一种用于应变梯度弹性的具有三阶张量变量的最低阶稳健混合有限元方法
A Lowest-Order Robust Mixed Finite Element Method with a Third-Order Tensor Variable for Strain Gradient Elasticity
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中文总结 AI 辅助
针对应变梯度弹性模型，开发最低阶混合有限元方法，以三阶双应力张量为主要变量，推导公式并近似相关量，建立多种误差估计，开发后处理和杂交公式，数值实验验证了理论结果。
中文摘要 AI 辅助
本文针对任意维度的应变梯度弹性（SGE）模型开发了一种最低阶混合有限元方法。以具有物理意义的三阶双应力张量$\boldsymbol{\Phi}:=\iota^2\operatorname{grad}\boldsymbol{\sigma}(\boldsymbol{u})\in\mathbb{S}\otimes\mathbb{R}^d$作为主要变量，推导了分布混合公式。双应力由最低阶Raviart-Thomas元的$\mathbb{S}\otimes\mathbb{R}^d$值扩展近似，位移由向量值线性Crouzeix-Raviart元近似。建立了参数稳健离散稳定性、固定参数下的最优一阶误差估计以及与大小参数$\iota$和拉梅系数$\lambda$无关的互补参数均匀$\mathcal{O}(\iota^{1/2}+h)$误差估计。还开发了局部二次后处理和杂交公式。二维和三维数值实验支持了理论结果。
英文摘要
A lowest-order mixed finite element method is developed for the strain gradient elasticity (SGE) model in arbitrary dimensions. We take the physically meaningful third-order double stress tensor $\boldsymbolΦ:=ι^2\operatorname{grad}\boldsymbolσ(\boldsymbol{u})\in\mathbb{S}\otimes\mathbb{R}^d$ as a primary variable and derive a distributional mixed formulation. The double stress is approximated by an $\mathbb{S}\otimes\mathbb{R}^d$-valued extension of the lowest-order Raviart--Thomas element, while the displacement is approximated by the vector-valued linear Crouzeix--Raviart element. Thus, the method avoids both high-degree bubble enrichment and a Nitsche-type treatment of the higher-order boundary condition. We establish parameter-robust discrete stability, an optimal first-order error estimate for fixed parameters, and a complementary parameter-uniform $\mathcal{O}(ι^{1/2}+h)$ error estimate with constants independent of both the size parameter $ι$ and the Lamé coefficient $λ$. In the boundary-layer regime $ι^{1/2}\lesssim h$, the latter retains a first-order convergence rate in $h$. We also develop a local quadratic post-processing and a hybridized formulation. Numerical experiments in two and three dimensions support the theoretical results.
