Iterative solution method

Iterative solution methods are more effective for problems arising in solid mechanics and are not a common feature of the finite element modelling of polymer processes. However, under certain conditions they may provide better computer economy than direct methods. In particular, these methods have an inherent compatibility with algorithms used for parallel processing and hence are potentially more suitable for three-dimensional flow modelling. In this chapter we focus on the direct methods commonly used in flow simulation models. 

The general structure of an iterative solution method for the linear system of Eq. (38) is given as... 

An iterative solution method for linear algebraic systems which damps the shortwave components of the iteration error very fast and, after a few iterations, leaves predominantly long-wave components. The Gauss-Seidel method [85] could be chosen as a suitable solver in this context. 

This section will illustrate the tools taught in the above sections in the form of examples applied to steady state problems. Example 8.3 applies the finite difference method to a simple one-dimensional fin cooling problem and illustrates the nature of the system of equations that is normally achieved. Example 8.4 present a 2D compression molding problem where an iterative solution method is introduced. 

The diagonal dominance of such a modified discretized equation set is increased since the coefficient of modified equation is larger than that in Eq. (6.31) while other coefficients remain the same. This formulation has a positive effect on many iterative solution methods and is, therefore, recommended. 

In what follows, we discuss the solution methods for polynomial equations, followed by the most commonly used iterative solution methods. 

Although the iterative solution method has its downfalls, it can be drastically improved through the use of averaging. In this method, the function is still solved for x in the form ... 

The first equation contains one unknown parameter 6. However, expanding the summation of NC terms and multiplying through all the denominator terms (aj 0) give a polynomial in 6 whose order is NC. Therefore, there are NC roots of this polynomial. One of these roots lies between the two relative volatility values otoc and q hk- This is found using some iterative solution method. It is substituted into the second equation, which can then be solved explicitly for the minimum reflux ratio. 

Axelsson, O., 1994, Iterative Solution Methods. Cambridge Univ.-Press, Cambridge... 

Due to the high pressures, Langmuir is used with fugacities determined from the virial equation of state. It was found that the predictions with the multi-component Langmuir model were better than with lAST for two binary gas mixtures (H2-CO and CO-CH4) at various temperatures, but lAST proved to be superior when modelling the systems CO-CO2 and CH4-CO2 and for aU the ternary and quaternary systems. However, overall both models proved to adequately predict the mixed gas data and the predictions from the two models were very similar. From a mathematical and computational point of view, the explicit Langmuir model is simpler, while lAST needs an iterative solution method ... 

Equation (8.10) is a nonlinear algebraic equation and is difficult to solve in general. Depending on the load magnitude and the stiffness of the structure, the nonlinea-rities may be too significant for any iterative solution method. That is why the load or... 

The components of the momentum equation are usually solved one by one in a sequential manner, meaning that the set of algebraic equations for each component of the momentum is solved in turn, treating the grid point values of its dominant velocity as the sole set of unknowns. Due to the non-linearity and coupling of the underlying partial differential equations, (12.171)-(12.173) cannot be solved directly as the coefficients, a, and the source term, S, depend on the unknown solution An iterative solution method is required. 

However, if the direct sparse matrix solution method is used in the solution, then the calculation will take approximately twice the time without indicating that the equation is linear as the SPM method will require a complete matrix solution at the second Newton iteration. The approximate COE iterative solution method is used here to illustrate another possible solution approach with the pde2fe() function. 

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