Numerical Computation

The complexity of polymeric systems make tire development of an analytical model to predict tlieir stmctural and dynamical properties difficult. Therefore, numerical computer simulations of polymers are widely used to bridge tire gap between tire tlieoretical concepts and the experimental results. Computer simulations can also help tire prediction of material properties and provide detailed insights into tire behaviour of polymer systems. A simulation is based on two elements a more or less detailed model of tire polymer and a related force field which allows tire calculation of tire energy and tire motion of tire system using molecular mechanisms, molecular dynamics, or Monte Carlo teclmiques 1631. 

D. Okunbor and R. D. Skeel. Explicit canonical methods for Hamiltonian systems. Working document. Numerical Computing Group, University of Illinois at UrbanarChampaign, 1991. 

A key observation for our purposes here is that the numerical computation of invariant measures is equivalent to the solution of an eigenvalue problem for the so-called Frobenius-Perron operator P M - M defined on the set M. of probability measures on F by virtue of... 

The many approaches to the challenging timestep problem in biomolecular dynamics have achieved success with similar final schemes. However, the individual routes taken to produce these methods — via implicit integration, harmonic approximation, other separating frameworks, and/or force splitting into frequency classes — have been quite different. Each path has encountered different problems along the way which only increased our understanding of the numerical, computational, and accuracy issues involved. This contribution reported on our experiences in this quest. LN has its roots in LIN, which... 

Apart from this simple result, comparison of stability predictions for the two limiting situations can be made only by direct numerical computation, and for this purpose a specific algebraic form must be assumed for the reaction rate function, and a specific shape for che catalyst pellet. In particular, Lee and Luss considered a spherical pellet and a first order... 

All numerical computations inevitably involve round-off errors. This error increases as the number of calculations in the solution procedure is increased. Therefore, in practice, successive mesh refinements that increase the number of finite element calculations do not necessarily lead to more accurate solutions. However, one may assume a theoretical situation where the rounding error is eliminated. In this case successive reduction in size of elements in the mesh should improve the accuracy of the finite element solution. Therefore, using a P C" element with sufficient orders of interpolation and continuity, at the limit (i.e. when element dimensions tend to zero), an exact solution should be obtaiiied. This has been shown to be true for linear elliptic problems (Strang and Fix, 1973) where an optimal convergence is achieved if the following conditions are satisfied ... 

The first quantitative model, which appeared in 1971, also accounted for possible charge-transfer complex formation (45). Deviation from the terminal model for bulk polymerization was shown to be due to antepenultimate effects (46). Mote recent work with numerical computation and C-nmr spectroscopy data on SAN sequence distributions indicates that the penultimate model is the most appropriate for bulk SAN copolymerization (47,48). A kinetic model for azeotropic SAN copolymerization in toluene has been developed that successfully predicts conversion, rate, and average molecular weight for conversions up to 50% (49). 

Erom a technology perspective, knowledge-based systems (KBS) represent a new software methodology for solving certain types of problems effectively. It is important to understand what is encoded in knowledge-based systems, and how KBS technology differs from conventional numeric computational techniques. 

By contrast, a numerical computer program for solving such integration problems would depend on approximating the mathematical expression by a series of algebraic equations over expHcit integration limits. 

The Prandtl mixing length concept is useful for shear flows parallel to walls, but is inadequate for more general three-dimensional flows. A more complicated semiempirical model commonly used in numerical computations, and found in most commercial software for computational fluid dynamics (CFD see the following subsection), is the A — model described by Launder and Spaulding (Lectures in Mathematical Models of Turbulence, Academic, London, 1972). In this model the eddy viscosity is assumed proportional to the ratio /cVe. 

The importance of emphasizing the essential difference between simple rate-dependent and path-dependent processes is that in the former case one does not have to follow the actual time-resolved deformation path in numerical computation, while in the latter case it is essential. 

Introduction of the surface-nucleation mechanism in numerical computation of elastic-plastic wave evolution leads to enhanced precursor attenuation in thin specimens, but not in thicker ones. Inclusion of dislocation nucleation at subgrain boundaries indicates that a relatively low concentration of subgrain boundaries ( 2/mm) and nucleation density (10"-10 m ) is sufficient to obtain predicted precursor decay rates which are comparable to those obtained from the experiments. These experiments are only slightly above the threshold necessary to produce enhanced elastic-precursor decay. 

Nelson, I., Constitutive Models for Use in Numerical Computations, in Proceedings of the Dynamical Methods in Soil and Rock Mechanics, Vol. 2, Plastic and Long-Term Effects in Soils (edited by Gudehus, G.), A.A. Balkema, Rotterdam, 1978, pp. 45-99. 

Flow fields resulting from these combustion modes were computed by means of similarity methods (Section 4.2.1) and used to provide initial conditions for numerical computations. The main conclusion was that blast waves at some distance from the charge were very similar, regardless of whether the combustion mode was detonation or strong deflagration. 

Chushkin, P. I., and L. V. Shurshalov. 1982. Numerical computations of explosions in gases. (Lecture Notes in Physics 170). Proc. 8th Int. Conf. on Num. Meth. in Fluid Dynam., 21-42. Berlin Springer Verlag. 

Launder, B. D., and D. B. Spalding. 1974. The numerical computation of turbulent flows. Comput. Meth. Appl. Mech. Eng. 3 269-289. 

The logarithm of a positive number N is the power to which the base (10 or e) must be raised to produce N. So, x = log N means that e = N, and x = log,(,N means that lO = N. Logarithms to the base 10, frequently used in numerical computation, are called common or denary logarithms, and those to base e, used in theoretical work, are called natural logarithms and frequently notated as In. In either case,... 

This is possible within the framework of the self-consistent field (SCF) approach to polymer configurations, described more completely elsewhere [18, 19, 51, 52]. Implementation of this method in its full form invariably requires numerical computations which are done in one of two equivalent ways (1) as solutions to diffusion- or Schrodinger-type equations for the polymer configuration subject to the SCF (in which solutions to the continuous-space formulation of the equations are obtained by discretization) or (2) as solutions to matrix equations resulting from a discrete-space formulation of the problem on a lattice. 

Criteria for R-A and IG, also shown in Table XI, were formulated in terms of dimensionless parameters e, a, B and b. They apply to both homopolymerizations and copolymerizations for various initiator systems at or near the condition Tr = Tq, and were developed through modified Semenov-type analyses (1, 2) numerous computer simulations. Owing to the fact that... 

All this is summarized in Fig. 3-12. The energy ordering of the free-ion terms is not determined by consideration of angular momentum properties alone and in general yields only to detailed numerical computation. The ground term - and only the ground term - may be deduced, however, from some simple rules due to Hund. 

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