Contents Introduction 导言
In recent years, there has been an increase in the demand for low-frequency (30–1000 MHz) electromagnetic absorbers [1] and [2]. This has spawned an emergence of new absorber designs and structures; two examples are alternating wedges and hollow pyramids [1], [3]. Other structures of great interest are honeycomb composites. These composites are widely used because of their high strength-to-weight ratio. Applications include substrates, shielding materials, low-weight antenna reflectors, aircraft composites, and others. The field interaction with these types of structures can be obtained by full numerical means. However, while this approach is accurate, it can be very time-consuming.
近年来，对低频（30-1000 MHz）电磁吸收器 [1] 和 [2] 的需求不断增加。这催生了新的吸收器设计和结构的出现；交替楔形和空心金字塔 [1] 、 [3] 就是两个例子。其他备受关注的结构还有蜂窝复合材料。这些复合材料因其高强度重量比而被广泛使用。应用领域包括基板、屏蔽材料、低重量天线反射器、飞机复合材料等。与这些类型结构的场相互作用可以通过全数值方法获得。不过，这种方法虽然精确，却非常耗时。
Recently, an efficient method for analyzing periodic structures, known as homogenization, has been used to solve problems of this type when the period of the structure is small compared to the wavelength. Only a few of these published results are applicable to electromagnetic problems: [4] and [5] for corrugated impedance surfaces, [6] and [7] for wire grids and conducting strips, [8]–[9] [10] for perfectly and nonperfectly conducting rough surfaces, [11] [12] [13] for pyramidal electromagnetic absorbers, and [14] for propagation through concrete block walls. Even though the homogenization technique assumes that the period of the structure is small compared to a wavelength, results given in [12] and [15]–[18] indicate that the homogenization models can be accurate for periods as large as
最近，当结构的周期与波长相比很小时，一种分析周期性结构的高效方法（即均质化）被用于解决此类问题。这些已发表的结果中只有少数适用于电磁问题： [4] 和 [5] 适用于波纹阻抗表面， [6] 和 [7] 适用于线栅和导电带， [8] - [9] [10] 适用于完全导电和非完全导电粗糙表面， [11] [12] [13] 适用于金字塔形电磁吸收体， [14] 适用于穿过混凝土砌块墙的传播。尽管均质化技术假定结构的周期与波长相比很小，但 [12] 和 [15] - [18] 中给出的结果表明，均质化模型可以精确到
Homogenization is a technique developed in the early 1970's, primarily by a group of French mathematicians (see [4] and [19]–[25]). This asymptotic technique is based on the method of multiple-scales associated with the microscopic and macroscopic field variations in the periodic structure. In most situations, only the average (slowly varying) fields are of interest, and not the microstructure of the fields. Homogenization allows the separation of the average field from the microstructure. It is then possible to show that the average fields of a composite periodic structure satisfy Maxwell's equations for a homogeneous medium. The effective (or equivalent) material properties of this homogeneous medium are related to the properties of the composite structure.
均质化是 20 世纪 70 年代初主要由一群法国数学家开发的一种技术（见 [4] 和 [19] - [25] ）。这种渐近技术基于与周期结构中微观和宏观场变化相关的多尺度方法。在大多数情况下，人们只关心平均（缓慢变化的）场，而不关心场的微观结构。均质化可以将平均场与微观结构分离开来。这样就可以证明，复合周期结构的平均场满足均质介质的麦克斯韦方程。这种均质介质的有效（或等效）材料特性与复合结构的特性相关。
In this paper, we present approximate closed-form expressions for the effective material properties of two-dimensionally periodic structures with unit cells that take three different forms: the cross-section of periodically alternating wedges, the cross-section of hollow pyramids, and honeycomb composites. An exact general expression for the effective properties for any arbitrary geometry is not possible. We would like stress that the purpose of the paper was to address three specific commonly used geometries. However, in the discussion of these three geometries, an approach which can be used for a variety of other geometries is presented. The approximate expressions presented here are compared to results obtained from a numerical solution of the actual cross-sectional geometry. We also present reflection coefficients calculated using the effective-material-property model and show comparisons of these results to a full-wave three-dimensional (3-D) finite-element (FE) solution of the reflection problem. A discussion is given in Section III on how the results presented here and elsewhere can be used to investigate other types of periodic structures.
