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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=714351
• 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一系列复杂的多物理场决定了宏观现象。该研究计划将重点关注两个问题，其中简单的宏观现象是较低尺度下复杂相互作用的结果。具体来说，我们会问：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，饱和土壤如何突然失去电阻。在这两种情况下，我们将从数学角度检查微米过程，以测量这些本质上不稳定的过程如何产生明确且稳定的宏观量。我们的重点将是系统的稳定性，以及较低尺度的不确定性如何相互抵消或相互配合，以促进宏观尺度的数量。 高温超导体是新一代超导体，可以用液氮冷却，而不是昂贵且难以管理的液氦。它们的生产成本已大幅下降，但其对 MRI 机器、磁悬浮列车、甚至变压器和故障限流器等高功率设备等应用带来革命性的承诺尚未实现，因为数值模型的计算量太大实现迭代和自适应设计的成本高昂。估计交流负载期间产生的交流损耗非常重要，因为这些损耗通常是必须最小化的主要设计参数。不幸的是，交流损耗与涡流柱、外部电场和超导体内的异质性之间的微妙相互作用有关。 尽管我们在由毫米颗粒形成的压实土壤上行走和建造，但总的来说，它们作为固体，可以承受令人难以置信的压力。然而，当某些土壤被水饱和时，微小的振动可能会导致孔隙之间的流体压力增加，从而将颗粒推开。这种现象称为液化，是圣劳伦斯山谷许多灾难性事件的根源，部分原因是该地区含有大量莱达粘土沉积物，很容易发生这种现象。我们打算开发一种新颖的颗粒组装随机模型，这将有助于分离出影响土壤宏观特性的颗粒几何形状的特性。这个新模型还将引入新方法来测量这些土壤对扰动的稳定性，特别是这种异质组件固有的不确定性。