Review and unification of methods for computing derivatives of multidisciplinary computational models

AIAA Journal

Volumn 51, Issue 11, 2013, Pages 2582-2599

  Martins, Joaquim R R A ^a   Hwang, John T ^a  

^a UNIVERSITY OF MICHIGAN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMIC DIFFERENTIATIONS; BUILDING BLOCKES; COMPLEX-STEP METHOD; COMPUTATIONAL MODEL; MATHEMATICAL FRAMEWORKS; MULTI-DISCIPLINARY SYSTEMS; PARTIAL DERIVATIVES; SYMBOLIC DIFFERENTIATION;

AEROSPACE ENGINEERING;

COMPUTATIONAL METHODS;

EID: 84882984792     PISSN: 00011452     EISSN: None     Source Type: Journal    
DOI: 10.2514/1.J052184     Document Type: Conference Paper

References (97)

1. Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., Saisana, M., and Tarantola, S., Global Sensitivity Analysis: The Primer, Wiley, Hoboken, NJ, 2008.
2. Barthelemy, J.-F., and Sobieszczanski-Sobieski, J., "Optimum Sensitivity Derivatives of Objective Functions in Nonlinear Programming," AIAA Journal, Vol. 21, No. 6, 1983, pp. 913-915. doi:10.2514/3.8172
3. Sherman, L. L., Taylor, A. C., III, Green, L. L., Newman, P. A., Hou, G. W., and Korivi, V. M., "First- and Second-Order Aerodynamic Sensitivity Derivatives via Automatic Differentiation with Incremental Iterative Methods," Journal of Computational Physics, Vol. 129, No. 2, 1996, pp. 307-331. doi:10.1006/jcph.1996.0252 (Pubitemid 126164762)
4. Zole, A., and Karpel, M., "Continuous Gust Response and Sensitivity Derivatives Using State-Space Models," Journal of Aircraft, Vol. 31, No. 5, 1994, pp. 1212-1214. doi:10.2514/3.46632
5. Haug, E. J., Choi, K. K., and Komkov, V., Design Sensitivity Analysis of Structural Systems, Vol. 177, Mathematics in Science and Engineering, Academic Press, London, U.K., 1986, pp. 1-49.
6. van Keulen, F., Haftka, R., and Kim, N., "Review of Options for Structural Design Sensitivity Analysis. Part 1: Linear Systems," Computer Methods in Applied Mechanics and Engineering, Vol. 194, Nos. 30-33, 2005, pp. 3213-3243. doi:10.1016/j.cma.2005.02.002 (Pubitemid 40771521)
7. Barcelos, M., Bavestrello, H., and Maute, K., "A Schur-Newton-Krylov Solver for Steady-State Aeroelastic Analysis and Design Sensitivity Analysis," Computer Methods in Applied Mechanics and Engineering, Vol. 195, Nos. 17-18, 2006, pp. 2050-2069. doi:10.1016/j.cma.2004.09.013 (Pubitemid 43193532)
8. Hicken, J., and Zingg, D., "A Parallel Newton-Krylov Solver for the Euler Equations Discretized Using Simultaneous Approximation Terms," AIAA Journal, Vol. 46, No. 11, 2008, pp. 2773-2786. doi:10.2514/1.34810
9. Biros, G., and Ghattas, O., "Parallel Lagrange-Newton-Krylov-Schur Methods for PDE-Constrained Optimization. Part I: The Krylov-Schur Solver," SIAM Journal on Scientific Computing, Vol. 27, No. 2, 2005, pp. 687-713. doi:10.1137/S106482750241565X
10. Biros, G., and Ghattas, O., "Parallel Lagrange-Newton-Krylov-Schur Methods for PDE-Constrained Optimization. Part II: The Lagrange-Newton Solver and its Application to Optimal Control of Steady Viscous Flows," SIAM Journal on Scientific Computing, Vol. 27, No. 2, 2005, pp. 687-713. doi:10.1137/S1064827502415661
11. Kennedy, G. J., and Martins, J. R. R. A., "Parallel Solution Methods for Aerostructural Analysis and Design Optimization," 13th AIAA/ISSMO Multidisciplinary Analysis Optimization Conference, AIAA Paper 2010-9308, Sept. 2010.