本文提出了具有三种不同形式单元格的二维周期结构的有效材料特性近似闭式表达式：周期交替楔形截面、空心金字塔截面和蜂窝复合材料。任何任意几何形状的有效特性都不可能有精确的一般表达式。我们想强调的是，本文的目的是讨论三种特定的常用几何形状。不过，在讨论这三种几何形状时，我们提出了一种可用于其他各种几何形状的方法。本文介绍的近似表达式与实际截面几何图形的数值求解结果进行了比较。我们还介绍了使用有效材料特性模型计算的反射系数，并将这些结果与反射问题的全波三维（3-D）有限元（FE）解决方案进行了比较。 Section III 中讨论了如何将此处和其他地方的结果用于研究其他类型的周期性结构。
More comments are in order on the need for effective-material model for the periodic structures analyzed in this paper. Obviously, the reflection and transmission properties of any one of the structures can also be obtained with full-wave numerical approaches. However, numerical approaches do not lend themselves well to design procedures due to the complexity (the entire structure would have to be spatially resolved) and time of computation. The use of approximate formulas for the effective material properties are very useful, especially when designing low frequency absorbers. Once the periodic structure is replaced by an effective medium the reflection properties can be calculated very efficiently [11]. In fact, most manufacturers use various forms of approximated formulas for the effective material properties when designing various types of absorbers for the 30 MHz to 3 GHz frequency range. These close form expressions for the different types of composite structures give individuals a means to determine and optimize the performance of a structure without the need to resort to time consuming full numerical approaches.
关于本文分析的周期性结构是否需要有效材料模型的问题，还需要做更多的评论。显然，任何一种结构的反射和透射特性都可以通过全波数值方法获得。然而，数值方法由于其复杂性（必须对整个结构进行空间解析）和计算时间，并不能很好地用于设计程序。使用有效材料特性的近似公式非常有用，尤其是在设计低频吸收器时。一旦用有效介质取代周期结构，就可以非常高效地计算出反射特性 [11] 。事实上，大多数制造商在设计 30 MHz 至 3 GHz 频率范围内的各类吸波材料时，都会使用各种形式的有效材料特性近似公式。这些针对不同类型复合结构的近似表达式为个人提供了确定和优化结构性能的方法，而无需采用耗时的全数值方法。
We need to also make a few comments here about the applications of the three structures analyzed in this paper. All three of the structures presented in the paper are at the present being used as absorbing structures for electromagnetic absorbers as well as other applications. Alternating wedge absorbers are often used in anechoic and semi-anechoic chambers that operate in the 30 MHz-to-5 GHz frequency range. In fact, one chamber manufacturer that designs and builds chambers in this frequency range uses various variations of an alternating absorber design. Hollow pyramid absorber are being investigated and used as an alternative to solid pyramids. When compared to solid pyramids, the hollow pyramid offers additional degrees of freedom in design parameters. This can be advantageous when optimizing the (broadband) performance of absorber geometries. Finally, new high power absorbers are being developed that are constructed out of pyramidal shapes with honeycomb cross-section. The reason for this cross-section is so that air can be blown through the pyramidal absorber in order for cooling. Also, honeycomb structures are used for other applications (e.g., antenna substrate materials).
