WO1995005013A1

WO1995005013A1 - Method for constructing an absorber and absorber structure

Info

Publication number: WO1995005013A1
Application number: PCT/US1994/008850
Authority: WO
Inventors: Kefeng Liu
Original assignee: Ray Proof Shielding Systems Corporation
Priority date: 1993-08-06
Filing date: 1994-08-04
Publication date: 1995-02-16
Also published as: AU7554994A

Abstract

An absorber (2) and a method for constructing such absorber is provided for use, for example, in an anechoic chamber. The absorber includes one or more backing layers (8) and at least one tapered section (4) extending therefrom. The total length of the absorber (10), the length (12) and shape of the tapered section (4) and the thickness of each backing layer (16) are based upon optimization of the absorber (2), including optimization of the tapered section (4).

Description

METHOD FOR CONSTRUCTING AN ABSORBER AND ABSORBER STRUCTURE

A portion of the disclosure of this patent document contains material which is subject to copyright protection. The copyright owner has no objection to the facsimile reproduction by anyone of the patent document or the patent disclosure, as it appears in the Patent and Trademark Office patent file or records, but otherwise reserves all copyright rights whatsoever.

BACKGROUND OF THE INVENTION Field of the Invention

The present invention relates to a method for constructing an absorber such as a pyramidal absorber for use, for example, in an anechoic chamber by which is meant a typical anechoic or semi- anechoic chamber. The absorber includes one or more backing layers and at least one tapered section, extending therefrom. By tapered section, sometimes also referred to herein as a tip, cone, pyramid, taper, or the like, is meant a structure having a closed plane base and a surface formed by line segments joining every point of the boundary of the base to a common vertex, or to the boundary of a plane spaced from the base and having an area smaller than the area of the base. Examples include, without limitation, a pyramid, a truncated pyramid, a cone, and a truncated cone. In the method, an absorber performance evaluaton is performed which includes the application of homogenization and a transmission line method to the tapered section. Optimization of the tapered section and the backing layers is also performed. The method results in an absorber design which optimizes the electromagnetic wave absorption of the absorber.

Description Of Prior Art

Anechoic chambers are useful in, for example, antenna measurement and testing, radar cross-section evaluation and electromagnetic compatibility and susceptibility testing. Such chambers provide a test environment substantially free of electromagnetic noise and echo. Anechoic chambers are particularly

SUBSTITUTE SHEET (RULE 2φ useful in the measurement of electromagnetic radiation of electrical and electronic equipment being tested in such a chamber. In addition to the evaluation of such equipment for the purpose of design improvement and the like, such testing of electrical and electronic equipment is required by the Federal Communications Commission (FCC) and the equivalent government agencies in many other countries. FCC regulations exist which limit the degree of electromagnetic radiation permitted to be emitted by electrical and electronic equipment. The FCC requires that testing be conducted over a frequency range of 30 to 1000 MHz under open field site conditions. Until recently such testing was performed at an open area test site. However, such tests are adversely affected by weather conditions such as rain and wind. Since test results and the scheduling of testing are weather dependent, a more reliable and efficient means of testing has been sought. Anechoic chambers provide a means of satisfying this need.

There are many advantages to indoor testing using anechoic chambers rather than out-door testing. In addition to being immune to weather conditions, testing in anechoic chambers provides an environment which is immune to ambient RF noise typically present at out-door test sites. In addition, whereas an open area test site tends to be constructed in an area remote from facilities which manufacture the equipment being tested, an anechoic chamber can be located in a shielded enclosure adjacent to the production site. This will greatly reduce the cost of conducting compliance measurements to meet FCC requirements. In addition, having the test facility in proximity to the manufacturing facility will reduce the amount of time required during product development since it will not be necessary to transport the equipment to a remote facility each time a test is required.

The FCC allows the substitution of an anechoic chamber for open field site testing but requires that there be a correlation between anechoic chamber and open field site test results. In response to the transition from open field sites to the use of anechoic chambers industry has attempted to provide optimum facilities having a measured site attenuation within 4 dB of that of an ideal open field test site.

In the construction of an anechoic chamber, absorbers are provided to absorb the electromagnetic radiation emitted by the electrical and electronic equipment disposed within the chamber. Typically, such absorbers line the walls and ceiling, and sometimes also the floor, of the chamber, and the efficiency of the chamber will be dependent upon the design of the chamber and absorber. One type of absorber is a pyramidal absorber which includes one or more tapered sections which extend from a base formed from one or more backing layers having different absorbing properties. Typically, such a pyramidal absorber will be fabricated from low density, flexible foam. Pyramidal absorbers known in the art include regular pyramidal absorbers (rectangular and square) and twisted pyramidal absorbers. The present invention relates to the optimization of the absorber design including the optimization of the design of the tapered section of the absorber.

In considering the present state of the art, carbon loaded polyurethane foams are widely used to design broadband absorbers to simulate free space site testing conditions in an anechoic chamber. It is known in the art that in applications involving microwave frequencies the length of the absorber is often many times the wavelength of the incident wave to which the absorber is subjected. In such applications a gradual transition from the free space to the lossy loading can be designed to extend up to several wavelengths, and therefore a broadband and high performance absorber design is possible without undue restriction caused by physical dimensions. Due to this physical length advantage, it is also possible to design high frequency absorbers which perform at a very broad angle of incidence.

Polyurethane foam absorbers are also widely used in anechoic chambers designed to measure the compatibility of electrical and electronic devices. A frequency range of 30 to 1000 MHz is a band with respect to which FCC regulations require that the fields within a chamber be within 4 dB deviation of corresponding open field site test results. Although this requirement may not seem overly demanding , it is somewhat difficult to achieve without a superior performance of the absorber, particularly if the same chamber is also used for testing at higher frequency. A poor performance of a semi-anechoic chamber is in the low band of the frequency range (30 - 200 MHz). Without limitation, the present invention is particularly applicable to absorbers for use in low frequency applications.

It is known in the art to provide absorbers which have a high absorbing performance at low frequencies. For example, the use of single and multi-layer dispersive material such as a typical ferrite tile has resulted in a low coefficient of reflection at low frequencies. One problem incurred in such applications is that since the dispersion effected by such material is not completely controllable, the bandwidth of the absorber is very limited. In an effort to overcome this deficiency, anechoic chambers have been lined with ferrite tiles in front of which has been disposed polyurethane foam absorbers. However, such efforts have resulted in the cost of the chamber being very high.

Other efforts made to satisfy performance requirements have involved the use of a variety of different shaped anechoic chambers. Such efforts have also been costly and have not contributed to improvement of absorber design.

