Multiscale boundary conditions in crystalline solids: Theory and application to nanoindentation

Abstract

This paper presents a systematic approach to treating the interfaces between the localized (fine grain) and peripheral (coarse grain) domains in atomic scale simulations of crystalline solids. Based on Fourier analysis of regular lattices structures, this approach allows elimination of unnecessary atomic degrees of freedom over the coarse grain, without involving an explicit continuum model for the latter. The mathematical formulation involves compact convolution operators that relate displacements of the interface atoms and the adjacent atoms on the coarse grain. These operators are defined by geometry of the lattice structure, and interatomic potentials. Application and performance are illustrated on quasistatic nanoindentation simulations with a crystalline gold substrate. Complete atomistic resolution on the coarse grain is alternatively employed to give the benchmark solutions. The results are found to match well for the multiscale and the full atomistic simulations.

A phonon heat bath approach for the atomistic and multiscale simulation of solids

“…Provided that there is no input or release of energy inside P or I , temperature in these domains will be approaching the value T utilized for the amplitudes (32). At the same time, the convolution integral in the boundary conditions (17) will be damping all the non-equilibrium oscillations, represented by the difference (u 0 − R 0 ), and the displacements u 1 will be approaching R 1 in a stochastic manner.…”

“…Numerical computation of the K -matrices is particularly reasonable for complex interatomic potentials, when the second-order derivatives of the lattice potential U in (5) are analytically cumbersome (even though U is written for the associate substructure only). Various methods for numerical computation of second-order derivatives of complex functions can be found elsewhere in literature; special techniques accounting for the specifics of the lattice structure application are discussed in [32].…”

“…Using these matrices, compute S N c normal frequencies (23), polarization vectors (21), and characteristic masses (28). Then evaluate S N c amplitudes (32), and sample independently 2S N c random phases ± . Create a subroutine for computing the summations in (20), using the lattice site indices n, m and l and physical time t as input parameters.…”

A domain-reduction approach to bridging-scale simulation of one-dimensional nanostructures

“…Based on the earlier development by Adelman [10,11], a class of interface treatment approach using the Langevin approach have been presented in [12,13]. These approaches are further extended in the bridging-scale method (BSM) [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29], which was first developed by Wagner and Liu [14] to couple atomistic and continuum scales. An important feature of the method is the decomposition of the field variables through the bridging-scale decomposition.…”

Multiscale modeling of nano/micro systems by a multiscale continuum field theory

“…The bridging-scale method was proposed by Liu and coworkers [11][12][13][14][15], in which FE approximation co-exists with atomistic description. By employing the harmonic approximation, Liu and his colleagues have derived a multiscale boundary condition analytically or semi-analytically so that they have exact matching impedance at the multiscale interface, which will eliminate the spurious phonon reflections.…”