Finite-Element Time-Domain Solution of the Vector ... - IEEE Xplore

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Finite-Element Time-Domain Solution of the Vector Wave Equation in Doubly Dispersive Media Using Möbius Transformation Technique Ali Akbarzadeh-Sharbaf, Student Member, IEEE, and Dennis D. Giannacopoulos, Member, IEEE

Abstract—Several finite-element time-domain (FETD) formulations to model inhomogeneous and electrically/magnetically/doubly dispersive materials based on the second-order vector wave equation discretized by the Newmark- scheme are developed. In contrast to the existing formulations, which employ recursive convolution (RC) approaches, we use a Möbius transformation method to derive our new formulations. Hence, the obtained equations are not only simpler in form and easier to derive and implement, but also do not suffer from the intrinsic limitations of the RC methods in modeling arbitrary high-order media. To obtain the formulations, we first demonstrate that the update equation for the electric field strength in the mixed Crank-Nicolson (CN) FETD formulation, which is based on expanding the electric and magnetic field in terms of the edge and face elements in space and discretizing the resultant first-order differential equations using Crank-Nicolson scheme in time, is equivalent to the unconditionally stable (US) second-order vector wave equation for the same variable ( ) discretized by the Newmark- method with . In addition, we show that the update equation for the magnetic flux density in CN-FETD is the same as the second-order vector wave equation for on the dual grid discretized again by a similar Newmark- method. Subsequently, thanks to the mixed FETD formulation properties, we derive update equations for the constitutive relations using a Möbius transformation method separately. In addition, we use the shown equivalence to derive formulations based on the vector wave equation. Finally, several numerical examples are solved to validate the developed formulations. Index Terms—Dispersive media, finite-element time-domain method.

I. INTRODUCTION

I

N many practical applications, such as electromagnetic interaction with biological tissues, snow, soil, rocks, metamaterials, and optical fibers, one has to take into account effects of medium dispersion in the governing equations [1], [2]. Efforts to consider this effect in finite-difference time-domain (FDTD) modeling of Maxwell equations were initiated in 1990 [3] and since then a considerable amount of papers have been published to extend different formulations of the FDTD to dispersive media [4]–[7]. There are three main approaches Manuscript received November 20, 2012; revised February 28, 2013; accepted April 08, 2013. Date of publication April 30, 2013; date of current version July 31, 2013. This work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. The authors are with the Department of Electrical and Computer Engineering, McGill University, Montreal, QC, Canada H3A 0E9 (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2013.2260716

to obtain update equations in frequency-dependent media, namely recursive convolution (RC) [3], auxiliary differential equation (ADE) [8], [9] and -transform [10]. Most of the other approaches, such as Möbius transformation method [11], can be categorized into those groups. Finite-element time-domain (FETD) is another powerful numerical method to solve electromagnetic phenomena. There are different formulations that are divided into two main categories [12]. In the first group, the vector wave equation is directly discretized using a single type of basis functions (most often Whitney 1-form, ), which results in a semi-discrete second-order ordinary differential equation (ODE) in time. The resultant ODE then has to be solved in time using an appropriate finite-difference (FD) scheme. The most popular method is the Newmark- method [13]. The choice of results in a provably US time-marching process with second-order accuracy in time [14] (for the sake of simplicity, we call the obtained formulation NB-FETD). This approach is much more popular than the others. The second group directly discretizes coupled first-order Maxwell equations. Both electric field strength and magnetic flux density are expanded using (most often) Whitney ) basis functions, which lead 1-form and Whitney 2-form ( to two semi-discrete coupled first-order ODEs in time (usually referred as mixed FETD). The leap-frog method is the most common approach to discretize them in time, which results in a conditionally stable method (LF-FETD). This formulation , which has been is equivalent to the NB-FETD with used to develop a stable hybridization of the FDTD and FETD [15]. Recently, Movahhedi has employed the Crank-Nicolson method to discretize coupled first-order ODEs (CN-FETD) [16]. Although unconditional stability of CN-FETD is not proved, due to the complex nature of the amplification matrix, numerical experiments show that it is stable for time-steps much larger than the stability limit of the LF-FETD method. It should be noted that although the NB-FETD and the CN-FETD have the advantage of being stable for large time-steps, it is not a clear-cut advantage because: • These methods have a combination of stiffness and mass matrices on the LHS (e.g., in (5) and (12a)). As the mesh and so the LHS is refined, the condition number of increases without bound, which slows convergence of iterative solvers. However, the LHS of the conditionally whose condistable LF-FETD is solely composed of tion number remains bounded during refinement as long as the quality of the mesh is preserved; thus, it can be solved faster using iterative solvers.

