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FINITE ELEMENT MODELING OF PERIODIC STRUCTURES
Zheng Lou and Jian-Ming Jin Center for Computational Electromagnetics, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801-2991 E-mail:
[email protected] Periodic structures have a variety of important applications in modern technologies and engineering due to their unique electromagnetic properties. Commonly used periodic structures include frequency selective surfaces, optical gratings, phased array antennas, photonic bandgap materials, and various metamaterials. The analysis of periodic structures has always been an important topic in computational electromagnetics. In this chapter, we describe an accurate and efficient numerical analysis, based on a higher-order finite element method (FEM), for characterizing the electromagnetic properties of periodic structures. Based on the Floquet theory, periodic boundary conditions and radiation conditions are first derived for the unit cell of a periodic structure. The FEM is then applied to solve Maxwell’s equations in the unit cell. To enhance the accuracy and efficiency of the FEM, curvilinear elements are employed to discretize the unit cell and higher-order vector basis functions are used to expand the electric field. The asymptotic waveform evaluation (AWE) is implemented to perform fast frequency and angular sweeps. To demonstrate the capability of the proposed FEM, we apply it to the analysis of periodic absorbers, frequency selective structures, and phased array antennas. For the antenna analysis, a rigorous waveguide port condition is developed to accurately model the antenna feed structures. In all the cases studied, satisfactory results are obtained.
1. Introduction Periodic structures have been extensively used in electromagnetic engineering. The periodicity in the geometry is often exploited to achieve certain desired electromagnetic properties, such as frequency selective behaviors, 1
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which are not obtainable in the case of a single element. Many microwave and optical devices, such as anechoic chamber absorbers, frequency selective structures (FSS), and phased array antennas, fall into this category. Numerical analysis of periodic structures has been carried out using a variety of numerical methods, such as the moment method (MoM), the finite-element method (FEM), and the finite-difference time-domain (FDTD) method. Among these methods, the FEM excels at modeling complicated geometry and material inhomogeneity. The method is also versatile in its ability to incorporate different types of boundary conditions and different excitation modes without significantly affecting its formulation. The FEM modeling of periodic structures has been reported in literature for both scattering [1]-[5] and radiation [6]-[8] analyses. In this chapter, we describe a robust, higher-order FEM formulation to model infinitely periodic array structures. By imposing appropriate radiation boundary conditions and periodic boundary conditions, the computation domain is confined to a single unit cell in the infinite array. The unit cell interior region is discretized with curvilinear tetrahedral elements for better geometry modeling. The electromagnetic field is then expanded with higher-order vector basis functions, which are more efficient than the conventional zeroth-order basis functions. For broadband calculations, the asymptotic waveform evaluation (AWE) technique can be combined with the FEM to perform fast frequency and angular sweeps. For radiation analysis, an accurate coaxial feed modeling has been implemented, employing a rigorous waveguide port condition. For the purpose of validation, numerical examples are given for both scattering and radiation analyses for various element configurations. The agreement with previously published data and measured results demonstrates the accuracy and capability of the proposed FEM formulation. 2. Formulation This section describes various aspects of the FEM formulation for the analysis of periodic structures. The fast sweep techniques are also outlined at the end of this section. 2.1. Discretization The structures under consideration are planar arrays, infinitely periodic in the x and y directions. Since all the elements in the array are identical, the FEM analysis is applied only to a single element, or a unit cell, as shown
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in Fig. 1. We start from Maxwell’s equations for the electromagnetic fields inside the computation domain: z y x
Unit cell
Fig. 1.
Unit cell in an infinite periodic structure.
∇ × E = −jωµ0 µr H
(1)
∇ × H = jω²0 ²r E + J.
(2)
Eliminating H in the two equations above results in the wave equation for E: µ ¶ 1 ∇× ∇ × E − k02 ²r E = −jk0 Zo J. (3) µr Assuming that E satisfies certain boundary conditions on the surface S that encloses the computation domain V , it can be shown that the original problem is equivalent to the following variational problem [9]: δF (E) = 0 where F (E) =
(4)
¸ 1 (∇ × E) · (∇ × E) − k02 ²r E · E dV V µr ZZZ +jk0 Z0 E · JdV 1 2
ZZZ ·
V
+ surface integral terms
(5)
where the surface integral terms are the results of imposing corresponding boundary conditions. In this subsection, we only consider the interior part
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of the problem. The boundary condition terms will be discussed in the following subsection. To solve the variational problem (4) numerically, the entire computation domain V is discretized into small elements. In the FEM discretization, commonly used elements are triangular elements for 2-D problems and tetrahedral elements for 3-D problems. To faithfully model curved surfaces, curvilinear elements are often utilized. For the curvilinear elements, the evaluation of the integrals in (5) is facilitated by mapping the integrals onto a corresponding straight element with unit length in a new coordinate system. For triangular elements, the new coordinate system is denoted as the ξη coordinate and the mapping is given by x=
6 X
Nie (ξ, η)xi
(6)
Nie (ξ, η)yi
(7)
i=1
y=
6 X i=1
∂x ∂y ∂ξ ∂ξ (8) dxdy = |J|dξdη [J] = ∂x ∂y ∂η ∂η where xi and yi are the coordinates of the ith node in the xy coordinate system and Nie (ξ, η) are scalar shape functions defined in the ξη coordinate system. The mapping for tetrahedral elements can be defined in a similar fashion with 10 X x= Nie (ξ, η, ζ)xi (9)
i=1
y=
10 X
Nie (ξ, η, ζ)yi
(10)
Nie (ξ, η, ζ)zi
(11)
i=1
z=
10 X i=1
dxdydz = |J|dξdηdζ
∂x ∂ξ ∂x [J] = ∂η ∂x ∂ζ
∂y ∂ξ ∂y ∂η ∂y ∂ζ
∂z ∂ξ ∂z ∂η ∂z ∂ζ
.
