There is probably a performance benefit to using a single data structure, but separating the submodels provides more clarity and provides a path to directing parallelization efforts towards only parts of a code. We review a methodology to design, implement and execute multi-scale and multi-science numerical simulations. We identify important ingredients of multi-scale modelling and give a precise definition of them. Our framework assumes that a multi-scale model can be formulated in terms of a collection of coupled single-scale submodels. Our approach has been successfully applied to an increasing number of applications from different fields of science and technology.
Multiple-scale Analysis
The results enable prediction of the macroscopic behavior by the macro structural analysis. Further, it is possible to predict the microscopic behavior by going back to the micro structure analysis again. Urban planners use multiple-scale analysis to design sustainable and resilient cities. W. Zhang, “Analysis of the heterogeneous multiscale method for dynamic homogenization problems,” preprint.
(b). Submodel execution loop and coupling templates
Multiple-scale analysis is a global perturbation scheme that is usefulin systems characterized by disparate time scales, such as weakdissipation in an oscillator. These effects could be insignificant on short time scales but become importanton long time scales. Classical perturbation methods generally breakdown because of resonances that lead to what are called secularterms. Usually, it is common to perform the material test in order to determine the material characteristics of the composite material. However, characteristics obtained from this test are actually characteristics of the macro-structure instead of the microstructure. In addition, there is a possibility that if the material could be on the design variables, product development can be performed with great features that did not exist before.
Hierarchy of Scales
Note that the SSM can give a quick estimate of the CPU time gained by the scale splitting process when it concerns a mesh-based calculation. The CPU time of a submodel goes as (L/Δx)d(T/Δt), where d is the spatial dimension of the model, and (Δx,L) and (Δt,T) are the lower-left and upper-right coordinates of the rectangle shown on the SSM. Therefore, the computational time of the system in figure 2a is likely to be much larger than those in figure 2b. Whilethe simulation and analysis technology for metal structures such as car framesis quite robust, the analysis of novel “advanced materials” how to hire a software developer is lagging. Theconsensus is that by using conventional techniques (standard FEA) it is notpossible to accurately simulate these materials without extensive experimentaland empirical “calibration” data.
The Multiscale Modelling and Simulation Framework
Thus, the introduction of new materials intoa structure results in increased time to market and costs. In the example of the growth of biological cells subjected to the blood flow shear stress, there is a clear time-scale separation between the two processes (see figure 7 and 22). Therefore, the converged flow field is first sent from the physical model BF to the biological one, in order to define the SMC proliferation rate in SMC (OBFf→SSMC). Then, the new geometry of the cells induces a new boundary condition for the flow, which must be recomputed (). The five possible relations between two submodels in the SSM. Then, with submodels C, D, E and F, we illustrate scale separation either in time or in space, or both.
Data Availability Statement
Multiple-scale analysis is a transformative approach that allows us to unravel the complexities of our world, from the behavior of subatomic particles to the dynamics of ecosystems. By embracing the hierarchy of scales, understanding interconnectedness, and exploring emergent properties, we gain profound insights into the systems that surround us. Whether in scientific research, engineering innovation, healthcare, or environmental conservation, multiple-scale analysis empowers us to make informed decisions and tackle complex challenges with confidence. As we stand on the cusp of an era driven by data and interdisciplinary collaboration, the significance of multiple-scale analysis in shaping our future cannot be overstated.
- According to our definitions, the sender of information is either Oi or Of.
- Supposing that the characteristics of the composite material can be homogenized, we could predict the behavior of the overall product.
- E, “Multiscale modeling of dynamics of solids at finite temperature,” J.
- The scale bridging will take place in the coupling, and this will be described by the multi-scale modelling language.
- Mappers are useful to optimize a coupling, for instance to avoid repeating twice the same data transformation for two different recipients.
Therefore, the coupling templates are Oi→S, Oi→B, Oi→finit and Of→S, Of→B, Of→finit. Finally, we define two observation operators, Oi and Of, which compute some desired quantities from the model variables. The subscript i and f are for intermediate and final observations, respectively. They are described in 14 and will not be discussed further here. However, a performance study of DMC can be found in another contribution in this Theme Issue 10.
The MML description of the in-stent restenosis model (see also figure 7). Here, BF stand for the blood flow submodel, SMC for the biological growth of smooth muscle cells, DD for drug diffusion and IC for injury score (the initial condition). As shown in figure 10, the extremities of conduits carry some symbols.
E, “Multiscale modeling of dynamics of solids at finite temperature,” J. Alternatively, modern approaches derive these sorts of models using coordinate transforms, like in the method of normal forms,3 as described next. Examples of utility of multi-scale analysis are shown below. Engquist, “The heterogeneous multi-scale method https://wizardsdev.com/en/news/multiscale-analysis/ for homogenization problems,” submitted to SIAM J. Multiscale Modeling and Simulations.