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AKBARZADEH-SHARBAF AND GIANNACOPOULOS: FETD SOLUTION OF THE VECTOR WAVE EQUATION

• Increasing the time-step not only decreases accuracy, especially in highly-refined portions of the mesh, but also makes the LHS matrix more ill-conditioned. • There are other time-scales that can play a role in modeling dispersive media determined by, e.g., relaxation times and resonant frequencies. Properly capturing them place additional constraints on the time-step, which are often stricter than the Courant-Friedrichs-Lewy (CFL) limit. Hence, a time-step much larger than the CFL limit is usually avoided in the US methods. Due to the rapidly growing usage of complex materials in practical applications, there is an ever-increasing demand for numerical methods capable of accurately taking into account dispersion of the media. Unfortunately, just a few papers are dedicated to the FETD formulation in such media [17]–[19]. To the best knowledge of authors, all FETD solutions of the vector wave equation for general electromagnetic problems in dispersive media rely on convolution-based approaches. These methods not only can’t model arbitrary frequency-dependent materials, but also become cumbersome for materials described with multiple poles. Although the first problem has been alleviated using recursive fast Fourier transform (FFT) [19], the computation cost of the recursive FFT is added to the simulation. In contrast to the finite-element (FE) formulation for the vector wave equation, dispersion of the media can be easily incorporated in the mixed FETD formulations, because constitutive relations can be derived separately. In [20], Donderici and Teixeira have applied the ADE method to leap-frog mixed FETD. However, as mentioned earlier, this formulation is conditionally stable. In this paper, we show that the update equations of the electric field strength and magnetic flux density in CN-FETD are equivalent to the corresponding vector wave equation (based on and ) discretized by the Newmark- method with on the primal grid (original FEM mesh) and the dual grid, respectively. In other words, CN-FETD is a mixed (coupled) version of the NB-FETD formulations. Subsequently, we derive a CN-FETD formulation in dispersive media using a Möbius transformation method, which is free from the aforementioned limitations of convolution-based methods. Afterwards, having utilized the proven equivalence, several FETD formulations based on the US vector wave equation are derived for general linear electrically/magnetically/doubly dispersive media. Furthermore, the developed formulations can be utilized to extend the US hybrid FETD-FDTD [21] to dispersive media. The rest of the paper is organized as follows. First, we briefly review different FETD formulations. Then, equivalence between CN-FETD and NB-FETD is revealed. Next, we develop update equations for the constitutive relations using Möbius transformation method. It is followed by deriving the FETD formulation in dispersive media based on the vector wave equation using the existing equivalence. Finally, several numerical examples are presented to show validity of the proposed formulations. II. FINITE-ELEMENT FORMULATIONS IN THE TIME-DOMAIN A. Vector Wave Equation The vector wave equation for the time-dependent electric field strength in a source-free medium is (1)

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Through a Galerkin testing procedure and expanding the electric field strength in terms of edge elements, the semi-discrete finiteelement solution in the time-domain can be obtained as (2) where

and (3) (4)

are mass and stiffness matrices, respectively. To obtain a timemarching process, we discretize (2) in time using the Newmark- integration technique [13]

(5) in which results in unconditional stability and highest possible order of accuracy [14]. B. Coupled First-Order Maxwell Equations The other approach involves directly solving two coupled curl equations. In this approach, electric field strength is expanded in terms of edge elements and magnetic flux density is expressed in terms of face elements. By denoting another unknown vector as , the semi-discrete formulation can be worked out as (6a) (6b) where

and represent the electric flux density and the magnetic field strength, respectively. The matrix is a sparse rectangular incident matrix consisting of non-zero entries of and (7)

