Integrating Topology and Numerics at CAD Interfaces

计算机辅助设计（CAD）中的一个典型几何模型被表示为一个有限的紧致2-流形集合，这些流形沿着它们的交集所形成的边界连接。 因此，这些表示严重依赖于交集算法。 为了最小化数据量并提高算法性能，仅近似交集。 这些近似误差的后果可能远非微不足道，并且可能导致被理解为相邻的表面之间的“间隙”或“裂缝”。 不幸的是，普遍的做法是，相交算法不给这些近似的任何误差范围。 本研究的重点是一个模型的拓扑结构和数值算法内的近似之间的相互依赖性的中心问题。 任何形式上的拓扑概念对于几何集合的计算表示都必须明确地承认近似和有限算术的作用。 这个项目将为所需的近似边界开发正式的定义和计算表示。 将进行计算实验，以测试跨编程接口的关键近似信息的通信。 预期的结果是一个正式的推理框架内的大型软件system.Three最近的研讨会（OPAAL '97在新奥尔良，CAD/CFD '99在戴维斯和MSRI '99在伯克利）的拓扑公差的传播已经注意到需要严格的数学基础的计算机辅助设计。 特别麻烦的是表面应该满足的区域，因为计算模型仅限于近似这种表面连接。 咖啡杯的计算模型必然会有小裂缝沿着其底部和手柄附近。 如果在计算机模拟中将流体放置在这个杯子中，流体将从杯子中泄漏。 当然，在飞机的计算模型中也会存在类似的裂缝，例如，机翼与机身的连接处。 这种接头在飞行过程中的性能特性对飞机安全至关重要。 在飞机模型上进行流体动力学模拟，以验证关键的工程性能特征。 即使几何模型中的小误差也会导致这些工程模拟中的大误差。 这些模拟通过允许直接从电子模型创建可靠的飞机而具有显著节省成本的前景。 然而，为了实现这一目标，建模和仿真之间的接口必须接受建模“裂缝”的不可避免性，并仍然产生有效的工程分析。 该项目将创建新的数学公式和计算表示，作为可靠工程模拟的必要步骤。 NIST最近的经济研究表明，这些工程问题每年花费约10亿美元，仅对汽车工业来说。 因此，这一新理论有可能产生重大的实际经济影响，因为它将改善必须在高性能计算环境中执行的复杂工程模拟的结果。 这些高性能计算资源的利用率将得到提高，更可靠的工程估算也将直接节省主要行业的材料和制造成本。

9985802A typical geometric model for Computer-Aided Design (CAD) is represented as a finite collection of compact 2-manifolds, joinedalong boundaries formed by their intersection sets. Hence, these representations critically depend upon intersection algorithms. To minimize data volume and improve algorithmic performance, the intersection sets are only approximated. The consequences of these approximation errors may be far from trivial and can lead to `gaps' or `cracks' between surfaces that are understood to be adjoining. Unfortunately, the prevalent practice is that intersection algorithms do not give any error bounds on these approximations. This research focuses upon the central issue of the interdependencies between topology of a model and the approximations within numerical algorithms. Any formal notion of topology for computational representations of geometric sets must specifically acknowledge the role of approximation and finite arithmetic. This project will develop formal definitions and computational representations for needed approximation bounds. Computational experiments will be performed to test the communication of the critical approximation information across programmatic interfaces. The expected result is a formal framework for reasoning about the propagation of topological tolerances within large software systems.Three recent workshops (OPAAL '97 in New Orleans, CAD/CFD '99 at Davis, and MSRI '99 at Berkeley) have directed attention to the need for rigorous mathematical foundations for computer-aided design. Particularly troublesome are regions where surfaces should meet, as computational models are limited to only approximating such surface joinings. A computational model of a coffee cup will necessarily have small cracks along its base and near its handle. If fluid were to be placed in this cup in a computer simulation, the fluid would leak out of the cup. Of course, similar cracks will exist in a computational model of an airplane; for example, where the wing joins the fuselage. The performance characteristics of this joint during flight are crucial to aircraft safety. Fluid dynamic simulations are performed on airplane models to verify critical engineering performance characteristics. Even small errors in the geometric model can cause large errors in these engineering simulations. These simulations hold the promise of significant cost savings by permitting creation of a reliable airplane directly from the electronic model. However, to attain that goal, the interface between modeling and simulation must accept the inevitability of modeling `cracks' and still produce valid engineering analyses. This project will create new mathematical formulations and computational representations as a necessary step for reliable engineering simulations. Recent economic studies done for NIST indicate that these engineering problems cost about 1 billion dollars annually, for the automotive industry alone. Hence, there is opportunity for significant practical economic impact from this new theory, as it will improve results from complex engineering simulations which must be executed within high-performance computing environments. Utilization of these high performance computing resources will be improved and the more reliable engineering estimates will also lead to direct savings in material and manufacturing costs for major industries.