Constraint Satisfaction Approach to the Design of Multi-Component, Multi-Phase Alloys

Abstract

The development of new materials must start with an understanding of their phase stability. Researchers have used the CALPHAD method to develop self-consistent databases encoding the thermodynamics of phases. In this forward approach, thermo dynamic conditions (processing conditions such as composition, temperature, pressure, etc.) are mapped to equilibrium states. In this research, we are instead interested in the inverse problem of mapping a set of desired phase constitutions to the set of thermodynamic conditions that give rise to them. Recently, search and optimization techniques have been used to determine thermodynamic conditions that yield a particular phase stability state (point-to-point mapping). In this research, we focus on a more general problem: mapping of specific regions in multi-dimensional phase constitution spaces to ranges in values of thermodynamic conditions(set-to-set mapping). In the context of search theory, we are interested in finding all solutions to a Continuous Constraint Satisfaction Problem (CCSP). The problem is typically multi-dimensional, highly nonlinear, and, importantly, contains non-isolated solutions (the solution is ranges of values rather than finite points). Numerical methods for finding all solutions to CCSPs typically rely on branch-and-prune methods, which interleave branching with pruning steps. These methods are mainly designed to address CCSPs with isolated solutions and would be inefficient if applied to the problem at hand. In this work, we describe a novel algorithm to search the thermodynamic phase field for the full set of thermodynamic conditions that result in user-specified phase constitutions. The approach combines techniques from computational materials science, evolutionary computation, and machine learning to approximate the non-isolated solution set to the CCSP. We investigate the performance of the algorithm on an Fe-Ti binary alloy system using ThermoCalc with TCFE7 database. For this system, the algorithm is able to generate solutions with low error rates.