Frictionless Motion of Diffuse Interfaces by Sharp Phase-Field Modeling

Abstract

Diffuse interface descriptions offer many advantages for the modeling of microstructure evolution. However, the numerical representation of moving diffuse interfaces on discrete numerical grids involves spurious grid friction, which limits the overall performance of the model in many respects. Interestingly, this intricate and detrimental effect can be overcome in finite difference (FD) and fast Fourier transformation (FFT)-based implementations by employing the so-called sharp phase-field method (SPFM). The key idea is to restore the discretization-induced broken translational invariance (TI) in the discrete phase-field equation by using analytic properties of the equilibrium interface profile. We prove that this method can indeed eliminate spurious grid friction in the three-dimensional space. Focusing on homogeneous driving forces, we quantitatively evaluate the impact of spurious grid friction on the overall operational performance of different phase-field models. We show that the SPFM provides superior degrees of interface isotropy with respect to energy and kinetics. The latter property enables the frictionless motion of arbitrarily oriented diffuse interfaces on a fixed 3D grid.

An Application-Driven Method for Assembling Numerical Schemes for the Solution of Complex Multiphysics Problems

“…A rather effective, alternative approach is the modification of the governing equations such that they become tailored to a specific numerical scheme. For instance, the wellknown phase field model has been adapted using specialized stencils to the FDM such that spurious grid friction effects are eliminated [61,62]. This approach, however, requires extensive knowledge about the numerics as well as the physical nature of a given problem.…”

“…The phase field variable ϕ can be understood as a coloring function that locally indicates the presence or absence of a certain phase or a certain material state within a given microstructure. For instance, in modeling of microstructure evolution during solidification, ϕ = 1 may denote the local presence of the solid and ϕ = 0 may denote the local presence of the liquid phase [61,62]. If applied to the description of crack propagation, the order-parameter field ϕ is understood as the local material state, which can be either broken ϕ = 1 or not ϕ = 0 [69,70].…”

“…The scenario is realized as a quasi 1D problem [0; L = 100] × [0; W = 1], where the interface normal direction is pointing in the x-direction and the use of simple von Neumann boundary conditions with zero phase field fluxes at the borders of the rectangular domain is legitimate. The realization of this scenario with tilted interface orientations, including the formulation of appropriate boundary conditions on the borders of the rectangular domain, is discussed in detail in [62]. In this highly symmetric quasi-1D case, the scenario can be quantitatively evaluated using the existing analytic solution for the phase field:…”

“…This extra physical length scale originates from the nonlinearity of the Allen-Cahn equation and complements the other length scales, such as the dimensions of the domain, as well as the grid spacing, both being more natural in the numerical solution of PDEs. This poses the issue of numerical resolution of the systems length scales, that is, both the domain dimensions, as well as the width ξ of the diffuse interface, need to be properly represented on the discrete numerical grid [61,62]. However, the fact that the problem is quasi-one-dimensional restricts the computational demands of the scenario.…”

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

“…With the aim to overcome the limitations discussed above, the so-called 'sharp phase-field method' has been developed by Finel et al [22], see also [23][24][25]. The method allows the computational grid to be larger than the theoretical interface thickness, hence significantly coarser meshes can be used compared to the conventional phase-field method, and thus larger physical domains can be effectively simulated.…”