Application of hill functions to circular plate problems
A type of finite element—hill functions—is applied to solve circular plate problems in conjunction with the method of Lagrange multipliers which is used to treat various constraint conditions. Results obtained compare very nicely wTith the exact solutions. Introduction. Finite-element methods have been widely applied to obtain numerical solutions in both engineering [1, 2] and applied mathematics [3, 4, 5], The popularity of the methods may be partially due to their ability to deal with more complex problems and to provide accurate numerical results. However, it should be noted that there are different kinds of finite-element methods and the use of hill functions as finite elements which are developed in applied mathematics [3, 4, 5, 6] is one of them. In [6], hill functions are utilized to solve one-dimensional string-beam problems in which the so-called "method of artificial parameters" is used for handling various boundary conditions. In the present paper, these same functions are employed to solve axisymmetric (polar coordinate) circular plate problems; the Lagrange multiplier method is used here for the treatment of constraint situations. The striking difference between these two types of applications is in the evaluation of matrix elements of the system equations; for string-beam problems, all the terms in these elements turn out to be simply multiples of hill function coefficients [6], but for circular plate problems, they are not so straight-forward and require, for evaluation, methods of numerical integration. Rayleigh Ritz method. For the sake of completeness and use in the following sections, key equations involved in the Rayleigh-Ritz method are presented in this section. Suppose there exists a sequence 2/1,2/2, • • ■ of admissible functions in the variational problem such that lim F(yn) = d, (1) n—>00 where d is the lower bound of the functional F(y)-. The Rayleigh-Ritz method is a recipe for the construction of such a sequence by choosing an arbitrary system of coordinate functions, «i , u2 , ■ ■ ■ , with the property that any linear combination yn = Cjo>1 + c2w2 + • • • + c„co„ (2) is admissible in the variational problem, and that the solution function y and its relevant * Received October 26, 1973. The research reported here was supported by the Office of Naval Research (N00014-67-A-0377-0011).