12. Kenway, G. K.W., Kennedy, G. J., and Martins, J. R. R. A, "A Scalable Parallel Approach for High-Fidelity Steady-State Aeroelastic Analysis and Derivative Computations," AIAA Journal (in press).
13. Balay, S., Buschelman, K., Eijkhout, V., Gropp, W. D., Kaushik, D., Knepley, M. G., McInnes, L. C., Smith, B. F., and Zhang, H., "PETSc Users Manual," Argonne National Lab. TR-ANL-95/11, Rev. 3.3, 2012.
14. Heroux, M. A., Bartlett, R. A., Howle, V. E., Hoekstra, R. J., Hu, J. J., Kolda, T. G., Lehoucq, R. B., Long, K. R., Pawlowski, R. P., Phipps, E. T., Salinger, A. G., Thornquist, H. K., Tuminaro, R. S., Willenbring, J. M., Williams, A., and Stanley, K. S., "An Overview of the Trilinos Project," ACM Transactions on Mathematical Software, Vol. 31, No. 3, 2005, pp. 397-423. doi:10.1145/1089014
15. Chung, H. S., and Alonso, J. J., "Using Gradients to Construct Response Surface Models for High-Dimensional Design Optimization Problems," 39th AIAA Aerospace Sciences Meeting, AIAA Paper 2001-0922, Jan. 2001.
16. Seyranian, A. P., Lund, E., and Olhoff, N., "Multiple Eigenvalues in Structural Optimization Problems," Structural and Multidisciplinary Optimization, Vol. 8, No. 4, 1994, pp. 207-227. doi:10.1007/BF01742705
17. James, K., Hansen, J. S., and Martins, J. R. R. A., "Structural Topology Optimization for Multiple Load Cases Using a Dynamic Aggregation Technique," Engineering Optimization, Vol. 41, No. 12, 2009, pp. 1103-1118. doi:10.1080/03052150902926827
18. Sigmund, O., "On the Usefulness of Non-Gradient Approaches in Topology Optimization," Structural and Multidisciplinary Optimization, Vol. 43, No. 5, 2011, pp. 589-596. doi:10.1007/s00158-011-0638-7
19. Lee, E., James, K. A., and Martins, J. R. R. A., "Stress-Constrained Topology Optimization with Design-Dependent Loading," Structural and Multidisciplinary Optimization, Vol. 46, No. 5, Nov. 2012, pp. 647-661. doi:10.1007/s00158-012-0780-x
20. Lee, E., and Martins, J. R. R. A., "Structural Topology Optimization with Design-Dependent Pressure Loads," Computer Methods in Applied Mechanics and Engineering, Vols. 233-236, 2012, pp. 40-48. doi:10.1016/j.cma. 2012.04.007
21. Martins, J. R. R. A., Alonso, J. J., and Reuther, J. J., "High-Fidelity Aerostructural Design Optimization of a Supersonic Business Jet," Journal of Aircraft, Vol. 41, No. 3, 2004, pp. 523-530. doi:10.2514/1.11478
22. Kenway, G. K.W., and Martins, J. R. R. A., "Multi-Point High-Fidelity Aerostructural Optimization of a Transport Aircraft Configuration," Journal of Aircraft (accepted for publication).
