6.1.1 MDA UNDER EPISTEMIC UNCERTAINTY
Chapter 2 presented a new methodology to systematically include both aleatory and epistemic uncertainty in the input variables, and the epistemic uncertainty due to model
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errors, within feedback-coupled MDA. This methodology offers a comprehensive framework for the representation and propagation of multiple sources of uncertainty within MDA, using the LAMDA concept, which estimates the probability of the the interdisciplinary compatibility condition being satisfied given a value of the coupling variable.
First, a likelihood-based approach is employed to represent both variability and data uncertainty in the input random variables (due to sparse and/or imprecise data) through non-parametric distributions, which is consequently propagated within the MDA framework using methods such as Monte Carlo sampling, FORM and SORM.
Then, an auxiliary variable method based on the probability integral transform is proposed to include the effect of stochastic model error in coupled MDA. This method brings the epistemic uncertainty to the same level of analysis as input variability such that the propagation of both aleatory and epistemic uncertainty can be implemented in a single loop manner. The proposed methodology provides a general formulation to include both model form error and numerical errors (e.g., discretization error, surrogate model error, etc.) within feedback coupled MDA. A mathematical problem and an electronic packaging application were used to illustrate the proposed methodology.
The auxiliary variable approach also provides a breakthrough in global sensitivity analysis, which previously was only used in the context of aleatory uncertainty and for feed-forward problems.
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6.1.2 MDA WITH HIGH-DIMENSIONAL FEEDBACK COUPLING
The original LAMDA approach was implemented using FORM (first-order reliability method), and has difficulty in solving high-dimensional problems, since the accuracy and computational efficiency deteriorate as the number of the coupling variables increases.
Therefore, Chapter 3 proposed an efficient implementation of the LAMDA concept using Bayesian network and copula sampling for high-dimensional MDA, in the presence of a large number of coupling variables.
The Bayesian network is adopted to estimate the joint probability distribution of the coupling variables given interdisciplinary compatibility. This is similar to the concept of Bayesian updating, and a Gaussian copula-based sampling technique is adopted to generate samples from the conditioned Bayesian network (i.e., conditioned on zero difference between two successive iteration values of the coupling variables) and to estimate the conditional joint distribution of the coupling variables.
When the dimension of the coupling is so large that incorporating all the coupling variables in one Bayesian network becomes computationally cumbersome, principal component analysis is adopted to decrease the dimensionality of the Bayesian network. A mathematical MDA example and an aeroelastic wing analysis example were used to demonstrate the efficiency and accuracy of this BNC-MDA approach. It can be seen from the example that, the PCA compresses a BN with hundreds or thousands of nodes into a BN with 30 to 60 nodes (i.e., using 10 to 20 principal components) without sacrificing too much accuracy.
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In the proposed methodology, each sample only requires a few iterations of the coupled physics analysis instead of a fully converged solution as in fixed point iteration.
Thus the proposed BNC-MDA approach is promising for high-dimensional problems.
6.1.3 MULTI-OBJECTIVE OPTIMIZATION UNDER UNCERTAINTY
Chapter 4 further develops the use of Bayesian network and vine copula sampling as a probabilistic surrogate model for multi-objective optimization under uncertainty and BN training point selection.
The first innovation is to construct the Bayesian network as a probabilistic surrogate model based on input-output samples from the original model. A vine copula- based sampling technique is used for efficient uncertainty propagation. A vehicle side impact problem is used to demonstrate the proposed methodology. For a given set of design values, the joint probability of multiple constraints and objectives are efficiently estimated using the proposed BNC approach, by exploiting the forward propagation capability of the Bayesian network.
The second novelty is the training point selection technique to construct the Bayesian network. Additional training points are generated in the desired region based on sculpting, which exploits the dependence relations among the inputs and outputs, and the inverse propagation capability of the Bayesian network. This sculpting further refines the BN model and improves the Pareto surface for multi-objective optimization.
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6.1.4 MULTIDISCIPLINARY OPTIMIZATION UNDER UNCERTAINTY
Chapter 5 developed a comprehensive framework for multidisciplinary design optimization under uncertainty. The BNC-MDA technique proposed in Chapter 3 and probabilistic graphical surrogate modeling introduced in Chapter 4 were integrated for the optimization of feedback coupled MDA under uncertainty.
In this framework, the Bayesian network is pursued for two purposes: (1) as a probabilistic surrogate model based on the dependence relations among the input, output and the coupling variables, and (2) to perform stochastic MDA/MDO with samples from only a few iterations of the feedback coupled analysis, without the fully converged physics analysis. The BNC approach simultaneously enforces interdisciplinary compatibility and evaluates the optimization objectives and constraints through conditional sampling, without any further evaluations of the original physics models. Further efficiency is achieved by adopting the vine copula-based technique for generating samples from the conditioned Bayesian network efficiently. A mathematical RBDO example, an electronic packaging design problem and an aeroelastic wing design problem were used to demonstrate the proposed methodology.
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