Elbow plot

Using efficient DFT codes, whole potential energy surfaces of the interaction of molecules with surface can be mapped out in great detail. Figure 2 presents two-dimensional cuts through the six-dimensional configuration space, so-called elbow plots of two benchmark systems, namely H2/Pd(l 0 0) and H2/Cu(l 0 0). In these plots, the PES is shown as a function of the hydrogen distance... 

It is important to note that DFT total-energy calculations do not provide a continuous potential energy surface, as one might naively assume from the inspection of Fig. 2. In fact, the elbow plots shown are based on a series of 50-100 DFT calculations with varying center of mass and H-H distance. The continuous representation is just a result of a contour plot routine that interpolates between the actually calculated energies. 

In contrast to H2/Pd, the vibrational effects in the adsorption of H2/Cu(l 0 0) are mainly caused by the curved reaction path. The basic mechanism can be discussed within a two-dimensional elbow plot shown in Fig. 2b. The PES corresponds to a so-called late barrier system which refers to the fact that the barrier is located after the curved region of PES. If the molecule is already initially vibrating, i.e. if it is oscillating back and forth in the d-direction, then the vibrational energy can be very efficiently used to make it around the curve and enter the dissociation channel. Nevertheless, adiabatic effects as just discussed in the context of the hydrogen dissociation on Pd(l 00) also contribute to the vibrational effects for H2/Cu(l 0 0). 

Figure 25.4 Two-dimensional representation ( elbow" plots) of the potential energy situation when a molecule interacts with an active metal surface. The coordinate describes the internuclear H-H distance, y is the distance of the H2 molecule from the surface. Two possible trajectories are indicated (1) represents a reflection trajectory (unsuccessful event) with no chemisorption, (2) a successful approach...

The plot is made up for sections of pipe which contain no valves, elbows, sudden contractions, sudden expansions, etc. These are probably present in all the actual situations described in Examples 6.3, 6.4, and 6.5. In Secs. 6.8 and 6.9 we discuss how to account for them. 

In our context, model selection is the selection of the number of components, the choice between a PARAFAC and a T2 model (e.g., is there energy transfer ), or the choice between parametric models for specific ways and components (e.g., is the time dependence a single exponential ). Most often, one will be interested in deciding on the number of components to use. There are a variety of statistical tests based on the decline of the sum of squared residuals with additional components. Other methods look at the relationship between residuals for evidence that they show no patterns and hence the model is adequate. However, the simplest and perhaps the most effective approach is simply to plot the logarithm of the sum of squared residuals versus the number of components and use the number of components at the elbow where the curve flattens out. " "... 

The results may be plotted as specific volume versus temperature (see Figure 8.9) (24). Since the elbow in volume-temperature studies is not sharp (all measurements of Tg show a dispersion of some 20-30 C), the two straight lines below and above the transition are extrapolated until they meet that point is usually taken as Tg. Dilatometric and otiher methods of measuring Tg are summarized in Table 8.5. 

Tamman and Jellinghaus (142) showed that a plot of volume versus pressure at a temperature near the transition shows an elbow reminiscent of the volume-temperature plot (see Figure 8.9). If the temperature is raised at elevated pressures, Tg will in fact show a corresponding increase (see Figure 8.33) (143). 

Stress plots were created for designed elbow prosthesis. Figure 3 shows an example of stress plot that were created during of this study. As is shown in figure 3 stress concentration in articulation surface is more in comparison to other area. Also in Humeral component, a significant stress concentration appeared in the upper part while a moderate stress was observed in middle. On the contrary, in Ulnar component, elements which are in middle or further parts of articulation surface suffer less range of stress than elements which are closer to articulation surface. 

As it was reported in previous experimental studies most total elbow replacement failure are happened in articulation surface and in bone-implant interface, also according to the stress plots that are resulted from this study, upper part of Humeral component, articulating surface and connecting pin suffer high magnitude of stress which may cause that two kinds of loosening and it shows the validation of results. 

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