Phase-field modelling and computing for a large number of phases

Abstract

We propose a framework to represent a partition that evolves under mean curvature flows and volume constraints. Its principle follows a phase-field representation for each region of the partition, as well as classical Allen–Cahn equations for its evolution. We focus on the evolution and on the optimization of problems involving high resolution data with many regions in the partition. In this context, standard phase-field approaches require a lot of memory (one image per region) and computation timings increase at least as fast as the number of regions. We propose a more efficient storage strategy with a dedicated multi-image representation that retains only significant phase field values at each discretization point. We show that this strategy alone is unfortunately inefficient with classical phase field models. This is due to non local terms and low convergence rate. We therefore introduce and analyze an improved phase field model that localizes each phase field around its associated region, and which fully benefits of our storage strategy. To demonstrate the efficiency of the new multiphase field framework, we apply it to the famous 3D honeycomb problem and the conjecture of Weaire–Phelan’s tiling.

Multiphase mean curvature flows with high mobility contrasts: A phase-field approach, with applications to nanowires

“…where, again, the Lagrange multiplier field λ encodes the partition constraint N k=1 u k = 1. The idea to localize the Lagrange multiplier λ near the diffuse interface has been recently proposed in [16] in order to improve the accuracy of the two-phases model. In our case, we will show that, at least formally,…”

Learning phase field mean curvature flows with neural networks

“…where the Lagrange multiplier λ is associated with the partition constraint L k=1 u k = 1. As explained in [68,45], this PDE can be for instance computed in two steps:…”

“…The second row of Figure 17 gives a similar numerical experiment with additional constraint on the volume of each phase. Here, following the approach developed in [69,68], the idea is to consider the Allen-Cahn system with volume conservation…”

Longest minimal length partitions

“…It is obvious that with (7) and ( 8) it is possible to implement a gradient-based optimization algorithm in order to minimize F ε and P ε . The software FreeFEM [20] is used for constructing the finite element framework and the algorithm LBFGS from the package Nlopt [23] is used for the minimization of (7). We address the question of handling the constraints in the next section.…”