035067-3-T HYBRID FINITE ELEMENT AND MOMENT METHOD SOFTWARE FOR THE SERAT ARRAY 3rd Quarterly Report Sanders, A Lockheed Martin Co. 95 Canal Street NCAI-6268 P.O. Box 868 Nashua, NH 030601-0868 35067-3-T = RL-2484

PROJECT INFORMATION PROJECT TITLE: REPORT TITLE: U-M REPORT No.: CONTRACT START DATE: END DATE: DATE: SPONSOR: SPONSOR CONTRACT No.: U-M PRINCIPAL INVESTIGATOR: Hybrid Finite Element Design Codes for the SERAT Array 3rd Quarterly Report 035067-3-T October 1996 September 1998 July 1, 1997 (3rd Quarterly Report) Roland Gilbert SANDERS, INC, A Lockheed Martin Co. MER 24-1583 PO Box 868 Nashua, NH 030601-0868 Phone: (603) 885-5861 Email: RGILBERT@mailgw.sanders.lockheed.com P.O. QP2047 John L. Volakis EECS Dept. University of Michigan 1301 Beal Ave Ann Arbor, MI 48109-2122 Phone: (313) 764-0500 FAX: (313) 747-2106 volakis @umich.edu ihttp://www-personal.engin.umich.edu/-volakis/ Y. Erdemli (UM), Thomas Eibert(UM) D. Jackson(UH), J. Volakis(UM) and D. Wilton(UH) CONTRIBUTORS TO THIS REPORT: 1

TABLE OF CONTENTS TABLE OF CONTENTS 2 LIST OF FIGURE CAPTIONS 3 CHRONOLOGY of Events (Updated Every Quarter)____________________ 4 MEETINGS 5 SUMMARY OF 3RD QUARTER PROGRESS 6 1. Previous Quarter Progress Summary 6 2. Summary of 3rd Quarter's Progress. 6 2.1 Geometry Driver for FSS-BRICK 6 2.2 Geometry Driver and Progress in the development of FSS-EIGER 8 2.3 Development and Validation of FSS-PRISM 10 3.1 Theory and Description of FSS-PRISM for Commensurate FSS: Upgrading FEMA-PRISM to Deal with Infinite Periodic Planar Structures 12 APPENDIX 1: Project Goals 2 2 APPENDIX 2: Presentation Given on June 25, 1997 at Nashua, N.H. 2 6 APPENDIX 3: Univ. of Houston (FSS-EIGER) Presentation on the May 30, 1997 review held in Ann Arbor. 5 8 APPENDIX 4: FSS-PRISMpresentation by T. Eibert given on the May 30, 1997 review held in Ann Arbor 9 1 APPENDIX 5: FSS-BRICK Presentation Given by Y. Erdemli on the May 30, 1997 review held in Ann Arbor ________________________________________ 9 2

LIST OF FIGURE CAPTIONS IN SUMMARY SECTION Figure 1. Illustration of the primitive elements used in FSS-BRICK. Note that the element must fit within the uniform grid. 8 Figure 2. Comparison of reflection coefficient for a slot FSS as computed by the finite array code FSSBRICK and the infinite array code FSS-EIGER. 8 Figure 3. Validation of the FSS-EIGER for slot arrays. 9 Figure 4. Broadside E-plane active reflection coefficient for a dipole array on a dual dipole FSS. 10 Figure 5. Comparison of FSS-PRISM, FSS-EIGER and FSS-BRICKfor computing the reflection coefficient of a slot FSS on a dielectric layer. 11 3

CHRONOLOGY of Events (Updated Every Quarter) * April 1996 * July 1996 * August 1996 * 20 Sept. 1996 UH) * October 1996 * October 1996 * 15 Nov. 1996 * 9 January 1997 * 28 February 1997 * 5 April 1997 ~ 30 May 1997 Proposal Submission Answers to Proposal Questions Began Contract Negotiations Kickoff meeting at Ann Arbor (attended by Sanders, UM and Contract Signed between U-M and Sanders in Mid October Subcontract to the Univ of Houston (formalized in early November) SERAT Review meeting (at Nashua) Submitted First Quarterly Report Report Described Code Plan and Progress on the Moment Method FSS Code. Specifically, a new scheme was developed to accelerate the convergence of the periodic Green's function Prepared viewgraphs on the project's progress review. Showed first validation results for the moment method FSS code with the new accelerated Green's function; showed results for a new algorithm to accelerate the boundary integral truncation of the planar and curved FSS hybrid FEM code using the Adaptive Integral Method(AIM) and CVSS, the new LU solver specialized to sparse matrices Submission of Second Quarterly Report. Report included the first validation results for the stand alone small array FEM code (with dipole FSS elements and dipole antenna elements). A similar validation was done for the moment method FSS developed at Houston. The fast AIM algorithm was described for boundary truncation and the TRIANGLE surface mesher was introduced to generate the aperture mesh, subsequently grown down to the FSS. Semi-annual review at the Univ. of Michigan (attended by all parties) Review covered progress up-to-date. At this meeting, emphasis was on the validation of the three codes which took 'shape and form' between March-May 1997 in accordance with the proposed schedule. Theory, validations and comparisons among the codes were presented. 4

25 June 1997 5 July 1997 SERAT review Meeting (Nashua) Submission of Third Quarterly Report The major component of this report was the description and validation of the periodic hybrid FEM code, FSS-PRISM. Comparisons among the FSS-EIGER, FSS-BRICK and FSS-PRISM were given for the first time. Also, the geometry drivers for the FSS-BRICK and FSS-EIGER were given. MEETINGS Two meetings were held this period See the Chronology list above 5

