C0 and C1 theories and test functions for FEM patch test in microstructures

Among many theories and categories in microstructures, rotation-displacement used as "independent" or "dependent" variables, is a noticeable topic. In FEM, it is called C-0 and C-1 theory. The convergence criteria of finite elements for microstructures are less mature than those for the conventional thin plate bending problem. In this paper, the patch test functions for assessing convergence of the C-0 and C-1 finite elements in microstructures is established based on the enhanced patch test theory. The author has further explored the C-0 and C-1 finite element theories and investigated the difference and correlation between their finite element formulations. Newly proposed finite element theories for microstructures are as follows: (1) the displacement-rotation dependent C-1 element that requires the element function satisfying both C-0 and C-1 continuity; (2) the displacement-rotation independent C-0 element which requires new convergence criteria, such as non-zero constant shear stress patch test and zero constant shear stress patch test for approximating C-1 element.