Towards a sharper phase-field method: a hybrid diffuse-semisharp approach for microstructure evolution problems

TL;DR

This paper addresses the computational challenge of simulating microstructure evolution with moving interfaces by marrying the phase-field (diffuse-interface) approach with the laminated element technique (LET) to create a hybrid LET-PF method. In LET-PF, the diffuse phase-field governs the interface position, while a semisharp LET treatment confines phase mixing to a thin laminate layer in interface-cut elements, yielding sharper mechanical responses compared to conventional PFMs. The authors demonstrate, through 2D small-strain elastic examples, that LET-PF delivers higher accuracy on coarser meshes especially when elastic (bulk) energy dominates, enabling significant computational savings. They provide a comprehensive numerical study, discuss regularization to improve Newton convergence, and outline extensions to multi-phase, 3D, and finite-deformation problems. Overall, LET-PF offers a promising, efficient framework for two-phase microstructure evolution problems with propagating interfaces.

Abstract

A new approach is developed for computational modelling of microstructure evolution problems. The approach combines the phase-field method with the recently-developed laminated element technique (LET) which is a simple and efficient method to model weak discontinuities using nonconforming finite-element meshes. The essence of LET is in treating the elements that are cut by an interface as simple laminates of the two phases, and this idea is here extended to propagating interfaces so that the volume fraction of the phases and the lamination orientation vary accordingly. In the proposed LET-PF approach, the phase-field variable (order parameter), which is governed by an evolution equation of the Ginzburg-Landau type, plays the role of a level-set function that implicitly defines the position of the (sharp) interface. The mechanical equilibrium subproblem is then solved using the semisharp LET technique. Performance of LET-PF is illustrated by numerical examples. In particular, it is shown that, for the problems studied, LET-PF exhibits higher accuracy than the conventional phase-field method so that, for instance, qualitatively correct results can be obtained using a significantly coarser mesh, and thus at a lower computational cost.

Figures (19)

• Figure 1: Laminated element technique (LET): (a) the elements cut by an interface are treated as simple laminates; (b) the laminated elements are grouped in subset ${\cal T}_{\rm int}$, the remaining elements belong to subset ${\cal T}_1$ or ${\cal T}_2$.
• Figure 2: LET-PF compared to the conventional phase-field method. The profile of the order parameter is sketched in the upper figures. Shading of the mesh in the bottom figures indicates the volume fraction of the phases that governs the bulk energy. In the conventional phase-field method, the transition layer is diffuse and spans several elements (left), while in the semisharp LET-PF method it is localized to only one layer of elements (right).
• Figure 3: Evolving circular inclusion: (a) scheme of the problem; (b) computational domain with a regular (non-conforming) mesh of quadrilateral elements. A coarse mesh of $11\times 11$ elements within the internal square part (element size $h=0.1$) is shown, the actual computations are carried out using significantly finer meshes.
• Figure 4: Thermodynamic driving forces $\hat{f}_{\rm bulk}$ and $\hat{f}_{\rm int}$ as a function of the inclusion radius $\rho$, see Eq. \ref{['eq:rhodot:forces']}.
• Figure 5: Time evolution of the mean inclusion radius $\bar{\rho}$ predicted by LET-PF (a,c) and PFM (b,d) for $\gamma=0.0001$ (a,b) and for $\gamma=0.003$ (c,d). The computations are performed using a coarse mesh, $h=0.02$, and small time increment, $\Delta t_{\rm max}=T_{\rm exact}/500$. As a reference, the analytical solution is depicted by a solid black line.