三维电孔隙弹性方程的锁定分析与无锁定高阶有限元方法
Locking analysis and high-order locking-free finite element method for three-dimensional electroporoelasticity equations
AI总结：
本文针对三维电孔隙弹性方程的Poisson锁定问题，引入五场格式与全离散高阶有限元方法，证明其对大Lamé常数的一致稳定性，建立无锁定格式并经数值实验验证了收敛阶与鲁棒性。
AI中文摘要：
电孔隙弹性方程将Maxwell方程与Biot孔隙弹性模型耦合，当Lamé常数较大时，标准协调有限元离散会出现严重的Poisson锁定现象。本文对三维准静态电孔隙弹性方程的Poisson锁定现象进行严格分析，表明在近不可压缩 regime 中，协调有限元近似的空间收敛阶会降低。为消除该锁定效应，引入五场格式和全离散高阶有限元方法，证明了其对大Lamé常数的一致稳定性，在此基础上通过推导关于Lamé系数的一致误差估计建立了无锁定格式。该分析涵盖完全耦合的电磁-孔隙弹性系统，适用于三维高阶单元，大量数值实验验证了理论收敛阶并展示了其对Lamé参数的鲁棒性。
英文摘要：
Electroporoelasticity equations couple Maxwell's equations with Biot's poroelasticity model and admit severe Poisson locking in standard conforming finite element discretizations when the Lamé constant is large. In this paper, we provide a rigorous analysis of the Poisson locking phenomenon for three-dimensional quasi-static electroporoelasticity equations and show that the spatial convergence order of conforming finite element approximations is reduced in the nearly incompressible regime. To eliminate this locking effect, we introduce a five-field formulation and a fully discrete high-order finite element method. We prove uniform stability with respect to large Lamé constant. Based on this, we establish a locking-free scheme by deriving its uniform error estimates with respect to the Lamé coefficient. The analysis covers the fully coupled electromagnetic-poroelastic system and applies to high-order elements in three dimensions. Extensive numerical experiments are presented to verify the theoretical convergence rates and demonstrate robustness with respect to the Lamé parameter.
