Fackeldey, Konstantin: The Weak Coupling Method for Coupling Continuum Mechanics with Molecular Dynamics. - Bonn, 2009. - Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn.
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5N-17231
@phdthesis{handle:20.500.11811/4062,
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5N-17231,
author = {{Konstantin Fackeldey}},
title = {The Weak Coupling Method for Coupling Continuum Mechanics with Molecular Dynamics},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2009,
month = apr,

note = {For the global behavior of solids in structural mechanics of nonlinear processes, local effects on the atomistic level play an important role. Often a direct numerical simulation of the macroscopic behavior by a complete resolution of the microscale is for computational reason not possible.
Thus, employing a multiscale strategy for an efficient and accurate modelling seems favorable since by separating the problem into two different frameworks, the accuracy of a fine scale model can be combined with the advantages of a computationally efficient model. More precisely a comparably small region of atoms e.g. surrounding the tip of a crack is modelled by molecular dynamics. Outside of this region, we take advantage of the fact that the displacement is almost homogeneous and can thus be modelled efficiently by a linear elastic continuum dynamical simulation.
Clearly, both scales offer fundamentally different descriptions of the matter and they use different simulation methods. Whereas on the continuum scale the finite element method and a function space setting is used, the molecular dynamics is based on the movement of particles in the Euclidean space. Additionally, dynamical simulations with a transition zone (handshake region) between atomistic systems and the coarser finite element mesh suffer from unwanted (spurious) reflections, since the finite element method can not represent short wave length vibrational modes.
Here a completely new approach is presented, which takes advantage of an infinite dimensional function space for the information transfer between the scales. Starting from a handshake region, the key idea is to construct a transfer operator between the different scales. This transfer operator is based on local averaging taken values. In order to construct the local weight functions, a partition of unity is assigned to the molecular degree of freedom. This allows us to decompose the micro scale displacement in the handshake region into a small and large wave number part by means of a weighted $L^2$ projection.
In the first instance, this function space oriented interpretation of the atomistic displacement is applied in the context of a completely overlapping decomposition. More precisely, we consider the case, when the domain of the handshake region is conform with the domain of the molecular dynamics. In order to identify the displacements pertaining to the atomistic or continuum level respectively, we employ a multiscale decomposition. In particular, we decompose the micro scale displacement into a "low frequency'' and a "high frequency'' part in a weak sense.
This new approach is also used in the context of a partly overlapping decomposition. Therein, the coarse and the fine scale simulation are matched by constraining the two displacements in the handshake region. The key issue in this context is, that our function space oriented approach allows us to interpret the constraints in a weak sense. Thus the "low frequent'' part can be captured by the coarse scale, whereas the "high frequent'' part of the displacement which has no meaning on the coarse scale is damped in the handshake region. Moreover numerical examples in 1d,2d and 3d show that this approach allows molecular displacements for entering into the continuum domain and the other way round flawlessly.},

url = {http://hdl.handle.net/20.500.11811/4062}
}

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