Inner iterations

For liquid-liquid separations, the basic Newton-Raphson iteration for a is converged for equilibrium ratios (K ) determined at the previous composition estimate. (It helps, and costs very little, to converge this iteration quite tightly.) Then, using new compositions from this converged inner iteration loop, new values for equilibrium ratios are obtained. This procedure is applied directly for the first three iterations of composition. If convergence has not occurred after three iterations, the mole fractions of all components in both phases are accelerated linearly with the deviation function... 

Each iteration requires only one call of the thermodynamic liquid-liquid subroutine LILIK. The inner iteration loop requires no thermodynamic subroutine calls thus is uses extremely little computation effort. 

Choose an initial solution x, an initial temperature T, a lower limit on temperature TLOW, and an inner iteration limit L. 

In every inner iteration step the objective function... 

Due to these inner iterations via IVP solvers and due to the need to solve an associated nonlinear systems of equations to match the local solutions globally, boundary value problems are generally much harder to solve and take considerably more time than initial value problems. Typically there are between 30 and 120 I VPs to solve numerous times in each successful run of a numerical BVP solver. 

Modification of Describing Equations. In the simple K-value and enthalpy models just described, there are Ng+6 parameters, a, A, B, C, D, E and F, which have characteristics that make them excellent choices for the iteration variables of an outer iteration loop. This is in fact the essence of the inside-out concept. Its success, however, rests on the ability to transform and rearrange the describing equations to properly accommodate these variables. The result should be an efficient and well-behaved inner iteration loop in which values of the primitive variables - now regarded as dependent variables - are calculated. When the inside loop is converged, the actual K-value and enthalpy models can be employed to calculate new values of the simple model parameters. 

The describing equations are expressed in terms of the simple models, and are rearranged in a novel way so that a complete solution for the primitive variables is possible. In most cases this is achieved by converging a N-dimensional inner iteration loop, where N is the number of stages. In multi-stage applications, this inner loop is particularly amenable to solution by... 

DEW P. The calculational scheme here is shown in Fig. 12.13. We read and store T and yk, along with appropriate constants. Since we can calculate neither the d> nor the yk, all values of each are set equal to unity. Values of P are found from the Antoine equation, and Eqs. (12.29) and (12.27) are then solved for P and xk. Evaluation of y now allows recalculation of P by Eq. (12.29). With this rather good estimate of P, we evaluate and enter an inner iteration loop that converges on values for xk and (yj. Subsequent recalculation of P by Eq. (12.29) leads to the outer iteration loop that establishes the final value of P. Since the xk calculated within the inner loop are not constrained to sum to... 

An effective residual test (RT)113 checks at each inner iteration whether the relative residual fk is sufficiently small ... 

An alternative quadratic truncation test (QT) has also been suggested with associated asymptotic superlinear convergence.116 This criterion monitors, instead of the relative residual, the sufficient decrease of the quadratic model, 4 (p). Specifically, it checks whether qk p) has decreased sufficiently from one inner iteration to the next, in relation to the progress realized per inner iteration ... 

In the PCG process of the inner loop, Hessian/vector multiplications (Hd) and linear solutions of the system Mz — r for the preconditioner M are required repeatedly (see the linear PCG Algorithm [A3]). The products Hd can generally be computed satisfactorily by the following finite-difference design of gradients, at the expense of only one additional gradient evaluation per inner iteration ... 

Truncated Newton methods can be competitive only with preconditioning. Thus, the operation count for obtaining p in TN reflects IT inner PCG iterations per Newton step. Each such inner iteration involves the following operations an Hd multiplication (an, for an additional gradient evaluation in this finite-difference approximation [58]) calculation of the PCG vectors and scalars (7 , Algorithm [A4]) and numerical solution of Mz = r by forward and backward substitution (0(1), see [61]). 

Inner iteration. In some cases explicit expressions can be produced for most QSSA species, but for some other species the QSSA equations are still implicit coupled non-linear expressions. These equations can be solved separately and the concentration of QSSA species calculated by an iteration cycle. This so-called inner iteration method has proved to be a successful technique for this purpose. See Chapter 6 in [163] for an example of its application in methane and ethylene flames. 

For unsteady flows the system of non-linear equations are linearized in the iteration process within each time step, since all the solvers are limited to linear systems. The iterative process is thus performed on two different levels. The solver iterations are performed on provisional linear systems with fixed coefficients and source terms until convergence. Then, the system coefficients and sources are updated based on the last provisional solution and a new linearized system is solved. This process is continued until the non-linear system is converged, meaning that two subsequent linear systems give the same solution within the accuracy of a prescribed criterion. A standard notation used for the different iterations within one time step is that the coefficient and source matrices are updated in the outer iterations, whereas the inner iterations are performed on provisionally linear systems with fixed coefficients. On each outer iteration, the equations solved are on the form ... 

Bove [16] proposed a different approach to solve the multi-fluid model equations in the in-house code FLOTRACS. To solve the unsteady multifluid model together with a population balance equation for the dispersed phases size distribution, a time splitting strategy was adopted for the population balance equation. The transport operator (convection) of the equation was solved separately from the source terms in the inner iteration loop. In this way the convection operator which coincides with the continuity equation can be employed constructing the pressure-correction equation. The population balance source terms were solved In a separate step as part of the outer iteration loop. The complete population balance equation solution provides the... 

The Lewis-Matheson method is also an equation-tearing procedure. It was formulated according to the Case I variable specification in. Table 6.2 to determine stage requirements for specifications of the separation of two key components, a reflux ratio and a feed-stage location criterion. Both outer and inner iterations are required. The outer loop tear variables are the mole fractions or flow rates of nonkey components in the products. The inner loop tear variables are the interstage vapor (or liquid) flow rates. The Lewis-Matheson method was widely used for hand calculations, but it also proved often to be numerically unstable when implemented on a digital computer. 

The proposed method combines the outer SCF iteration and the inner iteration required for diagonalization at each SCF step into one nonlinear subspace iteration. In this approach an initial subspace is progressively refined by a low degree Chebyshev polynomials filtering. This means that each basis vector is processed... 

For the important case in which the matrices Bij = 0 for jf > i, it follows that the matrix (7 — (/ tj2)Q) is a block triangular matrix, and in the case of one space variable where the diagonal blocks are tridiagonal, (7 — (A /2)Q) can be directly inverted. For two or more space variables, the process of solving (7.5) for large numbers of mesh points would involve again inner iterations. 

In a numerical algorithm the step parameter is found again by an iterative scheme. So, beside the outer iteration which computes the next approximation of the solution we have an inner iteration for the determination of the step parameter. 

Here the linear system of the Newton equation is solved by yet another inner iteration (e.g. conjugate gradients) until a prespecified tolerance is reached. This tolerance can be chosen adaptively to recapture quadratic local convergence (see Dembo and Steihaug [1]). The iterative approach makes the truncated Newton method especially attractive for large scale problems and it is a much better procedure than the simple rule of thumb Hake Newton if good else gradienf ... 

Similar calculations for the other steady state species need to be performed. In the next step then these relations are used to calculate the remaining set of differential equations which govern the non-steady state species. The overall reaction rate for each remaining species is thus determined by means of an inner iteration loop, a simple fixed-point iteration procedure is often employed. The remaining set of differential equations can be solved in different ways, and several numerical solvers for stiff differential equations are freely available. 

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