Microwave Finite Element Modeling Applications to Coupled Electromagnetic-Thermal Problems and Characterization of Dielectric Materials

Transcription

1 Micowave Finite Element Modeling Applications to Coupled Electomagnetic-Themal Poblems and Chaacteization of Dielectic Mateials Hulusi Acikgoz, Yann Le Bihan, Olivie Meye, and Lionel Pichon Laboatoie de Génie Electique de Pais, SUPELEC; UPMC Univesite Pais 6; Univesite Pais Sud-P11, Gif-su-Yvette Cedex, Fance This talk eviews eseach wok caied out in the Laboatoie de Génie Electique de Pais duing the last decade about finite element modeling fo micowave engineeing applications. The fist pat is devoted to coupled electomagnetic-themal poblems. In paticula the micowave heating of dielectic mateial located in a coaxial cell will be pesented. The second pat summaizes ecent wok dedicated to micowave chaacteization of dielectic mateial using a combination of finite element methods and neual netwoks. It is hoped that the mateial of this pape could help to evaluate the potential of the finite element technique fo micowave sinteing. Intoduction Computational methods ae playing an inceasingly impotant ole duing the design step in micowave engineeing. This is the case fo micowave sinteing. In [1,] fo example, the FDTD method was shown to povide an efficient appoach to electomagnetic modeling of multimodal cavities and micowave sinteing. Also in [, 4] the TLM was successfully applied in computational electomagnetics elevant to micowave sinteing. Among the diffeent numeical methods the finite element method is the most flexible. It allows complex geometies to be modeled and easily takes into account inhomogeneous mateials, while poviding an accuate local analysis when solving patial diffeential equations. In this talk we pesent eseach wok achieved duing the last decade in the Laboatoie de Génie Electique de Pais devoted to micowave engineeing applications involving the finite element method. In the fist pat the micowave heating pocess will be consideed. The time hamonic electomagnetic poblem is coupled with the tansient heat equation. The tempeatue dependence of electomagnetic and themal paametes is taken into account. Both bounded and unbounded poblems ae addessed. In the second pat a methodology combining finite elements and neual netwoks is developed in ode to chaacteize dielectic mateials while solving the invese poblem of econstucting pemittivity fom the measued eflection coefficient. Coupled Electomagnetic-Themal Analysis Let us conside a egion Ω containing lossy dielectic mateial and suounded by a bounday Γ. Stating fom the vecto wave equation and using Geen s identity the weak fom of Maxwell s equations in tems of the electic field e can be deived: 9

2 Ω 1 cule.cule d' Ω k ε e.e d' Ω + iωµ µ Ω Γ ( n h).e d' Γ = (1) whee µ is the elative pemeability, k εµ ω ε is the elative pemittivity, ω is the angula fequency and k is the wave numbe ( = ). The diffeent pats of Γ include electic walls (pefectly conducting walls), symmety planes, atificial boundaies suounding the computational domain (in the case of unbounded poblems) and pots (access planes in the case of a micowave cavity). The bounday tem in (1) is detemined on a diffeent pat of Γ accoding to the bounday condition type. In this wok two diffeent types of tangentially continuous elements have been consideed [5]. The tempeatue T of the heated volume V is detemined by solving the heat equation in weak fom: d dt V ρc T dv + p V k gadt.gadϕ dv V T k ϕdγ = n V QϕdV () whee ρ, C p, k, Q ae espectively the density, specific heat capacity, themal conductivity and heat powe density. Convective adiation conditions ae used on the bounday. Equation () is discetized using tetahedal nodal elements of fist ode. Equations (1) and () accounting fo bounday conditions ae solved by iteations. In the heating pocess the tempeatue changes occu much moe slowly than vaiation of the electomagnetic field, so we may assume that the electomagnetic time-hamonic steady state is eached fo each tempeatue distibution. The coupling is taken into account though the dissipated powe density pe unit volume: 1 Q = ωε ε " e () In a fist example [6], the coupled model is used to analyze a shot cicuited standad waveguide containing a lossy dielectic mateial such as a ceamic mateial o foodstuff. The convective bounday condition is applied on the face between ai and dielectic. We suppose that the faces against the waveguide walls ae themally isolated. A typical finite element mesh is shown in Fig.1. Symmety allows us to model only one quate of the stuctue. The same mesh is used fo the themal poblem in the dielectic egion. The cavity is excited by a TE 1 mode at.45 GHz. The electomagnetic and themal popeties of the mateial ae taken fom [7]. The powe geneated by the exciting souce is 1 kwh. The initial tempeatue is 5 C. The convective heat paamete is taken to be constant fo the entie simulation ( h = 5 W / m C ). Fig. shows the evolution of tempeatue in the lossy dielectic medium at diffeent time steps in case of aluminum nitide. The second example consists of a measuement cell used in contolling the micowave pocess fo polyme cuing; it is a potion of a cylindical waveguide shot-cicuited at one end and excited at the othe end by a cicula coaxial cable (Fig. ). This cell belongs to an instumentation that allows dielectic popety measuements (pemittivity, elaxation fequency) 1

