A graph theoretic approach to problem formulation for multidisciplinary design analysis and optimization

The formulation of multidisciplinary design, analysis, and optimization (MDAO) problems has become increasingly complex as the number of analysis tools and design variables included in typical studies has grown. This growth in the scale and scope of MDAO problems has been motivated by the need to incorporate additional disciplines and to expand the parametric design space to enable the exploration of unconventional design concepts. In this context, given a large set of disciplinary analysis tools, the problem of determining a feasible data flow between tools to produce a specified set of system-level outputs is combinatorially challenging. The difficulty is compounded in multi-fidelity problems, which are of increasing interest to the MDAO community. In this paper, we propose an approach for addressing this problem based on the formalism of graph theory. The approach begins by constructing the maximal connectivity graph (MCG) describing all possible interconnections between a set of analysis tools. Graph operations are then conducted to reduce the MCG to a fundamental problem graph (FPG) that describes the connectivity of analysis tools needed to solve a specified system-level design problem. The FPG does not predispose a particular solution procedure; any relevant MDO solution architecture could be selected to implement the optimization. Finally, the solution architecture can be represented in a problem solution graph (PSG). The graph approach is applied to an example problem based on a commercial aircraft MDAO study.