Multivariable terrain methods

A novel geometric terrain methodology was recently presented for finding all physically relevant solutions and singular points to chemical process simulation problems based on intelligently moving along special integral curves of the gradient vector field to guide both uphill and downhill movement. The terrain methodology is extended here to include multivariable problems and integral curve bifurcations. The integral curves of interest are those associated with valleys and ridges and characterized as a collection of extrema in the norm of the gradient over a set of level curves. It is shown that integral curves can undergo tangent, pitchfork, and other types of bifurcations and that terrain methods are clearly superior to differential arc homotopy-continuation methods on problems that exhibit parametric disconnectedness. Several examples, including a pair of CSTR problems, a retrograde flash calculation, and the task of finding all azeotropes for a heterogeneous ternary mixture, are used to show that terrain methods represent a reliable, efficient, and global way of solving multivariable process engineering simulation and optimization problems. Geometric illustrations are used whenever possible to clarify underlying ideas.