Modern engineering applications, such as computer-aided design and manufacturing of aircraft and auto parts, demand a surface representation methodology that can be used with so-called unstructured meshes, which are typically required for modeling of complex geometric shapes on a digital computer. This research project will address theoretical questions concerning recently introduced parametric surfaces for approximating solutions of partial differential equations. The research involves areas critical to many industries, such as the aircraft and car manufacturing industries and others, and it has the potential to lead to significant cost savings for these industries. Although the proposed research will be conducted mainly in the context of engineering applications, the fields of computer-aided design and finite element analysis span many other disciplines, including medicine, biology, art, architecture, scientific visualization, and crude oil and natural gas exploration. The results of this project will advance scientific discovery, which may contribute to increased competitiveness of U.S.-based industries.
The project will address theoretical questions concerning recently introduced parametric surfaces, called RAGS - Rational Geometric Splines, and their utility in Isogeometric Analysis for approximating solutions of partial differential equations. RAGS are piecewise rational functions that can be used to model surfaces of arbitrary topology from unstructured meshes. The project will investigate the applicability of RAGS in representing finite-element spaces and their effectiveness in approximating solutions of differential equations numerically. In particular, one goal of the proposed project will be to determine whether RAGS can successfully compete with and/or complement the current technology used in Isogeometric Analysis, namely NURBS (Non Uniform Rational B-splines) and their generalization, T-splines. The objective is to develop tools for Isogeometric Analysis that are more versatile and robust than NURBS. The PI and collaborators will develop and test new methods for spline-based surfaces and the associated finite-element spaces that are mathematically sound and computationally tractable. The project will also explore how a successful synergy of disciplines, ranging from the classical approximation theory to industrial computer-aided design and engineering, can be achieved to advance the understanding of geometric models and their analysis. The research will also have an important educational component: graduate students will be exposed to interesting and relevant problems of interdisciplinary nature.
