Benchmark Problems for Phase Field Methods
Advances in material science, while spawning development in engineering and other related fields, result in a need to address more computationally complicated problems that describe materials structures. One particular method for dealing with such problems is known as phase field modeling, which mathematically models processes in microstructural evolution of materials (such as coarsening and solidification). Because the method tends to treat sharp material transitions as “fuzzy,” allowing for smooth variations of physical quantities across interfaces, it has proved extremely useful for solving interfacial problems; its popularity has increased commensurate with advances in computing power. Like all physical models, however, phase field modeling can only provide approximations; in particular, phase field modeling replaces atomic structures that vary on atomic length scale with smooth microstructures that vary on nanometer length scales. Moreover, the numerical computational methods it employs prohibit exact (analytical) answers. To accommodate its growing popularity, there is a need for benchmark problems to ensure consistent and accurate solutions. This is to say that researchers seek to define a standard set of problems so that solutions across different codes may be compared.
Accordingly, researchers at The Center for Hierarchical Materials Design in a collaboration between Northwestern University, Argonne National Laboratory, and the National Institute of Standards and Technology have proposed two such benchmark problems. These problems cover the physics of solute diffusion, growth, and coarsening at two different levels of complexity. They were developed using data obtained from two workshops held to test initial formulations, and the final versions of the problems relied heavily on community input and workshop feedback.
Said Andrea Jokisaari at Northwestern University, lead author of the work: “We are excited about this work, and from the feedback that we have received from the community, we believe these benchmark problems will be very useful and helpful in evaluating new codes or numerical approaches.”
The first problem models what is known as spinodal decomposition. Spinodal decomposition occurs in the decomposition of a mixture of a materials, for example, the decomposition of a solid solution of two materials into its constituent materials.
The second models a phenomenon known as Ostwald ripening. Ostwald ripening occurs in materials whose microstructure changes over time because of, e.g., diffusion in a solid solution, and the final microstructure can have several variants or phases. For the benchmark problem, there are four phase field variables describing the various phases, and one phase field describing the composition.
Phase field modeling always begins with the assignment of a field (a mathematical variable) to each phase in the system. An energy expression is then constructed from the assigned fields so as to include terms that describe their variation (e.g., that which occurs at phase interfaces).
Spinodal decomposition requires only a single continuously alternating phase field variable. Ostwald ripening, while well-defined in terms of energy and time evolution, presents a more more complicated problem by virtue of its multivariable nature. The paper explores three distinct domains, all in two dimensions for simplicity: a square, a T-shape, and the surface of a sphere. Different domains and initial conditions (the specific starting point of a particular problem) were used to stress different numerical approaches.
In both scenarios the paper presents, the total energy will never increase when solved correctly: it will either remain constant or decrease. This serves as a first simple but important check on numerical solutions. The numerical approaches involve tracking the evolution of the systems’ microstructure; the microstructure itself is described by the phase distributions.
The problems in the paper were chosen after carefully considering several trade-offs between free energy formulations and initial conditions. Ultimately, free energies expressions that allow numerical tractability and straightforwardness were selected.
The authors of the paper have set up a website (https://pages.nist.gov/465 chimad-phase-field) to host the benchmark problems and the presented simulation data. The paper is published by Computational Materials Science and is available online at http://www.sciencedirect.com/science/article/pii/S0927025616304712 (DOI: 10.1016/j.cmmatsci.2016.09.022).
The authors of the paper are A.M. Jokisaari and P.W. Voorhees of Northwestern University, J.E. Guyer and J. Warren of National Institute of Standards and Technology, and O.G. Heinonen of Northwestern University and Argonne National Laboratory.
The work was funded by U.S. Department of Commerce, National Institute of Standards and Technology as part of the Center for Hierarchical Material Design (CHiMaD). Computation was completed using Blues and Fission, high-performance computing clusters operated by the Laboratory Computing Resource Center at Argonne National Laboratory and the High Performance Computing Center at Idaho National Laboratory, respectively.