The two most widely used error estimators for adaptive mesh re®nement are discussed and developed in the context of non-linear elliptic problems. The ®rst is based on the work of Babuska and Rheinboldt (1978) where the error norm is a function of the residual and the inter-element discontinuity of the stress ®eld. The discontinuous stress ®eld arises in the Finite Element formulation where C 0 continuity of the velocity ®eld is assumed.The second error estimator is based on the work of Zienkiewicz and Zhu (1987). This method also uses the discontinuous stress ®eld to measure the error, but results in a more simpli®ed expression for the error norm. In fact, the equivalence between the two error norms has been shown by Zienkiewicz.Finally, an error estimator which is based on the approximation velocity space only is proposed. This estimator has the advantage in that it does not require the a posteriori calculation of the pressure (or stress) ®eld. The method is applied to non-Newtonian Stokes¯ow which has a similar formulation to non-linear elasticity problems.
Problem. The study investigates the free vibration of a composite beam controlled by piezo patch sensors and actuators. The model simulating feedback control of a piezoelastic beam incorporates the electrostatic equation which is centred on the divergence free electric displacement. The basic formulation is given in terms of a differential equation which facilitates the computation of the eigenfrequencies and eigenfunctions of the beam.Method. The pseudospectral method is used to transform the differential eigenvalue problem to an approximate algebraic system of equations using a truncated series of basis functions and the Chebyshev collocation points.Results. The results show that the error is the smallest for the smallest eigenvalue and then increases rapidly until the matrix eigenvalues become useless as approximations to those of the differential equation. N/2 eigenvalues were obtained as a good approximation of the differential equation.
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