MoM-based topology optimization method for planar metallic antenna design

Shutian Liu·Qi Wang·Renjing Gao

MoM-based topology optimization method for planar metallic antenna design

Shutian Liu1·Qi Wang1·Renjing Gao1

?The Chinese Society of Theoretical and Applied Mechanics;Institute of Mechanics,Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2016

The metallic antenna design problem can be treated as a problem to find the optimal distribution of conductive material in a certain domain.Although this problem is well suited for topology optimization method,the volumetric distribution of conductive material based on 3D finite element method(FEM)has been known to cause numerical bottlenecks such as the skin depth issue,meshed“air regions”and other numerical problems.In this paper a topology optimization method based on the method of moments(MoM) for configuration design of planar metallic antenna was proposed.The candidate structure of the planar metallic antenna was approximately considered as a resistance sheet with position-dependent impedance.In this way,the electromagnetic property of the antenna can be analyzed easily by using the MoM to solve the radiation problem of the resistance sheet in a finite domain.The topology of the antenna was depicted with the distribution of the impedance related to the design parameters or relative densities.The conductive material(metal)was assumed to have zero impedance,whereas the non-conductive material was simulated as a material with a finite but large enough impedance.The interpolation function of the impedance between conductive material and non-conductive material was taken as a tangential function.The design of planar metallic antenna was optimized for maximizing the efficiency at the target frequency.The results illustrated the effectiveness of the method.

? Shutian Liu stliu@dlut.edu.cn

1State Key Laboratory of Structural Analysis for Industrial Equipment,Dalian University of Technology,Dalian 116024, China

Metallic antenna design·Topology optimization·Method of moments·Resistance sheet·Impedance boundary condition

DOI 10.1007/s10409-016-0584-0

1 Introduction

As more requirements,such as small size and maximal efficiency are incorporated into antennas,the design for application-specific antenna should be guided by use of advanced design methodology[1,2].The antenna design problem aims to find the optimal distribution of conductive material,which is well suited for the topology optimization method.However,it has been known that the determination of the volumetric distribution of conductive material based on the 3D finite element method(FEM)for electromagnetic analysis will cause numerical bottlenecks such as the skin depth problem[2].The main purpose of this paper is to investigate a design method of planar metallic antenna using gradient-based topology optimization method based on the method of moments(MoM).

Topology optimization method can be used to find optimal material distributions within a given design domain,such that the desired response of the system can be enhanced corresponding to a general initial configuration.Traditional design methods,such as the shape and size optimization methods, are carried out upon certain initial configurations subjected to the designers’experience and use the shape and size parameters as design variables resulting in no new layout can be generated in the design domain,whereas the topology optimization method has the ability to create new layout in the design domain,that is,the topology of the structure can be evolved automatically.Although topology optimization method was originally developed to design elastic structures[3–5],the method has been extended successfully to solve multidisciplinary structure design problems[6],such as the design problems of photonic crystal structures[7,8],meta-materials[9],radio frequency devices[10],aircraft structures [11],micro-mass sensors[12]and so on.In the topology optimization method,the design domain is divided into a series of small elements,and a design variable associated with each element value range from 0 to 1 to indicate the absence or presence of candidate materials.With this material distribution approach,topology optimization has the ability to change the configuration through creating and merging holes in the design domain,thus can get an optimal design even from a general initial configuration,and is especially well suited for numerical methods such as the FEM or the MoM.In the 3D FEM,the skin depth,which is a measure for the distance through which the current density decreases by a factor e?1in a conductor[2],calls for highly refined meshes to capture the real physics when an electromagnetic wave is propagating through finite-thickness and finite-conductivity metallic patches,and results in the optimization for microwave problems being inefficient[1]. Owing to the skin depth issue boundary conditions,such as the impedance boundary condition(IBC)[1]or the perfect electric condition(PEC)[13],were introduced in the optimization of microwave devices to simplify the numerical modeling and decrease the computational cost.Koulouridis et al.[13]proposed an optimization scheme based on genetic algorithms(GA)using the PEC to model the real conductor where the skin depth issue was eliminated.However,this optimization scheme based on GA and binary handling of the PEC leads to the sensitivities which cannot be determined.Furthermore,the optimization method based on GA requires an extensive analysis for the large scale population and the time-consumption limits its applicability in the design with a large number of independent variables.In order to improve the solving efficiency of metallic antenna design problems,an optimization scheme based on gradient algorithms should be developed.However,to the best of our knowledge,there are only a few articles(for examples Refs. [14–17])using the topology optimization method to design metallic antenna based on gradient algorithms.Erentok and Sigmund[14]used the 3D FEM as their numerical method to design a conductor-based sub-wavelength antenna.However,the extra introduced free space domain to simulate the absorbing boundary conditions results in the optimization being inefficient.Furthermore,the skin depth issue existed in the 3DFEM is difficult to deal with.Hassan et al.[15]realized ultra-wide-band monopole antenna design and dual-band microstrip antenna design based on the finite-difference time domain(FDTD)method,although the FDTD is well suited to simulate antennas operating at wideband frequencies,that method prefers to use large-scale regular meshes and the“airregions”around antennas is also needed to be meshed[18]. Zhou et al.[16]proposed a level-set based topology optimization method for dipole antenna design based on the MoM, and used adaptive meshes to capture the real physics.But the adaptive meshes will reduce the efficiency of optimizations and cause numerical oscillations in the sensitivity analysis. In our previous paper[17],a planar metallic antenna was optimized for miniaturization by minimizing the S11of the antenna at a lower frequency and an exponential interpolation function was used,where the Ohmic loss of the antenna was ignored.But a significant difference was found between the performances of the results before and after post-processing.

