Advanes in adio Siene 2003) 1: 99 104 Copernius GmbH 2003 Advanes in adio Siene A hybrid method ombining the FDTD and a time domain boundary-integral equation marhing-on-in-time algorithm A Beker and V Hansen Lehrstuhl für Theoretishe Elektrotehnik der Universität Wuppertal, ainer-gruenter-str 21, 42119 Wuppertal, Germany Abstrat In this paper a hybrid method ombining the FDTD/FIT with a Time Domain Boundary-Integral Marhing-on-in-Time Algorithm TD-BIM) is presented Inhomogeneous regions are modelled with the FIT-method, an alternative formulation of the FDTD Homogeneous regions whih is in the presented numerial example the open spae) are modelled using a TD-BIM with equivalent eletri and magneti urrents flowing on the boundary between the inhomogeneous and the homogeneous regions The regions are oupled by the tangential magneti fields just outside the inhomogeneous regions These fields are alulated by making use of a Mixed Potential Integral Formulation for the magneti field The latter onsists of equivalent eletri and magneti urrents on the boundary plane between the homogeneous and the inhomogeneous region The magneti urrents result diretly from the eletri fields of the Yee lattie Eletri urrents in the same plane are alulated by making use of the TD-BIM and using the eletri field of the Yee lattie as boundary ondition The presented hybrid method only needs the interpolations inherent in FIT and no additional interpolation A numerial result is ompared to a alulation that models both regions with FDTD 1 Introdution The FDTD-Method is a very effiient and aurate tehnique for solving bounded eletromagneti field problems For the analysis eg of sattering or antenna problems inluding the open spae the solution spae must be kept finite by introduing an absorbing boundary ondition ABC) An often applied ABC is obtained by introduing a perfetly mathed layer PML), the effet of whih is inorporated into the solution-proedure by regarding the PML-region as a part of the FDTD solution spae, that means, by treating it within the FDTD sheme As the FDTD is a loal method the PML Correspondene to: A Beker abeker@uni-wuppertalde) an be regarded as a loal ABC, too A global ABC is obtained if an integral formulation for a surfae inlosing the bounded FDTD solution spae is used as a starting point As an example for suh a global FDTD-ABC formulation de Moerlosse et al, 1993) shall be mentioned, where the magneti field is alulated just outside the FDTD solution spae by an integral equation As the Yee-sheme onsisting of two shifted latties does not provide the required eletri and magneti urrents in the same plane, one of these must be alulated by an additional interpolation A very interesting lass of TD-integral formulations are the so alled Time Domain Boundary-Integral Marhing-on-in-Time Algorithms TD-BIMs), whih in the last few years draw more an more attention By this, the two main drawbaks of TD- BIMs -stability problems and high omputational ost- have been more and more alleviated eg Shanker et al, 2000) Thus, it is reasonable to make use of these advantages and to develop a hybrid method that ombines the FDTD/FIT and a TD-BIM, in other words, to make use of the TD-BIM as a global ABC for the FDTD solution spae This paper is organized as follows: Setion 2 gives an overview of the TD-BIM that is used here to alulate the eletri urrent density at the boundary between the inhomogeneous and the homogeneous body Setion 3 shortly repeats the differenes between the FDTD and the FIT The latter is used here to model inhomogeneous bodies Setion 4 explains the algorithm of the hybrid method proposed here whih ombines both numerial tehniques Setion 5 shows a first numerial example to test the method In Set 6 a summary and an outlook are given
