Abstract
Based on the asymptotic solution for finite element predicted natural frequencies of a membrane vibration problem, the concept of asymptotic error and a practical error estimator are presented in this paper. The present practical error estimator contains two criteria: one is the error estimator criterion and the other is the finite element mesh design criterion. By using this practical error estimator, not only can the accuracy of a finite element solution for natural frequencies of a membrane vibration problem be directly evaluated without further finite element calculation, but also a new target finite element mesh for the solution of desired accuracy can be immediately designed from the relevant information of an original finite solution. Generally, for the purpose of designing a new target finite element mesh, this original finite element solution is obtainable from a very coarse mesh of a few elements and usually does not satisfy the accuracy requirement. Since the new target finite element mesh could result in a finite element solution of desired accuracy, the finite element solutions so obtained can be used for structural design in engineering practice. The related numerical results from vibration problems of five representative membranes of different shapes have demonstrated the correctness and applicability of the present practical error estimator.
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