Mathematical Problems in Engineering

Volume 2015, Article ID 834517, 10 pages

http://dx.doi.org/10.1155/2015/834517

## Challenges in Atomistic-to-Continuum Coupling

^{1}Institute for Mathematics, TU Berlin, Strasse des 17. Juni 135, 10623 Berlin, Germany^{2}Zuse Institute Berlin (ZIB), Takustrasse 7, 14195 Berlin, Germany

Received 18 March 2015; Revised 31 May 2015; Accepted 1 June 2015

Academic Editor: Xiao-Qiao He

Copyright © 2015 Konstantin Fackeldey. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper is concerned with the design, analysis, and implementation of concurrent coupling approaches where different (atomic and continuous) models are used simultaneously within a single simulation process. Thereby, several problems or pitfalls can happen, for example, the reflection of molecular movements at the “boundary” between the atomic and continuum regions which leads to an unphysical increase in energy in the atomic model. We investigate the problems with the aim of giving an introduction into this field and preventing errors for scientists starting their research towards multiscale methods.

#### 1. **Introduction**

Classical molecular dynamics provides often an adequate description of a physical phenomenon but is computationally expensive. On the other hand, a depiction by a continuum mechanical model combined with the finite elements method is a smaller computational burden but does not go into atomistic details and can be considered less accurate. Multiscale methods which aim at combining fine scale method such as MD with coarse models such as continuum mechanics have been developed. In particular, for elastic systems these methods have reached a considerable interest, since they promise to be a powerful tool in many engineering problems, where only in a highly localized region a precise model (fine scale) is needed and on a larger (surrounding) domain a less accurate model (coarse scale) suffices.

The development of different multiscale methods in different fields started decades ago and has been accelerated in the last few years. Along with this expansion several survey articles have been published in order to classify the multiscale methods by different aspects [1–5].

These methods vary not only in scope and the underlying assumptions but also in their approach to broader questions such as a hierarchical and concurrent multiscale approach. In the first class, the computations are performed on each scale separately. Often, the scale coupling is done by transferring problem parameters; that is, the results obtained on one scale determine the parameters for the computational model on another scale [6, 7]. Thus, for instance, a continuum model can be derived from the atomic information [8]. Another approach is pursued in the concurrent coupling techniques, where the behavior at each length scale depends strongly on the others and an appropriate model is solved on each scale simultaneously, while smooth coupling between the scales is introduced. In the following, we confine our considerations to concurrent methods.

#### 2. **Demands on Multiscale Methods and Domain Decompositions**

In the following, we show that Domain Decomposition (DD) methods serve as a good motivation for a classification of multiscale methods. To do so, we briefly explain some basic concepts from DD methods.

The term DD [9, 10] is often used to describe a data distribution, in which the local data of each process corresponds topologically to a subdomain of the whole computational domain .

In the following, we use a different approach by defining DD as certain numerical methods that split the computational domain into two or more subdomains. Although DD methods have been developed for the purpose of achieving concurrency, they can be used in sequential as well as concurrent computations.

In their origin DD techniques have been developed as a powerful iterative method for solving systems of algebraic equations stemming from the discretization of partial differential equations (PDE), that is, from a continuum description; see [11, 12] for an overview. Therein, DD is considered as a decomposition of the finite element space into a sum of subspaces. Then these subproblems are solved by a direct or iterative method. In a next step, projection operators are developed for the information transfer between the subspaces. As a matter of fact, the quality of the approximation on each subdomain depends on the corresponding properties of the approximation subspace. On the contrary, the DD method allows taking benefit of the presence of the subdomains in order to choose the discretization method, which is best adapted to the local behavior of the solution of the PDE which has to be approximated. Thus, the shape of the subdomains and their magnitude of overlap (interface) can be chosen problem dependent. Summing up, the choice of overlapping or nonoverlapping domains and the choice of the transfer operators deeply influence the performance of the DD method. In particular, the choice of an overlapping or nonoverlapping method directly influences the choice of the transfer operator.

Let us now consider the continuum/molecular coupling. Then, the following three points have to be clarified:(i)Definition of the domain.(ii)Design of the coupling region.(iii)Design of suitable transfer operators.

Let us elucidate this DD motivated classification.

##### 2.1. Definition of the Domain

Depending on the type of problem and on the considered domain, the regions with highly local interest have to be defined. For example, cracks and similar defects can involve a global simulation by finite elements and highly localized regions with strong deformations (e.g., crack tips) which are resolved by a molecular dynamics simulation. More precisely, the domain is decomposed bywhere in a fine resolution down to the atomistic scale is employed and in a coarser representation by finite elements is applied. As mentioned previously, since the simulation on is a higher computational burden, the size of plays an important role in the overall performance of any multiscale scheme. As a particular advantage of DD methods the size and shape of the subdomains and thus the interface or handshake region can be chosen arbitrarily. For the different types of interfaces we can distinguish between three cases (cf. Figure 1); these are the following:(i)*Interface Methods*: The two scales are separated by an interface, which is often a manifold of dimension ().(ii)*Handshake Region Methods*: The atomistic and the continuum part are matched in an overlap region , , and .(iii)*Completely Overlapping Methods*: The continuum mechanic description is on the whole domain and the molecular part is a portion of it; that is, the atomistic simulation is everywhere accompanied by the continuum simulation ().