Mixed Finite-Element Method for Resonant Cavity Problem With ...

IEEE TRANSACTIONS ON MAGNETICS, VOL. 52, NO. 2, FEBRUARY 2016

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Mixed Finite-Element Method for Resonant Cavity Problem With Complex Geometric Topology and Anisotropic Lossless Media Wei Jiang1 , Na Liu1 , Yongqing Yue1 , and Qing Huo Liu2 , Fellow, IEEE 1 Department

of Electronic Science, Institute of Electromagnetics and Acoustics, Xiamen University, Xiamen 361005, China of Electrical and Computer Engineering, Duke University, Durham, NC 27708 USA

2 Department

Electromagnetic eigenvalue problems are contaminated by nonphysical zero modes in the conventional finite-element method (FEM) with edge elements. Here, we investigate the cavities with anisotropic lossless media, complex geometry structure, and perfect electric conductor (PEC) walls and eliminate all nonphysical zero and nonzero modes successfully. We introduce a Lagrangian multiplier to deal with the constraint of divergence-free condition. Our method is based on the mixed FEM employing the first-order edge basis functions to expand electric field and linear element basis functions to expand Lagrangian multiplier. The validity of our method is confirmed by several numerical experiments. Meanwhile, the numerical experiments show that when the cavity has a connected boundary, there is no physical zero mode; when the cavity has several disconnected boundaries, then the number of physical zero modes is equal to one less than the number of disconnected PEC boundaries. Index Terms— Edge element, Maxwell eigenvalues, mixed finite-element, spurious modes.

I. I NTRODUCTION

I

T IS well known that a resonant cavity is one of the most important microwave devices, and the resonant frequency is the key parameter of the cavity. When the cavity is filled with an inhomogeneous medium and/or has complicated boundary geometry, numerical solution is required to find its resonant frequencies. This usually involves the computation of the eigenvalue problem about the curl–curl operator [1], [2]. For the application of the rectangular cavity resonators, Lin et al. [3] investigate complex permittivity and permeability of several commercially available yttrium iron garnet and nickel ferrite ceramics in the demagnetized state. Farahani and Konrad [4] introduce an improved finite difference time-domain method to obtain the resonant frequencies of a rectangular cavity filled with magnetic material. For using the numerical method to solve the resonant cavity problem, spurious modes will appear if the numerical solution cannot preserve the actual physical properties of the field. In the finite-element method (FEM) for electromagnetics, usually, the spurious modes can be classified into two kinds, i.e., spurious nonzero modes and zero modes. How to eliminate the two kinds of spurious modes is a very important problem in computational electromagnetics. To solve the waveguide problems filled with homogeneous isotropic medium, Silvester [5] applied the FEM with nodal basis functions to analyze the waveguide modes. Recently, Gomez-Revuelto et al. [6] proposed an automatic self-adaptive hp-version FEM for the characterization of waveguide discontinuities. For cavity problems filled with homogeneous isotropic medium, Yang et al. [7] adopted the two-scale discretization scheme of P2 − P1 element to obtain high-accuracy numerical solution without spurious modes, and the error analysis for the numerical solution is Manuscript received March 12, 2015; revised July 4, 2015; accepted August 20, 2015. Date of publication August 24, 2015; date of current version January 18, 2016. Corresponding author: Q. H. Liu (e-mail: [email protected]). Digital Object Identifier 10.1109/TMAG.2015.2472366

also provided. From the electromagnetic theory [8], we know that electric field E has the property of tangential continuity on the interface between two different media, but its normal component on this interface is usually discontinuous. For the problem of waveguide or cavity filled with an inhomogeneous medium, Csendes and Silvester [9] used the nodal basis function to discretize the electric field E, and they immediately found that there were some first kind of spurious modes in the numerical results. The main reason is that the nodal basis function not only has the tangential continuity but also keeps the normal continuity that, however, is actually not true for the behavior of electric field E in inhomogeneous media. Now, we have already known that the vector edge basis function introduced in [10] and [11] satisfies the actual physical property of the electric field E. Moreover, the vector edge basis function facilitates outer boundary conditions on the tangential components. Therefore, it is very suitable to employ the vector edge basis functions to expand electric field E. Wang and Ida [12] have used linear tetrahedral edge elements and curvilinear hexahedral edge elements to compute the eigenmodes in arbitrarily shaped electromagnetic cavities loaded with an anisotropic material. Nevertheless, it is found that it exists spurious zero modes. On the other hand, how to constrain the divergencefree condition is also an important problem. Rahman and Davies [13], Winkler and Davies [14], and Kobelansky and Webb [15] suggested a penalty function to enforce the divergence-free condition. Unfortunately, this method cannot completely eliminate all the spurious modes and leaves the task of selecting a parameter to users, although this method can make these spurious eigenmodes shift far into the visible spectrum. In recent years, Venkatarayalu and Lee [16] presented a novel method to deal with the constraint of divergence-free condition and to eliminate all the spurious modes successfully. The mixed FEM (MFEM) is an active branch in the FEM, which has become a very important numerical method for partial differential equations with constraint

