基于可杂交间断Galerkin方法的各向异性铁电材料数值建模
Numerical modeling of anisotropic ferroelectric materials with hybridizable discontinuous Galerkin methods
中文总结 AI 辅助
该研究重构各向异性铁电材料GLD模型的能量形式，证明其梯度流解的存在唯一性，提出能量稳定的半隐式时间格式与可杂交间断Galerkin空间离散方法，并通过数值测试验证了格式的稳定性、收敛性及铁电材料特性。
中文摘要 AI 辅助
我们通过重构能量形式，研究了各向异性铁电材料的Ginzburg--Landau--Devonshire（GLD，金兹堡-朗道-德文希尔）模型的梯度流结构。我们证明修正后的能量形式至少存在一个极小值点。在对电荷分布和初始极化场的若干正则性假设下，我们证明$L^2$梯度流结构存在唯一解。为对GLD模型进行数值模拟，我们提出了一种能量稳定的半隐式时间步进格式，以及用于空间离散的可杂交间断Galerkin方法。我们开展了若干数值测试，验证了所提数值格式的稳定性与收敛性，以及铁电材料的部分特性。
英文摘要
We investigate a gradient flow structure of the Ginzburg--Landau--Devonshire (GLD) model for anisotropic ferroelectric materials by reconstructing its energy form. We show that the modified energy form admits at least one minimizer. Under some regularity assumptions for the electric charge distribution and the initial polarization field, we prove that the $L^2$ gradient flow structure has a unique solution. To simulate the GLD model numerically, we propose an energy-stable semi-implicit time-stepping scheme and a hybridizable discontinuous Galerkin method for space discretization. Some numerical tests are provided to verify the stability and convergence of the proposed numerical scheme as well as some properties of ferroelectric materials.