U.S. Patent no. 5,016,185 to Kuester et al. describes a process using a model to optimize the coefficient of reflection of a pyramidal cone absorbing structure over a desired frequency range. In this process homogenization is used to equate the absorbing structure to an effective layer subsequent to which an equivalent transmission line description of such an effective layer for parallel and perpendicular polarization and for various incident angles of the fields is developed. In this process, Riccati's equation is used to obtain the magnitude of the reflection coefficient at a plurality of frequencies, a pyramidal cone absorbing structure then being constructed. In essence, Kuester et al. use Hashin-Shtrikman and Jackson-Coriell bounds as formulas to homogenize the absorber tip. Such bounds are discussed at Z. Hashin and S. Shtrikman, "A variational approach to the theory of the effective magnetic permeability of multiphase materials," Journal of Applied Physics, vol. 33, pp. 3125-3131, 1962. and S.R. Coriell and J.L. Jackson, "Bounds on transport coefficients of two-phase materials," Journal of Applied Physics, vol. 39,pp. 4733-4736, 1968. Further discussion of homogenization can be found at E. K. Kuester, "The fundamental properties of transversely periodic lossy waveguides," tech. rep., Electromagnetic Laboratory, University of Colorado, Boulder, CO 80309, IBM Research Contract Rep. No. 86, Aug. 1986; E. K Kuester, "Low-frequency properties of transversely periodic lossy waveguides," tech. rep., Electromagnetic Laboratory, University of Colorado, Boulder, CO 80309, IBM Research Contract Rep. No. 86, Oct. 1987; and E. K. Kuester and C. L. Holloway, "Plane-wave reflection from inhomogeneous uniaxially anisotropic absorbing dielectric layers," tech. rep., Electromagnetic Laboratory, University of Colorado, Boulder, CO 80309, IBM research Contract Rep. No. 97, May 1989. In the 5,016,185 patent Kuester et al. simulate a pyramidal absorber cone with a continuously tapered transmission line using the method of homogenization and solve its reflection coefficient using the Riccati equation. By combining the reflection coefficients of the cone with the reflection coefficients of the base, the total reflection coefficient of the absorber is obtained. However, the Riccati equation formulated for the pyramidal cone absorber does not have an analytic solution and a numerical solution inherently results in a certain level of numerical errors. In addition, Kuester et al. do not subject the tapered section of the pyramidal absorber to optimization. Since the tapered section typically forms about 70% of the total absorber length, excluding the tip design from the optimization process will limit the absorber design capability considerably.

It has been demonstrated that a combination of the homogenization analysis and a numerical optimization model makes it possible to achieve optimum design of absorber performance in the frequency range where the homogeneous theory is valid. In particular, reference is to H.T. Gibbons, "Design of backing layers for pyramidal absorbers to minimize low frequency reflection," tech. rep., Electromagetic Laboratory, University of Colorado, Boulder, CO 80309, IBM Research Contract Rep. No. 105, July 1990. However, in Gibbons' work, the length of the taper of the absorber is fixed, and the optimization is performed by varying parameters of the backing layers. In such a process, although the optimization procedure is stable and the computer time used is minimized, the design obtained by this process is not the most optimum design possible within the physical restriction of the absorber size. For example, the reported theoretical optimum designs do not exhibit any better than 14 dB performance for a 96 inch absorber using practical sets of dielectric materials, and better than 18 and 15 dB have been reported in corresponding prior art 96-inch absorber products in the frequency range where the homogeneous theory is valid at 30 MH_z. In addition, like Kuester, et al . , Gibbons does not attempt to optimize the cone or tip of the pyramidal absorber.

A rigorous periodical moment method (PMM) for analyzing absorber performance has been reported. In particular, reference is to C. F. Yang, W. D. Burnside, and R. C. Rudduck, "A periodical moment method solution for TM scattering from lossy dielectric bodies with application to wedge absorber," IEEE Trans. Antennas Propagat., vol. AP-40, pp. 652-660, June 1992 and W. D. Burnside, R.C. Rudduck, and C. F. Yang, "New wedge and pyramidal absorber designs," Proceeding of Antenna Measurement Techniques Association, vol. 14th Annual Symposium, pp. 16-9-16-14, Oct. 1992. However, such method relies upon a computationally intensive computer program and is therefore more suitable for analysis than absorber design. Nonetheless, it has been shown that the homogenization model compares well with the PMM prediction for a majority of conventional absorber designs and reference is to "Personal communication with Jeffrey Gao of absorber development program of Ohio State University," AMTA Touring of OSU, vol. 14th Annual

SUBSTITUTE SHEET(RULE 26} International AMTA Symposium, Oct. 1992. Compared with PMM, the homogenization model requires only negligible computer time for an analysis.

The present invention is based upon the belief that the tip parameters (material properties, length of the tip, and the shape of tip) play the most important role in the global performance of the absorber and that omitting tip parameters in the optimization of the absorber will greatly limit the design capability. However, by including the tip parameters in the optimization process, the feasibility of the optimum design becomes much more complicated. First, the performance of the tip has to be evaluated each time during every step of the optimization update, and many steps of the optimization updates must be carried out before an optimum conditon is reached. The computer time for analyzing the tip comprises over 95% of the total time needed for the whole absorber performance evaluation. This results in a significant increase of computer power needed to implement the optimization process. Secondly, since the overall performance of the absorber is very sensitive to the change of parameters in the tip, it becomes much more difficult to determine the optimum condition of the design.

It is an object of the present invention to provide a method for constructing an improved absorber having a design which optimizes electromagnetic wave absorption of the absorber.

It is another object of the present invention to provide such a method which optimizes the tapered section of the absorber.

A further object of the present invention is to provide such a method which includes an improved numerical optimization procedure.

Yet another object of the present invention is to provide such a method which optimizes the tapered section of the absorber without the need for extensive computer time.

It is another object of the present invention to provide such a method which optimizes the tapered section and each backing layer of the absorber. A further object of the present invention is to provide such a method wherein the optimization procedure is stabilized.

Another object of the present invention is to provide an improved absorber for use in an anechoic chamber.

SUMMARY OF THE INVENTION

This invention achieves these and other results by providing a method for constructing an absorber, the method including the optimization of the tip of the absorber as well as each backing layer. A simple transmission line model of a finite number of layered media is introduced to simulate a homogenized absorber tip in order to allow a computer code to analyze the absorber tip with various geometries. Such a simple model allows analysis of the change of the geometry of the absorber tip without having to code many new formulas into the program. Therefore, many geometrical shapes of the absorber tip can be fed into an optimization program in order to determine the optimum design of the absorber tip. An empirical formula is also developed to determine the number of layers needed to maintain accuracy of the finite layered transmission line model without having to sacrifice the efficiency of the analysis. Since the absorber performance is very sensitive to the change of parameters in the absorber tip, the convergent stability of the optimization problem becomes fragile and therefore a dedicated nonlinear numerical optimization program is developed to enhance the stability of the optimum condition.

The method of the present invention relates to constructing an absorber of the type having a plurality of absorber elements each including one or more backing layers and a tapered section or tip extending therefrom. The method comprises the steps of:

(a) selecting input data sets with respect to which absorber performance is to be optimized;

(b) performing an absorber performance evaluation which, with respect to the tapered section, uses a transmission line method and homogenization; (c) performing an iterative optimization of the absorber performance evaluation including at least optimization of the tapered section; and

(d) constructing an absorber element having shape, size and material properties identified by the iterative optimization.