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condition. In the MFEM, we usually need to construct two finite-element spaces, which is a main characteristic of MFEM. For the cavity problem with homogeneous isotropic medium and perfect electric conductor (PEC) boundary, it is well known that the traditional edge element method will lead to second kind of spurious modes [17] due to the dropping of divergence-free condition, and Boffi [18] provides numerical results to support this viewpoint and points out that the number of nonphysical zero modes is equal to the number of interior nodes of the computational mesh. Recently, Coyle and Ledger [19] present some numerical examples using the hp-version FEM, their numerical results show that the hp-version FEM has the property of exponential rates of convergence. The method in [19] is based on the MFEM. Liu et al. [20] use the mixed spectral-element method (SEM) to solve the 3-D Maxwell’s eigenvalue problem, and their method can suppress all nonphysical eigenvalues. Because SEM usually employs hexahedral partition, it is not good to approximate the complex geometry. Hence, this paper will adopt tetrahedral partition. In the community of computational mathematics, the vector Maxwell’s eigenvalue problem also arouses mathematical researchers’ interest. For the vector Maxwell’s eigenvalue problem with the isotropic medium, Levillain [21] has proved that under certain conditions, numerical eigenvalues have a second-order rate of convergence using the MFEM, and the zero modes exist in the cavity problem with a disconnected boundary. Boffi et al. [22] addressed some difficulties which arise in computing the eigenvalues of Maxwell’s system by the FEM and proposed a criterion to establish whether or not a finite-element scheme is well suited to approximate the eigensolutions. Hiptmair [23] pointed out that there is a close relationship between the second-order elliptic equations and the governing equations of electromagnetism, but the property of lacking of strong ellipticity introduces subtle new challenges. Moreover, applying the identity ∇ × ∇ × F = ∇(∇ · F) − ∇ 2 F, there is a definite relationship between the curl–curl operator and the Laplace operator when the medium is homogeneous and isotropic. To the best of our knowledge, much less is written about the numerical treatment without any spurious mode of cavity problem with anisotropic lossless media and complex geometry structure. Inspired by the above work, this paper will carry out the research about this aspect. We adopt the first-order edge element basis functions to expand the electric field E and employ the linear nodal basis functions to expand the Lagrangian multiplier, so that Gauss’ law is satisfied to remove the spurious zero modes. Numerical results show that the proposed method in this paper can eliminate all the nonphysical spurious modes. The outline of this paper is as follows. In Section II, we introduce some variational formulations of the 3-D resonant cavity problem with anisotropic lossless medium. The first-order edge element space, linear finiteelement space, and the forms of corresponding basis functions are given in Section III. Finally, we carry out some numerical experiments to verify that the MFEM can effectively suppress all the spurious modes.


II. R ESONANT C AVITY P ROBLEM IN E LECTROMAGNETISM In this section, we introduce the common 3-D resonant cavity problem in electromagnetics. It can be deduced from Maxwell’s system. In this paper, we only deal with the cavity problem with anisotropic lossless media and PEC boundary, and this method can also be extended to the case with other boundary conditions. Let ⊂ R3 be a connected, bounded domain with a Lipschitz-continuous boundary ∂. Let nˆ be the outward normal unit vector on ∂. The boundary ∂ may be disconnected. We usually denote by E : → C3 and H : → C3 , the electric field phasor and magnetic field phasor, respectively. The corresponding instantaneous fields are Re(Ee j ωt ) and Re(He j ωt ) with implied time-harmonic factor e j ωt , and ω is the angular frequency. Let 0 and μ0 be permittivity and permeability in the vacuum, respectively. Let r and μr be full 3 × 3 tensors for relative permittivity and permeability of the medium, respectively. Moreover, we assume that the medium in the resonant cavity is anisotropic, lossless, and piecewise homogeneous. In this case, r and μr in each region are two second-order positive Hermitian tensors †

