The bridging scale for two-dimensional atomistic/continuum coupling

Park, Harold S.; Karpov, Eduard G.; Liu, Wing Kam; Klein, Patrick A.

doi:10.1080/14786430412331300163

Cited by 145 publications

(109 citation statements)

References 23 publications

(13 reference statements)

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“…However, the concurrent modeling method reviewed in Section 5.3, borrowing ideas from the bridging scale method [344] and the bridging domain method [345], can exchange information and capture interaction between the fine scale and coarse scale in both directions. There is a tremendous amount of concurrent multiscale modeling methods developed in the recent ten years with rigorous mathematical foundations, but mostly for metals [344,[346][347][348][349][350] and carbon nanomaterials [345,[351][352][353][354]. These concurrent methods enable the information exchange between different length scales, capture their interactions and couple MD simulations with continuum simulations.…”

Section: From Atomistic To Macroscopic Scales and Backmentioning

confidence: 99%

Challenges in Multiscale Modeling of Polymer Dynamics

et al. 2013

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The mechanical and physical properties of polymeric materials originate from the interplay of phenomena at different spatial and temporal scales. As such, it is necessary to adopt multiscale techniques when modeling polymeric materials in order to account for all important mechanisms. Over the past two decades, a number of different multiscale computational techniques have been developed that can be divided into three categories: (i) coarse-graining methods for generic polymers; (ii) systematic coarse-graining methods and (iii) multiple-scale-bridging methods. In this work, we discuss and compare eleven different multiscale computational techniques falling under these categories and assess them critically according to their ability to provide a rigorous link between polymer chemistry and rheological material properties. For each technique, the fundamental ideas and equations are introduced, and the most important results or predictions are shown and discussed. On the one hand, this review provides a comprehensive tutorial on multiscale computational techniques, which will be of interest to readers newly entering this field; on the other, it presents a critical discussion of the future opportunities and key challenges in the multiscale geometric mapping matrix between all-atomistic and coarse-grained models N number of monomers per chain N e entanglement length, which is the number of monomers between two entanglements n v number of polymer chains per unit volume n ij (n 0 (r ij )) entanglement number (at equilibrium) between particle pairs i and j p r , P R configurational probability distributions in all-atomistic and coarse-grained models, respectively R ee end-to-end distance of polymer chain R G radius of gyration of polymer chain s segment index/contour length variable along primitive chain S(q) single chain coherent dynamic scattering function Z number of entanglements per chain, defined as Z = N/N e α exponent in standard Mittag-Leffler function σ

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Section: From Atomistic To Macroscopic Scales and Backmentioning

confidence: 99%

Challenges in Multiscale Modeling of Polymer Dynamics

et al. 2013

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“…The bridging scale method (BSM) was introduced recently by Liu et al through a series of papers, using this time history treatment essentially only for the fine fluctuation components in the displacement. See References [1,[20][21][22] and references therein.…”

Section: Introductionmentioning

confidence: 99%

A mathematical framework of the bridging scale method

Tang

Hou

Liu

2005

Numerical Meth Engineering

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SUMMARYIn this paper, we present a mathematical framework of the bridging scale method (BSM), recently proposed by Liu et al. Under certain conditions, it had been designed for accurately and efficiently simulating complex dynamics with different spatial scales. From a clear and consistent derivation, we identify two error sources in this method. First, we use a linear finite element interpolation, and derive the coarse grid equations directly from Newton's second law. Numerical error in this length scale exists mainly due to inadequate approximation for the effects of the fine scale fluctuations. An modified linear element (MLE) scheme is developed to improve the accuracy. Secondly, we derive an exact multiscale interfacial condition to treat the interfaces between the molecular dynamics region D and the complementary domain C , using a time history kernel technique. The interfacial condition proposed in the original BSM may be regarded as a leading order approximation to the exact one (with respect to the coarsening ratio). This approximation is responsible for minor reflections across the interfaces, with a dependency on the choice of D . We further illustrate the framework and analysis with linear and non-linear lattices in one-dimensional space.

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“…We also perform a MacrosCopic (MC) computation with a coarse grid over the whole domain, at a much lower computational cost. Motivated by the bridging scale method (BSM) [13,16,20,28], we develop a pseudo-spectral multiscale method with balanced accuracy on coarse and fine scales, as well as at the interfaces. More precisely, by a spectral decomposition, we split the displacements into the mean part and fine fluctuation part.…”

Section: Introductionmentioning

confidence: 99%

A pseudo-spectral multiscale method: Interfacial conditions and coarse grid equations

Tang

Hou

Liu

2006

Journal of Computational Physics

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In this paper, we propose a pseudo-spectral multiscale method for simulating complex systems with more than one spatial scale. Using a spectral decomposition, we split the displacement into its mean and fluctuation parts. Under the assumption of localized nonlinear fluctuations, we separate the domain into an MD (Molecular Dynamics) subdomain and an MC (MacrosCopic) subdomain. An interfacial condition is proposed across the two scales, in terms of a time history treatment. In the special case of a linear system, this is the first exact interfacial condition for multiscale computations. Meanwhile, we design coarse grid equations using a matching differential operator approach. The coarse grid discretization is of spectral accuracy. We do not use a handshaking region in this method. Instead, we define a coarse grid over the whole domain and reassign the coarse grid displacement in the MD subdomain with an average of the MD solution. To reduce the computational cost, we compute the dynamics of the coarse grid displacement and relate it to the mean displacement. Our method is therefore called a pseudo-spectral multiscale method. It allows one to reach high resolution by balancing the accuracy at the coarse grid with that at the interface. Numerical experiments in one-and two-space dimensions are presented to demonstrate the accuracy and the robustness of the method.

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The bridging scale for two-dimensional atomistic/continuum coupling

Abstract: 2005) The bridging scale for two-dimensional atomistic/continuum coupling, Philosophical Magazine, 85:1, 79-113,

Cited by 145 publications

References 23 publications

Challenges in Multiscale Modeling of Polymer Dynamics

Challenges in Multiscale Modeling of Polymer Dynamics

A mathematical framework of the bridging scale method

A pseudo-spectral multiscale method: Interfacial conditions and coarse grid equations

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