An absorber based upon the method of the present invention is also described herein.

BRIEF DESCRIPTION OF THE DRAWINGS

This invention may be clearly understood by reference to the attached drawings in which:

Figure 1A is a view of an absorber having one backing layer and a tapered section extending therefrom;

Figure IB is a view of an absorber having a plurality of backing layers and a plurality of tapered sections extending therefrom;

Figure 2 is a flow chart of the absorber optimization process of the present invention;

Figure 3 is a flow chart of the absorber performance evaluation program used in the process of Figure 2;

Figure 4 is a flow chart of the optimization algorithm used in the process of Figure 2;

Figures 5 to 11 are graphs illustrating the dielectric properties of nine select polyurethane foam materials for use in the manufacture of the absorber of the present invention;

Figure 12 is a graph illustrating the performance norm (in dB) vs. taper length (in feet) of seven 96-inch absorbers, each based upon one of the materials of Figures 5 to 13;

Figure 13 is a graph illustrating performance of optimized regular pyramidal absorbers, each based upon one of the materials of Figures 5 to 11, compared to 2B-prime;

Figure 14 is a graph illustrating a weighted design vs. best design possible;

Figure 15 is a configuration of a flat-top (truncated) pyramidal absorber design; Figure 16 is a graph depicting optimum performance of the design of Figure 15 compared to 2B-prime;

Figure 17 is a configuration of a curved flat-top (truncated) design;

Figure 18 is a graph depicting optimum performance of the design of Figure 17 compared to 2B-prime;

Figure 19 is a configuration of a curved sharp tip (pointed) absorber design;

Figure 20 is a graph depicting optimum performance of the design of Figure 19 compared to 2B-prime;

Figure 21 is a configuration of a bi-linear sharp tip (pointed) design;

Figure 22 is a graph depicting optimum performance of the design of Figure 21 compared to 2B-prime;

Figure 23 is a graph depicting optimum performance of a 96- inch absorber for 45-degree oblique incidence performance; and

Figure 24 is a graph depicting performance of an optimized 72- inch twisted pyramidal absorber vs. a 72 inch regular pyramidal absorber.

DESCRIPTION OF THE PREFERRED EMBODIMENT

Figures 1A and IB depict a typical broadband absorber element 2 comprising a plurality of tapered sections or tips 4, 4' and a base 6. The absorber may be fabricated of, for example carbon impregnated, polyurethane foam. Without limitation, Figure 1A depicts an absorber having one backing layer and one tapered section and Figure IB depicts an absorber having a plurality of backing layers and a plurality of tapered sections. Base 6 may have one (Figure 1A) or more (Figure IB) backing layers 8 having the same or different absorbing properties. Absorber 2 has a total length 10 which is equal to the physical length 12 of a tapered section 4, and the combined physical thickness 14 of the base 6 which is based upon the sum of the individual thicknesses 16 of each backing layer 8. In the process of the present invention, it is desired to construct an absorber 2 of optimum design by optimizing the electromagnetic wave absorption of the absorber. In the process, a variety of absorbing materials having measured dielectric constants are selected by the designer of the absorber. For example, in the embodiment discussed herein a plurality of sets of measured complex dielectric constants relating to various absorbing materials are selected by the designer. In addition, the designer selects the total physical length of the absorber, and the incident angle and frequency range for which optimum performance is desired. With reference to step 20 of the flow chart of Figure 2, these constraints serve as "givens" in the process of the present invention. At step 22 an initial combination of absorbing materials is chosen from the selected variety of absorbing materials for the tip and the backing layer(s) of the absorber. Such materials may be the same or different for the various layers and tapered section.

The designer also selects an initial design which includes a randomly selected shape and size of the absorber tip 4 and the size(s) of the backing layer(s) 8 for the initially chosen combination of absorbing materials. Such initial design is given in step 24.

The next process step involves absorber performance evaluation at step 26 and optimizing the absorber performance at step 28. The optimization process is an iterative procedure which will usually require a plurality of steps of optimization update before an optimum conditon is reached. In each update, at step 30 another combination of absorbing materials is chosen for the tapered section and backing layer(s) and the optimization procedure is repeated. This process will be repeated until the optimum conditon is satisfied, or if this is not possible, until the optimization procedure has been repeated as much as possible. The optimum condition accurs when the optimum minimum coefficient of reflection is satisfied.

The absorber performance evaluation step 26 is depicted in the flow chart of Figure 3. With reference to Figure 3, the absorber performance evaluation comprises the steps of (a) reading in complex dielectric constants for the tip and the backing layer(s) (step 30); (b) computing the cascaded wave impedance of each backing layer(s) for a given polarization and angle of incidence (step 32); (c) determining the number of subsections needed to analyze the absorber tip (step 34); (d) applying homogenization theory to compute the effective dielectric constants of each subsection; computing the wave impedance of each subsection; and combining the wave impedance of each subsection with the wave impedance of the backing layer(s) to obtain the total wave impedance (step 36); (e) computing the reflection coefficient at all given frequencies (step 38); and (f) constructing the objective function by applying frequency and angle weighting factors and constraint conditions (total physical length, physical shape, non- negative layer thickness) (step 40).

In considering the absorber performance evaluation of step 26, it is believed that optimization of the tip of the absorber is of primary importance in obtaining superior design results. Therefore, in process step (34) each tip is subdivided into a finite number of subsections of equal length. The number of subsections will control the accuracy of the computation of the reflection coefficient. If the number of subsections is large enough, the solution for a continuous cone will be approached. However, if the number of subsections is too large the efficiency of the analysis will be decreased. Therefore, it is important to determine the number of subsections which will satisfy the requirement of accuracy and also maintain the degree of efficiency desired. It is possible to calculate the optimum number of subsections N of the tapered section or tip 4 by use of the following equation:

N_s = Integer +10

where I ej is the magnitude of the complex permitivity, L is the length of the absorber tip, and λ₀ is the free space wave length. After the number of subsections of the tapered section is calculated, a method of homogenization is performed within each subsection as provided in process step (36) to simulate each subsection with a homogeneous layer of absorber material. In the homogenization process, the effective dielectric constants (real and imaginary) of each subsection are computed. In particular, in considering the homogenization process, it is known in the art that when the periods of a cone structure become small compared to wavelength, the average fields over a period rather than the exact variations of the fields within the microstructure of the period is a predominant indicator of the field problem. Homogenization involves the application of Maxwell's equations to the averaged fields in a homogenized medium to characterize wave reflection and wave propagation. The use of a homogenization method is well known in the art. For example, the homogenized longitudinal properties of pyramidal absorbers are known for regular pyramids. In the homogenization step, the effective longitudinal permitivity for regular pyramids can be expressed as: e_z = e . + A₂(e₂ - _βl) and for twisted pyramids, the effective longitudinal permitivity can be expressed as: e_z = e_χ + R(G₂ - e_x) where

2A_ if A₂ <i

R =

Similarly, the effective transverse material properties can be approximated using the Haskin-Shtrikman formula for regular geometries such as rectangular pyramids and the Jackson-Coriell formula for more general geometries such as twisted pyramids. In particular, in the homogenization step for regular pyramids the Haskin-Shtrikman bounds for regular pyramids are:

and the Jackson-Coriell bounds for twisted pyramids are:

In the above formula, e_]f e_? are pemitivities of two interfacing materials and A^ A_? = 1' _j, the fractional areas occupied by the two different materials.

