用变分不等式分析铁电材料的极化过程
The polarization process of ferroelectric materials analyzed in the framework of variational inequalities
摘要：
本文研究铁电介质中极化过程的数学模型。我们假设这个耗散过程是由两个本构函数，这是自由能函数和耗散函数。与耗散能密切相关的耗散函数通常是不可微的。因此，总能量的最小化条件包括耗散函数的次微分。这个条件也可以用应变、介质位移、反射极化和反射应变等未知场的变分不等式来表示。我们分析了这个问题的数学适定性。我们提供了时间离散更新方程的存在性和唯一性结果。在较强的假设下，我们可以证明这个含时变分不等式解的存在性。为了解决离散化的变分不等式，我们使用混合有限元，其中机械位移和介电位移是未知的，以及极化（和，如果包括在模型中，残余应变）。这样就有可能精确地满足高斯零自由电荷定律。我们建议正则化的耗散函数和解决所有的未知数在一次在一个单一的牛顿迭代。我们提出了在开源软件包Netgen/NGSolve中获得的数值例子。
英文摘要：
We are concerned with the mathematical modeling of the polarization process in ferroelectric media. We assume that this dissipative process is governed by two constitutive functions, which are the free energy function and the dissipation function. The dissipation function, which is closely connected to the dissipated energy, is usually non-differentiable. Thus, a minimization condition for the overall energy includes the subdifferential of the dissipation function. This condition can also be formulated by way of a variational inequality in the unknown fields strain, dielectric displacement, remanent polarization and remanent strain. We analyze the mathematical well-posedness of this problem. We provide an existence and uniqueness result for the time-discrete update equation. Under stronger assumptions, we can prove existence of a solution to the time-dependent variational inequality. To solve the discretized variational inequality, we use mixed finite elements, where mechanical displacement and dielectric displacement are unknowns, as well as polarization (and, if included in the model, remanent strain). It is then possible to satisfy Gauss' law of zero free charges exactly. We propose to regularize the dissipation function and solve for all unknowns at once in a single Newton iteration. We present numerical examples gained in the open source software package Netgen/NGSolve.
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