Convergent solutions

The described algorithm may not yield a converged solution in particular for values of power law index less than 0.5. To ensure convergence, in the iteration cycle (h + 1) for updating of the nodal pressures, an initial value found by... 

After a number of iterations, the energy from one iteration may be the same as from the previous iteration. This is what chemists desire a converged solution. 

There is a great difference between various simulators (5) in terms of how easily and how well the hypothetical calculation units can be incorporated in the simulation. The trial-and-error calculations, which ate called iterative calculations, do not always converge for every flow sheet being simulated. Test problems can be devised to be tried with various simulators to see if the simulator will give a converged solution (11). Different simulators could take different numbers of iterations to converge and take different amounts of computet time on the same computet. 

Example 7 Calculation of Inside Out Method For the conditions of the simple distillation column shown in Fig, 13-55, obtain a converged solution by the inside-out method, using the SRK eqiiation-of-state for thermodynamic properties (in the outer loop),... 

From these tables, it is seen that the stage temperatures and total liquid flows are already close to the converged solution after only one outer-loop iteration. However, the composition of the bottoms product, specifically with respect to the lightest component, C, is not close to the converged solution until after two iterations. The inside-out method does not always converge so dramatically, but is usually quite efficient,... 

Likewise, efficient interface reconstruction algorithms and mixed cell thermodynamics routines have been developed to make three-dimensional Eulerian calculations much more affordable. In general, however, computer speed and memory limitations still prevent the analyst from doing routine three-dimensional calculations with the resolution required to be assured of numerically converged solutions. As an example. Fig. 9.29 shows the setup for a test involving the oblique impact of a copper ball on a hardened steel target... 

It should also be remembered that the discretization scheme influences the accuracy of the results. In most CFD codes, different discretization schemes can be chosen for the convective terms. Usually, one can choose between first-order schemes (e.g., the first-order upwind scheme or the hybrid scheme) or second-order schemes (e.g., a second-order upwind scheme or some modified QUICK scheme). Second-order schemes are, as the name implies, more accurate than first-order schemes. However, it should also be remembered that the second-order schemes are numerically more unstable than the first-order schemes. Usually, it is a good idea to start the computations using a first-order scheme. Then, when a converged solution has been obtained, the user can continue the calculations with a second-order scheme. 

If the flow is isothermal, there is no need to solve for the temperature equation (Eq. (11.6)). In this case the last term in Eq. (11.5) is also dropped. If, however, the thermal comfort is simulated, then the temperature equation must be solved. In ventilation the temperature variations are normally small, which means that it is sufficient to account for density variation only in the gravitation term (the last term in Eq. (11.5)). The gravitation term acts in the vertical direction, and in Eq. (11.5) it is assumed that the xj coordinate is directed vertically upward. denotes a reference temperature, which should be constant. It does not influence the predicted results, except that the pressure level is changed. It could, however, affect convergence rate (i.e., increase the number of required iterations required to reach a converged solution), and it should be chosen to a reasonable value, such as the inlet temperature. 

A typical computation such as the ones described here used about 100 adaptively placed mesh points and required about 5 minutes on a Cray 1-S. Of course, larger reaction mechanisms take more time. Also, simpler transport models can be used to reduce computation time. Since the solution methods are iterative, the computer time for a certain simulation can be reduced by starting it from the solution of a related problem. For example, it may be efficient to determine the solution to a problem with a susceptor temperature of 900 K starting from a converged solution for a reactor with a susceptor temperature of 1000 K. In fact, it is typical to compute families of solutions by this type of continuation procedure. 

Step 6. When a converged solution has been achieved or at the minimum of the objective function check the variances of the parameters and the eigenvectors corresponding to the smallest eigenvalues to identify any highly correlated zones. Combine any adjacent zones of high variance. 

The converged solution may not be realistic in terms of realistic values of the parameters and variables involved. 

A converged solution is obtained when SAL — L to within a given tolerance. 

Boger, D.L., Jiang, W., and Goldberg, J. Convergent solution-phase combinatorial synthesis with multiplication... 

Relaxation methods involve iteratively seeking a convergent solution to the Laplace equation. In the present case, for instance, if we rewrite the coefficient matrix A = I + E, where the latter matrix consists of elements that are all small compared to 1, the matrix Laplace equation takes the form = EU + b. One begins the calculation with values U = b [or, equivalently, U = 0] and iteratively computes successive values The calculation terminates when a specified limit of accuracy is achieved. One such measure involves calculating the proportional differences ... 

Figures 2 and 3 present typical results obtained from a low plate count column and a high plate count column. The graphs present the calculated molar concentrations of macromolecular species as a function of their degree of polymerization. The straight lines are the theoretical, kinetic distributions. Inasmuch as convergent solutions are obtained, the algorithm is effective for correction for Imperfect resolution.

The results of the computer calculation are shown in the next two pages. (The program was altered to produce only a short output, omitting the mole fractions.) It is apparent that the number of iterations to the converged solution, 80, is large. The time of solution on an IBM 704 computer was 10.3 min., which when roughly translated into operation on a more modern machine at prevailing industrial rates costs about... 

Do not use steps 3-5 of the DC convergence solutions. Using these steps may not produce a valid DC operating point, which is essential for SPICE to linearize the circuit. See the AC analysis description. Once DC convergence is achieved, the AC analysis will also converge. 

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