A thorough numerical investigation of A/C **coupling** using the boundary element method should be undertaken to verify our results from Chapter 4. The efficiency gain due to removing the finite element regions is offset by the numerical cost of the boundary element method. The structure of the system matrix of the boundary element method is quite different from the sparse structure of finite element discretizations, and the cost for solving these systems can be non-negligible. For the 2D displacement problem studied here we only need to solve a problem on a curve. And since we can choose N ≈ K, the length of this curve is not too large. Therefore the size of the BEM problem will still be small enough to use dense direct solvers and more sophisticated strategies based, for example, on an H-matrix structure will not be required. But to really be able to meaningfully compare the A/C **coupling** using FEM and the method using BEM, a careful choice of the quantities to compare against is crucial. Simply choosing the degrees of freedom, as discussed in Chapter 3, is not a reliable indicator for the computational complexity of both problems. Comparing the run time directly requires careful tuning of the solver process, e.g., the choice of preconditioning.

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On the other hand, in the **continuum** region where the interactions are approxi- mated by the Cauchy–Born energy, we could increase the accuracy by using Pp-FEM with p > 1. In later sections we will review that the Cauchy–Born approximation yields a 2nd-order error, whereas employing the P1-FEM in the **continuum** region would reduce the accuracy to first order. In fact, we will show in that, with opti- mized mesh grading, P2-FEM is sufficient to obtain a convergence rate that cannot be improved by other choices of **continuum** discretisations. High-order Pp-FEM with p > 2 will increase the computational costs but yield the same error convergence rate (see § 3.5).

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Although a growing numerical analysis literature exists on the subject of a/c methods [10, 12,14,16, 29,31, 37, 40, 42], it has so far focused primarily on one-dimensional model problems. (A notable exception is the work of Lu and Ming on force-based hybrid methods [31]. However, the techniques used therein require large overlaps and cannot accommodate sharp interfaces.) Only speciﬁc methods are analyzed; the question whether absence of “ghost forces” (or, patch test consistency, as we shall call it) in general guarantees substantially improved accuracy has neither been posed nor addressed so far. The purpose of the present paper is to ﬁll precisely this gap. After introducing a general **atomistic** model and a general class of abstract a/c methods, and establishing the necessary analytical framework, it will be shown in Theorem 6.1, which is the main result of the paper, that in two dimensions patch test consistency together with additional technical assumptions implies ﬁrst-order consistency of energy-based a/c methods.

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In Figure 6(a) we observe a small but clear gap in the stability constants where they cross zero. Realistically, given the smallness of the gap, we must question whether it is genuine or a numerical error such as a domain size effect. The plots in Figure 6(b, c) suggest that the gap is genuine since the unstable QNL eigenmode is concentrated on the interface, and therefore of a different “type” than the unstable eigenmode of the **atomistic** model.

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In the present article we formulate and analyze one-dimensional a/c methods for an **atomistic** model that is deﬁned through an interaction ﬁeld satisfying a linear variational principle. Our results are related to two classes of a/c methods: First, our work can be viewed as an analysis of (a simpliﬁed version of) the a/c method proposed by Iyer and Gavini [9], who use ﬁeld-based versions of classical potentials to formulate their method. Second, the **atomistic** model we formulate can be considered a toy model of (orbital-free) density functional theory, and hence our work represents a preliminary step towards a rigorous analysis of the a/c methods described in [8, 6]. Our main results, stated in Theorems 5.5 and 6.6, are a priori error estimates for two closely related a/c couplings. While in a comparatively simple setting, the technical steps leading up these theorems address several important issues relevant for a/c **coupling** in the presence of ﬁelds, most prominently the dependence on the

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The recent article of Miller and Tadmor (2009) reviews and benchmarks a list of at least fourteen diﬀerent a/c couplings. Most of these schemes have at their core some aspects of the three fundamental a/c **coupling** con- cepts considered in the present article: energy-based blending, force-based **coupling** (and blending) and ghost force removal (QNL-type ideas). How- ever, they are presented in a primarily algorithmic format and it would be interesting to distinguish the various approximate models from nonlinear solver and implementation issues to provide a classiﬁcation more suitable to a numerical analysis of these methods.

