Volume element complex

The first discretization step of the finite integration method consists in the restriction of the electromagnetic field problem, which represent an open boundary problem, to a simply connected and bounded space domain fl containing the region of interest. The next step consists in the decomposition of the computational domain into a finite number of volume elements (cells). This decomposition yields the volume element complex G, which serves as computational grid. Assuming that i is brick-shaped, we have the volume element complex... 

The discrete curl-matrices C and C, and the discrete divergence matrices S and S are defined on the grids G and G, respectively, and depend only on the grid topology. The integral voltage- and flux state-variables allocated on the two different volume element complexes are related to each other by the discrete material matrix relations... 

By Max Bom s postulate, the produet of /(a ) and its complex conjugate r / (A ) times an infinitesimal volume element d x is proportional to the probability that a paitiele will be in the volume element d x... 

In quantum mechanics, the state of an atom or nucleus is described by a complex wave function i/r(ri, r2,..., t) such that xfnfr = ij/ 2 is the probability density of finding particles in volume elements d3rj centred on r, at time t. satisfies the Schrodinger equation... 

Classifications of data have two purposes (Hartigan, 1983 Gordon, 1981) data simplification (also called a descriptive function) and prediction. Simplification is necessary because there is a limit to both the volume and complexity of data that the human mind can comprehend and deal with conceptually. Classification allows us to attach a label (or name) to each group of data, to summarize the data (that is, assign individual elements of data to groups and to characterize the population of the group), and to define the relationships between groups (that is, develop a taxonomy). 

The representation of structure factors as vectors in the complex plane (Qr complex vectors) is useful in several ways. Because the diffractive contributions of atoms or volume elements to a single reflection are additive, each contribution can be represented as a complex vector, and the resulting structure factor is the vector sum of all contributions. For example, in Fig. 6.4, F represents a structure factor of a three-atom structure, in which f), f2, and f3 are the atomic structure factors. 

The ISV formulation is a means to capture the effects of a representative volume element and not all of the complex causes at the local level, and hence, an ISV will macroscopically average in some fashion the details of the microscopic arrangement. In essence, the complete microstructure arrangement is unnecessary as long... 

Given any complex system of heterogeneous catalytic first order reactions the mass balance on a differential volume element of the reactor at the height h yields the following system of differential equations for the j-th reaction component i) for the bubble phase... 

Cartesian coordinates are a convenient alternative representation for a spatial distribution function. Being uniform over the local space, the data structure obtained is easy to represent (access), to normalize, and to visualize. Use of a Cartesian representation becomes a necessity for complex or very flexible molecules. The principal drawbacks of this coordinate system are the size of the data structure it generates (typically about 1,000,000 elements), its inherent inefficiency (since the grid size is determined by the shortest dimension of the smallest feature one hopes to capture), and the fact that its sampling pattern is usually not commensurate with the structures one wants to represent (which can cause artificial surface features or textures when visualized). Obtaining sufficiently well-averaged results in more distant volume elements can be a problem if the examination of more subtle secondary features is desired. See Figures 7, 8 and 9 for examples of SDFs that have utilized Cartesian coordinates. 

Equation (21-139) reflects the evolution of granule size distribution for a particular volume element. When integrating this equation over the entire vessel, one is able to predict the granule-size distribution vs. time and position within the granulator. Lastly, it is important to understand the complexities of scaling rate processes on a local level to overall growth rate of the granulator. If such considerations are not... 

A complex set of equations, proposed by Riley, Stommel, and Bumpus (1949) (5) first introduced the spatial variation of the phytoplankton with respect to depth into the conservation of mass equation. In addition, a conservation of mass equation for a nutrient (phosphate) was also introduced, as well as simplified equations for the herbivorous and carnivorous zooplankton concentrations. The phytoplankton and nutrient equations were applied to 20 volume elements which extended from the surface to well below the euphotic zone. In order to simplify the calculations, a temporal steady-state was assumed to exist in each volume element. Thus, the equations apply to those periods of the year during which the dependent variables are not changing significantly in time. Such conditions usually prevail during the summer months. The results of these calculations were compared with observed data, and again the results were encouraging. 

Although we use ij in the text, it should strictly be written as ipip where 0 is the complex conjugate of ip. In the x-direction, the probability of finding the electron between the limits x and (x + dx) is proportional to ip x)ip (x) dx. In three-dimensional space this is expressed as ipip dr in which we are considering the probability of finding the electron in a volume element dr. For just the radial part of the wavefunction, the function is R r)R r). 

By resorting to a simulation, we are admitting our inability to handle the calculus for an electrochemical system described by complex continuous functions. So, we move backward one stage in sophistication and consider the electrolyte solution in terms of small, discrete volume elements. Throughout any element, the concentrations of all substances are... 

The photokinetic differential equations for the changes of the concentration of the reactant with time cannot be integrated in the given form. The reason is that the amount of the light absorbed I, of a reactant Aj per time and volume unit depends on the volume element where the reaction takes place and on the time of irradiation r in a very complex manner. This is one of the essential difficulties in photokinetics. This /, only can be calculated if the... 

Extension to more than four particles would be expeeted to generate even more complexity in the volume element, and would be subject to the additional difficulty that the number of distinet then exeeeds the number of... 

The theory presented in Section 3.3.1 is called the kinematic theory of scattering. Diffraction by a three-dimensional body is, however, more complex than suggested by Fig. 3.11. On the one hand, the primary radiation is attenuated by diffraction and the secondary beams may be rediffracted. Hence, the different volume elements do not all receive the same primary intensity for this reason, the kinematic theory does not obey the law of conservation of energy. On the other hand, the interference between the primary wave and the divers diffracted waves has been neglected. All these effects generally lead to diffracted intensities that are weaker than those predicted by the kinematic theory. These... 

As already mentioned, the form of the fundamental continuity equations is usually too complex to be conveniently solved for practical application to reactor design. If one or more terms are dropped from Eq. 7.2.a-6 and or integral averages over the spatial directions are considered, the continuity equation for each component reduces to that of an ideal, basic reactor type, as outlined in the introduction. In these cases, it is often easier to apply Eq. 7.1.a-l directly to a volume element of the reactor. This will be done in the next chapters, dealing with basic or specific reactor types. In the present chapter, however, it will be shown how the simplified equations can be obtained from the fundamental ones. 

Living tissue is a very heterogeneous and complex biomaterial. Impedance as a quantity is valid for a tissue volume and the contribution of the different volume elements (voxels) can be very different according to the electrode system sensitivity field. The results always represent some sort of mean values valid for a body region. 

It is important to be able to generalize the above approach to more complex stress distributions. For example, the body can be broken down into volume elements in which the principal stresses can be assumed constant. Equation (9.10) can then be used to describe the strength degradation of each element. Ritter (1992) has shown that the fatigue survival probability, Rf., for each element can be written... 

The complex interactions, as found in trickle-bed reactors in particular, between the chemical reaction in the catalyst particle and the simultaneous part processes for material, energy and momentum transport can be described mathematically by means of the balances of mass, energy and momentum. The mathematical models show different complexity depending on whether certain transport steps are considered or neglected. The balance of the mass or material change of a component j over time (known as material balance) is generally calculated for a defined reactor volume. For reactor-modeling purposes, an infinitesimal volume element (see Fig. 4.4) is assumed for said balance space. 

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