Improved boundary elements in torsion problems

Since the epoch-making "memoir" of Saint-Venant in 1855 the torsion of prismatic and cilindrical bars has reduced to a mathematical problem: the calculation of an analytical function satisfying prescribed boundary values. For over one century, till the first applications of the F.E.M. to the problem, the only possibility of study in irregularly shaped domains was the beatiful, but limitated, theory of complex function analysis, several functional approaches and the finite difference method. Nevertheless in 1963 Jaswon published an interestingpaper which was nearly lost between the splendid F. E.M. boom. The method was extended by Rizzo to more complicated problems and definitively incorporated to the scientific community background through several lecture-notes of Cruse recently published, but widely circulated during past years. The work of several researches has shown the tremendous possibilities of the method which is today a recognized alternative to the well established F .E. procedure. In fact, the first comprehensive attempt to cover the method, has been recently published in textbook form. This paper is a contribution to the implementation of a difficulty which arises if the isoparametric elements concept is applicated to plane potential problems with sharp corners in the boundary domain. In previous works, these problems was avoided using two principal approximations: equating the fluxes round the corner or establishing a binode element (in fact, truncating the corner). The first approximation distortes heavily the solution in thecorner neighbourhood, and a great amount of element is neccesary to reduce its influence. The second is better suited but the price payed is increasing the size of the system of equations to be solved. In this paper an alternative formulation, consistent with the shape function chosen in the isoparametric representation, is presented. For ease of comprehension the formulation has been limited to the linear element. Nevertheless its extension to more refined elements is straight forward. Also a direct procedure for the assembling of the equations is presented in an attempt to reduce the in-core computer requirements.