The Robin problem for the Helmholtz equation in normal-polar shells is addressed by using a suitable spherical harmonic expansion technique. Attention is in particular focused on the wide class of domains whose boundaries are defined by a generalized version of the so-called “superformula” introduced by Gielis. A dedicated numerical procedure based on the computer algebra system Mathematica? is developed in order to validate the proposed methodology. In this way, highly accurate approximations of the solution, featuring properties similar to the classical ones, are obtained.
 B. N. Khoromski, “Integro-Difference Method of Solution of the Dirichlet Problem for the Laplace Equation,” Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, Vol. 24, No. 1, 1984, pp. 53-64.
 D. M. Young, “Iterative Methods for Solving Partial Difference Equations of Elliptic Type,” Transactions on American Mathematical Society, Vol. 76, 1954, pp. 92-111. doi:10.1090/S0002-9947-1954-0059635-7
 P. Natalini, R. Patrizi, and P. E. Ricci, “The Dirichlet Problem for the Laplace Equation in a Starlike Domain of a Riemann Surface,” Numerical Algorithms, Vol. 28, No. 1-4, 2001, pp. 215-227. doi:10.1023/A:1014059219005
 D. Caratelli and P. E. Ricci, “The Dirichlet Problem for the Laplace Equation in a Starlike Domain,” Proceedings of International Conference on Scientific Computing, Las Vegas, 14-17 July 2008, pp. 160-166.
 D. Caratelli, B. Germano, J. Gielis, M. X. He, P. Natalini and P. E. Ricci, “Fourier Solution of the Dirichlet Problem for the Laplace and Helmholtz Equations in Starlike Domains,” Lecture Notes of Tbilisi International Centre of Mathematics and Informatics, Tbilisi University Press, Tbilisi, 2010.
 D. Caratelli, P. Natalini, P. E. Ricci and A. Yarovoy, “The Neumann Problem for the Helmholtz Equation in a Starlike Planar Domain,” Applied Mathematics and Computation, Vol. 216, No. 2, 2010, pp. 556-564. doi:10.1016/j.amc.2010.01.077
 D. Caratelli, J. Gielis, P. Natalini, P. E. Ricci and I. Tavkelidze, “The Robin Problem for the Helmholtz Equation in a Starlike Planar Domain,” Georgian Mathematical Journal, Vol. 18, No. 3, 2011, pp. 465-480.
 D. Caratelli, J. Gielis and P. E. Ricci, “Fourier-Like Solution of the Dirichlet Problem for the Laplace Equation in k-Type Gielis Domains,” Journal of Pure and Applied Mathematics: Advances and Applications, Vol. 5, No. 2, 2011, pp. 99-111.
 D. Caratelli, P. E. Ricci and J. Gielis, “The Robin Problem for the Laplace Equation in a Three-Dimensional Starlike Domain,” Applied Mathematics and Computation, Vol. 218, No. 3, 2011, pp. 713-719.
 J. Gielis, D. Caratelli, Y. Fougerolle, P. E. Ricci and T. Gerats, “Universal Natural Shapes from Unifying Shape Description to Simple Methods for Shape Analysis and Boundary Value Problems,” PLoS One, Vol. 7, No. 9, 2012, Article ID: e29324. doi:10.1371/journal.pone.0029324
 G. Dattoli, B. Germano, M. R. Martinelli and P. E. Ricci, “A Novel Theory of Legendre Polynomials,” Mathematical and Computer Modelling, Vol. 54, No. 1-2, 2011, pp. 80-87. doi:10.1016/j.mcm.2011.01.037