(12)
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Using the mapping defined in (6)–(12), the integration is carried out in the new coordinate system, where it can be evaluated easily using GaussianLegendre quadratures. It is also more convenient to define all the basis functions in the new coordinate system. The next step involves expanding the electromagnetic fields in terms of basis functions in each element. In the context of 3-D problems, it is more convenient to use vector basis functions instead of scalar basis functions. For tetrahedral elements, the vector basis function associated with the ith edge is given by Wie = (Lei1 ∇Lei2 − Lei2 ∇Lei1 )lie
(13)
where i1 and i2 denote the two vertices associated with the ith edge and lie is the length of the ith edge. Further, Lek (k = 1, 2, 3, 4) are the nodebased basis functions associated with the four vertices of the tetrahedral. It can be demonstrated easily that Wie automatically satisfies the divergence condition. Thus, the solutions obtained by using vector basis functions are free of spurious solutions. Also, it is easier to model dielectric interfaces and conducting edges and tips using vector basis functions than using scalar basis functions. The vector basis functions defined in (13) are zeroth-order interpolating functions. Higher-order basis functions can be constructed systematically by multiplying the zeroth-order interpolating functions with Lagrange interpolatory polynomials [10] given by 1 i2 Niijkl = γ Pˆip+2 (ξ1 )Pˆjp+2 (ξ2 )Pˆkp+2 (ξ3 )Pˆlp+2 (ξ4 )Wi1 i2
(14)
where (i1 , i2 ) are the combinations of two integers from (1, 2, 3, 4) and i3 and i4 are two integers among (1, 2, 3, 4) other than i1 and i2 . Also, i, j, k, and l satisfy i + j + k + l = p + 2 where p is the order of basis functions, γ is a normalization constant, Wi1 i2 are zeroth-order basis functions, and Pˆip+2 (ξ) are the shifted Silvester polynomials defined by Pˆip+2 (ξ) =
i−1 Y 1 [(p + 2)ξ − l]. (i − 1)!
(15)
l=1
The detailed discussion of higher-order basis functions can be found in [10]. It has widely been accepted that higher-order basis functions are more accurate and efficient in modeling the actual field solution for electromagnetic problems. As an example, Fig. 2 shows the relative error versus the number of total unknowns for using different order basis functions in a 2-D eigenvalue problem. The eigenvalue problem is to determine the cutoff
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wavenumber of the T E22 mode propagating in a rectangular waveguide. The relative error is obtained by comparing the numerical and analytical cutoff wavenumbers. It is easily seen from Fig. 2 that for a given number of unknowns, the higher-order basis functions produce a higher accuracy than do the lower-order basis functions. Moreover, different error curves in Fig. 2 exhibit different slopes, indicating that the relative error decreases much faster by choosing higher-order basis functions. In a general sense, one can always achieve a higher accuracy by using higher-order basis functions. Practically, however, the choice of interpolating order depends on specific problems. It is advisable to choose the order of basis functions such that it is not much higher than the order of geometrical transformation.
−1
10
−2
Relative Error
10
−3
10
−4
10
0th order 1st order 2nd order
−5
10
2
3
10
10
Number of Unknowns Fig. 2. Relative error in computing T E22 cutoff wavenumer of a 3 × 2 cm rectangular waveguide.
Once the basis functions are determined, the electric (or magnetic) field within each element can be expanded as e
E =
n X
Nei Eie
(16)
i=1
where n is the number of interpolating points (depending on the order of basis functions) within each element. Substituting (16) into (5), one arrives
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at the elemental matrices [K e ] with the entries given by ¸ ZZZ · 1 e Kij = (∇ × Nei ) · (∇ × Nej ) − k02 ²r Nei · Nej dV V µr and the right-hand-side (RHS) known vector {be } given by ZZZ bei = −jk0 Z0 Nei · JdV.
(17)
V
The [K e ] and {be } are then assembled to the system matrix [K] and system known vector {b}, respectively, according to the connectivity of the FEM mesh. The result of subdomain discretization, basis function expansion, and finite element assembling is a set of linear equations in the matrix form of [K] {E} = {b}.
(18)
Both direct and iterative methods can be employed to solve the linear system equations. Popular iterative methods include the conjugate gradient (CG) method and the generalized minimal residual (GMRES) methods. For direct methods, the frontal and multifrontal algorithms [11] are suitable solvers for linear systems resulting from the finite element assembling procedure. In the finite element assembling procedure, a large number of elemental equations are decoupled. Taking advantage of this fact, the frontal method performs Gaussian elimination as soon as an equation is completely decoupled from the rest of the equations. The multifrontal method is an extension of the frontal method, which employs multiple frontal matrices. The advantage of the frontal/multifrontal method is its capability to deal with arbitrary sparsity pattern, making it far more efficient than the standard LU decomposition method. A popular multifrontal package is called UMFPACK a , which is a sparse linear solver based on the unsymmetric-pattern multifrontal method. 2.2. Boundary Conditions Consider the unit cell in an infinite periodic structure, as shown in Fig. 3. The interior volume, denoted here as V , is enclosed by four side surfaces, a top surface, and a bottom surface. It may contain arbitrary dielectric and conducting structures. The four side surfaces Sx1 , Sx2 , Sy1 , and Sy2 are located at x = 0, x = Tx , y = 0, and y = Ty , respectively, where Tx and a www.cise.ufl.edu/research/sparse/umfpack/
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Ty are periodic lengths in the x and y directions. The top surface St is the interface between free-space and the unit cell region. The bottom surface Sb is usually a ground plane. It may also contain waveguide apertures Sw that provide excitation for the radiation case. In such a general configuration, four kinds of boundary conditions are involved. On the four side surfaces, periodic boundary conditions are imposed, relating the fields on the opposite side surfaces. On the top surface, a periodic radiation boundary condition is imposed, which simulates the radiation towards the free space in the presence of an infinite array. If a waveguide is present in the structure, then a waveguide port condition is imposed on Sw . Finally, on conducting surfaces, a perfectly electrically conducting (PEC) boundary condition is enforced explicitly as a homogeneous Dirichlet boundary condition.
