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. Likewise, the number of submodel instances may be known in advance (single or static) or be determined at runtime (dynamic). This Programming language last option means a runtime environment will need to instantiate, couple and execute submodels based on runtime information. It is clear that a well-established methodology is quite important when developing an interdisciplinary application within a group of researchers with different scientific backgrounds and different geographical locations. A multi-scale modelling framework and a corresponding modelling language is an important step in this direction.
Multiple-scale Analysis
As was declared by Dirac back in 1929 (Dirac, 1929), the right physical principle for most of what we are interested in is already provided by the principles of quantum mechanics, there is no need to look further. We simply have to input the atomic numbers of all the participating atoms, then we have a complete model which is sufficient for chemistry, much of physics, material science, biology, etc. Dirac also recognized the daunting mathematical difficulties with such an approach — after all, we are dealing with a quantum many-body problem. With each additional particle, the dimensionality of the problem is increased by three. For this reason, direct applications of the first principle are limited to rather simple systems without much happening at the macroscale. Multiscale modeling refers to a style of modeling in whichmultiple models at different scales are used simultaneously todescribe a system.
Code, Data and Media Associated with this Article
- For example, the densities ofconserved quantities such as mass, momentum and energy densities areGoldstone modes.
- The execution of B amounts to specifying the boundary conditions for the computation.
- For example, themodels used at the finest level might be molecular dynamics or MonteCarlo models whereas the effective models used at the coarse levelscorrespond to some continuum models.
- They sometimes originate from physical laws ofdifferent nature, for example, one from continuum mechanics and onefrom molecular dynamics.
- On the other hand, it is not possible to coarse grain everything, as it incurs a loss of information at each step.
The structure of such an algorithm follows that of the traditionalmulti-grid method. In a two-level setup, at any macro time step ormacro Multi-scale analysis iteration step, the procedure is as follows. Multiple-scale analysis is a very general collection of perturbation techniques that embodies the ideas of both boundary-layer theory and WKB theory. Multiple-scale analysis is particularly useful for constructing uniformly valid approximations to solutions of perturbation problems. These methods are certainly more accurate than their single-scale, isotropic predecessors, but fall short when trying to analyze novel parts/materials for which there is no historical correlations or empirical guide-posts. While heterogeneity offers huge advantages in performance (making airplanes, space shuttles and lightweight cars possible), it also introduces difficulties in the engineering design.
The need for multi-scale analysis
Traditional multi-grid method is a way of efficiently solving a largesystem of algebraic equations, which may arise from the discretizationof some partial differential equations. For this reason, theeffective operators used at each level can all be regarded as anapproximation to the original operator at that level. In recentyears, Brandt has proposed to extend the multi-grid method to caseswhen the effective problems solved at different levels correspond tovery different kinds of models (Brandt, 2002). For example, themodels used at the finest level might be molecular dynamics or MonteCarlo models whereas the effective models used at the coarse levelscorrespond to some continuum models.