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The Galerkin method for a regularised combined field integral equation without a dual basis function

Kazuki Niino, Shunpei Yamamoto

Abstract

We propose discretisation of a regularised combined field integral equation (regularised CFIE) only with the Rao-Wilton-Glisson (RWG) basis function. The CFIE is a formulation of integral equations, which avoids the so-called ficticious frequencies of integral equations. The most typical CFIE, which is a linear combination of the electric field integral equation (EFIE) and magnetic field integral equation (MFIE), is known to be ill-conditioned and requires many iterations when solved with iteration methods such as the generalised minimum residual (GMRES) method. The regularised CFIE is another formulation of the CFIE to solve this problem by applying a regularising operator to the part of the EFIE. In several previous studies the regularising operator is determined based on the Calderon preconditioning. This regularising operator however takes much more computatonal time than the standard CFIE since discretising the EFIE with the Calderon preconditioner requires the dual basis function. In this article we propose a formulation of the regularised CFIE, which can be discretised with the Galerkin method without the dual basis function.

The Galerkin method for a regularised combined field integral equation without a dual basis function

Abstract

We propose discretisation of a regularised combined field integral equation (regularised CFIE) only with the Rao-Wilton-Glisson (RWG) basis function. The CFIE is a formulation of integral equations, which avoids the so-called ficticious frequencies of integral equations. The most typical CFIE, which is a linear combination of the electric field integral equation (EFIE) and magnetic field integral equation (MFIE), is known to be ill-conditioned and requires many iterations when solved with iteration methods such as the generalised minimum residual (GMRES) method. The regularised CFIE is another formulation of the CFIE to solve this problem by applying a regularising operator to the part of the EFIE. In several previous studies the regularising operator is determined based on the Calderon preconditioning. This regularising operator however takes much more computatonal time than the standard CFIE since discretising the EFIE with the Calderon preconditioner requires the dual basis function. In this article we propose a formulation of the regularised CFIE, which can be discretised with the Galerkin method without the dual basis function.
Paper Structure (7 sections, 20 equations, 4 figures, 1 table)

This paper contains 7 sections, 20 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. formulations of CFIE
  3. Discretisation of the regularised CFIE based on Galerkin's method
  4. Conventional regularised CFIE
  5. Proposed method
  6. Numerical example
  7. Conclusion

Figures (4)

  • Figure 1: The RWG and BC basis functions. The thick lines forms a triangular mesh and thin lines are its barycentric refinement. The BC function (red arrows) is defined as a linear combination of the RWG on the refined mesh. The black arrow is the RWG function defined on the original mesh.
  • Figure 2: Relative errors of the numerical methods solving \ref{['eq:disc_reg_cfie_new']} (blue) and \ref{['eq:disc_reg_cfie_conv']} (red). The horizontal axis corresponds to the numbers of triangular meshes and the vertical axis to the $L_2$ relative error in \ref{['eq:relative_error']}. Note that the numerical results for the conventional regularised CFIE in \ref{['eq:disc_reg_cfie_conv']} are truncated up to $5780$ triangles since memory requirement for \ref{['eq:disc_reg_cfie_conv']} is much more than that for \ref{['eq:disc_reg_cfie_new']} due to the use of the BC function.
  • Figure 3: Relative error of the numerical methods.
  • Figure 4: Iteration numbers of the GMRES