Plane analysis approach

A coupled analysis need not be all encompassing. For example, a two dimensional plane frame analysis of a building employing two or more degrees of freedom is considered a coupled analysis approach. Separate plane frames for each orthogonal horizontal direction can be used in lieu of a single comprehensive three dimensional model. Refer to Section 6.6.2 for a discussion on modeling considerations for this type of structure. 

A general systematic technique applicable to second-order differential equations, of which (11.31) is a particular example, is that of phase plane analysis. We have seen this approach before (chapter 3) in the context of systems with two first-order equations. These two cases are, however, equivalent. We can replace eqn (11.31) by two first-order equations by introducing a new variable g, which is simply the derivative of the concentration with respect to z. Thus... 

Reducing the dynamics of a complex system to that of a two-variable system is the goal pursued in most studies devoted to periodic or excitable behaviour, in chemistry as in biology. The main impetus for such a reduction is that it allows us to study these phenomena by means of phase plane analysis. The latter clarifies the origin of the two modes of dynamic behaviour, and highlights basic features common to different systems. This approach was followed in the study of excitability and oscillations in the Belousov-Zhabotinsky reaction (Tyson, 1977), and in... 

It is important to mention one serious deficiency of the extended Drude analysis. This formalism is based on an isotropic version of Fermi-liquid theory which is highly questionable in cuprates where the electronic structure is in fact very anisotropic in the ab-plane. One approach that addresses this problem is to first calculate the conductivity, using a proper anisotropic theory with A -dependent Fermi velocity and scattering rate 1/r, and then use the real and imaginary parts of the calculated conductivity to find the effective scattering rates l/r(co) and m. Some steps in that direction have been made by Stojkovic and Pines (1996) and Branch and Carbotte (1998). 

Linear stability analysis provides one, rather abstract, approach to seeing where spatial patterns and waves come from. Another way to look at the problem has been suggested by Fife (1984), whose method is a bit less general but applies to a number of real systems. In Chapter 4, we used phase-plane analysis to examine a general two variable model, eqs. (4.1), from the point of view of temporal oscillations and excitability. Here, we consider the same system, augmented with diffusion terms a la Fife, as the basis for chemical wave generation ... 

Figure 7.7 Adsorption of a protein on a hydrophobic surface obtained by a Biacore 3000 surface plasmon resonance instrument. The data are fitted to the Langmuir isotherm. Notably, despite the apparent good fit to the Langmuir model, a more detailed understanding of the kinetics of the adsorption process relies on more rigorous approaches, e.g. phase plane analysis, as, for example, discussed by Hansen et al. (2012). Courtesy of Thomas Callisen, Novozymes...

It turns out that many surfaces (and many line patterns such as shown in Fig. XV-7) conform empirically to Eq. VII-20 (or Eq. VII-21) over a significant range of r (or a). Fractal surfaces thus constitute an extreme departure from ideal plane surfaces yet are amenable to mathematical analysis. There is a considerable literature on the subject, but Refs. 104-109 are representative. The fractal approach to adsorption phenomena is discussed in Section XVI-13. 

The previous section has illustrated a simple convenient means of analysing in-plane loading of symmetric laminates. Many laminates are of this type and so this approach is justified. However, there are also many situations where other types of loading (including bending) are applied to laminates which may be symmetric or non-symmetric. In order to deal with these situations it is necessary to adopt a more general type of analysis. 

This equation has been derived as a model amplitude equation in several contexts, from the flow of thin fluid films down an inclined plane to the development of instabilities on flame fronts and pattern formation in reaction-diffusion systems we will not discuss here the validity of the K-S as a model of the above physicochemical processes (see (5) and references therein). Extensive theoretical and numerical work on several versions of the K-S has been performed by many researchers (2). One of the main reasons is the rich patterns of dynamic behavior and transitions that this model exhibits even in one spatial dimension. This makes it a testing ground for methods and algorithms for the study and analysis of complex dynamics. Another reason is the recent theory of Inertial Manifolds, through which it can be shown that the K-S is strictly equivalent to a low dimensional dynamical system (a set of Ordinary Differentia Equations) (6). The dimension of this set of course varies as the parameter a varies. This implies that the various bifurcations of the solutions of the K-S as well as the chaotic dynamics associated with them can be predicted by low-dimensional sets of ODEs. It is interesting that the Inertial Manifold Theory provides an algorithmic approach for the construction of this set of ODEs. 

Mathematical approaches used to describe micelle-facilitated dissolution include film equilibrium and reaction plane models. The film equilibrium model assumes simultaneous diffusive transport of the drug and micelle in equilibrium within a common stagnant film at the surface of the solid as shown in Figure 7. The reaction plane approach has also been applied to micelle-facilitated dissolution and has the advantage of including a convective component in the transport analysis. While both models adequately predict micelle-facilitated dissolution, the scientific community perceives the film equilibrium model to be more mathematically tractable, so this model has found greater use. 

This approach is useful because it allows quantitative analysis via Walsh correlation diagrams to be made without extensive calculations. Figure 5.11 may clarify the approach. Initially the extra half-plane is at x = 0 and atom C is covalently bonded to atom A. When the half plane moves to the mid-glide position, x = b/2, the activation complex, ACB forms (Figure 5.10). Finally, when the half-plane moves to x = b, the pair CB forms a new covalent bond. Symbolically ... 

This analysis is limited, since it is based on a steady-state criterion. The linearisation approach, outlined above, also fails in that its analysis is restricted to variations, which are very close to the steady state. While this provides excellent information on the dynamic stability, it cannot predict the actual trajectory of the reaction, once this departs from the near steady state. A full dynamic analysis is, therefore, best considered in terms of the full dynamic model equations and this is easily effected, using digital simulation. The above case of the single CSTR, with a single exothermic reaction, is covered by the simulation examples THERMPLOT and THERM. Other simulation examples, covering aspects of stirred-tank reactor stability are COOL, OSCIL, REFRIG1 and REFRIG2. Phase-plane plots are very useful for the analysis of such systems. 

---