It has been reported that Haskin-Shtrikman's lower bound has the best agreement with numerical results (< 4% error) for a square rod of pure real dielectric constant. Although concerns have been raised over the extension to complex valued e and e , the validity of such extension has been demonstrated. For twisted pyramid designs, a geometrical means value of the Jackson-Coriell bounds has been recommended which gives:

where

The foregoing formulas are known in the art and are useful in determining effective permitivity. Effective permeability can be homogenized in a similar manner.

After each sub-section has been subjected to homogenization to simulate each sub-section with a homogeneous layer of material, each subsection of the tapered section is modelled as a small section of a transmission line. When the inhomogeneity of the tapered section of each subsection is modelled by its effective homogenized material property, the field problem is simplified into a plane wave transmission and reflection problem in a multi-layered lossy material representing the averaged fields in the problem region. When considering a plane wave incident on the periodical arrays at an angle Θ off normal, field components can be decomposed into a superposition of perpendicular (or E ),and parallel (or H ) polarizations. Without losing generality, it can be assumed that the waves propagate only on an x - z plane and that therefore Maxwell's equations for the fields can be simplified. In particular. Maxwell's equations for the field of perpendicular (or E ) polarization can be simplified into:

where j = -f- 1 , and

/J_e£t(Z) = r- . ( Z ) sin²Θ «_βrr(z) = c,(z)

For the field of perpendicular (or H ) polarization. Maxwell's equations can be simplified into:

dH(z) = -jwe₀e_eff(z)E(z) dz where j = -f-1 , and sin Θ μ_etf(z) = μ_t(z)

G,(Z)

Therefore, in modelling each subsection of the tapered section as a small section of a transmission line in step 36, for each subsection the effective propagation constant can be obtained by

and the effective wave impedance can be obtained by

Process step (36) then calls for combining the wave impedance for each subsection, and the wave impedance of the backing layer(s), which was obtained by process step 32, to obtain a total wave impedance. To this end, the taper of the absorber tip is approximated as a finite number of cascaded lossy layers and the wave impedance at the interface of the (i + l)-th layer is related to the i-th interface by a straight forward transmission line equation in the form of

₇, Z(z_{) + jZ_{z.)tan(fl,(z.-)Δ*)

^{[ *'+1 " i)} Z_e(z + jZ(z_i) n(β_e(z_i)Δz)

where

Z.'+l + *i

Subsequent to obtaining the total equivalent impedance of the absorber, the reflection coefficient of the absorber is computed at step 38 at all given frequencies using the following equation which provides an analytical formula to compute the reflection level of the absorber at any given frequencies and incident angles:

Z(L) - η₀ total

Z(L) + η₀

To run the optimization program, an objective function must be provided. In process step 40 the objective function is provided by applying frequency and weighting factors, and constraint conditions (total physical length, physical shape, and non-negative layer thickness). A general form of the objective function to minimize the overall reflectivity within the required frequency range and incident angle can be expressed as

where _α and W₂ are positive weighting functions which value the importance of the reflection level at different frequencies and incident angles. However, since reflection levels are computed at discrete frequencies and at discrete incident angles, it is more convenient to compute the objective function by

It is important to use care in selecting the function of reflection levels ₂ ( T ) . In particular, it is known in the art that if we take W₂ (T) = I rl , although it is possible that the objective function will be minimized to a low value, reflection levels at a few entries will be unacceptably high. On the other hand, if the worst case reflection level ("infinity" norm) is chosen as the value for the objective function entries, the objective function will be fragile and unsmooth from iteration to iteration. It has been reported that a power norm of 20-th order is better suited for this type of optimization problem. Therefore, by applying frequency and angle weighting factors, the objective function can be explicitly written as

with w(f_J/Θ_J) being a positive weight factor reflection level control.

In addition to providing an objective function, it is necessary to incorporate constraint conditions into the optimization process since the absorber design will always be subjected to a number of physical restrictions. One way of doing this is to introduce penalty functions. The penalty function method is a known procedure for approximating constrained optimization problems by unconstrained problems. Such an approximation is accomplished by adding a high cost penalty function for violating constraint conditions. The penalty functions are required to have continuous first derivatives. Therefore, by applying constraint conditions (total physical length, physical shape, and non-negative layer thickness) a typical penalty function can be constructed as

{max[0,r_m(x)]}² if r_m(x) < 0

Pm(x) = { m(x)|² if r_m(x) = 0

{min[0, r_m(χ)]}² if r_m(x) > 0

and a completely defined objective function can therefore be expressed as

where c_m's are positive penalty constants. In essence, all of the physical constraints such as the total length of the tapered section, the dielectric constant, etc. have been implemented into the objective function so that the objective function does not violate such physical constraints.

Mathematically, extre a found by using the penalty function approaches exact extrema as c_m -» ∞. By properly choosing the value of penalty constants, the approximated extrema is within negligible error of the exact extrema. Since most absorber designs can only be implemented in the production line with limited accuracy, this approximation should be accurate enough. However, a great saving of computation and programming effort can be achieved.

Step 28 requires optimizing absorber performance and involves changing the design (shape and size) of the tip and the size(s) of the backing layer(s). The process is an iterative process. Such a process continues until the optimum condition is obtained or it exceeds a pre-set maximum number of iterations. The same optimization is repeated with the next combination of absorbing materials for the tip and the backing layer (step 30). In each repetition of the optimization process, the tapered section of the absorber is subdivided into a finite number of subsections of equal length with respect to which the absorber performance evaluation is applied.

The optimization process essentially involves the use of numerical techniques in a search of extrema (often minima) of a nonlinear objective function of many variables while satisfying identified restrictions or constraints. Typically, such a task is referred to as numerical optimization or nonlinear programming. Generally, a numerical optimization algorithm involves an iterative process the running of which requires that an objective function be provided, and can be described as follows:

1. Given a set of an initial guess of variables to be optimized.

2. Find a search direction along which the objective function is decreasing.

3. Move a distance along the decreasing search direction.

4. Update the strategy matrix for a new search direction if necessary.

Most optimization algorithms require that the objective function be a smooth, single valued function, having continuous first derivatives. Optimization programs often require derivatives for updating the strategy matrix. Therefore, for an objective function computed numerically, the derivatives have to be estimated numerically. Since there is no universal optimization algorithm which will solve all types of nonlinear problems, different algorithms are used to solve different problems, one objective being to achieve the best stability.

