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各种各样的形状，从空气动力学机翼到汽车部件。这种多功能性源于一种称为非均匀有理Bezier样条（即，NURBS）。事实上，NURBS是当今最流行的曲线和曲面建模技术，并且是计算机辅助设计中事实上的标准。然而，NURBS的一个固有缺点是它无法捕捉尖锐的不连续性。这一缺点在包括制造规划、故障分析、流体流动建模等多个应用中显露出来。该项目将通过追求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更复杂的模型。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。 这些模块将针对K-12学生，包括代表性不足的少数民族学生。该项目将导致NURBS的一种新的推广，通过解耦NURBS沿着不同方向的基函数获得。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，特别是无法捕捉不连续性。该团队将首先建立推广的理论基础。接下来，使用解耦的权重作为额外的设计变量，将追求“增材制造”零件建模和等几何分析中的新应用。为了加速知识的传播，该小组还将开发一个工具包，以展示推广的潜在好处，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.