Topological metallic structure design for microwave applications using a modified interpolation scheme

Abstract

In structural design for microwave applications, metallic structures generate the skin effect that critically affects the performance of microwave devices. In finite element analysis (FEA), highly refined mesh generation is necessary to take the skin effect into account. To avoid the expensive fine meshing and computing process, the condition of the perfect electrical conductor (PEC) or the impedance boundary condition has been generally used in FEA based topology optimization. In this study, we proposed a modified penalization formulation using the shifted sigmoid function for the interpolation of the electric permittivity of conductive materials and applied it to microwave structural design through the phase field design method. The proposed approach is available in case of applications to structural design composed of non-ferromagnetic metals. Based on the derived optimal shape, a simple post-processing scheme is employed only once to determine the clear boundary by eliminating the gray scale area for the purpose of manufacturing feasibility. The validity of the proposed design approach is discussed in three numerical examples allowing the change of the target operation frequency.

Appendix

A Relative permittivity and refractive index for the artificial PEC.

In this study, the material property for design process is determined by the relative permittivity derived from the refractive index. In the analysis of electromagnetic wave propagation, the relative permittivity is a measure of the development of the electric field intensity occurred by the electric flux polarization effects inside the medium. The complex relative permittivity is applied for the conductive material and it can be derived from the complex refractive index of n = α − jβ. For metallic medium, the imaginary term of the extinction coefficient β is usually greater than the real term of the refractive index α as shown in Fig. 17(a). Therefore, the electric field intensity is exponentially declined by the large extinction coefficient in the medium. The refractive index n is defined as the relation of $n=\sqrt{\varepsilon_r{\mu}_r}$ and the permittivity is expressed as ${\varepsilon}_{cr}={\varepsilon}_r-j{\tilde{\varepsilon}}_r$. Fig. 17(a) shows that the extinction coefficient β increases according to the increase of the wavelength while the refractive index α is converged to zero. It means that the wavelength increase leads to the attenuation of the electric field in the metallic medium.

Fig. 17

Material property variation of Cu. a plot of the refractive index, b plot of the complex permittivity

In this study, the proposed artificial PEC has a huge negative permittivity calculated from the relation of ε _cr  = n ² in case of μ _r  ≈ 1 assuming non-ferromagnetic material such as copper (Cu) or aluminum. The negative permittivity provides the electric flux of 180° out of phase and its physical meaning is wave reflection. At high frequencies, the free electrons flow only on the surface of the metallic medium that is called as the skin effect. Therefore, the excitation of wave propagation cannot be represented inside of the metallic medium. Fig. 17(b) shows the permittivity of Cu with respect to the variation of the wavelength between 0.3 μm and 1.7 μm (Polyanskiy, 2017). The real part of the permittivity continuously decreases according to the increase of the wavelength while the imaginary part gives small real values as shown in Fig. 17(b). The property of Cu can be estimated having a large negative real permittivity according to the increase of the wavelength, especially in microwave ranges. Therefore, the relative permittivity of the artificial PEC provides the tangential electric field to be approximately 0. It means that the conductive characteristic of the artificial PEC can be represented with a large negative value of the real permittivity.