Transformations and Microstructures
Discussion of the Accuracy of the Multi-Phase-Field Approach to Simulate Grain Growth with Anisotropic Grain Boundary Properties
2020 Volume 60 Issue 8 Pages 1832-1834
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2020 Volume 60 Issue 8 Pages 1832-1834
Well-defined benchmark problems based on simple geometries and idealized assumptions are extremely useful, because they offer a precise analytical solution as reference for quantitative validation of alternative numerical simulation approaches. In a recently published paper,1) different phase-field approaches for anisotropic grain growth were validated by application to a tri-crystal benchmark problem. It was concluded that the multi-phase-field (MPF) approach by Steinbach and Pezzola2) can only be applied to predict anisotropic grain growth, if an error of around 10% is accepted. An extended phase-field approach3) with adjusted higher-order energy terms was claimed to allow for prediction with significantly improved accuracy. However, a wide-spread approximate solution to the tri-crystal problem was used as reference to validate the phase-field results. As the mathematical inaccuracy of this approximation widely exceeds the evaluated inaccuracy of the phase-field results, the conclusions of this validation have to be questioned. The present paper provides the accurate analytical solution to the tri-crystal problem and discusses the implications of the approximation on the accuracy evaluation. For a reevaluation, a series of own MPF simulations were performed. Comparison with the derived analytical solution proves the high-accuracy of the MPF2) formulation and demonstrates that additional higher-order energy terms in the free energy functional are not required, but can result in considerable deviation from the targeted sharp interface solution.
Applied phase-field models for microstructure simulation are based on a diffuse interface description, but commonly aim at most accurately reproducing the sharp interface limit. The accuracy of the diffuse interface prediction strongly depends on the underlying phase-field approach. In a recent study, Miyoshi et al.1) reviewed and validated four alternative phase-field approaches for anisotropic grain growth. For a quantitative comparison, they applied all approaches to a well-defined tri-crystal benchmark problem. The present note is meant to be a comment to this previous paper. It focusses on the validation of the multi-phase-field (MPF) approach by Steinbach and Pezzola,2) which forms the basis for many advanced phase-field models, e.g. the multicomponent multi-phase-field model (MMPF) by Eiken et al.,3) nowadays widely used for applied microstructure simulation. A major target of this paper is to show that the rather high inaccuracy (≈10%) reported for this approach in the former evaluation1) is not justified, but stems from the fact that a simplified approximate solution to the tri-crystal benchmark problem was used as reference. The paper starts with the deduction of the analytic solution. Afterwards, the approximation and its impact on the accuracy evaluation are discussed. Eventually, a reevaluation of the MPF approach is performed.
The tri-crystal problem defined by Miyoshi et al.1) with symmetric arrangement of grains 2 and 3 is sketched in Fig. 1. The height equals H = 128 μm and Zero-Neumann boundary conditions are applied. Here, all boundary mobilities are set equal (MB = 10−12 m4·J−1·s−1) and the boundaries GB12 and GB13 are defined with equal energy (σB = 1 J·m−2). These values are kept constant, while the energy of the straight boundary GB23 is systematically varied (σA = 0.1 ... 1.9 J·m−2). Simulations start from initially planar boundaries. After a short transient stage, a steady state is achieved during which the system evolves with constant shape and velocity in horizontal direction. Under assumption that the triple junction does not drag the boundaries and Young’s law can be applied, an unambiguous analytical solution for the steady state velocity can be derived.
Tri-crystal arrangement moving with steady-state velocity in horizontal direction. The existence of an unambiguous analytical solution enables a quantitative accuracy evaluation for anisotropic grain growth predictions. (Online version in color.)
The analytical solution is derived under the assumption that the steady-state junction angle θ is determined by Young’s law:
| (1) |
A co-moving coordinate system is defined with x-axis in positive direction of motion (Fig. 1). The velocity of GB12 in normal boundary direction vn(x) is calculated from the local curvature κ(x).5,6)
| (2) |
| (3) |
Combining Eqs. (2) and (3) gives the differential equation:
| (4) |
Consideration of the boundary conditions y(0) = 0, y′(0) = ∞, y′(xJ) = tan(θ) and y(xJ) = H/2 yields the solution for the steady state shape y(x) and the corresponding steady state velocity vx:
| (5) |
| (6) |
The correctness can be checked by insertion of Eqs. (5), (6) into Eq. (4), under consideration of the given boundary conditions. It is noteworthy that the condition y(xJ) = H/2 yields the junction coordinate xJ = −H/(π−2θ)·ln(sin(θ)), used by Gottstein and Shvindlerman6) in their equivalent formulation of the same solution:
| (7) |
It is important to note that the derived analytical velocity (Eq. (6)) is not an approximation, but the precise and unambiguous mathematical solution to the well-defined tri-crystal problem. This is an important prerequisite for using it as a reliable reference for a quantitative validation of the phase-field results.
