应变梯度弹性的一种杂交交错间断伽辽金-混合有限元方法
A Hybridized Staggered Discontinuous Galerkin--Mixed Finite Element Method for Strain Gradient Elasticity
- The Chinese University of Hong Kong（香港中文大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
提出一种杂交交错间断伽辽金-混合有限元方法求解应变梯度弹性模型，通过引入辅助应力变量并强加对称性，实现关于材料长度尺度和拉梅常数一致的最优收敛，数值实验验证了方法的有效性。
中文摘要 AI 辅助
我们针对应变梯度弹性模型提出了一种杂交交错间断伽辽金-混合有限元方法。该模型是一个由材料长度尺度参数$\iota$以及拉梅常数$\lambda$和$\mu$控制的四阶奇异摄动问题。通过引入总应力以及缩放的柯西应力和超应力作为辅助未知量，我们将该模型改写为一阶系统，并使用交错间断伽辽金空间对位移和总应力进行离散，使用Raviart-Thomas对两个缩放应力进行离散，同时对三个应力强加对称性。高阶狄利克雷边界条件自然地进入变分公式，因此该方案避免了限制基于位移方法的数值边界层。通过在对称子空间上建立inf-sup条件和离散Korn不等式，我们证明了代数稳定性和最优收敛性，且两者均关于$\iota$和$\lambda$一致。我们进一步开发了一种杂交方案，其中局部静态凝聚仅留下网格骨架上的两个乘子作为全局耦合未知量。数值实验证实了预测的收敛速率和参数鲁棒性。
英文摘要
We propose a hybridized staggered discontinuous Galerkin--mixed finite element method for the strain gradient elasticity model, a fourth-order singularly perturbed problem governed by a material length scale parameter $ι$ and the Lamé constants $λ$ and $μ$. Introducing the total stress together with the scaled Cauchy and hyper stresses as auxiliary unknowns, we recast the model as a first-order system and discretize the displacement and the total stress by staggered discontinuous Galerkin spaces, the two scaled stresses by Raviart--Thomas pairs, with symmetry imposed strongly on all three stresses. The higher-order Dirichlet boundary condition enters the variational formulation naturally, so that the scheme avoids the numerical boundary layer that limits displacement-based methods. By establishing an inf-sup condition on the symmetric subspace and a discrete Korn inequality, we prove algebraic stability and optimal convergence, both uniform in $ι$ and $λ$. We further develop a hybridized scheme in which local static condensation leaves only two multipliers on the mesh skeleton as globally coupled unknowns. Numerical experiments confirm the predicted convergence rates and the parameter robustness.
