三维准静态电孔隙弹性方程的无锁虚拟元方法

中文摘要 AI 辅助

我们开发并分析了一种用于三维准静态电孔隙弹性方程的无锁虚拟元方法。压力、电场、磁场和位移场分别由最低阶节点、边、面和Stokes-like虚拟元空间离散，使得该方案适用于一般多面体网格，而时间离散采用后向欧拉格式。通过在细化网格$\widetilde{\mathcal{T}}_h$上定义Stokes-like空间来克服锁死现象，在该网格上我们可以构造一个插值算子$\mathrm{I}_s$，满足$÷(\mathrm{I}_s\bm v)|_K=\Pi^K_0÷\bm v$，这使得误差估计不依赖于Lamé常数$\lambda$。此外，由于精确序列$\curl\bm V^e_h=\bm V^f_{h,0}$成立，磁场被显式更新，将四场系统简化为三场系统，降低了计算成本。我们提供了数值实验来展示所提出方法的性能。

英文摘要

We develop and analyze a locking-free virtual element method for the three-dimensional quasi-static electroporoelasticity equations. The pressure, electric, magnetic, and displacement fields are discretized by the lowest-order nodal, edge, face, and Stokes-like virtual element spaces, respectively, so that the scheme works on general polyhedral meshes, while the temporal discretization is carried out by a backward Euler scheme. The locking phenomenon is overcome by defining the Stokes-like space on a refined mesh $\widetilde{\mathcal{T}}_h$, on which we can construct an interpolation operator $\mathrm{I}_s$ satisfying $÷(\mathrm{I}_s\bm v)|_K=Π^K_0÷\bm v$, which renders the error estimates independent of the Lamé constant $λ$. Moreover, since the exact sequence $\curl\bm V^e_h=\bm V^f_{h,0}$ holds, the magnetic field is updated explicitly, reducing the four-field system to a three-field one and lowering the computational cost. Numerical experiments are provided to demonstrate the performance of the proposed method.

发表机构

• Shanghai Jiao Tong University（上海交通大学）

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