†

r = r , μr = μr

(1)

and ξ † r ξ > 0 ∀ 0 = ξ ∈ C3

(2)

ξ † μr ξ > 0 ∀ 0 = ξ ∈ C3

(3)

where K † is conjugate transpose of the complex matrix K . Note that (1) is the condition for anisotropic lossless medium [24], and (2)–(3) are valid for the types of lossless material in the nature [8]. By writing the source-free Maxwell’s equations for vector phasors, we obtain ∇ × E = − j ωμ0μr H in ∇ · ( r E) = 0 in

(4a) (4b)

∇ × H = j ω0 r E in

(4c)

∇ · (μr H) = 0 in nˆ × E = 0 on ∂

(4d) (4e)

nˆ · (μr H) = 0 on ∂.

(4f)

Our aim is to find a triplet (ω, E, H) with E = 0 and H = 0, such that the Maxwell’s system (4) holds. Note that when is simply connected, this cavity has no physical zero mode, i.e., ω = 0 is not a solution of this problem. As a matter of fact, if ω = 0 is a solution of problem (4), then this problem only has the trivial solution. According to (4a), we arrive at ∇ × E = 0. Since is simply connected, we can write E = −∇ϕ. Substituting it in (4b) and (4e), we obtain −∇ · ( r ∇ϕ) = 0, and ϕ is a constant on ∂; from this and (2), we can conclude that ϕ is a constant in , thus E = 0 in . Similarly one can also obtain H = 0. This is also confirmed by our numerical experiments. However, when ∂ is disconnected, then is not simply connected. We cannot deduce E = −∇ϕ in the whole domain from ∇ × E = 0, because the corresponding cohomology

JIANG et al.: MFEM FOR RESONANT CAVITY PROBLEM

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is not trivial. Therefore, the presence of ω = 0 is inevitable, and this is the case in the cavity with multiple conductors. An effective method for solving Maxwell’s system (4) is to reduce the first-order Maxwell’s system (4) to a second-order system by eliminating a field variable. For instance, we eliminate the magnetic field H and then get 3-D vector Maxwell’s eigenvalue problem for the electric field E: Seek ∈ R (eigenvalue) and E = 0 (eigenfunction), such that −1 (5a) ∇ × μr ∇ × E = r E in ∇ · r E = 0 in (5b) nˆ × E = 0 on ∂ (5c)

numerical eigenfunction with the norm · b in our numerical experiments. Using Green’s formula, we get the variational form of problem (5): find ∈ R and 0 = E ∈ Q, such that

where = ω2 0 μ0 is the square of wavenumber. As usual, we need to introduce some complex Hilbert spaces in the FEM associated with problem (5) 2 2 | f (x, y, z)| d x d ydz < +∞ L () = f :

(7)

H () = {v ∈ L 2 () : ∇v ∈ (L 2 ())3 } 1

H01() = {v ∈ H 1(): v|∂ = 0} H(curl, ) = {F ∈ (L 2 ())3 : ∇ × F ∈ (L 2 ())3 } H0 (curl, ) = {F ∈ H(curl, ) : nˆ × F|∂ = 0} H(div, , r ) = {F ∈ (L 2 ())3 : ∇ · ( r F) ∈ L 2 ()} H(div0 , , r ) = {F ∈ H(div, , r ) : ∇ · ( r F) = 0}. The usual norms in these spaces are defined as follows: v 0 =

2

|v(x, y, z)| d x d ydz

12

∀ v ∈ (L 2 ())d

1 v 1 = v 20 + ∇v 20 2 ∀ v ∈ H 1() 1 F curl = F 20 + ∇ × F 20 2 ∀ F ∈ H(curl, ) 1 F div,r = F 20 + ∇ · ( r F) 20 2 ∀ F ∈ H(div, , r ) where · 0 stands for the usual norm of (L 2 ())d (d = 1, 3). Denote Q = H0 (curl, ) ∩ H(div0 , , r ). Denote the continuous sesquilinear form a : H0 (curl, ) × H0 (curl, ) → C (E, F) →

