Date of Completion
Analytic Methods, Geometric Modeling, Computer-Aided Design, Design and Manuficaturing, Measure Theory, Convolution Algebra, Harmonic Analysis, Dirac Delta Calculus
Horea T. Ilies
Donald R. Sheehy
Julian A. Norato
Field of Study
Doctor of Philosophy
Current practice in mechanical product description and computational processing is rooted in the theory of solid modeling whose principles were established almost four decades ago. Despite providing a concrete foundation for computerizing the product development process at the time, the limited capacity of the traditional models to answer the growing design and manufacturing needs is only recently being recognized in the scientific and engineering communities.
I propose an alternative approach that employs spatial functions (i.e., 3D signals) instead of (or in addition to) topological pointsets in the Euclidean 3-space to abstract physical objects and their various geometric, physical, and material properties. The central benefit of this new approach is its great promise and potential extensibility in response to the emerging needs for more meaningful form-function correlations (e.g., for conceptual design) and more powerful tools for modeling new materials (e.g., knitted composites) and processes (e.g., additive manufacturing). This requires a paradigm shift from 'explicit' descriptions equipped with 'combinatorial' methods---e.g., combinatorial intersection test between objects modeled as r-sets, approximated by sphere trees, and tested using set-theoretic operations that exploit efficient tree traversal---to 'implicit' descriptions equipped with 'analytic' methods---e.g., analytic intersection test between objects modeled as density functions, approximated by frequency domain samples, and tested using measure-theoretic operations that are streamlined via fast Fourier transforms (FFT).
The results obtained so far suggest that the proposed approach is a powerful unifying alternative to the conventional approaches to geometric computing and overcomes some of the key shortcomings of the traditional models. It opens up new promising theoretical and computational directions for future research in the nascent but emerging field of analytic solid geometry. I hope that the early-stage results in this thesis will inspire other researchers to develop more rigorous formal models and that it will eventually encourage broad industrial adoption of the analytic techniques into future generations of the product life-cycle management (PLM) software.
Behandish, Morad, "Analytic Methods for Geometric Modeling" (2017). Doctoral Dissertations. 1400.