100 A Beker and V Hansen: A hybrid method ombining the FDTD and a time domain boundary-integral Fig 1 Temporal and spatial basis funtion used for the TD-BIM Fig 2 Spatial alloation of the eletri and the magneti grid voltages and the straightforward alloation of the magneti urrent 2 TD-BIM The time domain eletri field integral equation TD-EFIE) for a homogeneous infinite spae is given to: A r, t) E r, t) = φ r, t) 1 F r, t) 1) t ε When applying the equivalene priniple the potentials are related to the equivalent surfae soures of the field by Ar, t) = µ J r, t ) da, 2) 4π A φr, t) = 1 ρr, t ) da, 3) 4πε A F r, t) = ε Mr, t ) da 4) 4π A with = r r and the region outside of A soure free The surfae A is the surfae on whih the urrents flow, the so alled Huygens-surfae The substitution of Eq 2), Eq 3) together with the ontinuity equation J + t ρ = 0) and Eq 4) into Eq 1) results into a relationship between the eletri field and the eletri and the magneti urrent densities ao, 1999): E r, t) = + A [ µ J r, t ) 4π t t 0 J r, t ) dt Mr, t ) ] da 5) 4πε 4π For the numerial solution the surfae A is subdivided into small pathes The eletri and magneti urrent density is numerially approximated as a series of unknown oeffiients multiplied with basis funtions: N s N t J r, t) = J i,j) β i r) τ j t), i=0 j=0 N s N t M r, t) = M i,j) β i r) τ j t) 6) i=0 j=0 N s and N t are the numbers of spae samples and the time step in whih the eletri field is alulated, respetively The spatial and temporal basis funtions applied here are shown in Fig 1 The width of the temporal basis funtion is 2 T BEM We integrate the ombination of Eqs 5) and 6) multiplied with a test funtion β k over the domain A k of the test funtion Sarkar et al, 2000): N s N t E r, t) β k r)da = A k A k A i=0 j=0 [ µ J i,j) β k r) β i r ) τ t ) 4π t 0 J i,j) γ k r)γ i r ) τ t) dt 4πε M i,j)β k r) β i r ) τ t )] da da 7) 4π
A Beker and V Hansen: A hybrid method ombining the FDTD and a time domain boundary-integral 101 Fig 3 The alloation of the Huygens-surfae A in the Yee lattie The term γ represents the surfae-divergene of the basis funtion β Equation 7) onnets the tangential eletri field on surfae A k to the eletri and the magneti urrents flowing on surfae A The equivalene priniple relates the tangential eletri field on the surfae A k to the equivalent magneti urrent: β k n M) = β k n E n)) = β k [E n n) n n E)] = β k E 8) Substituting Eq 8) into Eq 7) leads to a relationship between the eletri and magneti urrent density: N M i,nt )β k n β i r)) da = A k i=1 A k [ N N t A i=1 j=1 µ 4π J i,j) β k r) β i r ) τ t ) t 0 J i,j) γ k r)γ i r ) τ t) dt 4πε M i,j)β k r) β i r ) τ t )] da da 9) 4π If the atual magneti urrent density on surfae A is known, the atual eletri urrent density an be alulated from the eletri urrent density retarded in time at least one time step and from the magneti urrent density by enforing ontinuity on all subdomains A k of A Eq 9) an be written as: Z 1,1 Z 1,N Z N,1 Z N,N }{{} = f = Z N i=1 N t 1 j=1 I 1,Nt I N,Nt ) ) N N t J i,j) + g M i,j) i=1 j=1 Ni=1 Nt 1 j=1 Y 1,i,j)J i,j) + N i=1 Nt j=1 X 1,i,j)M i,j) Ni=1 Nt 1 j=1 Y N,i,j)J i,j) + N Nt i=1 j=1 X N,i,j)M i,j) 10) The matrix Z desribes the atual eletri fields produed by the atual eletri urrents and thus is highly sparse In a time-invariant system both Z and the oeffiients X k,i,j) and Y k,i,j) have to be alulated just one, at the first time step If the tangential eletri field equals zero that means the material is a perfet eletri ondutor and there is no magneti urrent density) Eq 10) will be suffiient for alulating the eletri urrent density in a marhing-on-in-time algorithm MoT) Otherwise a seond equation is needed to alulate both urrent densities In our ase we will use the FIT to update the magneti urrent density and Eq 10) to update the eletri urrent density Thus the surfae A will enase the region modelled with FIT 3 FDTD/FIT The TD-FIT an be onsidered as a speial formulation of the FDTD-method Weiland, 1996) In ontrast to FDTD it is based on the Maxwell Equations in integral form By using eletri and magneti fluxes and grid voltages as unknown oeffiients and by using the Yee lattie to alloate them, these equations an be written as: Cê = ˆˆb 11) t Cĥ = ˆˆd + ˆĵ 12) t ê represents