Discontinuous profiles

A new one-dimensional mierowave imaging approaeh based on suecessive reeonstruetion of dielectrie interfaees is described. The reconstruction is obtained using the complex reflection coefficient data collected over some standard waveguide band. The problem is considered in terms of the optical path length to ensure better convergence of the iterative procedure. Then, the reverse coordinate transformation to the final profile is applied. The method is valid for highly contrasted discontinuous profiles and shows low sensitivity to the practical measurement error. Some numerical examples are presented. 

Discontinuous Profile Grinding With discontinuous profile grinding, the tooth flanks of a gear can be either ground separately or joint depending on the number and set-up of the grinding wheels. Fig. 4 (Abler et al. 2004). 

Gear Grinding, Fig. 4 Discontinuous profile grinding - tools and set-up (Karpuschewski et al. 2008)... 

A discontinuous profile grinding for work-pieces up to a diameter of 5,000 mm will be possible. 

The optimal control problem represents one of the most difficult optimization problems as it involves determination of optimal variables, which are vectors. There are three methods to solve these problems, namely, calculus of variation, which results in second-order differential equations, maximum principle, which adds adjoint variables and adjoint equations, and dynamic programming, which involves partial differential equations. For details of these methods, please refer to [23]. If we can discretize the whole system or use the model as a black box, then we can use NLP techniques. However, this results in discontinuous profiles. Since we need to manipulate the techno-socio-economic poHcy, we can consider the intermediate and integrated model for this purpose as it includes economics in the sustainabiHty models. As stated earlier, when we study the increase in per capita consumption, the system becomes unsustainable. Here we present the derivation of techno-socio-economic poHcies using optimal control appHed to the two models. 

A similar procedure exists for producing hollow discontinuous profiles. Tube Blowing RTM utilizes a preform with the textile component wrapped around a flexible blow tube. The preform is placed in a mold and internally pressurized to ensure a uniform draping at the tool wall. The resin is then injected into the preform with a pressure lower than the inner pressure (Figure 1.17) [58]. 

We consider now a continuous profile of specifically a uniformfy continuous profile of (the discontinuous profile is studied in Section 3.2.2). In Figure 3.2.2, the uniform continuous profile of is represented as a straight line varying from ( J ) ,3 at z = 0 to 0 at z =. Two simple examples of such a profile are a uniform electrical field of electrostatic potential a gravitational potential along the vertical axis. A nondimensional representation... 

We will now study the multicomponent separation capability of a discontinuous profile (shown in Figure 3.2.2 by the dashed line). The particular discontinuous profile we have chosen has a maximum value of at all values of 0 < z < Z, but at z = Z, abruptly drops to tbe value of zero and stays at that level beyond l It is thus a step function. Such a profile can be obtained in practice by having = 0 or F = 0 and maintaining two different phases or solvents in the two regions 0 < z < Z and Iz governing equation for the concentration distribution of species 1 is then obtained from equation (3.2.10) as the simpler... 

Figure 3.2.5A. Concentration profiles of species 1, 2 and 3 at any time t for a discontinuous profile in the vessel 0 < z <1. ...

As pointed out in Section 31-8, the radiation-mode contribution to e j and h j in Eq. (31-42) is valid only for continuous profiles, but e j and h j can be found directly from the second equations in Eq. (30-Sc) and (30-Sd) once and htj- are known. For a discontinuous profile, we will also obtain the correct results for e j and h j, provided we interpret the fields of Eq. (31-42) as the limit of a continuous, or smoothed-out, profile. However, it is crucial that the size of the region in which the smoothing occurs be much smaller than the perturbation region. 

The sets of coupled, integro-differential equations of Eqs. (31-50) and (31-51) are an exact restatement of Maxwell s equations for waveguides with continuous profiles. This restriction follows because we have used the expansions of Eq. (31-46). However, the coupled mode equations also apply to discontinuous profiles if we adopt the smoothing procedure described in Section 31-10. The coupled mode equations can be solved by perturbation analysis, provided n = n,for reasons given in Section 31-10. 

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