Analysis of Atomistic-to-Continuum Coupling Methods

Lead Research Organisation: University of Oxford
Department Name: Mathematical Institute


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.


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Dobson M (2010) Accuracy of quasicontinuum approximations near instabilities in Journal of the Mechanics and Physics of Solids

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Dobson M (2011) Iterative methods for the force-based quasicontinuum approximation: Analysis of a 1D model problem in Computer Methods in Applied Mechanics and Engineering

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Hudson T (2011) On the stability of Bravais lattices and their Cauchy-Born approximations in ESAIM: Mathematical Modelling and Numerical Analysis

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Ortner C (2012) Justification of the Cauchy-Born Approximation of Elastodynamics in Archive for Rational Mechanics and Analysis

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Li X (2012) Positive Definiteness of the Blended Force-Based Quasicontinuum Method in Multiscale Modeling & Simulation

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Luskin M (2013) Atomistic-to-continuum coupling in Acta Numerica

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Luskin M (2013) Formulation and optimization of the energy-based blended quasicontinuum method in Computer Methods in Applied Mechanics and Engineering

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Van Koten B (2013) Symmetries of 2-Lattices and Second Order Accuracy of the Cauchy--Born Model in Multiscale Modeling & Simulation

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Li X (2014) Theory-based benchmarking of the blended force-based quasicontinuum method in Computer Methods in Applied Mechanics and Engineering

Related Projects

Project Reference Relationship Related To Start End Award Value
EP/H003096/1 01/06/2010 01/09/2011 £292,960
EP/H003096/2 Transfer EP/H003096/1 02/09/2011 01/06/2013 £175,557
Description The project has achieved that a class of numerical coarse-graining schemes (quasicontinuum methods, atomistic/continuum multiscale methods), which has previously been intractable for mathematical analysis, is now well-understood. While theoretical open questions remain, there is now a clear pathway and mathematical framework for their study.
Exploitation Route Molecular Simulation, in particular in materials research. The project has placed the research group into the position to initiate collaborations with engineers, physicists and materials scientists to exploit the new ideas developed during this project in practical materials science research.
Sectors Energy,Manufacturing, including Industrial Biotechology,Other

Description The work on this project has fundamentally influenced the way atomistic multi scale methods are now viewed. The long-term effect will be a new class of robust multi-scale simulation tools based on rigorous mathematics. These tools will primarily be used in materials science and engineering research.
First Year Of Impact 2012
Sector Other
Description Mitch Luskin, University of Minnesota 
Organisation University of Minnesota
Country United States 
Sector Academic/University 
PI Contribution Intensive collaboration with the group of M Luskin on all aspects of atomistic/continuum multiscale methods.
Start Year 2009
Description Surface Cauchy Born Method 
Organisation Boston University
Country United States 
Sector Academic/University 
PI Contribution New collaboration with Harold Park (Boston University) on a new class of atomistic/continuum multiscale methods, focusing on surface effects in crystalline materials. Through workshops and discussions, we created a new research programme, starting from the achievements in the present project. We have one publication on the new project and are now in the process of applying for funding to take this further.
Start Year 2011