Estimating Dimensionality Other Methods

When using those spectral dimensionality reduction methods that do not exhibit a prominent gap in the spectrum of eigenvalues other intrinsic dimensionality estimation methods need to be employed. A coarse split in these algorithms can be made by examining whether they estimate the intrinsic dimensionality at a local or a global scale. The rest of this section will follow this grouping and discuss the advantages of each approach. 

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Results of calculations using Eq. [4.1.44] of three-dimensional dualistic model coincide with the experimental values of Xh and the Xh values calculated by other methods. Values A, D, and 5 can been obtained from IR-spectroscopy and Hansen s parame-ters. " Values of the TDM parameters are presented in Tables 4.1.3, 4.1.4. It should be noted that Hansen parameters are used for estimation of values of TDM parameters from equation 5 h- 5pi = 5, 5 = A D . 

Petrie and Ito (84) used numerical methods to analyze the dynamic deformation of axisymmetric cylindrical HDPE parisons and estimate final thickness. One of the early and important contributions to parison inflation simulation came from DeLorenzi et al. (85-89), who studied thermoforming and isothermal and nonisothermal parison inflation with both two- and three-dimensional formulation, using FEM with a hyperelastic, solidlike constitutive model. Hyperelastic constitutive models (i.e., models that account for the strains that go beyond the linear elastic into the nonlinear elastic region) were also used, among others, by Charrier (90) and by Marckmann et al. (91), who developed a three-dimensional dynamic FEM procedure using a nonlinear hyperelastic Mooney-Rivlin membrane, and who also used a viscoelastic model (92). However, as was pointed out by Laroche et al. (93), hyperelastic constitutive equations do not allow for time dependence and strain-rate dependence. Thus, their assumption of quasi-static equilibrium during parison inflation, and overpredicts stresses because they cannot account for stress relaxation furthermore, the solutions are prone to numerical instabilities. Hyperelastic models like viscoplastic models do allow for strain hardening, however, which is a very important element of the actual inflation process. 

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