St
z
S x1
S y1
S y2
V Ty
S x2
y
Sw
Tx
x
Sb Fig. 3.
Illustration of boundaries on a unit cell.
Periodic boundary conditions can be derived directly from the Floquet theorem. In accordance with the Floquet theorem, the electromagnetic fields inside and above the periodic media satisfy E(x + mTx , y + nTy ) = E(x, y)ej(mkxi Tx +nkyi Ty )
(19)
H(x + mTx , y + nTy ) = H(x, y)ej(mkxi Tx +nkyi Ty )
(20)
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where kxi = k0 sin θi cos φi i
i
kyi = k0 sin θ sin φ
(21) (22)
where (θi , φi ) are the incident or scan angles. To facilitate the implementation of periodic boundary conditions, identical surface meshes are created on the opposite side surfaces. Then for each unknown Ei on one side surface, a corresponding unknown Ej is identified on the opposite surface which has the same relative position as Ei . Applying the Floquet theorem (19), one can easily obtain the following relationship between Ei and Ej : Ej = Ei ejΨij where Ψij is a phase shift term kxi Tx Ψij = kyi Ty k T +k T xi x
yi y
(23)
given by ∀Ei ∈ Sx1 , Ej ∈ Sx2 ∀Ei ∈ Sy1 , Ej ∈ Sy2
(24)
∀Ei ∈ Sx1 ∩ Sy1 , Ej ∈ Sx2 ∩ Sy2 .
In the matrix context, (23) is enforced explicitly as an inhomogeneous Dirichlet boundary condition. That is, for each unknown pair (Ei , Ej ): Ej is eliminated; the matrix entries associated with Ei are modified as Kil = Kil + Kjl ejΨij
(25)
Kik = Kik + Kjl ej(Ψij −Ψkl )
(26)
for all El ∈ / Sx2 ∪ Sy2 , and
for all El ∈ Sx2 ∪ Sy2 which is related to Ek by phase shift Ψkl ; and finally, the RHS vector entry associated with Ei is modified as bi = bi + bj ejΨij .
(27)
The functional given in (5) contains a surface integral term over the top surface St with a form given by ZZ S(E) = −jk0 Z0 M · HdS (28) St
where M = −ˆ n × E is the equivalent magnetic current on St . Hence, there is a need to derive an equation that relates the magnetic current M and the magnetic field H. In order to do that, we first assume that a ground
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plane is placed on St . From the equivalence principle, the total magnetic field above St can be written as H(r) = Hinc (r) + Href (r) + Hscat (r)
(29)
where Hinc is the incident field and Href is the reflected field by the ground plane. Hscat is the magnetic field radiated by the magnetic current source on St in the presence of the ground plane. On invoking the image theory, Hscat can be written as ZZ scat ¯ H (r) = −2jk0 Y0 G(r, r0 ) · M(r0 )dS 0 (30) S∞
where S∞ denotes the infinite plane that coincides with the ground plane ¯ denotes the dyadic Green’s function for free space. Since M has a and G spacial periodic property, it can be expanded in terms of Fourier series ∞ X
M(x, y) =
˜ pq e−j(kxp x+kyq y) M
(31)
M(x, y) ej(kxp x+kyq y) dxdy
(32)
p,q=−∞
where ˜ pq = M
1 Tx Ty
ZZ St
where 2π p + kxi Tx 2π = q + kyi Ty
kxp =
(33)
kyq
(34)
are expansion wavenumbers, which are also referred to as the Floquet harmonics. By substituting (31) into (30), the integration over the infinite surface is reduced to an infinite summation in the spectral domain. Consequently, (29) can be written as H(x, y) = Hinc (r) + Href (r) ∞ X −j(kxp x+kyq y) ˜¯ ˜ G(k −2jk0 Y0 xp , kyq ) · Mpq e
(35)
p,q=−∞
˜¯ is the spectral version of dyadic Green’s function and it can be where G expressed explicitly as · 2 ¸ 1 k0 − kxp 2 −kxp kyq ˜ ¯ (36) G(kxp , kyq ) = 2 2 2j kz k0 2 −kxp kyq k0 − kyq
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FINITE ELEMENT MODELING OF PERIODIC STRUCTURES
where
11
q k0 2 − kxp 2 − kyq 2 .
kz =
(37)
The integral equation (35) relates the surface magnetic current to its magnetic field. It enforces the radiation condition on the top surface of the unit cell in the presence of an infinite array. It is therefore referred to as the periodic radiation boundary condition. By substituting (35) into (28) and then (5), the functional now becomes ¸ ZZZ · 1 1 2 F (E) = (∇ × E) · (∇ × E) − k0 ²r E · E dV 2 V µr ZZZ +jk0 Z0 E · JdV V
ZZ −2k02
M· St
∞ X
−j(kxp x+kyq y) ˜¯ ˜ G(k dS xp , kyq ) · Mpq e
p,q=−∞
ZZ
M · Hinc dS
−2jk0 Z0 St
+ other surface integral terms.
(38)
The discretization of the surface integral term (28) yields the corresponding matrix entries ∞ X ∗ ˜¯ ˜ pq · G(k ˜ pq Sij = −2k02 Tx Ty N (39) xp , kyq ) · Nj i p,q=−∞
where ˜ pq = N i
1 Tx Ty
ZZ Ni ej(kxp x+kyq y) dS
(40)
St
and the RHS known vector entries
ZZ Ni · Hinc dS.
bi = −2jk0 Z0
(41)
St
Numerically, the infinite summation in (39) is always truncated with −M < p < M and −N < q < N , where M and N are the maximum Floquet mode included in the summation. A sufficient number of Floquet modes needs to be included to obtain convergent result. One the other hand, the field expansion in the FEM determines that the variation in the FEM solution cannot exceed the maximum variation that the FEM mesh can resolve, which in turn depends on the mesh density and the element order. This fact imposes a limitation that M and N cannot be arbitrarily large.