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The damping in magnetic materials is a very complex process, that is not controlled by a single mechanism and incorporates a large amount of underlying physics. Here we conclude that in order to describe the ultrafast dynamics of amorphous TM-RE alloys it is necessary to allow two **coupling** channels. The first couples the TM spins to the conduction elec- tron temperature and the second couples the RE to the lattice. The former is responsible for the ultrafast demagnetization and the latter for the rather slower longitudinal macroscopic relaxation associated with FMR. In complex materials, it is evidently important not to expect a single damping parameter but to consider the energy transfer channel relevant to the technique and time scale of the measurement.

We present a new hybrid method for simulating dense ﬂuid systems that exhibit multiscale behaviour, in particular, systems in which a Navier–Stokes model may not be valid in parts of the computational domain. We apply molecular dynamics as a local microscopic reﬁnement for correcting the Navier–Stokes constitutive approximation in the bulk of the domain, as well as providing a direct measurement of velocity slip at bounding surfaces. Our hybrid approach differs from existing techniques, such as the heterogeneous multiscale method (HMM), in some fundamental respects. In our method, the individual molecular solvers, which provide information to the macro model, are not coupled with the **continuum** grid at nodes (i.e. point-wise **coupling**), instead **coupling** occurs over distributed heterogeneous ﬁelds (here referred to as ﬁeld-wise **coupling**). This affords two major advantages. Whereas point-wise coupled HMM is limited to regions of ﬂow that are highly scale-separated in all spatial directions (i.e. where the state of non-equilibrium in the ﬂuid can be adequately described by a single strain tensor and temperature gradient vector), our ﬁeld-wise coupled HMM has no such limitations and so can be applied to ﬂows with arbitrarily-varying degrees of scale separation (e.g. ﬂow from a large reservoir into a nano-channel). The second major advantage is that the position of molecular elements does not need to be collocated with nodes of the **continuum** grid, which means that the resolution of the microscopic correction can be adjusted independently of the resolution of the **continuum** model. This in turn means the computational cost and accuracy of the molecular correction can be independently controlled and optimised. The macroscopic constraints on the individual molecular solvers are artiﬁcial body-force distributions, used in conjunction with standard periodicity. We test our hybrid method on the Poiseuille ﬂow problem for both Newtonian (Lennard-Jones) and non-Newtonian (FENE) ﬂuids. The multiscale results are validated with expensive full-scale molecular dynamics simulations of the same case. Very close agreement is obtained for all cases, with as few as two micro elements required to accurately capture both the Newtonian and non-Newtonian ﬂowﬁelds. Our multiscale method converges very quickly (within 3–4 iterations) and is an order of magnitude more computationally eﬃcient than the full-scale simulation.

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In previous experimental and theoretical works [21 25], it was reported that the presence of interactions media in granular and ECC media can lead to devia- tion of the switching behaviour, in particular the angu- lar dependence of magnetic properties, from the coher- ent Stoner Wohlfarth theory[26]. Studies of the reversal process of such a complex structure as ECC/CGC me- dia are lacking and are still required in experiment and theory. Hence, it is important to investigate the efects of the complicated interaction as the hybrid ECC/CGC media in order to depth understand the complex physics in such new media design. Complex media, with rela- tively thin layers are not necessarily amenable to micro- magnetic calculations with a relatively crude spatial dis- cretisation. In this paper, **atomistic** spin simulation[27] based on the Landau-Lifshitz-Gilbert (LLG) equation of motion is chosen to study the complex reversal mech- anisms for hybrid ECC/CGC media due to the reduc- tion of magnetic granular grain and the layer thickness almost to the atomic level. It is shown that the quan- titative intra/inter exchange **coupling** between spins be- comes very signiicant in terms of the reversal behaviour. The magnetization curve and the angular dependence of the critical ield H cr are investigated to study the mag-

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polymer in a solvent, subjected to oscillatory flow (Barsky, 2004). Past studies have investigated the boundary conditions used in the flux **coupling** approaches attempting to smoothen any numerical artifacts and discontinuities induced at the HSI (Kalweit & Drikakis, 2008a; Kalweit & Drikakis, 2008b; Kalweit & Drikakis, 2010). Extensions to the flux **coupling** models have been proposed to take into account the fluctuations of state variables when transferring information from the molecular to the **continuum** region. This can be achieved by adapting the macroscopic equations as well as by implementing a relatively fine grid near the HSI. Such flux **coupling** methods have successfully simulated sound waves propagating through water and reflected by a lipid monolayer (Delgado- Buscalioni, 2005; De Fabritiis, 2007). Previous investigations have also used GD to couple fluxes from the **continuum** to the molecular domain in connection with the study of dynamic friction between crystal silver on copper at high pressure (Barton et al., 2011).