Figure 4 is a flow chart of the optimization algorithm used in optimization step 28. In the embodiment described herein, a Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm is used. A BFGS algorithm is a quasi-Newton method. A Goldstein-Armijo test is introduced in the line search procedure along the descending direction. The Goldstein-Armijo test helps to ensure a convergence of the algorithm in the presence of the inaccurate line search. The inaccurate line search is very common because it is impossible to obtain the exact minimum point along the descending direction. Therefore, a stable optimization algorithm should always consider its existence. The BFGS algorithm used herein can be outlined as the following steps, each of which corresponds to the steps of Figure 4:

1. Choose X₀; and given H_Q = I, g₀ = vf(x₀).

2. Let d = -H g .

3. Choose α_κ so that f(X_R + α_{κ κ}) meet Goldstein-Armijo test condition.

4. Update X_, = X + α d ; and P = αd

5 . Compute g = Vf (X_κtl ) ; and q_κ = g -g_κ

6 . Update H

7. If optimum condition not satisfied, go to step 2.

In the above algorithm, f(x) is the objective function; x is the n-dimensional control variable set; H is the Hessian matrix; I is the identity matrix; d is the descending direction vector; g is the gradient vector, and p and q are the difference vectors. It should be noted that most optimization procedures assume that the objective function behaves locally quadratic. Therefore, a good guess of initial values to be used in the process of determining the reflection coefficient of the pyramidal absorber will greatly speed up the optimization procedure. On the other hand, if no locally-quadratic valley is found, the optimization may be unsuccessful.

In the final method step, an absorber is constructed based upon the design parameters obtained as a result of the optimization process. In particular, an absorber such as, for example, the absorbers of Figure 1A and IB, may be constructed having (1) a specified tip shape; (2) a specified size, including length 12 of the tip or taper, thickness 16 of each backing layer 8, and total absorber length 10; and (3) specific dielectric constants for each base layer 8 and the tip 4.

In accordance with the method of the present invention, a number of absorber designs were investigated although actual pyramidal absorbers were not fabricated. In the investigations discussed herein dielectric constants were not included in the variable set. Rather, the optimization process was conducted using measured dielectric constants, the measured dielectric constants serving as one of the constraints implemented into the objective function. Seven sets of absorbing materials were considered and their dielectric constants are depicted in Figures 5 to 11. The frequency range for the optimization process was confined to 30 to 200 MHz.

Figure 12 depicts objective function behavior based upon the optimization process discussed herein, with respect to a regular pyramidal absorber having a 96-inch taper or tip length, for each of the sets of absorbing material identified. In each investigation, the value of the objective function corresponds to the worst case reflection level in the available frequency range. As shown in Figure 12, the objective function is extremely sensitive to the absorber taper length of the design which confirms that absorber taper is the most important variable in the parameters to be optimized. Figure 12 depicts performance differences of 5 to 15 dB between designs which demonstrates a great potential for optimizing design to enhance the performance of the absorber.

^ ^t s mu 2® With reference to Figures 13 to 22, optimization was performed, using the materials and their dielectric constants identified in Figures 5 to 11, by choosing a combination of materials for a particular design. The program then optimized the length of each backing layer and the length and shape of the taper or tip. The process being iterative, the program continued to optimize the next combination until all combinations of materials in all layers and the tip were computed. In this manner a complete collection of the optimum performance of the absorber with a variety of practical dielectric loadings was obtained. Some typical results are presented in Figures 13 to 22.

An effort was first devoted to design a 96-inch absorber using a regular pyramidal design. Such a design is the most frequently used absorber for lining a 10-meter range semianechoic chamber. Three backing layers and a tapered section were included in the optimization variables. Both uniform weighting and low frequency discrimination were used in the optimization process. Figure 13 depicts normal incidence performances of seven optimized 96-inch regular pyramidal absorber designs, identified with RAP prefixes, compared with a 102.2-inch twisted pyramidal absorber design 2B- prime. The 2B-prime design is an optimized design provided by IBM from the University of Colorado, based on IBM Research Contract Rep. No. 105, July 1990 referred to above. As demonstrated in Figure 13, the optimized regular pyramidal designs have better reflection levels than the 2-B prime design while enjoying a savings in length of 6.2 inches.

To demonstrate the effectiveness of the weighting function, a non-uniform weighting function was applied in the optimization process. The solid curve in Figure 14 depicts the performance of a 96-inch absorber design discriminating performance in a frequency -range of 30 to 45 MHz. The optimization process enables the user to tune the absorber performance according to a specific requirement within its physical capability. To determine the physical capability of the performance of the 96-inch regular pyramidal absorber, the dielectric constants were entered as part of the variables by specifying the frequency behavior of dielectric constants having the same function form as described in the IBM Reseach Contract Report No. 105. The dashed curve in Figure 14 shows the optimized performance.

The second design configuration is a truncated pyramidal absorber as depicted in Figure 15. By entering the percentage of the area on the flat top as an optimization variable, one more tuning dimension is entered in the optimization process. Again, the optimization program looks through all combinations of materials in different layers. Figure 16 presents two performance curves representative of flat or truncated pyramidal absorbers designated F227 and F1417. As one can observe, the truncated design improves the performance of the 96-inch absorber by at least 6 dB relative to a 2B-prime design.

The third design is a flat-top (truncated) absorber with a polynomial curved taper as depicted in Figure 17. Since the curvature adds a few more tuning dimension to the optimization, the performance of the absorber over the designed frequency range should be improved. Figure 18 presents the performance of two 96- inch absorbers designated CF227 and QF227 having such a design. As shown in Figure 18, the best performance of such a design has a better than 27 dB performance over the design frequency range, there being an additional 2 dB improvement of worst case reflection level over the flat-top pyramidal design.

Since the flat-top design might have a negative impact on the higher frequency performance above the design frequency range, sharp-tip designs were investigated. Figure 19 shows a sharp-tip design using the polynomial curve. Figure 20 presents two performance curves for absorbers designated Poly 6-227 and Poly 7- 127 each of which includes such a design. Although the results relating to this design are not as good as those relating to the flat-top curved taper-design, it is believed that this design will have better high frequency performance. However, since the cutting of the curve might add to the production cost of the absorber, a bisectional linear design with a sharp-tip was also investigated, and Figure 21 depicts such a design. The performance of two absorbers designated B1127 and B1115 having such a design is depicted in Figure 22. This family demonstrates about the same performance as that of the curved shape-tip design of Figure 19.

The foregoing analysis relates to rectangular pyramidal absorbers having a design obtained by the optimization process described herein based upon normal incidence. It is believed that the use of such absorbers to line the end walls (front and back) of a typical semi-anechoic chamber will substantially improve performance relative to prior art absorbers. Since performance is also dependent upon the performance of the two side walls and the ceiling, optimization may also be performed for oblique incidence. Figure 23 relates to an optimized 96-inch regular truncated pyramidal absorber for use with a 45-degree angle of incidence. For oblique incidence absorber performance, the parallel polarization is about 3 dB less important than the perpendicular polarization. As shown in Figure 23, the optimized absorber shows a better than 17 dB performance over the frequency range of 30 MHz to 200 MHz for perpendicular polarization which is a performance level required to design a semi-anechoic chamber having better than +4 dB deviation from an ideal open field test site required by the FCC regulation.