23. Mader, C. A., and Martins, J. R. R. A., "Derivatives for Time-Spectral Computational Fluid Dynamics Using an Automatic Differentiation Adjoint," AIAA Journal, Vol. 50, No. 12, 2012, pp. 2809-2819. doi:10.2514/1.J051658
24. Mader, C. A., and Martins, J. R. R. A., "Computation of Aircraft Stability Derivatives Using an Automatic Differentiation Adjoint Approach," AIAA Journal, Vol. 49, No. 12, 2011, pp. 2737-2750. doi:10.2514/1.J051147
25. Martins, J. R. R. A., and Lambe, A. B., "Multidisciplinary Design Optimization: A Survey of Architectures," AIAA Journal,Vol. 51, No. 9, 2013, pp. 2049-2075. doi:10.2514/1.J051895
26. Adelman, H. M., and Haftka, R. T., "Sensitivity Analysis of Discrete Structural Systems," AIAA Journal, Vol. 24, No. 5, 1986, pp. 823-832. doi:10.2514/3.48671
27. Peter, J. E. V., and Dwight, R. P., "Numerical Sensitivity Analysis for Aerodynamic Optimization: A Survey of Approaches," Computers and Fluids Journal, Vol. 39, No. 3, 2010, pp. 373-391. doi:10.1016/j.compfluid.2009.09.013
28. Sobieszczanski-Sobieski, J., "Sensitivity Analysis and Multidisciplinary Optimization for Aircraft Design: Recent Advances and Results," Journal of Aircraft, Vol. 27, No. 12, Dec. 1990, pp. 993-1001. doi:10.2514/3.45973
29. Sobieszczanski-Sobieski, J., "Sensitivity of Complex, Internally Coupled Systems," AIAA Journal, Vol. 28, No. 1, 1990, pp. 153-160. doi:10.2514/3.10366 (Pubitemid 20634794)
30. Sobieszczanski-Sobieski, J., "Higher Order Sensitivity Analysis of Complex, Coupled Systems," AIAA Journal, Vol. 28, No. 4, 1990, pp. 756-758. doi:10.2514/3.10458 (Pubitemid 20730560)
31. Gill, P., Murray, W., and Saunders, M., "SNOPT:An SQP Algorithm for Large-Scale Constraint Optimization," SIAM Journal on Optimization, Vol. 12, No. 4, 2002, pp. 979-1006. doi:10.1137/S1052623499350013
32. Perez, R. E., Jansen, P.W., and Martins, J. R. R. A., "pyOpt: a Python-Based Object-Oriented Framework for Nonlinear Constrained Optimization," Structural and Multidisciplinary Optimization, Vol. 45, No. 1, Jan. 2012, pp. 101-118. doi:10.1007/s00158-011-0666-3
33. Martins, J. R. R. A., Sturdza, P., and Alonso, J. J., "The Complex-Step Derivative Approximation," ACM Transactions on Mathematical Software, Vol. 29, No. 3, 2003, pp. 245-262. doi:10.1145/838250
34. Lyness, J. N., "Numerical Algorithms Based on the Theory of Complex Variable," Proceedings - ACM National Meeting, Thompson Book Co., Washington, D.C., 1967, pp. 125-133.
35. Lyness, J. N., and Moler, C. B., "Numerical Differentiation of Analytic Functions," SIAM Journal on Numerical Analysis, Vol. 4, No. 2, 1967, pp. 202-210. doi:10.1137/0704019
36. Squire, W., and Trapp, G., "Using Complex Variables to Estimate Derivatives of Real Functions," SIAM Review, Vol. 40, No. 1, 1998, pp. 110-112. doi:10.1137/S003614459631241X (Pubitemid 128628446)
37. Anderson, W. K., Newman, J. C., Whitfield, D. L., and Nielsen, E. J., "Sensitivity Analysis for Navier-Stokes Equations on Unstructured Meshes Using Complex Variables," AIAA Journal, Vol. 39, No. 1, Jan. 2001, pp. 56-63. doi:10.2514/2.1270 (Pubitemid 32872423)
38. Newman, J. C., III, Whitfield, D. L., and Anderson, W. K., "Step-Size Independent Approach for Multidisciplinary Sensitivity Analysis," Journal of Aircraft, Vol. 40, No. 3, 2003, pp. 566-573. doi:10.2514/2.3131
39. Martins, J. R. R. A., Alonso, J. J., and Reuther, J. J., "A Coupled-Adjoint Sensitivity Analysis Method for High-Fidelity Aero-Structural Design," Optimization and Engineering, Vol. 6, No. 1, 2005, pp. 33-62. doi:10.1023/B:OPTE.0000048536.47956.62 (Pubitemid 41529601)
40. Luzyanina, T., and Bocharov, G., "Critical Issues in the Numerical Treatment of the Parameter Estimation Problems in Immunology," Journal of Computational Mathematics, Vol. 30, No. 1, 2012, pp. 59-79. doi:10.4208/jcm
41. Sturdza, P., "An Aerodynamic Design Method for Supersonic Natural Laminar Flow Aircraft," Ph.D. Dissertation, Dept. of Aeronautics and Astronautics, Stanford Univ., Stanford, CA, 2004.
