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

在流体流动的数值模拟中，就像在实验中一样，我们经常质疑结果的准确性，我们构建误差条来反映数值解的准确性。然而，在许多情况下，由于模拟流动的物理参数、几何形状和操作条件不精确，存在更大的误差。随着计算流体力学领域达到一定程度的成熟，我们自然会提出更普遍的问题，即如何用数学方法对不确定性和随机输入进行建模，以及如何开发新的算法，从而产生准确反映不确定性传播的模拟结果。为此，可以采用蒙特卡罗方法，但它的计算成本很高，只能作为最后的手段使用。在这项拨款中，我们开发了一种类似于高阶有限元方法的新方法，但不是分解随机域。具体来说，我们扩展了Norbert Wiener在广义傅里叶级数中的开创性思想——所谓的多项式混沌展开——并将其局部应用于每个随机元素。所得到的控制方程系统形成了一组耦合的修正流动方程，这些方程是确定的，因此可以用标准数值方法求解。与蒙特卡罗方法的比较表明，新方法的速度平均快100到1000倍。我们建议系统地记录这种方法，并将其用于详细研究高速血流和人体动脉树血流建模中的重要问题。所提出的方法将从根本上影响我们设计新实验的方式和我们可以解决的问题类型，而模拟和实验之间的相互作用将变得更有意义和更动态。反过来，这将在流体系统设备的设计中找到它的方式，并将提供一个严格的可靠性框架。我们计划让研究生和本科生参与目前的研究，我们将制定一项具体的倡议，以吸引大学预科女生学习数学和计算科学。

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.