Jacobian tridiagonal block

Adding two PI controllers to the set of equations increases the number of dynamic equations the DAE solver must integrate by 2 and inserts 14 unstructured elements into the well-structured tridiagonal block Jacobian, as shown in Figure 5.4. 

Figure 5.4 Partially structured tridiagonal block Jacobian. The points in row 941 and 942 and in column 941 and 942 represent the unstructured elements Specifically, the circles are elements of matrices A, 2 and A2,i, while the diamonds belong to the matrix A2,2.

The unknowns are taken in a large vector in the order [Rq, Pq, Co,ifo,Ri,Pi,Ci, ifi,..., RN,PN,CN,ifNV but are lumped into the vector of four-point vectors U = [Ri, Pi, Ci, ifiY, i = 0... N, to prepare for the block-tridiagonal procedure for solving the system. The system of equations (13.37) is nonlinear, and the Newton method is used to solve it. At each index i we have three 4x4 blocks in the Jacobian matrix L the left-hand block for the elements at index / — 1 M,, the middle block for index /, and Q the right-hand block for index t -I-1, that symbol chosen here in order to avoid clashes with the concentration symbol R. They produce a tridiagonal block system. For this example, three-point BDF was used, started with one BI step. 

Hence, we can maintain the block tridiagonal structure of the Jacobian in (4.5) if we introduce the parameter a as a dependent variable at m of the m -h 1 grid points and if we specify a normalization condition at the remaining grid point that does not introduce nonzero Jacobian entries outside of the three block diagonals. The success of this procedure depends upon the choice of the normalization condition. 

In flame extinction studies the maximum temperature is used often as the ordinate in bifurcation curves. In the counterflowing premixed flames we consider here, the maximum temperature is attained at the symmetry plane y = 0. Hence, it is natural to introduce the temperature at the first grid point along with the reciprocal of the strain rate or the equivalence ratio as the dependent variables in the normalization condition. In this way the block tridiagonal structure of the Jacobian can be maintained. The flnal form of the governing equations we solve is given by (2.8)-(2.18), (4.6) and the normalization condition... 

The Jacobian of the system is a square matrix, but importantly, because the residuals at any mesh point depend only on variables at the next-nearest-neighbor mesh point, the Jacobian is banded in a block-tridiagonal form. Figure 16.10 illustrates the structure of the Jacobian in the form used by the linear-equation solution at a step of the Newton iteration,... 

Fig. 16.10 Illustration of the block-tridiagonal structure of the Jacobian matrix. The structure on the right would result if the mass flux were not defined as a variable at each mesh point.

Standard specifications for the Naphtali-Sandholm method are Q-(including zero values) at each stage at which heat transfer occurs and sidestream flow ratio Sj or Sj (including zero values) at each stage at which a sidestream is withdrawn. However, the desirable block tridiagonal structure of the Jacobian matrix can still be preserved when substitute specifications are made if they are associated with the same stage or an adjacent stage. For example, suppose that for a reboiled absorber, as in Fig. 13- it is desired to specify a boil-up ratio rather than reboiler duty. Equation (13-95) for function is removed from the N(2C + 1) set of equations and is replaced by the equation... 

For single, simple columns, the Jacobian matrix has the block tridiagonal structure... 

As a consequence, the Jacobian matrix has a block tridiagonal structure. All matrix blocks other than the three central diagonals are zero. With this structure the matrix can be readily solved by a Gaussian elimination scheme (Naphtali et al., 1971). 

The residues (Nb-kNp at each node) are reduced to zero (a small positive number fixed by specifying an error tolerance at input) iteratively by computing corrections to current values of the unknowns using the Newton-Raphson method (14). Elements of the Jacobian matrix required by this method are computed from analytical expressions. The system of equations to be solved for the corrections has block tridiagonal form and is solved by use of a published software routine (1.5b... 

Equations (15-58), (15-59), and (15-60) are solved simultaneously by the Newton-Raphson iterative method in which successive sets of the output variables are produced until the values of the M, E, and H functions are driven to within some tolerance of zero. During the iterations, nonzero values of the functions are called discrepancies or errors. Let the functions and output variables be grouped by stage in order from top to bottom. As will be shown, this is done to produce a block tridiagonal structure for the Jacobian matrix of partial derivatives so that the Thomas algorithm can be applied. Let... 

This Jacobian is of a block tridiagonal form like (15-12) because functions for stage / are only dependent on output variables for stages /-1, j, and j+l. Each A, B, or C block in (15-67) represents a (2C-I-1) by (2C+ 1) submatrix of partial derivatives, where the arrangements of output variables and functions are... 

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