Discrete directions

Note that this latter method differs from the midpoint method, where one would use r(q +i/2) = (qn+i + qn)/2 instead of (7c) for r +i/2 in (7b). For highly oscillatory systems with k e, this can be a significant difference, because r is discretized directly in (7). An example in 4 below shows that the midpoint method can become unstable while (7) and (6) remain. stable. 

Thus, the scattering of a periodic lattice occurs in discrete directions. The larger the translation vectors defining the lattice, the smaller a i=1 3, and the more closely spaced the diffracted beams. This inverse relationship is a characteristic property of the Fourier transform operation. The scattering vectors terminate at the points of the reciprocal lattice with basis vectors a i=1>3, defined by Eq. (1.21). 

Algorri, M. E., M. Zimmermann, M. Hofmann-Apitius, and C. Friedrich. 2007c. Turning a Binary Image into a Vector Graph by Analyzing the Texture of Discrete Directions. Fraunhofer SCAI Internal report. 

Compared to the results from 1982 to 1986, the maximum velocities appear to be higher than those of the older data. This is because the older data were based on hourly means, whereas most of the new observations are 10-min means. Magnitude and speed values agree with the previous measurements. The mean surface flow exhibits two major discrete directions with a net outflow. According to the ADCP data, the transition between the outflow in the surface layer and the inflow down below starts at about 15 m at the site. The mean flow direction above bottom is confined within a narrow sector toward 100°-120°. 

In finite-volume methods, the integral formulation of the conservation laws over a small physical control volume is discretized directly. FVM employs a conservative discretization, that is, each species is guaranteed to be conserved, even for coarse meshes. In contrast, many traditional FDMs are not conservative. For example, owing to the nonlinear nature of the constitutive flux equations of ionic species in an a priori unknown electric field, FDM is nonconservative, even when constant physical... 

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It must be noted that classical geometric optics does not provide scattering matrix elements at discrete directions but only for finite-sized scattering-angle bins. The backscattering phase matrix elements shown in Fig. 9 have been calculated for the angular bin [179.95°, 180°]. i.e. for a bin size of 0.05°. 

To describe the heat transfer by radiation, FLUENT provides the following radiation models Rosseland model, PI model, discrete transfer radiation model (DTRM), surface to surface model, and discrete ordinates (DOs) model [32, 36]. These are explained in more detail in [38]. Due to its suitability for the entire range of optical density and the justifiable cost of computation, the DOs model is used to model the reactors used to process fuel. In contrast to other radiation models, the DOs model does not track individual heat rays but solves the radiative transfer equation (RTE) in the discrete directions. 

Discrete Ord inates Model The discrete ordinates method is capable of resolving the nonisotropic directional characteristics of radiative heat transfer by subdividing the directional space into discrete solid angles. After preselecting a set of representative discrete directions the radiative-transfer equations as well as the corresponding boundary conditions can be written as a set of equations for each direction, which is then solved. However, the equations for each direction depend on each other if scattering and reflection is considered. 

With the atomic scattering represented by a set of phase shifts 5 , we now assemble atoms into layers and calculate the latter s coherent elastic scattering, that is, diffraction. This is according to the so-called muffin-tin potential displayed in Figure 3.2.1.21a. In the example shown, the muffin-tin constant - equivalent to the real part of the inner potential Vor - consists of a contribution of a possible adlayer and that of the subsequent substrate. As we are dealing with a two-dimensionally periodic arrangement, we will get only intensities in discrete directions as described by the layer s reciprocal lattice vectors gy = gj, of the layer. Accordingly, this will create... 

The procedure to calculate fiber orientation is the same as explained above, but their implementation into explicit solvers and non-linear material models is more complex than it is for quasi-static load-cases and purely elastic material models. The fiber orientation is characterized by a so called orientation distribution function (ODE) that describes the chance of a fiber being oriented into a certain direction. For isotropic, elastic matrix materials an integral of the individual stiffness in every possible direction weighted with the ODE provides the complete information about the anisotropic stiffness of the compound. However, this integral can not be solved in case of plastic deformation as needed for crash-simulation. Therefore it is necessary to approximate and reconstruct the full information of the ODE by a sum of finite, discrete directions with their stiffness, so called grains [10]. Currently these grains are implemented into a material description and different methods of formulation are tested. 

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