Sharp phase-field modeling of isotropic solidification with a super efficient spatial resolution

Abstract

The phase-field method provides a powerful framework for microstructure evolution modeling in complex systems, as often required within the framework of integrated computational materials engineering. However, spurious grid friction, pinning and grid anisotropy seriously limit the resolution efficiency and accuracy of these models. The energetic resolution limit is determined by the maximum dimensionless driving force at which reasonable model operation is still ensured. This limit turns out to be on the order of 1 for conventional phase-field models. In 1D, grid friction and pinning can be eliminated by a global restoration of Translational Invariance (TI) in the discretized phase-field equation. This is called the sharp phase-field method, which allows to choose substantially coarser numerical resolutions of the diffuse interface without the appearance of pinning. In 3D, global TI restricts the beneficial properties to a few specific interface orientations. We propose an accurate scheme to restore TI locally in the local interface normal direction. The new sharp phase-field model overcomes grid friction and pinning in three-dimensional simulations, and can accurately operate at dimensionless driving forces up to the order of $$10^{4}$$ 10 4 . At one-grid-point interface resolutions, exceptional degrees of isotropy can be achieved, if further the largely inhomogeneous latent heat release at the advancing solid-liquid interface is mitigated. Imposing a newly proposed source term regularization, the new model captures the formation of isotropic seaweed structures without spurious dendritic selection by grid anisotropy, even at one-grid-point interface resolutions.

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

“…Recall that η (ω) and n (ω) depend on the level-set function ϕ h , see Eqs. (25) and (26). The micro-macro transition for the laminated microstructure is discussed in Appendix A.…”

“…Preliminary computations have revealed that the convergence of the Newton method at the global level is severely affected by the non-differentiable functions (the absolute value and Macauley brackets) that are involved in the formula for the volume fraction η (ω) in element ω, see Eq. (25).…”

“…In order to improve the robustness of the method, the following regularization is applied to the absolute value function and Macauley brackets used in Eq. (25),…”

“…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.…”

3D Minimum Channel Width Distribution in a Ni-Base Superalloy

“…By numerical resolution issues, conventional phase-field models require the diffuse interface width to be chosen at least four times larger than the grid spacing. This limitation can be overcome by advanced discretization techniques [39] or by employing the newly proposed sharp phase-field method, which allows the operation with interface widths on the scale of a single distance between two neighboring grid points [40,41].…”