تجزیه و تحلیل حساسیت مبتنی بر الحاق برای مدل چند مولفه

In typical design calculations, a multi-component model (i.e. a chain of codes) is often employed to calculate the quantity of interest. For design optimization, sensitivity analysis studies are often required to find optimum operating conditions or to propagate uncertainties required to set design margins. This manuscript presents a hybrid approach to enable the transfer of sensitivity information between the various components in an efficient manner that precludes the need for a global sensitivity analysis procedure, often envisaged to be computationally intractable. The presented method has two advantages over existing methods which may be classified into two broad categories: brute force-type methods and amalgamated-type methods. First, the presented method determines the minimum number of adjoint evaluations for each component as opposed to the brute force-type methods which require full evaluation of all sensitivities for all responses calculated by each component in the overall model, which proves computationally prohibitive for realistic problems. Second, the new method treats each component as a black-box as opposed to amalgamated-type methods which requires explicit knowledge of the system of equations associated with each component in order to reach the minimum number of adjoint evaluations. The discussion in this manuscript will be limited to the evaluation of first-order derivatives only. Current work focuses on the extension of this methodology to capture higher order derivatives.

The challenges facing sensitivity analysis algorithms continue to grow as engineering design codes become more complex. In particular, a modeling strategy that has found wide application in many engineering disciplines is the so-called multi-scale multi-physics phenomena modeling. In this modeling strategy, several models are employed to describe system behavior starting with detailed first principles fine scale models and ending with coarse scale models to predict the system's macroscopic performance metrics. From a high level, this modeling strategy may be viewed as an assembly of numerous models coupled together in various manners to account for the different scales and physics that affect system behavior. The interconnectivity of the models complicates the manner in which sensitivity information is transferred between the models. In particular, we focus in this manuscript on adjoint sensitivity analysis. Although, powerful adjoint sensitivity analysis tools may exist for the individual scales and/or physics models (often referred to as single-physics or single-scale adjoints, and hereinafter denoted by single-component adjoints), there is often no generally accepted way to formulating a global adjoint for the multi-scale multi-physics model (denoted hereinafter by multi-component model). A global adjoint is often much more complicated to implement and must be planned in advance for the particular set of components’ models. Given the dynamic nature of code development and the need to utilize and exchange models frequently, it is paramount to design sensitivity analysis algorithms that can generate sensitivity information for multi-component models from the single-component adjoint. This is a challenging task since a global adjoint for a multi-component model depends on the manner in which the single-components models are connected. This manuscript proposes a new method to elucidate the coupling of adjoint sensitivity information between different components’ models in a multi-component model (Abdel-Khalik et al., 2011). The proposed method combines the advantages of two existing methods for evaluating sensitivity information for a multi-component model: the brute force methods1 (Jessee et al., 2009a) and the amalgamated methods (Dan Cacuci, 2003). The brute force method is simple to implement but as will be shown in the next section requires significant computational overhead. The amalgamated method minimizes the number of adjoint evaluations but requires great insight into the components’ models. Revealed in the next section are the differences between these two methods and the proposed method to combine their advantages.

This manuscript has presented a novel method for communicating adjoint information between different components in a multi-component model. The objective is to evaluate first order derivatives of the overall model responses (calculated by the last component) with respect to its basic input data (input to its first component). The method utilizes only single-component adjoints, treats each component model as a black-box like the brute force methods, and finds the minimum number of adjoint evaluations like the amalgamated methods. This is done by employing a pseudo response for each component's adjoint model. The pseudo response is a linear combination of all responses generated by the component's model. By utilizing this pseudo response, one can achieve two goals. First, determine the matrices’ product comprising the global sensitivity matrix directly without ever having to form the full components’ sensitivity matrices. Second, exploit the correlations between the responses by projecting the information transferred between the components along the components’ input- and response-effective subspaces. This work has focused on evaluation of first-derivatives information only. Our current research on nonlinear sensitivity analysis (published elsewhere (Abdel-Khalik et al., 2010)) has demonstrated the potential of employing subspace methods to determine higher order derivatives of responses with respect to input data, including cross-product terms, in a computationally efficient manner. We intend to employ the results of this work to extend the algorithms presented here to enable the evaluations of higher-order global adjoints for multi-component models.