Abstract
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.
Highlights
Multiple spatial scales are encountered in many complex physical problems of importance
For a typical multiscale computation, one performs Molecular Dynamics (MD) computation only in the localized region where the atomistic description is necessary to model the underlying physics properly, while a coarse grid description is used in the surrounding region
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
Summary
Multiple spatial scales are encountered in many complex physical problems of importance. A handshaking region is introduced to reduce interfacial reflections [2,4,14] In this region, one either uses a certain weighted average between the MD description and the coarse grid description, or an artificial damping to diffuse the outgoing fluctuations [3]. With a fixed domain decomposition and coarsening ratio, the mathematical formulation of the pseudo-spectral multiscale method gives a systematic derivation of the interfacial conditions and the coarse grid equations. An arbitrary error reduction is obtained due to the accurate time history treatment and coarse grid equations derived by the matching differential operator method This is a major advantage of the pseudo-spectral multiscale method compared with other multiscale methods, which rely heavily on empirical derivations due to lack of a systematic mathematical analysis. The internal force comes from atomistic interactions, described by a potential U as f 1⁄4 ÀruU ðuÞ: ð2Þ
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