Boundary artificial

If the simulated system uses periodic boundary conditions, the logical long-range interaction includes a lattice sum over all particles with all their images. Apart from some obvious and resolvable corrections for self-energy and for image interaction between excluded pairs, the question has been raised if one really wishes to enhance the effect of the artificial boundary conditions by including lattice sums. The effect of the periodic conditions should at least be evaluated by simulation with different box sizes or by continuum corrections, if applicable (see below). 

The translational diffusion coefficient in Eq. 11 can in principle be measured from boimdary spreading as manifested for example in the width of the g (s) profiles although for monodisperse proteins this works well, for polysaccharides interpretation is seriously complicated by broadening through polydispersity. Instead special cells can be used which allow for the formation of an artificial boundary whose diffusion can be recorded with time at low speed ( 3000 rev/min). This procedure has been successfully employed for example in a recent study on heparin fractions [5]. Dynamic fight scattering has been used as a popular alternative, and a good demonstra-... 

One of the basic elements of the computational algorithm is the determination of dependent variables at the inlet/outlet boundaries of a computational domain representing a finite length combustor. The essence of the problem lies in the fact that the nonstationary flow field has to be considered throughout a whole (unbounded) physical space, and only in this case the problem is mathematically well-posed. When solving a specific problem numerically, one has to consider a computational domain of a finite size, in which boundary conditions at artificial boundaries are to be imposed. 

Artificial boundaries have been defined as boundaries at which the occupation probability of one or more sites obeys special equations, not covered by the analytic expression r(n) and g(n) that apply to the other n. The variety of possible artificial boundaries is, of course, endless. A restricted class are the pure boundaries, defined as those in which only the end site requires a special equation. Another subdivision is in reflecting boundaries those which conserve total probability and absorbing boundaries, at which probability disappears. The latter definition requires comment. 

The site n = 0 is a pure, artificial boundary. The equations can be interpreted as a random walk on — oo < n < oo in which the transitions from — 1 to 0 are impossible a drunkard s walk with a bottomless pit on one side. The total probability is not conserved,... 

We shall now list some examples of artificial boundaries. One example with a reflecting pure boundary has already been encountered in (3.11). 

Only a few problems with artificial boundaries can be treated by the reflection principle. In this section the method of normal modes is expounded, which in principle is able to deal with artificial boundaries of all types. Rather than develop this method in full generality we demonstrate it on the example (ii) of section 7 the model for diffusion-controlled chemical reactions. 

In order to obtain an analytic solution they subsequently took the Wvfl belonging to a quantized harmonic oscillator, which is not unreasonable for the vibrations of a diatomic molecule ). Thus their M-equation has an artificial boundary ... 

The section of the table topped by the heavy bar indicates the "packed CEC" region and it is these areas which are to be covered in this chapter. There is considerable scope for overlap between the described cells, and many further divisions of these artificial boundaries are possible, for example pressure-assisted electrochromatography (PEC) describes a continuum from pure CEC to pure pLC. 

In the attempt to focus the volume on the fundamental chemical physics and biophysics of electromagnetic-field interactions, and to connect hitherto disparate areas of research, an artificial boundary had to be drawn to encompass well-defined aspects of such a focus and hence exclude a multitude of others. In order that the volume may not only depict the state of the art, within its stated objectives, but also stimulate its development, one may only hope that such exclusions will not have been pro-crustean. 

The geometry of the flow is supposed infinite i.e. very large in one direction), such as in the case of a flow around an obstacle, or in a long die. The flow at infinity is assumed to be known (uniform, PoiseuiUe flow,. ..). For computational purposes, one introduces artificial boundaries, at a finite, hopefully not too large, distance. The problem is to define which conditions to impose on the artificial boundaries in order to obtain a solution of the truncated problem, which is as close as possible to the solution of the original problem. 

From physical considerations (plane waves travelling through the fluid), one gets boundary conditions which make the artificial boundaries transparent to the waves leaving the computational domain and which absorb the waves entering the domain (other than those generated by the solution at infinity). 

We suppose that the artificial boundaries are far enough away in order to justify the linearizations around the uniform solution at infinity. The linearized problem reads... 

To be of any practical use, the artificial boundary conditions have to be stable in particular, they should not depend on rounding errors. The stability analysis is made on the linearized problem by computing the time evolution of some solution norms. 

L. Halpern, Artificial boundary conditions for incompletely parabohc perturbations of hyperbolic systems, SIAM J. Math. Anal., 87 (1985) 213-251. 

Cluster calculations can be made for any atomic arrangement without imposing artificial boundary conditions. The convergence of properties of interest with the cluster size can be explicitly revealed. This is extremely important when we deal with imperfection-induced states that are localized within some extent. 

First-principles energetics is very powerful for computational structure characterization. The main difficulty in applying this method to GO structure study is the amorphous nature of GO. When the number of atoms is very large, it is not feasible to compare energy for all possible structures. Therefore, first-principles calculations are typically limited to a small system. For this reason, an artificial periodic boundary condition is required for GO structure study. Such an artificial boundary condition of course will weaken the power of predictability of first-principles energetics for GO. However, useful insights can still be obtained from such calculations, especially in prediction of local building blocks. 

In principle and also in practice, the small quantity can be viewed as a legitimate means for introducing an optical potential that is used in practice to avoid the artificial boundary reflection. Such a view is promoted by Seideman and Miller (56). Thus (7 , ) can be calculated by a half Fourier transform on the time-dependent wave function evaluated at a large distance R,, ... 

Baier, R.E., Loeb, G.I., and Wallace, G.T., Role of an Artificial Boundary in Modifying Blood Proteins, Federation Proceedings, 1523-1538, 1971. 

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