Tucker models properties

To illustrate the effect of fiber orientation on material properties of the final part, Fig. 8.60 [5] shows how the fiber orientation distributions that correspond to 67 50 and 33% initial mold coverage affect the stiffness of the finished plates. The Folgar-Tucker model has been implemented into various, commercially available compression mold filling simulation programs and successfully tested with several realistic compression molding applications. 

An important question is how the PARAFAC and Tucker3 models are related. PARAFAC models provide unique axes, while Tucker3 models do not. A Tucker model may be transformed (rotated) and simplified to look more like PARAFAC models. This can sometimes be done with little or no loss of fit. There is a hierarchy e.g. within the family of Tucker models, Tucker3, Tucker2 and Tuckerl, which is worth studying in more detail. PARAFAC models may be difficult or impossible to fit due to so-called degeneracies (Section 5.4), in which case a Tucker3 model is usually a better a choice. Further, the statistical properties of the data - noise and systematic errors - also play an important role in the choice of model. 

The uniqueness properties of Tucker, constrained Tucker and PARAFAC models are discussed. A Tucker3 model finds unique subspaces, whereas a PARAFAC model finds unique axes. For constrained Tucker models, the situation is more complicated and no straightforward results are available. 

Scatter plots in PCA have special properties because the scores are plotted on the base P, and the columns of P are orthonormal vectors. Hence, the scores in PCA are plotted on an orthonormal base. This means that Euclidean distances in the space of the original variables, apart from the projection step, are kept intact going to the scores in PCA. Stated otherwise, distances between two points in a score plot can be understood in terms of Euclidian distances in the space of the original variables. This is not the case for score plots in PARAFAC and Tucker models, because they are usually not expressed on an orthonormal base. This issue was studied by Kiers [2000], together with problems of differences in horizontal and vertical scales. The basic conclusion is that careful consideration should be given to the interpretation of scatter plots. This is illustrated in Example 8.3. 

Closed-form expressions based on composite theory are especially useful in correlating and predicting the thermoelastic properties (moduli and coefficients of linear thermal expansion) of multiphase materials [1,2]. An article by Tucker et al [3] with emphasis on the internally consistent combination of a set of judiciously chosen techniques to predict the thermoelastic properties of a wide variety of multiphase polymeric systems, and the review articles by Ahmed and Jones [4] and by Chow [5], provide concise descriptions of micromechanical models and are recommended to readers interested in relatively brief discussions of several popular models. 

This chapter discusses some properties of three-way component models. First of all, hierarchical relationships exist between Tucker, constrained Tucker and PARAFAC models. These relationships have consequences for fit values, model complexity and model selection. Moreover, if there are no hierarchical relationships between different component models, then comparing fit values becomes problematic. 

The Baron model [15], which assumes considerable degradation of thermal conductivity by bumup, is used for the thermal conductivity of MOX fuel. It has been known to be more conservative than MATPRO-11 model [16] in most aspects of the fuel rod thermal behavior such as fuel centerline temperature and fission gas release [17]. The amount of fission gas generation in MOX fuel is assumed to be the same as that in UO2 fuel. Fission gas release is predicted by the White and Tucker-Speight model [18,19]. The Studsvik model [20] is adopted as it is a representative fuel pellet swelling model. Other material properties are taken from the MATPRO-11 model. 

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