2.3. Approximate SolutionIn the former accuracy evaluation the following approximation was used as reference:
| (8) |
An obvious shortcoming of this approximation compared to the analytical solution in Eq. (6) is that it does not consider the impact of boundary energy σB on the steady-state velocity. Despite this fact, the approximate solution is widespread in literature. Brosh and Shneck7) deduced it as a first order approximation, while Holm et al.8) and Moelans et al.9) derived the same velocity based on the simplifying assumption that the shapes of the symmetric boundaries GB12 and GB13 are circle segments with constant global curvature κθ:
| (9) |
Note that a global curvature κθ would result in an equal velocity vn in normal boundary direction, but in unequal velocities vx(x) = vn·sin (φ(x)) in horizontal direction (see Fig. 1 and Eq. (3)). Therefore, a curvature-driven circle segment cannot maintain its shape during horizontal motion and does not represent a valid solution to the tri-crystal problem. Figure 2 illustrates that the approximate velocity vx,approx (Eq. (8)) differs by up to 32% from the precise analytic velocity vx (Eq. (6)). This error widely exceeds the discussed inaccuracy of the phase-field results.
The approximate solution to the tri-crystal problem deviates by up to 32% from the accurate analytical solution (In the isotropic case: 4.75%). MPF simulations converge with increasing numerical resolution towards the correct solution. a) Standard FD solutions with only six interface cells (η = 6Δx = 6 μm) generally underestimate the velocity. b) Improved agreement is reached for η = 10Δx = 10 μm). c) A special FD-MPF formulation10,11) allows for a reduced resolution η = 5Δx = 5 μm without loss of accuracy. (Online version in color.)
The former accuracy study1) started with the isotropic case (σA = σB). It was observed that all approaches, except the XMPF3) approach, converged to a similar value with increasing number of numerical interface cells using a standard finite-differences (FD) scheme. This converged value is in good agreement with the here derived analytical solution (Eq. (6), Fig. 2), but deviates by around +5% from the approximate solution vx,approx (Eq. (8), Fig. 2). Since the systematically underestimated non-converged velocities better agreed with the approximate reference, a reduced numerical interface resolution of only six cells was used in the former accuracy study to simulate the anisotropic grain growth (σA ≠ σB). Previous studies of interface-controlled motion10,11) revealed that the standard FD scheme requires at least ten cells for convergence, even without additional complication by junction forces. Accordingly, the conclusions drawn in the former validation1) have to be questioned for two reasons: 1.) The MPF2) computations were performed with an insufficient number of interface cells. 2.) The inaccuracy was evaluated with reference to an oversimplifying approximate solution to the tri-crystal problem. The combination of both shortcomings eventually lead to the erroneous conclusions that the MPF2) approach is less accurate than the XMPF3) approach and that an acceptable accuracy below approx.10% can only be achieved by calibrated higher order terms in the free energy functional.
3.2. Reevaluation of the MPF ApproachTo demonstrate the true predictive capabilities of the MPF2) approach, a series of new simulations were performed using a commercially available phase-field code based on the MMPF approach by Eiken et al.4) Note that the MMPF4) approach reduces to the MPF2) approach in case of purely curvature-driven motion. Figure 2 shows that the new MPF predictions well align with the analytical solution deduced in this work (Eq. (6)). Moreover, it can be seen that the residual inaccuracy does not stem from the fundamental continuous MPF2) formulation, but from its numerical discretization. The discretization error can either be minimized by choosing a sufficiently high number of interface cells or by using the advanced FD discretization scheme,10,11) implemented in the applied phase-field code. Additional terms in the continuous free energy functional are not required and not adequate to compensate the discretization error. Higher-order contributions to the interface energy generally result in kinetic deviation from Young’s law and are only justified for modelling junction drag,5) which was however not assumed in the present benchmark. Comparison of the respective simulation results published by Miyoshi et al.1) with the here derived analytic solution, demonstrate that such higher-order contributions can result in considerable deviation from the targeted sharp interface solution.
A quantitative evaluation of the accuracy of phase-field results requires a well-defined benchmark problem with an unambiguous analytical reference solution. The tri-crystal example proposed by Miyoshi et al.1) is well suited to validate anisotropic grain growth predictions. However, a simplifying approximate solution to this problem is wide-spread in literature. This paper provides the derivation of the precise analytical solution. A reevaluation of the capability of the MPF2,4) approach to predict anisotropic grain growth revealed that the recently reported high inaccuracy (≈10%) is not justified, but mainly resulted from the inaccuracy of the approximate reference solution. The former accuracy evaluation was furthermore biased by an insufficient numerical resolution. The present reevaluation demonstrates that MPF predictions well converge to the analytical solution with improved numerical discretization. It is emphasized that the proposed fitting of simulation results by calibrated higher-order contributions to the interface energy is not necessary and not consistent with predictive phase-field simulation.
