Partial Fourier approximation of the Lamé equations in axisymmetric domains

Abstract

In this paper, we study the partial Fourier method for treating the Lame equations in three-dimensional axisymmetric domains subjected to non-axisymmetric loads. We consider the mixed boundary value problem of the linear theory of elasticity with the displacement u, the body force f ϵ (L2)3 and homogeneous Dirichlet and Neumann boundary conditions. The partial Fourier decomposition reduces, without any error, the three-dimensional boundary value problem to an infinite sequence of two-dimensional boundary value problems, whose solutions un (n = 0, 1, 2,…) are the Fourier coefficients of u. This process of dimension reduction is described, and appropriate function spaces are given to characterize the reduced problems in two dimensions. The trace properties of these spaces on the rotational axis and some properties of the Fourier coefficients un are proved, which are important for further numerical treatment, e.g. by the finite-element method. Moreover, generalized completeness relations are described for the variational equation, the stresses and the strains. The properties of the resulting system of two-dimensional problems are characterized. Particularly, a priori estimates of the Fourier coefficients un and of the error of the partial Fourier approximation are given. Copyright © 1999 John Wiley & Sons, Ltd.