Modified bounded homotopies to enable a narrow bounding zone

A New Homotopy-Based Strategy for the Robust Determination of All the Feasible Solutions for CSTR Systems

“…In this study, it will be shown that the target of finding all the feasible solutions for the investigated CSTR systems can be reached with the Newton homotopy-based solving method if it is equipped with the features of homotopy parameter and variables bounding presented in Malinen and Tanskanen [30][31]. The solving strategy of the present work extends the applicability of the method presented in Malinen et al [32] in the determination of multiple steady states of chemical engineering systems.…”

Nonlinear Homotopic Continuation Methods: A Chemical Engineering Perspective Review

“…A third strategy that has often been reported in the literature ,,,− is the “differential equation based approach”. In this method, an artificial additional parameter P is introduced and can be geometrically interpreted as the arc-length of the homotopic path.…”

“…Three alternatives have been proposed to date to overcome these situations. An obvious first solution is to maintain a very short step size of the predictor to ensure that predictions fall sufficiently close to the next homotopic step. , Another possible solution, proposed by Malinen and Tanskanen and Malinen, consists of a second mapping (the bounding approach is a mapping itself) to follow the unbounded homotopic paths in bounded artificial domains, which allows the construction of correct compressed homotopic paths. The third solution, proposed by Oliveros-Muñoz and Jiménez-Islas, implements a special path tracking methodology named “the hyperspherical path tracking methodology”.…”

A Novel Method To Find All Physical Solutions of Constrained Chemical Engineering Models in Polynomial Equations

“…Homotopy continuation is one of the most widely used methods for locating all mathematical solutions. This is a sophisticated global method for gradually approaching a solution set for the equations from a starting point that satisfies another simpler system of equations. , In practice, homotopy continuation methods are frequently used in finding all solutions of arbitrary nonlinear systems of equations. Finding all solutions can be guaranteed in some cases, for example, continuous polynomial systems .…”

“…In recent decades, constrained global solution methods have been studied to handle constrained simulation problems. Bounded homotopy methods have been developed to track multiple solutions within the physical bounds . Newton-like methods using interval arithmetic, also known as interval Newton methods, have attracted attention. , The main attractive feature of interval Newton methods is the mathematical guarantee of convergence.…”