Finite field model

Fermi contact term 308 Fermi correlation 186 Fermi energy 213 Fermi level 213 Field, scalar 7 Field, vector 7 Finite field model 289 Flux 10, 318 Force 11... 

Descriptions given in Section 4 of this chapter about the imposition of boundary conditions are mainly in the context of finite element models that use elements. In models that use Hermite elements derivatives of field variable should also be included in the set of required boundai conditions. In these problems it is necessary to ensure tluit appropriate normality and tangen-tiality conditions along the boundaries of the domain are satisfied (Petera and Pittman, 1994). 

The LST is a finitely parameterized model of the action of a given CA rule, >, on probability measures on the space of configurations on an arbitrary lattice. In a very simple manner - which may be thought of as a generalization of the simple mean field theory (MFT) introduced in section 3.1.3. - the LST provides a sequence of approximations of the statistical features of evolving CA patterns. 

Quantum mechanical models at different levels of approximation have been successfully applied to compute molecular hyperpolarizabilities. Some authors have attempted a complete determination of the U.V. molecular spectrum to fill in the expression of p (15, 16). Another approach is the finite-field perturbative technique (17) demanding the sole computation of the ground state level of a perturbated molecule, the hyperpolarizabilities being derivatives at a suitable order of the perturbed ground state molecule by application of the Hellman-Feynman theorem. 

The finite-temperature field theory has been the most popular approach to equilibrium phase transitions (L. Dolan et.al., 1974). The effective potential of quantum fluctuations around a classical background provides a convenient tool to describe phase transitions. The symmetry breaking or restoration mechanism can be illustrated by a scalar field model with broken symmetry... 

The above observation suggests an intriguing relationship between a bulk property of infinite nuclear matter and a surface property of finite systems. Here we want to point out that this correlation can be understood naturally in terms of the Landau-Migdal approach. To this end we consider a simple mean-field model (see, e.g., ref.[16]) with the Hamiltonian consisting of the single-particle mean field part Hq and the residual particle-hole interaction Hph-... 

An alternative approach is to apply stronger fields and only use energies calculated for positive field strengths in generating the polynomial fit. In this case the energy is a function of both odd and even powers in the polynomial fit. We will show that the dipole moments derived from our non-BO calculations with the procedure that uses only positive fields and polynomial fits with both even and odd powers match very well the experimental results. Thus in the present work we will show results obtained using interpolations with even- and odd-power polynomials. Methods other than the finite field method exist where the noise level in the numerical derivatives is smaller (such as the Romberg method), but such methods still do not allow calculation of odd-ordered properties in the non-BO model. 

Field models estimate the fire environment in a space by numerically solving the conservation equations (i.e., momentum, mass, energy, diffusion, species, etc.) as a result of afire. This is usually accomplished by using a finite difference, finite element, or boundary element method. Such methods are not unique to fire protection they are used in aeronautics, mechanical engineering, structural mechanics, and environmental engineering. Field models divide a space into a large number of elements and solve the conservation equations within each element. The greater the number of elements, the more detailed the solution. The results are three-dimensional in nature and are very refined when compared to a zone-type model. 

Thermal/structural response models are related to field models in that they numerically solve the conservation of energy equation, though only in solid elements. Finite difference and finite element schemes are most often employed. A solid region is divided into elements in much the same way that the field models divide a compartment into regions. Several types of surface boundary conditions are available adiabatic, convection/radiation, constant flux, or constant temperature. Many ofthese models allow for temperature and spatially dependent material properties. 

A. Crisanti and F. Ritort, Potential energy landscape of finite-size mean-field models for glasses. 

This approach of subdividing space into an increasing number of discrete pieces provides the basis for many numerical computer models (e.g., the so-called finite difference models) an example will be discussed in Chapter 23. Although these models are extremely powerful and convenient for the analysis of field data, they often conceal the basic principles which are responsible for a given result. Therefore, in the next chapter we will discuss models which are not only continuous in time, but also continuous along one or several space axes. In this context continuous in space means that the concentrations are given not only as steadily varying functions in time [QY)], but also as functions in space [C,(r,x) or C,(t,x,y,z)]. Such models lead to partial differential equations. A prominent example is Fick s second law (Eq. 18-14). 

Once a finite element formulation has been implemented in conjunction with a specific element type — either 1D, 2D or 3D — the task left is to numerically implement the technique and develop the computer program to solve for the unknown primary variables — in this case temperature. Equation (9.19) is a form that becomes very familiar to the person developing finite element models. In fact, for most problems that are governed by Poisson s equation, problems solving displacement fields in stress-strain problems and flow problems such as those encountered in polymer processing, the finite element equation system takes the form presented in eqn. (9.19). This equation is always re-written in the form... 

Figure 6. Remanence enhancement in a two-phase Nd2Fe 4B/Fe3B magnet containing 343 grains. Left Finite element model of the grain structure. Right Magnetization distribution in a slice plane for zero applied field. The arrows denote the magnetization direction projected on a slice plane.

Several hybrid simulations on crystal growth can be found in recent literature. Examples include dendritic solidification by coupling finite-different discretization of a phase field model to a MC simulation (Plapp and Karma, 2000), coupling a finite difference for the melt with a cellular automata for the solidification (Grujicic et al., 2001), a DSMC model for the fluid phase with a Metropolis-based MC for the surface to address cluster deposition onto substrates (Hongo et al., 2002 Mizuseki et al., 2002), a step model for the surface processes coupled with a CFD simulation of flow (Kwon and Derby, 2001) (two continuum but different feature scale models), an adaptive FEM CVD model coupled with a feature scale model (Merchant et al., 2000), and one-way coupled growth models in plasma systems (Hoekstra et al., 1997). Some specific applications are discussed in more detail below. 

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