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

DMS-0138098托马斯·J·彼得斯研究人员将为计算机辅助设计(CAD)几何对象中相交的样条线曲面定义新的表示法。他们还将创建算法来计算表示拓扑边界的近似交点的误差界。许多用于几何设计的建模技术依赖于可靠的交集算法。相交算法将被设计成提供基于样条线的CAD几何模型，这些模型可以在各种工程计算中稳健地使用。这些交集表示和误差界将显著提高工程设计和应力分析、流体力学和电磁学计算应用程序之间的稳健性。没有一种单一的交集表示或方法适用于所有的交集问题，类似于计算线性代数中关于矩阵求逆的成熟观点。首先分析矩阵是否奇异、病态(如Hilbert矩阵)、正定等。对于这些逼近的样条线交点，将得到新的保证误差界，以及相应的基于样条线几何模型中形成的交点边界语义一致性的新准则。例如，本项目对近似样条线模型的重大改进将为科学、工业和国防带来明显的好处。波音777飞机的交付在大众媒体上被誉为首个无纸飞机的生产，这意味着计算机化的模型取代了工程设计室传统的纸质图纸。虽然这是一个令人印象深刻的步骤，但下一个雄心勃勃的步骤是在放飞飞机之前淘汰许多用于破坏性测试和工程分析的物理原型。这些对物理原型的测试成本很高。一个吸引人的、经济高效的替代方案是可靠地模拟这些飞机的计算机化几何模型上的湍流、应力和应变的复杂行为。这一目标的进展由于这些模拟的几何输入中引入的不准确而被推迟，特别是在交叉点边界上，而改善此类交叉点是本项目的主要重点。该项目将通过波音公司的科学家和工程师的参与，确保与这些当代航空设计问题相关。

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.