42. Blackwell, B. F., and Dowding, K. J., "Sensitivity Analysis and Uncertainty Propagation of Computational Models," Handbook of Numerical Heat Transfer, edited by Minkowycz,W. J., Sparrow, E. M., and Murthy, J. Y., Wiley, Hoboken, NJ, 2009, pp. 443-469, Chap. 14.
43. Ridout, M. S., "Statistical Applications of the Complex-Step Method of Numerical Differentiation," The American Statistician, Vol. 63, No. 1, 2009, pp. 66-74. doi:10.1198/tast.2009.0013
44. Al-Mohy,A. H., and Higham, N. J., "The Complex Step Approximation to the Fréchet Derivative of a Matrix Function," Numerical Algorithms, Vol. 53, No. 1, 2010, pp. 133-148. doi:10.1007/s11075-009-9323-y
45. Forssen, P., Arnell, R., and Fornstedt, T., "An Improved Algorithm for Solving Inverse Problems in Liquid Chromatography," Computers and Chemical Engineering, Vol. 30, No. 9, 2006, pp. 1381-1391. doi:10.1016/j. compchemeng.2006.03.004
46. Lai, K.-L., Crassidis, J., Cheng, Y., and Kim, J., "New Complex-Step Derivative Approximations with Application to Second-Order Kalman Filtering," AIAA Guidance, Navigation, and Control Conference and Exhibit, AIAA Paper 2005-5944, Aug. 2005.
47. De Pauw, D. J.W., and Vanrolleghem, P. A., "Using the Complex-Step Derivative Approximation Method to Calculate Local Sensitivity Functions of Highly Nonlinear Bioprocess Models," Proceedings 17th IMACS World Congress on Scientific Computation, Applied Mathematics and Simulation (IMACS 2005), École centrale de Lille, Paris, France, July 2005.
48. Roberts, J. L., Moy, A. D., van Ommen, T. D., Curran, M. A. J.,Worby, A. P., Goodwin, I. D., and Inoue, M., "Borehole Temperatures Reveal a Changed Energy Budget at Mill Island, East Antarctica, over Recent Decades," The Cryosphere, Vol. 7, No. 1, 2013, pp. 263-273. doi:10.5194/tc-7-263-2013
49. Lantoine, G., Russell, R. P., and Dargent, T., "Using Multicomplex Variables for Automatic Computation of High-Order Derivatives,"ACM Transactions on Mathematical Software,Vol. 38, No. 3, 2012, pp. 16:1-16:21. doi:10.1145/2168773
50. Fike, J. A., and Alonso, J. J., "Automatic Differentiation Through the Use of Hyper-Dual Numbers for Second Derivatives," Recent Advances in Algorithmic Differentiation, edited by Forth, S., Hovland, P., Phipps, E., Utke, J., and Walther, A., Vol. 87, Lecture Notes in Computational Science and Engineering, Springer-Verlag, Berlin, 2012, pp. 163-173.
51. Lambe, A. B., and Martins, J. R. R. A., "Extensions to the Design Structure Matrix for the Description of Multidisciplinary Design, Analysis, and Optimization Processes," Structural and Multidisciplinary Optimization, Vol. 46, No. 2, Aug. 2012, pp. 273-284. doi:10.1007/s00158-012-0763-y
52. Griewank, A., Evaluating Derivatives, SIAM, Philadelphia, 2000.
53. Naumann, U., The Art of Differentiating Computer Programs - An Introduction to Algorithmic Differentiation, SIAM, Philadelphia, PA, 2011.
