Applications of Graph Embedding in Mesh Untangling

The subject of this thesis is mesh untangling through graph embedding, a method of laying out graphs on a planar surface, using an algorithm based on the work of Fruchterman and Reingold[1]. Meshes are a variety of graph used to represent surfaces with a wide number of applications, particularly in simulation and modelling. In the process of simulation, simulated forces can tangle the mesh through deformation and stress. The goal of this thesis was to create a tool to untangle structured meshes of complicated shapes and surfaces, including meshes with holes or concave sides. The goals of graph embedding, such as minimizing edge crossings align very well with the objectives of mesh untangling. I have designed and tested a tool which I named MUT (Mesh Untangling Tool) on meshes of various types including triangular, polygonal, and hybrid meshes. Previous methods of mesh untangling have largely been numeric or optimizationbased. Additionally, most untangling methods produce low quality graphs which must be smoothed separately to produce good meshes. Currently graph embedding techniques have only been used for smoothing of untangled meshes. I have developed a tool based on the Fruchterman-Reingold algorithm for force-directed layout[1] that effectively untangles and smooths meshes simultaneously using graph embedding techniques. It can untangle complicated meshes with irregular polygonal frames, internal holes, and other complications that previous methods struggle with. The MUT does this by using several different approaches: untangling the mesh in stages from the frame in and anchoring the mesh at corner points to stabilize the untangling

A two-stage design framework for optimal spatial packaging of interconnected fluid-thermal systems

Optimal spatial packaging of interconnected subsystems and components with coupled physical (thermal, hydraulic, or electromagnetic) interactions, or SPI2, plays a vital role in the functionality, operation, energy usage, and life cycle of practically all engineered systems, from chips to ships to aircraft. However, the highly nonlinear spatial packaging problem, governed by coupled physical phenomena transferring energy through highly complex and diverse geometric interconnects, has largely resisted automation and quickly exceeds human cognitive abilities at moderate complexity levels. The current state-of-the-art in defining an arrangement of these functionally heterogeneous artifacts still largely relies on human intuition and manual spatial placement, limiting system sophistication and extending design timelines. Spatial packaging involves packing and routing, which are separately challenging NP-hard problems. Therefore, solving the coupled packing and routing (PR) problem simultaneously will require disruptive methods to better address pressing related challenges, such as system volume reduction, interconnect length reduction, ensuring non-intersection, and physics considerations. This dissertation presents a novel automated two-stage sequential design framework to perform simultaneous physics-based packing and routing (PR) optimization of fluid-thermal systems. In Stage 1, unique spatially-feasible topologies (i.e., how interconnects and components pass around each other) are enumerated for given fluid-thermal system architecture. It is important to guarantee a feasible initial graph as lumped-parameter physics analyses may fail if components and/or routing paths intersect. Stage 2 begins with a spatially-feasible layout, and optimizes physics-based system performance with respect to component locations, interconnect paths, and other continuous component or system variables (such as sizing or control). A bar-based design representation enables the use of a differentiable geometric projection method (GPM), where gradient-based optimization is used with finite element analysis. In addition to geometric considerations, this method supports optimization based on system behavior by including physics-based (temperature, fluid pressure, head loss, etc.) objectives and constraints. In other words, stage 1 of the framework supports systematic navigation through discrete topology options utilized as initial designs that are then individually optimized in stage 2 using a continuous gradient-based topology optimization method. Thus, both the discrete and continuous design decisions are made sequentially in this framework. The design framework is successfully demonstrated using different 2D case studies such as a hybrid unmanned aerial vehicle (UAV) system, automotive fuel cell (AFC) packaging system, and other complex multi-loop systems. The 3D problem is significantly more challenging than the 2D problem due to vastly more expansive design space and potential features. A review of state-of-the-art methods, challenges, existing gaps, and opportunities are presented for the optimal design of the 3D PR problem. Stage 1 of the framework has been investigated thoroughly for 3D systems in this dissertation. An efficient design framework to represent and enumerate 3D system spatial topologies for a given system architecture is demonstrated using braid and spatial graph theories. After enumeration, the unique spatial topologies are identified by calculating the Yamada polynomials of all the generated spatial graphs. Spatial topologies that have the same Yamada polynomial are categorized together into equivalent classes. Finally, CAD-based 3D system models are generated from these unique topology classes. These 3D models can be utilized in stage 2 as initial designs for 3D multi-physics PR optimization. Current limitations and significantly impactful future directions for this work are outlined. In summary, this novel design automation framework integrates several elements together as a foundation toward a more comprehensive solution of 3D real-world packing and routing problems with both geometric and physics considerations