Iterative solution technique

However, each of these forms possesses a spurious root and has other characteristics (maxima or minima) that often give rise to convergence problems with common iterative-solution techniques. 

Thermodynamic design, for example, can be relieved of repetitive hand calculation, iterative solution techniques, and two-way interpolations. Beyond this is a pervasive cultural preference of this generation towards interacting with computers. 

As a result, an iterative solution technique is required. Initial values for the fiber volume fraction ly and resin pressure are assumed and compared with calculated values found by solution of Equations 13.9-13.13. This iterative process is described in detail in Reference [11]. 

Iterative solution techniques are commonly required to match the conversion of the gas and solids, which must satisfy the stoichiometry of reactions like that described by Equation (42). While some fairly general models have been proposed for gas-solid reactions in fluidized beds (15,20,83), most models for reactions of this type are specific to a particular reaction or reactor. 

Figure 12.67. Surface plot of solution obtained from the ADI iterative solution technique for the example in Listing 12.26. The potential value is zero on all four boundaries.

The only generally applicable methods are CISD, MP2, MP3, MP4, CCSD and CCSD(T). CISD is variational, but not size extensive, while MP and CC methods are non-variational but size extensive. CISD and MP are in principle non-iterative methods, although the matrix diagonalization involved in CISD usually is so large that it has to be done iteratively. Solution of the coupled cluster equations must be done by an iterative technique since the parameters enter in a non-linear fashion. In terms of the most expensive step in each of the methods they may be classified according to how they formally scale in the large system limit, as shown in Table 4.5. 

A multipass marching solution is used in COBRA IIIC (Rowe, 1973). The inlet flow division between subchannels is fixed as a boundary condition, and an iterated solution is obtained to satisfy the other boundary solution of zero pressure differential at the channel exit. The procedure is to guess a pattern of subchannel boundary pressure differentials for all mesh points simultaneously, and from this to compute, without further iteration, the corresponding pattern of crossflows using a marching technique up the channel. The pressure differentials are updated during each pass, and the overall channel iteration is completed when the fractional change in subchannel flows is less than a preset amount. 

Without a solution, formulated mathematical systems (models) are of little value. Four solution procedures are mainly followed the analytical, the numerical (e.g., finite different, finite element), the statistical, and the iterative. Numerical techniques have been standard practice in soil quality modeling. Analytical techniques are usually employed for simplified and idealized situations. Statistical techniques have academic respect, and iterative solutions are developed for specialized cases. Both the simulation and the analytic models can employ numerical solution procedures for their equations. Although the above terminology is not standard in the literature, it has been used here as a means of outlining some of the concepts of modeling. 

Obviously this approach is not easily extended to cascades containing more than three reactors and, in those cases, an alternative trial and error procedure is preferable. One chooses a reactor volume and then determines the overall fraction conversion that would be obtained in a cascade of N reactors. When one s choice of individual reactor size meets the specified overall degree of conversion, the choice may be regarded as the desired solution. This latter approach is readily amenable to iterative programming techniques using a digital computer. 

In this section we consider how Newton-Raphson iteration can be applied to solve the governing equations listed in Section 4.1. There are three steps to setting up the iteration (1) reducing the complexity of the problem by reserving the equations that can be solved linearly, (2) computing the residuals, and (3) calculating the Jacobian matrix. Because reserving the equations with linear solutions reduces the number of basis entries carried in the iteration, the solution technique described here is known as the reduced basis method. ... 

It must choose as the variables for iteration those for which the sensitivity of the system is appropriate and consistent with the particular solution technique chosen for the system. 

Among the several 2-RDM-oriented methods that have been developed for the study of chemical systems, one of the most recent and promising techniques is based on the iterative solution of the second-order contracted Schrodinger equation (2-CSE) [1, 6, 15, 18, 36, 45-60, 62-65, 68, 70, 79-85, 103-111]. The 2-CSE was initially derived in 1976 in first quantization in the works of Cho [103], Cohen and Erishberg [104, 105], and Nakatsuji [106] and later on deduced in second quantization by Valdemoro [45] through the contraction of... 

A two-level optimisation solution technique as presented in Chapter 6 and 7 can be used for a similar optimisation problem. For a given product specifications (in terms of purity of key component in each Task) and considering ReTi as the only outer level optimisation variable, the above MDO problem (OP) can be decomposed into a series of independent minimum time problem (Single-period Dynamic Optimisation (SDO) problem) in the inner level. For each iteration of the outer level optimisation, the inner-level problems are to be solved. As mentioned in the earlier chapters, the method is efficient for simultaneous design and operation optimisation especially with multiple separation duties. 

Models for the reacting polydispersed particles contain stiff ordinary differential equations. The stiffness is due partly to the wide range of thermal time constants of the particles and partly to the high temperature dependence of reactions like combustion and devolatilization. As an alternative to the established solution techniques based on Gear s method an iterative approach is developed which uses the finite difference representations of the differential equations. The finite differences are obtained by... 

There are two basic families of solution techniques for linear algebraic equations Direct- and iterative methods. A well known example of direct methods is Gaussian elimination. The simultaneous storage of all coefficients of the set of equations in core memory is required. Iterative methods are based on the repeated application of a relatively simple algorithm leading to eventual convergence after a number of repetitions (iterations). Well known examples are the Jacobi and Gauss-Seidel point-by-point iteration methods. 

Our remarks so far concern the speeding of convergence in iterative solutions of the SSOZ equation with various closures, and are applicable to any site-site pair potentials. Additional considerations arise when some of the sites carry charges. Special techniques then have to be adopted to overcome difficulties associated with the long range of the Coulomb potential. The method conventionally used for this purpose is the renormalization introduced by Hirata and Rossky, which is analogous to Allnatt s method for electrolytes. This has been combined with the Gillan method by Morriss and Monson and was later used to obtain solutions for... 

The initial condition may be determined by an iterative solution of equation (2.93) just once at the beginning of the simulation, or, indeed, by a prior, off-line calculation. Alternative techniques based on integrating an artificial prior transient are given in Chapter 18, Sections 18.7 to 18.9, where a more detailed worked example is given. 

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