在此，我们还需要对本文所分析的三种结构的应用做一些评论。本文介绍的三种结构目前都被用作电磁吸收器的吸收结构以及其他应用。交替楔形吸波材料通常用于在 30 MHz 至 5 GHz 频率范围内工作的消声室和半消声室。事实上，一家在此频率范围内设计和制造电波暗室的制造商就使用了各种不同的交替吸收器设计。目前正在研究空心金字塔吸收器，并将其用作实心金字塔的替代品。与实心金字塔相比，空心金字塔为设计参数提供了更多的自由度。这在优化吸收器几何结构的（宽带）性能时非常有利。最后，目前正在开发的新型高功率吸收器是由具有蜂窝状横截面的金字塔形状构成的。采用这种横截面的原因是，空气可以吹过金字塔形吸收器进行冷却。此外，蜂窝结构还可用于其他用途（如天线基板材料）。
Theory 理论
When electromagnetic fields interact with periodic structures, the fields exhibit microscopic and macroscopic variations. In a large class of periodic electromagnetic problems, only the macroscopic (or global) variations is of interest. The technique of homogenization allows for the investigation of this global field variation. Homogenization uses asymptotic expansions and the concept of multiple-scales to expand the
当电磁场与周期性结构相互作用时，场会呈现微观和宏观变化。在一大类周期性电磁问题中，只有宏观（或全局）变化令人感兴趣。均质化技术可以研究这种全局场变化。均质化利用渐近展开和多尺度概念，在具有慢（宏观）和快（微观）变化的渐近幂级数中展开
The zeroth-order fields
\eqalignno{ \nabla\times(\bar{E}^{o})_{\rm av}&= -j\omega[\mu^{h}]\cdot (\bar{H}^{o})_{\rm av} \cr \nabla\times(\bar{H}^{o})_{\rm av}&= j\omega[\epsilon^{h}]\cdot (\bar{E}^{o})_{\rm av}. &\hbox{(1)} } These equations state that the average fields satisfy Maxwell's equations in an anisotropic homogeneous medium characterized by the tensors \eqalignno{ (\epsilon\bar{E}^{o})_{\rm av}&\equiv [\epsilon^{h}] \cdot (\bar{E}^{o})_{\rm av} \cr (\mu\bar{H}^{o})_{\rm av}&\equiv [\mu^{h}] \cdot (\bar{H}^{o})_{\rm av} &\hbox{(2)} }零阶场
\eqalignno{ \nabla\times(\bar{E}^{o})_{\rm av}&= -j\omega[\mu^{h}]\cdot (\bar{H}^{o})_{\rm av} \cr \nabla\times(\bar{H}^{o})_{\rm av}&= j\omega[\epsilon^{h}]\cdot (\bar{E}^{o})_{\rm av}. &\hbox{(1)} } 这些方程表明，在以张量 \eqalignno{ (\epsilon\bar{E}^{o})_{\rm av}&\equiv [\epsilon^{h}] \cdot (\bar{E}^{o})_{\rm av} \cr (\mu\bar{H}^{o})_{\rm av}&\equiv [\mu^{h}] \cdot (\bar{H}^{o})_{\rm av} &\hbox{(2)} } ，其中 Assuming a medium which is invariant in the
\eqalignno{ [\epsilon^{h}]&= \left[\matrix{ \epsilon_{x}&0&0\cr 0&\epsilon_{y}&0\cr 0&0&\epsilon_{z}}\right]\cr [\mu^{h}]&=\left[\matrix{ \mu_{x}&0&0\cr 0&\mu_{y}&0\cr 0&0&\mu_{z}}\right]. &\hbox{(3)} } For the lattices and element geometries studied in this paper, symmetry renders the transverse effective material parameters identical; i.e., \eqalignno{ \nabla_{\xi}\times\bar{E}^{o} &= 0 \cr \nabla_{\xi}\times\bar{H}^{o} &= 0 &\hbox{(4)}\cr \noalign{\rm and} \nabla_{\xi}\cdot(\epsilon\bar{E}^{o}) &= 0 \cr \nabla_{\xi}\cdot(\mu\bar{H}^{o}) &= 0 &\hbox{(5)} }\bar{\xi} = {1\over p}(x\bar{a}_{x}+y\bar{a}_{y}) \eqno{\hbox{(6)}}假设介质在
