Analysis of Atomistic-to-Continuum Coupling Methods

原子到连续耦合方法分析

• 批准号: EP/H003096/1
• 负责人: Christoph Ortner
• 金额: $ 37.33万
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2010
• 资助国家: 英国
• 起止时间: 2010 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FH003096%2F1
• 关键词: Analysis; Atomistic; Continuum; Coupling; Methods

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.

在新工程材料的开发中，目标通常是制造具有规定机械性能（弹性模量，韧性等）的材料。人们早就知道，这种“宏观”性质的设计只能通过理解和控制材料在纳米和微米尺度上的细节（晶格结构、缺陷分布、微观结构等）来实现。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。虽然我们首要的建模工具应该是分析，但在纳米尺度上精确描述材料所需的复杂模型的定量结果只能通过数值模拟来获得。本提案侧重于在数千至数百万原子间距的尺度上模拟材料，特别是缺陷。尽管晶格中的许多类型的缺陷是高度局部化的，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。这正是困难所在。对于缺陷的模拟，应使用精确但昂贵的原子材料模型，而对于远场的描述，连续模型便宜得多，但足够精确。因此，自然的挑战是将原子模型（例如，准连续体（quasicontinuum，QC）方法是粗粒化技术的一个范例，它实现了晶体固体静态（和准静态）模拟的这种耦合。它的主要特点是，而不是一个原子模型耦合到一个连续模型，它也使用原子模型在远场区域，但通过有限元方法删除的自由度。然后进行额外的近似，使粗粒度的问题可计算，导致不同类的QC methods.A效果共享的几乎所有的方法耦合固有的不同的物理模型，包括QC方法，是一个有缺陷的力平衡的接口区域。主要的研究重点是了解和消除这种错误的模拟。目前，严格的分析提供了很好的理解这个问题的一维模型问题。尽管这些结果的初步性质，从这个数学研究可以得出一些重要的结论：（i）它表明，在实践中使用的某些类别的方法是非常不准确的，应该被丢弃;（ii）至少有两个令人兴奋的和新颖的想法，关于准确地处理界面区域准确，被发现：“基于力的质量控制方法”和“几何一致的质量控制方法”目前的主要挑战以及本研究提案的目的是将这一分析工作推广到实际相关的二维和三维情况，在这种情况下，几何和分析变得更具挑战性。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QC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，以检验分析预测。严格的数值分析的主要好处是保证在无法通过实验检验的情况下，模拟是可靠的。此外，这项研究将确定那些最有潜力的QC方法，并将提供新的见解，以指导新的和改进的方法的发展。

项目成果

Positive Definiteness of the Blended Force-Based Quasicontinuum Method

• DOI: 10.1137/110859270
• 发表时间: 2011-12
• 期刊: Multiscale Model. Simul.
• 作者: X. Li;M. Luskin;C. Ortner

Atomistic-to-Continuum Coupling Approximation of a One-Dimensional Toy Model for Density Functional Theory

密度泛函理论一维玩具模型的原子连续耦合近似

• DOI: 10.1137/110857787
• 发表时间: 2013
• 期刊: Multiscale Modeling & Simulation
• 影响因子: 1.6
• 作者: Langwallner B

A posteriori error control for a quasi-continuum approximation of a periodic chain

• DOI: 10.1093/imanum/drt011
• 发表时间: 2012-11
• 期刊: Ima Journal of Numerical Analysis
• 影响因子: 2.1
• 作者: C. Ortner;Hao Wang

The Spectrum of the Force-Based Quasicontinuum Operator for a Homogeneous Periodic Chain

• DOI: 10.1137/110825704
• 发表时间: 2010-04
• 期刊: Multiscale Model. Simul.
• 作者: M. Dobson;C. Ortner;A. Shapeev

An Analysis of Surface Relaxation in the Surface Cauchy--Born Model

表面柯西-玻恩模型中的表面弛豫分析

• DOI: 10.48550/arxiv.1112.0683
• 发表时间: 2011
• 期刊: arXiv e-prints
• 作者: Jayawardana K.