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Fortunately, numerical experiments show that the magnitude of Floquet harmonics decreases rapidly with increasing p and q and an inclusion of seven to nine lowest Floquet harmonics is typically sufficient to produce convergent results. Since irregular elements are used, the evaluation of the integral in (40) requires numerical methods such as the Gaussian-Legendre quadrature. When St is large, the evaluation of the integrals may take a long time. Assembling of (39) leads to a linear system that is nonsymmetric and partially full, partially sparse. Such linear systems can be factorized and solved efficiently using the UMFPACK package. To show that the periodic boundary condition and the periodic radiation boundary condition correctly model the field behavior, a simple example is tested. The test example consists of a 20-cm-thick uniform dielectric layer composed of lossy material (²r = 3 − j) and backed with a ground plane. A plane wave is incident from above with incident angle θ = 60o . The TE and TM power reflection coefficients, shown in Fig. 4, are calculated over 0.1 ∼ 0.7 GHz frequency band. In the FEM calculation, the unit cell is modeled as a 20 × 20 × 20 cm homogeneous dielectric box with a ground plane placed on its bottom surface. The good agreement between the FEM results and the analytical solutions validates the proper enforcement of the periodic boundary condition and the periodic radiation boundary condition.
1 0.9 TE case, Analytical TE case, FEM TM case, Analytical TM case, FEM
Power Reflection
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
frequency (GHz)
Fig. 4. Power reflection coefficients for the 20-cm-thick uniform layer (²r = 3−j) backed with a ground plane at θ = 60o .
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In active array structures, such as array antennas, waveguides are often used to feed each antenna element. The waveguide considered here is a homogenous waveguide with an arbitrary cross section. For hollow waveguides, propagating modes are TE and TM modes. For coaxial lines, an additional TEM mode exists. Without losing generality, the electric field on the waveguide aperture Sw can be written as a sum of distinct waveguide modes: ∞ X E(r) = Einc (r) + Cm em (r)ejkzm z (42) m=0
where
q kzm =
2 k02 − kcm
(43)
where kcm and em are cut-off wavenumbers and transverse mode functions associated with the mth mode. They can be obtained numerically by solving the corresponding 2-D eigenvalue problem using the FEM. The mode index m includes all possible modes (TE, TM, and TEM if possible), and Einc is the incident field. For simplicity, we assume fundamental mode incidence; that is, Einc = e0 . The Cm is the reflection coefficient for each mode. Using orthogonality between different modes, Cm can be expressed as ZZ −jkzm z Cm = e em · (E − Einc )dS. (44) Sw
A boundary condition can be derived by substituting (44) into (42), and then applying n ˆ × ∇× operator on both sides of the equation, yielding ZZ X n ˆ × ∇ × E − jk0 Z0 Ym em em · EdS = −2jk0 Z0 Y0 e0 (45) m
Sw
where Ym are modal admittances given by kzm for TE modes k 0 Z0 −k0 Ym = for TM modes k zm Z0 1 for TEM mode. Z0
(46)
Applying boundary condition (45) on Sw yields an additional surface integral term in functional (5): Ã ! ZZ ZZ X jk0 Z0 E· Ym em em · EdS − 2Y0 E · e0 dS. (47) S(E) = 2 Sw Sw m
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After discretization and field expansion, (47) can be cast into the following matrix format as X ˆ mN ˆm Sij = jk0 Z0 Ym N (48) i j m
ˆi0 bi = 2jk0 Z0 Y0 N where
(49)
ZZ ˆim = N
Ni · em dS.
(50)
Sw
Assembling the above matrix entries results in a full submatrix in the system matrix [K], as in the case of the periodic radiation boundary condition. The evaluation of (48) requires the evaluation of the integral (50), which is also calculated numerically. In computer implementation, the infinite sum in (47) must be truncated. The number of waveguide modes included in (47) should be large enough to represent the actual field distribution on the waveguide aperture. But again, the variation of the highest mode included should not exceed the variation that the FEM mesh can resolve. To validate the waveguide port condition, a waveguide junction discontinuity problem is considered. The waveguide junction consists of two sections of concentric circular waveguides, with a radius of 1.5 m and 0.95 m, respectively. A T E11 wave is incident from the larger section and the power reflection coefficient for the T E11 mode observed at the larger section is calculated. In this example, the waveguide port conditions are imposed on the waveguide cross sections to truncate the computation domain, and 25 higher modes plus the dominant mode are included in the boundary condition. Figure 5 shows the calculated reflection coefficient, which agrees well with the solution obtained by the mode matching method. The numerical example shows clearly that the waveguide port condition correctly models the field behavior on the interface of the waveguide junction. 2.3. Fast Sweep Techniques The formulation presented in the previous two subsections is a frequencydomain method, which means that the system matrix has to be generated and solved for each specific frequency. In many practical applications, broadband frequency responses need to be calculated. One can imagine that the total time required for such frequency sweeps will become undesirably
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1.1
Power Reflection
0.9
Mode Matching FEM
0.7 0.5 0.3 0.1
R 2=0.95m
−0.1 60
R 1=1.5m
80
100
120
140
160
180
200
frequency (MHz)
(a)
(b)
Fig. 5. (a) Geometry of a circular waveguide junction. (b) Power reflection coefficient for T E11 mode.
long. In this subsection, the AWE method is combined with the FEM formulation to perform fast sweeps. The AWE technique was first introduced in circuits analysis [12]. Its applications in the frequency-domain FEM and MoM have been introduced recently [13]-[15]. In the context of electromagnetic problems, the linear system of equations to be solved is [A] {x} = {b}
(51)
where both the matrix and the vectors are functions of wavenumber (or frequency). First we want to expand {x} in terms of a Taylor series {x} =
n X {x}(i) i=1
i!
