Quadrature point coordinates

It is not necessary to have an equation for the KEO. One must only be able to calculate matrix elements of the KEO which can be done (with quadrature) as long as it is possible to evaluate (coordinate dependent) coefficients of differential operators in the KEO at the quadrature points. This numerical approach is very useful. It has a long history [35-39], but its utility has been greatly enhanced by the TNUM program of Lauvergnant [40]. One way to implement a numerical approach is to determine... 

C TRO, DTRO coordinates and their z derivatives at quadratures points do k=l,ndvr do ic=l,ncor... 

If we pick an arbitrary element we can see that it is represented by the xy-coordinates of the four nodal points, as depicted in Fig. 9.16. The figure also shows a -coordinate system embedded within the element. In the r/, or local, coordinate system, we have a perfectly square element of area 2x2, where the element spreads between —1 > < 1 and — 1 > rj < 1. This attribute allows us to easily allows us to use Gauss quadrature as a numerical integration scheme, where the limits vary between -1 and 1. The isoparametric element described in the //-coordinate system is presented in Fig. 9.17. 

Exercise 3.6 Consider a bivariate distribution (M = 2) with two internal coordinates and 2, and let us construct a four-point quadrature approximation, resulting from univariate quadratures of order N = N2 = 2. Knowledge of the first, 2N = 2N2 = 4, pure moments with respect to the first, f, and second, 2, internal coordinates, suffices for... 

Figure 3.1. Positions in the internal-coordinate plane of the four nodes of the bivariate tensor-product QMOM (M = 2) obtained with two-point univariate quadratures N =...

The system formed by Eqs. (3.48)-(3.51) can be solved to find the weights. It can be seen that the four-point bivariate quadrature approximation accommodates eight moments, namely the moment of order zero with respect to both the internal coordinates, six pure moments, and one mixed moment. Table 3.8 reports in matrix form these eight moments, and comparison with Table 3.5 clearly shows that this is a subset of the optimal moment set. 

W v is the total number of points in the quadratures of the interaction potential over the vibrational coordinate. In the present calculations the quadratures are carried out by dividing the mass-scaled vibrational coordinate r into ArKvs segments extending from to Each segment is then integrated by a N 1 -point Gauss-... 

---