Nonlinear convergence test

Nonlinearity typically arises because of material or geometric nonlinearities. Nonlinearity makes the problem more difficult because the geometry, support conditions, and material properties required for the equilibrium equations are not known until the solution is known. The solution cannot be obtained in a single step and some sort of iterative solution must be used, together with a relevant convergence test. In a nonlinear analysis, the principle of superposition cannot be applied and a separate analysis is required for each load case. 

If LINAIOD is zero, the model is treated as nonlinear, and a test is made for convergence of the parameter estimation. Convergence is normally declared, and final statistics of Section 6.6 are reported, if no free parameter is at a CHMAX limit and if j for each basis parameter is less than one-tenth of the 95% posterior probability half-interval calculated from Eq. (6.6-10). If the number of basis parameters equals the number of events, leaving no degrees of freedom for the estimation, this interval criterion is replaced by Aef < RPTOL... 

Newton-Raphson method. This is a trial-and-error method for solving simultaneous, nonlinear, algebraic equations. For our VLLE problem we would guess the two unknowns, L and V, use (11.1.31) and (11.1.32) to calculate values for the Rachford-Rice functions, Fj and and then test for convergence. If our convergence criteria are... 

One now needs to evaluate how well the inhibition constant (.Sri) can afford a fit to the pattern predicted for the inhibited enzyme (upper curve in Fig. 14.14a) of the two-enzyme model. One approach would be to apply a nonlinear regression (which in this case did not allow for convergence or a good fit). An alternative approach is a more pencil-and-paper type of exercise to test the inhibited enzyme (of the two-enzyme model) for fit by rearranging Eq. (14.12) to solve for A l by calculating A l for the inhibited enzyme component for each datum point or observation made ... 

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