Equation-free modeling

Equation-Free Modeling For Complex Systems loannis G. Kevrekidis, Princeton University... 

With the fomi of free energy fiinctional prescribed in equation (A3.3.52). equation (A3.3.43) and equation (A3.3.48) respectively define the problem of kinetics in models A and B. The Langevin equation for model A is also referred to as the time-dependent Ginzburg-Landau equation (if the noise temi is ignored) the model B equation is often referred to as the Calm-Flilliard-Cook equation, and as the Calm-Flilliard equation in the absence of the noise temi. 

With due deference to the myriad mathematics dissertations and journal articles on the subject of optimization, f will briefly mention some of the general approaches to finding an optimum and then describe the recommended methods of experimental design in some detail. There are two broad classes that define the options systems that are sufficiently described by a priori mathematical equations, called models, and systems that are not explicitly described, called model free. Once the parameters of a model are known, it is often quite trivial, via the miracles of differentiation, to find the maximum (maxima). 

The form of the free energy functional G appearing in the Polarizable Continuum Model is discussed in refs [35-37], Recently, Mennucci and Cammi have extended their integral equation formalism model for medium effects on shielding to the NMR shielding tensor for solutions in liquid crystals [38,39],... 

The resulting set of 10 equations, assuming toroidal symmetry and replacing the radial component of the ion momentum balance equation by an ad hoc diffusions ansatz (likewise the other radial transport coefficients are replaced by ad hoc anomalous expressions) is the basis for most current edge plasma simulation models. These anomalous ad-hoc coefficients are free model parameters. They, and their empirical scalings, can be determined by comparison with experimental plasma profile data, if one can be sure that all other terms in the equations, and in particular the source terms Sm resulting from atomic and molecular processes, are accurately known and implemented. 

To parameterize the polymersome model, the identification of the virial coefficients, vaij and wapv is driven by the requirement that the amphiphiles described by (10) and (12) should create a stable bilayer with given material properties. Assuming that the hydrophobic interior should be in a melt state, the coefficients vaa and Waaa are determined such that (12) enforces the A-blocks to create a melt in equilibrium with its vapor which, in a solvent free model, represents the surrounding water. It can be shown, from (12), that the equation of state of such a homogeneous melt, within mean-field approximation, is [138]... 

Finite element analysis, free equation-based modeling, general periodic boundary conditions, evaluation of material, energy balances... 

Consider a simplified illustration of the foregoing QSAR examples. Consider a list of normal alkanes together with their water solubility and boiling points. A plot shows that solubility (in logarithmic form) is linear with number of carbon atoms and that boiling point is nonlinear. Such a relation is a QSAR based on the simple structure feature, number of carbon atoms. A linear equation captures all the structure information available in this data set. (The structure information could, of course, be represented in other ways, such as number of methylene groups, number of hydrogen atoms, number of carbon-carbon bonds, etc.) It is important to note here that no assumption has been made about the relation between water solubility and number of carbon atoms. This is an example of what Adamson has called a mechanism-free model. 

The resonator with the highest symmetry is the spherical resonator. A treatment of oscillations in a spherical cavity was already given by Rayleigh in 1894 A recent discussion may be found in Ref. In the simple loss-free model, sound is described by the homogeneous wave equation ... 

In the above equation 6 is an adjustable parameter to achieve a suitable fit, while the parameter f encapsulates the driving force of the diffusion-controlled mechanism by means of the Doolittle (14) equation, to model the free volume as follows ... 

In Zeytounian (1998) I have considered, for a thin film provblem, three main situations in the relation with the magnitude of the characteristic Reynolds number and discussed various model equations - these model equations are analyzed from various point of view but the central intent of the my review paper is to elucidate the role of the Marangoni number on the evolution of the free-surface in space-time. 

Fares, E., Schrooder, W. A general one-equation turbulence model for free shear and waUbounded flows. Flow, Turbrflence and Combustion 73, 187-215 (2004)... 

Flow rate problem with skin. Equation 18-20d provides the boundary condition for the flow rate problem without skin effects, and an exact solution can be obtained in closed analytical form (Proett and Chin, 1998). However, it is possible to obtain an exact solution for the more difficult problem including skin effects (Proett and Chin, 2000). Since this more general solution is available, we will not discuss the skin-free model in this book. Perhaps the greatest difficulty in formulating the problem correctly lies in the form of the skin model used. Conventionally, the ad hoc skin model p, = p - SR 9p/ar is... 

The brief discussion above shows that the structure of a polymer electrolyte and the ion conduction mechanism are complex. Furthermore, the polymer is a weak electrolyte, whose ions form ion pairs, triple ions, and multidentate ions after its ionic dissociation. Currently, there are several important models that attempt to describe the ion conduction mechanisms in polymer electrolytes Arrhenius theory, the Vogel-Tammann-Fulcher (VTF) equation, the Williams-Landel-Ferry (WLF) equation, free volume model, dynamic bond percolation model (DBPM), the Meyer-Neldel (MN) law, effective medium theory (EMT), and the Nernst-Einstein equation [1]. 

Within the framework of the same dielectric continuum model for the solvent, the Gibbs free energy of solvation of an ion of radius and charge may be estimated by calculating the electrostatic work done when hypothetically charging a sphere at constant radius from q = 0 q = This yields the Bom equation [13]... 

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