Generally, optimization of twisted pyramid designs did not produce results comparable to those discussed above regarding regular pyramidal designs, the twisted pyramidal designs being substantially less favorable than the regular pyrimidal designs for the given dielectric sets and for a 96-inch absorber length. However, Figure 24 depicts the performance of a 72-inch absorber of twisted pyramidal design. As can be seen, such absorber exhibits a slightly better performance than a 72-inch regular pyramidal design.

The computer program noted below sets forth a process that identifies input and output data sets. The input data sets include (1) 8 sets of measured complex dielectric constants of a special family of absorbing materials; (2) the incident angle for which performance is to be optimized; (3) the total physical length of the absorber (in inches); (4) the frequency range for which optimum performance is desired; and (5) an initial design of the absorber, including shape and size of the tip and the size of each backing layer for the chosen absorbing material. The output data sets include (1) the shape and size of the absorber for a given combination of dielectric constants in the base and tip; (2) a performance index for the optimum design; and (3) an optimum condition number. The performance index is merely a measure of how the design performs in dBs. The optimum condition number advises the user that the design is optimized or that the iteration has proceeded as far as possible. The computer program is diagrammatically depicted in the flow charts of Figures 2 to 4.

The embodiments which have been described herein are but some of several which utilize this invention and are set forth here by way of illustration but not of limitation. It is apparent that many other embodiments which will be readily apparent to those skilled in the art may be made without departing materially from the spirit and scope of this invention.

SUBSTITUTE SHEET (RULE 2ty C

C Program FLAT-TOP in FORTRAN

C Version 2.0 January 1993

C

C by

C Kefeng Liu

C 38 Water Street

C A esbury, MA 01913

C

0=============================================================================

C This program optimizes the electromagnetic wave absorption of an EMC absorber

C element with a flat-top pyramidal tip design for normal incidence or off-angle

C performance. The program list shown below designs the optimum performance

C of an eight-foot EMC absorber for 45-degree incidence angle performance.

C However, the basic program structure and its supporting sub-programs can

C be easily modified to design a great variety of EMC absorbers with different

C tip designs. Although the program list only shows a design example of a

C flat-top pyramidal tip design with 3 backing layers, the program can be easily

C modified to design any number of backing layers or other tip designs without

C any significant modification.

C

C Two fundamental elements are included in this program: The numerical

C optimization algorithm and the absorber performance evaluation program with

C the transmission line and homogenization methods in the analysis of the

C absorber tip. The numerical optimization program was developed in close

C reference to the BFGS algorithm outlined in ''Linear and Nonlinear

C Programming' ' by D. G. Luenberger (Addison and Wesley, 1989). The analysis

C of the absorber tip is performed by using transmission line method and

C Hashin-Shtrikman bounds recommended by E. F. Keuster in ''Low-frequency

C properties of transversly periodical lossy waveguides'' (Scientific Report

C No. 86 of University of Colorado, 1987). Five subprograms are included

C in this program listing.

C

C FUNCTION GNORM(XP)—A functional subprogram to evaluate the absorber

C performance and translate into an objective function. At the same

C time, the physical constraints are also encoded in this unit.

C

C COMPLEX FUNCTION CTAN(Z)—This functional subprogram provides evaluation

C of tangent for complex argument i.e. tan(z) .

C

C SUBROUTINE GLDARMζG.D.X.P.FO.EPS.N)—Subroutine used to determine the

C most appropriate update step size along the descending direction by

C using Goldstein Ar ijo's rules.

C

C SUBROUTINE BFGS(P,q,H,HQ,Pq,N)--This subroutine is developed to perform C Broyden-Fletcher-Gold arb-Shanno update of the Hessian strategy matrix .

C

C SUBROUTINE GRADF(DX,FO,X,G,N)~This subroutine is developed to perform

C numerical derivatives of objective function to provide the BFGS

C algorithm with gradient of the objective function.

C

C Input data sets

C 1) . 8 sets of measured complex dielectric constants of a special family

C of Ray Proof's absorbing materials.

C 2). Incident angle for which the performance is to be optimized.

C 3). Total physical length of the absorber (in inches).

C 4). The frequency range for which optimum performance is desired.

C 5). Initial absorber design (shape and size of tip, and size of

C each backing layer) .

C

C Outptut data sets

C 1). Size (total absorber length) and design (shape and size of tip, and

C size of each backing layer) of the absorber for a given combination

C of dielectric constants in the base and the tip.

C 2). Performance index for the optimum design.

C 3). Optimum condition number.

C

PARAMETER (NN=8, F=51,NV=5,NT=NV*NV)

COMPLEX J,ER

C0MM0N/RST/ER(NF,NN),IP(4),N0RM,TL,AMP,STH,CTH,FS,DF

C0MM0N/CST/J,PI,SCAL

REAL XP(NV) ,P(NV) ,q(NV) ,H(NV,NV) ,Hq(NV)

REAL D(NV),G(NV),WK,T0L,Pq,F0 C. Some mathematical and physical constants

PI=3.1415926536

J=CMPLX(0.,1.)

SCAL=0.532349E-3

IMAX=19*NV

N0RM=20 C. EPS is the computer machine epsilon which is used to determine the C. optimum step size in taking numerical derivative.

EPS=1.0 2 EPS=EPS*0.5

WK=1.0+EPS

IF(WK.GT.l.O) GOTO 2

T0L=SqRT(EPS)

DX=T0L C. Amplification factor, angle of incidence, and absorber length C. The amplification factor is used to enhance the sensitivity of C. the absorber objective function so that optimum condition can be C. reached at a better accuracy and efficiency.

AMP=10.

ANG=45. TL=96.

C.. Starting frequency FS and the step DF(in MHz) FS=30. DF=3.4

STH=(SIN(ANG*PI/180.))**2 CTH=SqRT(l.-STH)

C. Telling the program where to get the specified 8 sets of measured

C. dielectric constant files. Then read in to a complex array

C.. ER(NF.NN).

OPEN(ll,FILE='/u/liu/rap/eps/a07ssl6.eps'₎STATUS='OLD') 0PEN(12,FILE='/u/liu/rap/eps/aO8ssl6.eps',STATUS=^,0LD') 0PEN(13,FILE='/u/liu/rap/eps/aOlssl6.eps\STATUS='0LD') OPEN(14,FILE='/u/liu/rap/eps/a02ssl6.eps' ,STATUS='0LD') 0PEN(15,FILE='/u/liu/rap/eps/aO4ssl6.eps',STATUS='0LD') 0PENC16,FILE='/u/liu/rap/eps/a05ssl6.eps' ,STATUS='OLD') OPEN(17,FILE='/u/liu/rap/eps/a09ssl6.eps',STATUS='OLD') 0PEN(18,FILE='/u/liu/rap/eps/al0ssl6.eps' ,STATUS='OLD') DO 4 N=11,10+NN DO 4 1=1,3

4 READ(N,*) DO 5 N=1,NN M=N+10

DO 5 1=1, F READCM,*) F.A.B ER(I,N)=CMPLX(A,B)

5 CONTINUE

C. The optimization process starts from here

C. The following four loops provide all possible combinations of C. different absorbing materials in the tip and in the three backing C. layers. For a practical reason, the tip was given only six C. different dielectric materical options.