\eqalignno{ [\epsilon^{h}]&= \left[\matrix{ \epsilon_{x}&0&0\cr 0&\epsilon_{y}&0\cr 0&0&\epsilon_{z}}\right]\cr [\mu^{h}]&=\left[\matrix{ \mu_{x}&0&0\cr 0&\mu_{y}&0\cr 0&0&\mu_{z}}\right]. &\hbox{(3)} } 给出。对于本文研究的晶格和元素几何结构，对称性使得横向有效材料参数相同，即 \eqalignno{ \nabla_{\xi}\times\bar{E}^{o} &= 0 \cr \nabla_{\xi}\times\bar{H}^{o} &= 0 &\hbox{(4)}\cr \noalign{\rm and} \nabla_{\xi}\cdot(\epsilon\bar{E}^{o}) &= 0 \cr \nabla_{\xi}\cdot(\mu\bar{H}^{o}) &= 0 &\hbox{(5)} } 其中 \bar{\xi} = {1\over p}(x\bar{a}_{x}+y\bar{a}_{y}) \eqno{\hbox{(6)}} 其中 A. Bounds for Effective Material Parameters
A.有效材料参数的界限
In the past, there has been a great deal of attention toward determining the effective properties of composite regions; a survey of this work can be found in [26]. The present paper deals exclusively with two-phase media; i.e.,
\eqalignno{ \epsilon_{\rm HS}^{L}&\equiv \epsilon_{0} {(1+g)\epsilon_{a}+(1-g)\epsilon_{0}\over (1-g)\epsilon_{a}+(1+g)\epsilon_{0}} &\hbox{(7a)}\cr \epsilon_{\rm HS}^{U}&\equiv \epsilon_{a} {(2-g)\epsilon_{0}+g\epsilon_{a}\over g\epsilon_{0}+(2-g)\epsilon_{a}}. &\hbox{(7b)} } and when 过去，人们对确定复合区域的有效特性给予了极大关注；有关这方面工作的概述，请参阅 [26] 。本文只讨论两相介质；即
\eqalignno{ \epsilon_{\rm HS}^{L}&\equiv \epsilon_{0} {(1+g)\epsilon_{a}+(1-g)\epsilon_{0}\over (1-g)\epsilon_{a}+(1+g)\epsilon_{0}} &\hbox{(7a)}\cr \epsilon_{\rm HS}^{U}&\equiv \epsilon_{a} {(2-g)\epsilon_{0}+g\epsilon_{a}\over g\epsilon_{0}+(2-g)\epsilon_{a}}. &\hbox{(7b)} } ，当 
\eqalignno{ \epsilon_{\rm Li}^{L} &= \int_{0}^{p} {dy\over \int_{0}^{p} {dx\over \epsilon(x,y)}} &\hbox{(8a)}\cr \epsilon_{\rm Li}^{U} &= {1\over \int_{0}^{p} {dx\over \int_{0}^{p}\epsilon(x,y)dy}} &\hbox{(8b)} }Lichtenecker (Li) 边界 [28] 、 [29] 取决于单元格的具体几何形状。对于二维晶格，当双周期结构的单元格包含对称性，使
\eqalignno{ \epsilon_{\rm Li}^{L} &= \int_{0}^{p} {dy\over \int_{0}^{p} {dx\over \epsilon(x,y)}} &\hbox{(8a)}\cr \epsilon_{\rm Li}^{U} &= {1\over \int_{0}^{p} {dx\over \int_{0}^{p}\epsilon(x,y)dy}} &\hbox{(8b)} } ，其中 Using a finite element (FE) program, the
使用有限元 (FE) 程序，可以获得周期单元（特定周期结构）的
Once the composite periodic structure is reduced to an effective medium, the interaction of the fields with this composite structure can be determined with classical layered-media approaches or transmission line methods. In the case where the cross-section varies with
一旦将复合周期结构简化为有效介质，就可以用经典的分层介质方法或传输线方法确定场与这种复合结构的相互作用。在横截面随
Results 成果
We analyzed the effective material properties of periodic structures based on three specific geometries: honeycomb composites, alternating wedges, and hollow pyramids (see Figs. 1 –3). Finite element (FE) results are compared to the theoretical upper and lower bounds for the effective material properties defined in the previous section. For each geometry, the theoretical formula (or combinations of formulas) that gives a “best-fit” representation of the effective material properties was determined.
我们根据蜂窝复合材料、交替楔形结构和空心金字塔（见 Figs. 1 –3 ）这三种特定几何结构分析了周期性结构的有效材料特性。将有限元（FE）结果与上一节中定义的有效材料特性的理论上下限进行比较。针对每种几何形状，确定能 "最佳拟合 "有效材料特性的理论公式（或公式组合）。
All results were calculated for two-phase media, with one of the phases being vacuum
