Higher-order, partial hybrid stress, finite element formulation for laminated plate and shell analyses by using preconditioned iterative procedures

A finite element formulation for arbitrarily curved orthotropic composite plate and shell analyses is presented by using a higher order, partial hybrid stress method. The governing equation of the laminated plate is variationally derived from the Hellinger-Reissner principle, and the flexural stress components are separated from the transverse shear stress components so that the continuity of interlaminar stress is enforced in the transverse shear stresses only. A generalized transverse shear traction continuities are set up to be valid for any number of layers and any thickness ratios of the layers. A general formulation is developed by using a shell geometry to suitably transform the plate equations into the shell equations. The partial hybrid stress method satisfies interface traction continuity conditions exactly in the transverse shear stress, and avoids the complexity of formulation that the normal hybrid stress method has.; Efficient numerical algorithms for solving large size eigenvalue problems are developed in the next for the materials with or without piezoelectric effect. Multi-mesh, preconditioned iterative methods are proposed in this study. The generalized eigenvalue problems are solved by iterative methods with a preconditioner which is a partially factorized stiffness matrix. Initial trial eigenvectors for the iterative methods are obtained by interpolation using the eigenvectors obtained from a coarser mesh. The employment of these trial eigenvectors is found to significantly increase the rate of convergences of the methods, and also to prevent slow convergence/convergence failure in problems with closely spaced eigenvalues and repeated eigenvalues. A conjugate gradient iterative algorithm is used for the mechanical eigenvalue problems, while Rayleigh quotient interation scheme is adapted in the piezoelectric eigenvalue problems. In order for these iterative methods to be effective, an eigenvector of interest in the fine mesh must resemble an eigenvector in the coarse mesh. Hence, the methods are effective for finding the set of eigenpairs in the low frequency range. And also, an algorithm involving the use of Lanczos eigensolver is presented, in which the piezoelectric eigenvalue problem is solved by maintaining skyline storage and the structural pattern of the consistent mass matrix is exploited to gain savings in both memory and solution time.