Uncertainty Quantification in PDE-Based Modelling and Optimization

在许多当代工程挑战中，发生在不同长度尺度上的复杂多物理场级联决定了宏观现象。本研究计划将集中于两个问题，其中简单的宏观现象是在较低尺度上复杂相互作用的结果。具体来说，我们将问1)超导体内部的交流损耗是如何由表面附近的涡旋柱的微米级相互作用引起的，以及2)当小振动导致孔隙压力增加时，饱和土壤是如何突然失去阻力的。在这两种情况下，我们将从数学角度研究微米过程，以测量这些固有不稳定的过程如何导致定义良好且稳定的宏观量。我们的重点将放在系统的稳定性特性上，以及低尺度的不确定性如何相互抵消或相互合作，从而对宏观量做出贡献。***高温超导体是可以用液氮冷却的新一代超导体，而不是昂贵且难以管理的液氦。它们的生产成本已经大幅下降，但它们在核磁共振成像仪、磁悬浮列车、甚至变压器和故障电流限制器等大功率设备等革命性应用方面的承诺尚未实现，因为数值模型的计算成本太高，无法实现迭代和自适应设计。估计交流负载期间产生的交流损耗是很重要的，因为这些通常是必须最小化的主要设计参数。不幸的是，交流损耗与涡旋柱、外电场和超导体内部的非均质性之间的微妙相互作用有关。虽然我们行走和建筑在由毫米颗粒组成的夯实土壤上，但总的来说，它们就像固体一样，可以承受难以置信的压力。然而，当某些土壤达到水饱和状态时，微小的振动可能会导致孔隙之间流体压力的增加，从而使颗粒分开。这种现象被称为液化，它是圣劳伦斯山谷许多灾难性事件的根源，部分原因是该地区含有大量易受这种现象影响的勒达粘土沉积物。我们打算为颗粒组合开发一种新的随机模型，这将有助于分离颗粒几何形状的特性，这些特性负责土壤的宏观尺度特性。这个新模型还将引入新的方法来测量这些土壤在扰动下的稳定性，特别是在这种异质组合中固有的不确定性。

In many of the contemporary engineering challenges, a cascade of complex multiphysics occurring at different length scales determines macroscopic phenomena. This research program will focus on two problems where simple macroscopic phenomena are the result of complex interactions at the lower scales. Specifically we will ask 1) how AC losses inside superconductors are caused by the micrometer interaction of vortex columns near the surface, and 2) how saturated soils may suddenly lose their resistance when small vibrations lead to an increase in pore pressure. In both cases, we will be examining the micrometer processes for a mathematical perspective in order to measure how those inherently unstable processes lead to well-defined and stable macroscale quantities. Our focus will be on the stability properties of the systems and how uncertainties at the lower scales either negate each other, or cooperate with each other, to contribute to macroscale quantities. ***High-temperature superconductors are a new generation of superconductors that can be cooled with liquid nitrogen, rather than expensive and difficult to manage liquid helium. Their production costs have been decreasing dramatically, but their promise of revolutionizing applications such as MRI machines, MagLev trains, or even high-power devices such as transformers and fault-current limiters, has yet to come to fruition because the numerical models are too computationally costly to enable iterative and adaptive design. It is important to estimate the AC losses generated during AC loading because these will often be the main design parameter that must be minimized. Unfortunately, AC losses are related to subtle interactions between vortex columns, the external electric field, and heterogeneities within the superconductor. ***Although we walk and build on compacted soils formed of millimeter grains, overall they act as solids and can sustain incredible amounts of stress. Yet, when certain soils become water-saturated, small vibrations may lead to increases in the pressure of the fluid between the pores, hence pushing the grains apart. This phenomena is called liquefaction and has been at the origin of many catastrophic events in the St-Lawrence Valley, in part because the region contains large deposits of Leda clay which are susceptible to such phenomena. We intend to develop a novel stochastic model for assemblies of grains that will help to isolate the properties of grain geometries that are responsible for the macroscale properties of soils. This new model will also introduce new methods to measure the stability of these soils to perturbations, and in particular to the uncertainties inherent in such heterogeneous assemblies.