Multivariable Terrain Methods

A no®el geometric terrain methodology was recently presented for finding all physically rele®ant solutions and singular points to chemical process simulation problems based on intelligently mo®ing along special integral cur®es of the gradient ®ector field to guide both uphill and downhill mo®ement. The terrain methodology is extended here to include multi®ariable problems and integral cur®e bifurcations. The integral cur®es of interest are those associated with ®alleys and ridges and characterized as a collection of extrema in the norm of the gradient o®er a set of le®el cur®es. It is shown that integral cur®es 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. Se®eral 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 sol®ing multi®ariable process engineering simulation and optimization problems. Geometric illustrations are used whene®er possible to clarify underlying ideas.