1. More accurate interpolation of flux
In the current project, we used the flux information of the old elements to get an approximation of that of the new elements. The special one that was newly added after the merge then didn’t have a parent element for this information inherence. To deal with this situation, we used the flux of the neighboring nodal point directly as an approximate number. This choice wouldn’t cause errors more than O(10^-3) in the homogeneous isotropic material simulation. But it would cause larger errors in the anisotropic cases. To improve this approximation, we could use the interpolation of the flux of several neighboring nodes as the flux information for the new element. A polynomial interpolation would be easy to implement.
2. Implement with real parameters
In the current project, we used imaginary numbers for the parameters. We could look up to the literatures for the real reasonable parameters and see if we could get similar results with them.
3. Multiple particle system
The ultimate goal would be to build a multiple particle system. But it would take us more effort in dealing with the case when multiple particles meet and merge together. This situation could become very complicated when particles are not lying in the same line. Another complex thing is how to implement the global mass conservation law for the whole system.
4. Reverse process: refining
There is a reverse process called refinement in particle formation. Particles would tend to have a uniformly distributed critical size in the system. Then there would be particle separation happening with a big particle. We could include that case in our code by adding detection and element rearrangement algorithms.
A phase field model is a mathematical model for solving interfacial problems. The method substitutes boundary conditions at the interface by a partial differential equation for the evolution of an auxiliary field (the phase field) that takes the role of an order parameter. This phase field takes two distinct values (for instance +1 and −1) in each of the phases, with a smooth change between both values in the zone around the interface, which is then diffuse with a finite width. A discrete location of the interface may be defined as the collection of all points where the phase field takes a certain value (e.g., 0).[5]
After struggling with these challenges in dealing with particle merging using interface tracking, we realize phase field simulation is the better method to deal with coarsening process. By solving the concentration field, we don’t have to deal with the judging and rearranging for the nodes and elements any more, since concentration is continuous. All the changes of the particle sizes will show up as the concentration varies across the domain over time. We believe a proper implementation of the phase field simulation could deal with both refining and coarsening process in a multiple particle system.
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