Structural and Multidisciplinary Optimization

, Volume 58, Issue 2, pp 349–377 | Cite as

An automated selection algorithm for nonlinear solvers in MDO

  • Shamsheer S. Chauhan
  • John T. Hwang
  • Joaquim R. R. A. Martins


There are two major types of approaches that are used for the multidisciplinary analysis (MDA) of coupled systems: fixed-point-iteration-based approaches and coupled Newton-based approaches. Fixed-point-iteration approaches are easier to implement, but can require a large number of iterations or diverge for strongly coupled problems. On the other hand, coupled-Newton approaches have superior convergence orders, but generally require more effort to implement and have more expensive iterations. Additionally, these two major approaches have many variations, including hybrid approaches where the MDA begins with a fixed-point iteration and then switches to a coupled-Newton approach after a certain number of iterations. However, there is a lack of criteria to govern how to select between these approaches, and when to switch between them in a hybrid approach. This paper compares these approaches and provides an algorithm that can be used to automatically select and switch between them. The proposed algorithm is implemented using OpenMDAO, NASA’s open-source framework for multidisciplinary analysis and optimization, and is tested using OpenAeroStruct, an open-source low-fidelity tool for aerostructural optimization. The results show that the proposed algorithm provides a balance of improved robustness and speed.


Complex systems Coupling strength Hybrid MDA Newton Nonlinear block Gauss–Seidel 



This work was supported by the National Science Foundation (award number 1435188). The authors would also like to thank Justin Gray for his support related to OpenMDAO, and John Jasa for his support related to OpenAeroStruct.


