Time Multiscale Modeling

Bridging the gap between the production and detection times thus requires using various computational strategies and connecting them carefully in suitable time domains. This chapter aims to present the time multiscale modeling approach in more details in the context of atomic and molecular... 

Our second application of the time multiscale modeling approach aims to describe the sequential emission of carbon dimers from a purely theoretical perspective, without any adjustment. Because the absolute dissociation rates... 

Multiscale modeling of process operations. The description of process variables at different scales of abstraction implies that one could create models at several scales of time in such a way that these models communicate with each other and thus are inherently consistent with each other. The development of multiscale models is extremely important and constitutes the pivotal issue that must be resolved before the long-sought integration of operational tasks (e.g., planning, scheduling, control) can be placed on a firm foundation. 

Multiscale process identification and control. Most of the insightful analytical results in systems identification and control have been derived in the frequency domain. The design and implementation, though, of identification and control algorithms occurs in the time domain, where little of the analytical results in truly operational. The time-frequency decomposition of process models would seem to offer a natural bridge, which would allow the use of analytical results in the time-domain deployment of multiscale, model-based estimation and control. 

Multiscale modeling can reduce the product development time by alleviating costly trial and error iterations. 

The recent growth of multiscale modeling has permeated every material type known to mankind regarding structural members. Although most of the work has been focused on metal alloys as they have been used the most over time as reliable structural materials, multiscale modeling has also been employed for ceramics and polymer systems (both synthetic and biological). 

The need for multiscale modeling of biological networks in zero-dimensional (well mixed) systems has been emphasized in Rao et al. (2002). The multiscale nature of stochastic simulation for well-mixed systems arises from separation of time scales, either disparity in rate constants or population sizes. In particular, the disparity in species concentrations is commonplace in biological networks. The disparity in population sizes of biological systems was in fact recognized early on by Stephanopoulos and Fredrickson (1981). This disparity in time scales creates slow and fast events. Conventional KMC samples only fast events and cannot reach long times. 

One question that arises is if one uses multiscale simulation to predict systems behavior from first principles, then why does one need to carry out parameter estimation from experimental data The fact is that model predictions using even the most accurate QM techniques have errors. In the foreseeable future, one would have to refine parameters from experiments to create a fully quantitative multiscale model. Furthermore, for complex systems, QM techniques may be too expensive to carry out in a reasonable time frame. As a result, one may rely on estimating parameters from experimental data. Finally, an important, new class of problems arises when one has to estimate parameters of... 

As the structure, dynamics and properties are determined by phenomena on many length and time scales physical modelling is subdivided into the quantum mechanical, atomistic, mesoscale, microscale and continuum levels, while research into the way in which these levels are linked is known as hierarchical or multiscale modelling. The typical structural levels arising in the polymer field are shown Figure 1. 

The major limitation of the approaches to multiscale modeling discussed thus far is the timescale. In each of these examples, there are atomic vibrations (on the order of 10 seconds) that need to be followed. This pins down the total simulation time to 0(10 seconds for reasonable calculations. There are many clever multiple time step methods for improving efficiency (e.g., Nakano 1999) by using a quatemion/normal mode representation for atoms that are simply vibrating or rotating, but this buys only a factor of 0(10). 

For each phenomenon, there are also many elements involved which determine the behaviour of each phenomenon. These phenomena are described by a wide range of characteristic time and length values. For the case of CVI fabrication of fibre-reinforced ceramic-matrix composites, the diameter of a molecule and the thickness of the interfacial phase are about 10 1 run and 102nm respectively, whilst the sizes of the substrate/component and the reaction are around 1 m. In addition, elementary chemical reactions occur in a time range of 10 " to 10 4 s, the time for heat transfer and mass transfer is around 1 s to 10 min. By contrast, the total densification time for one CVI run is as long as approximately 102 h. In such cases, it is necessary to establish multiscale models to understand and optimise a CVD process. 

Historic Examples of Multiscale Modeling The treatment of discrete and continuous representations of the vibrating string in Mathematical Thought from Ancient to Modern Times by Morris Kline, Oxford University Press, New York New York 1972, is enlightening. Here it is evident that both the continuous and discrete representations had features that recommended them as the basis for further study. 

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