所有结果都是针对两相介质计算得出的，其中一相为真空
A. Honeycomb Composites A.蜂窝复合材料
图 4.蜂窝结构的有效横向介电常数
蜂窝复合材料是六边形晶格几何的一种实现形式（见 Fig. 1 ）。我们只考虑蜂窝是包围空六角形圆柱体（真空）的高密度介质的结构。Perrins 等人的研究 [30] 指出，这种类型的几何结构具有（宏观上的）旋转不变性，我们也通过数值验证了这一点。因此，对于这种结构，
We calculated the effective transverse permittivity
\epsilon_{t, {\rm Honeycomb}}\simeq \epsilon_{\rm HS}^{U} \eqno{\hbox{(9)}} as a proper best-fit.我们计算了
\epsilon_{t, {\rm Honeycomb}}\simeq \epsilon_{\rm HS}^{U} \eqno{\hbox{(9)}} 作为合适的最佳拟合。B. Alternating Wedges B.交替楔形
图 5.交替楔形几何的有效横向介电常数
如 Fig. 2 所示，交替楔形吸收器是一种基于标准楔形吸收器的几何形状，其中一半楔形以交替方式旋转 90°。本文使用的交替楔形几何体的单元格的合适选择如 Fig. 2 所示。填充因子
Fig. 5 shows the calculated FE results for the effective transverse permittivity as a function of
\eqalignno{ \epsilon_{x}^{-1}&= (1-g)\epsilon_{0}^{-1}+g\epsilon_{a}^{-1}\cr \epsilon_{y}&=(1-g)\epsilon_{0}+g\epsilon_{a} &\hbox{(10)} } where \epsilon_{t}=\sqrt{\epsilon_{x}\epsilon_{y}}. \eqno{\hbox{(11)}}Fig. 5 显示了在
\eqalignno{ \epsilon_{x}^{-1}&= (1-g)\epsilon_{0}^{-1}+g\epsilon_{a}^{-1}\cr \epsilon_{y}&=(1-g)\epsilon_{0}+g\epsilon_{a} &\hbox{(10)} } 给出，其中 \epsilon_{t}=\sqrt{\epsilon_{x}\epsilon_{y}}. \eqno{\hbox{(11)}} 这个表达式可以从 [32] , [33] 中各向异性介电常数棋盘的结果中获得。 Fig. 5 显示了基于 (11) 的结果。这些结果也能合理地估算出 
\epsilon_{t,{\rm AltWedge}}= \left[(\epsilon_{x}\epsilon_{y})^{{N-1\over 2}}\epsilon^{L}_{\rm HS}\right]^{{1\over N}}. \eqno{\hbox{(12)}}在周期性双相介质中，假设材料参数为实值，"密度较大 "的相
\epsilon_{t,{\rm AltWedge}}= \left[(\epsilon_{x}\epsilon_{y})^{{N-1\over 2}}\epsilon^{L}_{\rm HS}\right]^{{1\over N}}. \eqno{\hbox{(12)}} ，该表达式的结果也显示在 Fig. 5 中。这些数据针对的是介电常数的纯真实值， C. Hollow Pyramids C.空心金字塔
The hollow pyramidal absorber, in its simplest form, consists of four planar slabs of absorbing material joined together to make a pyramid with constant wall thickness [3] (see Fig. 3). Compared to the solid pyramid, the hollow pyramid offers an additional degree of freedom. This can be advantageous when optimizing the broadband performance of an absorber geometry.
最简单的空心金字塔形吸收器由四块平面吸收材料板组成，它们连接在一起，形成一个壁厚 [3] 不变的金字塔（见 Fig. 3 ）。与实心金字塔相比，空心金字塔提供了额外的自由度。这在优化吸收器几何形状的宽带性能时非常有利。
The fill factor of a hollow pyramid depends on the relative linear size of the dielectric-filled region. Using the parameters defined in Fig. 3, the fill factor is given by
\displaylines{ \epsilon_{\rm Li}^{U}\simeq \left[{2t/p\over \epsilon_{0}+2d(\epsilon_{a}- \epsilon_{0})/p}+ {1-2(t+d)/p\over \epsilon_{0}}\right.\hfill\cr + \left.{2d/p\over \epsilon_{0}+2(t+d)(\epsilon_{a}- \epsilon_{0})/p}\right]^{-1}. \quad{\hbox{(13)}} }Figs. 7 and 8 show 空心金字塔的填充因子取决于介质填充区域的相对线性尺寸。使用 Fig. 3 中定义的参数，填充因子由
\displaylines{ \epsilon_{\rm Li}^{U}\simeq \left[{2t/p\over \epsilon_{0}+2d(\epsilon_{a}- \epsilon_{0})/p}+ {1-2(t+d)/p\over \epsilon_{0}}\right.\hfill\cr + \left.{2d/p\over \epsilon_{0}+2(t+d)(\epsilon_{a}- \epsilon_{0})/p}\right]^{-1}. \quad{\hbox{(13)}} } Figs. 7 和 8 显示了根据该表达式得到的 
