Hybrid Multiscale Simulations

the various types of multiscale simulation are elaborated and various examples are provided. The presentation on coarse graining is mainly focused on stochastic (KMC) simulations to provide the underlying foundations and ideas in some depth. Coarse graining of other atomistic, e.g., MD, and mesoscopic tools will be covered in a forthcoming communication. Some excellent reviews on coarse graining in soft-matter physics problems are available (e.g., Kremer and Muller-Plathe, 2001 Muller-Plathe, 2002, 2003 Nielsen et al., 2004). 

Hybrid multiscale simulation is the most developed branch of multiscale simulation and will be covered in this section. The onion-type hybrid simulation 

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A. Onion-type Hybrid Multiscale Simulations and Algorithms 15... 

B. Application of Onion-type Hybrid Multiscale Simulation to Growth of Materials 17... 

A hierarchy of models can often be derived from a more detailed model under certain assumptions. This approach was discussed above in the case of deterministic, continuum models (see Fig. 3a). Such hierarchical models can be valuable in multiscale modeling. Let us just mention two cases. First, one could use different models from a hierarchy of models for different situations or length scales. This approach plays a key role in hybrid multiscale simulation discussed extensively below. Second, one could easily apply systems tasks to a simpler model to obtain an approximate solution that is then refined by employing a more sophisticated, accurate, and expensive model from the hierarchy. 

Fig. 5. Schematic of onion-type hybrid multiscale simulation. At each scale a different model is used. Consecutive scale models are simultaneously solved in the overlap region where exchange of information occurs.

There have been many hybrid multiscale simulations published recently in other diverse areas. It appears that the first onion-type hybrid multiscale simulation that dynamically coupled a spatially distributed 2D KMC for a surface reaction with a deterministic, continuum ODE CSTR model for the fluid phase was presented in Vlachos et al. (1990). Extension to 2D KMC coupled with ID PDE flow model was described in Vlachos (1997) and for complex reaction networks studied using 2D KMC coupled with a CSTR ODEs model in Raimondeau and Vlachos (2002a, b, 2003). Other examples from catalytic applications include Tammaro et al. (1995), Kissel-Osterrieder et al. (1998), Qin et al. (1998), and Monine et al. (2004). For reviews, see Raimondeau and Vlachos (2002a) on surface-fluid interactions and chemical reactions, and Li et al. (2004) for chemical reactors. 

The materials community has made significant advances in predicting mechanical properties of materials and initiation of defects using hybrid multiscale simulation. This is one of the application areas where multiscale simulation has advanced the most. Several nice reviews and perspectives have already been published (Maroudas, 2000, 2003 Miller and Tadmor, 2002 Rudd and Broughton, 2000). Therefore, it suffices to give only a brief account of the evolution of multiscale simulation in this area here. One of the earlier and... 

Recently, there has been strong interest in multigrid-type hybrid multiscale simulation. As depicted in Fig. 6, a coarse mesh is employed to advance the macroscopic, continuum variable over macroscopic length and time scales. At each node of the coarse mesh, a microscopic simulation is performed on a finer mesh in a simulation box that is much smaller than the coarse mesh discretization size. The microscopic simulation information is averaged (model reduction or restriction or contraction) to provide information to the coarser mesh by interpolation. On the other hand, the coarse mesh determines the macroscopic variable evolution that can be imposed as a constraint on microscopic simulations. Passing of information between the two meshes enables dynamic coupling. 

E. An Example of Multigrid-type Hybrid Multiscale Simulation for Growth under Large Length Scale Gradients... 

The major issue in hybrid multiscale simulation is patching of models used in different subdomains (Nie et al., 2003 Raimondeau and Vlachos, 2002a). In... 

The exposition in Schulze s (2004) recent paper underscores in an excellent manner some additional difficulties encountered in hybrid multiscale simulation (not just of crystal growth problems) when overlapping subdomains are used. The replacement of KMC on terraces with the continuum model Eq. (2) reduces... 

The discussion above focused on onion-type hybrid multiscale simulation. Finally, even though there are a limited number of examples published, I expect that the multigrid-type hybrid simulations share the same problems with onion-type hybrid multiscale models. In addition, appropriate boundary conditions for the microscopic grid model need to be developed to increase the accuracy and robustness of the hybrid scheme. Furthermore, the inverse problem of mapping coarse-grid information into a microscopic grid is ill posed. Thus, it is... 

Doyle and co-workers have used sensitivity and identifiability analyses in a complex genetic regulatory network to determine practically identifiable parameters (Zak et al., 2003), i.e., parameters that can be extracted from experiments with a certain confidence interval, e.g., 95%. The data used for analyses were based on simulation of their genetic network. Different perturbations (e.g., step, pulse) were exploited, and an identifiability analysis was performed. An important outcome of their analysis is that the best type of perturbations for maximizing the information content from hybrid multiscale simulations differs from that of the deterministic, continuum counterpart model. The implication of this interesting finding is that noise may play a role in systems-level tasks. 

In between the aforementioned levels, in both length and timescales are hybrid (multiscale) simulation strategies that combine the theoretical approaches from the neighboring levels in different ways, thereby bridging the couesponding scales. Such coupled (e.g., quantum-to-atomistic or atomistic-to-continuum) approaches have been developed and employed in recent years. For each scale, we review the simulation techniques and tools, as well as discuss important recent contributions (see Box 2). 

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