Using the leap-frog method to obtain a fully discrete form of (6) is the most popular approach. In this manner, an explicit and conditionally stable formulation with the stability criteria of [22] (8)

denotes the spectral radius of . Howis obtained where ever, due to the unstructured and complex nature of the FE mesh in real-life problems, an US method is highly desired. The Crank-Nicolson method is a widely used implicit scheme in numerical analysis [23]. It has been successfully applied to develop several FDTD formulations in electromagnetics and has been proven to be US [24]–[26]. Recently, the CN method has

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been employed to discretize (6) (CN-FETD) [16]. Although unconditional stability of CN-FETD is not proved, numerical experiments show that it is stable for time-steps much larger than the stability limit of the leap-frog method indicated by (8). To derive CN-FETD, we discretize the coupled curl (6a) by applying the CN method

. Comparing (13) with (5) reveals that they are exactly the same. Hence, the update equation for the electric field strength in CN-FETD is exactly the same as the vector wave equation for discretized by the Newmarkwith , which is US. Following the same procedure, from (11a)-(11b) to reach an update process we eliminate for . Denoting , we obtain

(9) to them, which yields (14)

(10a) (10b) and Substituting constitutive relations into the above equations, one obtains (11a) (11b) Replacing

in (11a) with (11b), we reach (12a)

(12b) (12c) which involves an implicit update process for and an ex. Similarly, one can replace in (11b) plicit one for with (11a) to obtain an implicit update formulation for . However, the latter formulation is not computationally efficient because, has to be calculated explicitly, which is not generally a sparse matrix. C. Equivalence Between CN and Newmark- Methods in FETD Consider (11a)-(11b) again. By subtracting (11a) in the th th time-step, the time-step from (11a) in the term appears on the RHS. The same term can be reached on the LHS of (11b) by summing it in the th and the th time-steps. Eliminating this term, one has

(13) where In addition, we know that are face elements connected to edge . So, we can write

Apparently, the LHS is the discretized form of using the central difference method and the RHS is the convex combination of during three consecutive time-steps. Hence, (14) discretized by Newmark- method is with . This equation is known as the dual FEM formulation (in contrast to (2) known as the primal FEM formulation) in which and can be considered as dual forms of the conventional mass and stiffness matrices, respectively [27]. It’s worth mentioning that regardless of the shown equivalence, there are still some differences between the CN-FETD and the NB-FETD formulations. For example, (1) supports a non-physical pure-gradient solution of the form in a source-free region where denotes a scalar potential and and are constants, which result in a late-time linear growth in the solution. This non-trivial solution can not be avoided in the discrete case since the cumulative round-off errors during the time-marching process, due to the finite precision of the machine and iterative algorithms, can produce non-zero components in the space of such spurious solutions. However, it is not often a problem because it tends to appear after very late times. Nonetheless, in very long simulations, such as low-frequency problems or structures with very sharp resonances, a remedy should be pursued to prevent exciting such spurious solutions. These solutions could be avoided only if proper initial/boundary conditions are utilized and exact arithmetic is assumed. Although these conditions can not usually be satisfied in practice, it could be delayed by increasing the termination criteria of the iterative solvers at the expense of more computational cost. Fortunately, mixed FETD formulations, actually the Maxwell curl equations, have the advantage of being free from these spurious solutions and can be considered as a suitable substitute for the NB-FETD in long time simulations. Some other approaches to alleviate late-time linear growth in the NB-FETD can be found in [28]–[30]. All in all, we can conclude that CN-FETD is nothing but a mixed version of the primal and dual FEM formulations discretized by Newmark- method with . This may explain the US behaviour of the CN-FETD in numerical studies. III. DISCRETIZATION OF CONSTITUTIVE RELATIONS DISPERSIVE MEDIA

IN

Consider a 3D computational domain consisting of distinct regions with different material properties ( ) which are constant within each element. In addition, we assume that the permittivity and the permeability of

AKBARZADEH-SHARBAF AND GIANNACOPOULOS: FETD SOLUTION OF THE VECTOR WAVE EQUATION

each region ( ) can be written in the following general form in the Laplace domain

