THE TORSION OF AN IRREGULAR POLYGON

摘要

This paper is the first in a series presenting a new exact method of solving linear boundary value problems, of elliptic type, for regions bounded externally and, possibly, internally by polygons - problems previously attacked in some simplified cases, by finite-difference and relaxation methods. The torsion problem is chosen as the most suitable medium to develop and exhibit the method, when the boundary conditions do not involve derivatives. A subsequent paper on the corresponding elastic plate problems will remove this limitation. The method is based on representing the boundary effects by a series of exponentially decaying harmonics associated with each edge of the polygon. At the outset it is developed for the basic polygon - a solid polygon without re-entrant angles the boundary conditions being satisfied by equating to zero the corresponding Fourier Sine harmonics.