Research Papers: Design Automation

A Spatial-Random-Process Based Multidisciplinary System Uncertainty Propagation Approach With Model Uncertainty

[+] Author and Article Information
Zhen Jiang

Department of Mechanical Engineering,
Northwestern University,
Evanston, IL 60208
e-mail: zhenjiang2015@u.northwestern.edu

Wei Li

School of Aeronautics,
Northwestern Polytechnical University,
Xi'an, Shaanxi 710072, China
e-mail: liwiair@gmail.com

Daniel W. Apley

Department of Industrial Engineering
and Management Sciences,
Northwestern University,
Evanston, IL 60208
e-mail: dapley@northwestern.edu

Wei Chen

Department of Mechanical Engineering,
Northwestern University,
Evanston, IL 60208
e-mail: weichen@northwestern.edu

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received December 9, 2014; final manuscript received July 1, 2015; published online August 10, 2015. Assoc. Editor: Gary Wang.

J. Mech. Des 137(10), 101402 (Aug 10, 2015) (13 pages) Paper No: MD-14-1779; doi: 10.1115/1.4031096 History: Received December 09, 2014

The performance of a multidisciplinary system is inevitably affected by various sources of uncertainties, usually categorized as aleatory (e.g., input variability) or epistemic (e.g., model uncertainty) uncertainty. In the framework of design under uncertainty, all sources of uncertainties should be aggregated to assess the uncertainty of system quantities of interest (QOIs). In a multidisciplinary design system, uncertainty propagation (UP) refers to the analysis that quantifies the overall uncertainty of system QOIs resulting from all sources of aleatory and epistemic uncertainty originating in the individual disciplines. However, due to the complexity of multidisciplinary simulation, especially the coupling relationships between individual disciplines, many UP approaches in the existing literature only consider aleatory uncertainty and ignore the impact of epistemic uncertainty. In this paper, we address the issue of efficient uncertainty quantification of system QOIs considering both aleatory and epistemic uncertainties. We propose a spatial-random-process (SRP) based multidisciplinary uncertainty analysis (MUA) method that, subsequent to SRP-based disciplinary model uncertainty quantification, fully utilizes the structure of SRP emulators and leads to compact analytical formulas for assessing statistical moments of uncertain QOIs. The proposed method is applied to a benchmark electronic packaging design problem. The estimated low-order statistical moments of the QOIs are compared to the results from Monte Carlo simulations (MCSs) to demonstrate the effectiveness of the method. The UP result is then used to facilitate the robust design optimization of the electronic packaging system.

Copyright © 2015 by ASME
Your Session has timed out. Please sign back in to continue.


