Numerical regularization parameters

Contact conditions add even more difficulty and complexity to an already very complex and difficult analysis of rubber products and tires. Contact conditions are unilateral and need to be constantly checked during the incremental nonlinear analysis. In addition, they are not smooth, thus degrading the performance of nonlinear solvers. A number of numerical regularization parameters need to be introduced to prevent chattering and ensure robustness of a finite element analysis (FEA) with frictional contact. 

The basic principles used for determining the regularization parameter a were discussed in the previous sections of the book. We introduced in Chapter 2 a simple numerical method to determine the parameter a. Consider for example the progression of numbers... 

Mead [23] noted that to overcome the ill-posedness it is necessary to provide additional information and to use specialized numerical methods. In the case of the step function choice for F( t,M) he was able to find an analytical solution to the inversion problem, while for other choices, he recommended the use of the CONTIN software, originally designed for use in treating light scattering data [28]. This makes use of a non-linear regularization parameter that is adjusted in accord with the noise level in the data. 

In this way and by numerical evaluation, Driessens (2) proved that the experimental activities could be explained on the basis of substitutional disorder, according to Equation (27), within the limits of experimental error. It seems, therefore, that measurements of distribution coefficients and the resulting activities calculated by the method of Kirgintsev and Trushnikova (16) do not distinguish between the regular character of solid solutions and the possibility of substitional disorder. However, the latter can be discerned by X-ray or neutron diffraction or by NMR or magnetic measurements. It can be shown that substitutional disorder always results in negative values of the interaction parameter W due to the fact that... 

From the physics point of view, the system that we deal with here—a semiflexible polyelectrolyte that is packaged by protein complexes regularly spaced along its contour—is of a complexity that still allows the application of analytical and numerical models. For quantitative prediction of chromatin properties from such models, certain physical parameters must be known such as the dimensions of the nucleosomes and DNA, their surface charge, interactions, and mechanical flexibility. Current structural research on chromatin, oligonucleosomes, and DNA has brought us into a position where many such elementary physical parameters are known. Thus, our understanding of the components of the chromatin fiber is now at a level where predictions of physical properties of the fiber are possible and can be experimentally tested. 

The influence of steric effects on the thermodynamic parameters of extraction has been discussed in numerous publications (see the review in Ref. 12 and references therein). For a small series of selected extractants, a decrease of log Kex as a function of the length of alkyl substituents has been discovered.6970 However, in other cases, this dependence passes through a maximum or is not regular at all.71 The point is that the increase in the volume and branching of the substituents leads to some changes of molecular properties decrease of solubility in the aqueous phase, increase of steric hindrance upon the complexation, decrease of the extractant s aggregation, etc.12 Some of these factors strengthen extraction, but others weaken it. 

The solubility parameter is valid only for regular solutions (where the excess entropy is equal to zero) and mainly for nonpolar classes of substances. Of the numerous suggested improvements that have been made, the one by Hansen is worth mentioning. Here the solubility parameter is the sum of three parts (Barton, 1983) corresponding to a nonpolar or pure dispersive (8,/), polar (8P) and hydrogen bonding (8/,) based interactions ... 

In Figs. 3 and 4, we show numerical results for the quantum kicked rotor as in Section II. A. The system parameters are chosen to pick up a regular dynamics (Fig. 3) and a chaotic dynamics (Fig. 4), and the others are T = 400 and a = 1. The optimal field after 100 iterations for the regular case (Fig. 3a) is much simpler than that for the chaotic case (Fig. 4a). (See also Figs. 3b and 4b.) This is because more states are involved in the latter chaotic process. 

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