Newton/sparse matrix methods

The Newton/sparse matrix methods now used by electrical engineers have become the solution method of choice. Hutchison and his students at Cambridge were among the first chemical engineers to publish this approach, in the early 1970s. They used a quasi-linear model rather than a Newton one, but the ideas were really very similar. (It appears that the COPE flowsheeting system of Exxon was Newton based it existed in the mid-1960s but slowly evolved into a sequential modular system. One must assume the Newton method failed to compete.)... 

Sparse Matrix Methods. In order to get around the limitations of the sequential modular architecture for use in design and optimization, alternate approaches to solving flowsheeting problems have been investigated. Attempts to solve all or many of the nonlinear equations simultaneously has led to considerable interest in sparse matrix methods generally as a result of using the Newton-Raphson method or Broyden s method (22, 23, 24 ). ... 

A method for solving individual models such as Newton or quasi-Newton methods combined with sparse matrix methods to convert the nonlinear alge-... 

Two extremes are encountered in flowsheeting software. At one extreme, the entire set of equations (and inequalities) representing the process is employed. This representation is known as the equation-oriented method of flowsheeting. The equations can be solved in a sequential fashion analogous to the modular representation described below or simultaneously by Newton s method, Broyden s method, or by employing sparse matrix techniques to reduce the extent of matrix manipulations. Refer to the review by Evans and Chapter 5. ... 

After the Broyden correction for the independent variables has been computed, Broyden proposed that the inverse of the jacobian matrix of the Newton-Raphson equations be updated by use of Householder s formula. Herein lies the difficulty with Broyden s method. For Newton-Raphson formulations such as the Almost Band Algorithm for problems involving highly nonideal solutions, the corresponding jacobian matrices are exceedingly sparse, and the inverse of a sparse matrix is not necessarily sparse. The sparse characteristic of these jacobian matrices makes the application of Broyden s method (wherein the inverse of the jacobian matrix is updated by use of Householder s formula) impractical. 

Approximating the coupling waveforms Uj = (Vj,Vj) and treating them as inputs we can generate an iteration process, where at each iteration step the block systems can be solved concurrently on time windows using general known methods (for instance BDF, Newton s method and sparse matrix solver). 

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. 

For the solution of the entire system (2.1), (2.3), (2.4) it suggests itself to use the Newton iteration. But due to the structure of (2.4) the resulting Jacobian matrix would be non-sparse The oxygen concentration in the upper part of the reactor depends on the values of both the carbon concentration and the temperature in the layers lying underneath. Thus (2.1), (2.3) have been discretized by means of the finite element method and afterwards been solved each individually. The latter was realized through two fixed point iterations for (2.1), (2.4) with fixed temperature T and for (2.3) with fixed concentrations Cc, Coaj respectively. For this (2.1) (including boundary conditions) is written as... 

For a source term with no field-dependence, (6.228) is a sparse linear system. If not, it must be solved with Newton s method however, the Jacobian matrix is also sparse. 

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