Euler-Lagrange method

The minimization of Wlost (=Elost) entails the minimization of the integral in Eq. (4.182) with the constraint of constant production rate J (mol/h) and use of the Euler-Lagrange method... 

The arbitrary Euler-Lagrange method. It consists of moving the finite element mesh nodes with time. This method works well as long as the mesh is not too distorted. In practice, remeshing is usually required after a few time steps. 

It should be emphasized once again the important point confining considerably the application field of the calculus of variations. When determining extrema of the target functional the Euler-Lagrange method does not take into account the possibility for the existence of limitations imposed on the control parameters and phase coordinates. 

Using the Euler-Lagrange method (which will be presented in Chapter 5) to minimize the free energy we get... 

Using Euler-Lagrange method to minimize the elastic energy, we obtain... 

Instead of using the Euler-Lagrange method to find the exact solutions for 0 and a, we use approximations for them. For 0 f), we use the same solution found in the last section. For a(r), because the boundary conditions are a(r=0) = 0, we use the approximation... 

A hierarchy of computational models is available to simulate dispersed gas-liquid-solid flows in three-phase slurry and fluidized bed reactors [84] continuum (Euler-Euler) method, discrete particle/bubble (Euler-Lagrange) method, or front tracking/capturing methods. While every method has its own... 

Lapin, A., Muller, D., and Reuss, M. (2004) Dynamic behaviour of microbial populations in stirred bioreactors simulated with Euler-Lagrange method traveling along the lifelines of single cells. Ind. Eng. Chem. Res., 43, 4647-4656. 

Fiere, we will explore Euler—Lagrange methods for the case of relatively dense multiphase flows where both two-way coupling between the discrete and continuous phase and interactions between individual members of the discrete phase are important. A number of reviews have already appeared in... 

An alternative procedure is the dynamic programming method of Bellman (1957) which is based on the principle of optimality and the imbedding approach. The principle of optimality yields the Hamilton-Jacobi partial differential equation, whose solution results in an optimal control policy. Euler-Lagrange and Pontrya-gin s equations are applicable to systems with non-linear, time-varying state equations and non-quadratic, time varying performance criteria. The Hamilton-Jacobi equation is usually solved for the important and special case of the linear time-invariant plant with quadratic performance criterion (called the performance index), which takes the form of the matrix Riccati (1724) equation. This produces an optimal control law as a linear function of the state vector components which is always stable, providing the system is controllable. 

Simulations of multiphase flow are, in general, very poor, with a few exceptions. Basically, there are three different kinds of multiphase models Euler-Lagrange, Euler-Euler, and volume of fluid (VOF) or level-set methods. The Euler-Lagrange and Euler-Euler models require that the particles (solid or fluid) are smaller than the computational grid and a finer resolution below that limit will not give a... 

Using the Euler-Lagrange variational method, one can find an analytical solution for the optimal modulation phase (p t), given a Markovian bath at long times (see Suppl. Info.). This yields... 

The computational effort of solving orbital Euler-Lagrange (OEL) equations is significantly reduced if the generally nonlocal exchange-correlation potential vxc can be replaced or approximated by a local potential vxc(r). A variationally defined optimal local potential is determined using the optimized effective potential (OEP) method [380, 398]. This method can be applied to any theory in which the model... 

Attempts to extend RANS formulation to LES of two-phase combustion may be found in [318 354 317 255 292]. They are all based on a Euler-Lagrange (EL) description of the dispersed phase in which the flow is solved using an Eulerian method and the particles are tracked with a Lagrangian approach. An alternative is the Euler-Euler (EE) description, also called two-fluid approach, in which both the gas and the dispersed phases are... 

We present how to treat the polarization effect on the static and dynamic properties in molten lithium iodide (Lil). Iodide anion has the biggest polarizability among all the halogen anions and lithium cation has the smallest polarizability among all the alkaline metal cations. The mass ratio of I to Li is 18.3 and the ion size ratio is 3.6, so we expect the most drastic characteristic motion of ions is observed. The softness of the iodide ion was examined by modifying the repulsive term in the Born-Mayer-Huggins type potential function in the previous workL In the present work we consider the polarizability of iodide ion with the dipole rod method in which the dipole rod is put at the center of mass and we solve the Euler-Lagrange equation. This method is one type of Car-Parrinello method. 

If we now look back at Vcin der Waals treatment, sec. 2.5. several steps can be recognized. In a sense he anticipated the present method. His function to be minimized is [2.5.19] he eliminated boundary condition [A1.3] by working grand canonically and [2.5.25] is his Euler-Lagrange equation. From this. F could be written as [2.5.30] ). 

The reason is that the finite element method is not well suited to problems like this, with convection but no diffusion. Your job is to use enough artificial diffusion to eliminate the oscillations in the solution but without obscuring the essential details. A variety of specialized methods are available to do that, as described by Finlayson (1992). The specialized methods include Random Choice, Euler-Lagrange, MacCormack, and Taylor-Galerkin. 

The free energy is at its minimum when the liquid crystal is at equilibrium. One can obtain the minimum by means of the Euler-Lagrange functional method. 

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