DO 100 MA=1,6

IP(1)=MA

DO 100 MB=1,8

IP(2)=MB

DO 100 MC=1,8

IP(3)=MC

DO 100 MD=1,8

IP(4)=MD C. Provide an initial design of the absorber C. XP(1)—The length of the tip (in inches)

C. XP(2)—The thickness of the first backing layer (in inches) C. XP(3)—The thickness of the second backing layer (in inches) C. XP(4)—The thickness of the third backing layer (in inches) C. XP(5)—The portion of the flat-top design (in ratio to the C. size of the base)

XP(1)=78.

XP(2)=6. XP(3)=6.

XP(4)=6.

XP(5)=0.125

IT=0 C. Initilize the Hessian matrix

DO 6 1=1,NV

DO 6 N=1,NV 6 H(N,I)=0. C. Computing the gradient of the objective function

F0=GN0RM(XP)

CALL GRADF(DX,F0,XP,G,NV)

DO 8 1=1,NV C. Determining the initial descending direction.

D(I)=-G(I) 8 H(I,I)=1.0 10 IT=IT+1 C. Checking Goldstein Armijo test condition to determine C. the step size of the update

CALL GLDARM(G,D,XP,P,FO,EPS,NV)

F0=GN0RM(XP)

IF(IT.GT.IMAX) GOTO 50 C.. Calling the gradient of the objective function again

CALL GRADF(DX,FO,XP,q,NV) C. Computing the optimum condition number Pq

DO 15 1=1,NV wκ=q(i)

15 G(I)=WK pq=o.

DO 16 1=1,NV

16 pq=pq+p(i)*q(i)

IF(Pq.GT.O.O.AND.Pq.LT.TOL) GOTO 50 C. If optimum condition not reached yet, calling the BFGS C. update of the Hessain matrix

CALL BFGS(P,q,H,Hq,Pq,NV)

DO 20 1=1,NV

WK=0.

DO 18 N=1,NV 18 WK=WK+H(N,I)*G(N) 20 D(I)=-WK C. Go back to the next iteration of optimization algorithm.

GOTO 10 C. Program jump to here after a completion of optimum design. 50 XL=XP(1)+XP(2)+XP(3)+XP(4)

DD=AMIN1(0.,TL-XL)

XA=AMIN1(0.,XP(D)

XB=AMIN1(0.,XP(2))

XC=AMIN1(0.,XP(3)) XD=AMIN1(0.,XP(4))

PX=128.*(XA*XA+XB*XB+XC*XC+XD*XD)+16.*DD*DD C.. Evaluate the performance norm by extracting the contribution C. of the constraint conditions.

DB=20.*ALOG10((FO-PX)/AMP) C. Output the optimum design to the output devices 100 WRITE(6,200) IP,XP,XL,DB,IT 200 F0RMAT(4I2,4F6.2,F8.5,F8.2,' DB=',F6.2,' IT=',I4)

STOP

END

FUNCTION GNORM(XP)

PARAMETER (NN=8,NF=51,NV=5)

C0MM0N/RST/EPS(NF,NN),IT(4),N0RM,TL,AMP,STH,CTH,FS,DF

C0MM0N/CST/J,PI,SCAL

COMPLEX J,EPS,EP,CRTA,CRTB,DE,CTAN,ZA,ZB,EZ,BA,BB

REAL XP(NV) .GNORM,GAMMA,DP,TH,FS,DF,DD,GS,AA C. Reset the initial value of the objective function.

GAMMA=0. C.. Start a loop to cover all frequencies in the optimization range

DO 50 K=1,NF C. Provide the operation frequency for this analysis

F=FS+(K-1)*DF C. The frequency scaling factor includes beta

FSC=F*SCAL C. CRTA-Perpendicular polarization, CRTB-Parallel Polarization

CRTA=CMPLX(0.,0.)

CRTB=CMPLX(0.,0.) C. Perform a combination of the wave impedance starting from the C. last backing layer to the layer next to the tip.

DO 5 1=4,2,-1

DE=EPS(K,IT(D)

BA=CSqRT(DE-STH)

ZA=1./BA

ZB=BA/DE

DD=XP(I)*FSC

BA=CTAN(BA*DD)

CRTA=ZA*(CRTA+J*ZA*BA)/(ZA+J*CRTA*BA) 5 CRTB=ZB*(CRTB+J*ZB*BA)/(ZB+J*CRTB*BA) C. Analysis of the absorber tip start here.

DP=XP(1)

DE=EPS(K,IT(D)

DD=DP*FSC C. Empirical formula to determine the number of subsections C. needed to analyze the tip of the absorber.

MM=INT(3.*CABS(DE)*DD)+10

DD=DD/MM

AA=XP(5)

DX=(1.-AA)/MM C. Combination of wave impedance with the backing layers

DO 8 I=MM,1,-1

FX=AA+(I-0.5)*DX

GS=FX**2 C. Formula to perform homogenization of each individual subsection

EP=1.+2.*GS*(DE-1.)/((1.+GS)+(1.-GS)*DE)

EZ=1.+(DE-1.)*GS

BA=CSqRT(EP-STH)

BB=CSqRT(EP*(l.-STH/EZ))

ZA=1./BA

ZB=BB/EP

BA=CTAN(BA*DD)

BB=CTAN(BB*DD)

CRTA=ZA*(CRTA+J*ZA*BA)/(ZA+J*CRTA*BA) 8 CRTB=ZB*(CRTB+J*ZB*BB)/(ZB+J*CRTB*BB) C. Total effective wave impedance is done here C. Computing reflection coefficients

CRTA=CRTA*CTH

CRTA=(CRTA-1.)/(CRTA+1.)

CRTB=(CRTB-CTH)/(CRTB+CTH) C. Applying the amplification factor, weighting factor, and 3db C.. discrimination factor on parallel polarization performance.

GA=CABS(CRTA)*AMP*SqRT(F/30.)

GB=CABS(CRTB)*AMP*SqRT(F/30.)*0.7071

GAMMA=GAMMA+GA**N0RM+GB**N0RM 50 CONTINUE C. Applying physical constraint conditions by introducing penalty C. functions into the objective function.

