Simulation Data

The comparison between measured data and simulated data are good for both real and imaginary parts. The measured signal has a low resolution due to the low interaction between the eddy current and the slot. 

The comparison between measured data and simulated data are good for the imaginary part, but differences appear for the real part. The ratio between simulated data and measured data is about 0.75 for TRIFOU calculation, and 1.33 for the specialised code. Those differences for the real part of the impedance signal can be explained because of the low magnitude of real part compared to imaginary part signal. 

The comparisons between measured and simulated data lead to the same conclusion as in case 2. The simulated data show more details on the curves, especially in the slot edges zones. This is linked most probably to the measured data resolution. 

This paper is structured as follows in section 2, we recall the statement of the forward problem. We remind the numerical model which relates the contrast function with the observed data. Then, we compare the measurements performed with the experimental probe with predictive data which come from the model. This comparison is used, firstly, to validate the forward problem. In section 4, the solution of the associated inverse problem is described through a Bayesian approach. We derive, in particular, an appropriate criteria which must be optimized in order to reconstruct simulated flaws. Some results of flaw reconstructions from simulated data are presented. These results confirm the capability of the inversion method. The section 5 ends with giving some tasks we have already thought of. 

Figure 2 Comparison between simulated data and experimental data.

In Fig. 3a,b are shown respectively the modulus of the measured magnetic induction and the computed one. In Fig. 3c,d we compare the modulus and the Lissajous curves on a line j/ = 0. The results show a good agreement between simulated data and experimental data for the modulus. We can see a difference between the two curves in Fig. 3d this one can issue from the Born approximation. These results would be improved if we take into account the angle of inclination of the sensor. This work, which is one of our future developpements, makes necessary to calculate the radial component of the magnetic field due to the presence of flaw. This implies the calculation of a new Green s function. 

Figure 4 Results of reconstruction from simulated data.

The model is meant to be relatively open to the evolution of NDT techniques. Thus, a normal evolution of the standard is to include, in future revisions, as "standard devices" some devices which have proved to be of current use. Two other axes of evolution are the handling of processed data and of simulated data. 

Figure Bl.9.9. Comparison of the distance distribution fiinction p(r) of a RNA-polymerase core enzyme from the experimental data (open circle) and the simulation data (using two different models). This figure is duplicated from [27], with pennission from Elsevier Science.

A systematic comparison of two sets of data requires a numerical evaluation of their likeliness. TOF-SARS and SARIS produce one- and two-dhnensional data plots, respectively. Comparison of sunulated and experimental data is accomplished by calculating a one- or two-dimensional reliability (R) factor [33], respectively, based on the R-factors developed for FEED [34]. The R-factor between tire experimental and simulated data is minimized by means of a multiparameter simplex method [33]. 

Liidemann et al., 1997] Liidemann, S. K., Carugo, O., and Wade, R. C. Substrate access to cytochrome P450cam A comparison of a thermal motion pathway analysis with moleculM dynamics simulation data. J. Mol. Model. 3 (1997) 369-374... 

Comparison of the measured isotope distributions with simulation data is carried out within the framework of the models corresponding to various developments, accompanied accident of 1986. Estimations of characteristics of physical influence (temperature, etc.) to which ChAPS s concrete stmcture have been undergone are discussed. 

Because the datay are random, the statistics based on y, S(y), are also random. For all possible data y (usually simulated) that can be predicted from H, calculate p(S(ysim) H), the probability distribution of the statistic S on simulated data y ii given the truth of the hypothesis H. If H is the statement that 6 = 0, then y i might be generated by averaging samples of size N (a characteristic of the actual data) with variance G- = G- (yacmai) (yet another characteristic of the data). 

More generally, the connection between the free energy surface and the simulation data can be made by the relation [81 ]... 

To illustrate the relationship between the microscopic structure and experimentally accessible information, we compute pseudo-experimental solvation-force curves F h)/R [see Eq. (22)] as they would be determined in SEA experiments from computer-simulation data for T z [see Eqs. (93), (94), (97)]. Numerical values indicated by an asterisk are given in the customary dimensionless (i.e., reduced) units (see [33,75,78] for definitions in various model systems). Results are correlated with the microscopic structure of a thin film confined between plane parallel substrates separated by a distance = h. Here the focus is specifically on a simple fluid in which the interaction between a pair of film molecules is governed by the Lennard-Jones (12,6) potential [33,58,59,77,79-84]. A confined simple fluid serves as a suitable model for approximately spherical OMCTS molecules confined... 

A successful method to obtain dynamical information from computer simulations of quantum systems has recently been proposed by Gubernatis and coworkers [167-169]. It uses concepts from probability theory and Bayesian logic to solve the analytic continuation problem in order to obtain real-time dynamical information from imaginary-time computer simulation data. The method has become known under the name maximum entropy (MaxEnt), and has a wide range of applications in other fields apart from physics. Here we review some of the main ideas of this method and an application [175] to the model fluid described in the previous section. 

