An energy-based model of the ferroelectric polarization process is presented in the current contribution. In an energy-based setting, dielectric displacement and strain (or displacement) are the primary independent unknowns. As an internal variable, the remanent polarization vector is chosen. The model is then governed by two constitutive functions: the free energy function and the dissipation function. Choices for both functions are given. As the dissipation function for rate-independent response is non-differentiable, it is proposed to regularize the problem. Then, a variational equation can be posed, which is subsequently discretized using conforming finite elements for each quantity. We point out which kind of continuity is needed for each field (displacement, dielectric displacement and remanent polarization) is necessary to obtain a conforming method, and provide corresponding finite elements. The elements are chosen such that Gauss' law of zero charges is satisfied exactly. The discretized variational equations are solved for all unknowns at once in a single Newton iteration. We present numerical examples gained in the open source software package Netgen/NGSolve.

中文翻译：

---

基于能量的铁电材料极化过程模型的高阶混合有限元

当前的贡献中提出了基于能量的铁电极极化过程模型。在基于能量的设置中，介电位移和应变（或位移）是主要的独立未知数。选择剩余偏振矢量作为内部变量。该模型由两个本构函数控制：自由能函数和耗散函数。给出了两个函数的选择。由于速率无关响应的耗散函数是不可微的，因此建议对问题进行正则化。然后，可以提出一个变分方程，随后使用每个量的一致有限元对其进行离散化。我们指出每个场（位移、介电位移和剩余极化）是获得符合方法所必需的，并提供相应的有限元。选择元素使得高斯零电荷定律完全满足。离散化的变分方程在单个牛顿迭代中一次性求解所有未知数。我们展示了在开源软件包 Netgen/NGSolve 中获得的数值例子。