Multiscale coupling in function space—weak coupling between molecular dynamics and continuum mechanics

Fackeldey, Konstantin; Krause, Rolf

doi:10.1002/nme.2626

Cited by 19 publications

(21 citation statements)

References 44 publications

(73 reference statements)

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“…For this operation, a scale transfer operator is used to define the transmission of information, such as displacement and velocity, from one length scale to another without corrupting the system dynamics in either of the scales. For example, Fackeldey et al [89,90] have developed an atomistic-continuum coupled approach using a weighted least square projection as a scale transfer operator to decompose the atomic displacements into low and high frequency components. They provided numerical examples demonstrating the seamless transmission of displacements from MD to FE at zero Kelvin.…”

Section: Scale Transfer Operatormentioning

confidence: 99%

MD/FE Multiscale Modeling of Contact

Ramisetti

Anciaux

Molinari

2014

Fundamentals of Friction and Wear on the Nanoscale

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Limitations of single scale approaches to study the complex physics involved in friction have motivated the development of multiscale models. We review the state-of-the-art multiscale models that have been developed up to date. These have been successfully applied to a variety of physical problems, but that were limited, in most cases, to zero Kelvin studies. We illustrate some of the technical challenges involved with simulating a frictional sliding problem, which by nature generates a large amount of heat. These challenges can be overcome by a proper usage of spatial filters, which we combine to a direct finite-temperature multiscale approach coupling molecular dynamics with finite elements. The basic building block relies on the proper definition of a scale transfer operator using the least square minimization and spatial filtering. Then, the restitution force from the generalized Langevin equation is modified to perform a two-way thermal coupling between the two models. Numerical examples are shown to illustrate the proposed coupling formulation.

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Section: Scale Transfer Operatormentioning

confidence: 99%

MD/FE Multiscale Modeling of Contact

Ramisetti

Anciaux

Molinari

2014

Fundamentals of Friction and Wear on the Nanoscale

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“…Some hybrid methods that eliminate reflections for effective dynamic simulation have been proposed; for example, the bridging scale method (BSM), 6,19,20) the perfectly matched multiscale simulation (PMMS), 21,22) and the bridging domain method with the perfectly matching layer. 23) These methods focus on eliminating incident large-wavenumber waves from the atomistic to coarse region, and realize transmission of small-wavenumber waves and elimination of the reflection of largewavenumber waves at the interface. However, it is still unclear whether these methods can effectively eliminate the spurious reflections for the middle-wavenumber waves near the Brillouin zone (BZ) edge of the coarse system where group velocities of the atomistic and coarse systems are quite different.…”

Section: Introductionmentioning

confidence: 99%

A Coupled Molecular Dynamics/Coarse-Grained-Particle Method for Dynamic Simulation of Crack Growth at Finite Temperatures

2011

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A hybrid molecular dynamics/coarse-grained-particle (MD-CGP) method is proposed for dynamic simulation of crack growth at finite temperatures. In the present method, the MD method is applied for the non-linear elastic region such as the crack tip and dislocation core, while the CGP method is applied for the surrounding linear elastic region. For coupling the atomistic and coarse regions, extra atoms and particles are placed, respectively, beyond the interface of the atomistic and coarse systems. They move according to the Langevin-type equation. The dissipation term in the Langevin-type equation contains the velocity difference between the atomistic and coarse systems. Hence, the coupling is achieved through the damped oscillation of the difference between the two systems. It is shown in a one-dimensional simulation that the present hybrid method achieves the non-reflecting boundary condition at the interface for the entire wavenumber range at finite temperatures. Twodimensional crack growth simulations are performed using both the present hybrid MD-CGP method and full MD method for accuracy check. In the hybrid simulation, the location of the atomistic region shifts to follow the moving crack tip. In spite of such drastic operations, the results obtained by the present hybrid simulation agree very well with those by the full MD simulation. The hybrid MD-CGP method allows us to simulate very large systems without losing atomistic information at the singular point and non-linear region, with much lesser degrees of freedom than required for the full MD method.

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“…We refer to [1] for a more detailed explanation of this process. As a consequence the resulting "micro scale displacement" can be expressed as…”

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confidence: 99%

“…In [1], different choices for the multiplier space are discussed, which has consequences for the extension of the displacement outside of the coupling region. In order to compute the algebraic representation of π in (2), we need to assemble two (scaled) mass matrices.…”

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confidence: 99%

CM/MD Coupling ‐ A Function Space Oriented Multiscale‐Coupling Approach

Fackeldey

Krause

2008

Proc Appl Math and Mech

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Many non-linear effects in elasticity, as, e.g., crack-propagation, contact, or friction, take place on different scales, i.e. a micro scale and a macro scale. Whereas the micro-scale is usually bound to the non-linear effect itself, the macro-scale is connected to the response of the bulk of the material. Therefore, often for the numerical simulation of these multi scale effects different simulation techniques on different scales are employed. One possibility is to use molecular dynamics on the microscale and finite elements on the macro-scale. In this case, due to the large differences in the length scale, a sound coupling between the two discretization schemes has to be developed. Here, we present a new scale-transfer operator, which is based on a weak coupling approach. The key idea is to construct the transfer operator between the micro and the macro scale on the basis of weighted local averaging instead of using point wise taken values, which is widely applied. The local weight functions are constructed by assigning a partition of unity to the molecular degrees of freedom. This allows for decomposing the micro scale displacements into a low frequency and a high frequency part by means of a weighted L 2 -projection. Thus, the entire formulation is given in a function space setting.Since our operator is based on a weak formulation of the transfer conditions, quadrature is necessary for the assembling of its algebraic representation. To resolve this quadrature problem, a new methodology has been developed.For the derivation of our new weak coupling approach, in a first step, we follow Hughes et al.[3] by applying scale separation techniques. Starting from a total displacement field ν, we separate the fine scale parts of ν from the coarse scale parts by means of the decompositionsee [3]. Here,ν is the coarse part of the total displacement and ν refers to the fine scale displacements. In our multiscale context, this frequency decomposition is intended to separate the high frequency atomic interactions from the low frequency parts of the solution on the micro scale. Any scale decomposition like (1) has to deal with the difficulty that the atomistic displacements are given as point-values in the "discrete" space R 3N , whereas the macro-scale displacements are usually assumed to be some functions in, e.g., a Sobolev space. Thus, in order to formulate the scale decomposition (1) properly, a suitable space has to be chosen, which contains the atomistic displacements as well as the macro scale displacements. As a matter of fact -at least to our knowledge-in all existing multiscale methods using this decomposition approach, the scale separation is performed in a completely discrete setting. This is possible, if the macro scale displacements are discretized by means of, e.g., finite elements. Then, the discretized displacements can be regarded as an element of R 3N and the scale decomposition (1) can easily be performed. In contrast, our new weak coupling approach provides a scale decomposition of the total displaceme...

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Multiscale coupling in function space—weak coupling between molecular dynamics and continuum mechanics

Cited by 19 publications

References 44 publications

MD/FE Multiscale Modeling of Contact

MD/FE Multiscale Modeling of Contact

A Coupled Molecular Dynamics/Coarse-Grained-Particle Method for Dynamic Simulation of Crack Growth at Finite Temperatures

CM/MD Coupling ‐ A Function Space Oriented Multiscale‐Coupling Approach

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