  1. Allison J, Kokkolaras M, Papalambros P (2005) On the impact of coupling strength on complex system optimization for single-level formulations. In: ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, pp. 265–275.
  2. Arian E (1997) Convergence estimates for multidisciplinary analysis and optimization. Technical Report NASA/CR-97-201752, NAS 1.26:201752, ICASE-97-57 Institute for Computer Applications in Science and Engineering. Hampton, VA United StatesGoogle Scholar
  3. Baharev A, Schichl H, Neumaier A, Achterberg T (2015) An exact method for the minimum feedback arc set problem. University of Vienna, ViennaGoogle Scholar
  4. Balay S, Gropp WD, McInnes LC, Smith BF (1997) Efficient Management of Parallelism in Object Oriented Numerical Software Libraries. Birkhäuser Press, Birkhäuser, pp 163–202. zbMATHGoogle Scholar
  5. Balling R, Wilkinson C (1997) Execution of multidisciplinary design optimization approaches on common test problems. AIAA J 35(1):178–186. CrossRefzbMATHGoogle Scholar
  6. Barcelos M, Bavestrello H, Maute K (2006) A Schur-Newton-Krylov solver for steady-state aeroelastic analysis and design sensitivity analysis. Comput Methods Appl Mech Eng 195(17–18):2050–2069. CrossRefzbMATHGoogle Scholar
  7. Bazilevs Y, Calo VM, Zhang Y, Hughes TJR (2006) Isogeometric fluid–structure interaction analysis with applications to arterial blood flow. Comput Mech 38(4):310–322. MathSciNetCrossRefzbMATHGoogle Scholar
  8. Bloebaum CL (1995) Coupling strength-based system reduction for complex engineering design. Struct Optim 10(2):113–121. CrossRefGoogle Scholar
  9. Cervera M, Codina R, Galindo M (1996) On the computational efficiency and implementation of block-iterative algorithms for nonlinear coupled problems. Eng Comput 13(6):4–30. CrossRefzbMATHGoogle Scholar
  10. Chauhan SS, Hwang JT, Martins JRRA (2017) Benchmarking approaches for the multidisciplinary analysis (MDA) of complex systems using a Taylor series-based scalable problem. a presentation for the 12th World Congress of Structural and Multidisciplinary Optimization, Braunschweig, Germany, June, 2017
  11. Chauhan SS, Hwang JT, Martins JRRA (2018) Benchmarking approaches for the multidisciplinary analysis of complex systems using a Taylor series-based scalable problem. In: Advances in Structural and Multidisciplinary Optimization: Proceedings of the 12th World Congress of Structural and Multidisciplinary Optimization (WCSMO12). Springer International Publishing, Cham, pp 98–116.
  12. Fernández MÁ, Moubachir M (2005) A Newton method using exact Jacobians for solving fluid-structure coupling. Comput Struct 83 (2–3):127–142. CrossRefGoogle Scholar
  13. Gill PE, Murray W, Saunders MA (2005) An SQP algorithm for large-scale constrained optimization. Society for Industrial and Applied Mathematics 47(1):99–131. MathSciNetzbMATHGoogle Scholar
  14. Gray J, Moore KT, Naylor BA (2010) OpenMDAO: An open source framework for multidisciplinary analysis and optimization. In: Proceedings of the 13th AIAA/ISSMO Multidisciplinary Analysis Optimization Conference, Fort Worth.
  15. Gray J, Hearn T, Moore K, Hwang JT, Martins JRRA, Ning A (2014) Automatic evaluation of multidisciplinary derivatives using a graph-based problem formulation in OpenMDAO. In: 15th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, AIAA AVIATION, AIAA.
  16. Gundersen T, Hertzberg T (1983) Partitioning and tearing of networks applied to process flowsheeting. Model Identif Control: A Nor Res Bullet 4(3):139–165. CrossRefGoogle Scholar
  17. Haftka RT, Sobieszczanski-Sobieski J, Padula SL (1992) On options for interdisciplinary analysis and design optimization. Struct Optim 4(2):65–74. CrossRefGoogle Scholar
  18. Heil M (2004) An efficient solver for the fully coupled solution of large-displacement fluid-structure interaction problems. Comput Methods Appl Mech Eng 193(1–2):1–23. MathSciNetCrossRefzbMATHGoogle Scholar
  19. Heil M, Hazel AL, Boyle J (2008) Solvers for large-displacement fluid–structure interaction problems: segregated versus monolithic approaches. Comput Mech 43(1):91–101. CrossRefzbMATHGoogle Scholar
  20. Hu X, Chen X, Lattarulo V, Parks GT (2016) Multidisciplinary optimization under high-dimensional uncertainty for small satellite system design. AIAA J 54(5):1732–1741. CrossRefGoogle Scholar
  21. Hulme K, Bloebaum C, Nozaki Y (2000) A performance-based investigation of parallel and serial approaches to multidisciplinary analysis convergence. In: 8th Symposium on Multidisciplinary Analysis and Optimization, AIAA
  22. Hulme KF, Bloebaum CL (1997) Development of a multidisciplinary design optimization test simulator. Struct Optim 14(2):129–137. CrossRefGoogle Scholar
  23. Hwang JT, Martins JRRA (2016) Allocation-mission-design optimization of next-generation aircraft using a parallel computational framework. In: 57th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, AIAA SciTech, AIAA.
  24. Hwang JT, Martins JRRA (2018) A computational architecture for coupling heterogeneous numerical models and computing coupled derivatives. ACM Transactions on Mathematical Software (In press)Google Scholar
