Volume 27, Issue 3
A Consistency Study of Coarse-Grained Dynamical Chains through a Nonlinear Wave Equation of Mixed Type

Mingjie Liao & Ping Lin

Commun. Comput. Phys., 27 (2020), pp. 921-948.

Published online: 2020-02

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  • Abstract

A dynamical atomistic chain to simulate mechanical properties of a onedimensional material with zero temperature may be modelled by the molecular dynamics (MD) model. Because the number of particles (atoms) is huge for a MD model,in practice one often takes a much smaller number of particles to formulate a coarsegrainedapproximation. Weshall mainlyconsiderthe consistency of the coarse-graine dmodel with respect to the grain (mesh) size to provide a justification to the goodness of such an approximation. In order to reduce the characteristic oscillations with very different frequencies in such a model, we either add a viscous term to the coarse-grained MD model or apply a space average to the coarse-grained MD solutions for the consistency study. The coarse-grained solution is also compared with the solution of the(macroscopic) continuum model (a nonlinear wave equation of mixed type) to show how well the coarse-grained model can approximate the macroscopic behavior of the material. We also briefly study the instability of the dynamical atomistic chain and the solution of the Riemann problem of the continuum model which may be related to the defect of the atomistic chain under a large deformation in certain locations.

  • Keywords

Mixed type wave equation, conservation law, Riemann problems,instability, coarsegrained approximation.grained approximation.

  • AMS Subject Headings

65M06, 35L20, 35L67, 74J40, 74H15, 35Q74

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

mliao@xs.ustb.edu.cn (Mingjie Liao)

plin@maths.dundee.ac.uk (Ping Lin)

  • BibTex
  • RIS
  • TXT
@Article{CiCP-27-921, author = {Liao , Mingjie and Lin , Ping }, title = {A Consistency Study of Coarse-Grained Dynamical Chains through a Nonlinear Wave Equation of Mixed Type}, journal = {Communications in Computational Physics}, year = {2020}, volume = {27}, number = {3}, pages = {921--948}, abstract = {

A dynamical atomistic chain to simulate mechanical properties of a onedimensional material with zero temperature may be modelled by the molecular dynamics (MD) model. Because the number of particles (atoms) is huge for a MD model,in practice one often takes a much smaller number of particles to formulate a coarsegrainedapproximation. Weshall mainlyconsiderthe consistency of the coarse-graine dmodel with respect to the grain (mesh) size to provide a justification to the goodness of such an approximation. In order to reduce the characteristic oscillations with very different frequencies in such a model, we either add a viscous term to the coarse-grained MD model or apply a space average to the coarse-grained MD solutions for the consistency study. The coarse-grained solution is also compared with the solution of the(macroscopic) continuum model (a nonlinear wave equation of mixed type) to show how well the coarse-grained model can approximate the macroscopic behavior of the material. We also briefly study the instability of the dynamical atomistic chain and the solution of the Riemann problem of the continuum model which may be related to the defect of the atomistic chain under a large deformation in certain locations.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2018-0206}, url = {http://global-sci.org/intro/article_detail/cicp/13920.html} }
TY - JOUR T1 - A Consistency Study of Coarse-Grained Dynamical Chains through a Nonlinear Wave Equation of Mixed Type AU - Liao , Mingjie AU - Lin , Ping JO - Communications in Computational Physics VL - 3 SP - 921 EP - 948 PY - 2020 DA - 2020/02 SN - 27 DO - http://dor.org/10.4208/cicp.OA-2018-0206 UR - https://global-sci.org/intro/cicp/13920.html KW - Mixed type wave equation, conservation law, Riemann problems,instability, coarsegrained approximation.grained approximation. AB -

A dynamical atomistic chain to simulate mechanical properties of a onedimensional material with zero temperature may be modelled by the molecular dynamics (MD) model. Because the number of particles (atoms) is huge for a MD model,in practice one often takes a much smaller number of particles to formulate a coarsegrainedapproximation. Weshall mainlyconsiderthe consistency of the coarse-graine dmodel with respect to the grain (mesh) size to provide a justification to the goodness of such an approximation. In order to reduce the characteristic oscillations with very different frequencies in such a model, we either add a viscous term to the coarse-grained MD model or apply a space average to the coarse-grained MD solutions for the consistency study. The coarse-grained solution is also compared with the solution of the(macroscopic) continuum model (a nonlinear wave equation of mixed type) to show how well the coarse-grained model can approximate the macroscopic behavior of the material. We also briefly study the instability of the dynamical atomistic chain and the solution of the Riemann problem of the continuum model which may be related to the defect of the atomistic chain under a large deformation in certain locations.

Mingjie Liao & Ping Lin. (2020). A Consistency Study of Coarse-Grained Dynamical Chains through a Nonlinear Wave Equation of Mixed Type. Communications in Computational Physics. 27 (3). 921-948. doi:10.4208/cicp.OA-2018-0206
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