Using Different FEM Formulations in Calculations of Thin-Walled Structures

A comparative analysis of the accuracy of finite element solutions of a thin-walled structure in the form of an ellipsoidal shell using displacement-based and mixed FEM is presented. The discretization element of the thin-walled structure is a four-node fragment of the middle surface with displacement components and their first-order partial derivatives with respect to curvilinear coordinates as the nodal unknowns. When implementing the mixed FEM formulation, strains and curvatures of the middle surface of the thin-walled structure are chosen as the force-type nodal unknowns. The stiffness matrix of the discretization element of dimension 36×36 according to the displacement method was obtained by minimizing the Lagrange functional. The finite element stiffness matrix in the mixed formulation was compiled by minimizing the mixed functional with respect to the kinematic and force nodal unknowns. The use of the substitution method when solving the system of matrix equations of the mixed FEM made it possible to maintain the optimal dimension of the stiffness matrix of the discretization element 36×36, the same as in the case of the displacement-based FEM. Test examples of calculations of a cylindrical shell with circular and elliptical cross sections show that the proposed version of the mixed FEM has significant advantages in terms of the accuracy of finite element solutions compared to the displacement-based FEM. Moreover, these advantages improve as the curvature of the surface of the analyzed shell structure increases.