图 7.空心金字塔结构的有效横向介电常数
图 8.空心金字塔结构的有效横向介电常数
A few comments are in order about these results. We note first that for the case shown in Fig. 7(a), the HS “upper bound” is actually below the FE data, while the HS “lower bound” is above the FE result. As mentioned above, this is in agreement with the fact that only when the permittivities of the constituent phases are real do the HS results actually form bounds. When one of the permittivities is complex, reversal of the HS results such as we see here sometimes occurs. As for the accuracy of the various approximations, notice that the Li upper bound tends to have roughly the same graphical form as the FE results, although offset by a small amount. We observe that, for
对于这些结果，我们有必要做一些评论。我们首先注意到，对于 Fig. 7(a) 中所示的情况，HS 的 "上限 "实际上低于 FE 数据，而 HS 的 "下限 "则高于 FE 结果。如上所述，这与只有当组成相的介电常数为实数时，HS 结果才会实际形成边界这一事实相一致。当其中一个介电常数是复数时，HS 结果有时会发生逆转，比如我们在这里看到的情况。至于各种近似值的准确性，请注意 Li 上限的图形形式与 FE 结果大致相同，只是偏移量较小。我们注意到，对于
Guided by these observations, we have determined that the effective properties of this structure can be better approximated overall by a fitted version of the Li upper bound, given by
\displaylines{ \epsilon_{{\rm t, Hollow Pyr}}\simeq \left[{2t/p\over \mathhat{\epsilon}_{1}+2d(\mathhat{\epsilon}_{2}- \mathhat{\epsilon}_{1})/p} + {1-2(t+d)/p\over \mathhat{\epsilon}_{1}}\right.\hfill\cr + \left.{2d/p\over \mathhat{\epsilon}_{1}+2(t+d)(\mathhat{\epsilon}_{2}- \mathhat{\epsilon}_{1})/p}\right]^{-1}. \quad{\hbox{(14)}} } Here, \eqalignno{ &\left.\epsilon_{{\rm t, Hollow Pyr}}^{U}\right\vert_{t=0} = \left.\epsilon_{\rm HS}^{L}\right\vert_{t=0} \cr &\left.\epsilon_{{\rm t, Hollow Pyr}}^{U}\right\vert_{t=p/2-d} = \left.\epsilon_{\rm HS}^{U}\right\vert_{t=p/2-d} &\hbox{(15)} }\eqalignno{ \mathhat{\epsilon}_{1}&= \epsilon_{\rm HS}^{L} \left[\vphantom{\sqrt{\left({\epsilon_{\rm HS}^{L}}^2+{\epsilon_{\rm HS}^{U}}^2\right) \left(1-d_{0}+d_{0}^2\right)^2- 2\epsilon_{\rm HS}^{L} \epsilon_{\rm HS}^{U} \left(1-2d_{0}+d_{0}^2+2d_{0}^3-d_{0}^4\right)}}\epsilon_{\rm HS}^{U} \left(2-5d_{0}+5d_{0}^2-d_{0}^3\right) - \epsilon_{\rm HS}^{L} \left(d_{0}-d_{0}^2+d_{0}^3\right)\right.\cr &\quad - \left. d_{0}\left.\sqrt{\left({\epsilon_{\rm HS}^{L}}^2+{\epsilon_{\rm HS}^{U}}^2\right) \left(1-d_{0}+d_{0}^2\right)^2- 2\epsilon_{\rm HS}^{L} \epsilon_{\rm HS}^{U} \left(1-2d_{0}+d_{0}^2+2d_{0}^3-d_{0}^4\right)}\right]\right/\cr &\quad \left\{2(1-d_{0})\left[\epsilon_{\rm HS}^{U} (1-d_{0}) -\epsilon_{\rm HS}^{L} d_{0}\right]\right\}\cr \mathhat{\epsilon}_{2}&= \epsilon_{\rm HS}^{U} \left[\vphantom{\sqrt{\left({\epsilon_{\rm HS}^{L}}^2+{\epsilon_{\rm HS}^{U}}^2\right) \left(1-d_{0}+d_{0}^2\right)^2 - 2\epsilon_{\rm HS}^{L} \epsilon_{\rm HS}^{U} \left(1-2d_{0}+d_{0}^2+2d_{0}^3-d_{0}^4\right)}}\epsilon_{\rm HS}^{U} \left(1-2d_{0}+2d_{0}^2-d_{0}^3\right) - \epsilon_{\rm HS}^{L} \left(1-2d_{0}+2d_{0}^2+d_{0}^3\right)\right.