(k − k0 )i
(52)
where k0 is the expansion wavenumber. In order to calculate {x}(i) , i = 1, 2, · · · , n, one can take the nth derivative of the both sides of (51) to give ([A] {x})
(n)
= {b}(n) .
The left-hand side of the above equation can be easily expanded as n X n! [A](i) {x}(n−i) = {b}(n) . i!(n − i)! i=0
(53)
(54)
From (54), a recursive formula can be derived to calculate {x}(n) , which is given by à ! n X n! (n) −1 (n) (i) (n−i) {x} = [A] {b} − [A] {x} . (55) i!(n − i)! i=1
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Once the Taylor coefficients are obtained, the Taylor series is then converted to the corresponding Pad´e series PL
{x} =
i i=0 {a}i (k − k0 ) . PM 1 + j=1 {b}j (k − k0 )j
(56)
The Pad´e expansion is more capable of capturing the poles of the transfer function and thus yields a larger convergence radius. In cases where the convergence range of one single expansion point is not enough to provide an accurate solution over the entire frequency band, a multiple point expansion technique called complex frequency hopping (CFH) can be employed [13]. In CFH, a binary search algorithm is used to determine the expansion points adaptively until the entire frequency band is covered by the total convergence areas of the expansion points. The algorithm is outlined as follows. Given a sweep range [kmin , kmax ] and an error tolerance ε: (1) Apply AWE at kmin and kmax ; obtain the corresponding expansion functions x1 (k) and x2 (k). (2) Let kmid = (kmin + kmax )/2 and calculate x1 (kmid ) and x2 (kmid ). (3) If |x1 (kmid ) − x2 (kmid )| < ε, the algorithm stops; otherwise, the above steps are repeated for subregions [kmin , kmid ] and [kmid , kmax ]. In general, the fast sweep technique described above is much more efficient in generating broadband frequency responses compared to the pointby-point approach. Figure 6 shows the reflection coefficient for a 1-D periodic dielectric structure over frequency band 130 ∼ 176 MHz. The AWE solver gives the same result as the direct calculation at each frequency point. Table 1 shows the comparison of computation time for using the direct and AWE solvers. It is seen from the table that the total computation time required by the AWE solver is much less than that required by the direct solver. The convergence radius of an expansion point is mainly determined by two factors: the highest order of the derivatives and the accuracy of the derivatives. Because of the enforcement of the periodic boundary condition and the periodic radiation boundary condition, the derivatives of matrix entries with respect to wavenumber generally cannot be calculated analytically in closed forms, which is different from [16] where analytical derivatives are obtainable. Therefore, most derivatives are calculated numerically. This imposes limitations on both the highest order and the accuracy of the derivatives.
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Power Reflection
1
q
... h
e r1 e r2 e r1 e r2 e r1
...
X
0.8
AWE Direct
0.6
0.4
0.2
Y d/2
Z 0 130
d
140
150
160
170
frequency (MHz)
(a)
(b)
Fig. 6. (a) Geometry of the dielectric layer with periodically varying permittivity with ²r1 = 1.44, ²r2 = 2.56, d = 1 m, h = 2.037 m, θ = 45o . (b) Specular power reflection coefficient. Table 1. Solver type Direct AWE
Comparison of computation time for frequency sweep. Time for a single frequency point (s) 53.2 116.7
Number of points
Total time (s)
461 6
24525 700.2
Many important parameters of array structures, such as scan pattern, are angular responses. It is easily seen that the system matrix entries are dependent on incident angles (or scan angles) as a result of the enforcement of the periodic boundary condition (23) and the periodic radiation condition (35). This is different from the nonperiodic case, in which only the RHS known vector is angle-dependent and the system matrix is factorized only once for all angles of incidence. However, for periodic structures, the system matrix must be regenerated and refactorized at each incident angle. To accelerate wide-ranged angular sweeps, the AWE technique described above for frequency sweep can be formulated for angular sweep in a very similar fashion. In most scan analyses, θ is the scan angle. It is therefore convenient to define an angle-dependent wavenumber kt = k0 cosθ. Then every quantity in (51) can be written as a function of kt and its derivatives with respect to kt can be calculated. Following the same procedures as described in this section, one will finally obtain a set of expansion coefficients at several expansion points kt1 , kt2 , · · · , ktn , from which the desired quantity over the entire angular range can be extrapolated. Figure 7 shows the angular sweep of the specular reflection coefficient
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for the same example as shown in Fig. 6 at f = 140 MHz. The two curves calcuated by the AWE solver and the direct solver superimpose upon each other. Table 2 shows the comparison of computation time for calculating the entire angular response. It is evident that the AWE solver outperforms the direct solver in terms of total computation time. 1 0.9 AWE Direct
Magnitude Reflection
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
10
20
30
40
50
60
70
80
90
θ (degrees) Fig. 7. Specular magnitude reflection coefficient versus incident angle for the dielectric layer in Fig. 6 at f = 140 MHz.
Table 2. Solver type Direct AWE
Comparison of computation time for angular sweep. Time for a single frequency point (s) 53.2 116.7
Number of points
Total time (s)
90 8
4788 933.6
3. Numerical Results This section presents examples of FEM analysis of periodic structures including periodic absorbers, frequency selective structures, and phased array antennas. The numerical examples validate the FEM formulation outlined in Section 2 and demonstrate its capabilities.