Shubin, G. R. , 1994, “Optimization Problem Formulation for Multidisciplinary Design,” Proceedings of the Conference Inverse Problems and Optimal Design in Industry, Philadelphia, July 8–10, Vieweg+Teubner Verlag, Springer, Wiesbaden, pp. 213–216.
Alexandrov, N. M. , and Lewis, R. M. , 2000, “Algorithmic Perspectives on Problem Formulations in MDO,” AIAA Paper No. 2000-4719.
Belytschko, T. , 1980, “Fluid-Structure Interaction,” Comput. Struct., 12(4), pp. 459–469. [CrossRef]
McNamara, J. J. , and Friedmann, P. P. , 2011, “Aeroelastic and Aerothermoelastic Analysis in Hypersonic Flow: Past, Present, and Future,” AIAA J., 49(6), pp. 1089–1122. [CrossRef]
Yao, W. , Chen, X. Q. , Luo, W. C. , van Tooren, M. , and Guo, J. , 2011, “Review of Uncertainty-Based Multidisciplinary Design Optimization Methods for Aerospace Vehicles,” Prog. Aerosp. Sci., 47(6), pp. 450–479. [CrossRef]
Matthies, H. G. , 2007, “Quantifying Uncertainty: Modern Computational Representation of Probability and Applications,” Extreme Man-Made and Natural Hazards in Dynamics of Structures, Springer, Dordrecht, the Netherlands, pp. 105–135.
Kiureghian, A. D. , and Ditlevsen, O. , 2009, “Aleatory or Epistemic? Does It Matter?” Struct. Saf., 31(2), pp. 105–112. [CrossRef]
Kennedy, M. C. , and O'Hagan, A. , 2001, “Bayesian Calibration of Computer Models,” J. R. Stat. Soc., Ser. B, 63(3), pp. 425–464. [CrossRef]
Zaman, K. , and Mahadevan, S. , 2013, “Robustness-Based Design Optimization of Multidisciplinary System Under Epistemic Uncertainty,” AIAA J., 51(5), pp. 1021–1031. [CrossRef]
Helton, J. C. , Johnson, J. D. , Sallaberry, C. J. , and Storlie, C. B. , 2006, “Survey of Sampling-Based Methods for Uncertainty and Sensitivity Analysis,” Reliab. Eng. Syst. Saf., 91(10–11), pp. 1175–1209. [CrossRef]
Landau, D. P. , and Binder, K. , 2005, A Guide to Monte Carlo Simulations in Statistical Physics, 2nd ed., Cambridge University Press, New York.
Robert, C. , and Casella, G. , 1999, Monte Carlo Statistical Methods, Springer, New York.
Thoft-Cristensen, P. , and Baker, M. J. , 1982, Structural Reliability Theory and Its Applications, Springer, New York.
Green, L. L. , Lin, H.-Z. , and Khalessi, M. R. , 2002, “Probabilistic Methods for Uncertainty Propagation Applied to Aircraft Design,” AIAA Paper No. 2002.3140.
Cao, H. , and Duan, B. , 2004, “Uncertainty Analysis for Multidisciplinary Systems Based on Convex Models,” AIAA Paper No. 2004-4504.
Du, X. , Guo, J. , and Beeram, H. , 2008, “Sequential Optimization and Reliability Assessment for Multidisciplinary Systems Design,” Struct. Multidiscip. Optim., 35(2), pp. 117–130. [CrossRef]
Guo, J. , and Du, X. P. , 2010, “Reliability Analysis for Multidisciplinary Systems With Random and Interval Variables,” AIAA J., 48(1), pp. 82–91. [CrossRef]
Gu, X. , and Renaud, J. E. , 2002, “Implementation Study of Implicit Uncertainty Propagation (IUP) in Decomposition-Based Optimization,” AIAA Paper No. 2002-5416.
Gu, X. S. , Renaud, J. E. , and Penninger, C. L. , 2006, “Implicit Uncertainty Propagation for Robust Collaborative Optimization,” ASME J. Mech. Des., 128(4), pp. 1001–1013. [CrossRef]
Du, X. , and Chen, W. , 2002, “Collaborative Reliability Analysis for Multidisciplinary Systems Design,” AIAA Paper No. 2002-5474.
Du, X. , and Chen, W. , 2005, “Collaborative Reliability Analysis Under the Framework of Multidisciplinary Systems Design,” Optim. Eng., 6(1), pp. 63–84. [CrossRef]
Xiong, F. , Yin, X. , Chen, W. , and Yang, S. , 2010, “Enhanced Probabilistic Analytical Target Cascading With Application to Multi-Scale Design,” Eng. Optim., 42(6), pp. 581–592. [CrossRef]
Sankararaman, S. , and Mahadevan, S. , 2012, “Likelihood-Based Approach to Multidisciplinary Analysis Under Uncertainty,” ASME J. Mech. Des., 134(3), p. 031008. [CrossRef]
Liang, C. , and Mahadevan, S. , 2013, “Stochastic Multidisciplinary Analysis With High Dimensional Coupling,” Tenth World Congress on Structural and Multidisciplinary Optimization, Orlando, FL, May 19–24.
Sankararaman, S. , and Mahadevan, S. , 2011, “Likelihood-Based Representation of Epistemic Uncertainty Due to Sparse Point Data and/or Interval Data,” Reliab. Eng. Syst. Saf., 96(7), pp. 814–824. [CrossRef]
Zhang, S. L. , Zhu, P. , Chen, W. , and Arendt, P. , 2013, “Concurrent Treatment of Parametric Uncertainty and Metamodeling Uncertainty in Robust Design,” Struct. Multidiscip. Optim., 47(1), pp. 63–76. [CrossRef]
Gu, X. , Renaud, J. E. , Batill, S. M. , Brach, R. M. , and Budhiraja, A. S. , 2000, “Worst Case Propagated Uncertainty of Multidisciplinary Systems in Robust Design Optimization,” Struct. Multidiscip. Optim., 20(3), pp. 190–213. [CrossRef]
