A Systematic Procedure for Analysis and Modification of Nonlinear Equation Sets to Enhance Convergence and Reduce Computational Effort

      Many practical problems in chemical engineering give rise to the need to solve systems of non-linear algebraic equations (NLE). Typical examples include phase equilibrium and chemical equilibrium computations, steady state process simulation, flow distribution in pipeline networks etc. Nowadays powerful mathematical software packages are available for solving systems of NLEs. Shacham and Cutlip^[1]  evaluated and compared six state of the art software packages. In this work only the POLYMATH^[2] and MATLAB[^3] packages are employed. Based on our experience, these packages can serve as good representative of the additional ones. 

      Some of the library functions for solving NLEs employ "globally convergent" methods (see, for example p. 176 in Press et al^[4]), but even such methods are often unable to find the solution from certain sets of initial estimates. In such cases some trial and error with regard to the initial estimates may be required in order to enable the algorithm to converge to the correct solution. A common cause of such difficulties is the presence of discontinuities and/or regions where some of the functions are undefined (Shacham and Brauner^[5]). The impact of such discontinuities becomes most severe if they lie in the vicinity of the solution. Consider, for example the "Combustion of Propane in Air" test problem of Hiebert^[6] shown in Table 1.

Table 1. Hiebert’s^[6]  "Combustion of Propane in Air" NLE Solver Test Problem

No.	Equations	Variable	x0	x*	f(x*)
1	R = 40
2	f1 = x1+x4-3 = 0	x1	2	2.9976	-3.55E-05
3	f2 = 2*x1+x2+x4+x7+x8+x9+2*x10-R = 0	x2	5	3.9664	3.37E-04
4	f3 = 2*x2+2*x5+x6+x7-8 = 0	x3	40	80	-5.34E-05
5	f4 = 2*x3+x5-4*R = 0	x4	1	0.0023645	6.04E-04
6	f5 = x1*x5-0.193*x2*x4 = 0	x5	0	6.04E-04	-1.99E-08
7	f6 = x6*sqrt(x2)-0.002597*sqrt(x2*x4*xs) = 0	x6	0	0.0013659	-1.00E-07
8	f7 = x7*sqrt(x4)-0.003448*sqrt(x1*x4*xs) = 0	x7	0	0.064573	2.57E-08
9	f8 = x8*x4-1.799e-5*x2*xs = 0	x8	0	3.5308	-3.59E-08
10	f9 = x9*x4-0.0002155*x1*sqrt(x3*xs) = 0	x9	0	26.432	1.45E-06
11	f10 = x10*x4^2-3.846e-5*xs*x4^2 = 0	x10	5	0.0044998	-1.56E-13
12	xs = x1+x2+x3+x4+x5+x6+x7+x8+x9+x10			117.00021

      This example comprises of 10 implicit algebraic equations with 10 unknowns (x_i, i=1…10), requiring 10 initial guesses. The x_i's represent the number of moles of the various components,  thus all x_i must be ≥0. Four variables are constrained to non-negative values (x₁, x₂, x₃, x₄ and x_s). Using a constrained NLE solver several solutions to this problem can be obtained, some of them are physically infeasible. The solution shown in Table 1 (x*) was obtained by CONLES, the constrained NLE solver algorithm of Shacham^[7] , which is available in POLYMATH and also for MATLAB (see pp. 198-199 in ^[8]). 

      To alleviate the solution of NLE systems which contain discontinuities the following systematic procedure is proposed.

1.   Identify expressions with discontinuities (like logarithm, square root, division by unknowns, etc.). Modify such expressions as to eliminate the discontinuities (replacing log by exp., for example), if possible.

2.   Select a unique "output" variable from each of the equations, where an output variable only appears on the left hand side (l.h.s) of one (and only one) of the equations. Leave the equations from which no output variable can be selected in an implicit form.

3.   Mark the unknowns that could not be selected as output variables as of "known" (value) and arrange the explicit equations in a sequence so that only "known"  variables appear in the r.h.s. of the equations and the "output" variable on the l.h.s. Explicit equations that cannot be arranged in this form should be rewritten as implicit equations.

4.   Solve the revised system of equations with an NLE solver program.

5. Substitute the solution obtained into the original NLE system to verify that the reformulated and rearranged system is identical (in terms of the solution obtained) to the original system.

      Applying the proposed procedure to the example problem yields the results of Table 1.

Table 1. Revised Formulation of the Example Problem

No.	Equations	Variable	x0	x*	f(x*)
1	R = 40
2	x1 = 3 - x4	x1	-	2.9976	-
3	x5 = 4 * R - (2 * x3)	x2	-	3.966427	-
4	x2 = x1 * x5 / (.193 * x4)	x3	40	79.9997	-
5	x6 = .002597 * sqrt(x4 * xs)	x4	1	0.0023645	-
6	x7 = .003448 * sqrt(x1 * xs)	x5	-	6.04E-04	-
7	x8 = .00001799 * x2 * xs / x4	x6	-	0.0013659	-
8	x9 = .0002155 * x1 * sqrt(x3 * xs) / x4	x7	-	0.0645727	-
9	x10 = 0 - ((2 * x1 + x2 + x4 + x7 + x8 + x9 - R) / 2)	x8	-	3.530816	-
10	f1 = 2 * x2 + 2 * x5 + x6 + x7 – 8 = 0	x9	-	26.43155	1.15E-10
11	f2 = x10 * x4 ^ 2 - (.00003846 * x4 ^ 2 * xs) = 0	x10	-	0.0044998	-3.02E-16
12	f3= xs - (x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10) = 0	xs	100	116.9995	-5.409E-11

      Observe that the use of the proposed procedure alleviated the problem solution by reducing the dimension of the problem (3 implicit equations, instead of 10), reduction of the number of initial estimates that need to be provided (3 instead of 10) that enables providing good initial estimates based on physical considerations, and reducing the number of variables that are associated with discontinuity (x₂ is removed from the list).

      In the presentation the proposed procedure for analysis and modification of NLE systems will be described in more detail and several examples where the use of this procedure alleviates considerably the solution of NLE systems that contain regions where some of the functions are undefined, will be presented.

 

References

1.     Shacham, M. and M.B. Cutlip, “A Comparison of Six Numerical Software Packages for Educational Use in the Chemical Engineering Curriculum”, Computers in Education Journal, IX(3), 9-15 (1999)

2.     POLYMATH is a product of Polymath Software (http://www.polymath-software.com).

3.     MATLAB is a trademark of The Math Works, Inc. (http://www.mathworks.com).

4.     Press, W.H., P.B. Flannery, S.A. Teukolsky and W.T. Vetterling, Numerical Recipes, 2nd Ed., Cambridge University Press, Cambridge (1992)

5.     Shacham, M. and  N.  Brauner, ”Numerical Solution of Nonlinear Algebraic Equations with Discontinuities”, Comp. & Chem. Eng., 26, 1449-1457 (2002)

6.     Hiebert, K. L. "An evaluation of mathematical software that solves systems of nonlinear equations". ACM Transactions on Mathematical Software, 8, 5–20 (1983).

7.     Shacham, M. Numerical solution of constrained nonlinear equations. International Journal of Numerical Methods in Engineering, 23, 1455–1481 (1986)..

8.     Cutlip, M. B., and M. Shacham, Problem solving in chemical and biochemical engineering with POLYMATH, Excel and MATLAB. 2nd Ed., Prentice Hall, Upper Saddle River, N. J.  (2008)