−1

μr ∇ × E · ∇ × F∗ d x d ydz

b : (L 2 ())3 × (L 2 ())3 → C (E, F) →

r E · F∗ d x d ydz

c : H0 (curl, ) × H01() → C (E, q) → r E · ∇q ∗ d x d ydz

where ∗ denotes the complex conjugate. According to the above notations, it is easy to know that b(·, ·) is also an inner product in L 2 ()3 . Therefore, we can define the norm F b = (b(F, F))1/2 , which is equivalent to the usual norm · 0 in L 2 ()3 . We will normalize the

a(E, F) = b(E, F) ∀ F ∈ Q.

(6)

It is very difficult to construct a conforming finite-element space of the Hilbert space Q because of the constraint of the divergence-free condition (5b). Therefore, the conforming finite-element discretization for variational form (6) is difficult (see [13], [25], [26]). If we ignore the constraint (5b), then we will get the following variational form: find ∈ R, E ∈ H0 (curl, ), and E b = 1, such that: a(E, F) = b(E, F) ∀ F ∈ H0 (curl, ).

However, in this way, = 0 will be added to the spectrum of the operator corresponding to the variational form (6), and the corresponding eigenspace is ∇ H01(), which is an infinite-dimensional space. Obviously, these zero modes are nonphysical modes (denoted as nonphysical zero modes) and do not satisfy the divergence-free condition. On the other hand, the eigenmodes associated with = 0 remain unchanged. The conventional FEM with the edge element method (including those in powerful commerical software, such as COMSOL Multiphysics and High Frequency Structure Simulator) deals with the 3-D cavity problem (5) using variational form (7), which results in the presence of nonphysical zero modes. The number of these spurious modes is equal to the number of internal nodes of the mesh [2]. Therefore, the numerical treatment for the resonant cavity problem using the conventional edge element method is not perfect. We try to remedy this disadvantage in this work for anisotropic lossless media. When 3-D resonant cavity problem (5) does have several actual physical zero eigenvalues or the eigenvalues are very close to zero, then it is very difficult to distinguish the actual physical zero eigenvalues from all the numerical eigenvalues which converge to zero [27]. The presence of physical zero modes relates to topological structure of the cavity. From our numerical results, we can obtain the following well-known conclusions: when ∂ is connected, then there is no physical zero mode; when ∂ is not connected, exists physical zero mode. Assume that N then it

j = ∅(i = j ), then the number ∂ = k=1 k , i of these physical zero modes is N − 1. When the medium in the cavity is isotropic, this conclusion has been proven in [21]. For the anisotropic and lossless medium, this conclusion can also be proven by modifying the process of proof in [21] sightly. Hence, if N > 1, then actual physical zero modes and nonphysical zero modes obtained by the traditional edge element method will mix together. Eliminating these nonphysical zero modes is very significant and important. Fortunately, we can successfully suppress these nonphysical zero modes by employing the MFEM [26]. Moreover, our method is computationally more efficient than the conventional edge element method if the eigenvalues close to zero are required.

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First, we introduce the following variational form: seek ∈ R, E ∈ H0 (curl, ), and E b = 1, such that: a(E, F) = b(E, F) ∀ F ∈ H0 (curl, )

(8a)

c(E, q) = 0 ∀ q ∈

(8b)

H01().

It is clear that the variational form (8) is equivalent to the original problem (5). We must solve variational form (8a) under the constraint of (8b). Based on [28], we try to change the variation form (8) to the mixed variational form by means of a Lagrangian multiplier. Let us now introduce the mixed variational form: seek ∈ R, E ∈ H0 (curl, ), E b = 1, and p ∈ H01(), such that a(E, F) + c∗ (F, p) = b(E, F) ∀ F ∈ H0 (curl, ) c(E, q) = 0 ∀ q ∈ H01()

