Rational Geometric Splines for Isogeometric Analysis

现代工程应用，如飞机和汽车零件的计算机辅助设计和制造，需要一种表面表示方法，可以与所谓的非结构化网格一起使用，这通常需要在数字计算机上对复杂的几何形状进行建模。本研究计画将针对最近引进的参数曲面来近似解偏微分方程式的理论问题。该研究涉及对许多行业至关重要的领域，例如飞机和汽车制造业等，并有可能为这些行业节省大量成本。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跨越许多其他学科，包括医学，生物学，艺术，建筑，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的竞争力。基础产业。该项目将解决有关最近推出的参数曲面的理论问题，称为RAGS -有理几何样条，以及它们在等几何分析中用于近似偏微分方程的解的效用。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表面。该项目将研究RAGS在表示有限元空间中的适用性及其在数值上近似微分方程解的有效性。特别是，拟议项目的一个目标将是确定RAGS是否可以成功地竞争和/或补充目前使用的技术在等几何分析，即NURBS（非均匀有理B样条）和他们的泛化，T样条。目标是开发比NURBS更通用和更强大的等几何分析工具。PI和合作者将开发和测试基于样条的曲面和相关有限元空间的新方法，这些方法在数学上是合理的，在计算上是易于处理的。该项目还将探讨如何成功地实现从经典近似理论到工业计算机辅助设计和工程的学科协同作用，以促进对几何模型及其分析的理解。研究也将有一个重要的教育组成部分：研究生将接触到有趣的和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.