Generalization of Non-Uniform Rational Bezier Splines: Theory and Applications

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Computer models today can accurately capture a wide variety shapes, ranging from aerodynamic wings, to automotive components. This versatility stems from a particular modeling technique called non-uniform rational Bezier-splines (i.e., NURBS). Indeed, NURBS are the most popular curve and surface modeling technique today, and is the de facto standard in computer aided design. However, an inherent short-coming of NURBS is its inability to capture sharp discontinuities. This shortcoming reveals itself in several applications including manufacturing planning, failure analysis, fluid flow modeling, etc. This project will address this long-standing deficiency by pursuing a generalization of NURBS, and establishing its effectiveness through targeted applications. The project will have a wide societal impact by allowing engineers to create and simulate much more complex models than possible today. The research team will disseminate the research findings and software to the scientific community. Teaching modules will be created to excite young students about computer modeling, scientific computing, and engineering. These modules will target K-12 students, including underrepresented minority students.The project will lead to a novel generalization of NURBS that is obtained by decoupling the basis functions in NURBS along different directions. This simple, yet unexplored, concept can help resolve many of the fundamental challenges in NURBS, specifically, its inability to capture discontinuities. The team will first establish the theoretical foundations of the generalization. Next, using the decoupled weights as extra design variables, new applications in modeling of "as additively manufactured" parts, and iso-geometric analysis, will be pursued. To accelerate dissemination of knowledge, the team will also develop a tool-kit to demonstrate the potential benefits, and highlight possible drawbacks, of the generalization.