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Remark 5 Our formulation of Proposition 3 in terms of the **continuum** and **atomistic** stress functions reveals that the interface terms in the weak form of the QCF method, first observed in [9], are simply jumps of the stress. In multiphysics **continuum** mechanics one usually requires that the (normal component of) the jump of the stress vanishes, which would correspond to removing the interface terms from the variational formulation. The analysis in [9] has shown that these terms are the origin of the poor stability properties of the QCF method. Hence, in Section 5 we will investigate a new **coupling**

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An alternative approach to a/c **coupling** is the force-based quasicontinuum (QCF) approximation [7, 12, 13, 23, 24], but the nonconservative and indeﬁnite equilibrium equations make the iterative solution and the determination of lattice stability more challenging [13, 14, 15]. Indeed, it is an open problem whether the (sharp-interface) QCF method is stable in dimensions greater than one.

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In Fig. 3(a) (for bulk node), the sudden jumps in the measured values between the constrained and buffer region bins are due to the fact that the cell is periodic and, consequently, the two buffer ‘slices’ are contiguous. Fig. 3(b) (for a wall boundary node) shows that on the non-periodic boundaries the density oscillates. This is due to molecular stratiﬁcation and it is the correct behaviour in the proximity of a wall. In our boundary nodes, however, only one side has a real wall; the other side has an artiﬁcial wall that is only needed to conﬁne the molecules. On the artiﬁcial wall, therefore, the oscillations are not physical and must be excluded from the core region. Sometimes, however, it is not easy to estab- lish beforehand how deep into the cell the effects of these oscilla- tions propagate. For this reason, the buffer of the boundary nodes should normally be larger. The method we propose (see Section 4) handles the boundary and the internal nodes differently. At the boundary nodes, all the necessary **coupling** information is not ex- tracted from the core, but from bins close to the wall. 5 Therefore,

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The objective of this paper is to give a reader insight into the changes caused by hydrogen residing in materials, a process that can have catastrophic effects. We mainly focus on iron and steels; other metal systems are briefly introduced to explain additional effects that the presence of hydrogen can cause. Following catastrophic fractures related to hydrogen we will then discuss where hydrogen is likely to reside and how hydrogen moves inside the material. The detection of hydrogen inside the material is extremely challenging. We believe that combining experimental, numerical and theoretical techniques is the way forward to make a break- through in understanding hydrogen embrittlement in metals. We discuss relevant experimental methods available in the literature, such as thermal desorption analysis (TDA) and permeation experiments used to monitor ingress and egress of hydrogen. We then discuss innovative experimental techniques used to detect hydrogen inside the material, such as atom probe tomography (APT) or high-resolution electron microscopy. **Atomistic** calculations are then intro- duced as they provide insight into stable positions where hydrogen resides and microscopic mecha- nisms of hydrogen diffusion. ‘‘Description of the failure mechanisms affecting steels’’ section is devo- ted to a review of accepted hydrogen embrittlement mechanisms using recent experimental evidence and modelling techniques at different length scales. It is widely accepted that in most cases a combination of mechanisms, activated under different conditions, is responsible for HE. The key point is to understand under which conditions these mechanisms are acti- vated. Despite the amount of research dedicated to this topic, this point remains elusive. This is an essential point from a steel design point of view, i.e. how can we design HE resistant steels if we do not have a clear picture of what are the mechanisms of failure caused by hydrogen in a given steel? We dedicate ‘‘Hydrogen embrittlement mitigation strategies and design of new steels’’ section to hydrogen mitigation strategies. We give a detailed overview on strategies adopted in order to mitigate HE. We emphasise the use of engineered

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As numerical stability may arise at the **coupling** inter- face, we have tested diﬀerent initial ﬂow conditions with random noises with the magnitude close to the moving wall velocity for simple Couette ﬂow (see Fig.11). For this test case, we do not observe numerical stability prob- lem. However, it may become an important issue for complicated ﬂows and high-order numerical schemes. In this work, the ﬁrst-order extrapolation scheme is used in the interface for exchanging non-equilibrium informa- tion. For more complicated ﬂows, we may need to con- sider higher-order scheme or **coupling** overlapping zone to improve numerical accuracy. Finally, we have also tested the **coupling** scheme for a pressure-driven 2D ﬂow in micro-channel. The Knudsen number is 0.03 at the channel outlet, and the channel length L and hight H are 100 and 1 respectively. The extrapolated bound- ary conditions are used at the inlet and outlet. And the densities at the inlet and outlet are renormalized to be ρ in = 1.3 and ρ out = 1 (see Ref.[62] for detail). For M-