\cr &\quad + \left.(1-d_{0})\left.\sqrt{\left({\epsilon_{\rm HS}^{L}}^2+{\epsilon_{\rm HS}^{U}}^2\right) \left(1-d_{0}+d_{0}^2\right)^2 - 2\epsilon_{\rm HS}^{L} \epsilon_{\rm HS}^{U} \left(1-2d_{0}+d_{0}^2+2d_{0}^3-d_{0}^4\right)}\right] \right/\cr &\quad \left\{2d_{0}\left[\epsilon_{\rm HS}^{U} (1-d_{0})-\epsilon_{\rm HS}^{L}d_{0}\right]\right\}, &\hbox{(16)} }在这些观察结果的指导下，我们确定这种结构的有效特性可以通过李氏上界的拟合版本得到更好的近似，其值为
\displaylines{ \epsilon_{{\rm t, Hollow Pyr}}\simeq \left[{2t/p\over \mathhat{\epsilon}_{1}+2d(\mathhat{\epsilon}_{2}- \mathhat{\epsilon}_{1})/p} + {1-2(t+d)/p\over \mathhat{\epsilon}_{1}}\right.\hfill\cr + \left.{2d/p\over \mathhat{\epsilon}_{1}+2(t+d)(\mathhat{\epsilon}_{2}- \mathhat{\epsilon}_{1})/p}\right]^{-1}. \quad{\hbox{(14)}} } 这里， \eqalignno{ &\left.\epsilon_{{\rm t, Hollow Pyr}}^{U}\right\vert_{t=0} = \left.\epsilon_{\rm HS}^{L}\right\vert_{t=0} \cr &\left.\epsilon_{{\rm t, Hollow Pyr}}^{U}\right\vert_{t=p/2-d} = \left.\epsilon_{\rm HS}^{U}\right\vert_{t=p/2-d} &\hbox{(15)} } 的要求决定、
\eqalignno{ \mathhat{\epsilon}_{1}&= \epsilon_{\rm HS}^{L} \left[\vphantom{\sqrt{\left({\epsilon_{\rm HS}^{L}}^2+{\epsilon_{\rm HS}^{U}}^2\right) \left(1-d_{0}+d_{0}^2\right)^2- 2\epsilon_{\rm HS}^{L} \epsilon_{\rm HS}^{U} \left(1-2d_{0}+d_{0}^2+2d_{0}^3-d_{0}^4\right)}}\epsilon_{\rm HS}^{U} \left(2-5d_{0}+5d_{0}^2-d_{0}^3\right) - \epsilon_{\rm HS}^{L} \left(d_{0}-d_{0}^2+d_{0}^3\right)\right.\cr &\quad - \left. d_{0}\left.\sqrt{\left({\epsilon_{\rm HS}^{L}}^2+{\epsilon_{\rm HS}^{U}}^2\right) \left(1-d_{0}+d_{0}^2\right)^2- 2\epsilon_{\rm HS}^{L} \epsilon_{\rm HS}^{U} \left(1-2d_{0}+d_{0}^2+2d_{0}^3-d_{0}^4\right)}\right]\right/\cr &\quad \left\{2(1-d_{0})\left[\epsilon_{\rm HS}^{U} (1-d_{0}) -\epsilon_{\rm HS}^{L} d_{0}\right]\right\}\cr \mathhat{\epsilon}_{2}&= \epsilon_{\rm HS}^{U} \left[\vphantom{\sqrt{\left({\epsilon_{\rm HS}^{L}}^2+{\epsilon_{\rm HS}^{U}}^2\right) \left(1-d_{0}+d_{0}^2\right)^2 - 2\epsilon_{\rm HS}^{L} \epsilon_{\rm HS}^{U} \left(1-2d_{0}+d_{0}^2+2d_{0}^3-d_{0}^4\right)}}\epsilon_{\rm HS}^{U} \left(1-2d_{0}+2d_{0}^2-d_{0}^3\right) - \epsilon_{\rm HS}^{L} \left(1-2d_{0}+2d_{0}^2+d_{0}^3\right)\right.\cr &\quad + \left.(1-d_{0})\left.\sqrt{\left({\epsilon_{\rm HS}^{L}}^2+{\epsilon_{\rm HS}^{U}}^2\right) \left(1-d_{0}+d_{0}^2\right)^2 - 2\epsilon_{\rm HS}^{L} \epsilon_{\rm HS}^{U} \left(1-2d_{0}+d_{0}^2+2d_{0}^3-d_{0}^4\right)}\right] \right/\cr &\quad \left\{2d_{0}\left[\epsilon_{\rm HS}^{U} (1-d_{0})-\epsilon_{\rm HS}^{L}d_{0}\right]\right\}, &\hbox{(16)} } ，其中 
图 9.三种壁厚的空心金字塔结构的有效横向介电常数
In this way, the first equation in (15) anchors the fitted Li lower bound to the HS lower bound and the second equation in (15) anchors the fitted Li lower bound to the HS upper bound.