the grid voltage above one Line L i of the Yee ell The term ˆˆd represents the flux through one Area A i of the Yee ell The unknown oeffiients an be onsidered as the analyti and thus numerially error-free values The appliation of Eq 12) is illustrated in Fig 2a A seond order aurate entral differene approximation for the time derivative leads to the leap frog algorithm The definition of grid voltages as unknown oeffiients is essential if using the FIT instead of FDTD for our hybrid method 4 Hybrid method In this setion we will introdue our hybrid method The TD- BIM ats as a global absorbing boundary ondition enlosing the inhomogeneous bodies whih are modelled by FIT
102 A Beker and V Hansen: A hybrid method ombining the FDTD and a time domain boundary-integral Fig 4 The temporal alloation of the magneti urrents and possible alloations of the eletri urrents Fig 5 The boundary of the FIT-domain and the resulting minimal retardation in the alulation of the magneti grid voltages 41 Basi features We interonnet the inhomogeneous body and the surrounding homogeneous region by alulating tangential magneti grid voltages just outside the inhomogeneous body using a TD magneti field integral equation TD-HFIE) whih uses eletri and magneti urrents flowing on the Huygenssurfae A whose loation is shown in Fig 3 The magneti urrent on the surfae A results diretly from the eletri grid voltages of the FIT, whih will be shown beneath Due to the fat that the magneti grid voltages are spatially and temporally) separated from the eletri grid voltages the eletri urrents on the Huygens-surfae A an not be alulated from the magneti grid voltages Therefore the eletri urrent will be alulated by using a TD-BIM -as desribed in Set 2 whih is apable of onsidering bodies outside the FIT volume To develop the algorithm of the hybrid method we start with the FIT and a Cartesian grid The unknowns in FIT are grid voltages defined as: ê i = Eds, ĥ i = H ds 13) L i L i As already mentioned we use the rooftop basis funtion for the TD-BIM beause the disretization for FIT leads to a path model of the surfae A onsisting of retangular elements see Fig 2) if Cartesian Coordinates are used The unknowns in the TD-BIM are proportional to the flux orthogonal to these edges With the loation in time of the magneti urrent density aording to Fig 4 this results into the following relationship between the unknown ê i of the FIT and the magneti urrent of the TD-BIM see Fig 2b): ) M Nt e 3 = e2 E Nt e3 ds = L i L i E Nt e 1 ds = ê i,n 14) t L i Now the magneti urrent density an be alulated from the eletri grid voltages by using Eq 14) The eletri urrents are loated at the same time step like the magneti urrents see Fig 4a) It is possible to loate the eletri urrents at the same time step like the magneti grid voltages and even to use a time step T BEM for the eletri urrents whih is larger than the time step given by FIT see Fig 4b) The latter dereases alulation time and improves stability beause the time step given by the FIT is extremely small 42 Time step t = N t T The alulation of the magneti grid voltages interonneting the inhomogeneous and the homogeneous region see Fig 3) with the TD-HFIE is equivalent to the use of the so alled pulse-test-funtion: H ds = H β pulse da, with β pulse = ds ds 15) rɛs S A The eletri and magneti urrent density is approximated aording to Set 2 Thus the magneti grid voltages an be alulated by ombining Eq 7) and the Fitzgeraldtransformation/duality The use of a pulse-test-funtion allows us to alulate the magneti grid voltages without the need to alulate a term proportional to 1/r 2 As seen in Fig 5a the minimum distane between boundary plane and magneti grid voltages is 05L min whih is half of the smallest spatial disretization In Fig 5a the distane between the Huygens-surfae and the magneti grid voltages is half a ellsize In our numerial example we will use one and a half ellsizes between Huygens-surfae and the grid voltages alulated by the TD-HFIE see Fig 6a) Due to the Courant ondition the minimal temporal retardation an be alulated to: τ min = L min 3 T = = 3 T > 05 T 16)
