Improving the robustness with modified bounded homotopies and problem-tailored solving procedures
Malinen, Ilkka (2011-01-11)
The aim of this work is to improve the overall robustness in equation-oriented chemical engineering simulation work. Because the performance of locally convergent solving methods is strongly dependent on a favourable initial guess, bounded homotopy methods were investigated as a way to enlarge the domain of convergence. Bounded homotopies make it possible to keep the homotopy path inside a feasible problem domain. Thus the fatal errors possibly caused by unfeasible variable values in thermodynamic subroutines can be avoided.
To enable the utilization of a narrow bounding zone, modifications were proposed for bounded homotopies. The performance of the modifications was studied with simple test problems and several types of distillation systems in the MATLAB environment.
The findings illustrate that modified bounded homotopies with variables mapping make it possible to bound the homotopy path strictly to run inside a feasible problem domain. The homotopy path can be tracked accurately and flexibly also inside a narrow bounding zone.
It was also noticed that by utilizing the concept of bounding the homotopy path with respect to the homotopy parameter, the possibility of approaching starting point and solution multiplicities is increased in cases where the traditional problem-independent homotopy method fails. The concept aims to connect separate homotopy path branches thus offering a trackable path with real space arithmetic.
Even though the modified bounded homotopies were found to overcome several challenges often encountered with traditional problem-independent homotopy continuation methods, alone they are not enough to guarantee that the solution is approached from an arbitrary starting point. Therefore, problem-tailored solving procedures were implemented in the consideration of complex column configurations. Problem-tailored solving procedures aim to offer feasible consecutive sub-problems and thus direct the solving towards the state distribution that fulfils exact product purity specifications.
As a whole, the modified bounded homotopies and problem-tailored solving procedures were found to improve the overall robustness of an equation-oriented solving approach. Thus the threshold for designing and implementing complex process systems such as complex distillation configurations for practical use could be lowered.
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