_{1}

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A numerical simulation scheme is proposed to analyze domain tessellation and pattern formation on a spherical surface using the phase-field method. A multi-phase-field model is adopted to represent domain growth, and the finite-difference method (FDM) is used for numerical integration. The lattice points for the FDM are distributed regularly on a spherical surface so that a mostly regular triangular domain division is realized. First, a conventional diffusion process is simulated using this lattice to confirm its validity. The multi-phase-field equation is then applied, and pattern formation processes under various initial conditions are simulated. Unlike pattern formation on a flat plane, where the regular hexagonal domains are always stable, certain different patterns are generated. Specifically, characteristic stable patterns are obtained when the number of domains, n, is 6, 8, or 12; for instance, a regular pentagonal domain division pattern is generated for n = 12, which corresponds to a regular dodecahedron.

The phase field model has been developed to simulate microstructures in materials, and various complicated patterns such as dendrite, cellular, lamellar, and polycrystalline structures have been successfully regenerated [

Now, our next target is region division on a curved surface. First, we focus on a spherical surface. In a flat plane, the most preferred unit is a regular hexagon because of its symmetry. However, region division on a spherical surface is not straightforward; it is impossible to divide the spherical surface into equally shaped hexagons. This is revealed by reviewing polyhedra or platonic solids. For instance, a regular dodecahedron consists of 12 pentagons, and a regular icosahedron consists of 20 regular triangles. This geometrical feature means that the surface division with regular pentagons or triangles is possible if slight distortion along the curved surface is allowed. However, these are only the exceptions, and it is well known that regular polyhedra are realized only for the tetra-, hexa-, and octahedron in addition to the dodeca- and the icosahedron. Similar types of problem have been investigated in geophysics and meteorology and developed as the discrete global grid system [

The multi-phase-field model is applied in this study. This model was originally proposed by Steinbach et al. to simulate phase transformation among multiple phases [

where

and

Here, _{ij}, a_{ij}, w_{ij}, and Δf_{ij} are parameters depending on the combination of i and j, and n is the number of phases on site. The last term, _{i} is the area of the i-th domain and _{i} has a relative value with respect to the total surface area of the large sphere. This term is introduced to prevent domain overgrowth and to keep all domain area equal, because domain coarsening occurs due to the nature of the phase field model [

The multi-phase-field Equation (1) is solved numerically using the finite difference method (FDM). Grids are usually arranged on a rectangular lattice in 2-D FDM, but such gridding is not applicable on a curved face. Especially for a spherical surface, rectangles are unable to fill the surface completely. For example, the grid arrangement based on latitude and longitude division unavoidably creates distortion around the polar area. Triangular division has greater flexibility, and it is possible to fill the surface with mostly equally spaced grid lines, although precisely equal spacing is impossible. Even though the difference is negligible, grid symmetry cannot be realized. In a planar problem, each grid in the regular triangular division has six neighboring grids, but some of the grids on a spherical surface must have five neighbors, according to Euler’s theorem on the polyhedron. In this study, we create a triangular lattice by accepting this irregularity, and solve the conventional diffusion equation using the grids to check their validity before applying them to the phase-field equation.

The discretized equation for the simple diffusion equation

where n is the number of neighboring grids, Δt is the time increment, and Δr_{ij} is the distance between the central i-th grid and the neighboring j-th grid. This is a type of isotropic integration and is the case of forward difference with respect to the time increment. This form is reduced for usual FDM calculations with a regular square lattice when n = 4 and Δr_{ij} = Δx. In the current stage of our study, the influence of the curvature on the numerical calculation is not imposed.

The presented numerical integration scheme is validated by solving a conventional diffusion problem. The non- dimensional heat conduction equation is solved for an initial temperature T = 0.0 in entire domain. Then, the temperature in one of the polar domains, i.e., the area where the normal vector is inclined up to five degrees in the positive direction of the z axis, is fixed at T = 1.0.