Du, X. , and Chen, W. , 2000, “An Efficient Approach to Probabilistic Uncertainty Analysis in Simulation-Based Multidisciplinary Design,” AIAA Paper No. 2000-0423.
Du, X. , and Chen, W. , 2000, “Methodology for Managing the Effect of Uncertainty in Simulation-Based Design,” AIAA J., 38(8), pp. 1471–1478. [CrossRef]
Du, X. , and Chen, W. , 2002, “Efficient Uncertainty Analysis Methods for Multidisciplinary Robust Design,” AIAA J., 40(3), pp. 545–552. [CrossRef]
Jiang, X. M. , and Mahadevan, S. , 2009, “Bayesian Hierarchical Uncertainty Quantification by Structural Equation Modeling,” Int. J. Numer. Methods Eng., 80(6–7), pp. 717–737. [CrossRef]
Jiang, X. M. , and Mahadevan, S. , 2009, “Bayesian Structural Equation Modeling Method for Hierarchical Model Validation,” Reliab. Eng. Syst. Saf., 94(4), pp. 796–809. [CrossRef]
Sankararaman, S. , McLemore, K. , Mahadevan, S. , Bradford, S. C. , and Peterson, L. D. , 2013, “Test Resource Allocation in Hierarchical Systems Using Bayesian Networks,” AIAA J., 51(3), pp. 537–550. [CrossRef]
Allaire, D. , He, Q. , Deyst, J. , and Willcox, K. , 2012, “An Information-Theoretic Metric of System Complexity With Application to Engineering System Design,” ASME J. Mech. Des., 134(10), p. 100906. [CrossRef]
Sacks, J. , Welch, W. J. , Mitchell, T. J. , and Wynn, H. P. , 1989, “Design and Analysis of Computer Experiments,” Stat. Sci., 4(4), pp. 409–423. [CrossRef]
Conti, S. , Gosling, J. P. , Oakley, J. E. , and O'Hagan, A. , 2009, “Gaussian Process Emulation of Dynamic Computer Codes,” Biometrika, 96(3), pp. 663–676. [CrossRef]
Conti, S. , and O'Hagan, A. , 2010, “Bayesian Emulation of Complex Multi-Output and Dynamic Computer Models,” J. Stat. Plann. Inference, 140(3), pp. 640–651. [CrossRef]
Higdon, D. , Kennedy, M. , Cavendish, J. C. , Cafeo, J. A. , and Ryne, R. D. , 2004, “Combining Field Data and Computer Simulations for Calibration and Prediction,” SIAM J. Sci. Comput., 26(2), pp. 448–466. [CrossRef]
Higdon, D. , Gattiker, J. , Williams, B. , and Rightley, M. , 2008, “Computer Model Calibration Using High-Dimensional Output,” J. Am. Stat. Assoc., 103(482), pp. 570–583. [CrossRef]
Arendt, P. D. , Apley, D. W. , and Chen, W. , 2012, “Quantification of Model Uncertainty: Calibration, Model Discrepancy, and Identifiability,” ASME J. Mech. Des., 134(10), p. 100908. [CrossRef]
Arendt, P. D. , Apley, D. W. , Chen, W. , Lamb, D. , and Gorsich, D. , 2012, “Improving Identifiability in Model Calibration Using Multiple Responses,” ASME J. Mech. Des., 134(10), p. 100909. [CrossRef]
Jiang, Z. , Chen, W. , Fu, Y. , and Yang, R.-J. , 2013, “Reliability-Based Design Optimization With Model Bias and Data Uncertainty,” SAE Int. J. Mater. Manuf., 6(3), pp. 502–516. [CrossRef]
Rasmussen, C. E. , and Williams, C. K. I. , 2006, Gaussian Processes for Machine Learning, The MIT Press, Cambridge, MA.
Kennedy, M. C. , and O'Hagan, A. , 2000, “Predicting the Output From a Complex Computer Code When Fast Approximations Are Available,” Biometrika, 87(1), pp. 1–13. [CrossRef]
Hasselman, T. K. , Yap, K. , Lin, C.-H. , and Cafeo, J. A. , 2005, “A Case Study in Model Improvement for Vehicle Crashworthiness Simulation,” 23rd International Modal Analysis Conference (IMAC-XXIII), Orlando, FL, Jan. 31–Feb. 3.
Chen, W. , Xiong, Y. , Tsui, K. L. , and Wang, S. , 2008, “A Design-Driven Validation Approach Using Bayesian Prediction Models,” ASME J. Mech. Des., 130(2), p. 021101. [CrossRef]
Qian, P. Z. G. , and Wu, C. F. J. , 2008, “Bayesian Hierarchical Modeling for Integrating Low-Accuracy and High-Accuracy Experiments,” Technometrics, 50(2), pp. 192–204. [CrossRef]
Wang, S. C. , Chen, W. , and Tsui, K. L. , 2009, “Bayesian Validation of Computer Models,” Technometrics, 51(4), pp. 439–451. [CrossRef]
Bayarri, M. J. , Berger, J. O. , Paulo, R. , Sacks, J. , Cafeo, J. A. , Cavendish, J. , Lin, C. H. , and Tu, J. , 2007, “A Framework for Validation of Computer Models,” Technometrics, 49(2), pp. 138–154. [CrossRef]
Liu, F. , Bayarri, M. J. , Berger, J. O. , Paulo, R. , and Sacks, J. , 2008, “A Bayesian Analysis of the Thermal Challenge Problem,” Comput. Methods Appl. Mech. Eng., 197(29–32), pp. 2457–2466. [CrossRef]
Salas, A. O. , 1995, “MDO Test Suite,” University at Buffalo, Buffalo, NY, http://www.eng.buffalo.edu/Research/MODEL/mdo.test.orig/class2prob3.html
Renaud, J. , and Gabriele, G. , 1994, “Approximation in Nonhierarchic System Optimization,” AIAA J., 32(1), pp. 198–205. [CrossRef]
Padula, S. L. , Alexandrov, N. , and Green, L. L. , 1996, “MDO Test Suite at NASA Langley Research Center,” AIAA Paper No. 96-4028.
Kodiyalam, S. , and Sobieszczanski-Sobieski, J. , 2001, “Multidisciplinary Design Optimization: Some Formal Methods, Framework Requirements, and Application to Vehicle Design,” Int. J. Veh. Des., 25(1–2), pp. 3–22. [CrossRef]