(9a) (9b)

where p is the Lagrangian multiplier, and c∗ (F, p) stands for the complex conjugation of the continuous sesquilinear form c(F, p). The weak form (9) is a saddle point problem in the finite-element analysis. Now, we prove the equivalence between the variational form (8) and the variational form (9). Obviously, any eigenpair of (8) with p = 0 satisfies (9). Conversely, by taking F = ∇ p in (9a) and q = p in (9b), we get c∗ (∇ p, p) = b(E, ∇ p) = c(E, p) = 0. From (2), we can deduce that ∇ p = 0. According to Poincaré inequality p 1 ≤ C ∇ p 0 [29] in H01(), then p = 0, which shows that any eigenpair of the variational form (9) satisfies (8) as well. In Section III, we utilize the MFEM to discretize the mixed variational problem (9). III. M IXED F INITE -E LEMENT D ISCRETIZATION The tetrahedral elements are simplex in 3-D and they can approximate the curved objects perfectly, thus we only consider the finite-element spaces in this mesh. Let Jh be a regular tetrahedral partition [29] of with mesh parameter h. The usual definition of the linear element space S h is S h = {φ ∈ H 1() : φ| K ∈ P1 (K ) ∀K ∈ Jh } where P1 (K ) is the first-order polynomial space on K . Set S0h = S h ∩ H01(). We know that S0h is a finite-element subspace of H01(), which is a scalar function space. The first-order edge element on tetrahedral mesh Jh is usually defined as follows. Definition: The first-order edge element is a triple (K , P, N ) having the following properties. 1) K is a tetrahedron (element shape). 2) P = {F : F = α + β × r } (shape function), where α and β are two constant vectors.

3) N = li F · τi ds, 1 ≤ i ≤ 6 (degree of freedom), where τi is the tangential unit vector of edge li . Based on the above definition, we can derive the concrete expressions of six local edge basis functions [23], [30] on K ,

Fig. 1. Local nodal numbering for the element K and the local reference direction for the edge is chosen by means of local nodal numbering.

that is N1K = ϕ1K ∇ϕ2K − ϕ2K ∇ϕ1K , N2K = ϕ2K ∇ϕ3K − ϕ3K ∇ϕ2K N3K = ϕ1K ∇ϕ3K − ϕ3K ∇ϕ1K , N4K = ϕ3K ∇ϕ4K − ϕ4K ∇ϕ3K N5K = ϕ1K ∇ϕ4K − ϕ4K ∇ϕ1K , N6K = ϕ2K ∇ϕ4K − ϕ4K ∇ϕ2K where ϕiK (1 ≤ i ≤ 4) are the four local nodal basis functions on the element K . Fig. 1 shows the local numbering of the nodes and edges of a tetrahedron. The definition of the first-order edge element space is

V h = F ∈ H(curl, ) : F| K ∈ span N Kj , j = 1, 2, . . . , 6 for ∀ K ∈ Jh . We know that the vector function in V h has tangential continuity on the edge between two adjoint elements. Set V0h = V h ∩ H0 (curl, ). We employ the first-order edge element space V0h and the linear element space S0h to approximate the space H0 (curl, ) and H01(), respectively. Restricting (9) on V0h × S0h , we get the discrete mixed variational form: seek h ∈ R, Eh ∈ V0h , Eh b = 1, and ph ∈ S0h , such that a(Eh , F) + c∗ (F, ph ) = h b(Eh , F) ∀ F ∈ V0h c(Eh , q) = 0 ∀ q ∈

S0h .

(10a) (10b)

Scheme (10) can be expressed in a matrix form. Let n be the number of interior edges and m be the number of interior nodes in tetrahedral mesh. Then, V0h and S0h can be expressed as V0h = span{N1 , N2 , . . . , Nn }, S0h = span{ψ1 , ψ2 , . . . , ψm } where Nk (k = 1, 2, . . . , n) are the global basis functions in V0h , and ψk (k = 1, 2, . . . , m) are the global basis functions in S0h . Since Eh ∈ V0h and ph ∈ S0h , we write Eh =

n k=1

ξk Nk , ph =

m

ζk ψk .

