Asymptotic homogenisation via the method of multiple scales is considered for problems in which the microstructure comprises inclusions of one material embedded in a matrix formed from another. In particular, problems are considered in which the interface conditions include a global balance law in the form of an integral constraint; this may be zero net charge on the inclusion, for example. It is shown that for such problems care must be taken in determining the precise location of the interface; a naive approach leads to an incorrect homogenised model. The method is applied to the problems of perfectly dielectric inclusions in an insulator, and acoustic wave propagation through a bubbly fluid in which the gas density is taken to be negligible.

For a general class of atomistic-to-continuum coupling methods, coupling multi-body interatomic potentials with a P1-finite element discretisation of Cauchy–Born nonlinear elasticity, this paper adresses the question whether patch test consistency (or, absence of ghost forces) implies a first-order error estimate. In two dimensions it is shown that this is indeed true under the following additional technical assumptions: (i) an energy consistency condition, (ii) locality of the interface correction, (iii) volumetric scaling of the interface correction, and (iv) connectedness of the atomistic region. The extent to which these assumptions are necessary is discussed in detail.

The dynamics of dendritic growth of a crystal in an undercooled melt isdetermined by macroscopic diffusion-convection of heat and by capillary forcesacting on the nanometer scale of the solid-liquid interface width.Its modelling is useful for instance in processing techniques based on casting.The phase-field method is widely used to study evolution of such microstructural phase transformations ona continuum level; it couples the energy equation to a phenomenological Allen-Cahn/Ginzburg-Landauequation modelling the dynamics of an order parameter determining the solid and liquid phases,including also stochastic fluctuations to obtain the qualitatively correctresult of dendritic side branching.This work presents a method to determine stochastic phase-field models from atomisticformulations by coarse-graining molecular dynamics. It has three steps: (1) a precisequantitative atomistic definition of the phase-field variable, based on the localpotential energy;(2) derivation of its coarse-graineddynamics model, from microscopic Smoluchowski molecular dynamics (that is Brownian or over damped Langevin dynamics);and (3) numerical computation of the coarse-grained model functions. The coarse-grained model approximates Gibbs ensemble averages of the atomistic phase-field, bychoosing coarse-grained drift and diffusion functions that minimize the approximation error of observables in thisensemble average.