A note on the finite element method with singular basis functions

Abstract

SUMMARYIn this note, we make a few comments concerning the paper of Hughes and Akin (Int. J. Numer. Meth. Engng., 15, 733-751 (1980)). Our primary goal is to demonstrate that the rate of convergence of numerical solutions of the ÿnite element method with singular basis functions depends upon the location of additional collocation points associated with the singular elements. Copyright ? 1999 John Wiley & Sons, Ltd.KEY WORDS: ÿnite element method with singular basis functions; the collocation method; singular interpolation elements FINITE ELEMENT METHOD WITH SINGULAR BASIS FUNCTIONSIn the paper [1], Hughes and Akin made an interesting observation concerning the ÿnite element analysis that incorporates singular element functions. A need for introducing some singular elements as part of basis functions in certain ÿnite element analysis arises out of the following considerations. The solution of certain problems, such as a ÿeld problem [2], exhibits highly singular behaviour due to geometric features of the spatial domain. On the other hand, in other circumstances, the solution is overwhelmingly a ected by the nature of loading and the problem of singularity can be ignored. To satisfy both situations just described, it is thought that an incorporation of singular elements that emulate the solution with the standard polynomial elements may perhaps be desirable. This is the point that was exploited by Hughes and Akin [1]. In order to make the computations of the ÿnite element method with singular elements more e cient, they consider the following algorithm for constructing interpolation functions. A construction of such algorithm was motivated by the idea that 'it is of practical interest to develop techniques for systematically deÿning shape functions for singularity modeling (and for developing special elements in general), which circumvent the interpolation problem ' [3; p:176]. The algorithm that they developed is as follows:

Variable Singularity Power Wedge Element for Multilayered Composites

“…As shown by Kaneko and Padilla [46], elements based on the shape functions defined above could have a convergence rate depending on the location of the interpolation points. Namely, locations can exist in which they could not work.…”