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is the where updated using

th auxiliary variable in region

and is

(15) (21) which are quite general and can take into account any linear dispersive material. In the following sections we first derive appropriate expressions to update constitutive relations of the electric field and magnetic field using Möbius transformation. Then, they are substituted into discretized coupled curl (10a)-(10b) to reach the final formulations for modeling electrically/magnetically/doubly dispersive materials. A. Update Equation for Electric Field Constitutive Relation Since material properties are constant in each region, we can rewrite the constitutive relation for the electric field in the following form

Denoting

and , one can rewrite (20) in the following

compact form (22) B. Update Equation for Magnetic Field Constitutive Relation Following a similar procedure, the permeability of the th medium can be written as follows in the -domain (23)

(16) where in region

is the assembled mass matrix of elements residing

(17) is the complex permittivity in that region. The same and approach can be utilized to model an anisotropic medium. In this case, we should replace with where and . term should reduce to in non-disObviously, the persive medium. For the sake of simplicity, we introduce auxiliary variables and . Now, we map the latter one onto the -domain using Möbius transformation [11]. The appropriate Möbius transformation consistent with the CN method (9) can be easily obtained using -transform as

and are related to and where coefficients ( , and ). Having defined auxiliary variables and in which , we can write the update equation in the time-domain as (24) or briefly (25) in which and

are

, auxiliary

variables

updated using

(18) (26)

substituting (18) into (15) results in the following permittivity in the -domain (19) where the coefficients ( , Using

and

are related to and and ). , the difference form of can be reached simply. However, to implement it more efficiently, we utilize an implementation approach named transposed direct form II, which is common in digital filters [11], [31] (20)

IV. FETD FORMULATIONS IN DISPERSIVE MEDIA Having derived update equations for constitutive relations in dispersive media, one can obtain FETD formulations by plugging them into discretized Maxwell equations. There are two possible strategies: 1) substituting them into discretized coupled curl (10a)-(10b) and solving them directly; 2) taking advantage of equivalence between CN-FETD and conventional FETD based on vector wave equation shown in Section II-C to reach a formulation for the dispersive media based on the curl-curl equation and discretized by the Newmark- method. We adopt the latter approach here, because the resultant formulation is more widely used than the other one.

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A. Electrically Dispersive Media

C. Doubly Dispersive Media

For the sake of simplicity, we imagine that permeability across the computational domain is uniform and constant, so the update equation for magnetic field constitutive relation can be written as where . We substitute the above-mentioned update equation and (22) into (10a)-(10b). Eliminating the magnetic flux density from the resultant equations, one can reach the following equation for electrically dispersive media

In a similar manner, using (22) and (25) as update equations for constitutive relations of the electric field and magnetic field and eliminating magnetic flux density, we obtain

(30)

(27)

Equations (12b), (21) and (26) have to be used together with this one to reach a working formulation in doubly dispersive media. Finally, if we eliminate electric field strength, we obtain the following equation for doubly dispersive materials based on

This equation together with (21) forms a complete update process for the electric field strength. B. Magnetically Dispersive Media Following a similar procedure, we obtain an equation to update the electric field strength in magnetically dispersive media, which is (31) where and . This equation has to be used with (11a), (21) and (26) to reach a complete time-stepping. (28) where . Clearly, we need values of the magnetic flux density at each time-step. So, (28) has to be used along with (12b) and (26) to form a complete update process. Alternatively, we can eliminate electric field strength, instead of magnetic flux density, from the resultant CN-FETD formulation. In this way, we obtain an update equation based on as

V. NUMERICAL RESULTS In order to verify the formulations described earlier, we present several numerical examples based on two test systems. The first one is wave transmission through a 2D metamaterial slab loaded with dielectric defects and the other one involves computing reflection and transmission coefficients of a dispersive dielectric slab. A. Matched Impedance Zero-Index Material Matched impedance zero-index materials (MIZIMs) are metamaterials in which both and vanish over a narrow frequency range [32], [33]. Consider Fig. 1 in which a MIZIM slab is located inside a free-space domain [34]. The MIZIM is characterized by a doubly dispersive Drude model as: (32)

(29) and . In conwhere trast to the previous one, this formulation does not need values of the electric field strength during the update process and only has to be accompanied with (26) to complete time-stepping.