This paper presents a design methodology for the topology optimization of planar metallic antennas based on the MoM. The design domain of a planar metallic antenna was considered to be composed of a set of resistance sheets.In this way, the electromagnetic property of the antenna can be analyzed easily by using the MoM to solve the radiation problem of the resistance sheet in a finite domain.Furthermore,the body of conductive material often discretized with 3D finite elements in the FEM is transferred to a sheet with zero thickness,and a design-dependent element impedance boundary condition was introduced to resolve the skin depth issue.The topology of the antenna was depicted with the distribution of the impedance related to the design variables.The conductive material was assumed to have zero impedance,whereas the non-conductive material was simulated as a material with a finite but large enough impedance value.The interpolation function of the impedance between conductive material and non-conductive material was taken as a tangential function. The design of planar antennas is optimized for maximizing the efficiency at the target frequency.The optimization problem was solved by the method of moving asymptotes(MMA) [19],and binary results can be obtained by setting a threshold value for all design variables meanwhile the responses of the designs before and after post-processing show a good agreement.

2 Formulations of the optimization problems

2.1 Optimization formulation

This section aims to establish the optimization formulation of metallic antenna design problems.The energy balance of an antenna system is illustrated in Fig.1,whererepresents the input energy in the transmission line,represents the reflection energy,is the energy fed in the antenna which equalsWΩrepresents the Ohmic loss of the antenna,and Wradis the radiation energy which equals.Generally,antennas are desired to be designed to match with its transmission line and have no Ohmic loss,which means theand the WΩshould be optimized as small as possible,thus the objective can be

Fig.1 An illustration of energy balance of a transmitting antenna system,where the input energy Win,transequals the radiation energy Wradplus the reflection energy Wout,transand the Ohmic loss of the antenna WΩ

taken as maximizing the antenna efficiency,which is given as

whereηis the antenna efficiency,ρ are the design variables,I represent the state variables(surface currents)of the antenna system.

Then the optimization formulation can be formulated as find ρ=(ρ1,ρ2,...,ρN)T,

where N and Nfare the total numbers of design variables and target frequencies respectively.The second constraint is not always necessary in antenna designs,but is added to speed up the convergence[1].viis the volume of each element, Vfis the ratio of the volume occupied by conductor to that of the total design domain.fjs represent a series of target frequencies.

