Interpolating functions

Analogous intei-polation procedures involving higher numbers of sampling points than the two ends used in the above example provide higher-order approximations for unknown functions over one-dimensiona elements. The method can also be extended to two- and three-dimensional elements. In general, an interpolated function over a multi-dimensional element Q is expressed as... 

A very convenient indirect procedure for the derivation of shape functions in rectangular elements is to use the tensor products of one-dimensional interpolation functions. This can be readily explained considering the four-node rectangular element shown in Figure 2.8. 

Depending on the type of elements used appropriate interpolation functions are used to obtain the elemental discretizations of the unknown variables. In the present derivation a mixed formulation consisting of nine-node bi-quadratic shape functions for velocity and the corresponding bi-linear interpolation for the pressure is adopted. To approximate stres.ses a 3 x 3 subdivision of the velocity-pressure element is considered and within these sub-elements the stresses are interpolated using bi-linear shape functions. This arrangement is shown in Edgure 3.1. 

The interpolating function and its derivatives should have as simple an algebraic form as possible consistent with the desired goodness of fit. 

The interpolating function takes on specified values at the knots and has continuous first and second derivatives at the knots. Within the ith intei val, the function is... 

Next the equatio(ns and variables are placed within NDSolve and solved over a range of positions ( z-values) and times. Then we assign the resultant interpolation functions to the appropriatefunctionnames ... 

Different interpolation functions have been used for determining the influence coefficients, as summarized briefly in the following. 

If a linear interpolation function n(x) is applied to approximate the pressure distribution within the element, the influence coefficients can be obtained by performing the integration in Eq (29), which results in... 

For two-dimensional problems, if a bilinear interpolation function is employed, the influence coefficients can be computed likewise in analytical form [31]. 

Fig. 8.2. Values of A, B, and B for the B-dot (modified Debye-Huckel) equation at 0 °C, 25 °C, 60 °C, 100 °C, 150 °C, 200 °C, 250 °C, and 300 °C (squares) and interpolation functions (lines). Values correspond to I taken in molal and a in A. Data from the LLNL database, after Helgeson (1969) and Helgeson and Kirkham (1974).

An interpolating function between the Poisson and the Wigner distribution is given by the Brody distribution (Brody, 1973) reading for the... 

Figure 3. Computer plot obtained by isochronous interpolation of the experimental light-scattering data (O) of HEC during endocellulase attack. The Langrangian interpolation functions are given by the horizontal curves ana the isochronous interpolated Kc/Re values by A. The quadratic least squares extrapolations to zero angle ( ) are given by the...

Since only the principal value of the phase can be measured, provision must also be made for unwrapping the phase subject to the constraint that the cubic phase function and its derivative equal the measured frequency and phase at frame k + 1. An explicit solution can be obtained for interpolation and phase unwrapping by invoking an additional constraint requiring that the unwrapped cubic phase function be maximally smooth . The problem then reduces to finding that multiple of 2n that leads to the smoothest phase interpolation function while meeting the constraints on the frequency and phase at frame k + 1. It can be shown that these constraints are met for values of a and P that satisfy the relations [McAulay and Quatieri, 1986b]... 

After solving for the constants we obtain the following second order interpolation function,... 

According to eqns. (7.6), (7.7) and (7.12), we can generalize for an arbitrary polynomial interpolation of order n — 1 (n number of points), by saying that the function u (x) is approximately equal to a linear combination of the known values Ui multiplied by interpolating functions, Niy... 

Depending on the nature of the interpolation functions, the different types of schemes are reached. For the example in Fig. 7.2, the interpolation functions were polynomials. In eqn. (7.13), n is the number of points used in the interpolation, and therefore, n — 1 is the order of the interpolation. 

Global basis functions. Common global basis functions, where the interpolation functions for multi-dimensional domains can be obtained, come from expansions of Pascal s triangle. In 2D, Pascal s triangle is defined by,... 

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