AMC-SS: A Multi-Element Generalized Polynomial Chaos Method for Modeling Uncertainty in Flow Simulations

AMC-SS：一种用于流体仿真中不确定性建模的多元素广义多项式混沌方法

• 批准号: 0510799
• 负责人: George Karniadakis
• 金额: $ 31.18万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-09-01 至 2010-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0510799&HistoricalAwards=false
• 关键词: AMC; SS; Multi; Element; Generalized

In numerical simulations of fluid flows, just as in experiments, we often question the accuracy of the results, and we construct error bars that reflect the numerical accuracy of the solution. In many cases, however, there exists a much larger error associated with the fact that the physical parameters, the geometry, and the operating conditions of the simulated flow are not precisely known. With the computational fluid dynamics field reaching now some degree of maturity, we naturally pose the more general question of how to model uncertainty and stochastic input mathematically, and how to develop new algorithms that will yield simulation results that reflect accurately the propagation of uncertainty. To this end, the Monte Carlo approach can be employed but it is computationally expensive and it is only used as the last resort.In this grant we develop a new approach similar to high-order finite element methods but instead decomposing the random domain. Specifically, we extend the pioneering ideas of Norbert Wiener in generalized Fourier series -- the so-called polynomial chaos expansion -- and apply it locally to each random element. The resulting system of governing equations forms a set of coupled modified flow equations, which are deterministic and thus can be solved with standard numerical methods. Comparisons with the Monte Carlo method show that the new method is faster by a factor of 100 to 1000 on the average. We propose to document systematically this method and use it to study in detail important problems in high-speed flows and in modeling blood flow in the human arterial tree. The proposed approach will affect fundamentally the way we design new experiments and the type of questions that we can address, while the interaction between simulation and experiment will become more meaningful and more dynamic. This, in turn, will find its way into the design of flow systems equipment and will provide a rigorous reliability framework.We plan to involve graduate and undergraduate students in the current research and we will develop a specific initiative to attract pre-college female students to mathematics and computational science.

在对流体流的数值模拟中，就像在实验中一样，我们经常质疑结果的准确性，并构建了反映溶液数值准确性的误差线。 但是，在许多情况下，存在一个更大的错误，即物理参数，几何形状和模拟流的工作条件尚不清楚。 随着计算流体动力学领域现在达到一定程度的成熟度，我们自然而然地提出了如何在数学上对不确定性和随机输入进行建模的更一般的问题，以及如何开发新算法将产生准确反映不确定性传播的新算法。 为此，可以采用蒙特卡洛方法，但它在计算上很昂贵，并且仅用作最后的手段。在此赠款中，我们开发了一种类似于高阶有限元方法的新方法，而是分解随机域。 具体而言，我们扩展了诺伯特·维纳（Norbert Wiener）在广义傅立叶系列中的开创性思想 - 所谓的多项式混乱扩展 - 并将其局部应用于每个随机元素。 最终的管理方程系统形成了一组耦合的修改流程方程，这些方程是确定性的，因此可以使用标准的数值方法来求解。 与蒙特卡洛方法的比较表明，新方法的平均速度更快为100至1000倍。 我们建议系统地记录这种方法，并使用它来详细研究高速流中的重要问题以及对人动脉树的血流进行建模。提出的方法将从根本上影响我们设计新实验的方式以及我们可以解决的问题的类型，而模拟与实验之间的相互作用将变得更有意义，更具动态性。 反过来，这将进入流程度设备的设计中，并将提供严格的可靠性框架。我们计划让研究生和本科生参与当前的研究，我们将开发一项特定的计划，以吸引前校学生进入数学和计算科学。