Dimensional interpolation

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. 

The objective here is to construct the equilibrium surface in the T-P-x space from a set of available experimental VLE data. In general, this can be accomplished by using a suitable three-dimensional interpolation method. However, if a sufficient number of well distributed data is not available, this interpolation should be avoided as it may misrepresent the real phase behavior of the system. 

In practice, VLE data are available as sets of isothermal measurements. The number of isotherms is usually small (typically 1 to 5). Hence, we are often left with limited information to perform interpolation with respect to temperature. On the contrary, one can readily interpolate within an isotherm (two-dimensional interpolation). In particular, for systems with a sparingly soluble component, at each isotherm one interpolates the liquid mole fraction values for a desired pressure range. For any other binary system (e.g., azeotropic), at each isotherm, one interpolates the pressure for a given range of liquid phase mole fraction, typically 0 to 1. 

A one-dimensional interpolation consists of constructing an interpolation function f such that f(f,) — xit whatever i is. The unknown parameters of function f are determined from these relationships. It is worth noting that the interpolating curve passes rigorously through the experimental points, although these points are not free from stochastic errors. For this reason, and also to get equally spaced points, a smoothing of experimental points is often done before interpolation. 

A different approximation is obtained by an interpolation method that uses selected points on subsequent spherical surfaces to define lines along which one-dimensional interpolations are carried out for charge density. For this purpose, on each sphere S j of the set t=0, a set of m points. 

First-Order Interpolation. The dimension dependence is dominated by singularities at i = 1, arising from a second-order pole (like the hydrogenic atom but with a different residue) and a confluent first-order pole. Deducting the readily calculable contributions from these poles markedly improves the efficacy of dimensional interpolation. The simplest approximation of this kind yields... 

Table 1. Exact and Hartree energies for the system of N equivalent gravitating particles for D = 1 and D oo, and the dimensionally interpolated estimates at D = 3.

Table 2. Estimates of bond length and ground state energy for and Hartree-Fock H2 as calculated by dimensional interpolation actual values are listed for comparison.

In this and the following section we consider two alternatives to dimensional interpolation for obtaining the D = Z results from the dimensional limit calculations. First is the 1/D expansion. The zeroth-order term in this expansion is just the D oo limit solution, as given by the minimum of an effective potential, such as Eq. (35). The first-order or 1/D term arises from harmonic vibrations about the minimimi of the effective potential. Together, the first two terms of the 1 /D expansion for axially symmetric systems take the form... 

Because of the uniform procedure used to derive the Umiting forms, the solutions obtained from them can be expected to bracket the D = Z solutions. In fact, simple dimensional interpolation (linear interpolation in 1/i between dimensional limits evaluated in the above scaling) allows one to obtain rough estimates of i = 3 energies for this and a variety of other simple systems, as shown in Fig. 1. It is important to note that it is only because of the use of a uniform scaling that this is true. For example, if one used units of (I —1)/2 Bohr radii for the D limit and ( >—1)2/4 Bohr radii for the D— oo limit [12], one would obtain Eq. (5) without the factors of and in firont of the cavity radius, and the solutions obtained from these hamiltonians would not be useful for quantitative calculations. 

A second possibility would be to improve systematically upon the D— oo model within the dimensional scaling framework. This might be done by considering the electronic behavior at low D (specifically D = 1), and using a dimensional interpolation strategy. However, it seems more likely that progress could be made through consideration... 

Dimensional interpolation is used to approximate the configuration space integrals required in the computation of higher-order hard sphere virial coefficients. Simple analytic results can he obtained at D =... 

D — and D oo the smooth and generic dimension-dependence of the integrals enables one to interpolate reasonably accurate D = 3 values (rms error 1%) from the dimensional limit results. The interpolated integrals can be used either on their ovm, or in conjunction with an integral equation approximation which sums some subset of the required integrals exactly (such as the hypemetted-chain or Percus-Yevick methods) the combination methods are invariably better than either dimensional interpolation or integral equations alone. Interpolation-corrected Percus-Yevick values can be computed quite easily at arbitrary order however, errors in higher-order values are... 

We will now use dimensional interpolation to proceed from the known results, namely those at low n, at low B, and at high B, to the desired results at higher n and at D = 2 or D = 3. The interpolation will actually be performed on the cluster integral ratios p nk] D), since these quantities had finite but non-zero dimensional limits. Given interpolated ratios, one can simply step up from the known integrals for hard points or rods to those for hard disks or spheres (or even higher-dimensional fluids), since... 

It would probably be a waste of time to try to apply the procedure just described at higher order. This is because the errors associated with dimensional interpolation for any given integral (of the order of 1% at D = 3) would render the final sum, which is quite small compared to the contribution of any given integral, unreliable. Indeed, for Be, the final sum is only about of the sum of absolute values, and the dimensionally interpolated result does not even have the right sign. 

The following table compares hard sphere virial coefiicients through Bq obtained in three different ways by using dimensionally interpolated integrals to compute the second Ree- Hoover partial sum. 

Eq. (21) by using the PYc approximation, Eq. (38) and by starting with the PYc approximation and then correcting it using a dimensionally interpolated value for the leading-order correction, Eq. (23). Where comparisons with exact veilues can be made, the last of these procedures is the best, though the errors are still somewhat large. 

The dimensional interpolation strategy provides an alternative to Monte Carlo calculations for the evaluation of cluster integrals. Compared to the Monte Carlo technique, interpolation has several disadvantages. In particular, the errors associated with the method are difficult to estimate, and cannot be reduced through further calculations. On the... 

J.I. Agbinya, Two dimensional interpolation of real sequences using the DCT Electron. Lett., 29(2), 204-205 (1993)... 

FIGURE 6.1 (a) One-dimensional interpolation of the potential V[qo(z)] as a function of the parameter z along the instanton path qo(z) for three ab initio methods in the case of malon-aldehyde. The points represent the ab initio data. The lines are obtained by piecewise cubic interpolation, (b) One-dimensional interpolation of the elements of Hessian 9 V [q(z)]/9 along the instanton path for MP2/cc-pVDZ ab initio method. Two examples are shown. (Taken from Reference [122] with permission.)... 

MATLAB has several functions for interpolation. The function = interpl(x, y, x) takes the values of the independent variable x and the dependent variable y (base points) and does the one-dimensional interpolation based on x, to find yj. The default method of interpolation is linear. However, the user can choose the method of interpolation in the fourth input argument from nearest (nearest neighbor interpolation), linear (linear interpolation), spline (cubic spline interpolation), and cubic (cubic inteipolation). If the vector of independent variable is not equally spaced, the function interplq may be used instead. It is faster than interpl because it does not check the input arguments. MATLAB also has the function sp/ine to perform one-dimensional interpolation by cubic. splines, using nat-a- not method, ft can also return coefficients of piecewise poiynomiais, if required. The functions interp2, inte.rp3, and interpn perform two-, three-, and n-dimensional interpolation, respectively. 

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