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3.1. Periodic Absorbers Periodic absorbers are extensively used as low-reflection covering of the metallic shielding walls in electromagnetic compatibility measurements. Scattering analysis of periodic absorbers initially resorted to analytical methods. At very high frequencies, asymptotic methods such as the uniform theory of diffraction (UTD) [17] can be applied. At very low frequencies, the original problem can be transformed into a multilayer problem and approximate solutions can be obtained [18], [19]. Recent advances in full-wave numerical methods enable a more general and more accurate analysis. Periodic version of the MoM has been formulated and widely used [20]-[23]. There also exists frequency- and time-domain finite difference modeling of periodic absorbing structures [24]-[26]. The FEM is most suitable for modeling complicated absorber geometry and composition and has been utilized in the analysis and design of periodic absorbers [5], [27]. Figure 8 shows the unit cell in a periodic absorber array. The absorbing material is typically shaped into wedges for 2-D arrays and pyramid-cones for 3-D arrays. In the FEM analysis, the unit cell contains only one single absorber. On the four side surfaces of the unit cell, the periodic boundary conditions (23) are imposed. The top surface St is placed somewhere above the tip of the absorber and the periodic radiation boundary condition (35) is imposed there. The bottom surface Sb is placed on the physical boundary of the absorber, which is usually a PEC ground plane that serves as an electromagnetic shield. After the electric field is solved everywhere inside the unit cell, the specular reflection coefficient R (p = 0, q = 0) can be calculated from the aperture fields on the top surface St , giving scat Href t + Ht inc Ht ˜¯ ˜ = 1 − 2jk0 Y0 G(k x0 , ky0 ) · M00
R=
(57)
˜¯ ˜ pq and G(k where Y0 = 1/Z0 and M xp , kyq ) are defined in (32) and (36), respectively. First, we investigate the absorption characteristics of a metal-backed planar stratified absorber, which has been analyzed with the surface integral equation method [22]. The absorber is made up of three infinite layers, each with a distinct material property. The geometry is shown in the inset of Fig. 9. For comparison, the material properties of the three layers have been chosen to be similar to those in [22]. The calculated specular reflection coefficients from 2 to 18 GHz are plotted in Fig. 9. For this problem, the
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Absorbing material
z
y
Ground plane
x Fig. 8.
Unit cell in a periodic absorber array.
analytical solution is obtained by cascading the propagator matrix for each layer. It is observed that the numerical result agrees well with the analytical solution. 0
R (dB)
−5
−10
3.2mm
εr1
3.2mm
εr2
3.2mm
εr3
−15
−20
−25 2
Analytical FEM
4
6
8
10
12
14
16
18
frequency (GHz) Fig. 9.
Specular power reflection for a stratified absorber at normal incidence.
To evaluate the performances of higher-order vector elements, we reconsider the stratified absorber example by evaluating the root-mean-square
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(RMS) errors for using different orders. To compare different orders, two different mesh sizes are used and for each of them the RMS errors are calculated using zeroth-, first-, and second-order tetrahedral elements, respectively. The RMS errors versus the number of unknowns are shown in Fig. 10. In the calculation, the frequency is fixed to be 30 GHz and the reflection coefficients are calculated versus incident angles. The RMS error is defined as v u Ns u 1 X 2 t RMS = |Ri − Ra | (58) Ns i=1 where Ra and Ri are the analytical and calculated reflection coefficients, respectively, and Ns is the number of incident angles. It is obvious from Fig. 10 that the accuracy improves with increasing element order. It is also noted that, for the same number of unknowns, more accurate results can be obtained by using higher-order elements. This example demonstrates the advantage of using higher-order FEM. 1
10
0th
0
h/λ=0.13 h/λ=0.1
0th
RMS error (dB)
10
1st
−1
10
1st 2nd
−2
10
2nd 2
10
Fig. 10.
3
10 Unknowns (N)
4
10
Comparison of RMS errors for different mesh sizes and element orders.
Second, we examine the reflection from a wedge absorber versus incident angles. The absorber is singly periodic in the x direction and uniform in the y direction, and has a relative permittivity ²r = 1.45 − 0.4j . The detailed geometry is shown in the inset of Fig. 11(a). The plane of incidence
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is assumed to be the xz plane. Figures 11(a) and (b) compare the results by the FEM code and those by a low-frequency approximation method (or the so-called homogenization method) for f = 100 and 300 MHz, respectively. In the low frequency approximation, the absorber is replaced with several homogeneous layers, whose effective permittivity and permeability are determined from the profile and material composition of the original absorber. The approximation is valid as long as the period of the absorber is small compared to the wavelength. The comparisons show that at lower frequencies the two methods predict a very similar reflection pattern, while at higher frequencies some discrepancies are observed. In the latter case, the period is slightly larger than one half of the wavelength. In this example, the absorber is not backed by a conducting plane. 0
0 −5
Low Frequency, TE FEM, TE Low Frequency, TM FEM, TM
−10
R (dB)
R (dB)
−10 −15 z θ
−20
−15 −20 −25 −30
x
−35
40’’
−25
Low Frequency, TE FEM, TE Low Frequency, TM FEM, TM
−5
10’’
−40 20’’
−30 0
20
40
60
θ (degrees)
(a)
80
100
−45 0
10
20
30
40
50
θ (degrees)
60
70
80
90
(b)
Fig. 11. Specular power reflection for the wedge absorber with ²r = 1.45 − 0.4j . (a) f = 100 MHz. (b) f = 300 MHz.
