Global terrain methods for process simulation
The task of finding physically relevant solutions to mathematical models of physical systems remains an important and challenging area of active research in many branches of science and engineering. While there are several useful ‘global’ methods for finding one or more solutions to models for chemical and other processes, it is the solutions that are generally the primary focus. Singular points are often viewed as something to be avoided, and no use is made of the natural connectedness that exists between solutions and singular points.
This thesis describes a completely different, novel and general geometric methodology for finding physically meaningful solutions and singular points to mathematical models of physical systems by intelligently moving up and down the landscape of the least-squares function using the natural connectedness between solutions and singular points. Differential geometry is used to provide theoretical support for this approach. Based on this, we developed a new family of algorithms for finding physically meaningful solutions and singular points called Global Terrain Methods, which consist of a series of downhill, equation-solving computations, uphill, predictor-corrector calculations and a termination criterion based on limited connectedness.
A variety of numerical results and geometric illustrations for chemical process models are used to make clear key theoretical concepts and to demonstrate the reliability and efficiency of Global Terrain Methods on small and large-scale process problems. All examples clearly demonstrate that Global Terrain Methods represent a reliable, efficient and global way of solving process engineering simulation and optimization problems. It is also shown that Global Terrain Methods are superior to differential arc homotopy-continuation methods on problems that exhibit parametric disconnectedness.