is the plasma frequency and is related to the mean where free path. It has been shown that by producing an arbitrary number of homogeneously-filled circular defects anywhere inside the MIZIM with appropriate radii ( ) and constitutive properties ( and ), one can obtain total reflection or total transmission through the MIZIM. To have a total reflection or transmission behaviour, the radii and constitutive values of each

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Fig. 1. Plane wave impinges normally on a 2D MIZIM loaded with arbitrary number of circular defects. Appropriate boundary conditions are assigned to make plane wave propagation possible. Sampling points for electric field strength are marked with . Fig. 3. Electric field strength recorded during simulation in Region 1 and 2. After transient response, two samples match with each other very well.

Fig. 2. Magnetic field strength inside computational domain in steady state at . Plane wave has penetrated into the Region 2 with no significant attenuation.

defect have to satisfy or condition, respectively. In our numerical example, a 4 cm-wide MIZIM is placed in the middle of a computational domain with 12 cm width and 6 cm height. Four defects with radii of , 7.8 mm, 6.3 mm and 9.2 mm are selected. The relative permittivity of the defects in the total transmission case can be obtained as , 8.19, 3.74 and 12.37 and in the total reflection case as , 5.07, 1.47 and 8.95. The permeability of in all cases. The parameters of the the defects is equal to MIZIM are selected as and . The plane wave is excited by a sine source with a raised cosine ramp [35] and frequency of . Average edge length in the mesh is . Fig. 2 shows a snapshot of the magnetic field taken at . As can be seen, the magnetic field amplitude in Region 2 is as strong as Region 1, which demonstrates total transmission through the MIZIM layer. To investigate it more accurately, we have recorded electric field strength on the left and right walls (indicated in Fig. 1). As depicted in Fig. 3, after reaching steady state, no considerable difference between two samples can be observed. It should be noted that both samples are in phase, because the distance between sampling points is . A similar snapshot has been taken for the total reflection (see Fig. 4). As expected, the magnetic case at

Fig. 4. Magnetic field strength inside computational domain in steady state at . Magnetic field strength in Region 2 is very close to zero.

Fig. 5. Electric field strength recorded during simulation in Region 1 and 2. Amplitude of the electric field strength in Region 2 is almost negligible.

field amplitude in Region 2 is approximately zero. The recorded electric field strength shown in Fig. 5 verifies this result.

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Fig. 6. Plane wave normally incident on an infinitely large dispersive dielectric slab. Three different dispersive materials are considered in this example, namely a magnetically dispersive slab characterized by three Debye poles, an electrically dispersive slab characterized by two Lorentz pole-pairs and a doubly dispersive slab whose and are characterized by two Lorentz pole-pairs.

B. Reflection and Transmission Coefficients of A Dispersive Dielectric Slab

Fig. 7. Numerical and exact values of the reflection and transmission coefficients amplitude of a magnetically dispersive dielectric slab characterized by three Debye poles.

We have computed reflection and transmission coefficients, represented by and , of a dispersive dielectric slab. The slab is infinitely large and uniform with the thickness of 5 cm as shown in Fig. 6. To truncate the computational domain, we have applied the perfect electric conductor (PEC) boundary condition (BC) on the two walls perpendicular to the electric field and the two other side surfaces parallel to the electric field satisfy the perfect magnetic conductor (PMC) BC. The waveguide is excited by a plane wave and the other end is terminated by a first-order absorbing boundary condition (ABC). The time-step size is given by in which is the shortest edge length of the mesh and is the speed of light in vacuum. Here, this time-step is more than 4 times greater that the restriction enforced by (8). For the first example, we have considered a mag) characterized by three netically dispersive slab ( Debye poles and a static magnetic conductivity represented by (33) where the parameters are given by

Fig. 8. Numerical and exact values of the reflection and transmission coefficients amplitude of an electrically dispersive dielectric slab characterized by a fourth-order Lorentz model.