Next, we present some examples of doubly periodic absorbers. Figure 12(b) shows the specular power reflection coefficient for a doubly periodic absorber array whose unit cell geometry is shown in the inset. In this example, the permittivity of the absorbing material is a function of frequency, as shown in Fig. 12(a). To implement the fast frequency sweep, it is more convenient to expand the permittivity as a set of known functions of frequency, such as polynomial functions ¶n µ N X f (59) ²(f ) = an 1 − fmax n=0 where an are the expansion coefficients determined from the material’s per-
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mittivity characteristics and fmax is the maximum sweeping frequency. Typically, N = 4 is sufficient for a good modeling of realistic permittivities. The specular reflection coefficient is calculated for frequencies between 30 and 200 MHz. Results from three different methods are compared: the FEM, the finite-difference frequency-domain (FDFD) method [24], and the lowfrequency approximation method. It is observed that the three methods agree very well with each other for lower frequencies and differ by about 2 dB at higher frequencies. As another example, Fig. 13(b) shows the geometry and calculated specular reflection for a doubly periodic absorber of similar pyramid shape. The frequency-dependent permittivity used in this calculation is given in Fig. 13(a). The FEM result is compared with the low-frequency approximation and with the results by Jiang and Martin [5] for the 50 MHz to 1 GHz frequency range. Again at lower frequencies, the three results agree better. The differences at higher frequencies may due to the low power level of the reflected field (less than −50 dB). Also the condition for the low-frequency approximation to be valid may not be satisfied at these high frequencies. 0
0 εr ε
i
3
εr
−1
2.5
−2
2
−3
εi
Power Reflection (dB)
3.5
FDTD FEM Low frequency approximation
−5 −10 −15 −20 −25
1.5
0.05
0.1
0.15
frequency (GHz)
(a)
−4 0.2
−30
0.05
0.1
0.15
0.2
frequency (GHz)
(b)
Fig. 12. (a) Complex permittivity and (b) specular power reflection for a metal-backed pyramid absorber with period of 24 in and height of 96 in at normal incidence.
Finally, we investigate the effect of the pyramid profile on the absorber performance. Figure 14 shows three different pyramid profiles that are under consideration. They all have the same height and base dimensions, but the side surfaces of the pyramid are flat, concave, and convex, respectively. The radius of curvature R is 70 in for both concave and convex surfaces. The calculated specular reflection is shown in Fig. 15. The results show that,
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0 20
0 εr εi −5
ε
R (dB)
15
εi
r 10
−10
−10
18’’
−20
6’’ 8’’
−30
8’’
−40 5
−50
−15
−60 0
0.2
0.4
0.6
0.8
−20 1
Jiang et al. FEM Low Frequency 0.2
0.4
0.6
0.8
1
frequency (GHz)
frequency (GHz)
(a)
(b)
Fig. 13. (a) Complex permittivity and (b) specular power reflection for a metal-backed pyramid absorber with period of 8 in and height of 24 in at normal incidence.
R=70 18
,,
,, R=70
6
,,
8
Fig. 14.
,,
,,
Side view of pyramids with flat, concave, and convex side surfaces.
except at very low frequencies, the flat-surface pyramid outperforms both the the concave- and the convex-surface pyramids. 3.2. Frequency Selective Structures Frequency selective structures (FSSs) are 2-D arrays of metal and/or dielectric configurations, whose scattering response is characterized by a passband or stopband, at which total transmission or reflection occurs. FSSs have been analyzed using different methods. Analytical study of 1-D periodic dielectric layers was made in [28] based on a guided wave theory. Concerning numerical methods, integral equation based methods, such as the MoM, can be employed [29]-[31]. Time-domain analysis of FSSs was reported recently using the FDTD method [32]. Since the FEM can easily model irregular shapes, it is often preferred for more complicated structures. Successful FEM analysis of FSSs has been reported in [4] and [33]. In addition, a hybrid method that combines the FEM with the general-
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0
−10
Flat Concave Convex
R (dB)
−20
−30
−40
−50
−60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
frequency (GHz) Fig. 15. Specular power reflection for metal-backed pyramid absorbers with flat, concave, and convex side surfaces at normal incidence.
ized scattering matrix (GSM) method through a cascading procedure was proposed to maximize efficiency [34]. Figure 16 shows a generic FSS unit cell. It may consist of one or several dielectric layers which contain metal surfaces or structures. For simplicity, the metal surfaces are considered infinitesimally thin and perfectly conductive, but this is not required in the formulation. In fact, because of the volumetric discretization, the metal and dielectric can have any shape and composition. The unit cell is enclosed by four side surfaces and a top and a bottom surface. On the four side surfaces, the periodic boundary conditions (23) are imposed. On the top and bottom surfaces, the periodic radiation boundary conditions (35) are imposed with the incident term in (35) excluded for the bottom surface Sb . After the electric field is solved for everywhere inside the unit cell, the reflection coefficient can be calculated using (64) and the transmission coefficient can be calculated similarly as T =
Hscat t Hinc t
˜¯ ˜ = −2jk0 Y0 G(k x0 , ky0 ) · M00
(60)
˜ 00 is calculated from (32) with the surface integration carried out where M on Sb . First, we consider a 1-D periodic dielectric layer whose permittivity
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Unit Cell
Conducting Surfaces Fig. 16.
Front view of a generic frequency selective structure.
alternates between ²r1 = 1.44 and ²r2 = 2.56 along the x direction, as illustrated in Fig. 6(a). A TE plane wave is incident from one side of the slab at the incident angle θ = 45o . Two cases are calculated, with normalized slab height h/d chosen to be 1.713 and 2.037, as specified in [28]. The computed reflection coefficients are shown in Figs. 17(a) and 17(b). For comparison, the analytical solutions by Bertoni et al. [28] are also plotted. It is shown that the FEM accurately predicts the analytical resonant bands in both cases. We also note that only five to six expansion points are used in the fast frequency sweep, but all the detailed features have been captured. 1
1 FEM Bertoni
Magnitude Reflection
Power Reflection
0.8
0.6
0.4
0.2
0 5.9
6.2
6.5
6.8
7.1
Normalized wavenumber k0h
(a)
7.4
0.8
FEM Bertoni
0.6
0.4
0.2
0
2.5
3
3.5
4
4.5
5
Normalized wavenumber k0h
5.5
(b)
Fig. 17. (a) Specular magnitude reflection coefficient for h/d = 1.713. (b) Specular power reflection coefficient for h/d = 2.037.
