Joint modelling

It is worth recalling that any of the molecular force laws given by Eqs. (13-16) are derived within the framework of the freely-jointed model which considers the polymer chain as completely limp except for the spring force which resists stretching thus f(r) is purely entropic in nature and comes from the flexibility of the joints which permits the existence of a large number of conformations. With rodlike polymers, the statistical number of conformations is reduced to one and f(r) actually vanishes when the chain is in a fully extended state. 

To explain the difference between the experimental results and theory, Doherty et al. (4J have given an empirical and a theoretical hypothesis. The theoretical hypothesis concerns the question of the meaning to be attached to the concept of the "equivalent random link" in the statistical theory of the randomly-jointed chain. According to Doherty et al., the assumption that the optical properties of the chain are describable by a randomly jointed model, using the same value of n, as for the description of stress has no strictly logical foundation. 

AMJhj is a function 6f a) the orientation of the inter-crosslink vector h. with the respect to the magnetic field B0 and b) the time-averaged value of the sum over k, which is the actual measure of the motional restrictions induced by crosslinking. In lightly crosslinked networks, presented by the freely-jointed model of the polymer chain 108), the residual part also can be described by the number of statistical segments in the chain section between crosslinks (Z) (Eq. (24)) ... 

The real polymer chain may be usefully approximated for some purposes by an equivalent freely jointed chain. It is obviously possible to find a randomly jointed model which will have the same end-to-end distance as a real macromolecule with given molecular weight. In fact, there will be an infinite number of such equivalent chains. There is, however, only one equivalent random chain which will lii this requirement and the additional stipulation that the real and phantom chains also have the same contour length. 

Hu C, Sale ME. A joint model for nonlinear longitudinal data with informative dropout. J Pharmacokinet Pharmacodyn 2003 30 83-103. 

There is, however, one application when the overall desirability D can be rather safely used, namely as a tool in conjunction with response surface modelling. In this context, it can be used to explore the joint modelling of several responses so that a near-optimum region can be located by simulations against the response surface models. The search for conditions which incrase D can be effected either by simplex techniques or by the method of steepest ascent. For the steepest ascent, a linear model for D is first determined from the experiments in the design used to establish the response surface models. The settings which increase D can be translated back into the individual responses by using the response surface models. Thus, it is possible to establish immediately whether the simulated reponse values correspond to suitable experimental conditions. Such results must, of course, be verified by experimental runs. 

Estimation methods that are based on simulation platforms, such as Markov chain Monte Carlo (MCMC), also allow for model discrimination to be based on predictive or posterior distributions. When using MCMC, competing models can be fitted simultaneously as a joint model with an added mixing parameter to indicate which model is preferred (42, 43). The posterior distribution of the mixing parameter will provide both the weight of evidence and the posterior probability in favor of one model. The expectation of the prediction from m models and a the mixing parameter can then be evaluated ... 

The valence angle model, though more realistic than the freely jointed model, still underestimates the true dimensions of polymer molecules, because it ignores restrictions upon bond rotation arising from short-range steric interactions. Such restrictions are, however, more difficult to quantify theoretically. The usual procedure is to assume that the conformations of each sequence of three backbone bonds are restricted to the rotational... 

Mould et al. (2002), instead of using a conditional model to account for missing covariates, used a joint model that modeled the missing covariate simultaneously within the context of the overall pharmacokinetic model. In their analysis of topotecan pharmacokinetics in subjects with solid tumors, weight was known to be an important covariate but was missing in 20% of the database. The function for males... 

Table 8.8 Results from missing data analysis using the joint model. ...

The objectives of the present paper are a) to illustrate for which tasks modelling can support the monitoring programmes and the WFD implementation and b) to illustrate joint modelling and monitoring approaches in case studies. 

The Compliant Joint Model (CJM) was chosen for the mechanical calculations for this analysis. The CJM uses an equivalent continuum approach to model the behavior of jointed media. An equivalent continuum approach captures the average response of a jointed rock mass by distributing the response of the individual joints throughout the rock mass. The CJM in JAS3D can model up to four joint sets of arbitrary orientation, with the fractures in each set assumed to be parallel and evenly spaced. The intact rock between joints is treated as an isotropic linear-elastic material. More detailed descriptions of the CJM model can be found in Chen (1991). 

For these calculations, three different material models were investigated a linear elastic, an elasto-brittle, and elasto-plastic ubiquitous joint model. The ubiquitous joint, elasto-plastic model, is defined by a two-dimensional yield criterion, composed of two Mohr-Coulomb criteria, along two predefined directions characterised by their normal vectors ni and n2. 

In Figure 5 and 6, calculated displacements for the two nonlinear models are compared to the measured ones for three different multiple-point borehole extensometers. One general trend concerning the calculated results is an increasing relative displacement, from Anchor 1 to Anchor 4, with a delay in the development of the latter caused by the delay in the propagation of the thermal front. Non-linearities are more pronounced when using the ubiquitous joint model, whereas the brittle model leads to weakly nonlinear behaviour, close to the pure elastic one. 

For borehole BH155, the best agreement is obtained for Anchors 1, 2, and 3 using the ubiquitous joint model. However, both models overestimate the displacements in Anchor 4. 

For extensometers BH156, a difference can be observed in the response of Anchor 4 at very early times the ubiquitous joint model predicts, as is observed in the experiment, a very limited negative displacement, smaller than the elastic model, which indicates that some plastic deformation occurs. This is confirmed by looking at calculated contours of plastic zones plastic shear develops at the drift boundary as early as 3 months. For this extensometer, the right order of magnitude is predicted for Anchor 3 and 4, with an excellent agreement for Anchor 4 using the ubiquitous joint model. 

Figure 7 shows that LBNL s elastic and CEA s elasto-brittle models are within the range of maximum and minimum measured values, except in areas close to the drift wall. Near the drift wall, there appears to be an additional shift in the displacement field, possibly caused by inelastic behaviour near the open wall surface. The CEA s ubiquitous joint model overpredicts the displacement in the 60° inclined boreholes (Figure 7a) possibly by an overprediction of near-wall inelastic deformations. 

In general, the predicted displacement using both LBNL s elastic and CEA s elasto-brittle (weakly inelastic) models are within the ranges of field measurements, except for very close to the drift wall. However, in a few individual anchors, displacement values are more than 50% larger than predicted by the elastic material behaviour. The increased displacement in these anchors may be explained by inelastic responses leading to a better agreement with the ubiquitous joint model (e.g. Anchor 4 in Figure 5a). 

The mechanical and hydromechanical properties of the horizontal rock fractures were estimated using the Barton-Bandis Joint model (Barton and Bakhtar, 1983). In this context, the void aperture, by, is defined as the accessible volume per unit area of a fracture. The void aperture at a given effective normal stress is... 

The non-linear empirical Barton-Bandis (BB) joint model describes the Joint behaviour. The mean input values are given in Table 2. The initial mechanical aperture (o/ ,) at zero normal stress is calculated with the following empirical relationship (Bandise/a/. 1983) ... 

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