Grant Keady's Project on Exact Solutions of
Elastic Torsion Problem and CAS implementations

Recent effort includes work, with Steven Richardson, for ACMSM18 Conference at UWA, end Nov 2004.

Introduction

The elastic torsion function, u, for a plane domain D is a function whose laplacian is -1 and which vanishes on the boundary of D. A particular functional of interest, the torsional rigidity S, is the integral of u over D.

A "Structural Mechanics Pack", written in Mathematica, was released in Sept 1999. The Pack contains a list of exact solutions for the elastic torsion problem implemented in Mathematica. The original author, from 95-97, was Cetin Cetinkaya.

The Pack provides a framework into which further exact solutions can be added, and into which further improvements might be made. E.g. an omission from the 1999 version of the Pack is the exact solution for the torsional rigidity of a sector.

I have a few publications concerning exact solutions for torsion problem.

• Here is the html version of a 7 page account, in a Mathematica 3 notebook which appeared in the Proceedings of the Australian Engineering Mathematics Conference, July 96, Sydney. (Contains original work on symmetric centre-on-circumference lens shapes). The Mathematica 3 Notebook version is hyperlinked at my list of papers.
• I have original work on the hypergeometric/LerchPhi representations of the torsion function for a sector. This is published, with other items by Cetin Cetinkaya, in the Electronic Proceedings of the Asian Technology Conference in Mathematics, Penang, June 97. Here is this paper in pdf.
• Papers such as that with Sever Dragomir in Applicable Analysis (2000) concern bounds, not exact solutions.

What I would like to do (as a hobby!), over a period of years, is to build a list of exact solutions which is at least a little bit more complete than, say, that in the Structural Mechanics Pack. (I suspect that nothing much will come of this though, as I have other commitments and interests and very little time to devote to this 'exact solution' hobby.) There seems little point in me confining myself to just the one Computer Algebra System: maple and matlab (the latter with its symbolic toolbox) also have engineering users.
It may be that a more modest start, within the context of some wider collection of information, is more appropriate than a purely personal effort. A modest start, in time (decades?), might lead on to my original hope of a large collection of exact solutions. Fitting in with these wider collections might provide some method of organizing the torsion related information.
Perhaps ScienceWorld?
wikipedia?
Perhaps one could aim to get the coverage of (the simply-connected) domains in the survey paper of Higgins (1942), Am. Jnl of Physics. Click here for references.

Click for a list of shapes for which I have in

• Mathematica
• Maple
• Matlab

Here are remarks on the problem in general domains, where exact closed form solutions are not possible.

• One can prove properties. Concerning this, see the following. I have a 1993 survey paper with McNabb.
  I have in a 2000 paper, with Sever Dragomir, another item on the elastic torsion problem in convex domains. Click here to see
  • a list of my publications from which you will be able to get electronic forms of these papers.
  • a list of electronically available technical reports and papers from which you will be able to get the dvi file for the Report with Dragomir.
• One can solve the problem numerically.
  • While UWA Dept of Maths and Stats had the NAG library, I used NAG routine D03EAF. Work is likely to be done involving MathLink links for this, perhaps just for circular-arc polygons. InterCall may be a possibility too: see general remarks on InterCall, on genmex, etc.. (Arrigo Triulzi had InterCall going at QMW. See write-up in 1997 Asian Technology Conference in Mathematics paper. As at 2003, InterCall seems to have been discontinued.)
  • It seems within the bounds of present computers to have a Matlab implementation of something like the algorithm used in D03EAF. This would fit in with software used by colleagues at UWA.

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