Accuracy improvement of the second-kind Fredholm integral equations in computational electromagnetics

Accuracy analysis;Buffa-Christiansen functions;extinction theorem;first-kind integral equations;identity operator;magnetic-field integral equation;method of weighted residuals;near-singularity extraction;N-Müller integral equations;numerical accuracy;Rayleigh-Ritz scheme;second-kind integral equations;testing scheme.

Yan, Su ; Jin ; Jianming

关键词: Accuracy analysis;    Buffa-Christiansen functions;    extinction theorem;    first-kind integral equations;    identity operator;    magnetic-field integral equation;    method of weighted residuals;    near-singularity extraction;    N-Müller integral equations;    numerical accuracy;    Rayleigh-Ritz scheme;    second-kind integral equations;    testing scheme.;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/34442/Yan_Su.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

In computational electromagnetics, the second-kind Fredholm integral equations (IEs) are known to have very fast iterative convergence but rather poor solution accuracy compared with the first-kind Fredholm integral equations. The loss of the numerical accuracy is mainly due to the discretization error of the identity operators involved in second-kind IEs. In the past decade, although much effort has been made to improve the numerical accuracy of the second-kind integral equations, no conclusive understandings and final resolutions are achieved.In this thesis, the widely used surface integral equations in computational electromagnetics are first presented along with the discussions of their respective mathematical and numerical properties. The integral operators involved in these integral equations are investigated in terms of their mathematical properties and numerical discretization strategies. Based on such discussions and investigations, a numerical scheme is presented to significantly suppress the discretization error of the identity operators by using the Buffa-Christiansen (BC) functions as the testing function, leading to much more accurate solutions to the second-kind integral equations for smooth objects in both perfect electric conductor (PEC) and dielectric cases, while maintaining their fast convergence properties. This technique is then generalized for generally shaped objects in both PEC and dielectric cases by using the BC functions as the testing functions, and by handling the near-singularities in the evaluation of the system matrix elements carefully. The extinction theorem is applied for accurate evaluation of the numerical errors in the calculation of scattering problems for generally shaped objects. Several examples are given to investigate and demonstrate the performance of the proposed techniques in the accuracy improvement of the second-kind surface integral equations in both PEC and dielectric cases. The reasons for the accuracy improvement are explained, and several important conclusive remarks are made.