We propose two parameter-robust mixed finite element methods for linear Cosserat elasticity. The Cosserat coupling constant $\mu_c$, connecting the displacement $u$ and rotation vector $\omega$, leads to possible locking phenomena in finite element methods. The formal limit of $\mu_c\to\infty$ enforces the constraint $\frac{1}{2}\operatorname{curl}u = \omega$ and leads to the fourth order couple stress problem. Viewing the linear Cosserat model as the Hodge-Laplacian problem of a twisted de Rham complex, we derive structure-preserving distributional finite element spaces, where the limit constraint is fulfilled in the discrete setting. Applying the mass conserving mixed stress (MCS) method for the rotations, the resulting scheme is robust in $\mu_c$. Combining it with the tangential-displacement normal-normal-stress (TDNNS) method for the displacement part we obtain additional robustness in the nearly incompressible regime and for anisotropic structures. Using a post-processing scheme for the rotations, we prove optimal convergence rates independent of the Cosserat coupling constant $\mu_c$. Further, we propose a mixed method for the couple stress problem based on the MCS scheme. We demonstrate the performance of the proposed methods in several numerical benchmark examples.

翻译：本文针对线性Cosserat弹性问题提出了两种参数鲁棒的混合有限元方法。连接位移$u$与旋转矢量$\omega$的Cosserat耦合常数$\mu_c$可能导致有限元方法中的闭锁现象。当$\mu_c\to\infty$时，形式极限强制约束条件$\frac{1}{2}\operatorname{curl}u = \omega$，并导出四阶偶应力问题。通过将线性Cosserat模型视为扭曲de Rham复形的Hodge-Laplacian问题，我们推导了保持结构特性的分布有限元空间，其中极限约束在离散设置中得以满足。对旋转部分采用质量守恒混合应力（MCS）方法，所得格式对$\mu_c$具有鲁棒性。将其与位移部分的切向位移-法向法向应力（TDNNS）方法结合，我们进一步获得了在近不可压缩区域和各向异性结构中的附加鲁棒性。通过采用旋转量的后处理方案，我们证明了与Cosserat耦合常数$\mu_c$无关的最优收敛速率。此外，基于MCS格式，我们提出了一种针对偶应力问题的混合方法。通过多个数值基准算例，我们验证了所提方法的性能。