Skip to main content


Log in

Interval prediction of responses for uncertain multidisciplinary system

  • Published:
Structural and Multidisciplinary Optimization Aims and scope Submit manuscript


Considering that numerous sample data points are required in the probabilistic method, a non-probabilistic interval analysis method can be an alternative when the information is insufficient. In the paper, new strategies, which are iterative algorithm based interval uncertainty analysis methods (IA-IUAMs), are developed to acquire the bounds of the responses in multidisciplinary system. Two iterative processes, Jacobi iteration and Seidel iteration, are applied in the new methods respectively. The Jacobi iteration based interval uncertainty analysis method (JI-IUAM) utilizes the strategy of concurrent subsystem analysis to improve computational efficiency while the Seidel iteration based interval uncertainty analysis method (SI-IUAM) can accelerate convergence by utilizing the newest information. Both IA-IUAMs are able to evaluate the bounds of responses accurately and quickly. The presented methods are compared with general sensitivity analysis based interval uncertainty analysis method (SIUAM) and conventional Monte Carlo simulation approach (MCS). The validity and efficiency of the new methods are demonstrated by two numerical examples and two engineering examples. Results show that, on the one hand, IA-IUAMs are more efficient than MCS by avoiding hundreds of system analyses, on the other hand, IA-IUAMs are more accurate and have a wider range of application than SIUAM by avoiding linear approximation and global sensitivity calculation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18

Similar content being viewed by others


  • Alexandrov NM, Lewis RM (1999) Comparative properties of collaborative optimization and other approaches to MDO. Technical Report, NASA

  • Ashuri T, Zaaijer MB, Martins JR, Bussel GJ, Kuik GA (2014) Multidisciplinary design optimization of offshore wind turbines for minimum levelized cost of energy. Renew Energ 68:893–905

    Article  Google Scholar 

  • Balling RJ, Sobieszczanski-Sobieski J (1995) An algorithm for solving the system-level problem in multilevel optimization. Struct Optimization 9:168–177

  • Batill S, Renaud J, Gu X (2000) Modeling and simulation uncertainty in multidisciplinary design optimization. 8th Symposium on Multidisciplinary Analysis and Optimization, Long Beach, CA, U.S.A., AIAA 2000–4803

  • Ben-Haim Y (1994) A non-probabilistic concept of reliability. Struct Saf 14(4):227–245

    Article  Google Scholar 

  • Bloebaum C, Hajela P, Sobieszczanski-Sobieski J (1992) Non-hierarchic system decomposition in structural optimization. Eng Optimiz 19(3):171–186

    Article  Google Scholar 

  • Cao H, Duan B (2004) Uncertainty analysis for multidisciplinary systems based on convex models. 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Albany, New York, U.S.A., AIAA 2004–4504

  • Dribusch C, Missoum S, Beran P (2010) A multifidelity approach for the construction of explicit decision boundaries: application to aeroelasticity. Struct Multidisc Optim 42(5):693–705

  • Du X, Chen W (2000a) Methodology for managing the effect of uncertainty in simulation-based design. AIAA J 38(8):1471–1478

    Article  Google Scholar 

  • Du X, Chen W (2000b) Concurrent subsystem uncertainty analysis in multidisciplinary design. 8th Symposium on Multidisciplinary Analysis and Optimization, Long Beach, CA, U.S.A., AIAA 2000–4928

  • Du X, Chen W (2002) Efficient uncertainty analysis methods for multidisciplinary robust design. AIAA J 40(3):545–552

    Article  Google Scholar 

  • Du X, Wang Y, Chen W (2000) Methods for robust multidisciplinary design. 41st Structures, Structural Dynamics, and Materials Conference and Exhibit, Atlanta, GA, U.S.A., AIAA 2000–1785

  • Elishakoff I (1994) A new safety factor based on convex modelling, in: Ayyub, BM, Gupta, MM (eds) Uncertainty Modeling and Analysis: Theory and Applications, pp 145–171, North-Holland, Amsterdam

  • Gu X, Renaud JE, Batill SM (1998) An investigation of multidisciplinary design subject to uncertainty. 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, St. Louis, MO, U.S.A., AIAA 98–4747

  • Gu XS, Renaud JE, Penninger CL (2006) Implicit uncertainty propagation for robust collaborative optimization. J Mech Des 128(4):1001–1013

    Article  Google Scholar 

  • Huang H, Yu H, Zhang X, Zeng S, Wang Z (2010) Collaborative optimization with inverse reliability for multidisciplinary systems uncertainty analysis. Eng Optimiz 42(8):763–773

    Article  MathSciNet  Google Scholar 

  • Huang H, Zhang X, Yuan W, Meng D, Zhang X (2011) Collaborative reliability analysis under the environment of multidisciplinary design optimization. Concurrent Eng-Res A 19(3):245–254

    Article  Google Scholar 

  • Jiang Z, Chen W, German BJ (2014) Statistical sensitivity analysis considering both aleatory and epistemic uncertainties in multidisciplinary design. 15th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Atlanta, GA, U.S.A., AIAA 2014–2870

