Hyperbolic paraboloid surface

Almost two decades previous to the Doering papers a reasonable model for substituent rate effects was proposed that was based on a geometric model for the MOE-J energy surface for the 3,3-shift. Thus, a hyperbolic paraboloid surface equation could be differentiated to obtain coordinates and the activation free energy for the saddle point (the transition state) cast in terms of the relative free energies for formation of the diyl and the two allyl radicals, the same independent variables of Eqs. (7.1) and (7.2). Equation 7.3, which relates the independent variables by the harmonic mean is based on the simplest hyperbolic paraboloid surface, that is, one with linear edge potentials. Slightly more realistic models were also explored. 

Figure 5-25. Sections through a hyperbolic paraboloid energy surface constructed over an RIP diagram. The intrinsic barrier AG of the main reaction and the intrinsic well AC of the disparity reaction are shown.

The computational domain is the unit square in u and v, and this was divided into a 15 x 15 mesh i.e., 225 elements, and 16 x 16 = 256 nodes, so 256 basis functions and 256 residual equations. The Jacobian matrix was banded with a total bandwidth of 35. The first solution computed was the minimal surface, for which the initial estimate was an hyperbolic paraboloid. The nonlinear system of residual equations was solved by Newton iteration on a Cyber 124, each iteration using about 1 second cpu time. For nearly all the surfaces calculated, the mesh was an even mesh over the entire unit square. However, for the surfaces just near the close-packed spheres (CPS) limit, the nodes were evenly spaced in the u-direction but placed as follows in the i -direction i = 0,1/60,1/30,0.05,0.075,0.1,0.15,0.2,0.3,0.4,0.5,0.6,0.7,... 

The important matter is the fact that when values of function of two variables are analyzed, the range of material variables (coded values) is narrowed to <-l, 1>. The analysis of graphs (especially in case of hyperbolic paraboloid) shows pairs of maxima and minima (or very clear tendency to the pair of extrema. Considering the mathematics - such result is correct but considering the technical properties - it is necessary to find the extremum that would be correct and rational in an engineering sense. Some of the extrema need to be rejected as they are reached with combinations of material variable values for the polymer-cement coating which are not relevant. Such result was reached in case of the flexibility index in function of coded values polymer to Portland cement ratio (P/C) and hydrophobic agent to Portland cement ratio (H/C). The shape of surface described by this relation was a hyperbolic paraboloid (refer with Fig. 5). 

Compound curve n. A surface having curvature in two principal directions. Simply curved surfaces, such as cylinders and cones, having only one direction of curvature, may be cut along an element and laid flat. Compound curves, such as spheres and hyperbolic paraboloids, carmot be laid flat without distortion no matter how... 

All plane sections of surfaces of the second order are either circular, parabolic, hyperbolic, or elliptical, and are comprised under the generic word conicoids, of which spheroids, paraboloids, hyperboloids and ellipsoids are special cases. 

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