这样， (15) 中的第一个方程将拟合的 Li 下限锚定到 HS 下限， (15) 中的第二个方程将拟合的 Li 下限锚定到 HS 上限。
Figs. 7 and 8 also show results based on (14). The accuracy of
Figs. 7 和 8 也显示了基于 (14) 的结果。
Finally, using
最后，使用
图 10.空心金字塔形吸收体的反射系数
Discussion and Conclusion
讨论与结论
In this paper, we have presented closed-form expressions for the effective material properties of honeycomb composites and alternating-wedge and hollow-pyramidal absorbers. The expressions given here have been compared to and closely agree with calculated finite element results. These expressions can be used as material properties in effective-layer models to efficiently calculate the field interaction with the composite structures considered here.
在本文中，我们提出了蜂窝复合材料以及交替楔形和空心金字塔形吸波材料有效材料特性的闭式表达式。本文给出的表达式与有限元计算结果进行了比较，结果非常吻合。这些表达式可用作有效层模型中的材料属性，以有效计算场与本文所考虑的复合结构之间的相互作用。
For the results presented here, and those in [1], [11], [30], and [34], some general comments about approximations of composite materials can be made. In general, when a solid denser medium
对于本文以及 [1] 、 [11] 、 [30] 和 [34] 中的结果，可以就复合材料的近似值发表一些一般性评论。一般来说，当密度较大的固体介质
It needs to be stressed that the expressions for the effective material properties presented here are valid as long as the wavelength in the material is small compared to the spatial period of the structures, see [12] and [15]–[18]. Once either the frequency of operation or the material properties of the structures becomes too high, the traditional concept of an effective medium as described by quasistatic homogenization (in [11], for example) begins to break down. At the very least, effective material property models must be modified in order to capture the true physical behavior of the problem, and such a modification may not even be possible at all (see [10], [16], [18], and [37]). If the conductivity becomes too large (i.e., small skin-depth) the rapid and localized field behavior cannot be adequately described using quasistatic effective properties and more detailed analysis is needed. One reason for this is that standard effective medium models do not take into account the boundary layer fields that are present near the surface of a highly conducting structure. This point is illustrated in [10], where electromagnetic wave interaction of a periodic interface between a dielectric and highly conducting medium was analyzed. In [10] a comparison for the reflection coefficient obtained from an effective material property model to a more detailed boundary layer analysis is presented. This comparison shows that for larger skin-depth, effective material properties model give accurate results, whereas for small skin-depths, the effective medium model fails. The work in [10] showed that with a so-called “stiff” homogenization analysis, the rapid and localized field behavior for small skin-depth can be modeled in terms of electric and magnetic polarizability densities. We are presently investigating the use of this higher-order (stiff) homogenization analysis for modeling highly conducting composite structures. In fact, we have found that for highly conducting composite structures the effective material properties can become bianisotropic or chiral in nature. Details will be published elsewhere.
需要强调的是，只要材料中的波长与结构的空间周期相比较小，此处给出的有效材料特性表达式就有效，参见 [12] 和 [15] - [18] 。一旦工作频率或结构的材料特性变得过高，准静态均质化所描述的有效介质的传统概念（例如在 [11] 中）就会开始崩溃。为了捕捉问题的真实物理行为，至少必须修改有效材料属性模型，而这种修改甚至根本不可能实现（见 [10] 、 [16] 、 [18] 和 [37] ）。如果电导率变得过大（即表皮深度较小），则无法使用准静态有效特性充分描述快速和局部场行为，需要进行更详细的分析。原因之一是标准有效介质模型没有考虑到高导电结构表面附近的边界层场。 [10] 中说明了这一点，其中分析了介电质和高导电性介质之间周期性界面的电磁波相互作用。在 [10] 中，比较了从有效材料特性模型和更详细的边界层分析中获得的反射系数。比较结果表明，对于较大的表皮深度，有效材料特性模型能给出准确的结果，而对于较小的表皮深度，有效介质模型则失效。 [10] 中的工作表明，通过所谓的 "刚性 "均质化分析，可以用电极化率密度和磁极化率密度来模拟小表皮深度的快速局部场行为。我们目前正在研究如何使用这种高阶（刚性）均质化分析来模拟高导电复合结构。事实上，我们已经发现，对于高导电复合结构，有效的材料特性可以是各向异性的，也可以是手性的。详细内容将在其他地方发表。
The tensor representation of
ACKNOWLEDGMENT 致谢
The authors thank D. A. Hill at the National Institute of Standards and Technology for his helpful technical discussions.
作者感谢美国国家标准与技术研究院的 D. A. Hill 进行的有益的技术讨论。