  • Kang Z, Luo Y (2009) Non-probabilistic reliability-based topology optimization of geometrically nonlinear structures using convex models. Comput Methods Appl Mech Eng 198(41–44):3228–3238

  • Kang Z, Zhang W (2016) Construction and application of an ellipsoidal convex model using a semi-definite programming formulation from measured data. Comput Methods Appl Mech Eng 300:461–489

  • Koch PN, Simpson TW, Allen JK, Mistree F (1999) Statistical approximations for multidisciplinary design optimization: the problem of size. J Aircraft 36(1):275–286

    Article  Google Scholar 

  • Kroo I, Altus S, Braun R, Gage P, Sobieski I (1994) Multidisciplinary optimization methods for aircraft preliminary design. 5th Symposium on Multidisciplinary Analysis and Optimization, Panama City Beach, FL, U.S.A., AIAA 94–4325

  • Li Z (2008) Complex system uncertainty analysis based on convex models. Coal Mine Machinery 29(5):70–72

  • Liu G, Tan G, Li G, Rong YK (2013) Multidisciplinary design optimization of a milling cutter for high-speed milling of stainless steel. Int J Adv Manuf Tech 68(9-12):2431–2438

    Article  Google Scholar 

  • Lombardi M (1998) Optimization of uncertain structures using non-probabilistic models. Comput Struct 67(1-3):99–103

    Article  MATH  Google Scholar 

  • Madsen HO, Krenk S, Lind NC (2006) Methods of structural safety. Courier Dover Publications

  • Matsuno Y, Tsuchiya T, Imamura S, Taguchi H (2014) Multidisciplinary design optimization of long or short range hypersonic aircraft. T Jpn Soc Aeronaut S 57(3):143–152

    Article  Google Scholar 

  • Moens D, Vandepitte D (2004) An interval finite element approach for the calculation of envelope frequency response functions. Int J Numer Methods Eng 61(14):2480–2507

    Article  MATH  Google Scholar 

  • Moens D, Vandepitte D (2005) A survey of non-probabilistic uncertainty treatment in finite element analysis. Comput Methods Appl Mech Eng 194(12):1527–1555

  • Moore RE (1966) Interval analysis. Prentice-Hall

  • Qiao X, Qiu Y, Cao H (2008) Application of interval analysis method and convex models to multidisciplinary systems. Acta Armamentarii 29(7):844–848

  • Qiu ZP (2003) Comparison of static response of structures using convex models and interval analysis method. Int J Numer Methods Eng 56(12):1735–1753

    Article  MATH  Google Scholar 

  • Qiu ZP, Wang XJ, Chen JY (2006) Exact bounds for the static response set of structures with uncertain-but-bounded parameters. Int J Solids Struct 43(21):6574–6593

    Article  MathSciNet  MATH  Google Scholar 

  • Qiu ZP, Xia YY, Yang HL (2007) The static displacement and the stress analysis of structures with bounded uncertainties using the vertex solution theorem. Comput Methods Appl Mech Eng 196(49-52):4965–4984

  • Qiu Z, Ma L, Wang X (2009) Non-probabilistic interval analysis method for dynamic response analysis of nonlinear systems with uncertainty. J Sound Vib 319(1):531–540

    Article  Google Scholar 

  • Tappeta R, Renaud J (1997) Multiobjective collaborative optimization. J Mech Design 119(3):403–411

    Article  Google Scholar 

  • Wang X, Qiu Z, Elishakoff I (2008) Non-probabilistic set-theoretic model for structural safety measure. Acta Mech 198(1-2):51–64

    Article  MATH  Google Scholar 

  • Wang XJ, Wang L, Elishakoff I, Qiu ZP (2011) Probability and convexity concepts are not antagonistic. Acta Mech 219(1-2):45–64

    Article  MATH  Google Scholar 

  • Yao W, Chen X, Luo W, Tooren M, Guo J (2011) Review of uncertainty-based multidisciplinary design optimization methods for aerospace vehicles. Prog Aerosp Sci 47(6):450–479

    Article  Google Scholar 

  • Zhang X, Huang H, Zeng S, Wang Z (2009) Possibility-based multidisciplinary design optimization in the framework of sequential optimization and reliability assessment. ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, San Diego, California, pp 745–750

Download references


The authors would like to thank the major research project (No. MJ-F-2012-04), Defense Industrial Technology Development Program (No. JCKY2013601B001, No. JCKY2016601B001), the China Postdoctoral Science Foundation (2016 M591038) and National Nature Science Foundation of the P. R. China (No. 11372025, No. 11432002, No. 11572024) for the financial supports. Besides, the authors wish to express their sincere thanks to the reviewers for their useful and constructive comments.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Lei Wang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, X., Wang, R., Chen, X. et al. Interval prediction of responses for uncertain multidisciplinary system. Struct Multidisc Optim 55, 1945–1964 (2017).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: