Theory and Applications of Coupled Mathematical Models for Environmentally-friendly Multiscale Materials Systems

环境友好型多尺度材料系统耦合数学模型理论与应用

• 批准号: RGPIN-2020-06958
• 负责人: Melnik, Roderick
• 金额: $ 1.97万
• 依托单位: Wilfrid Laurier University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=741513
• 关键词: Theory; Applications; Coupled; Mathematical; Models

Many current advanced technologies rely heavily on environmentally harmful components. One of such components is lead whose contents may exceed 60 weight percent in commercially available cutting-edge materials such as piezoelectrics. These materials and their composites demonstrate excellent performance in many applications where strong coupling between mechanical and electric fields is required. However, there is a strong impetus and increasingly strict environmental regulations to eliminate the environmentally harmful components without compromising the materials performance. This task is connected with a multitude of challenges, some of which require much more close attention of applied mathematicians. Building on our successful work in the development of coupled multiscale mathematical models, the proposed research program expands it to new areas, crucial for both the development of state-of-the-art mathematical theories and innovative applications. The program is aimed at further advancement in the development and applications of coupled mathematical models for the analysis of environmentally-friendly multiscale materials systems. The main goal is to account systematically for coupling and multiscale nature of the problem in this development. Application-wise, major focus will be given to several key classes of multiscale environmental-friendly materials systems such as piezoelectric composites. In the first approximation, elegantly simple mathematical concepts and formulations can be applied to some such systems, e.g., by viewing them as those consisting of two main components - a matrix, considered as a background material, and crystalline-particle-based inclusions. However, many mathematical challenges in this field still are on only scarcely explored horizons. In addressing its long-term goals, the program will, firstly, allow a systematic study of properties of such systems, accounting for nonlinear and nonlocal coupled effects, and for the polycrystalline structure of inclusions. Secondly, it will carry out mathematical and numerical analysis of the developed models with microstructures, as well as with performance-enhancing nano-additions. Thirdly, it will systematically study an important class of geometrically-tailored systems. Finally, given their ubiquitous nature, it is expected that the models and tools developed in this program will assist in addressing other challenging problems of mathematics and its applications. Indeed, the results may not be restricted to just the exemplifications given and can be useful in studying other important systems in science and engineering. The program's core includes fundamental and applied research in mathematical modelling that will lead to state-of-the-art scientific advances and will entail positive impacts on important global challenges. Training of highly qualified personnel and expanding international collaboration will be effectively integrated with the development of the program.

许多当前的先进技术在很大程度上依赖于环境有害的组件。此类组件之一是铅，其内容物在市售的尖端材料（例如压电）中可能超过60％的重量。这些材料及其复合材料在需要机械和电场之间牢固耦合的许多应用中表现出卓越的性能。但是，在不损害材料性能的情况下消除环境有害的组件，可以消除环境有害的组成部分。这项任务与许多挑战有关，其中一些需要更加关注应用数学家。拟议的研究计划以我们在耦合多尺度数学模型开发的成功开发的基础上，将其扩展到了新领域，这对于最先进的数学理论和创新应用的开发至关重要。该计划旨在进一步进步，以在耦合数学模型的开发和应用中进行分析，以分析环保的多尺度材料系统。主要目标是系统地考虑该问题中问题的耦合和多规定性质。在应用方面，将重点放在多个多尺度环境友好型材料系统（例如压电复合材料）的几个关键类别中。在第一个近似值中，可以将优雅的简单数学概念和公式应用于某些这样的系统，例如，将它们视为由两个主要组件组成的系统（被视为背景材料）和基于晶体粒子的夹杂物的矩阵。但是，该领域的许多数学挑战仍然几乎没有探索地平线。在解决其长期目标时，该计划将首先允许对此类系统的性质进行系统的研究，计算非线性和非局部耦合效应，以及夹杂物的多晶结构。其次，它将对带有微观结构的开发模型以及增强性能的纳米辅助进行数学和数值分析。第三，它将系统地研究一类几何定制的系统。最后，鉴于其无处不在的性质，预计该计划中开发的模型和工具将有助于解决其他具有挑战性的数学问题及其应用。实际上，结果可能不仅限于给出的例证，并且对于研究科学和工程中的其他重要系统可能是有用的。该计划的核心包括数学建模中的基本和应用研究，这将导致最先进的科学进步，并将对重要的全球挑战产生积极影响。对高素质的人员和扩大国际合作的培训将有效地与该计划的发展融合。