Interpolation formula

Bogner, F. K., Fox, F. L. and Scliinit, L. A., 1965. The generation of interelemenl-coinpatible stiffness and mass matrices by the use of interpolation formulae. Proc. Conf. on Matrix Methods in Structural Mechanics, Air Force Institute of Technology, Wright-Patterson AF Base, OH. 

Algebraic methods - in these techniques calculation of grid coordinates is based on the use of interpolation formulas. The algebraic methods are fast and relatively simple but can only be used in domains with smooth and regular boundaries. 

If the accuracy afforded by a linear approximation is inadequate, a generally more accurate result may be based upon the assumption thedfix) may be approximated by a polynomial of degree 2 or higher over certain ranges. This assumption leads to Newtons fundamental interpolation formula with divided differences... 

Lagrange Interpolation Formulas A global polynomial is defined over the entire region of space... 

Use of Interpolation Formula If the data are given over equidistant values of the independent variable x, an interpolation formula such as the Newton formula (see Refs. 143 and 18.5) may be used and the resulting formula differentiated analytically. If the independent variable is not at equidistant values, then Lagrange s formulas must be used. By differentiating three- and five-point Lagrange interpolation formulas the following differentiation formulas result for equally spaced tabular points ... 

Implicit Methods By using different interpolation formulas involving y, it is possible to cferive imphcit integration methods. Implicit methods result in a nonhnear equation to be solved for y so that iterative methods must be used. The backward Euler method is a first-order method. 

Starting point for evaluating the settling characteristics of suspended solids for dilute systems. Note that from the definition of the Reynolds number, we can readily determine the settling velocity of the particles from the application of the above expressions (u, = /xRe/dpp). The following is an interpolation formula that can be applied over all three settling regimes ... 

T = 0.5,0.6,0.7,0.8, and 0.9. Despite some statistical fluctuations at late times after the T-jump, it is evident from Fig. 19 that the different curves collapse onto a single one if time is scaled by a single. As for the system of rate equations, (26), we again find = (I.SSLqo) where the power 5 is determined with an accuracy of 2%. An interpolation formula for the scaling function /(jc — = (0.215 + 8jc) appears to account well... 

The approxmations reviewed so far were all developed for the low-piston Mach number regime. Cambray and Deshaies (1978), on the other hand, developed a solution of the similarity equations by asymptotic expansions in powers of high-piston Mach numbers. These solutions are supposed to hold for piston Mach numbers higher than 0.7. Finally, Cambray et al. (1979) suggested an interpolation formula to cover the intermediate-piston Mach number range. 

The correlation energy of a uniform electron gas has been determined by Monte Carlo methods for a number of different densities. In order to use these results in DFT calculations, it is desirable to have a suitable analytic interpolation formula. This has been constructed by Vosko, Wilk and Nusair (VWN) and is in general considered to be a very accurate fit. It interpolates between die unpolarized ( = 0) and spin polarized (C = 1) limits by the following functional. 

The vapour-pressures of ice at various temperatures have recently been carefully determined Scheel (1905) finds that they may be represented by the interpolation formula ... 

The classical result for the image potential is -q/4x, independent of the metal, but various theories of the metal which assume an infinite potential barrier for the metal electrons give potentials which are reduced in size near the metal boundary, so that the interaction energy is actually finite24 at x = 0. An interpolation formula which reproduces this behavior is... 

SMla, L., and P. Pancoska. 1988. Interpolation Formula for Physical Properties of Polypeptides as a Function of the Number of Amino Acid Residues. Chem. Phys. 125, 21-30. 

Orthogonal Collocation The orthogonal collocation method has found widespread application in chemical engineering, particularly for chemical reaction engineering. In the collocation method, the dependent variable is expanded in a series of orthogonal polynomials. See "Interpolation and Finite Differences Lagrange Interpolation Formulas. ... 

This is the Menzel-Minnaert-Unsold interpolation formula (often used assuming Roo = 1). It gives a better approximation to stellar absorption-line profiles (which are definitely not flat-bottomed) than does the exponential formula the shape of the corresponding curve of growth is much the same, but its use leads to a b-parameter that is about 25 per cent higher for the same observational data. Denoting the central value of p by po, the Doppler part of the curve is given by... 

Next we compare Eq. (11) with an interpolation formula for the eight glitches of the Vela pulsar [16] ... 

Butler and Pillingf) calculated an exact numerical solution of the diffusion equation. They showed that the interpolation formula proposed by Gosele et al.e) reproduces the numerical solution with high precision. 

As pointed out by Shatkay [152], there was no theory of the response time before 1975, except for the work of Markovic and Osborn [99], and the time dependence of ise on a change in the concentration was characterized by interpolation formulae, such as the exponential,... 

Zhang and Davis proposed an interpolation formula based on Churchill and Usagi s (1972) method of combining asymptotic solutions to obtain a correlation valid over the whole range of variables. Their correlation is... 

Finlayson and Olson (1987) used the Galerkin finite element numerical method to explore heat transfer to spheres at low to intermediate Reynolds numbers (1 < Re < 100) and for Prandtl numbers in the range 0.001-1,000. They found that the best correlation of their data was an interpolation formula of the form proposed by Zhang and Davis their correlation is... 

Method of Lines. The method of lines is used to solve partial differential equations (12) and was already used by Cooper (I3.) and Tsuruoka (l4) in the derivation of state space models for the dynamics of particulate processes. In the method, the size-axis is discretized and the partial differential a[G(L,t)n(L,t)]/3L is approximated by a finite difference. Several choices are possible for the accuracy of the finite difference. The method will be demonstrated for a fourth-order central difference and an equidistant grid. For non-equidistant grids, the Lagrange interpolation formulaes as described in (15 ) are to be used. 

The best way to use the Kozeny-Carman model and other permeability models (e.g. the anisotropic model by Gebart) [18], is to use them as interpolation formulas for intermediate volume fractions between known values. Extrapolation should be done with extreme caution because the models are developed for idealized reinforcements. Typical values for the permeability of different types of reinforcement are given in Table 12.1. 

Recently, Hentschke [45] and also DuPre and Yang [46] proposed empirical interpolation formulas on the basis of Khokhlov and Semenov s a(N). The former used the Pade approximation, while the latter modified Odijk s a(N) so as to agree with Khokhlov and Semenov s in the asymptotic limits of N 1 and N 1 and derived... 

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