Fenfen Xiong, Xiaolei Yin, Wei Chen, Shuxing Yang

[Steve Yenisch]

Paper: www.mae.ufl.edu/nkim/ISSMO/FenfenXiongPaper.pdf
Presentation:www.mae.ufl.edu/nkim/ISSMO/FenfenXiongPresentation.ppt

Enhanced Probabilistic Analytical Target Cascading with Application to Multiscale Design
Fenfen Xiong¹, Xiaolei Yin², Wei Chen^3* and Shuxing Yang⁴

1 School of Aerospace Science and Engineering, Beijing Institute of Technology, 100081 & Predoctoral Visiting Fellow, Department of Mechanical Engineering, Northwestern University, IL 60208
2,3 Integrated Design Automation Laboratory (IDEAL), Mechanical Engineering, Northwestern University, IL 60208
4 School of Aerospace Science and Engineering, Beijing Institute of Technology, 100081

ABSTRACT
Probabilistic Analytical Target Cascading (PATC) is an approach for multilevel multidisciplinary design optimization under uncertainty. In the existing PATC approach, only the mean and variance of each individual interrelated response and linking variable are matched in a multilevel hierarchy. However, due to the existence of random linking variables or parameters, the interrelated responses from lower-level subsystems are statistically correlated and have a direct influence on the statistical performance of an upper-level subsystem. The ignorance of response correlation introduces difficulties in finding optimal solutions especially when the covariance of interrelated responses has a significant impact. In this paper, an enhanced PATC (EPATC) approach is proposed to improve the performance of PATC by considering the correlations in optimization cycles. With the EPATC approach, in addition to matching the first two statistical moments, the covariance between the interrelated responses is also considered by applying a modified updating strategy for estimating the statistical performance of an upper-level subsystem. A mathematical example and a multiscale design problem are used to demonstrate the effectiveness and efficiency of the proposed EPATC approach. The results using the Probabilistic All-In-One (PAIO) method are used as references to verify the accuracy of the EPATC approach. It is observed that the effectiveness of EPATC highly depends on the impact strength of the covariance on optimal solutions. Our study shows that the EPATC approach outperforms the original PATC by providing more accurate optimal solutions; the multilevel optimization that allows distributed design activities is highly applicable to multiscale design problems.

Keywords: Probabilistic analytical target cascading, Multilevel optimization, Uncertainty, Correlated response, Multiscale design

*Corresponding author, [e-mail]

[haftka]

Referring to slide 9, since the objective function is random, are you minimizing its mean? What is the mean value of Y?

Referring to slide 10, What value of STD is the top table for? What does constraint violation mean since it is a random constraint? I was expecting to see a deficiency in the probability of failure.

[EPATC]

May 18, 2009 7:39:09 GMT -5 haftka said:

Referring to slide 9, since the objective function is random, are you minimizing its mean? What is the mean value of Y?

Reply: We minimize the summition of mean and variance of the objective funtion, just like that in robust design. Here Y is the linking design variable shared by the two subsystems. The Y values in each subsystem are different and will become closer and closer with the optimization process.


Referring to slide 10, What value of STD is the top table for? What does constraint violation mean since it is a random constraint? I was expecting to see a deficiency in the probability of failure.

Reply: STD=0.1 Sorry for my carelessness.
In our paper, we plug the optimal design variables to the probabilistic all-in-one formulation to get the real values of objective and constraints. You can look at the paper(page 5, on the top). In our tests all the constraint has been changed to derterministic formulation by the moment matching method below:
mean(g)+k*std(g)<=0
k=3 means reliability level is 99.865% for all probabilistic constraints.
Here our goal is for design. We only need to satisfy the constraint and don't care too much about the probability of failure.
Of course, we can also calculate the probability of failure. It is a good point.

Hope it can explain well for you. Thanks so much for your suggestion and good points.