Integration of the modified double layer potential of the vector boundary element method for eddy current problems

• Published in: European journal of applied mathematics Vol. 34; no. 2; pp. 385 - 407
• Main Authors: SIVAK, S., STUPAKOV, I., ROYAK, M., ROYAK, S.
• Format: Journal Article
• Language: English
• Published: Cambridge Cambridge University Press 01.04.2023
• Subjects: Applied mathematics; Boundary element method; Eddy currents; Electric double layer; Exact solutions; Far fields; Linear algebra; Matrices (mathematics); Matrix algebra; Permeability; Singularity (mathematics)
• ISSN: 0956-7925; 1469-4425
• DOI: 10.1017/S0956792522000183

The boundary element method for the eddy current problem (BEM-ECP) was proposed in a number of papers and is applicable to important tasks such as the problem of inductive heating and transmission of electromagnetic energy. BEM-ECP requires the construction of a system of linear algebraic equations in which the matrix is inherently dense and is constructed out of element matrices. For the process of the element matrix computation, two cases are normally considered: far-field interaction and near-field interaction, because the construction of element matrices requires integration of a singular function. In this article, we suggest a transform that allows computing the matrix components of the near-singular interaction part while implementing only the single and double layer potentials. The previously suggested modified double layer potential (MDLP) can be integrated by means of this transform, which simplifies the program implementation of BEM-ECP significantly. Solving model problems, we analyse the drawbacks of the previously suggested approach. This analysis includes the proof of the MDLP singularity that makes the integration of this potential a rather difficult task without the help of our transform. The previously suggested approach does not work well with surfaces that are not smooth. Our approach does consider such cases, which is its main advantage. We demonstrate this on the model problems with known analytical solutions.