Predicted natural frequencies of two‐dimensional elastodynamic problems: a practical error estimator

Based on the asymptotic solution for predicted natural frequencies of a two‐dimensional elastodynamic problem from the finite element analysis, presents the concept of the asymptotic error, which is an approximate error but tends to the exact error when the characteristic length of elements approaches zero, and a practical error estimator. The present practical error estimator contains two criteria: one is the error estimator criterion, the other the finite element mesh design criterion. Using this practical error estimator, not only can the accuracy of a finite element solution for natural frequencies of a two‐dimensional elastodynamic problem be directly evaluated without any further finite element calculation, but also a new target finite element mesh for the desired accuracy of solution can be immediately designed from the relevant information of an original finite element 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 with a desire accuracy, the finite element solution so obtained can be used for a structural design in engineering practice. The related numerical results from vibration problems of three representative plates of different shapes under plane stress conditions have demonstrated the correctness and applicability of the present practical error estimator.