3 Fig. 1. Mesh of the dielectic waveguide Fig.. Distibution of the tempeatue in Aluminum Nitide (1/4 th of dielectic) in a boad fequency ange (1 MHz 1 GHz). The sample unde micowave cuing is a mixtue of epoxy esin and hadene. The cell is excited by a TEM mode. Consideing the axial symmety of the stuctue, an axi-symmetic model was developed [8]. It uses the azimuthal component of the magnetic field. To illustate the computation, Fig. 4 shows the evolution of 11

4 Fig.. Coaxial cell Fig. 4. Vaiation of tempeatue vesus time Table 1. Physical Paametes of Tissue Used in the Computation ε C p (J / kg C) ρ (kg / m ) k (W/m C) Fat 7 i.x1 Muscle 5 i4.55 x1.9 x x1.4 tempeatue in the middle of the sample vesus heating time at 44 MHz fo an incident powe of 7 W. The elaxation fequency is also plotted as a compaison: numeical modeling gives some insight into the elationship between the simulated macoscopic tempeatue and the mesoscopic elaxation fequency elated to the behavio of the molecula dipoles. The thid example was studied in ode to test the ability of the coupled electomagnetic model to deal with hypethemia poblems and to addess the case of an unbounded computational egion [9]: a lossy dielectic sphee is illuminated by an incident plane wave with a fequency f = 4MHz (usual hypethemia fequency). It has a adius R =. 6λ ( λ is the wavelength in fee space). This geomety of the sphee was pefeed because in this case analytical solutions exist fo the electomagnetic equation and allowed us to validate the numeical method. The values of the physical paametes ae those of fat and muscle (Table 1). When the micowave souce is switched on (t = ), the tempeatue is the equilibium tempeatue of the human body ( T = 7 C). Due to symmety, only a quate of the sphee is studied. Fig. 5 shows a typical mesh used fo the computation whee ae shown the boundaies of the dielectic egion and the oute bounday located at R =. 45λ. In Fig. 6, we show the computed tempeatue distibution in fat tissue afte 6 s of electomagnetic heating. This distibution fits that of the heating powe. The maximum tempeatue eached is 44.5 C and is located nea the suface of the dielectic sphee. Themal gadients have the highest values in the hot egion. Micowave Dielectic Chaacteization Detemination of dielectic constant and loss facto of dielectic mateials is a difficult poblem. In this pape the chaacteization technique is based on the coaxial cell pesented 1

