Numerical algorithms for boundary problems with disturbed axial symmetry

The axial symmetry in the real devices of image electron optics is always disturbed by small defects in manufacturing and assembly. We present a complete method for the numerical simulation of problems with such defects, which includes the algorithms for singularity extraction in a numerical solution of the boundary problem described by the Helmholtz equation. The effective recurrent formulas for evaluation of the kernels of integral representations and their derivatives are constructed. New modification of the well-known Bruns–Bertein (Crelles J. Math. 81 (1876) 349; Ann. de Radioelectr. 2 (7) (1947) 379) method is given, and correlation of this method with an integral equation in variations is investigated. The algorithms are implemented in the codes POISSON-2 (CAD Methods in Physical Electronics—2 parts, Inst. Math. RAS, Novosibirsk, 1986; Proceedings of the 2nd International Conference on Computations in Electromagnetism, Notingham, UK, May 13–15, 1994; Proceedings of the 11th IEEE International Pulsed Power Conference, Baltimore, MD, June 29–July 2, 1997) and OPTICS-2 (Proceedings of the XV Conference on High Speed Photography, Photonics and Metrology of High-Speed Processes, Moscow, 26–29 November, 1991). The results of the numerical simulation for various test problems with different kinds of boundary deformation are given.