(11)

k=1

Substituting (11) into (10) and taking F = Nk (k = 1, 2, . . . , n), q = ψk (k = 1, 2, . . . , m), we obtain the generalized matrix eigenvalue problem A T ξ M O ξ = (12) h ζ O O ζ T† O


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TABLE I F IRST F IVE S MALLEST E IGENVALUES F ROM A R ECTANGLE I SOTROPIC H OMOGENOUS C AVITY

where ξ = [ξ1 , ξ2 , . . . , ξn ]T , ζ = [ζ1 , ζ2 , . . . , ζm ]T A = (ai j ) ∈ Cn×n , T = (ti j ) ∈ Cn×m M = (m i j ) ∈ Cn×n , ai j = a(N j , Ni ) ti j = c∗ (Ni , ψ j ), m i j = b(N j , Ni ) where M is a positive definite Hermitian matrix, but A is only a positive semidefinite Hermitian matrix. If h = 0, then it turns out that the condition T † ξ = 0 is equivalent to requiring ξ to be M-orthogonal to the nullspace of A. Hence, the positive eigenpairs of (12) are precisely the eigenpairs of Aξ = h Mξ corresponding to positive eigenvalues. The computational cost in solving problem (12) is very large when the mesh is fine. Here, we adopt the computational method proposed in [31], and let us consider the generalized eigenvalue problem (A + T DT † )ξ = h Mξ

(13)

where D is a positive definite Hermitian matrix, such that the nonphysical zero eigenvalues are filtered out and do not disturb the computations of the positive eigenvalues. As suggested in [32], we take D = α I , where I is identity matrix. In our numerical experiments, we will take α = 100/ h. From the theory of linear algebra, it is well known that the eigenvalues of (13) are nonnegative real numbers, which is clearly in agreement with the fact that the cavity with lossless media and PEC wall has real resonant frequencies. Note that when the medium in the cavity is lossy, the resonant frequencies are usually complex numbers. The numerical solver for the cavity problem with lossy media is much more difficult than the one with lossless media. IV. N UMERICAL E XPERIMENTS Here, we carry out four numerical experiments to confirm the validity of the method in this paper. The goals are to support that our method is free of all the spurious modes. A. Rectangle Cavity With Homogeneous Isotropic Medium We first make a simulation for the simplest cavity. Let the length, width, and height of the rectangle cavity be π m, π/2 m, and π/4 m, respectively. The medium in this cavity is vacuum. This problem can be solved analytically. The exact eigenvalues are mnl = m 2 + 4n 2 + 16l 2 (m = 1, 2, 3, 4, . . . , n, l = 0, 1, 2, 3, . . . , and n = l = 0) [8]. We compute the first five eigenvalues using (12) and present the numerical results in Table I.

Fig. 2.

Toric cavity.

From Table I, we can see that the numerical eigenvalues approximate the exact eigenvalues from below, and the accuracy of the first smallest numerical eigenvalue is much higher than the rest ones. The physical interpretation is that sampling density in each wavelength for the dominant mode is bigger than the ones for rest physical modes under the same mesh. Levillain [21] gives the following error estimate: |k − k,h | < C(k)h 2 where C(k) is a constant independent of h, but dependent on k. Moreover, C(k) usually increases with the increase of k. The above estimate is in agreement with the physical interpretation that the number of sample points needs to increase for higher accuracy of the physical modes with higher frequencies. B. Toric Cavity With Homogeneous Anisotropic Medium Suppose that we have a torus with the major radius a = 1 m and the minor radius b = 0.5 m. The geometry of this cavity is shown in Fig. 2. Suppose that the medium parameters in this torus are ⎡ ⎡ ⎤ ⎤ 1 2j 0 1 0 0 r = ⎣ −2 j 5 0 ⎦, μr = ⎣ 0 2 0 ⎦. 0 0 4 0 0 3 The first five smallest eigenvalues obtained from (13) are listed in Table II. In the above two examples, we find that all nonphysical zero modes are eliminated by our method, and there is no physical zero mode in numerical results for these cavities, which have connected PEC boundary. The numerical eigenvalues are all convergent as h → 0. C. Cavity With Two Disconnected Boundaries Assume that 1 = (−a, a) × (−b, b) × (−c, c), 2 = {(x, y, z) : x 2 + y 2 + z 2 ≤ r 2 }, and = 1 \ 2 .