5 Fig. 5. Typical electomagnetic mesh. Fig. 6. Distibution of tempeatue (6 s). befoe. Nevetheless, fo a coaxial waveguide, the elation between the admittance measued at the discontinuity plane of the waveguide and the complex pemittivity (diect model) can aely be inveted. An altenative is then to use a diect model in an iteative pocedue to educe the diffeence between the calculated admittance and the measued one. Howeve, in many cases an analytical solution does not exist and numeical solutions ae computationally expensive when using an iteative pocedue. In ode to avoid the dawbacks of an iteative pocedue, a moe efficient appoach is to use a paametic model adjusted with a database. A good kind of candidate fo this is atificial neual netwoks (ANNs) since they ae univesal and pasimonious models and can allow fo appoximation of a wide ange of functions povided that they ae peviously tained. The taining consists of adjusting the paametes of the ANNs so that it coectly appoximates the physical behavio of the system. Ou appoach is based on the combination of ANNs and the finite element method [1]. The necessay data sets ae ceated by the finite element method. The inputs of an ANN ae the value of complex impedance (eal pat G, imaginay pat B) and the measuement fequency (f). The output ae the (, ) pais (Fig. 7). The use of ANNs is inteesting since they do not equie analytical solutions and can be implemented without any limitation on the fequency. The fist invesion esults obtained with ANNs (fom 1 MHz to 1.8 GHz) ae compaed with those obtained fom a potocol (SupeMit) used fo micowave chaacteization and having an analytical solution on a wide band of fequency (up to 18 GHz with an APC7 standad). The invese model of this second potocol can be easily obtained by an iteative pocedue using the gadient method (Fig. 8). Conclusion We showed in this pape that the finite element method povides an efficient tool fo dealing with coupled electomagnetic-themal poblems and that a numeical electomagnetic model can be efficiently combined with neual netwoks in ode to solve invese poblems involved in 1

6 Fig. 7. Stuctue of an ANN (Multi-Laye Pecepton) Fig. 8. Reconstuction of the pemittivity of ethanol micowave dielectic chaacteization. Capabilities of the models could be developed in the futue in ode to addess micowave sinteing. Refeences [1] M. F. Iskande, R.L. Smith, A.O.M. Andade, H. Kimey, J., and L.M. Wal, FDTD simulation of micowave sinteing of ceamics in multimode cavities, IEEE Tans. Micowave Theoy and Techn., vol. 4, no. 5, pp. 79-8, [] J. White, M. F. Iskande, and Z. Huang, Development of a multigid FDTD code fo theedimensional applications, IEEE Tans. Antennas and Popag., vol. 45, no. 1, pp , 1997, [] A. Ami and A. Saidane, TLM simulation of micowave hybid sinteing of multiple samples in a multimode cavity, Inten. J. Num. Modelling, vol. 16, no., pp ,. [4] A. Ami and A. Saidane, TLM modelling of micowave sinteing of multiple alumina samples, IEE Poc., Science, Measuement and Technology, vol. 151, no. 4, pp , 4. [5] J. Jin, The Finite Element Method fo Electomagnetics, nd Ed., John Wiley and Sons,. [6] A. Sekkak, L. Pichon, and A. Razek, D FEM magneto-themal analysis in micowave ovens, IEEE Tans. Magnetics, vol., no. 5, pp. 47-5, [7] F.M. Clak, Insulating Mateials fo Design and Engineeing Pactise, John Wiley and Sons, 196. [8] L. Pichon and O. Meye, Coupled themal-electomagnetic simulation of a micowave cuing cell, IEEE Tans. Magnetics, vol. 8, no., pp ,. [9] A. Sekkak A., V.N. Kanellopoulos, L. Pichon, and A. Razek,. An electomagnetic and themal analysis in biological bodies using D FE and absobing bounday conditions, IEEE Tans. Magnetics, vol. 1, no., pp , [1] H. Acikgoz, Y. Le Bihan, O. Meye and L. Pichon, Neual netwoks fo boad-band evaluation of complex pemittivity using a coaxial discontinuity, EPJ Applied Physics Jounal, vol. 9, no., pp. 97-1, 7. 14