Phase stability testing near the stability test limit

The phase stability analysis problem is highly important in phase equilibrium calculations. The stability test limit locus (STLL) is an important underlying property of multi-component system phase diagrams, because in its vicinity the number of iterations for phase stability testing dramatically increases and divergence may occur. The cause of convergence problems in phase stability calculations, as well as of the existence of a discontinuity of the TPD function in the single-phase state, is the topology of the TPD surface. At the STLL, the stationary point of the TPD function is a saddle point (the Hessian is indefinite), and for pressures just above the STLL the Hessian matrix is ill-conditioned in a domain of the hyperspace that must be “crossed” by iterates starting from one of the initial guesses. This makes stability testing in the vicinity of the STLL really challenging, and any algorithm will experience difficulties in this (fortunately tiny in most cases) region. A change of variables would not eliminate this problem, since the TPD function in the new hyperspace inherits certain properties from the original one. In this work we propose a modified objective function which exhibits multiple global minima corresponding to the stationary points of the original (TPD) function. The highly desirable feature of the modified objective functions is that the Hessian matrix is positive definite in the vicinity of the STLL (the nature of the singularity is changed). One additional derivative level is required for minimizing the new objective function, but a normal termination of the iterative sequence in a reasonable number of iterations is worth this effort. The minimization is performed by a quasi-Newton BFGS method with line search, using a suitable change of variables which avoids improper scaling, as well as by Newton iterations. Criteria to switch from the original TPD formulation to the new one are required, since the use of the more complex formulation is justified only near the STLL. Results show how application of the proposed stability testing method for a number of typical extremely difficult situations ensures convergence within tens of iterations, while all other iterative methods for minimizing the TPD function are extremely slow, unstable or even divergent.