An unconstrained formulation for thermodynamic complex solution phase minimization

While the last decade has seen significant progress in thermo-mechanical modelling of complex multiphase systems, the coupling with petrological thermodynamic modelling approaches, when addressed at all, remains a difficult task. First, most phase equilibria modeling tools have been developed with the primary focus to produce phase diagrams (e.g., Perple_X, Theriak_Domino, geoPS, MELTS) and do not offer useful interfaces for (parallel) geodynamic codes. Second, phase equilibrium modelling is generally achieved by solving a Gibbs energy minimization problem. This problem is computationally challenging as it involves solving a nested optimization problem subject to both equality and inequality constraints. As a result, the single point calculation of stable phase equilibrium is slow, and to our knowledge, >150 milliseconds for a compositional system involving a large number of chemical components. This limitation effectively precludes direct coupling of phase equilibria calculation with geodynamic models, which requires performing 1000s to 100'000s of such calculations every timestep.

We have recently developed a new open-source code, MAGEMin, that improves on part of this. In MAGEMin 75 to 90% of the computation time is dedicated to local minimization of solution models. Therefore, it becomes critically important to improve the minimization time of individual solution phase models to further speed-up phase-equilibria computational time.

Here, we present a reformulation of the solution phase model from Holland et al., (2018) that eliminates both equality and inequality constraints. Eliminating these constraints allows the utilization of faster unconstrained optimization methods, thus yielding much higher performance and stability. We compare the accuracy and performance of several unconstrained gradient-based optimization methods namely the conjugated gradient (CG), the Broyden-Fletcher-Goldfarb-Shanno (BFGS) methods and a hybrid combination (CG-BFGS).