Superposition, method

Global superposition Local superposition Method accuracy... 

Implicit in all these solutions is the fact that, when two spherical indentors are made to approach one another, the resulting deformed surface is also spherical and is intermediate in curvature between the shape of the two surfaces. Hertz [27] recognized this concept and used it in the development of his theory, yet the concept is a natural consequence of the superposition method based on Boussinesq and Cerutti s formalisms for integration of points loads. A corollary to this concept is that the displacements are additive so that the compliances can be added for materials of differing elastic properties producing the following expressions common to many solutions... 

A sound is generally not a pure tone, as the latter is only emitted from particular sources. It can be demonstrated that a sound can be divided into different pure tones (superposition method). The waves at different frequencies give the spectrum of the sound, which also describes its energy distribution. In frequency analysis, the spectrum is divided into octave bands. An octave band is defined as the frequency range with its upper boundary twice the frequency of its lower boundary. For every octave band, a central band frequency ( f. ) is defined as follows ... 

As a first approximation, a superposition method has been used that can provide a guide to predicting mold shrinkage (Fig. 3-27). However, problems arise in measuring the influencing variables, because they... 

The rigid-body superposition method RigFit[133] in FlexS is based on some of the above Fourier transform principles. Other examples of the usefulness of Fourier transformation in virtual screening and modeling can be found in [134-138]. 

Figure 4.4 Superposition method for constructing localized source for one-dimensional...

Use the superposition method to find the time-dependent solution. 

The calculation of residual stresses in the polymerization process during the formation of an amorphous material was formulated earlier.12 The theory was based on a model of a linear viscoelastic material with properties dependent on temperature T and the degree of conversion p. In this model the effect of the degree of conversion was treated by a new "polymerization-time" superposition method, which is analogous to the temperature-time superposition discussed earlier. 

The bond lengths found in the molecule are compared in Table 6 with the bond lengths predicted by the simple MO method (Pritchard and Sumner, 1954) and by the Pauling superposition method. The largest... 

Most of the factors affecting the rate of corrosion can be understood from a graphical superposition of the current-potential curves for the metal-dissolution and electronation reactions. The principle of the graphical superposition method is straightforward. 

Patterson and Symmetry Superposition Methods. An older bootstrap method, based on searches of the Patterson function and variants thereof (vector superposition and symmetry superposition functions), should present significant advantages for noncentric structures. Much recent progress has been made in such alternative algorithms, which should be used when direct methods fail. 

Equation (4.39) can be directly integrated because the term dThldz is constant. The linearity of the energy equation allows the use of the superposition method to build solutions for asymmetric hearing by adding the two fundamental solutions (1) the outer wall heated with the inner wall insulated and (2) the inner wall heated with the outer wall insulated... 

The linearity of the energy equation suggests that the superposition method may be applied to build solutions by adding two fundamental solutions for the top and bottom walls. For a constant heat flux, a simple energy balance is... 

The time-temperature superposition method can be also applied to viscosity data (Ferry, 1980). For any viscoelastic parameter, exact matching of the adjacent curves is an important criterion for the applicability of the method. In addition, when possible, the same values of oy must superpose all the viscoelastic parameters and the temperature dependence of ar should have a reasonable form based on experience. One advantage of the method is that the range of frequencies are extended beyond those available experimentally. The time-temperature method has been also referred to as thermorheological simplicity (Plazek, 1996). 

Superposition method Analysis of the Patterson map by setting the origin of the Patterson map in turn on the positions of certain atoms whose positions may already be known, and then recording those areas of the superposed maps in which peaks appear that are derived from both maps. As a result, it may be possible to derive the atomic arrangement. 

This is now generally replaced by a normalized structure factor (q.v.). Vector superposition map See Superposition method. 

There are many theories on the mechanism of the segregation effect that suggest either a chemical or an electronic mechanism or both types of mechanisms. However, it seems that the most reliable mechanism is electronic as proposed by Mukheijee and Moran [35]. This electronic model calculates the chemical properties of the pure constituents from their physical parameters and then estimates those of the alloys. It employs the tight-binding electronic theory, the band filling of the density of states, and the bandwidth of the pure components for the calculations. However, it seems that the 2D Monte Carlo simulations produce better results by using the embedded atom and superposition methods. The latter allows for the calculation of the compositions from the relative atom positions, and the strain and the vibrational energies has been reviewed for 25 different metal combinations in [36]. It was also possible to predict composition oscillations as a consequence of the size mismatch. 

The superposition method extends Stillinger and Weber s division of the PES into catchment basins for each minimum to calculate thermodynamic properties. In this approach the configurational part of the phase space integral in the definition of the density of states, Q E), or partition function, Z T), is divided into separate integrals for each minimum, giving... 

In their formulation of the superposition method Hoare and Mclnnes omitted the effect of symmetry [132]. This omission explains why they predicted that for LJ the cluster would reside in the icosahedral global minimum up to a reduced temperature of 0.6. In fact, LJ13 melts at r 0.29. Without the inclusion of symmetry, the density of states of the icosahedron is overestimated by a factor of 120. 

To obtain more accurate thermodynamic properties from the superposition method there are a number of more technical issues that must be addressed. Firstly, the sum in Eq. (1.25) is over all the minima. For small enough systems this range is not a problem because (virtually) all the minima can be catalogued. The size limit at which it becomes unfeasible to catalogue all the minima depends on the system of interest for example, the current limit for Lennard-Jones clusters lies not far beyond A = 13. However, Eq. (1.25) can also be applied to larger systems if the number of thermodynamically relevant minima is small enough to be catalogued. This... 

The second difficulty with the superposition method is obtaining expressions for 0, and Z,. As we noted earlier, the natural starting point is to truncate a Taylor expansion for the potential energy around each minimum at second order giving the harmonic approximation. This truncation... 

Figure 1.10. Thermodynamic properties of LJ55 in the canonical ensemble, (a) The heat capacity, C,. (b) The Landau free energy, Al( c)- (c) The probability of the cluster being in the Mackay icosahedron, with one or two defects, and liquid-like regions of configuration space, (d) Gn,( c). the number of minima with energy less than c. All these properties were calculated using the anharmonic form of the superposition method from a sample of 1153 minima generated from a microcanonical MD simulation. In (b) the results are compared with simulation data obtained by Lynden-Bell and Wales (data points) [218] with the zero of free energy chosen for clarity.

Figure 1.11. Equilibrium thermodynamic properties of LJ38 in the canonical ensemble, (a) The heat capacity, Cv (b) The probability of the cluster being in the fee, icosahedral and liquid-like regions of bound configuration space. These results were obtained using an anharmonic form of the superposition method (Section III.C).

The large free energy barriers between the nanocrystals prevent an easy determination of the relative stabilities of the different nanocrystals by conventional simulations. Therefore, we use the harmonic superposition method (Section III.C) to examine this question. It can be seen from Figure 1.28 that... 

Figure 1.28. Equilibrium occupation probabilities of the (NaCOssCP nanocrystals computed using the harmonic superposition method.

This example is used to illustrate the direct application of the superposition method and the use of the Green s function approach. For the superposition method, the loading is viewed as being the difference between a cracked body under uniform remote traction a and one in which the crack is held shut by uniform traction a along the crack surfaces, with the magnitude of the traction equal to the pressure p along the crack (see Fig. 3.6). 

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