3 Design parameterization

Fig.2 An illustration of the design parameterization for patch metallic antenna design problems,where the design domain Ωdes is assumed as a series of resistance sheets

Topology optimization of metallic antenna design can be treated as an optimization problem to find optimal distribution of conductive materials,so that the desired response (objective)of the antenna can be optimized.An illustration of the proposed design parameterization for patch metallic antenna design problems is shown in Fig.2,where the body of the antenna is transferred to a resistance sheet with zero thickness.The resistance in position r is denoted as ZS(r). When ZS(r)is small or large enough,corresponding to a conductor or non-conductor exists in position r,the layout of the conductive patch can be determined by ZS(r)with a series of design variables ρ∈[0,1].In order to establish an appropriate relationship between conductor and non-conductor, the interpolation function of ZSwith ρ should be developed to have the following properties:the interpolation function has to be monotonous;the change in the objective function caused by design variables could be detected.Besides,a difficulty of the infinite difference between the impedance of conductorand that of non-conductorshould also be resolved.To meet the above requirements and resolve the difficulty,it’s useful to express the sheet impedance ZSin tangential scale,which is given by

where Zatanis the arctangent of ZS,p is the penalty parameter.The proposed tangential interpolation function guarantees that ZShas the ability to value the total range of[0,+∞)when Zatanvalues the range of[0,π/2]by interpolating with design variables ρ∈[0,1].An appropriate threshold value ofhas been determined in our previous work[17],which is suggested to value the range from 106to 1010Ω/m2.The tangential interpolation function can be seen in Fig.3 with=108Ω/m2and Zmetal=0Ω/m2.

Fig.3 (Color online)Plots of the tangential interpolation function with Z non-metal=108Ω/m2 and Z metal=0Ω/m2.a Z S(ρ)with p=0. b Ztan(ρ)with p=1,10,40

Fig.4 Models used to evaluate the properties of candidate interpolation functions.a The antenna with impedance values Z S(r,ρ),for r∈ΩC,and ZS(r)=0,for r∈Ω/ΩC.b The standard fractal antenna without modeling ΩC

A fractal antenna is selected to evaluate the property of the proposed tangential interpolation function and determine an appropriate value of the penalty parameter p.The antenna is simulated based on its transmitting mode,where a delta gap voltage generator is set at the central position of the antenna. As shown in Fig.4a,the green regionΩCis filled with a series of resistance sheets where the impedance values ZS(ρ). The frequency is set to 400 MHz,Znon-metal=108Ω/m2and Zmetal=0Ω/m2.Because the exponential interpolation function cannot get to exact zero(i.e.,ex→ 0 when x→?∞),the impedance of metal is set to a small enough value(10?6Ω/m2)instead for that case.

The comparisons are shown in Fig.5.As stated before, there are two requirements for an appropriate interpolation function,one is the interpolation function has to be monotonous,the other is the change in the objective caused by the design variables must be detectable.As can be seen from Fig.5,for the linear interpolation function case,the antenna efficiency appears so insensitive with the change of ρ within a large range of ρ∈[0,1]that the change in the objective(η) cannot be detected;for the exponential interpolation function case,the sensitivities of the objective with design variables approach to zero in a large range when ρ closes to 0 and 1,which leads to difficulties in converging the ρ to 0 or 1; the proposed tangential interpolation function with p=40 has the ability to satisfy the above two requirements,which shows smoothing change within the range of ρ from 0 to 1.

Fig.5 Comparisons of the linear,exponential and tangential interpolation functions,the responses of the bowtie antenna and the fractal antenna are used as the reference solutions

4 Solutions of the optimization problems

4.1 Governing equations

Using the electric field integral equation(EFIE),the MoM develops a simple and efficient numerical procedure for treating electromagnetic problems of scattering by arbitrarily shaped objects.The scattered field at position r caused by any source at position r′is given by

When the conductivity of the conductive patch is finite,the total electric field on the conductive patch equals the product of the sheet impedance and the surface current.This IBC can be represented by the following equation

where ZS(ρ)is the design-dependent impedance of the resistance sheets with design variables ρ.When ZS(ρ)=0,the right-hand side of Eq.(5)disappears and the IBC becomes the standard PEC where the skin depth equals zero;when ZS(ρ)→+∞,the resistance sheet no longer presents and the surface current J will be calculated to be approximately zero,then the skin depth issue is no longer involved.Thus,if the design methodology is devised such that the final design is fully black and white,the skin depth is resolved.This is a commonly used trick in topology optimization and will also be used in the work presented here.

The MoM[20,21]begins by expressing the unknown J in terms of a set of basis functions Bns as

where the Ins are the unknown coefficients needed to be determined.One of the most frequently used sub-domain basis functions is the RWG basis function proposed by Rao et al.[22]in 1982.