54. Juedes, D., "A Taxonomy of Automatic Differentiation Tools," Automatic Differentiation of Algorithms: Theory, Implementation, and Application, edited by Griewank, A., and Corliss, G. F., SIAM, Philadelphia, 1991, pp. 315-329.
55. Pryce, J. D., and Reid, J. K., "AD01, a Fortran 90 Code for Automatic Differentiation," Rutherford Appleton Lab. Rept. RAL-TR-1998-057, Chilton, England, U.K., 1998.
56. Bischof, C. H., Roh, L., and Mauer-Oats, A. J., "ADIC: An Extensible Automatic Differentiation Tool for ANSI-C," Software: Practice and Experience, Vol. 27, No. 12, 1997, pp. 1427-1456. doi:10.1002/(SICI)1097- 024X(199712)27:12<1427::AID-SPE138>3.3.CO;2-H (Pubitemid 127632476)
57. Utke, J., "OpenAD: Algorithm Implementation User Guide," Argonne National Lab. TM-274, Argonne, IL, April 2004.
58. Hascoët, L., and Pascual, V., "TAPENADE 2.1 User's Guide," institut national de recherche en informatique et en automatique (INRIA) TR-300, Villers-lès-Nancy, France, 2004.
59. Griewank, A., Juedes, D., and Utke, J., "Algorithm 755: ADOL-C: A Package for the Automatic Differentiation of Algorithms Written in C/C++," ACM Transactions on Mathematical Software, Vol. 22, No. 2, 1996, pp. 131-167. doi:10.1145/229473.229474 (Pubitemid 126418651)
60. Bendtsen, C., and Stauning, O., "FADBAD, a Flexible C++Package for Automatic Differentiation-Using the Forward and Backward Methods," Technical Univ. of Denmark TR-IMM-REP-1996-17, Lyngby, Denmark, 1996.
61. Straka, C. W., "ADF95: Tool for Automatic Differentiation of a FORTRAN Code Designed for Large Numbers of Independent Variables," Computer Physics Communications, Vol. 168, No. 2, 2005, pp. 123-139. doi:10.1016/j.cpc.2005.01.011 (Pubitemid 40643865)
62. Carle, A., and Fagan, M., "ADIFOR 3.0 Overview," Rice Univ. TRCAAM-TR-00-02, 2000.
63. Utke, J., Naumann, U., and Lyons, A., "OpenAD/F: User Manual," Argonne National Lab. TR, Dec. 2012.
64. Giering, R., and Kaminski, T., "Applying TAF to Generate Efficient Derivative Code of Fortran 77-95 Programs," Proceedings of GAMM 2002, Gesellschaft für Angewandte Mathematik und Mechanik (GAMM), Dresden, Germany, 2002.
65. Pascual, V., and Hascoët, L., "Extension of TAPENADE Towards Fortran 95," Automatic Differentiation: Applications, Theory, and Tools, edited by Bücker, H. M., Corliss, G., Hovland, P., Naumann, U., and Norris, B., Lecture Notes in Computational Science and Engineering, Springer-Verlag, Berlin, 2005, pp. 1-5.
66. Shiriaev, D., "ADOL-F Automatic Differentiation of Fortran Codes," Computational Differentiation: Techniques, Applications, and Tools, edited by Berz, M., Bischof, C. H., Corliss, G. F., and Griewank, A., SIAM, Philadelphia, 1996, pp. 375-384.
67. Rhodin, A., "IMAS-Integrated Modeling and Analysis System for the Solution of Optimal Control Problems," Computer Physics Communications, Vol. 107, Nos. 1-3, 1997, pp. 21-38. doi:10.1016/S0010-4655(97)00107-0 (Pubitemid 127381500)
68. Bartholomew-Biggs, M., OPFAD - A Users Guide to the OPtima Forward Automatic Differentiation Tool, Numerical Optimization Centre, Univ. of Hertfordshire, 1995.
