Lagrangian models

Attempts to apply finite element techniques to machining have been made by many researchers. In these studies, we can distinguish basic differences due to the different approaches used by researchers. In particular, we can quote several finite element models used in the study of cutting processes Eulerian model Lagrangian explicit model Lagrangian implicit model... 

In order to obtain a correlation, the outflow of the effervescent spray was simulated by a numerical model based on the Navier-Stokes equations and the particle tracking method. The external gas flow was considered turbulent. In droplet phase modeling, Lagrangian approach was followed. Droplet primary and secondary breakup were considered in their model. Secondary breakup consisted of cascade atomization, droplet collision, and coalescence. The droplet mean diameter under different operating conditions and liquid properties were calculated for the spray SMD using the curve fitting technique [43] ... 

For ease of reference we here summarize the physical content of that part of the full standard model Lagrangian involving the interaction of the gauge bosons, the quarks and the Higgs. The full Fe5uiman rules are given in Appendix 2.3. Here we give just the vertices. All results are independent of the quark colour. 

As mentioned in Chapter 1, in general, the solution of the integral viscoelastic models should be based on Lagrangian frameworks. In certain types of flow... 

Lagrangian trajectory models can be viewed as foUowing a column of air as it is advected in the air basin at the local wind velocity. Simultaneously, the model describes the vertical diffusion of poUutants, deposition, and emissions into the air parcel as shown in Eigure 4. The underlying equation being solved is a simplification of equation 5 ... 

Trajectory models require spatiaUy and temporaUy resolved wind fields, mixing-height fields, deposition parameters, and data on the spatial distribution of emissions. Lagrangian trajectory models assume that vertical wind shear and horizontal diffusion are negligible. Other limitations of trajectory and Eulerian models have been discussed (30). 

Eig. 4. Schematic diagram of a Lagrangian trajectory model where Lf(/) represents the air column height in both Eulerian and Lagrangian (, Tj, ... 

The second model evaluated was the Lagrangian Photochemical Model LPM (54), a trajectory model. Backward trajectories were first determined so that starting positions could be used which would allow trajectories to reach station locations at the times of measurement. Measured concentrations ranged from 0.20 to 0.26 ppm and estimated concentrations from 0.05 to 0.53 ppm. 

MESOPUFF is a Lagrangian model suitable for modeling die transport, diffusion and removal of air pollutants from multiple point and area sources at transport distances beyond 10-50 KM. 

Taylor, J. A. (1989). A stochastic Lagrangian atmospheric transport model to determine global CO2 sources and sinks - a preliminary discussion, Tellus, Ser. B, 41,272-285. 

Because this method avoids iterative calculations to attain the SCF condition, the extended Lagrangian method is a more efficient way of calculating the dipoles at every time step. However, polarizable point dipole methods are still more computationally intensive than nonpolarizable simulations. Evaluating the dipole-dipole interactions in Eqs. (9-7) and (9-20) is several times more expensive than evaluating the Coulombic interactions between point charges in Eq. (9-1). In addition, the requirement for a shorter integration timestep as compared to an additive model increases the computational cost. 

An important alternative to SCF is to extend the Lagrangian of the system to consider dipoles as additional dynamical degrees of freedom as discussed above for the induced dipole model. In the Drude model the additional degrees of freedom are the positions of the moving Drude particles. All Drude particles are assigned a small mass mo,i, taken from the atomic masses, m, of their parent atoms and both the motions of atoms and Drude particles (at positions r, and rdj = r, + d, ) are propagated... 

Leip A, Lammel G (2004) Indicators for persistence and long-range transport potential as derived from multicompartment chemistrytransport modelling. Environ Poll 128 205-221 Lin SJ, Rood RB (1996) Multidimensional flux form semi-Lagrangian transport. Mon Wea Rev 124 2046-2068... 

The methods used for modeling pure granular flow are essentially borrowed from that of a molecular gas. Similarly, there are two main types of models the continuous (Eulerian) models (Dufty, 2000) and discrete particle (Lagrangian) models (Herrmann and Luding, 1998 Luding, 1998 Walton, 2004). The continuum models are developed for large-scale simulations, where the controlling equations resemble the Navier-Stokes equations for an ordinary gas flow. The discrete particle models (DPMs) are typically used in small-scale simulations or... 

In the two-fluid formulation, the motion or velocity field of each of the two continuous phases is described by its own momentum balances or NS equations (see, e.g., Rietema and Van den Akker, 1983 or Van den Akker, 1986). In both momentum balances, a phase interaction force between the two continuous phases occurs predominantly, of course with opposite sign. Two-fluid models therefore belong to the class of two-way coupling approaches. The continuum formulation of the phase interaction force should reflect the same effects as experienced by the individual particles and discussed above in the context of the Lagrangian description of dispersed two-phase flow. 

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