On the use of conformal transformations to calculate the eigenvalues of the Helmholtz equation for cylinders with polygonal cross section

Solutions of the Helmholtz equation, as it arises in reactor criticality calculations, have been obtained in cylinders of polygonal cross section. The method used employs the Schwarz–Christoffel transformation which maps the vertices of the polygon onto the circumference of the unit circle. Thus the polygonal geometry is converted to r–θ geometry thereby allowing a solution to be constructed which satisfies the symmetry and boundary conditions of zero flux exactly. Three cases have been studied numerically; the equilateral triangle, the square and the hexagon. The square is particularly useful because an exact solution can be obtained by separation of variables in x–y geometry and hence allows the accuracy of the conformal mapping procedure to be assessed. Numerical results are obtained in the r–θ domain using a variational method and appropriate trial functions. The accuracy of the eigenvalues is very high and by comparison with the exact value in the square it is shown that only one term in the trial function leads to an accuracy in the eigenvalue of 0.1%. The lowest order approximation to the current at the square centre edge gives an accuracy of 4%. Details of convergence for all cases are given, as well as some useful approximate analytical formulae for eigenvalues and edge current. ©