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，这种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。本研究计划将探讨有关最近引入的参数曲面近似偏微分方程解的理论问题。这项研究涉及到许多行业的关键领域，如飞机和汽车制造业等，它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涉及许多其他学科，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。该项目的成果将促进科学发现，这可能有助于提高美国产业的竞争力。该项目将解决有关最近引入的参数曲面（称为RAGS -有理几何样条）的理论问题，以及它们在偏微分方程近似解的等几何分析中的应用。破布是一种分段有理函数，可用于从非结构化网格中对任意拓扑曲面建模。该项目将研究rag在表示有限元空间中的适用性及其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rag是否能够成功地与等几何分析中使用的当前技术，即NURBS（非均匀合理b样条）及其推广，t样条竞争和/或补充。目标是开发比NURBS更通用和健壮的等几何分析工具。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，在计算上是可处理的。该项目还将探索如何成功地协同学科，从经典近似理论到工业计算机辅助设计和工程，以促进对几何模型及其分析的理解。这项研究也将有一个重要的教育组成部分：研究生将接触到跨学科性质的有趣和相关问题。

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.