69. Brown, S., OPRAD-A Users Guide to the OPtima Reverse Automatic Differentiation Tool, Numerical Optimization Centre, Univ. of Hertfordshire, 1995.
70. Bischof, C., Lang, B., and Vehreschild, A., "Automatic Differentiation for MATLAB Programs," Proceedings in Applied Mathematics and Mechanics, Vol. 2, No. 1, 2003, pp. 50-53. doi:10.1002/pamm.200310013
71. Bischof, C., Bucker, H., Lang, B., Rasch, A., and Vehreschild, A., "Combining Source Transformation and Operator Overloading Techniques to Compute Derivatives for MATLAB Programs," Proceedings of the 2nd IEEE International Workshop on Source Code Analysis and Manipulation, IEEE, New York, NY, 2002, pp. 65-72.
72. Jameson, A., "Aerodynamic Design via Control Theory," Journal of Scientific Computing, Vol. 3, No. 3, 1988, pp. 233-260. doi:10.1007/ BF01061285
73. Sirkes, Z., and Tziperman, E., "Finite Difference of Adjoint or Adjoint of Finite Difference?," Monthly Weather Review,Vol. 125, No. 12, 1997, pp. 3373-3378. doi:10.1175/1520-0493(1997)125<3373:FDOAOA>2.0.CO;2 (Pubitemid 127667546)
74. Jameson, A., Martinelli, L., and Pierce, N. A., "Optimum Aerodynamic Design Using the Navier-Stokes Equations," Theoretical and Computational Fluid Dynamics, Vol. 10, Nos. 1-4, 1998, pp. 213-237. doi:10.1007/s001620050060 (Pubitemid 128027931)
75. Anderson, W. K., and Venkatakrishnan, V., "Aerodynamic Design Optimization on Unstructured Grids with a Continuous Adjoint Formulation," Computers and Fluids Journal, Vol. 28, Nos. 4-5, 1999, pp. 443-480. doi:10.1016/S0045-7930(98)00041-3 (Pubitemid 29346194)
76. Giles, M. B., and Pierce, N. A., "An Introduction to the Adjoint Approach to Design," Flow, Turbulence and Combustion, Vol. 65, Nos. 3-4, 2000, pp. 393-415. doi:10.1023/A:1011430410075 (Pubitemid 32819926)
77. Liu, S., and Canfield, R. A., "Continuum Sensitivity Method for Aeroelastic Shape Design Problems," 14th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, AIAA Paper 2012-5480, Sept. 2012.
78. Nadarajah, S., and Jameson, A., "A Comparison of the Continuous and Discrete Adjoint Approach to Automatic Aerodynamic Optimization," 38th AIAA Aerospace Sciences Meeting and Exhibit, AIAAPaper 2000-0667, 2000.
79. Dwight, R. P., and Brezillon, J., "Effect of Approximations of the Discrete Adjoint on Gradient-Based Optimization," AIAA Journal, Vol. 44, No. 12, 2006, pp. 3022-3031. doi:10.2514/1.21744 (Pubitemid 46065079)
80. Kennedy, G. J., and Hansen, J. S., "The Hybrid-Adjoint Method: A Semi-Analytic Gradient Evaluation Technique Applied to Composite Cure Cycle Optimization," Optimization and Engineering, Vol. 11, No. 1, 2010, pp. 23-43. doi:10.1007/s11081-008-9068-9
81. Pironneau, O., "On Optimum Design in Fluid Mechanics," Journal of Fluid Mechanics, Vol. 64, No. 01, 1974, pp. 97-110. doi:10.1017/ S0022112074002023
82. Nemec, M., Zingg, D. W., and Pulliam, T. H., "Multipoint and Multi-Objective Aerodynamic Shape Optimization," AIAA Journal, Vol. 42, No. 6, 2004, pp. 1057-1065. doi:10.2514/1.10415
83. Driver, J., and Zingg, D. W., "Numerical Aerodynamic Optimization Incorporating Laminar-Turbulent Transition Prediction," AIAA Journal, Vol. 45, No. 8, 2007, pp. 1810-1818. doi:10.2514/1.23569 (Pubitemid 47436960)
84. Lyu, Z., and Martins, J. R. R. A., "RANS-Based Aerodynamic Shape Optimization of a Blended-Wing-Body Aircraft," In 21st AIAA Computational Fluid Dynamics Conference, San Diego, CA, July 2013.
