ASYMPTOTIC SOLUTIONS FOR PREDICTED NATURAL FREQUENCIES OF TWO‐DIMENSIONAL ELASTIC SOLID VIBRATION PROBLEMS IN FINITE ELEMENT ANALYSIS

In order to assess the discretization error of a finite element solution, asymptotic solutions for predicted natural frequencies of two-dimensional elastic solid vibration problems in the finite element analysis are presented in this paper. Since the asymptotic solution is more accurate than the original finite element solution, it can be viewed as an alternative solution against which the original finite element solution can be compared. Consequently, the discretization error of the finite element solution can be evaluated. Due to the existence of two kinds of two-dimensional problems in engineering practice, both the plane stress problem and the plane strain problem have been considered and the corresponding asymptotic formulae for predicted natural frequencies of two-dimensional solids by the finite element method have been derived from the fact that a discretized finite element system approaches a continuous one if the finite element size approaches zero. It has been demonstrated, from the related numerical results of three examples, that the present asymptotic solution, which can be obtained by simply using the corresponding formula without any further finite element calculation, is indeed more accurate than the original finite element solution so that it can be considered as a kind of corrected solution for the discretization error estimation of a finite element solution.