Macroscale modeling techniques

A main distinction has been made between deterministic and stochastic modeling techniques. A further distinction has been proposed based on the scale for which the mathematical model must be derived (eg, micro-, meso-, and/or macroscale). Notably, the complexity of the model approach depends on the desired model output. Detailed microstractural information is only accessible using advanced modeling tools but these are associated with an increase high in computational cost. The advanced models allow one to directly relate macroscopic properties to the polymer synthesis procedure and, thus, to broaden the application market for polymer products, based on a fundamental understanding of the polymerization kinetics and their link with polymer processing. 

Empirical approaches are useful when macroscale HRR measurements are available but little or no information is available regarding the thermophysical properties, kinetic parameters, and heats of reaction that would be necessary to apply a more comprehensive pyrolysis model. Although these modeling approaches are crude in comparison with some of the more refined solid-phase treatments, one advantage is that all required input parameters can be obtained from widely used bench-scale fire tests using well-established data reduction techniques. As greater levels of complexity are added, establishing the required input parameters (or material properties ) for different materials becomes an onerous task. 

Thus far, the discussion has assumed that the materials engineer has a desired micro-stmcture in mind. But how is it known which microstructure is optimal for a given application Microstructure design is an emerging field that focuses on producing microstructures that meet, or exceed, specified macroscale properties or performance criteria. This requires a combination of empirical techniques, mathematical modeling, and numerical simulation. A complicating factor has always been the enormous number... 

The most powerful property of the detailed microbalance models, especially in combination with visualization techniques, is the a priori prediction of (observable) macroscale phenomena. This can be particularly helpful in reducing the required experimental effort. Important problems are the amount of detailed information required for the microscale transport equations and the large progranuning and computational efforts required to solve specific problems. Nevertheless, these types of models, by generating insight in the micro- and... 

In multiscale modeling approaches the microscopic behavior of a system is linked by a compression operator to the macroscopic state variable. The strategy is then to use the microscopic model to provide necessary information for extracting the macroscale behavior of the system. Such a combined macro-microscale technique is supposed to be much more efficient than solving the full microscopic... 

Control theory is well developed for systems well described by linear models (see, e.g., Zhou [8]). (A linear model is, essentially, one in which doubling the inputs, the actuator actions, will double the outputs, the measurements.) All systems are nonlinear, but for small excursions about equilibria or prescribed trajectories, many systems can be linearized and effectively controlled using linear control techniques. For systems that cannot be approximated as linear, nonlinear control techniques [9] are available to varying degrees of success for subclasses of systems for Hamiltonian systems (e.g., [10]), for nonlinear systems that can be made linear by a clever choice of variable transformation (e.g., [11], Chap. 10 in [12]), or other system subclasses. Nonlinear control design has been motivated by systems that commonly appear in applications these have traditionally been in electronic circuits and mechanical, fluid, and chemical systems on the macroscale. 

The most advanced material model presently available for UHMWPE is the HM. This model focuses on creating a mathematical representation of the deformation resistance and flow characteristics for conventional and highly crosslinked UHMWPE at the molecular level. The physics of the deformation mechanisms establish the framework and equations necessary to model the behavior on the macroscale. As already mentioned, to use the constitutive model for a given material requires a calibration step where material-specific parameters are determined. A variety of numerical methods may be used to determine the material-specific parameters for a constitutive theory. In the previous section we employed a numerical optimization technique to identify the material parameters for the constitutive theory. 

SOFC modelling, be it electrochemical or mechanical, is multiscale. The knowledge gained at the microscale must be implemented at the stack macroscale, through homogenisation techniques or combined homogenisation/localisation procedures, for instance. This is currently seldom achieved in thermo-electro-chemical models [47] and almost never applied to mechanical aspects. Therefore, a quick overview of the modelling approaches at the smaller scale is worthwhile. 

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