85. Lyu, Z., Kenway, G. K., Paige, C., and Martins, J. R. R. A., "Automatic Differentiation Adjoint of the Reynolds-Averaged Navier-Stokes Equations with a Turbulence Model," In 21st AIAA Computational Fluid Dynamics Conference, San Diego, CA, July 2013.
86. Mani, K., and Mavriplis, D. J., "Unsteady Discrete Adjoint Formulation for Two-Dimensional Flow Problems with Deforming Meshes," AIAA Journal, Vol. 46, No. 6, 2008, pp. 1351-1364. doi:10.2514/1.29924
87. Wang, Q., Moin, P., and Iaccarino, G., "Minimal Repetition Dynamic Checkpointing Algorithm for Unsteady Adjoint Calculation," SIAM Journal on Scientific Computing, Vol. 31, No. 4, 2009, pp. 2549-2567. doi:10.1137/080727890
88. Rumpfkeil, M., and Zingg, D., "The Optimal Control of Unsteady Flows with a Discrete Adjoint Method," Optimization and Engineering, Vol. 11, No. 1, 2010, pp. 5-22. doi:10.1007/s11081-008-9035-5
89. Bloebaum, C., "Global Sensitivity Analysis in Control-Augmented Structural Synthesis," 27th AIAA Aerospace Sciences Meeting, AIAA Paper 1989-0844, Jan. 1989.
90. Martins, J. R. R. A., Alonso, J. J., and Reuther, J. J., "Aero-Structural Wing Design Optimization Using High-Fidelity Sensitivity Analysis," Proceedings of the CEAS Conference on Multidisciplinary Aircraft Design and Optimization, edited by Hölinger, H., Council of European Aerospace Societies, Cologne, Germany, June 2001, pp. 211-226.
91. Maute, K., Nikbay, M., and Farhat, C., "Coupled Analytical Sensitivity Analysis and Optimization of Three-Dimensional Nonlinear Aeroelastic Systems," AIAA Journal, Vol. 39, No. 11, 2001, pp. 2051-2061. doi:10.2514/2.1227 (Pubitemid 33084937)
92. Nykbay, M., Öncü, L., and Aysan, A., "Multidisciplinary Code Coupling for Analysis and Optimization of Aeroelastic Systems," Journal of Aircraft, Vol. 46, No. 6, Nov. 2009, pp. 1938-1944. doi:10.2514/1.41491
93. Johnson, E.H., "Adjoint Sensitivity Analysis in MSC/NASTRAN," MSC 1997 Aerospace Users' Conference, Paper 2897, 1997.
94. ANSYS FLUENT Adjoint Solver, ANSYS, Canonsburg, PA, Nov. 2011.
95. Poon, N. M. K., and Martins, J. R. R. A., "An Adaptive Approach to Constraint Aggregation Using Adjoint Sensitivity Analysis," Structural and Multidisciplinary Optimization, Vol. 34, No. 1, 2007, pp. 61-73. doi:10.1007/s00158-006-0061-7 (Pubitemid 46828877)
96. Marriage,C. J., and Martins, J. R. R. A., "Reconfigurable Semi-Analytic Sensitivity Methods and MDO Architectures Within the π MDO Framework," 12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, AIAA Paper 2008-5956, Sept. 2008.
97. Hwang, J. T., Lee, D.Y., Cutler, J., and Martins, J. R. R. A., "Large-Scale MDO of a Small Satellite Using a Novel Framework for the Solution of Coupled Systems and Their Derivatives," Proceedings of the 54th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, April 2013.