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TABLE II F IRST F IVE S MALLEST E IGENVALUES F ROM A T ORIC C AVITY W ITH H OMOGENOUS A NISOTROPIC M EDIUM

TABLE III F IRST F IVE S MALLEST E IGENVALUES F ROM THE C AVITY W ITH T WO D ISCONNECTED B OUNDARIES

Fig. 3.

Cavity with two disconnected boundaries.

In this example, we take a = π, b = π, c = π, and r = π/2. The geometry of this cavity is shown in Fig. 3. The medium parameters are ⎡ ⎡ ⎤ ⎤ 1 0 0 2 1−2j −j 4 j ⎦. r = ⎣ 0 1 0 ⎦, μr = ⎣ 1 + 2 j 0 0 1 j −j 5 Here, we adopt the matrix equation (13) to solve the numerical eigenvalues, and the results are listed in Table III. From Table III, we find that our method can also eliminate all nonphysical zero modes, and there is a physical zero mode in this model, because this cavity has two disconnected PEC boundaries.

Fig. 4.

Cavity with three disconnected boundaries.

The geometry of this cavity is shown in Fig. 4. The medium parameters in this cavity are ⎡ ⎡ ⎤ ⎤ 1 0 0 20 3j 1−5j (1) (1) r =⎣0 1 0⎦, μr =⎣ −3 j 40 j ⎦ in (1) 0 0 1 1+5j −j 5 ⎡ ⎡ ⎤ ⎤ 1 0 0 1 0 0 (2) (2) r = ⎣ 0 2 0 ⎦, μr = ⎣ 0 1 0 ⎦ in (2) . 0 0 3 0 0 1

Here, we also employ (13) to solve the numerical eigenvalues. From Table IV, we see that our method can also eliminate all nonphysical zero modes. We see that there are D. Cavity With Three Disconnected Boundaries two physical zero modes in this model for this cavity with Finally, we use our program to stimulate an example more three disconnected PEC boundaries. complex than the above three ones. Assume that In order to compare computational efficiency of our MFEM 3 and the conventional first-order edge element method, we use 1 = (0, π) , 2 = (π, 2π) × (0, π) × (0, π) our MATLAB program to simulate the numerical example 2 2 2 π π π above, respectively. The CPU time is listed in Table V. Here, 3 = (x, y, z) : x − + y− + z− ≤1 2 2 2 t , t , and t3 stand for CPU time using our MFEM, conven 1 2 2 2 2 tional first edge element method, and COMSOL Multiphysics, 3π π π 4 = (x, y, z) : x − + y− + z− ≤ 1 respectively. The time t1 and t2 is obtained from MATLAB 2 2 2 programs, and these time does not include post-processing (1) (2) = 1 \ 3 , = 2 \ 4 . time. The time t3 is from the COMSOL Multiphysics


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TABLE IV F IRST F IVE S MALLEST E IGENVALUES F ROM THE C AVITY W ITH T HREE D ISCONNECTED B OUNDARIES

TABLE V CPU T IME U SING O UR M ETHOD AND COMSOL M ULTIPHYSICS

(using the first-order edge element method), and it includes post-processing time. In this testing, in order to get the first five smallest eigenvalues, we only need to compute five numerical eigenvalues using our MFEM; however, in the conventional first edge element method, we need to compute N p + 5 numerical eigenvalues, where N p is the number of internal nodes in the mesh. When the mesh is fine, N p is very large. Our program can correctly stimulate all the resonant cavity problems with arbitrary shape and piecewise anisotropic lossless media. Because of making use of the mesh data obtained by COMSOL Multiphysics, we can contrast numerical eigenvalues from our approach and COMSOL Multiphysics (using the first edge element method). When the eigenvalue is nonzero, we verify the correctness of our numerical solutions by means of COMSOL Multiphysics. We find that the numerical results obtained by our method are in excellent accordance with the ones from COMSOL Multiphysics. When the mesh is not fine, we also compute function value of Lagrangian multiplier ph on mesh nodes using matrix equation (12) and find that these function values are very small, which is consistent with the theory in Section II. V. C ONCLUSION The electromagnetic eigenvalue problem for a cavity with complex geometry and anisotropic lossless media can be solved using the MFEM. The proposed method can eliminate all the spurious modes by employing a Lagrangian multiplier to enforce the divergence-free condition. In this paper, we use the first-order edge basis functions, but it will be extended to higher order edge basis functions in our future work. ACKNOWLEDGMENT This work was supported in part by the Fujian Province Natural Science Foundation under Grant 2013J05060 and in part by the National Natural Science Foundation of China under Grant 11101381, Grant 11371357, and Grant 41390453. R EFERENCES [1] K. Zhang and D. Li, Electromagnetic Theory for Microwaves and Optoelectronics. Berlin, Germany: Springer-Verlag, 2008. [2] J.-M. Jin, The Finite Element Method in Electromagnetics. New York, NY, USA: Wiley, 1993.

