Uncertainty Quantification in PDE-Based Modelling and Optimization

基于 PDE 的建模和优化中的不确定性量化

• 批准号: RGPIN-2018-06592
• 负责人: Laforest, Marc
• 金额: $ 1.31万
• 依托单位: École Polytechnique de Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=676967
• 关键词: Uncertainty; Quantification; PDE; Based; Modelling

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.

在许多当代工程挑战中，发生在不同长度尺度上的一系列复杂多物理学决定了宏观现象。该研究计划将重点放在两个问题上，其中简单的宏观现象是较低尺度上复杂相互作用的结果。具体而言，我们将问1）1）超导体内部的交流损失是如何由表面附近涡流柱的千分尺相互作用引起的，2）当小振动导致孔隙压力增加时，饱和土壤可能会突然失去抵抗力。在这两种情况下，我们都将检查数学观点的千分尺过程，以衡量那些固有的不稳定过程如何导致定义明确且稳定的宏观计量。我们的重点将放在系统的稳定性上，以及在较低尺度上的不确定性如何彼此否定，或者相互合作，以促进宏观量的数量。 ***高温超导体是新一代的超导体，可以用液氮冷却，而不是昂贵且难以管理液态氦气。他们的生产成本一直在急剧下降，但是他们对革新应用程序（例如MRI机器，Maglev火车甚至高功率设备（例如变形金刚和故障电流限制器））的承诺尚未实现，因为数值模型在计算上太计算了昂贵的迭代和适应性设计。重要的是要估计AC负载期间产生的交流损耗，因为这些通常是必须最小化的主要设计参数。不幸的是，AC损失与超导体内部涡流柱，外部电场和异质性之间的微妙相互作用有关。 ***尽管我们走路并建立在由毫米谷物形成的压实土壤上，但总体而言，它们充当固体，可以承受令人难以置信的压力。然而，当某些土壤变得饱和水时，小振动可能会导致毛孔之间的液体压力增加，从而将谷物隔开。这种现象被称为液化，在St-Lavrence山谷中许多灾难性事件的起源，部分原因是该地区包含大量的Leda粘土沉积物，这些粘土易受这种现象。我们打算开发一种新的随机模型，以用于晶粒组装，这将有助于分离负责土壤宏观特性的晶粒几何形状的性质。该新模型还将引入新方法，以测量这些土壤对扰动的稳定性，尤其是这种异质组装中固有的不确定性。