AN ASYMPTOTIC FORMULA FOR CORRECTING FINITE-ELEMENT PREDICTED NATURALFREQUENCIES OF MEMBRANE VIBRATION PROBLEMS

In the paper an asymptotic formula has been developed to correct the d iscretization error for the finite element predicted natural frequenci es of membrane transverse vibration problems. The general idea behind deriving this asymptotic formula is that, when the finite element size approaches zero, a discretized finite element system approaches a con tinuous system and the predicted natural frequencies of the system fro m the finite element analysis therefore approach the exact solutions o f the system. Without losing generality, several different finite elem ent mesh patterns have been considered and the same asymptotic formula for correcting the finite element predicted natural frequency has bee n obtained for all the different mesh patterns because of the uniquene ss of the exact solution to the natural frequency of a real structure. The usefulness, effectiveness and efficiency of the present asymptoti c formula have been assessed by a simple but critical problem, for whi ch the exact solution is available for comparison. In order to investi gate the applicability of the asymptotic formula to practical engineer ing problems, two challenging membrane vibration problems of irregular shapes, an L-shape and a tapered shape with a circular hole in the ce ntre, have also been analysed. The related numerical results have demo nstrated that the asymptotic formula provides a very useful post-proce ssing error corrector for the finite element predicted natural frequen cies of membrane transverse vibration problems, even though the proble m domains are of irregular shape. The greatest advantage in using the present asymptotic formula is that it yields a solution of higher accu racy, by simply using the formula to correct the rough solution obtain ed from a much coarser finite element mesh with fewer degrees of freed om, without any further finite element calculation.