New Error Estimation for C° Eigenproblems in Finite Element Analysis

Abstract

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.

Accuracy Estimates in Free Vibration Analysis

“…Assuming a harmonic behaviour u(t) = ueiax (6) equation (5) leads to (7) where [Kj is the stiffness matrix and [MJ is the consistent mass matrix of the structure. Equation (7) is of the form of a generalised eigenvalue problem, with eigenvalues l 1 } being equal to the squares of the eigenfrequencies w!…”

“…Avrashi and Cook [7] present an approach for the error estimation for C 0 eigenproblems by smoothing gradient (stress) and primary variable (displacement) fields. The improved eigenfrequency for the error estimate is obtained from the Rayleigh quotient with the modified field of the primary variables and its gradients by use of some users defined parameters.…”

Improvement of C0 vibration problems using Helmholtz equation to recover nodal gradients

“…• 1-D quadratic fit: the interpolation function for a two-node isoparametric element is listed in (Avrashi and Cook, 1993b) and shown in the Appendix.…”

Accuracy Estimates in Free Vibration Analysis/Paklaidų Nustatymas Laisvųjų Svyravimų Uždaviniuose

“…equation (5) leads to (7) where [Kj is the stiffness matrix and [MJ is the consistent mass matrix of the structure. Equation (7) is of the form of a generalised eigenvalue problem, with eigenvalues l 1 } being equal to the squares of the eigenfrequencies w!…”

“…which approximate the relative change in an eigenfrequency. Avrashi and Cook [7] present an approach for the error estimation for C 0 eigenproblems by smoothing gradient (stress) and primary variable (displacement) fields. The improved eigenfrequency for the error estimate is obtained from the Rayleigh quotient with the modified field of the primary variables and its gradients by use of some users defined parameters.…”