2024/10/18 by Andrea Dziubek, Dziubek, Andrea, Kaibo Hu +5
Computer Science · Engineering · #65N30 (Primary) 74S05 (Secondary) #Composite Structure Analysis and Optimization #Contact Mechanics and Variational Inequalities #FOS: Mathematics #Numerical Analysis (math.NA) #Vibration and Dynamic Analysis
paper · pdf · doi:10.48550/arxiv.2410.14176
openalex publication_date 2024/10/18 · openalex created_date 2024/11/02 · openalex updated_date 2026/07/28
We propose two parameter-robust mixed finite element methods for linear Cosserat elasticity. The Cosserat coupling constant μc, connecting the displacement u and rotation vector ω, leads to possible locking phenomena in finite element methods. The formal limit of μc→∞ enforces the constraint (1)/(2)curl u = ω 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 μc. 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 μc. We demonstrate the performance of the proposed methods in several numerical benchmark examples.