Error analysis of finite element results on plates with nonuniform grids

Abstract

D ISCRETE Dirac delta functions in two dimensions and the governing plate differential equations are used to produce continuous approximations from discrete finite-element data on nonuniform grids. The continuous approximation can be differentiated to compute continuous stresses/moments at any point in the plate and, when substituted in the differential equations, provides a The paper presents application examples of the procedure and studies the use of the solution in a Zienkiewicz-Zhu error estimator to assess the performance of the present analysis. In an example involving the linear response of a clamped square plate under uniform transverse load, the present procedure dramatically corrected and smoothed the discrete finite element results. Contents The finite element method is a powerful tool that provides approximate solutions to engineering problems. Hence a reliable estimate of the accuracy of stresses and deflections computed by the method is necessary for efficient and dependable designs. An earlier paper1 outlined a general approach to the error analysis and correction of results from the finite element method (FEM). The approach suggested using Newton's method for solving the Euler equations of shell theory for components of shell structures. Continuity of stresses and deformations between shell components are enforced by using substructuring. References 2 and 3 addressed and corrected an oscillatory behavior of the continuous solution between boundary nodes of rectangular plate sections (caused by a Gibbs' phenomenon in the Fourier sine series used in the numerical analysis) and applied the general approach of Ref. 1 to postbuckled, stiffened, rectangular, composite plates with initial imperfections. The continuous smoothed solution obtained from the error analysis1'3 when substituted back into the governing differential equations yields a residual error. This error provides a measure of how well the smoothed solution, and by extension, the finite element solution, satisfies shell theory. Earlier work1'3 was limited to the error analysis of FEM results from uniform rectangular grids. In general, the discretization of structural members for analysis with the finite element method can involve nonuniform grids and is often the case when adaptive refinement is used in the finite element analysis. The present paper extends the approach of earlier work on error analysis of finite element results for postbuckled plates to the general case of nonuniform grids. As in earlier work, the continuous solution for the transverse deflection W consists of two series solutions JT/and WE, W=W!+WE (la)