Technical Report PHD-2019-01

Title: Trivariate Volumes: Algorithms and Applications
Authors: Fady Massarwi
Supervisors: Gershon Elber
PDFCurrently accessibly only within the Technion network
Abstract: This work investigates algorithms and data structures for volumetric representation (V-reps) of 3D objects, representing the interior of the object in addition to its boundaries, extending the contemporary Boundary representation (B-rep) common scheme. In recent years, there is a growing emerging need for a volumetric representation of 3D objects. Specifically, with the development of Iso-geometric Analysis (IGA) and advanced manufacturing technologies employing heterogeneous materials, such as 3D-printing and additive manufacturing (AM) of functionally graded material. We employ the B-spline trivariate basis functions for the V-reps.

In Chapter 1, an introduction on geometrical representation and applications of volumetric objects is provided. Some required preliminary material is briefly presented in Chapter 2. In Chapter 3, we start by proposing a volumetric representation (V-rep) for geometric modeling that is based on trimmed B-spline trivariates and introduce its supporting volumetric modeling framework. The framework includes various volumetric models (V-model) construction schemes, from basic (non-singular) volumetric primitives to high level constructors, such as volumes of revolutions, as well as Boolean operations’ support for V-models. Further, this framework is also a seamless extension to existing boundary representations (B-reps) common in all contemporary geometric modeling systems, and allows a simple migration path of existing B-rep data, tools and algorithms. Then, we propose two untrimming algorithms. An algorithm for converting trimmed B-spline surfaces into tensor products is presented in Chapter 4, and an algorithm for converting trimmed trivariates into tensor products is presented in Chapter 5. The untrimming algorithms can be utilized to simplify algorithms and applications using the proposed framework, such as the integration process for IGA. Finally, in Chapter 6, we propose two algorithms for modeling volumetric microstructures using functional composition over V-reps. The first algorithm generates random microstructures with connectivity and smoothness guarantees, and the second algorithm can be used to construct micro-structures with bifurcations, that compensates for the non-isometric behavior of the V-rep trivariate. Chapter 7 concludes the work with some discussion on possible future work and extensions.

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