Analysis of Atomistic-to-Continuum Coupling Methods

在开发新的工程材料时，通常的目标是制造具有规定的力学性能(弹性模数、韧性等)的材料。长期以来，人们都知道，只有在纳米和微观尺度上了解和控制材料的细节(晶格结构、缺陷分布、微结构等)，才能实现这种“宏观”性能的设计。理解微观效应产生的宏观性质的愿望导致了多尺度建模领域的发展，这是一个充满活力的跨学科研究领域。虽然我们的首要建模工具应该是分析，但精确描述纳米级材料所需的复杂模型的定量结果只能通过数值模拟的方式获得。这项建议侧重于对材料的模拟，特别是对缺陷的模拟，其规模为数千到数百万个原子间距。尽管晶格中的许多类型的缺陷是高度局域化的，但它们受到所谓的“远场”(相对而言，远离缺陷的原子的位移)的强烈影响和强烈影响。这正是出现困难的地方。对于缺陷的模拟，应该使用精确但昂贵的原子材料模型，而对于远场的描述，连续介质模型要便宜得多，但也足够准确。因此，自然的挑战是将原子模型(例如缺陷)耦合到晶体弹性的连续模型，从而将现代纳米科学与经典的固体力学模型联系起来。准连续(QC)方法是实现这种耦合的粗粒化技术的范例，用于静态(和准静态)晶体固体的模拟。它的主要特点是，不是将原子模型耦合到连续介质模型，而是在远场区域使用原子模型，但通过有限元方法去除了自由度。然后进行额外的近似以使粗粒度问题可计算，从而导致不同类别的QC方法。几乎所有耦合内在不同物理模型的方法(包括QC方法)共享的一个影响是界面区域的力平衡有缺陷。主要的研究重点是了解并从模拟中消除这一错误。目前，严谨的分析已经很好地理解了这一问题中的一维模型问题。尽管这些结果是初步的，但从这项数学研究中可以得出一些重要的结论：(I)表明在实践中使用的某些方法是非常不准确的，应该被丢弃；(Ii)至少发现了关于准确处理界面区域的两个令人兴奋的新想法：基于力的QC方法和‘几何一致的QC方法’。目前的主要挑战和本研究提案的目的是将这种分析工作推广到实际相关的二维和三维情况，其中几何和分析变得更加具有挑战性。为此，将开发一系列简单但具有代表性的二维和三维模型，以便通过严格的数学分析对不同风格的质量控制方法进行基准测试。此外，还将创建一个用于快速成型的实验QC软件，以验证分析预测。严格的数值分析的主要好处是确保在无法通过实验测试的情况下模拟是可靠的。此外，本研究将找出最具潜力的质量控制方法，并将为指导新的和改进的方法的发展提供新的见解。

In the development of new engineering materials, the goal is generally to manufacture materials with prescribed mechanical properties (elastic moduli, toughness, etc.). It has long been known that the design of such `macroscopic' properties can only be achieved by understanding and controlling the details of the material at the nano- and micro-scale (lattice structure, defect distribution, microstructure, etc.). The desire to understand macroscopic properties arising from microscopic effects has led to the development of the field of multi-scale modelling, a vibrant interdisciplinary research area. While our first and foremost modelling tool should be analysis, quantitative results for the complex models required for an accurate description of materials at the nano-scale can only be obtained by means of numerical simulation.This proposal focuses on the simulation of materials, with particular emphasis on defects, on the scale of several thousand to several million atomic spacings. Even though many types of defects in lattices are highly localized, they are strongly affected by, and strongly influence, the so-called `far field' (the displacement of atoms which, in relative terms, are situated far from the defect). This is precisely where the difficulties arise. For the simulation of a defect, an accurate but expensive atomistic material model should be used while, for a description of the far field, continuum models are much cheaper yet sufficiently accurate. The natural challenge, therefore, is to couple atomistic models (e.g., of defects) to continuum models of crystal elasticity, thus, connecting modern nano-science with the classical models of solid mechanics.The quasicontinuum (QC) method is a paradigm example of a coarse-graining technique to achieve this coupling for static (and quasi-static) simulations of crystalline solids. Its key feature is that, instead of coupling an atomistic model to a continuum model, it also uses the atomistic model in the far field region but removes degrees of freedom by means of finite element methodology. Additional approximations are then performed to render the coarse-grained problem computable, leading to different classes of QC methods.An effect shared by virtually all methods coupling inherently different physical models, including the QC method, is a defective force balance in the interface region. The primary research focus is to understand and remove this error from the simulation. At present, rigorous analysis has provided good understanding of this issue in one-dimensional model problems. Despite the preliminary nature of these results, some significant conclusions from this mathematical research can be drawn: (i) it was shown that certain classes of methods that are used in practice are grossly inaccurate and should be discarded; (ii) at least two exciting and novel ideas concerning the accurate treatment of the interfacial region accurately, were discovered: the `force-based QC method' and the `geometrically consistent QC method'.The main challenge at present, and the aim of this research proposal, is to generalize this analytical work to the practically relevant two- and three-dimensional situations, where geometry and analysis become significantly more challenging. To this end, a hierarchy of simple, yet representative two- and three-dimensional models will be developed in order to benchmark different flavours of QC methods by means of rigorous mathematical analysis. In addition, an experimental QC software for rapid prototyping will be created, in order to test the analytical predictions.The main benefit of rigorous numerical analysis is the guarantee that simulations are reliable in situations which are not experimentally testable. Furthermore, this research will identify those QC methods with the greatest potential and will provide new insights to guide the development of new and improved methods.