XL=XP(1)+XP(2)+XP(3)+XP(4)

DD=AMIN1(0.,TL-XL)

XA=AMIN1(0.,XP(1))

XB=AMIN1(0.,XP(2))

XC=AMIN1(0.,XP(3))

XD=AMIN1(0.,XP(4))

PX=128.*(XA*XA+XB*XB+XC*XC+XD*XD)+16.*DD*DD

GN0RM=GAMMA**(1./FLOAT(NORM))+PX

RETURN

END

COMPLEX FUNCTION CTAN(Z)

COMPLEX Z.J.CRT

C0MM0N/CST/J,PI,SCAL C. Straight forward tangent function evaluation for complex argument

CRT=CEXP(2.*J*Z)

CTAN=J*(1.-CRT)/(1.+CRT)

RETURN

END

SUBROUTINE GLDARM(G,D,X,P,F0,EPS,N) C. Line search algorithm by using Goldstein + Armijo rules

SUBSTITUTE SHFFT /RUL E 28 REAL G(N) ,D(N) ,X(N) ,P(N) ,F0,DF,FRED,ALPHA,GNORM.GD.EPS C.. Setting the initial step size in the search direction DN=0.5 XN=FL0AT(N) C. Normilize the step size with respect to the magnitude of the C. current design parameter to ensure that there is no huge C. jump in the search direction DO 2 1=1,N DN=DN+ABS(D(D) 2 XN=XN+ABS(X(I)) SC=AMIN1(1.,XN/DN) IF(SC.LT.l.) THEN DO 4 1=1,N 4 D(I)=D(I)*SC ENDIF C. Evaluate the Frechet derivative GD=0.0 DO 6 1=1, 6 GD=GD+G(I)*D(I)

ALPHA=1.0 10 CONTINUE

DO 15 1=1,N 15 P(I)=X(I)+ALPHA*D(I) DF=GN0RM(P)-F0 FRED=4.*DF/(ALPHA*GD) C. Check the Goldstein + Ar ijo Test condition, if C. not satisfied, reduce the step size by a half. IF(FRED.LT.l.O.AND.ALPHA.GT.EPS) THEN ALPHA=ALPHA*0.5 GOTO 10 ENDIF C.. Output the step size and the updated parameters DO 30 1=1,N P(I)=ALPHA*D(I) 30 X(I)=X(I)+P(I) RETURN END

SUBROUTINE BFGS(P,q,H,Hq,Pq,N) REAL P(N) ,q(N) ,H(N,N) ,Hq(N) .SUM.Pq.FA C. Broyden-Goldfard-Fletcher-Shanno updating algorithm of C.. Hessian stategy matrix. DO 10 J=1,N SUM=0. DO 8 1=1,N 8 SUM=SUM+H(J,I)*q(I) 10 Hq(J)=SUM FA=0. DO 15 1=1, 15 FA=FA+Q(I)*Hq(I)

FA=(ι.+FA/pq)/pq

DO 30 1=1,N

DO 30 J=1,N 30 H(J,I)=H(J,I)+FA*P(J)*P(I)-(P(I)*Hq(J)+Hq(I)*P(J))/Pq

RETURN

END

SUBROUTINE GRADF(DX,FO,X,G,N) C. Use the step size given based on the machine epsilon to C. perform numerical derivative of the objective function

REAL G(N),X(N),DX,GN0RM.F0,FP

DO 10 1=1,N

X(I)=X(I)+DX

FP=GN0RM(X)

G(I)=(FP-F0)/DX C.. Restore vector X 10 X(I)=X(I)-DX

RETURN

END

SUBSTI7WΣSHEET(R_{ULE 26)}

Claims

I claim:

1. A method for constructing an absorber of the type having a plurality of absorber elements each including at least one backing layer and a tapered section extending therefrom, comprising the steps of:

(b) performing an absorber performance evaluation which, with respect to said tapered section, uses a transmission line method and homogenization;

(c) performing an iterative optimization of said absorber performance evaluation including at least optimization of said tapered section; and

(d) constructing an absorber element having shape, size and material properties identified by said iterative optimization.

2. The method of claim 1 wherein step (b) includes absorber performance evaluation of said at least one backing layer and step (c) includes optimization of said at least one backing layer.

3. The method of claim 2 wherein step (b) includes determining a number of subsections for analyzing said tapered section.

4. The method of claim 3 wherein said input data sets comprises measured complex dielectric constants, and further wherein step (b) includes:

(b reading in select complex dielectric constants for said tapered section and said at least one or more backing layer and

(b₂) computing the effective dielectric constant of each subsection during homogenization.

5. The method of claim 4 wherein step (b) further includes: (b₃) computing the wave impedance of each subsection.

6. The method of claim 5 wherein step (b) includes computing the cascaded wave impedance of said at least one backing layer for a given polarization and angle of incidence, and following step (b₃) includes: (b₄) combining the wave impedance of each subsection with the wave impedance of said at least one backing layer to obtain the total wave impedance.

7. The method of claim 6 wherein step (b) further includes: (b_) computing the reflection coefficient of said absorber using the total wave impedance.

8. The method of claim 7 wherein step (b) further includes: (b₆) constructing an objective function by applying frequency and angle weighting factors and constraint conditions.

9. A method for constructing an absorber of the type having a plurality of absorber elements each including one or more backing layer and a tapered section extending therefrom, comprising the steps of:

(a) selecting input data sets with respect to which absorber performance is to be optimized, said input data sets including a plurality of sets of measured complex dielectric constants for said one or more backing layer and said tapered section, an incident angle, a total physical length of said absorber, a frequency range, and parameters identifying an initial absorber shape and size of said tapered section and size of each backing layer;

(b) performing an absorber evaluation comprising the steps of

(b^ reading in complex dielectric constants for said tapered section and said backing layer(s);

(b₂) computing the cascaded wave impedance of all backing layers for a given polarization and angle of incidence;

(b₃) determining the number of subsections needed to analyze said tapered section;

(b₄) applying homogenization theory to compute the effective dielectric constants of each subsection;

(b₅) computing the wave impedance of each subsection;

(b₆) combining the wave impedance of each subsection with the wave impedance of each backing layer to obtain total wave impedance; (b_?) computing the reflection coefficient of said absorber at all given frequencies based upon said total wave impedance; and

(b₈) constructing an objective function by applying frequency and angle weighting factors and constraint conditions;

(c) optimizing said absorber evaluation; and

(d) constructing an absorber element having a total length, a size and shape of said absorber element, and size of each backing layer, resulting from said optimization.

10. A method for constructing an absorber of the type having a plurality of absorber elements each including one or more backing layer and at least one tapered section extending therefrom, comprising the steps of:

(a) selecting a plurality of measured dielectric constants of a corresponding plurality of absorbing materials;

(b) selecting an initial combination of dielectric constants for said tapered section and each backing layer;

(c) selecting an initial shape and size of said tapered section, and an initial size of each backing layer, for said combination of dielectric constants;

(d) performing absorber performance evaluation including using a transmission line method and homogenization theory for performance evaluation of said tapered section;

(e) performing optimization of said absorber performance evaluation; and

(f) constructing an absorber element based upon the results of said optimization.

11. An absorber, comprising a base which includes at least one backing layer, each backing layer having a thickness, and at least one tapered section extending from said base, said tapered section having a length, said absorber having a total length equal to the length of said tapered section plus the combined thickness of each backing layer, wherein said thickness, said length, said total length and the shape of said tapered section are based upon an optimization of said absorber, said optimization comprising optimization of said tapered section.

12. The absorber of claim 11 wherein said optimization further comprises optimization of each backing layer.

13. The absorber of claim 12 wherein each tapered section is truncated.

14. The absorber of claim 12 wherein each tapered section is pointed.