Abstract
Several subsystems with hierarchical relationship compose a multilevel system, each of which may have uncertainties in material properties and structural geometric parameters. Compared with the integrated strategy, hierarchical decomposition method is more effective in the investigation of the performance of multilevel systems. However, the process of hierarchical propagation of uncertainty may result in many challenging problems, such as multidimensional correlations and complex coupling of uncertainties. In this paper, a general integrated procedure of multilevel system design optimization by means of the integration of the hierarchical uncertainty analysis (HUA), hierarchical sensitivity analysis (HSA) and uncertainty-based design method is innovatively proposed. Firstly, a hierarchical framework combining R-vine copula with sparse polynomial chaos expansions is adopted to solve the problems of uncertainty quantification and propagation. After that, a map**-based hierarchical sensitivity analysis (MHSA) method is employed to obtain sensitivity indexes of multilevel systems with multidimensional correlations. At last, the probabilistic target cascade analysis method is used to accomplish the multilevel design optimization considering multilevel uncertainty. The proposed procedure is then applied to solve the material–structure integrated design problem of an automotive fiber composite shock tower. Results show that the proposed procedure can achieve a weight reduction compared with the initial design under the premise of meeting the structural performance requirements.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Balesdent M, Berend N, Depince P, Chriette A (2012) A survey of multidisciplinary design optimization methods in launch vehicle design. Struct Multidiscip Optim 45(5):619–642
Bedford BT, Cooke RM (2002) Vines-a new graphical model for dependent random variables. Ann Stat 30(4):1031–1068
Blatman G, Sudret B (2011) Adaptive sparse polynomial chaos expansion based on least angle regression. J Comput Phys 230(6):2345–2367
Borgonovo E, Plischke E (2016) Sensitivity analysis: a review of recent advances. Eur J Oper Res 248(3):869–887
Bostanabad R, Liang B, Gao J, Liu WK (2018) Uncertainty quantification in multiscale simulation of woven fiber composites. Comput Methods Appl Mech Eng 338:506–532
Chakraborty S, Chowdhury R (2017) A hybrid approach for global sensitivity analysis. Reliab Eng Syst Saf 158:50–57
Chehouri A, Younes R, Ilinca A, Perron J (2015) Review of performance optimization techniques applied to wind turbines. Appl Energy 142:361–388
Chen W, Yin X, Lee S, Liu WK (2010) A multiscale design methodology for hierarchical systems with random field uncertainty. ASME J Mech Des. https://doi.org/10.1115/1.4001210
De S, Hampton J, Maute K, Doostan A (2020) Topology optimization under uncertainty using a stochastic gradient-based approach. Struct Multidiscip Optim 62:2255–2278
Han J, Papalambros PY (2010) An SLP filter algorithm foe probabilistic analytical target cascading. Struct Multidiscip Optim 41(6):935–945
Hu J, Zhou Q, Mckeand A, **e T, Choi S-K (2021) A model validation framework based on parameter calibration under aleatory and epistemic uncertainty. Struct Multidiscip Optim 63:645–660
Jiang C, Zhang W, Wang B, Han X (2014) Structural reliability analysis using a copula function-based evidence theory model. Compos Structs 143:19–31
Jiang C, Zhang W, Han X et al (2015) A vine-copula based reliability analysis method for structures with multidimensional correlation. ASME J Mech Des 137(6):061405
Jiang Z, Chen W, German BJ (2016) Multidisciplinary statistical sensitivity analysis considering both aleatory and epistemic uncertainties. AIAA J 54(4):1326–1338
Jung Y, Kang N, Lee I (2018) Modified augmented Lagrangian coordination and alternating direction method of multipliers with parallelization in non-hierarchical analytical target cascading. Struct Multidiscip Optim 58:555–573
Kim HM, Michelena NF, Papalambros PY (2003) Target cascading in optimal system design. ASME J Mech Des 125(3):474–480
Li LS, Liu JH, Liu SH (2014) An efficient strategy for multidisciplinary reliability design and optimization based on CSSO and PMA in SORA framework. Struct Multidiscip Optim 49(2):239–252
Liu H, Chen W, Kokkolaras M, Kim HM (2006) Probabilistic analytical target cascarding: a moment matching formulation for multilevel optimization under uncertainty. ASME J Mech Des 128(4):991–1000
Liu Y, Yin X, Arendt P, Chen W (2010) A hierarchical statistical analysis method for multilevel systems with shared variables. ASME J Mech Des 132(3):031006
