Multiple-scale Analysis

Quasicontinuum method (Tadmor, Ortiz and Phillips, 1996; Knap and Ortiz, 2001)is a finite element type of method for analyzing the mechanicalbehavior of crystalline solids based on atomistic models. Atriangulation of the physical domain is formed using a subset of theatoms, the representative atoms (or rep-atoms). The rep-atoms areselected using an adaptive mesh refinement strategy. In regions wherethe deformation gradient is large, more atoms are selected. Typically,near defects such as dislocations, all the atoms are selected.

Sequential multiscale modeling

An example of such problems involve the Navier–Stokes equations for incompressible fluid flow. Alternatively, modern approaches derive these sorts of models using coordinate transforms, like in the method of normal forms,3 as described next. A classical example in which matched asymptotics has been used isPrandtl’s boundary layer theory in fluid mechanics. In the language used below, the quasicontinuum method can be thought of asan example of domain decomposition methods.

Straightforward perturbation-series solution

This is a general strategy ofdecomposing functions or more generally signals into components atdifferent scales. A well-known example is the wavelet representation(Daubechies, 1992). This is a strategy for choosing thenumerical grid or mesh adaptively based on what is known about thecurrent approximation to the numerical solution. Usually one finds alocal error indicator from the available numerical solution based onwhich one modifies the mesh in order to find a better numericalsolution.

Code, Data and Media Associated with this Article

• Thus, the introduction of new materials intoa structure results in increased time to market and costs.
• MUSCLE 2 can couple submodels written in different programming languages, e.g.
• The vegetation submodels would have mD interactions, exchanging only boundary information, but they would have sD interactions with the fire submodel.
• As a result, in most of the multi-scale applications found in the literature, methodology is entangled with the specificity of the problem and researchers keep reinventing similar strategies under different names.
• Therefore, it is necessary to grasp the material characteristics of microstructure first of all in order to understand the behavior of the overall product.
• In acyclic coupling topologies, each submodel is started once and thus has a single synchronization point, while in cyclic coupling topologies, submodels may get new inputs a number of times, equating to multiple synchronization points.

A classification of multi-scale problems, based on the separation of the submodels in the SSM, and based on the relation between their computational domain (see 18 for more details). As discussed in the next section, only a few couplings seem to occur in these examples. Usually, it is common to perform the material test in order to determine Multi-scale analysis the material characteristics of the composite material.

• They correspond to an exchange of data, often supplemented by a transformation to match the difference of scales at both extremities.
• For example, Booth et al. 4 discuss a ‘boxed dynamics’ approach to accelerate atomistic simulations for capturing the thermodynamics and kinetics of complex molecular-dynamics systems.
• Compartmentalizing a model as proposed in MMSF means having fewer within-code dependencies, thereby reducing the code complexity and increasing its flexibility.
• They represent the data transfer channels that couple submodels together.
• Some examples of possible a priori estimates are discussed in the contribution by Abdulle & Bai 3 in applications to continuum fluid dynamics equations with multiscale coefficients based on homogenization theory.

Partly forthis reason, the same approach has been followed in modeling complexfluids, such as polymeric fluids. The first problem is that simplicity is largely lost.In order to model the complex rheological properties of polymer fluids,one is forced to make more complicated constitutive assumptions withmore and more parameters. For polymer fluids we are often interested inunderstanding how the conformation of the polymer Software engineering interacts with theflow.

• Homogenization methods can be applied to many other problems of thistype, in which a heterogeneous behavior is approximated at the largescale by a slowly varying or homogeneous behavior.
• It also means that for complex systems, the guessing game can be quite hard and less productive, as we have learned from our experience with modeling complex fluids.
• Finally, in the fourth step of the pipeline shown in figure 1, the different submodels are executed on a computing infrastructure.
• Brandt noted that there is noneed to have closed form macroscopic models at the coarse scale sincecoupling to the models used at the fine scale grids automaticallyprovides effective models at the coarse scale.

For example, Booth et al. 4 discuss a ‘boxed dynamics’ approach to accelerate atomistic simulations for capturing the thermodynamics and kinetics of complex molecular-dynamics systems. In the area of biological fluid flows, examples of multiscale models are discussed in the contribution by Li et al. 5 in application to multicomponent blood cell interactions in small capillary vessels. Validation is also discussed in the contribution by Wu et al. 6, who consider the interactions of platelets, blood flow and vessel walls that occur during blood clotting. Despite the differences in the application methods, there is a good deal of similarity found in the application of scale separation and computational implementations in many multiscale problems.