The next example concerns a circular slot FSS whose unit cell geometry
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is shown in Fig. 18(a). Figures 18(b) and 18(c) show the power transmission coefficients at normal incidence for the free-standing FSS and the FSS printed on a 0.3-µm-thick dielectric substrate with permittivity ²r = 16, respectively. Both results are compared with those by Volakis et al. [4]. When a coarser mesh is used in the calculation of Fig. 18(c), the peak of our result shifts toward a longer wavelength. 2mm
2mm
1.6mm
Power Transmission
1 This method Volakis
0.8
0.6
0.4
0.2
0
3
4
1.8mm
5
6
7
8
9
10
Wavelength (µm)
(a)
(b)
Power Transmission
1 This method Volakis
0.8
0.6
0.4
0.2
0
10
15
20
Wavelength (µm)
25
30
(c) Fig. 18. (a) Top view of a circular slot FSS. (b) Specular power transmission coefficient for the free-standing FSS. (c) Specular power transmission coefficient for the FSS with substrate.
The next example involves an embedded FSS structure. The unit cell is a rectangular patch laid between two 1-mm-thick dielectric layers with permittivity ²r = 2.0, which is shown as the inset of Fig. 19. The power reflection coefficient for normal incidence is shown in Fig. 19. The result agrees well with the MoM result by Mittra et al. [29].
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1 0.9
2.5mm 0.8
Power Reflection
10mm
5mm
0.7 0.6
10mm
0.5
1mm 1mm
0.4 0.3 0.2
FEM Mittra, MoM
0.1 0
8
10
12
14
16
18
20
22
24
frequency (GHz)
Fig. 19.
Specular power reflection coefficient for the embedded patch FSS.
Finally, we present a comparison between numerical results and measured data. The structure of this example is illustrated in Fig. 20(a). Two rectangular microstrip patches reside on the top and bottom surfaces of a uniform dielectric slab with permittivity ²r = 2.5. A PEC screen is placed in the middle of the slab and a rectangular slot is cut on the screen so that the two patches are coupled. The transmission coefficient for this FSS structure has been measured using the waveguide simulation measurement technique in [31]. The comparison between the FEM result and the measured data is shown in Fig. 20(b). In the calculation, the incident angle varies according to the measurement setup described in [31]. 3.3. Phased Array Antennas Phased array antennas are of great importance in modern radar and communication systems. Recent integrated antenna technology has brought large arrays into widespread applications, and a good understanding of such arrays generally requires numerical analysis. Accurate prediction of array parameters using numerical methods not only reduces development costs and design time but also renders invaluable information to the design engineers. Available numerical methods include the MoM, the FEM, and some hybrid methods [3], [7], [8], [35], [36]. Among these methods, the MoM has been most extensively employed for numerical analysis, and verification has been made by comparing with experiment results for various
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28mm 36.07mm 28mm
2mm 1.6mm
8mm
Magnitude Transmission
1
0.8
FEM Measured
0.6
0.4
0.2
1.6mm 0 2.5
34.04mm
(a)
2.75
3
3.25
3.5
3.75
4
Frequency (GHz)
(b)
Fig. 20. (a) Geometry of the aperture-coupled patch FSS. Dielectric filling is not shown in the figure. (b) Calculated and measured specular magnitude transmission coefficients.
antenna configurations. Increasing demands on array performances often require complicated designs with inhomogeneous material composition and irregularly shaped conducting patches. To accurately model the electromagnetic field behavior in each array element and the mutual coupling between the elements, a 3-D full wave analysis is necessary. As another aspect of numerical modeling, an accurate analysis of radiation problems is impossible without a careful modeling of the feeding structures, which itself can be complicated for a successful design. In view of these reasons, the FEM appears to be most suitable as an analysis tool due to its general formulation and versatility in geometry modeling. This subsection describes the use of the FEM for a 3-D radiation analysis of infinite array antennas. The FEM discretization is confined to a single unit cell whose side surfaces and top surface are imposed with appropriate periodic boundary conditions. The coaxial line feed is modeled in an accurate manner using the FEM mesh. On the coaxial port aperture, a waveguide port condition is imposed to truncate the mesh. The numerical results obtained from this precise coaxial modeling are compared with the results obtained using the short current probe feed. For wide-band calculations, the AWE technique is used to accelerate frequency and angular sweeps. Numerical examples are given for microstrip-patch arrays and flared notch antenna arrays.
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3.3.1. Feed Modeling An important numerical issue in the analysis of radiation problems is the proper modeling of the feeding structures. For the MoM, a precise modeling of the detailed feeding structure is difficult, due to the 3-D nature of the feeding structures. In [35], an idealized current probe was used to model the coaxial feed. As shown in Fig. 21(a), the simplified probe-feed model consists an idealized current filament with an infinitesimal radius and a short electrical length. A constant current amplitude and phase are assumed on the filament. Although the simplified model predicts fairly accurate results for microstrip array antennas with a thin substrate and reveals very useful information on scan characteristics, it does not take into account the effects of the probe radius and the variation of the current along the probe, and thus is not applicable to microstrip-patch arrays with electrically thick substrates. On the other hand, accurate feed modeling can be easily realized using the FEM approaches. By applying a very fine mesh to the feed region, one can accurately model the detailed geometries of the feeding structures. For a coaxial feed, a conducting probe is modeled as the extension of the central conductor of the coaxial line, as shown in Fig. 21(b), and the PEC boundary condition is imposed on all the conducting surfaces. On the coaxial aperture, the waveguide port condition given in Subsection 2.5 is imposed. Using the above approach, we are able to include the effects of the probe radius as well as the current variation along the probe. The rapid field variation around the coaxial-substrate junction region is also captured.
I0 sw (a) Fig. 21.
(b)
Illustration of (a) short current probe feed and (b) coaxial line feed.
Both simplified and accurate feed models can be easily implemented in a FEM code. For the simplified model, the source current distribution Js is assumed to be Js = I0 δ(x0 , y0 )
0
