I-TANGO: Intersections --- Topology, Accuracy and Numerics for Geometric Objects (in Computer Aided Design)

研究人员将在计算机辅助设计（CAD）几何对象中定义相交样条曲面的新表示。他们还将创建算法来计算表示拓扑边界的近似交叉点的误差界限。许多几何设计的建模技术依赖于可靠的相交算法。交叉算法将被设计为提供基于样条的CAD几何模型，可以在各种工程计算中健壮地使用。这些交叉表示和误差界限将显著提高工程设计与应力分析、流体动力学和电磁学计算应用之间的鲁棒性。没有一个单一的交集表示或方法将适用于所有的交集问题，类似于计算线性代数相对于矩阵反演的成熟观点。在这里，首先分析一个矩阵是否奇异、是否病态（如希尔伯特矩阵）、是否正定等。对于这些近似的样条相交，将开发新的保证误差边界，以及在基于样条的几何模型中形成的相交边界的语义一致性的新标准。例如，该项目对近似样条模型的重大改进将为科学、工业和国防提供明显的好处。波音777飞机的交付被大众媒体誉为首架“无纸化”飞机，这意味着计算机模型取代了工程设计室的传统纸质图纸。虽然这是一个令人印象深刻的里程碑，但下一个雄心勃勃的步骤是在飞机发布之前消除许多用于破坏性测试和工程分析的物理原型。这些物理原型的测试是昂贵的。一种吸引人的、经济有效的替代方法是在这些飞机的计算机几何模型上可靠地模拟湍流、应力和应变的复杂行为。实现这一目标的进展一直被这些模拟的几何输入引入的不准确性所推迟，特别是沿着交叉路口的边界，而改进这种交叉路口是本项目的主要重点。该项目将通过波音科学家和工程师的参与确保与这些当代航空设计问题的相关性。

DMS-0138098Thomas J. PetersThe investigators will define new representations for intersecting spline surfaces within computer-aided design (CAD) geometric objects. They will also create algorithms to compute error bounds for approximated intersections representing topological boundaries. Many modelingtechniques for geometric design depend upon reliable intersection algorithms. Intersection algorithms will be designed to deliver spline-based CAD geometric models that can be robustly used within a variety of engineering calculations. These intersection representations and error bounds will significantly improve robustness between engineering design and computational applications for stress analysis, fluid dynamics and electromagnetics. No single intersection representation or method will be appropriate for all intersection problems, similar to the mature viewtaken in computational linear algebra relative to matrix inversion. There, a matrix is first analyzed as to whether it is singular, ill-conditioned (as in the case of a Hilbert matrix), positive-definite, etc. For these approximated spline intersections, new guaranteed error bounds will be developed, as well as corresponding new criteria forsemantic consistency of the intersection boundaries formed within spline-based geometric models.For example, the significant improvements to approximated spline models from this project will provide clear benefits to science, industry, and defense. The delivery of the Boeing 777 airplane has been hailed in the popular press as the first production of a 'paperless' airplane, meaning that computerized models supplanted the traditional paperdrawings of the engineering design room. While an impressivemilestone, the next ambitious step is to eliminate many of the physical prototypes used for destructive testingand engineering analyses prior to releasing an airplane. These tests on physical prototypes are expensive. An appealing, cost-effective alternative is to reliably simulate the complex behavior of turbulence, stress and strain on computerized geometric models of these aircraft. Progress towards that goal has been delayed by inaccuracies introduced in the geometric input to these simulations, specifically along intersection boundaries, and improvement of such intersections is the primary focus of this project. This project will ensure relevance to these contemporary aeronautical design problems by the participation ofBoeing scientists and engineers.