A simple situation is first considered to validate the present method. Grids shown in _{max} = 0.50, and the minimum interval is then Δx_{min} = 0.33. The sphere radius R is set as R = 50Δx_{max}.

The original phase and growing phases are expressed by

m_{ij} | a_{ij}^{ } | w_{ij} | Δf_{ij} | Δx | Δt | |||
---|---|---|---|---|---|---|---|---|

for i ≠ j | 0.14 | 2.38 | 1.15 | for i or j = 0 | 2.55 | 0 | 0.50 | 0.02 |

for i = j | 0 | 0 | 0 | for others | 0 | 0 in Section 3 250.0 in Section 4 |

tween the i-th and j-th phases; however, in this study, identical values are applied for each combination. Parameter Δf_{ij} represents the difference in free energy between two phases, and this value is assumed non-zero only for the interaction between the original phase and growing phases. First, the effect of the size-controlling term is excluded to observe the stability of the formed pattern. Then, to investigate the instable cases, this term is turned on by applying a non-zero value to

The nuclei of the growing phases are set as follows: Case 1: intersection points of the spherical surface with the x, y, and z axes (6 points). Case 2: intersection with the equivalent eight directions denoted by vector (±1, ±1, ±1) (8 points). Case 3: Case 1 plus Case 2 (14 points). The phase-field value _{n} = 0.01 from the i-th nuclei point and

The simulation results for Case 1 are shown in

equals 0 in the original phase (the region where

field model. In this simulation, as shown in

The simulation results for Case 2 are presented in

The simulation results for Case 3 are shown in

In the Case-3 simulation, the small triangular domains disappear soon after formation. This result reflects the natural tendency of the phasefield model, but this feature sometimes limits its application. In this section, simulations are demonstrated by imposing a condition to prevent the domain size from reducing. The other conditions and parameters are set the same as those in the previous section, except that an extra term

The simulation results are presented in

Finally, the pattern formation initiated with randomly arranged domain nuclei is simulated.

A variation in the domain size and area distribution is drawn in _{max} and R_{min}, respectively. The value length is expressed in Δx unit. The maximum and minimum areas S_{max} and S_{min} are also plotted in the same figure, where the area is expressed in the ratio to the average area of all domains. Variation of the domain area shows that each domain area increases by the 1000-th time step, when free domain growth is completed. Then, the maximum area starts decreasing, while the minimum area keeps increasing. This means that large domains shrink and small ones keep enlarging, and hence, the size adjustment works effectively. The domain size variation does not necessarily follow this trend if the shape is not the same even if the area is equivalent. In

Numerical analyses were carried out to investigate the space-dividing pattern on a spherical surface. The phasefield model was applied to simulate the domain distinction and its growth. A simple method was introduced for the finite-difference calculation on a spherical surface, and its validity was confirmed by solving a conventional diffusion equation resulting in concentric isolines. Pattern formation simulation was subsequently demonstrated and unique patterns were obtained; a stable barrel-shaped pattern with six faces was formed, whereas triangular division with cross-boundaries was meta-stable. The size-controlling term allowed the reproduction of a wider

variety of patterns, and a dodecahedral division with 12 pentagons was exhibited as an example. Consequently, the present method reveals to be valid and effective for pattern formation analysis. In this study, only a limited number of models were investigated, i.e., N = 6, 8, 14 for regular nuclei arrangement, and N = 12 for random arrangement. Nevertheless, many other patterns have been obtained for different values of N. These data will be analyzed in our next study based on geometrical theory in relation to the surface area and boundary or edge lengths. Furthermore, this method will be applicable for some practical purposes. For example, this method is expected to be utilized in materials design by coupling the geometrical features with interfacial energy and some other physical and chemical properties.

Takuya Uehara, (2016) Numerical Simulation of a Domain-Tessellation Pattern on a Spherical Surface Using a Phase Field Model. Open Journal of Modelling and Simulation,04,24-33. doi: 10.4236/ojmsi.2016.42003