A Beker and V Hansen: A hybrid method ombining the FDTD and a time domain boundary-integral 103 Fig 6 Geometry and eletri fields at the boundary of the FIT-volume Fig 7 The eletri fields at the boundary of the FIT-volume As seen in Fig 5b the atual magneti grid voltages are not funtions of the atual eletri and magneti urrents Thus the grid voltages an be alulated by using expired urrents only 43 Time step t = N t + 05) T If the tangential magneti grid voltages are known the FITalgorithm an be ompleted in the interior and on the boundary as usual t = N t + 05) T ) Consequently, the eletri grid voltages are known in the whole interior and on the boundary With Eq 14) the magneti urrents an be alulated in a very effiient way Anyway, the FIT does not give a relationship between the magneti and the eletri fields in one ommon plane At the next time-step t = N t + 1) T ) the magneti as well as the eletri urrents at time steps t = N t + 05) T are needed The latter are alulated by applying Eq 10) at t = N t + 05) T As boundary ondition we use the eletri fields alulated by FIT proportional to the magneti flux as related by Eq 14), ie the fields alulated by the TD-IE must equal the fields alulated by the FIT Now both eletri and magneti urrents are known on the surfae A 5 Numerial results As a first numerial evaluation of the hybrid method we simulate an eletri soure sinusoidal, 500 MHz) in the interior of an air filled ube see Fig 6a) by impressing one eletri grid voltage This soure reflets all inoming waves and so it ats like a small wire The spatial and temporal disretization is determined by the Courant ondition, assuming a 11 GHz soure and 20 ells per wavelength The ube is 6 7 8-ells big Between the Huygens-surfae and the FIT-boundary we put one layer As an example we alulate one eletri grid voltage on the Huygens-surfae The result is ompared to a alulation in whih a ube with PMCboundary ondition is simulated This simulation ontains so many ells in eah diretion that waves at the boundary an be negleted The results agree well despite of the extremely small distane between boundary and soure However, after approximately 12 periods the alulation with the hybrid method beomes unstable Suh instabilities are typial for TD-BIMs and the auray and stability strongly depends on the numerial evaluation of the matrix Z and the oeffiients X k,i,j) and Y k,i,j) So it an expeted be that these instabilities an be postponed in time far enough for pratial purposes, eg by using semi-analyti integration
104 A Beker and V Hansen: A hybrid method ombining the FDTD and a time domain boundary-integral 6 Summary and outlook The presented hybrid method ombines the loal FIT and a global TD-BIM The TD-BIM ats like an absorbing boundary ondition and allows waves to pass bidiretional The method an be diretly transformed into a frequeny domain tehnique and needs no additional interpolation than the interpolations inherent in FIT The auray and stability an be improved by using semi-analyti integration instead of the implemented numerial integration The effiieny an be drastially inreased by using tehniques like the plane wave time domain algorithm Shanker, 2000) eferenes De Moerlosse, J and De Zutter, D: Surfae Integral epresentation adiation Boundary Condition for the FDTD-method, IEEE Trans AP, 41, 890 896, 1993 ao, S M: Time Domain Eletromagnetis, Aademi Press, 132, 1999 Sarkar, T K, Lee, W, and ao, S M: Analysis of Transient Sattering From Composite Arbitrarily Shaped Complex Strutures, IEEE Trans AP, 48, 1625 1634, 2000 Shanker, B, Ergin, A, Aygü, K, and Mihielssen, E: Analysis of Transient Eletromagneti Sattering Phenomena Using a Two- Level Plane Wave Time-Domain Algorithm, IEEE Trans AP, 48, 510 523, 2000 Weiland, T: Time Domain Eletromagneti Field Computation With Finite Differene Methods, International Journal of Numerial Modelling: Eletroni Networks, Devies and Fields, 9, 295 319, 1996