Liu Z, Lu J, Zhu P (2016) Lightweight design of automotive composite bumper system using modified particle swarm optimizer. Compos Structs 140:630–643
Liu Z, Zhu C, Zhu P, Chen W (2018) Reliability-based design optimization of composite battery box based on modified particle swarm optimization algorithm. Compos Structs 204:239–255
Mara TA, Tarantola S (2012) Variance-based sensitivity indices for models with dependent inputs. Reliab Eng Syst Saf 107:115–121
Othman MF, Silva GHC, Cabral PH et al (2019) A robust and reliability-based aeroelastic tailoring framework for composite aircraft wings. Compos Structs 208:101–113
Ouyang Q, Chen X, Yao W (2014) Sequential probabilistic analytical target cascading method for hierarchical multilevel optimization under uncertainty. Struct Multidiscip Optim 49(2):267–280
Ouyang Q, Yao W, Chen X (2018) Mixed uncertainty based analytical target cascading: an approach foe hierarchical multilevel optimization under probabilistic and interval mixed uncertainties. Struct Multidiscip Optim 57(4):1475–1493
Pepper N, Montomili F, Sharma S (2019) Multiscale uncertainty quantification with arbitrary polynomial chaos. Comput Methods Appl Mech Eng 357:112571
Shang X, Ma P, Yang M, Chao T (2021) An efficient polynomial chaos-enhanced radial basis function approach for reliability-based design optimization. Struct Multidiscip Optim 63:789–805
Talgorn B, Kokkolaras M, Deblois A, Piperni P (2017) Numerical investigation of non-hierarchical coordination for distributed multidisciplinary design optimization with fixed computational budget. Struct Multidiscip Optim 55:205–220
Tang XS, Li DQ, Zhou CB et al (2013) Impact of copulas for modeling bivariate distributions on system reliability. Struct Saf 44:80–90
Tao W, Liu Z, Zhu P et al (2017) Multiscale design of three dimensional woven composite woven composite automobile fender using modified particle swarm optimization algorithm. Compos Structs 181:73–83
Torre E, Marelli S, Embrechts P, Sudret B (2018) A general framework for data-driven uncertainty quantification under complex dependencies using vine copulas. Probab Eng Mech 55:1–16
Wang F, Li H (2018) Distribution modeling for reliability analysis: impact of multiple dependences and probability model selection. Appl Math Model 59:483–499
Wang P, Lu ZZ, Zhang KC et al (2018) Copula-based decomposition approach for the derivative-based sensitivity of variance contributions with dependent variables. Reliab Eng Syst Saf 169:437–450
**ong F, Yin X, Chen W, Yang S (2010) Enhanced probabilistic analytical target cascading with application to multi-scale design. Eng Optimiz 42(6):581–592
Xu C, Liu Z, Tao W, Zhu P (2020a) A vine copula-based hierarchical framework for multiscale uncertainty analysis. ASME J Mech Des 142(3):1–12
Xu C, Liu Z, Zhu P, Li MS (2020b) Sensitivity-based adaptive sequential sampling for metamodel uncertainty reduction in multilevel systems. Struct Multidiscip Optim 62(3):1473–1496
Xu C, Zhu P, Liu Z, Tao W (2021) Map**-based hierarchical sensitivity analysis for multilevel systems with multidimensional correlations. ASME J Mech Des 143(1):1–26
Yao W, Chen XQ, Luo WC, Tooren MV, Jian G (2011) Review of uncertainty-based multidisciplinary design optimization methods for aerospace vehicles. Prog Aeronaut Sci 47(6):450–479
Yin X, Chen W (2008) A hierarchical statistical sensitivity analysis method for complex engineering systems design. ASME J Mech Des 130(7):071402
Yin X, Lee S, Chen W, Liu WK (2009) Efficient random field uncertainty propagation in design using multiscale analysis. ASME J Mech Des 131(2):021006
Funding
The authors would like to acknowledge the support from Key National Natural Science Foundation of China (Grant No. U1864211), National Natural Science Foundation of China (Grant No.11772191).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Replication of results
A detailed procedure and flowchart of the proposed method has been presented in Sec. 3.4 and one can follow them and reproduce the results. The used toolbox UQLab is completely free for academic users (https://www.uqlab.com). The author will help interested researchers reproduce the results given in the article. It also needs to be emphasized that the simulation model is restricted and it cannot be shared.
Additional information
Responsible Editor: Shikui Chen
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Liu, Z., Zhai, Q., Song, Z. et al. A general integrated procedure for uncertainty-based design optimization of multilevel systems by hierarchical decomposition framework. Struct Multidisc Optim 64, 2669–2686 (2021). https://doi.org/10.1007/s00158-021-03021-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-021-03021-y