. 141–149 Google Scholar
  3. 3.
    Cai, X.-C.; Farhat, C.; Sarkis, M. 1998: A Minimum Overlap Restricted Additive Schwarz Preconditioner and Applications in 3D Flow Simulations. The Tenth International Conference on Domain Decomposition Methods for Partial Differential Equations, ed. by Mandel, X.-C.C.J.; Farhat, C. Google Scholar
  4. 4.
    Cesare, M.; Sues, R. 1999: ProFES Probabilistic Finite Element System – Bringing Probabilistic Mechanics to the Desktop. AIAA-99-1607, 40th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit, April 12–15, 1999/St. Louis, MO Google Scholar
  5. 5.
    Chandu, S.; Grandhi, R. 1995: General Purpose Procedure for Reliability Based Structural Optimization under Parametric Uncertainties. Adv Eng Softw 23, pp. 7–14 Google Scholar
  6. 6.
    Chen, W.; Allen, J.; Mistree, F.; Tsui, K. 1996: A Procedure for Robust Design: Minimizing Variations Caused by Noise Factors and Control Factors. ASME J Mech Des 118, pp. 478–485 Google Scholar
  7. 7.
    Cornell, C. 1969: A Probability-Based Structural Code. J Am Concrete Inst 66(12), pp. 974–985 Google Scholar
  8. 8.
    Das, I.; Dennis, J. 1997: A Closer Look at Drawbacks of Minimizing Weighted Sums of Objectives for Pareto Set Generation in Multicriteria Optimization Problems. Struct Optim 14, pp. 63–69 Google Scholar
  9. 9.
    DeLaurentis, D.; Mavris, D. 2000: Uncertainty Modeling and Management in Multidisciplinary Analysis and Synthesis. 38th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV Google Scholar
  10. 10.
    Enevoldsen, I.; Sorensen, J. 1994a: Reliability-Based Optimization in Structural Engineering. Struct Saf 15, pp. 169–196 Google Scholar
  11. 11.
    Enevoldsen, I.; Sorensen, J. 1994b. Reliability-Based Optimization as an Information Tool. Mech Struct Mach 22(1), pp. 117–135 Google Scholar
  12. 12.
    Engelund, S.; Rackwitz, R. 1993: A Benchmark Study on Importance Sampling Techniques in Structural Reliability. Struct Saf 12, pp. 255–276 Google Scholar
  13. 13.
    Farhat, C.; Lesoinne, M.; Maman, N. 1995: Mixed Explicit/Implicit Time Integration of Coupled Aeroelastic Problems: Three–Field Formulation, Geometric Conservation and Distributed Solution. Int J Numer Methods Fluids 21, pp. 807–835 Google Scholar
  14. 14.
    Farhat, C.; Degand, C.; Koobus, B.; Lesoinne, M. 1998a: Torsional Springs for Two-Dimensional Dynamic Unstructured Fluid Meshes. Comput Methods Appl Mech Eng 163, pp. 231–245 Google Scholar
  15. 15.
    Farhat, C.; Lesoinne, M.; LeTallec, P. 1998b: Load and Motion Transfer Algorithms for Fluid/Structure Interaction Problems with Non-Matching Discrete Interfaces: Momentum and Energy Conservation, Optimal Discretization and Application to Aeroelasticity. Comput Methods Appl Mech Eng 157, pp. 95–114 Google Scholar
  16. 16.
    Frangopol, D.; Corotis, R. 1996: Reliability-Based Structural System Optimization: State-of-the-Art verse State-of-the-Practice. Analysis and Computation: Proceedings of the Twelfth Conference held in Conjunction with Structures Congress XIV, ed. by Cheng, pp. 67–78 Google Scholar
  17. 17.
    Ghattas, O.; Li, X. 1998: Domain Decomposition Methods for Sensitivity Analysis of a Nonlinear Aeroelastic Problem. Int J Comput Fluid Dyn 11, pp. 113–130 Google Scholar
  18. 18.
