Concurrent optimization using probabilistic analysis of distributed multidisciplinary architectures for design under uncertainty

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Ghosh, Sayan
Mavris, Dimitri N.
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Uncertainty based Multidisciplinary Optimization (UMDO) relies on propagation of uncertainties across several disciplines. A typical aircraft design process involves collaboration of multiple and diverse teams involving high-fidelity disciplinary tools and experts. Therefore, traditional methods such as All-In-One (AIO), which integrates all the disciplines and treats the entire multidisciplinary analysis process as a black box becomes infeasible for uncertainty propagation and analysis. If all the disciplines cannot be tightly integrated, then it is helpful to use a method that conducts uncertainty propagation in each discipline and combines their results into a system level uncertainty. Distributed UMDO methods based on Collaborative Optimization (CO), Concurrent Subs-Space Optimization (CSSO), Analytical Target Cascading (ATC), Bi-Level Integrated System Synthesis (BLISS), etc. use the strategy of decomposition and coordination to carry out distributed uncertainty analysis and optimization by preserving disciplinary autonomy. However, there are shortcomings in these methods which leads to inaccurate quantification of uncertainty at system level. One such disadvantage is the inability to handle statistical dependencies among coupling variables. In most cases, the statistical dependencies manifests due the underlying functional relationship between the variables. Most of the existing distributed UMDO methods in literature assume that the coupling variables are independent of each other. Although, under certain conditions this assumption is valid, nonetheless it may lead to inaccurate estimation of uncertainty quantification at system level if the dependencies of coupling variables are significant and if the system level metric is sensitive to the dependencies. Another limitation in the existing distributed UMDO literature is related to interdisciplinary compatibility. One of the common strategies to achieve interdisciplinary compatibility is by moment matching method. Since, only marginal distributions of coupling variables are considered in the moment matching, it works well when coupling variables are statistically independent. However, when coupling variables are dependent, this strategy does not guarantee that interdisciplinary compatibility is satisfied for every instantiation of uncertain variables. Also, most of these methods assume that the uncertain coupling variables have fixed functional form of probability density function, most commonly a Gaussian density function. This assumption breaks down when the local uncertainties in disciplines are non-Gaussian and disciplines are non-linear functions of input variables which lead to non-Gaussian coupling variables. To overcome these limitations, Probabilistic Analysis of Distributed Multidisciplinary Architectures (PADMA) is developed. PADMA is a bi-level distributed uncertainty based multidisciplinary analysis (UMDA) method which allows each discipline to carry out uncertainty propagation independently and concurrently. It is a non-iterative method in which dependence and interdisciplinary compatibility is handled by evaluating the probability of Event of Interdisciplinary Compatibility (EIC). Probability of EIC is evaluated using conditional probability density functions of disciplinary metrics. A quantile copula regression method is developed which is used to model conditional probability density functions. In quantile copula regression, the probability density functions are modeled by regressing multiple level of quantiles of disciplinary metric, allowing a comprehensive representation of overall distribution without any assumption of functional form of probability density function. Also, quantile copula regression models the dependency between disciplinary metric using copula functions when disciplines have multiple outputs. Finally, a distributed UMDO method, Concurrent Optimization using Probabilistic Analysis of Distributed Multidisciplinary Architectures (CO-PADMA), has been developed using PADMA and quantile copula regression. CO-PADMA is a bi-level distributed UMDO method which allows distributed analysis and optimization, while handling the dependencies and interdisciplinary compatibility, to find optimum design and quantify the uncertainty of system metric accurately. The advantages of the methods developed in this thesis have been demonstrated by their application on analytical and physics based problems.
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