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This article deals with improvement of eigenvalues obtained by finite element analysis of C¹ eigenproblems. The proposed method employs high order gradient smoothing at nodal points to derive improved high order interpolation functions for the single element of each mode. Two different schemes were developed for 1–D C¹ eigenproblems (free vibration of beams) and for 2–D quasi C¹ eigenproblems (transverse vibrations of thin plates). High order Hermitian polynomials are used for the beam problem together with some boundary node corrections, while a combination of high‐order and low‐order approximations are used for the modified formulation of the plate problem. Several smoothing options are proposed for both schemes. Numerical results for both schemes are used as examples to demonstrate the accuracy of the present approach.

This paper presents a new approach for estimating the discretization error of finite element analysis of generalized eigenproblems. The method uses smoothed gradients at nodal points to derive improved element‐by‐element interpolation functions. The improved interpolation functions and their gradients are used in the Rayleigh quotient to obtain an improved eigenvalue. The improved eigenvalue is used to estimate the error of the original solution. The proposed method does not require any re‐solution of the eigenproblem. Results for 1‐D and 2‐D C° eigenproblems in acoustics and elastic vibrations are used as examples to demonstrate the accuracy of the proposed method.

The 2D frequency domain boundary integral equation is solved by the boundary spectral strip method. Using an expansion for frequency domain elastodynamics kernel we reduce its singularity and present analytical solutions for the required integrals in the singular case when the integration path is a straight line. The method is illustrated by two different problems, both over a range of excitation frequencies. The first problem is a rectangular bar under a longitudinal excitation, which has an analytical solution. The other problem is a trapezoidal dam loaded by a transverse excitation at its base. The solution for the second problem is compared with a finite elements model. The results obtained from these tests show a good agreement between the results of the boundary strip method and analytical or finite elements results.

Gives a bibliographical review of the error estimates and adaptive finite element methods from the theoretical as well as the application point of view. The bibliography at the end contains 2,177 references to papers, conference proceedings and theses/dissertations dealing with the subjects that were published in 1990‐2000.