Petroval

Seifert F, Petrov V and Woerner M 1994 Solid-state laser system for the generation of midinfrared femtosecond pulses tunable from 3.3-Mu-M to 10-Mu-M Opt. Lett. 19 2009-11... 

Baranov A V, Bobovich Y S and Petrov V I 1993 Surface-enhanced resonance hyper-Raman (SERHR) spectroscopy of photochromic molecules J. Raman Spectrosc. 24 695-7... 

The first order derivative in Equation (2.80) corresponds to the convection in a field problem and the examples shown in Figure 2.26 illustraTes the ina bility of the standard Galerkin method to produce meaningful results for convection-dominated equations. As described in the previous section to resolve this difficulty, in the solution of hyperbolic (convection-dominated) equations, upwind-ing or Petrov-Galerkin methods are employed. To demonstrate the application of upwinding we consider the case where only the weight function applied to the first-order derivative in the weak variational statement of the problem, represented by Equation (2.82), is modified. 

Brooks, A. N, and Hughes, T. J.R., 1982. Streamline-upwind/Petrov Galerldn formulations for convection dominated hows with particular emphasis on the incompressible Navier -Stokes equations. Cornput. Methods Appl Meek Eng. 32, 199-259. 

Petera, J., Nassehi, V. and Pittman, J.F.T., 1989. Petrov-Galerkiii methods on isoparametric bilinear and biquadratic elements tested for a scalar convection-diffusion problem. Ini.. J. Numer. Meth. Heat Fluid Flow 3, 205-222,... 

Algorithms based on the last approach usually provide more flexible schemes than the other two methods and hence are briefly discussed in here. Hughes et al. (1986) and de Sampaio (1991) developed Petrov-Galerkin schemes based on equal order interpolations of field variables that used specially modified weight functions to generate stable finite element computations in incompressible flow. These schemes are shown to be the special cases of the method described in the following section developed by Zienkiewicz and Wu (1991). 

Derivation of the working equations of upwinded schemes for heat transport in a polymeric flow is similar to the previously described weighted residual Petrov-Galerkm finite element method. In this section a basic outline of this derivation is given using a steady-state heat balance equation as an example. 

In the consistent streamline upwind Petrov-Galerkin (SUPG) scheme all of the terms in Equation (3.52) are weighted using the function defined by Equation (3.53) and hence Wjj = Wj. 

Extension of the streamline Petrov -Galerkin method to transient heat transport problems by a space-time least-squares procedure is reported by Nguen and Reynen (1984). The close relationship between SUPG and the least-squares finite element discretizations is discussed in Chapter 4. An analogous transient upwinding scheme, based on the previously described 0 time-stepping technique, can also be developed (Zienkiewicz and Taylor, 1994). 

Hughes, T. J. R., Franca, L. P. and Balestra, M., 1986. A new finite-element formulation for computational fluid dynamics. 5. Circumventing the Babuska-Brezzi condition - a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal order interpolations. Cornput. Methods Appl. Meek Eng. 59, 85-99. 

Petera, J., Nassehi, V. and Pittman,. T.F.T., 1993. Petrov-Galerkin methods on... 

Least-square.s and streamline upwind Petrov-Galerkin (SUPG) schemes... 

Retaining all of the terms in the w eight function a least-squares scheme corresponding to a second-order Petrov-Galerkin formulation will be obtained. 

Petrov-Galerkin scheme - to discretize the energy Equation (5.25) for the calculation of T. 

A. D. Petrov, B. E. Mironov, V. A. Ponomarenko, and E. A. Chernyshev, Synthesis of Organosilicon Monomers, Consultants Bureau, New York, 1964,... 

Two important widely used sulfonic acids are known as TwitcheU s reagents, or as in Russia, the Petrov catalysts. These reagents are based on benzene or naphthalene ( ) and (12), [3055-92-3] and [82415-39-2] respectively. The materials are typically made by the coupling of an unsaturated fatty acid with benzene or naphthalene in the presence of concentrated sulfuric acid (128). These sulfonic acids have been used extensively in the hydrolysis of fats and oils, such as beef tallow (129), coconut oil (130,131), fatty methyl esters (132), and various other fats and oils (133—135). TwitcheU reagents have also found use as acidic esterification catalysts (136) and dispersing agents (137). 

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