Integrating Topology and Numerics at CAD Interfaces

9985802a计算机辅助设计(CAD)的典型几何模型被表示为紧凑的二维流形的有限集合，这些流形沿着它们的交集形成的边界连接在一起。因此，这些表示在很大程度上依赖于相交算法。为了最大限度地减少数据量和提高算法性能，只对交集进行近似。这些近似误差的后果可能远远不是微不足道的，并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，普遍的做法是交集算法不会给出这些近似的任何误差界。这项研究的中心问题是模型的拓扑结构与数值算法中的近似之间的相互依赖关系。对于几何集合的计算表示，任何形式的拓扑学概念都必须明确承认近似和有限算术的作用。这个项目将为所需的近似界限开发形式定义和计算表示。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。最近的三次研讨会(新奥尔良的OPAAL‘97，戴维斯的CAD/CFD’99，以及伯克利的MSRI‘99)将注意力集中在计算机辅助设计需要严格的数学基础上。尤其麻烦的是曲面应该相交的区域，因为计算模型仅限于近似这样的曲面接合。咖啡杯的计算模型在其底部和手柄附近必然会有小裂缝。如果在计算机模拟中将液体放入这个杯子中，液体就会从杯子中泄漏出来。当然，类似的裂缝也会存在于飞机的计算模型中；例如，机翼与机身的连接处。这种关节在飞行过程中的性能特性对飞机的安全至关重要。在飞机模型上进行了流体动力学模拟，以验证关键的工程性能特性。在这些工程模拟中，即使几何模型中的小误差也会导致大误差。这些模拟允许直接从电子模型创建可靠的飞机，从而有望显著节省成本。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“裂纹”的必然性，并仍然产生有效的工程分析。这个项目将创建新的数学公式和计算表示，作为可靠的工程模拟的必要步骤。最近为NIST所做的经济研究表明，仅就汽车行业而言，这些工程问题每年就耗资约10亿美元。因此，这一新理论有机会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，更可靠的工程估计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.