Keywords: multi-scale modelling, nonlocal discrete model, bridging scale method, quasi-continuum method, ghost forces, dynamics, damping.

Nonlocal discrete models are well established tools in computational mechanics for describing fibre reinforced concrete as well as composite or nanometerials. Indeed, they enable the solution of complex physical phenomena such as solid fracture, plasticity or damage. The nonlocal aspect arises as a result of the interaction range defined for each particle. However, such models are limited to localised regions in which the particle scale is important. In the remainder of the domain, a reduction method based on continuum simulations is used, which allows simulations at larger length and time scales. Consequently, the difficulty lies with the accurate coupling so that the critical region behaves as if the entire model were nonlocal and discrete.

In order to couple a discrete nonlocal model with the continuum local model in dynamics, two steps are defined: (i) the first is to couple the nonlocal discrete model with the local discrete model (ii) the second is to couple local discrete model with the continuum local model. Hence, two approaches to couple different models can be identified:

• surface coupling using a discrete interface: mortar method, quasi-continuum method [1], etc.
• volume coupling using an overlaping zone: bridging domain method [2], bridging scale method, etc.

The work presented in this paper considers both approaches, approach (i) by means of the quasi-continuum (QC) method and approach (ii) by means of a method based on the bridging scale method.

The QC method used for approach (i) generates a local-nonlocal interface where the local discrete model is homogenized from the nonlocal model by using the Cauchy-Born rule. Nevertheless, undesired forces called "ghost forces" [3] appear in this method, which are responsible, in dynamics, for spurious reflections at the interfaces that pollute the whole simulation. Consistant schemes [4] for the QC methods have been provided where the local-nonlocal interfaces between models remain free from "ghost forces". Approach (ii) leads to the coupling of macro and micro scales. In dynamics, a part of the micro scale is not represented by the macro scale which also generates spurious reflexions at the interfaces. Hence, a new discrete-continnum interface is developed, based on the bridging scale method and the perfectly matched layer [5] permeable for macro waves and absorbing for micro waves.

1

E.B. Tadmor, M. Ortiz, R. Phillips, "Quasicontinuum analysis of defects in solids", Philosophical Magazine, 73(6), 1529-1564, 1996. doi:10.1080/01418619608243000

2

S.P. Xiao, T. Belytschko, "A bridging domain method for coupling continua with molecular dynamics", Computer methods in applied mechanics and engineering, 193(17-20), 1645-1669, 2004. doi:10.1016/j.cma.2003.12.053

3

W. E, J. Lu, J.Z. Yang, "Uniform accuracy of the quasicontinuum method", Physical Review B, 74(21), 1-12, 2006.

4

J. Marchais, C. Rey, L. Chamoin, "Geometrically consistent approximations of the energy in the quasi-continuum framework". (submitted)

5

A.C. To, S. Li, "Perfectly matched multiscale simulations", Physical Review B, 72(3), 035414, 2005. doi:10.1103/PhysRevB.72.035414

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £65 +P&P)