Date of Award
Master of Science (MS)
Joshua D Summers
Cameron J Turner
The idea of design spaces in engineering has appeared in many forms and served a variety of purposes in practice, research, and literature. Yet very few of the definitions put forth have a concrete mathematical structure that can be practically applied by the designer in real-time. This research seeks to address this gap by taking advantage of tools and techniques in point-set topology, a field that has been used successfully in a number of different areas. The primary objective of this undertaking is to formalize definitions for design spaces as topological structures that will encapsulate many of the relevant characteristics of both the problem to be solved and the designs that are being considered.
Three separate spaces are presented: the problem space, the solution space, and the quality space. The problem space is defined by the requirements that pertain to the problem and represents the target that designs must hit to be considered a solution. The solution space is the collection of design embodiments for a given concept that meet the specified constraints. Finally, the objective space is a space that allows different design concepts to be compared to one another based on common criteria that any solution would exhibit. Along with these definitions, several methods are also proposed to operate on the design spaces to assist in their analysis and comparison. Measures are introduced for assessing the similarity of spaces as they evolve and for quantifying how sensitive solution spaces are to changes in requirements. Also, a process for gauging the relative utility of different concepts is presented. Two examples are included to demonstrate implementation, one simplistic for explanatory value and the second more complex to show scalability.
Topology has been demonstrated to be a versatile and extensible lens for data interpretation and exploration. Given this adaptability, it is hoped that this thesis will serve as a foundation upon which future work can build so that a wide array of novel capabilities can be established for engineers, designers, researchers to draw upon in their pursuits.
Ortiz, Joshua Bronson, "Applications of Set-Theoretic Topology in the Construction and Analysis of Engineering Design Spaces" (2021). All Theses. 3609.