    Gill, P.; Saunders, M.; Murray, W. 1997: SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization. Report no 97-2m Department of Mathematics, University of California, San Diego Google Scholar
  19. 19.
    Grandhi, R.; Wang, L. 1998: Reliability-Based Structural Optimization Using Improved Two-Point Adaptive Nonlinear Approximations. Finite Elem Anal Des, pp. 35–48 Google Scholar
  20. 20.
    Hasofer, A.; Lind, N. 1974: Exact and Invariant Second-Moment Code Format, J Eng Mech 100, pp. 111–121 Google Scholar
  21. 21.
    Hohenbichler, M.; Rackwitz, R. 1986: Sensitivities and Importance Measures in Structural Reliability. Civ Eng Syst 3, pp. 203–209 Google Scholar
  22. 22.
    Hou, G.-W.; Satyanarayana, A. 2000: Analytical Sensitivity Analysis of a Statical Aeroelastic Wing. AIAA 2000–4824, 8th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, September 6–8, 2000/Long Beach, CA Google Scholar
  23. 23.
    Imai, K.; Frangopol, D.M. 2000a: Geometrically Nonlinear Finite Element Reliability Analysis of Structural Systems. I: Theory. Comput Struct 77(6), pp. 677–691 Google Scholar
  24. 24.
    Imai, K.; Frangopol, D.M. 2000b: Response Prediction of Geometrically Nonlinear Structures. J Struct Eng 126(11), pp. 1348–1355 Google Scholar
  25. 25.
    Kahn, H. 1956: Use of Different Monte Carlo Sampling Techniques. Proc. Symp. on Monte Carlo Methods, New York, pp. 149–190 Google Scholar
  26. 26.
    Kalos, M.; Whitlock, P. 1986: Monte Carlo Methods (Volume I: Basics). New York: John Wiley & Sons Google Scholar
  27. 27.
    Karamchandani, A.; Cornell, C. 1992: Sensitivity Estimation within First and Second Order Reliability Methods. Struct Saf 11(2), pp. 95–107 Google Scholar
  28. 28.
    Kim, S-H.; Ra, K-W. 2000: Adaptive Importance Sampling Method for the Stochastic Finite Element Analysis. 8th ASCE Joint Specialty Conference on Probabilistic Mechanics and Structural Reliability, South Bend, IN Google Scholar
  29. 29.
    Li, W.; Huyse, L.; Padula, S. 2002: Robust Airfoil Optimization to Achieve Consistent Drag Reduction over a Range of Mach Numbers. Struct Multidisc Optim 24, pp. 38–50 Google Scholar
  30. 30.
    Luo, X.; Grandhi, R.V. 1997: ASTROS for Reliability-Based Multidisciplinary Structural Analysis and Optimization. Comput Struct 62, pp. 737–745 Google Scholar
  31. 31.
    Madsen, H.; Krenk, S.; Lind, N. 1986: Methods of Structural Safety. Prentice-Hall, Inc., Englewook Cliffs, NJ, USA Google Scholar
  32. 32.
    Martins, J.; Alonso, J. 2002: High-Fidelity Aero-Structural Design Optimization of a Supersonic Business Jet. AIAA 2002-1483, 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, April 22–25, 2002, Denver, CO Google Scholar
  33. 33.
    Maute, K.; Nikbay, M.; Farhat, C. 2000: Analytically Based Sensitivity Analysis and Optimization of Nonlinear Aeroelastic Systems. AIAA 2000–4825, 8th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, September 6–8, 2000, Long Beach, CA Google Scholar
  34. 34.
    Maute, K.; Nikbay, M.; Farhat, C. 2001: Coupled Analytical Sensitivity Analysis and Optimization of Three-Dimensional Nonlinear Aeroelastic Systems. AIAA J 39(11), pp. 2051–2061 Google Scholar
  35. 35.