The a posteriori probability P[A G] for having the spectral density A(uj), given the simulation data G, is... 

FIG. 3 The functions g r) and y r) for a hard sphere fluid. The broken curve gives PY results and the sohd curve gives the results of a fit of the simulation data. The circle gives the simulation results. The point at r = 0 gives the result obtained from Eq. (36), using the CS equation of state. 

Fig. 10(a) presents a comparison of computer simulation data with the predictions of both density functional theories presented above [144]. The computations have been carried out for e /k T = 7 and for a bulk fluid density equal to pi, = 0.2098. One can see that the contact profiles, p(z = 0), obtained by different methods are quite similar and approximately equal to 0.5. We realize that the surface effects extend over a wide region, despite the very simple and purely repulsive character of the particle-wall potential. However, the theory of Segura et al. [38,39] underestimates slightly the range of the surface zone. On the other hand, the modified Meister-Kroll-Groot theory [145] leads to a more correct picture. 

In Fig. 10(b) one can see the density profiles calculated for the system with /kgT = 5 and at a high bulk density, p = 0.9038. The relevant computer simulation data can be found in Fig. 5(c) of Ref. 38. It is evident that the theory of Segura et al, shghtly underestimates the multilayer structure of the film. The results of the modified Meister-Kroll-Groot theory [145] are more consistent with the Monte Carlo data (not shown in our... 

In Sec. 3 our presentation is focused on the most important results obtained by different authors in the framework of the rephca Ornstein-Zernike (ROZ) integral equations and by simulations of simple fluids in microporous matrices. For illustrative purposes, we discuss some original results obtained recently in our laboratory. Those allow us to show the application of the ROZ equations to the structure and thermodynamics of fluids adsorbed in disordered porous media. In particular, we present a solution of the ROZ equations for a hard sphere mixture that is highly asymmetric by size, adsorbed in a matrix of hard spheres. This example is relevant in describing the structure of colloidal dispersions in a disordered microporous medium. On the other hand, we present some of the results for the adsorption of a hard sphere fluid in a disordered medium of spherical permeable membranes. The theory developed for the description of this model agrees well with computer simulation data. Finally, in this section we demonstrate the applications of the ROZ theory and present simulation data for adsorption of a hard sphere fluid in a matrix of short chain molecules. This example serves to show the relevance of the theory of Wertheim to chemical association for a set of problems focused on adsorption of fluids and mixtures in disordered microporous matrices prepared by polymerization of species. 

However, before proceeding with the description of simulation data, we would like to comment the theoretical background. Similarly to the previous example, in order to obtain the pair correlation function of matrix spheres we solve the common Ornstein-Zernike equation complemented by the PY closure. Next, we would like to consider the adsorption of a hard sphere fluid in a microporous environment provided by a disordered matrix of permeable species. The fluid to be adsorbed is considered at density pj = pj-Of. The equilibrium between an adsorbed fluid and its bulk counterpart (i.e., in the absence of the matrix) occurs at constant chemical potential. However, in the theoretical procedure we need to choose the value for the fluid density first, and calculate the chemical potential afterwards. The ROZ equations, (22) and (23), are applied to decribe the fluid-matrix and fluid-fluid correlations. These correlations are considered by using the PY closure, such that the ROZ equations take the Madden-Glandt form as in the previous example. The structural properties in terms of the pair correlation functions (the fluid-matrix function is of special interest for models with permeabihty) cannot represent the only issue to investigate. Moreover, to perform comparisons of the structure under different conditions we need to calculate the adsorption isotherms pf jSpf). The chemical potential of a... 

Let us describe the results for the case of a matrix made of chains with four beads, i.e., m = M = 4. We observe that both the PY and the HNC approximation quite accurately describe the adsorption isotherms in matrices of different microporosity (Fig. 5). However, a discrepancy between the HNC results and simulation data increases with increasing chemical potential, especially at higher matrix densities. The structural... 

The simulation data, represented in Figs. 27(a-c), confirms Eq. (57) in the region of weak /tot BN < 4) for A = 8, 16, 32 at dilute regimes whereby the values for Dq have been measured for the same system in equilibrium [20] (cf. Sec, VIIB). Substantial discrepancies emerge only for high density of the medium. 

Because Eq. (2-41) was derived by means of approximations, its use may lead to approximate estimates of order, but its advantage is that it makes use of data taken at many time points. Figure 2-6 is a plot according to Eq. (2-41) with the simulated data used to construct Fig. 2-4. The slope of the line is 1.0, so the order is 2.0, in agreement with Fig. 2-4. [Experience with Eq. (2-41) applied to experimental data for first-order reactions indicates that Eq. (2-41) slightly overestimates the order for these reactions, leading to values n = 1.2-1.3.]... 

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