with the following parameters Fig. 7 shows computed absolute values of reflection and transmission coefficients along with the exact solutions. The calculated relative error is less than 1.8% in the considered frequency range (0.03–5 GHz) for both coefficients. This error is mainly enforced by the low-frequency components, which need much longer simulation to get resolved. Away from those lower frequencies, e.g., 0.2–5 GHz, the error drops to less than 0.05%. In the second example, the following model with two Lorentz pole-pairs is considered to describe the permittivity of the slab ( )

(34)

Numerical and analytical results for the electrically dispersive slab can be found in Fig. 8. The obtained relative accuracy is better than 0.7% and 0.06% in the frequency range of 0.03–5 GHz for reflection and transmission coefficients, respectively. Finally, the previous example is extended to a doubly dispersive one with the same permittivity. We have considered the same model for permeability with the parameters defined as follows:

AKBARZADEH-SHARBAF AND GIANNACOPOULOS: FETD SOLUTION OF THE VECTOR WAVE EQUATION

Fig. 9. Numerical and exact values of the reflection and transmission coefficients amplitude of a doubly dispersive dielectric slab. Both permittivity and permeability are characterized by a fourth-order Lorentz model.

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Fig. 11. Absolute error for the transmission coefficient of the doubly dispersive dielectric slab characterized by two Lorentz pole-pairs.

the equivalence between CN-FETD and NB-FETD formulations that are expanded in space using Whitney forms. Although our formulations are based on the Möbius transformations technique, other approaches like ADE or -transform are completely applicable. Hence, the developed formulations are more flexible and easier to implement than RC-based methods introduced in previous papers. They have been applied to several numerical examples and highly accurate results have been obtained. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their constructive suggestions and useful comments. REFERENCES

Fig. 10. Absolute error for the reflection coefficient of the doubly dispersive dielectric slab characterized by two Lorentz pole-pairs.

As shown in Fig. 9, this media exhibits extremely sharp resonances. However, numerical results match excellently with the exact solutions. Since the values of or are extremely close to zero for some frequencies, their absolute errors are plotted in Figs. 10 and 11 within 0.03–5 GHz, respectively. As expected, in both cases the worst accuracy is obtained in the low-frequency region and frequencies around points with sharp resonances or rapid changes. The worst obtained error for reflection and transmission coefficients is 7.3 and 0.01, respectively. Except for those regions, the absolute error is less than in both cases. More accurate results can be obtained during a longer simulation. VI. CONCLUSION In this paper, we have described several novel FETD formulations based on the second-order vector wave equation to take into account dispersion of the medium. The key point is

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Ali Akbarzadeh-Sharbaf (S’09) received the M.Sc. degree in electrical engineering from Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran in 2011. He is currently working toward the Ph.D. degree at the Computational Analysis and Design Laboratory (CAD Lab), department of electrical and computer engineering, McGill University. Mr. Akbarzadeh-Sharbaf is a recipient of the Iran Telecommunication Research Center (ITRC) grant to support his M.Sc. thesis. He is also a recipient of the McGill Engineering Doctoral Award (MEDA) and the Eric. L. Adler fellowship in electrical engineering, McGill University. His research interests include computational electromagnetics, especially differential-based techniques.

Dennis D. Giannacopoulos (’90–M’92) received the B.Eng.and Ph.D. degrees in electrical engineering from McGill University, Montreal, QC, Canada, in 1992 and 1999, respectively. He has been with the Department of Electrical and Computer Engineering at McGill University since 2000, where he is currently an Associate Professor and a member of the Computational Electromagnetics Group. He has been the recipient of his department’s Professor of the Year Award twice. His research interests include adaptive finite element analysis for electromagnetics and the acceleration of computational electromagnetics algorithms on emerging parallel architectures. He has authored or coauthored more than 90 referred journal and conference publications. His students have received three best-paper/presentation awards at international conferences and symposia. His research has been sponsored by the Natural Sciences and Engineering Research Council of Canada (NSERC), the Fonds de recherche du Québec – Nature et technologies (FQRNT), and the Canada Foundation for Innovation (CFI). He has served on the editorial boards and technical program committees of several major international conferences and served as Co-Chair of the editorial board for the 14th Conference on the Computation of Electromagnetic Fields. He is a member of the IEEE, the International Compumag Society, and the Ordre des Ingénieurs du Québec.