Decreasing shape errors

The above discussion of different DDA formulations implicitly assumed that particle shape can be exactly described by a set of cubes. Hence, all errors were due to the discretization of the integral equation and related simplifications. In this section so-called shape errors [68] are discussed, which are caused by violation of the above assumption by the real particle shape. First, it is important to note that none of the existing improvements is implemented in the production codes, partly because of the existing internal data structure capable of handling only several different values for polarizabilities of all dipoles [65], Hence, the main goal of this section is to put forward promising ideas for further development. 

A standard way to improve description of the particle shape in numerical solution of integral equations is adaptive discretization, using smaller dipoles near the particle surface. Application of this idea to the DDA is discussed in Ref. [40], but it is incompatible with the FFT acceleration (Sec. 2.4.2.2). Therefore, the only practically viable option is to keep the regular grid of cubical dipoles, but adjust the properties of the boundary dipoles. 

Evans and Stephens [50] proposed to modify the susceptibility of the boundary dipole using the Lorentz-Lorenz mixing rule  

A more advanced averaging, called the weighted discretization (WD), was proposed by Filler [116]. It modifies both the susceptibility and the self-term of the boundary dipoles. The particle surface, crossing the subvolume F,, is assumed linear and divides the subvolume into two parts the principal volume that contains the center and a secondary volume F/ with susceptibilities xf, xf and electric fields Ef = E/, Ef, respectively. Electric fields are considered constant inside each part and related to each other via a boundary condition tensor T,  

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The results of simulation experiments show that variations of the initial data by 100% change the stabilization time by no more than 30%, so that the distributions take shape in 4-8 years. One unstable parameter is river flow into the Arctic Basin. Figure 6.7 shows variations in simulation results under a change in river flow to the Arctic Basin. Radionuclear pollution is reduced by 80% when river flow decreases by 50%. While river flow increases by 50% the radionuclear pollution of the Arctic basin increases by only 58%. Hence, a 50% error in river flow estimate can cause a <100% deviation of the simulation results for radionuclear pollutants. As follows from the other curves of Figure 6.6, such deviations are less for heavy metals and oil hydrocarbons. 

With constant random error, the precision of an RIA increases as the slope of the dose-response curve increases and decreases as the error increases with constant slope. Future availability of monoclonal antibodies may greatly increase the steepness and improve the shape of the resulting dose-response curve. As with any analytical techniques, it is crucial to appreciate the confidence intervals which one has at various points of the dose-response curve, in addition to the many measurement and collection errors which may be made before the immunoassay is employed. 

The dependence of 50 on E for molecules in different vibrational and rotational states [19, 20] shown in Fig. 2 are characterised by an S-shape curve, where 50 saturates at high energy and falls exponentially as the energy decreases before flattening out at low energy. The sticking function is represented by a sigmoid curve based on an error function form ... 

The relation between the peroxide concentration on the X-ray-opaque powder and the gel time of bone cement could be seen in Figure 30.27, in which the time at maximal temperature was used instead of gel time or set time to avoid unnecessary error, since the temperature profiles seemed to have identical shape. As the DPPH consumption increased, the gel time decreased (i.e, faster polymerization) regardless of the types of plasma treatment, as seen in the relation between the fatigue properties and the peroxide concentration. The HMDSO plasma-based treatments showed a more pronounced effect than that by the methane plasma-based treatments. 

The variation of the value of for a powdery chemical of the TD type with the value of is also of the same order of magnitude as that with the value of ate. That is, a decrease of 0.25 in the value of the magnitude of which is the maximum error conceivable in the assignment of the value of dc to the shape of the chemical and corresponds to about 10 % of the value of for the shape of an average fiber drum used to carry a powdery chemical of the TD type, causes a decrease of about 0.5 K in the value of Tc calculated by applying Eq. (79), keeping the values of the other variables in the equation constant. 

In this example, lattice parameters and the zero shift correction have a substantial impact on the quality of the fit and the weighted profile residual, Rwp, decreases nearly two-fold (from 24 to 12 %), while the refinement of peak shape parameters decreases R p by only 4 %. Therefore, in this case lattice parameters should have been refined first. However, it is not always obvious which parameters are more important and should be released at a particular stage of the least squares refinement. Because of this, in complex cases a trial-and-error approach is often employed. ... 

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