Locking-free Formulations of Beams, Plates and Shells Through Stabilization
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Numerical locking is a critical challenge in thin structural elements due to parasitic stiffening as structural thickness diminishes, resulting in physically incorrect results. This work presents a numerical framework designed to overcome numerical locking of thin-walled structures through stabilization based on the Variational Multiscale (VMS) framework, specifically addressing Timoshenko beams and Reissner-Mindlin plates, and solid-shell elements For beams and plates, the stabilization explicitly addresses the unresolved numerical deficiencies responsible for locking \cite{aguirre2023VariationalMultiscaleb}. In contrast, the stabilization of solid-shell elements utilizes a mixed displacement-stress formulation that effectively suppresses all locking modes, including shear, membrane, trapezoidal, and thickness locking by controlling the specific parasitic stress components that trigger them \cite{aguirre2024StressDisplacement}. Furthermore, the framework is successfully extended to finite strain hyperelasticity and dynamics, proving its generality. The stabilized formulation not only eliminates locking but also significantly enhances the accuracy of the computed stress field, a critical outcome for reliable failure assessment in thin structures, and is stable and optimally convergent independently of the structural thickness. Notably, the Orthogonal Sub-Grid Scale (OSGS) approach guarantees optimal convergence in the $L^2$ norm without sensitivity to stabilization parameters \cite{aguirre2024StressDisplacementa}. A key advantage is the formulation's flexibility, as it circumvents the inf-sup conditions required by classical mixed methods. This allows for the use of arbitrary orders of interpolation of the unknowns: displacements/rotations in beams and plates, and displacements/stresses in solid-shells, simplifying their implementation. Moreover, the method retains its accuracy in structured and unstructured meshes of quadrilateral and simplex elements in 2D and 3D. Research demonstrates that VMS-based stabilization provides a robust and effective framework, providing reliable locking-free simulations of thin-walled structures across linear and nonlinear regimes in static and dynamic analyses.