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Wei Jiang received the bachelor’s degree in mathematics and applied mathematics from Hubei Normal College, Huangshi, China, in 2008, and the M.S. degree in computational mathematics from Guizhou Normal University, Guiyang, China, in 2011. He is currently pursuing the Ph.D. degree with the Institute of Electromagnetics and Acoustics, Xiamen University, Xiamen, China. His current research interests include finite-element method for PDE eigenvalue problems, applied numerical algebra, and computational electromagnetics, in particular, for eigenvalue problems in electromagnetics.

Na Liu received the bachelor’s degree in information and computation science from Henan University, Kaifeng, China, in 2008, and the M.S. and Ph.D. degrees in computational mathematics from the University of Chinese Academy of Sciences, Beijing, China, in 2013. She was a Visiting Student with the Department of Electrical and Computer Engineering, Duke University, Durham, NC, USA, from 2012 to 2013. Since 2013, she has held a post-doctoral position with the Institute of Electromagnetics and Acoustics, Department of Electronic Science, Xiamen University, Xiamen, China. Her current research interests include computational electromagnetics, in particular, the fast and efficient methods for complex media and their applications in cavities and optical waveguide problems.


Yongqing Yue received the B.S. degree in communication engineering from Shandong University, Jinan, China, in 2013. She is currently pursuing the M.S. degree in electronics and communication engineering with Xiamen University, Xiamen, China. She takes a part in the research project—air transient electromagnetic system, and makes her contribution as a Researcher. Her current research interests include computational electromagnetics.

Qing Huo Liu (S’88–M’89–SM’94–F’05) received the B.S. and M.S. degrees in physics from Xiamen University, Xiamen, China, in 1983 and 1986, respectively, and the Ph.D. degree in electrical engineering from the University of Illinois at Urbana–Champaign, Champaign, IL, USA, in 1989. He was with the Electromagnetics Laboratory, University of Illinois at Urbana–Champaign, as a Research Assistant from 1986 to 1988, and a PostDoctoral Research Associate from 1989 to 1990. He was a Research Scientist and Program Leader with Schlumberger-Doll Research, Ridgefield, CT, USA, from 1990 to 1995. From 1996 to 1999, he was an Associate Professor with New Mexico State University, Las Cruces, NM, USA. Since 1999, he has been with Duke University, Durham, NC, USA, where he is currently a Professor of Electrical and Computer Engineering. He has authored over 300 papers in refereed journals and 400 papers in conference proceedings. His current research interests include computational electromagnetics and acoustics, inverse problems, and their application in nanophotonics, geophysics, biomedical imaging, and electronic packaging. Dr. Liu is a fellow of the Acoustical Society of America, a member of Phi Kappa Phi and Tau Beta Pi, and a Full Member of the U.S. National Committee of URSI Commissions B and F. He was elected as a fellow of the Acoustical Society of America for his contributions to computational acoustics and elasticity. He was a recipient of the 1996 Presidential Early Career Award for Scientists and Engineers from the White House, the 1996 Early Career Research Award from the Environmental Protection Agency, and the 1997 CAREER Award from the National Science Foundation. He serves as the Deputy Editor-in-Chief of Progress in Electromagnetics Research, an Associate Editor of the IEEE T RANSACTIONS ON G EOSCIENCE AND R EMOTE S ENSING, and an Editor of the Journal of Computational Acoustics. He served as a Guest Editor of the P ROCEEDINGS OF THE IEEE. He serves as the Editor-in-Chief of the new IEEE J OURNAL ON M ULTISCALE AND M ULTIPHYSICS C OMPUTATIONAL T ECHNIQUES . He serves as an IEEE Antennas and Propagation Society Distinguished Lecturer from 2014 to 2016.