This paper describes a finite element method to obtain an accurate solution of the scalar Helmholtz equation with field singularities. It is known that the spatial derivatives of the eigenfunction of the scalar Helmholtz equation become infinite under certain conditions. These field singularities under mine the accuracy of the numerical solutions obtained by conventional finite element methods based on piecewise polynomials. In this paper, a regularized eigenfunction is introduced by subtracting the field singularities from the original eigenfunction. The finite element method formulated in terms of the regularized eigenfunction is expected to improve the accuracy and convergence of the numerical solutions. The finite element matrices for the present method can be easily evaluated since they do not involve any singular integrands. Moreover, the Dirichlet-type boundary conditions are explicitly imposed on the variables using a transform matrix while the Neumann-type boundary conditions are implicitly imposed in the functional. The numerical results for three test problems show that the present method clearly improves the accuracy of the numerical solutions.

This paper reports a novel 3D topology optimization method based on the finite difference time domain (FDTD) method for a dielectric lens antenna. To obtain an optimal lens with smooth boundary, we apply normalized Gaussian networks (NGnet) to 3D topology optimization. Using the proposed method, the dielectric lens with desired radiation characteristics can be designed. As an example of the optimization using the proposed method, the width of the main beam is minimized assuming spatial symmetry. In the optimization, the lens is assumed to be loaded on the aperture of a waveguide slot antenna and is smaller compared with the wavelength. It is shown that the optimized lens has narrower beamwidth of the main beam than that of the conventional lens.