Precomputing approach for a two-scale phase transition model

TL;DR

This work addresses computational challenges in two-scale phase transition models with moving microstructures by developing a precomputing offline strategy for the macro conductivity and a semi-implicit time-stepping method to handle nonlinear coupling. The authors establish local-in-time well-posedness using the Hanzawa transformation to fixed domains, derive Lipschitz bounds for the transformed cell problems, and quantify how interpolation errors propagate to the coupled macro/micro system. They prove that the precomputing error can be bounded and that the time-stepping scheme converges linearly under suitable regularity and step-size constraints, with numerical experiments validating the theory. The approach offers significant speedups through offline computation and parallelization, making large-scale two-scale phase transition simulations more feasible for applications involving evolving micro-geometries.

Abstract

In this study, we employ analytical and numerical techniques to examine a phase transition model with moving boundaries. The model displays two relevant spatial scales pointing out to a macroscopic phase and a microscopic phase, interacting on disjoint inclusions. The shrinkage or the growth of the inclusions is governed by a modified Gibbs-Thomson law depending on the macroscopic temperature, but without accessing curvature information. We use the Hanzawa transformation to transform the problem onto a fixed reference domain. Then a fixed-point argument is employed to demonstrate the well-posedness of the system for a finite time interval. Due to the model's nonlinearities and the macroscopic parameters, which are given by differential equations that depend on the size of the inclusions, the problem is computationally expensive to solve numerically. We introduce a precomputing approach that solves multiple cell problems in an offline phase and uses an interpolation scheme afterward to determine the needed parameters. Additionally, we propose a semi-implicit time-stepping method to resolve the nonlinearity of the problem. We investigate the errors of both the precomputing and time-stepping procedures and verify the theoretical results via numerical simulations.

Figures (6)

• Figure 1: A potential setup of a highly heterogeneous two-phase system with evolving phase interface.
• Figure 2: Representation of the two-scale system at some time $t>0$. At the left an example for the initial micro-domain $Y_0=Y(0)$. In this picture, $h(t,x_1)<0$ (shrinking ball) and $h(t,x_2)>0$ (growing ball). Please note that the mathematical setup and the theoretic result allows for any $C^3$-inclusion (not just balls) but in our numerical simulations, we only look at balls.
• Figure 3: The effective conductivity $K$ for changes in height $h$. At the right two cell solutions for different $h$.
• Figure 4: Convergence curves for the precomputing strategy. (a) shows the error for linear interpolation of $K$, (b) the error for quadratic spline interpolation of $K$. For $\Theta$ the plotted error is in the $L^2(S;H^1(\Omega))$ norm, for $\vartheta$ the $L^2(S\times \Omega; H^1(Y_0))$ norm is used and the error for $h$ is computed in the $L^2(S\times \Omega)$ norm. In (c) the influence of discretization $H_\text{cell}$ of the cell solutions on the accuracy of the quadratic interpolation is demonstrated. Here, only the error for $\Theta$ is shown.
• Figure 5: Convergence order of the interpolation of $K$ depending on the size of the height interval $I_h$.

Theorems & Definitions (27)

• Remark 1
• Definition 1: Weak solution in moving coordinates
• Definition 2: Weak solution in fixed coordinates
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4: Existence for prescribed height functions