  25. Hwang JT, Lee DY, Cutler JW, Martins JRRA (2014) Large-scale multidisciplinary optimization of a small satellite’s design and operation. J Spacecr Rocket 51(5):1648–1663. CrossRefGoogle Scholar
  26. Irons BM, Tuck RC (1969) A version of the Aitken accelerator for computer iteration. Int J Numer Methods Eng 1(3):275–277. CrossRefzbMATHGoogle Scholar
  27. Jasa JP, Hwang JT, Martins JRRA (2018) Open-source coupled aerostructural optimization using Python. Structural and Multidisciplinary Optimization,
  28. Jones E, Oliphant T, Peterson P et al (2001) SciPy: Open source scientific tools for Python.
  29. Joosten MM, Dettmer WG, Perić D (2009) Analysis of the block Gauss-Seidel solution procedure for a strongly coupled model problem with reference to fluid-structure interaction. Int J Numer Methods Eng 78 (7):757–778. MathSciNetCrossRefzbMATHGoogle Scholar
  30. Kenway GKW, Kennedy GJ, Martins JRRA (2014) Scalable parallel approach for high-fidelity steady-state aeroelastic analysis and adjoint derivative computations. AIAA J 52(5):935–951. CrossRefGoogle Scholar
  31. Keyes D, McInnes L, Woodward C, Gropp W, Myra E, Pernice M (2012) Multiphysics simulations: Challenges and opportunities. The International Journal of High Performance Computing Applications, 10.1177/1094342012468181Google Scholar
  32. Kodiyalam S, Yuan C (1998) Evaluation of methods for multidisciplinary design optimization phase I. Technical report, National Aeronautics and Space AdministrationGoogle Scholar
  33. Küttler U, Wall WA (2008) Fixed-point fluid-structure interaction solvers with dynamic relaxation. Comput Mech 43(1):61–72. CrossRefzbMATHGoogle Scholar
  34. Lambe AB, Martins JRRA (2012) Extensions to the design structure matrix for the description of multidisciplinary design, analysis, and optimization processes. Struct Multidiscip Optim 46(2):273–284. CrossRefzbMATHGoogle Scholar
  35. Martins JRRA, Hwang JT (2013) Review and unification of methods for computing derivatives of multidisciplinary computational models. AIAA J 51 (11):2582–2599. CrossRefGoogle Scholar
  36. Martins JRRA, Lambe AB (2013) Multidisciplinary design optimization: a survey of architectures. AIAA J 51(9):2049–2075. CrossRefGoogle Scholar
  37. Martins JRRA, Sturdza P, Alonso JJ (2003) The complex-step derivative approximation. ACM Trans Math Softw 29(3):245–262. MathSciNetCrossRefzbMATHGoogle Scholar
  38. Maute K, Nikbay M, Farhat C (2001) Coupled analytical sensitivity analysis and optimization of three-dimensional nonlinear aeroelastic systems. AIAA J 39(11):2051–2061. CrossRefzbMATHGoogle Scholar
  39. McCulley C, Bloebaum CL (1996) A genetic tool for optimal design sequencing in complex engineering systems. Struct Optim 12(2):186–201. CrossRefGoogle Scholar
  40. Mosher T (1999) Conceptual spacecraft design using a genetic algorithm trade selection process. J Aircr 36 (1):200–208. CrossRefGoogle Scholar
  41. Ning A, Petch D (2016) Integrated design of downwind land-based wind turbines using analytic gradients. Wind Energy 19(12):2137–2152. CrossRefGoogle Scholar
  42. Padula S, Alexandrov N, Green L (1996) MDO Test suite at NASA Langley research center. In: 6th Symposium on Multidisciplinary Analysis and Optimization. Multidisciplinary Analysis Optimization Conferences, AIAA.
  43. Peterson P (2009) F2PY: A tool for connecting Fortran and Python programs. Int J Comput Sci Eng 4(4):296–305. Google Scholar
  44. Saad Y (1993) A flexible inner-outer preconditioned GMRES, algorithm. SIAM J Sci Comput 14(2):461–469. MathSciNetCrossRefzbMATHGoogle Scholar
  45. Saad Y (2003) Iterative Methods for Sparse Linear Systems. Society for Industrial and Applied Mathematics.
  46. Sheldon JP, Miller ST, Pitt JS (2014) Methodology for comparing coupling algorithms for fluid-structure interaction problems. World J Mech 4(2):54–70. CrossRefGoogle Scholar
  47. Steward DV (1981) The design structure system: A method for managing the design of complex systems. IEEE Trans Eng Manag EM-28(3):71–74. CrossRefGoogle Scholar
  48. Tedford NP, Martins JRRA (2010) Benchmarking multidisciplinary design optimization algorithms. Optim Eng 11(1):159–183. MathSciNetCrossRefzbMATHGoogle Scholar
  49. Tosserams S, Etman LFP, Rooda JE (2010) A micro-accelerometer MDO benchmark problem. Struct Multidiscip Optim 41(2):255–275. CrossRefGoogle Scholar
  50. Trefethen LN, Bau D III (1997) Numerical linear algebra. SIAM:Society for Industrial and Applied Mathematics, IllinoisCrossRefzbMATHGoogle Scholar
  51. Turek S, Hron J (2006) Proposal for Numerical Benchmarking of Fluid-Structure Interaction between an Elastic Object and Laminar Incompressible Flow. Springer Berlin Heidelberg, Berlin. CrossRefzbMATHGoogle Scholar
  52. Yi SI, Shin JK, Park GJ (2008) Comparison of MDO methods with mathematical examples. Struct Multidiscip Optim 35(5):391–402. CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Aerospace EngineeringUniversity of MichiganAnn ArborUSA
  2. 2.Peerless Technologies Corp. (NASA Glenn Research Center)ClevelandUSA

Personalised recommendations