Motivation

Mathematicians, statistical mechanists and computational materials scientists are interested in studying the spatiotemporal evolutionary aspects of multi-phased partitioning of an n-dimensional space. We give four examples for this.

The 1st example is from mathematics where researchers are interested in the chaotic partitioning of a n-D bounded spatial domain and its spatiotemporal evolution under some governing rules.

The 2nd example is from statistical mechanics, the very well-known Ising model of the importance sampling Monte-Carlo techniques studying the spatiotemporal evolution of the kinetics and thermodynamics of the distribution of two phases in a lattice. Exact solutions have been developed for such simple models involving 2 states [1], but for more complex models like the Q-state Pott’s model, an exact model is impractical due to the vastness of the solution space.

The 3rd example is from fundamental computational materials science where researchers are interested in grain growth [2], where the temporal evolution of the spatial and thermodynamical parameters of multi-phase grain structures [3,4] is studies. A part of this research also touches upon understanding the kinematic and kinetic behaviour of insoluble 2nd phase particles in grain structures [5] and how they interact with the grain boundaries. Some of these studies have tried to validate empirical models of grain structure geometry such as the Zener equation [6]. As the shape of the particles influence the Zener drag working against grain boundary evolution during grain growth [7], and that nature presents irregularly shaped particles, computer models which can consider such particle shape and their spatial distribution becomes very essential.

The 4th example is from applied computational materials science where researchers need poly-crystalline grain structures to be used in techniques such as crystal plasticity based finite element analysis in order to study material’s phase-partitioned thermo-mechanical response and texture evolution under applied thermo-mechanical loads [8]. Though Voronoi tessellated geometries of grain structures have been used before in crystal plasticity-based simulations, they are simplifications and do not accurately represent the geometric irregularities presented by nature.

1. R.J. Baxter, Exactly Solved Model in Statistical Mechanics, Academic press, Harcourt Brace Jovanovich, 1989.
2. D. Weaire, S. Mcmurry, Some Fundamentals of Grain Growth, Solid State Phys. - Adv. Res. Appl. 50 (1996) 1–36. https://doi.org/10.1016/S0081-1947(08)60603-7.
3. M.P. Anderson, D.J. Srolovitz, G.S. Grest, P.S. Sahni, Computer simulation of grain growth-I. Kinetics, Acta Metall. 32 (1984) 783–791. https://doi.org/10.1016/0001-6160(84)90151-2.
4. M.P. Anderson, G.S. Grest, R.D. Doherty, K. Li, D.J. Srolovitz, Inhibition of grain growth by second phase particles: Three dimensional Monte Carlo computer simulations, Scr. Metall. 23 (1989) 753–758. https://doi.org/10.1016/0036-9748(89)90525-5.
5. D.J. Srolovitz, M.P. Anderson, G.S. Grest, P.S. Sahni, Computer simulation of grain growth-III. Influence of a particle dispersion, Acta Metall. 32 (1984) 1429–1438. https://doi.org/10.1016/0001-6160(84)90089-0.
6. P.A. Manohar, M. Ferry, T. Chandra, Five Decades of the Zener Equation, ISIJ Int. 38 (1998) 913–924. https://doi.org/10.2355/isijinternational.38.913.
7. W.B. Li, K.E. Easterling, The influence of particle shape on zener drag, Acta Metall. Mater. 38 (1990) 1045–1052. https://doi.org/10.1016/0956-7151(90)90177-I.
8. F. Roters, P. Eisenlohr, L. Hantcherli, D.D. Tjahjanto, T.R. Bieler, D. Raabe, Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: Theory, experiments, applications, Acta Mater. 58 (2010) 1152–1211. https://doi.org/10.1016/j.actamat.2009.10.058