Rate of Convergence of General Phase Field Equations in Strongly Heterogeneous Media Toward Their Homogenized Limit
Rate of Convergence of General Phase Field Equations in Strongly Heterogeneous Media Toward Their Homogenized Limit
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DOI：
10.1137/16m1079646
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发表时间：
2013-10
期刊：
影响因子：
--
通讯作者：
M. Schmuck;S. Kalliadasis
中科院分区：
文献类型：
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作者：
M. Schmuck;S. Kalliadasis
我们研究相场方程的基础上扩散界面近似的一般均匀的自由能密度显示不同的局部极小值的可能的平衡配置在穿孔/多孔域。在过去的几十年里，对均匀环境中这种自由能的研究引起了广泛的兴趣，因此现在在科学和工程中被广泛接受和应用。在这里，我们专注于强非均质材料与穿孔，如多孔介质。据我们所知，我们提出了一个一般的正式推导的upscaled相场方程的任意自由能密度，并给出了严格的理由，误差估计为广泛的一类多项式自由能。对于给定特性不均匀性的材料，新的升尺度公式的有效宏观解和微观相场问题的解之间的误差为1/2。我们的新的，有效的，可靠的宏观多孔介质制定的一般相场方程开辟了新的建模方向和计算的角度在强异质环境中的界面传输。
We study phase field equations based on the diffuse-interface approximation of general homogeneous free energy densities showing different local minima of possible equilibrium configurations in perforated/porous domains. The study of such free energies in homogeneous environments found a broad interest over the last decades and hence is now widely accepted and applied in both science and engineering. Here, we focus on strongly heterogeneous materials with perforations such as porous media. To the best of our knowledge, we present a general formal derivation of upscaled phase field equations for arbitrary free energy densities and give a rigorous justification by error estimates for a broad class of polynomial free energies. The error between the effective macroscopic solution of the new upscaled formulation and the solution of the microscopic phase field problem is of order $\epsilon^1/2$ for a material given characteristic heterogeneity $\epsilon$. Our new, effective, and reliable macroscopic porous media formulation of general phase field equations opens new modelling directions and computational perspectives for interfacial transport in strongly heterogeneous environments.