    Maute, K.; Nikbay, M.; Farhat, C. 2003: Sensitivity Analysis and Design Optimization of Three-Dimensional Nonlinear Aeroelastic Systems by the Adjoint Method. Int J Numer Methods Eng 56, pp. 911–933 Google Scholar
  36. 36.
    Mavris, D.; Macsotai, N.; Roth, B. 1998: A Probabilistic Design Methodology for Commercial Aircraft Engine Cycle Selection. 3rd World Aviation Congress and Exposition, Anaheim, CA, SAE-985510 Google Scholar
  37. 37.
    Mavris, D.; DeLaurentis, D. 1998: A Stochastic Design Approach for Aircraft Affordability. 21st ICAS Congress of the International Council on the Aeronautical Sciences, Melbourne, Australia Google Scholar
  38. 38.
    Melchers, R. 1990: Radial Importance Sampling for Structural Reliability. J Eng Mech 116(1), pp. 189–203 Google Scholar
  39. 39.
    Meric, R. 1985: Coupled Optimization in Steady-State Thermoelasticity. J Therm Stresses 8, pp. 333–347 Google Scholar
  40. 40.
    Meric, R. 1986: Material and Load Optimization of Thermoelastic Solids. Part I: Sensitivity Analysis. Part II: Numerical results. J Therm Stresses 9, pp. 359–372, 373–388 Google Scholar
  41. 41.
    Meric, R. 1990: Optimal Cross-Sectional Shape for MHD Channel Flows. Int J Numer Methods Eng 30, pp. 919–929 Google Scholar
  42. 42.
    Messac, A.; Ismail-Yahaya, A. 2002: Multiobjective Robust Design using Physical Programming. Struct Multidisc Optim 23, pp. 357–371 Google Scholar
  43. 43.
    Michaleris, P.; Tortorelli, D.A.; Vidal, C.A. 1995: Analysis and Optimization of Weakly Coupled Thermoelastoplastic Systems with Application to Weldment Design. Int J Numer Methods Eng 38, pp. 1259–1285 Google Scholar
  44. 44.
    Militello, C.; Felippa, C. 1991: The First ANDES Elements: 9-dof Plate Bending Triangles. Comput Methods Appl Mech Eng 91, pp. 217–246 Google Scholar
  45. 45.
    Møller, H.; Lund, E. 2000: Shape Sensitivity Analysis of Strongly Coupled Fluid–Structure Interaction Problems. AIAA 2000–4823, 8th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, September 6–8, 2000, Long Beach, CA Google Scholar
  46. 46.
    Padmanabhan, D.; Batill, S. 2000: An Iterative Concurrent Subspace Robust Design Framework, AIAA 2000-4841, 8th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, September 6–8, Long Beach, CA Google Scholar
  47. 47.
    Parkinson, A.; Sorensen, C.; Pourhassan, N. 1993: A General Approach for Robust Optimality. Trans ASME 115, pp. 74–80 Google Scholar
  48. 48.
    Pettit, C.; Grandhi, R. 2000: Multidisciplinary Optimization of Aerospace Structures with High Reliability. 8th ASCE Joint Specialty Conference on Probabilistic Mechanics and Structural Reliability, South Bend, IN Google Scholar
  49. 49.
    Pettit, C.; Grandhi, R. 2001: Reliability Optimization of Aerospace Structures for Gust Response and Aileron Effectiveness. 8th International Conference on Structural Safety and Reliability, June 2001, Newport Beach, CA Google Scholar
  50. 50.
    Riha, D.S.; Thacker, B.H.; Millwater, H.R.; Wu, Y.-T.; Enright, M.P. 2000: Probabilistic Engineering Analysis Using the NESSUS Software. AIAA 2000-1512, 41st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit, Atlanta, GA Google Scholar
  51. 51.
    Roe, P. 1981: Approximate Riemann Solvers, Parameter Vectors and Difference Schemes. J Computat Phys 43, pp. 357–372 Google Scholar
  52. 52.