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Theoretical evidences of hydrogen kinetics and hydrogen preferred sites **Atomistic** simulation techniques allow one to question the favourability of a given arrange- ment of atoms within a given system. Thus, these techniques are perfectly placed to assess where hydrogen is likely to reside. It is worth noting that since iron is magnetic, and hydrogen is for all intents and purposes a quantum particle, great care must be taken in carrying out such simulations [40]. It is also worth noting that, due to the high computational cost of carrying out these quantum-mechanical simula- tions, there is a limitation on the size of the system that can be analysed. Furthermore, the choice of the simulations to be carried out often requires insight from experiments. This is due to the fact that the combinatorial scaling of the total number of possible simulations is extremely high. When assessing the effect of hydrogen in steels there are several iron phases of interest; the body-centred cubic phase (al- pha iron or ferrite), the body-centred tetragonal phase (martensite), the face-centred cubic phase (austenite) and HCP martensite. Within the phases of interest of bulk iron, there are two high-symmetry sites; the octahedral and the tetrahedral. A number of authors have used density functional theory (DFT) to assess which of these sites is preferable for a given phase. This is achieved straightforwardly through calculation of the total energy of the system of iron with the hydrogen atom at the site of interest (the zero of energy can be made to be equivalent in both cases). These DFT simulations have shown that the hydrogen prefers the tetrahedral site over the octa- hedral site in both ferrite and martensite [31, 74, 85]. This is confirmed in the path-integral simulations by

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Current developments in micro and nanoﬂuidics have created the need for new computational methods that can concurrently and efﬁciently handle different time and length scales. At these scales in fact the **continuum**-ﬂuid hypothesis loses its validity and the behaviour of the ﬂuid should be calculated, at least in theory, from the (averaged) motion of its constitutive molecules. In prac- tice, however, most of the time the **continuum** formulation can still be employed to describe the overall behaviour of the ﬂuid, but certain ‘adjustments’ must be introduced. The standard no-slip boundary conditions, for instance, cannot always be employed in microﬂows, while conﬁnement in nanochannels creates anisotro- pies in a ﬂuid’s density and alterations of the molecular distribution function, which, in turn, affect all the macroscopic properties of the ﬂuid. This phenomenon has been clearly demonstrated for water in carbon nanotubes [3,4,5,6,28,29], where self-diffusivity, hydrogen bonding, freezing point, viscosity, etc. are not only very different from those of bulk water, but also non-uniformly distributed in the nanotube. If we consider, for instance, the case of transport properties (e.g. viscosity, diffusivity, and thermal conductivity), the attempt to provide a classical correlation of the type ﬂux = f(gradient) (viz. ﬂux of momentum, heat or mass as a function of, respectively, velocity, temperature or concentration gradient)

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The Cauchy–Born rule [12] establishes a connection between the deformation of the lattice vector of an **atomistic** system and that of a **continuum** displacement field, and plays an important role in the development of **continuum** constitutive models of atomic lattices. In this section, we describe a hyperelastic constitutive model that is derived from the first-order Cauchy–Born rule by using the coarse- grained Helmholtz free energy density to describe the atomic interactions.

The estimate (7.9) follows immediately from Lemma 4.4. 7.2. Optimal mesh design. In this section we develop heuristics on the choice of **atomistic** region sizes and coarsening rates of the finite element mesh, in order to obtain error estimates in terms of the number of degrees of freedom. For the sake of generality we will slightly deviate from the assumptions and results of our analysis. Throughout this section, we will liberally make use of the symbols . and h to indicate bounds up to constants that are independent of the mesh parameters (but may depend on the shape regularity).

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Abstract. We present a new variant of the geometry reconstruction approach for the formu- lation of **atomistic**/**continuum** **coupling** methods (a/c methods). For many-body nearest-neighbour interactions on the 2D triangular lattice, we show that patch test consistent a/c methods can be constructed for arbitrary interface geometries. Moreover, we prove that all methods within this class are first-order consistent at the **atomistic**/**continuum** interface and second-order consistent in the interior of the **continuum** region.

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