Thus the governing equations of an antenna system described by Eq.(5)can be calculated by using the standard Galerkin method[18]as

where

where Eincrepresents the source of the antenna system,ZRand V represent the impedance matrix and the source matrix occupied by ZSand Eincrespectively.It is worth mentioning that Z is fixed during the optimization process and only needed to be calculated once when the optimization begins.

4.2 Objective function and sensitivity analysis

Based on the linear system described by Eq.(7),the antenna efficiency can be calculated as

where? represents the real part operator.?Iis a vector where the number in the position corresponded to the input terminal of the antenna system equals the length of the feeding edge linand all others equal zero.represents the characteristic impedance of the transmission line,which is set to 50 Ω.

In order to update the design variables through a gradient-based algorithm,the sensitivities of the antenna efficiency with the design variables must be evaluated.According to the chain rule,the derivative of η with ρ can be expressed as

Then the sensitivity analysis can be done by the adjoint method as

where

Fig.6 The flowchart of the proposed design method for patch metallic antenna design

The optimization problem is solved by the MMA algorithm with the maximum change in ρ less than 1%as the stopping criteria if nothing else is stated.The flowchart of the proposed design method for patch metallic antenna design can be seen in Fig.6.

5 Numerical examples

In this section,the planar metallic antenna is designed based on its transmitting mode,and Fig.7 gives the design model. Parameter Vf=0.25,parameter p is set to 40 according to the above numerical tests,and the target frequencies are set to 370 and 420 MHz,Zmetalis set to 0 Ω/m2and Znon-metal is set to 108Ω/m2.A uniform distribution with ρi=0.25 for all ρi∈Ωdesis used as the initial configuration.The total number of design variables is 1680.The optimization problem is solved by a computer with four 2.80 GHz cores CPU and 4 GB RAM.

The design result is obtained after40 iterations with about 7.3 minutes running time which is shown in Fig.8a,as can be seen,there are some of gray scale elements remaining in the design result which should be removed.By setting a threshold value ρ0for all the design variables with ρi≥ρ0equal one and all others equal zero,fully black and white design can be obtained.The post-processed result with ρ0=0.2 is shown in Fig.8b.The efficiency of the designs before and after post-processing show a good agreement as shown in Fig.9.

Fig.7 The design model of a transmitting antenna

Fig.8 Topology optimization of planar metallic antenna design based on the tangential interpolation function.a The design result.b The post-processed design result with ρ0=0.2

Fig.9 Comparisons of the efficiency of the designed antennas before and after post-processing

Fig.10 (Color online)Comparisons of the performances(η and S11) of the design result simulated by MoM and Ansoft HFSS

It is necessary to claim that some corners of triangles (discretization elements)in Fig.8b touch sides of other triangles.In MoM analysis,there is no current flowing through these points,but in practice,there will be electrical contact.This point-connection feature is equivalent to that the relevant points are not connected in practice,which could be probably avoided by using an appropriate regularization method.In order to verify the effectiveness of the design result,the designed antenna is cross-verified by a commercial software(Ansoft HFSS)as shown in Fig.10,and the difference between the responses simulated by MoM and Ansoft HFSS is acceptable.

6 Conclusions and future work

In this paper,a methodology based on topology optimization for the design of planar metallic antennas through a gradient based optimization method is presented.A design parameterization associated with conductors and non-conductors suited for electromagnetic optimization problems involving MoM analysis has been built.The designs of planar metallic antennas are optimized for antenna efficiency improvement at the target frequency,and binary results can be obtained by setting a threshold value for all design variables meanwhile the responses of the designs before and after post-processing show a good agreement.In the future work,an appropriate regularization method(filtering method,perimeter constraint,or other methods)should be developed to ensure that solutions without point-connection feature can be obtained,and the efficiency improvement within a wide frequency sweep will be more actual.

Acknowledgments This project was supported by the National Natural Science Foundation of China(Grants 11332004,11372063,and 11572073),111 Project(Grant B14013),and the Fundamental Research Funds for the Central Universities(Grant DUT15ZD101).

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21 October 2015/Revised:15 April 2016/Accepted:3 May 2016/Published online:13 September 2016