The modelling, analysis and optimization of natural processes characterized by a confluence of partial differential equations of differing type

以不同类型偏微分方程汇合为特征的自然过程的建模、分析和优化

• 批准号: RGPIN-2014-04148
• 负责人: Bohun, Christopher
• 金额: $ 0.8万
• 依托单位: University of Ontario Institute of Technology
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=587630
• 关键词: modelling; analysis; optimization; natural; processes

Many modern physical processes in the natural sciences and engineering fields are the result of the unique set of circumstances that occurs when materials with differing properties and behaviours are brought into contact, and the resulting boundary is allowed to evolve in time. The primary objective of this program of research is to develop an understanding of the mathematical structure that underpins this class of problems in the effort to either impart some control over the associated processes, or achieve some form of optimality. Although the specifics of the underlying mechanisms depend strongly on the particular process, the evolution within a particular region is typically modelled with a partial differential equation (pde). What distinguishes the mathematical structure of interest of this proposal is that different regions are associated with different types of pdes, invalidating much of the classical theory. In other words, what characterizes the structure of these problems, known as mixed-type, is a decomposition into disjoint regions characterized by the type of pde (parabolic, hyperbolic, or elliptic), with different types in different regions. This contrasts sharply with more standard free boundary problems in solidification/melting processes where, for example, two parabolic equations are separated by a solidification front. There is a rich history of the separate study of pdes of each of the three distinct types (parabolic, hyperbolic, elliptic) and many mathematical tools have been developed to study the variety of solution properties. Having made these inroads, the scientific aim of this proposal is to initiate a comparable systematic attack on problems of mixed-type. Past progress in the study of these equations has been interdisciplinary combining experiments, computing and analysis, both asymptotic and rigorous. By examining the mathematical techniques, ideas and approaches to a unified set of tractable mixed-type problems, new ideas and new techniques can be identified and developed that apply to this important class that lies at the core of many innovative natural processes. The profound value of interpreting rigorous analysis within the context of particular applications has historically led to rapid advances by providing a framework to compare and contrast physical processes that are different on the surface, yet share the same underlying mathematical structure. To encourage this cross-pollination of ideas and techniques, this research will touch on many novel related processes that are only partially understood and as a result, no known necessary and sufficient conditions for controllability and optimality have been identified for them. A substantial, yet far from exhaustive list includes: i) Contrasting elastic/plastic behaviour in a growing crystal with the process of tunnelling through soil; ii) Flash sintering in which a ceramic powder becomes superplastic and densified by the passage of a large current; iii) Laser ablation and iv) laser polishing whereby a heat equation (parabolic) is intimately coupled with either a (hyperbolic) system of gas dynamics equations (ablation) or (elastic) thermoelastic equations (polishing). In many of these processes of particular interest is controllability and the optimal use of power. As evidenced by the involvement of the industries themselves, these processes of contemporary interest to Canadian industry and require a deep understanding of not only the physical processes, but also nonlinear pdes of mixed-type. While driving forward the research edge of the mathematics, the research is also important to industrial sectors that benefit from this innovation (cost savings, controllability) with progress benefiting Canada by impacting Canadian industries.

自然科学和工程领域中的许多现代物理过程是当具有不同性质和行为的材料接触时发生的独特情况的结果，并且允许由此产生的边界随时间演变。该研究计划的主要目标是了解支撑这类问题的数学结构，以便对相关过程进行控制，或实现某种形式的最优性。 尽管潜在机制的细节强烈依赖于特定的过程，但特定区域内的演化通常用偏微分方程（pde）建模。这个提议的数学结构的区别在于，不同的区域与不同类型的偏微分方程相关联，这使得许多经典理论失效。换句话说，这些问题的结构特征，被称为混合型，是分解成不相交的区域，其特征是偏微分方程的类型（抛物型，双曲型或椭圆型），在不同的区域有不同的类型。这与凝固/熔化过程中的更标准的自由边界问题形成鲜明对比，例如，两个抛物方程被凝固前沿分开。 对三种不同类型的偏微分方程（抛物型、双曲型和椭圆型）的单独研究有着丰富的历史，并且已经开发了许多数学工具来研究解的各种性质。在取得这些进展之后，这个建议的科学目标是对混合型问题发起一个类似的系统攻击。过去的进展，在这些方程的研究一直是跨学科的结合实验，计算和分析，渐近和严格。通过研究数学技术，思想和方法，以统一的一组易于处理的混合型问题，新的想法和新技术可以被识别和开发，适用于这个重要的类，位于许多创新的自然过程的核心。 在特定应用的背景下解释严格分析的深刻价值，通过提供一个框架来比较和对比表面上不同的物理过程，但共享相同的底层数学结构，从而在历史上导致了快速的进步。为了鼓励这种思想和技术的交叉授粉，本研究将涉及许多新的相关过程，这些过程仅被部分理解，因此，没有已知的可控性和最优性的必要和充分条件。一个实质性的，但远不是详尽的清单包括：i）对比弹性/塑性行为的增长晶体与隧道通过土壤的过程; ii）闪速烧结，其中陶瓷粉末成为超塑性和致密的通过一个大电流; iii）激光烧蚀和iv）激光抛光，由此热方程（抛物线）与气体动力学方程（烧蚀）或（弹性）热弹性方程（抛光）的（双曲线）系统紧密耦合。在许多这些过程中，特别感兴趣的是可控性和最佳使用功率。正如工业本身的参与所证明的那样，这些过程对加拿大工业具有当代意义，不仅需要深入了解物理过程，还需要深入了解混合型非线性偏微分方程。 在推动数学研究前沿的同时，这项研究对受益于这项创新的工业部门也很重要（成本节约，可控性），其进展通过影响加拿大工业而使加拿大受益。