Global homotopy continuation and stability analysis: optimal application to chemical engineering design

The design of chemical engineering process equipment routinely necessitates the solution of large systems of non-linear equations. The processing speed of standard desktop computers have allowed for equation-oriented process simulators to become commercially viable. For a fully equation-oriented simulator to be competitive with standard iterative solvers, a robust and reliable non-linear equation solver is required and standard Newton methods fail to provide the necessary reliability. Homotopy continuation methods have been proposed as a way to greatly expand the domain of convergence of the process simulator\u27s equation solving algorithm. This research expands upon existing homotopy continuation algorithms, investigates bounding algorithms, corrector and predictor step-length algorithms, with all techniques described being made available in the developed software package HOMES3. Two new cubic homotopies are introduced, as are alternative solution deflation homotopies that generate reliable methods to find additional roots after at least one has been located with a standard homotopy. Robustness of the code is demonstrated with chemical engineering design problems such as reactive distillation and two-phase flashing flow. A valuable property of the homotopy continuation methods is that they are global in nature and multiple solutions can be found from a single starting point. In the case of the reactive flash and reactive distillation models investigated, dynamic stability of the steady states is defined by the asymptotic stability of the corresponding index-2 differential-algebraic equation. Methods to numerically evaluate the stability are presented, including an analysis of how to retrieve lost stability information after standard index reductions --Abstract, page iii