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Abstract Designing the 3D layout of interconnected systems (SPI2), which is a ubiquitous task in engineered systems, is of crucial importance. Intuitively, it can be thought of as the simultaneous placement of (typically rigid) components and subsystems, as well as the design of the routing of (typically deformable) interconnects between these components and subsystems. However, obtaining solutions that meet the design, manufacturing, and life-cycle constraints is extremely challenging due to highly complex and nonlinear interactions between geometries, the multi-physics environment in which the systems participate, the intricate mix of rigid and deformable geometry, as well as the difficult manufacturing and life-cycle constraints. Currently, this design task heavily relies on human interaction even though the complexity of searching the design space of most practical problems rapidly exceeds human abilities. In this work, we take advantage of high-performance hierarchical geometric representations and automatic differentiation to simultaneously optimize the packing and routing of complex engineered systems, while completely relaxing the constraints on the complexity of the solid shapes that can be handled and enable intricate yet functionally meaningful objective functions. Moreover, we show that by simultaneously optimizing the packing volume as well as the routing lengths, we produce tighter packing and routing designs than by focusing on the bounding volume alone. We show that our proposed approach has a number of significant advantages and offers a highly parallelizable, more integrated solution for complex SPI2 designs, leading to faster development cycles with fewer iterations, and better system complexity management. Moreover, we show that our formulation can handle complex cost functions in the optimization, such as manufacturing and life-cycle constraints, thus paving the way for significant advancements in engineering novel complex interconnected systems. 

Systematic enumeration and identification of unique 3D spatial topologies (STs) of complex engineering systems (such as automotive cooling systems, electric power trains, satellites, and aero-engines) are essential to navigation of these expansive design spaces with the goal of identifying new spatial configurations that can satisfy challenging system requirements. However, efficient navigation through discrete 3D ST options is a very challenging problem due to its combinatorial nature and can quickly exceed human cognitive abilities at even moderate complexity levels. This article presents a new, efficient, and scalable design framework that leverages mathematical spatial graph theory to represent, enumerate, and identify distinctive 3D topological classes for a generic 3D engineering system, given its system architecture (SA)—its components and their interconnections. First, spatial graph diagrams (SGDs) are generated for a given SA from zero to a specified maximum number of interconnect crossings. Then, corresponding Yamada polynomials for all the planar SGDs are generated. SGDs are categorized into topological classes, each of which shares a unique Yamada polynomial. Finally, within each topological class, 3D geometric models are generated using the SGDs having different numbers of interconnect crossings. Selected case studies are presented to illustrate the different features of our proposed framework, including an industrial engineering design application: ST enumeration of a 3D automotive fuel cell cooling system (AFCS). Design guidelines are also provided for practicing engineers to aid the application of this framework to different types of real-world problems such as configuration design and spatial packaging optimization. 

Systematic enumeration and identification of unique 3D spatial topologies of complex engineering systems such as automotive cooling layouts, hybrid-electric power trains, and aero-engines are essential to search their exhaustive design spaces to identify spatial topologies that can satisfy challenging system requirements. However, efficient navigation through discrete 3D spatial topology options is a very challenging problem due to its combinatorial nature and can quickly exceed human cognitive abilities at even moderate complexity levels. Here we present a new, efficient, and generic design framework that utilizes mathematical spatial graph theory to represent, enumerate, and identify distinctive 3D topological classes for an abstract engineering system, given its system architecture (SA) — its components and interconnections. Spatial graph diagrams (SGDs) are generated for a given SA from zero to a specified maximum crossing number. Corresponding Yamada polynomials for all the planar SGDs are then generated. SGDs are categorized into topological classes, each of which shares a unique Yamada polynomial. Finally, for each topological class, one 3D geometric model is generated for an SGD with the fewest interconnect crossings. Several case studies are shown to illustrate the different features of our proposed framework. Design guidelines are also provided for practicing engineers to aid the utilization of this framework for application to different types of real-world problems.