Grahic Jump Location
Fig. 1

Illustration of SRP modeling

Grahic Jump Location
Fig. 2

A notional multidisciplinary system and its possible sources of uncertainties: block A—input variability (aleatory) and block B—model uncertainty (epistemic)

Grahic Jump Location
Fig. 3

The procedure of the proposed SRP-based MUA method

Grahic Jump Location
Fig. 4

Electronic packaging problem

Grahic Jump Location
Fig. 5

(a) Model prediction (after model bias correction) and (b) estimation of bias function for linking variable y11

Grahic Jump Location
Fig. 6

Histograms of system QOIs: (a) y1 and (b) y4

Grahic Jump Location
Fig. 7

Scatter plots of y6 and y11: (a) considering only aleatory uncertainty (for which they are positively correlated) and (b) considering both uncertainties (for which they are negatively correlated)

Grahic Jump Location
Fig. 8

Variances of (a) y1, (b) y11, and (c) y12 contributed by aleatory and epistemic uncertainties in three different scenarios

Grahic Jump Location
Fig. 9

Relationship between a multiplicative increase in model uncertainty and the resulting increase in the system QOIs' SD

Grahic Jump Location
Fig. 10

Relationship between a multiplicative increase in model uncertainty and the resulting increase in the proportion of total variance of system QOIs due to model uncertainty

Grahic Jump Location
Fig. 11

Contour plots of the variances of (a) y11 and (b) y12 induced by model uncertainty over the design region of x1 and x2




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In