Rational Geometric Splines for Isogeometric Analysis

现代工程应用，如飞机和汽车零部件的计算机辅助设计和制造，需要一种可以与所谓的非结构化网格一起使用的表面表示方法，而非结构化网格通常是在数字计算机上建模复杂几何形状所必需的。这个研究项目将解决最近引入的用于逼近偏微分方程解的参数曲面的理论问题。这项研究涉及对许多行业至关重要的领域，如飞机和汽车制造业以及其他行业，它有可能为这些行业带来显著的成本节约。虽然拟议的研究将主要在工程应用的背景下进行，但计算机辅助设计和有限元分析领域涉及许多其他学科，包括医学、生物学、艺术、建筑学、科学可视化以及原油和天然气勘探。该项目的结果将推动科学发现，这可能有助于提高美国工业的竞争力。该项目将解决最近引入的参数曲面的理论问题，称为碎屑-有理几何样条线，以及它们在偏微分方程解的等距分析中的应用。RAGS是分段有理函数，可用于从非结构化网格建模任意拓扑的曲面。该项目将研究RAGS在表示有限元空间方面的适用性，以及它们在数值逼近微分方程解方面的有效性。特别是，拟议项目的一个目标将是确定RAGS是否能够成功地与等距分析中使用的当前技术竞争和/或补充，即NURBS(非均匀有理B-样条线)及其推广T-Splines。目标是开发比NURBS更通用和更健壮的等轴测分析工具。PI和合作者将开发和测试基于样条线的曲面和相关的有限元空间的新方法，这些方法在数学上是合理的，在计算上容易处理。该项目还将探讨如何实现从经典近似理论到工业计算机辅助设计和工程的各学科之间的成功协同，以促进对几何模型及其分析的理解。这项研究还将有一个重要的教育组成部分：研究生将接触到有趣的、相关的跨学科性质的问题。

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.