    Rubinstein, R. 1981: Simulation of the Monte Carlo Method. New York: Wiley Google Scholar
  53. 53.
    Schittkowski, K. 1985: NLPQL: A FORTRAN Subroutine for Solving Constrained Nonlinear Programming Problems. Ann Oper Res 5, pp. 485–500 Google Scholar
  54. 54.
    Shinozuka, M. 1983: Basic Analysis of Structural Safety. J Am Concrete Inst 109, pp. 721–740 Google Scholar
  55. 55.
    Sobieszczanski-Sobieski, J. 1990: Sensitivity of Complex, Internally Coupled Systems. AIAA J 28, pp. 153–160 Google Scholar
  56. 56.
    Su, J.; Renaud, J. 1997: Automatic Differentiation in Robust Optimization. AIAA J 35(6), pp. 1072 Google Scholar
  57. 57.
    Sunderasan, S.; Ishii, K.; Houser, D. 1993: A Robust Optimization Procedure with Variation on Design Variables and Constraints. Adv Des Autom 65(1), pp. 379–386 Google Scholar
  58. 58.
    Taguchi, G.; Wu, Y. 1979: Introduction to Off-Line Quality Control. Quality Control Associate Google Scholar
  59. 59.
    Taguchi, G. 1986a: Introduction to Quality Engineering. Krauss International Publications Google Scholar
  60. 60.
    Taguchi, G. 1986b: Systems of Experimental Design. Krauss International Publications Vol. 1 & 2 Google Scholar
  61. 61.
    Taguchi, G. 1993: Taguchi on Robust Technology Development. ASME Press Google Scholar
  62. 62.
    Tortorelli, D.; Subramani, G.; Lu, S.; Haber, R. 1991: Sensitivity Analysis for Coupled Thermoelastic Systems. Int J Solids Struct 27, pp. 1477–1497 Google Scholar
  63. 63.
    Wu, Y.-T.; Millwater, H.; Cruse, T. 1990: Advanced Probabilistic Structural Analysis Method for Implicit Performance Functions. AIAA J 12, pp. 255–276 Google Scholar
  64. 64.
    Wu, Y.-T. 1992: Reliability Technology – 1992 28, New York: The American Society of Mechanical Engineers Google Scholar
  65. 65.
    Wu, Y.-T. et al. 2000: Probabilistic methods for design assessment of reliability with inspection (DARWIN). 41st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit, Atlanta, GA Google Scholar
  66. 66.
    Yang, J.; Nikolaidis, E. 1991: Design of Aircraft Wings Subjected to Gust Loads: A Safety Index Based Approach. AIAA J 29(5), pp. 804–812 Google Scholar
  67. 67.
    Yu, X.; Choi, K.; Chang, K. 1997a: A Mixed Design Approach for Probabilistic Structural Durability. J Struct Optim 14(2)–(3), pp. 81–90 Google Scholar
  68. 68.
    Yu, X.; Choi, K.; Chang, K. 1997b: Reliability and Durability Based Design Sensitivity Analysis and Optimization. Tech. Rep. R97-01, Center for Computer Aided Design University of Iowa Google Scholar
  69. 69.
    Yu, X.; Chang, K.; Choi, K. 1998: Probabilistic Structural Durability Prediction. AIAA J 36(4), pp. 628–637 Google Scholar
  70. 70.
    Zink, P.; Mavris, D.; Love, M.: Karpel, M. 1998: Robust Design for Aeroelastically Tailored/Active Aeroelastic Wing. 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, St. Louis, MO Google Scholar
  71. 71.
    Zink, P.; Raveh, D.; Mavris, D. 2000: Robust Structural Design for Active Aeroelastic Wing with Aerodynamic Uncertainties. 41st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit, Atlanta, GA Google Scholar

Copyright information

© Springer-Verlag 2004

Authors and Affiliations

  1. 1.Center for Aerospace StructuresUniversity of Colorado at BoulderBoulderUSA

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