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

今天的计算机模型可以准确地捕捉到各种各样的形状，从空气动力机翼到汽车部件。这种多功能性源于一种称为非均匀有理Bezier样条线(即NURBS)的特定建模技术。事实上，NURBS是当今最流行的曲线和曲面建模技术，也是计算机辅助设计的事实上的标准。但是，NURBS的一个固有缺陷是它无法捕获尖锐的不连续。这一缺陷在制造计划、故障分析、流体流动建模等几个应用中暴露出来。本项目将通过追求NURBS的泛化来解决这一长期存在的缺陷，并通过有针对性的应用来建立其有效性。该项目将产生广泛的社会影响，使工程师能够创建和模拟比今天复杂得多的模型。研究小组将向科学界传播研究成果和软件。将创建教学模块，以激发年轻学生对计算机建模、科学计算和工程的兴趣。这些模块将面向K-12学生，包括未被充分代表的少数族裔学生。该项目将导致一种新的NURBS泛化，该泛化是通过将NURBS中的基函数沿不同方向解耦而获得的。这个简单但尚未探索的概念可以帮助解决NURBS中的许多基本挑战，特别是它无法捕获间断。该团队将首先建立泛化的理论基础。下一步，使用解耦后的重量作为额外的设计变量，将在“附加制造”零件的建模和等几何分析中寻求新的应用。为了加快知识的传播，该小组还将开发一套工具包，以展示推广的潜在益处，并突出可能的缺点。

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.