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，在不同长度尺度上发生的复杂多物理级联决定了宏观现象。这项研究计划将集中在两个问题上，其中简单的宏观现象是较低尺度上复杂相互作用的结果。具体地说，我们将问1)超导体内部的交流损耗是如何由表面附近涡柱的微米相互作用引起的，以及2)当微小的振动导致孔压增加时，饱和土壤如何突然失去阻力。在这两种情况下，我们将从数学的角度研究微米过程，以测量这些内在不稳定的过程如何导致定义明确和稳定的宏观量度。我们的重点将放在系统的稳定性特性上，以及低尺度上的不确定性如何相互否定，或相互协作，以贡献宏观尺度的量。*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。它们的生产成本一直在大幅下降，但它们给核磁共振机、磁悬浮列车甚至变压器和故障电流限制器等高功率设备带来革命性应用的承诺尚未实现，因为数值模型的计算成本太高，无法进行迭代和自适应设计。估计交流负载期间产生的交流损耗很重要，因为这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、外部电场和超导体内部的不均一性之间的微妙相互作用有关。*虽然我们在由毫米颗粒组成的压实土壤上行走和建筑，但总的来说，它们像固体一样，可以承受难以置信的压力。然而，当某些土壤变得水饱和时，微小的振动可能会导致孔隙之间流体压力的增加，从而将颗粒推开。这种现象被称为液化，是圣劳伦斯山谷许多灾难性事件的根源，部分原因是该地区蕴藏着大量的勒达粘土，很容易发生这种现象。我们打算开发一种新的颗粒聚集的随机模型，该模型将有助于分离导致土壤宏观尺度特性的颗粒几何特性。这个新的模型还将引入新的方法来测量这些土壤对扰动的稳定性，特别是这种不均匀组合中固有的不确定性。