Multiscale methods

More recently, the development of wavelets has allowed the development of fast nonlinear or multiscale filtering methods that can adapt to the nature of the measured data. Multiscale methods are an active area of research and have provided a formal mathematical framework for interpreting existing methods. Additional details about wavelet methods can be found in Strang (1989) and Daubechies (1988). 

Hughes, T. J. R., A. A. Oberai, and L. Mazzei (2001a). Large eddy simulation of turbulent channel flows by the variational multiscale method. Physics of Fluids 13, 1784-1799. 

In this section, we give a brief overview of theoretical methods used to perform tribological simulations. We restrict the discussion to methods that are based on an atomic-level description of the system. We begin by discussing generic models, such as the Prandtl-Tomlinson model. Below we explore the use of force fields in MD simulations. Then we discuss the use of quantum chemical methods in tribological simulations. Finally, we briefly discuss multiscale methods that incorporate multiple levels of theory into a single calculation. 

Processing conditions are known to play a critical role in establishment of morphology and final properties of the materials. Balazs and coworkers [245, 246] designed a multiscale method (coarse-grained Cahn-Hilliard approach and Brownian dynamics) and found that addition of solid particles significantly... 

Mechanical properties of PNCs can also be estimated by using computer modeling and simulation methods at a wide range of length and time scales. Seamless movement from one scale to another, for example, from the molecular scale (e.g., MD) and microscale (e.g., Halphin-Tsai) to macroscale (e.g., finite element method, FEM), and the combination of scales (or the so-called multiscale methods) is the most important prerequisite for the efficient transfer and extrapolation of calculated parameters, properties, and numerical information across length scales. 

In the next sections, the multiscale modeling methods are presented from the different disciplines perspectives. Clearly one could argue that overlaps occur, but the idea here is to present the multiscale methods from the paradigm from which they started. For example, the solid mechanics internal state variable theory includes mathematics, materials science, and numerical methods. However, it clearly started from a solid mechanics perspective and the starting point for mathematics, materials science, and numerical methods has led to other different multiscale methods. 

Others then went on to study various aspects of quasi-continuum concurrent multiscale methods. Lidirokas et al. [57] studied local stress states around Si nanopixels using this method. Bazant [58] argued that these atomistic-finite element multiscale methods cannot really capture inelastic behavior like fracture because the softening effect requires a regularization of the local region that is not resolved. 

Concurrent multiscale methods have also been employed to address fatigue. Oskay and Fish [82] and Fish and Oskay [83] introduced a nonlocal temporal multiscale model for fatigue based upon homogenization theory. Although these formulations were focused on metals, Fish and Yu [84] and Gal et al. [85] used a similar concurrent multiscale method for analyzing fatigue of composite materials. 

Next to metals, probably the synthetic polymer-based composites have been modeled most by hierarchical multiscale methods. Different multiscale formulations have been approached top-down internal state variable approaches, self-consistent (or homogenization) theories, and nanoscale quantum-molecular scale methods. 

The earliest works of trying to model different length scales of damage in composites were probably those of Halpin [235, 236] and Hahn and Tsai [237]. In these models, they tried to deal with polymer cracking, fiber breakage, and interface debonding between the fiber and polymer matrix, and delamination between ply layers. Each of these different failure modes was represented by a length scale failure criterion formulated within a continuum. As such, this was an early form of a hierarchical multiscale method. Later, Halpin and Kardos [238] described the relations of the Halpin-Tsai equations with that of self-consistent methods and the micromechanics of Hill [29],... 

Wescott et al. [260] in a hierarchical sense used atomistic simulations, meso-scale simulations, and finite elements to examine the effect of nanotubes that were dispersed in polymers. This bottom-up multiscale method employed percolation notions in trying to bridge the effects of the subscale nanotubes. 

W.K. Liu et al Introduction to computational nanomechanics and materials. Nano Mechanics and Materials -Theory, Multiscale Methods and Apllication. John Wiley and Sons, NewYork (2006)... 

Kinetic Monte Carlo and hyperdynamics methods have yet to be applied to processes involved in thermal barrier coating failure or even simpler model metal-ceramic or ceramic-ceramic interface degradation as a function of time. A hindrance to their application is lack of a clear consensus on how to describe the interatomic interactions by an analytic potential function. If instead, for lack of an analytic potential, one must resort to full-blown density functional theory to calculate the interatomic forces, this will become the bottleneck that will limit the size and complexity of systems one may examine, even with multiscale methods. 

Before starting this review, there first needs to be a definition of what is meant by a multiscale method. There are several different definitions that vary depending on the type of coupling between the different modelling levels. This review will adopt perhaps the most broad definition of a multiscale method, namely that it is any method that involves a flow of information from one modelling level to another. By definition, if there is a flow of information from one level to another, then there must... 

Using a broad definition, QM/MM multiscale methods are those that involve a transfer of information across an interface between the QM and MM levels of modelling. There are several different types of interface in use, which fall into four... 

Christen and van Gunsteren ° have developed a novel multiscale method that they call multigraining , which aims to use the CG model to enable both relaxation of large molecular systems and sampling of slow processes with concurrent atomic detail representation of the results. In this method, both an atomistic and a CG model of a molecule are used simultaneously. Each molecule in the simulation has... 

Two-way embedded interfacing methods. These involve embedding an atomistic or CG model within a continuum representation. Implicit solvent models fall into this category. New multiscale methods, which capture hydrodynamic and mechanical effects have now also been developed. 

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