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Lagrangian sectional approach

Since the algebraic equations f = 0 are not expected to be satisfied with an initially assumed control, we penalize the Lagrangian. The approach is similar to that in Section 7.2.1 (p. 201). We set = Wf at each time in the interval [0, tf]. Thus, the objective functional... [Pg.210]

With this approach, when an element becomes severly distorted, it is eliminated from the computational grid and becomes a free mass point. Clearly, care must be taken to avoid eliminating elements that could potentially influence the problem at some later time. An example of a three-dimensional Lagrangian calculation that uses the eroding element scheme is presented in the next section. [Pg.346]

The first approach is based on introducing simple velocity or position rescaling into the standard Newtonian MD. The second approach has a dynamic origin and is based on a refonnulation of the Lagrangian equations of motion for the system (so-called extended Lagrangian formulation.) In this section, we discuss several of the most widely used constant-temperature or constant-pressure schemes. [Pg.58]

In the next section we derive the Taylor expansion of the coupled cluster cubic response function in its frequency arguments and the equations for the required expansions of the cluster amplitude and Lagrangian multiplier responses. For the experimentally important isotropic averages 7, 7i and yx we give explicit expressions for the A and higher-order coefficients in terms of the coefficients of the Taylor series. In Sec. 4 we present an application of the developed approach to the second hyperpolarizability of the methane molecule. We test the convergence of the hyperpolarizabilities with respect to the order of the expansion and investigate the sensitivity of the coefficients to basis sets and correlation treatment. The results are compared with dispersion coefficients derived by least square fits to experimental hyperpolarizability data or to pointwise calculated hyperpolarizabilities of other ab inito studies. [Pg.114]

Owing to the sensitivity of the chemical source term to the shape of the composition PDF, the application of the second approach to model molecular mixing models in Section 6.6, a successful model for desirable properties. In addition, the Lagrangian correlation functions for each pair of scalars (( (fO fe) ) should agree with available DNS data.130 Some of these requirements (e.g., desirable property (ii)) require models that control the shape of /, and for these reasons the development of stochastic differential equations for micromixing is particularly difficult. [Pg.312]

The dynamic treatment of the charges is quite similar to the extended Lagrangian approach for predicting the values of the polarizable point dipoles, as discussed in the previous section. One noteworthy difference between these approaches, however, is that the positions of the shell charges are ordinary physical degrees of freedom. Thus the Lagrangian does not have to be extended with fictitious masses and kinetic energies to encompass their dynamics. [Pg.105]

This article is organized as follows. In Section 2 ab initio molecular dynamics methods are described. Specifically, in Section 2.1 we discuss the extended Lagrangian atom-centered density matrix (ADMP) technique for simultaneous dynamics of electrons and nuclei in large clusters, and in Section 2.2 we discuss the quantum wavepacket ab initio molecular dynamics (QWAIMD) method. Simulations conducted and new insights obtained from using these approaches are discussed in Section 3 and the concluding remarks are given in Section 4. [Pg.334]

The fluid dynamics of bubble column reactors is very complex and several different CFD models may have to be used to address critical reactor engineering issues. The application of various approaches to modeling dispersed multiphase flows, namely, Eulerian-Eulerian, Eulerian-Lagrangian and VOF approaches to simulate flow in a loop reactor, is discussed in Chapter 9 (Section 9.4). In this chapter, some examples of the application of these three approaches to simulating gas-liquid flow bubble columns are discussed. Before that, basic equations and boundary conditions used to simulate flow in bubble columns are briefly discussed. [Pg.332]

In general, it may be concluded that it is possible to develop appropriate Eulerian-Eulerian models to simulate complex gas-liquid (solid) flows, with some support from the experimental data. Some of the possible applications of such models are discussed in the next section. Before discussing these applications, recent simulations carried out with Eulerian-Lagrangian and volume of fluid (VOF) approaches are briefly reviewed here. [Pg.353]

The approach in this section is lagrangian i.e., the model is for a drying object (particle, drop, sheet, etc.) as it moves through the drying process in time. More complicated models can use a eulerian frame of reference by simulating the dryer with material moving into and out of the dryer. [Pg.1351]


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