A linear finite element procedure for the Naghdi shell model
For computational mechanics researchers, this method provides a robust and accurate approach to simulating thin shells, addressing the long-standing issue of locking in finite element methods.
The paper presents a mixed finite element method for the Naghdi shell model that reduces membrane/shear locking, achieving optimal accuracy for general shell geometries and robustness with respect to shell thickness. The method uses piecewise linear functions and a parameter to handle different shell behaviors, with uniform accuracy when geometrical coefficients are piecewise constants.
We prove the accuracy of a mixed finite element method for bending dominated shells in which a major part of the membrane/shear strain is reduced, to free up membrane/shear locking. When no part of the membrane/shear strain is reduced, the method becomes a consistent discontinuous Galerkin method that is proven accurate for membrane/shear dominated shells and intermediate shells. The two methods can be coded in a single program by using a parameter. We propose a procedure of numerically detecting the asymptotic behavior of a shell, choosing the parameter value in the method, and producing accurate approximation for a given shell problem. The method uses piecewise linear functions to approximate all the variables. The analysis is carried out for shells whose middle surfaces have the most general geometries, which shows that the method has the optimal order of accuracy for general shells and the accuracy is robust with respect to the shell thickness. In the particular case that the geometrical coefficients of the shell middle surface are piecewise constants the accuracy is uniform with respect to the shell thickness.