SUMMARY OF 3RD QUARTER PROGRESS 1. Previous Quarter Progress Summary During the previous quarter we reported on * Small array code (FSS-BRICK) along with validation data for dipoles on multilayered FSS. It was primarily noted that this code was fast because it employed mesh compression algorithms for the boundary integral truncation. However, its geometrical generality was limited due to the brick elements used for mesh modeling. * Simple moment method code FSS-EIGER for simulation of antenna elements on commensurate FSS structures. Some preliminary validations were given. * Method for extracting reflection and transmission coefficient parameters in connection with the FSS-BRICK * Performance of a matrix compression scheme based on the Adaptive Integral (AIM) method. A ten-fold decrease in the CPU and memory requirements of this fast integral method was shown. Therefore, AIM is particularly attractive for simulating the doubly curved FSS structure where it would be necessary to treat the entire array surface without making of the periodic Green's function. 2. Summary of 3rd Quarter's Progress. This quarter has been extremely productive and pushed us ahead of schedule with the successful completion of the hybrid FEM code FSS-PRISM. The latter was developed by Dr. T. Eibert, a new addition to the SERAT code development team. Dr. Eibert will be responsible for the doubly curved array simulation during the second year. More details on FSS-PRISM are given in the appropriate section of this report. Below we summarize this quarter's activity. However, the reader is also referred to Appendix 2 which contains the entire slide presentation given at the June 25, 1997 review of the project. These viewgraphs give a concise overview of the code development and validation status. Also, Appendices 3-5 include more detailed viewgraphs of the presentation given on May 30, 1997 during the review held in Ann Arbor. Appendices 3 -5 cover the capabilities of the codes FSS-EIGER, FSS-PRISM and FSS-BRICK in this respective order. 2.1 Geometry Driver for FSS-BRICK A geometry Driver was developed for FSS-BRICK. As noted above and in the previous report, FSS-BRICK is our small array code and its function is to provide a fast finite array analysis tool. As note in the attached short description of the FSS-BRICK, the geometry Driver developed this quarter uses antenna and FSS element primitives to simplify its utility and a need to interface with external meshing packages. The meshing is done automatically and is based on the primitive element choices, FSS layers and their 6

composition. Because the code uses bricks as the basic element for volume and element modeling, it may be necessary to make approximation for modeling the entire SERAT panel. Here is a list of its present capabilities: * Uses slots, dipoles, crossed-slots and crossed-dipoles for primitives (see Figure 1) * Can accommodate resistive cards of rectangular shape, placed at any layer interface * Feeds are horizontal (x or y directed) current probes placed at any location on the uniform grid. Vertical probes are also available but not through the present Driver. * Lumped loads can be placed at any node location between two horizontal or two vertical nodes. * Conducting posts can be placed at any node location in much the same way done for the lumped loads. These posts can be concatenated to form long feeds and wires running from the base of the FSS panel to the surface. FSS-BRICK has been validated for slots and gives the same results as the FSS-EIGER and the FSS-PRISM. Figure 2 shows such as a comparison of all the codes under development. Comparisons among dipole arrays is still in progress. We are improving the reflection/transmission coefficient extraction process for finite array apertures. So far, good agreement is obtained for near resonance computations and this is to be expected because of the finite aperture. In the next three months, we will 1. Complete testing of the FSS-BRICK and its geometry Driver for dipole array simulations (input impedance vs. scan angle, FSS reflection/transmission coeff., etc.), resistive cards and lumped loads 2. Package the code with a short manual and plotting using XMGR The graduate student Y. Erdemli will then continue to implement a similar geometry Driver for FSS-PRISM. Depending on the success of FSS-PRISM and FSS-EIGER for modeling non-commensurate FSS, we may also decide to extend the application of FSSBRICK to non-commensurate FSS and periodic array structures by adding phase boundary conditions and mesh truncation using periodic Green's function. Coupled with the FFT for performing the matrix-vector products in the solver, FSS-BRICK may become our fastest code. 7

.................................................................................. 4 Cross-Stua................................................................................ - ----- IAIll i'. MAIL., -X.......................... "I T I."T SIDI Dipole Antenna Element =...= - -:.:_-.:!..:.i:.:::i.. =..... - ~~..... -........... ~~~~~~~~::: -.,:..::-.:-:-::-::.: =:'-::Slot Antenna Element [-. Figure 1. Illustration of the primitive elements used in FSS-BRICK. Note that the element must fit within the uniform grid. Single Layer FSS: Slot Element tLx FSS... h=lcm (:)(4.0,1.0) L............................................................. Slot 0.2x0.75cm2) Unit Cell / (xlcm2) / / II Power Reflection Coefficient FSS-Brick Code: TM incidence,O =1~, i =0~o FEM sample size: dx = 0.1, dy = 0.05 cm f= 13 GH 2x2 Array 3x3 Array No. of unknowns: 6795 15600 FFT pad size: 64x128 64x128 No. of iterations: 98 202 CPU time (sec): 139 247.......................-...................I....e............................ —.........-.-............................. Frequency, GHz Figure 2. Comparison of reflection coefficient for a slot FSS as computed by the finite array code FSS-BRICK and the infinite array code FSS-EIGER. 2.2 Geometry Driver and Progress in the development of FSS-EIGER As noted in earlier reports, FSS-EIGER is a generalization of the general purpose moment method code EIGER developed by the Univ. of Houston and Lawrence Livermore Labs. under Navy sponsorship. During the first quarter of this contract, a fast 8

periodic Green's function was developed which delivered two orders of magnitude in improved speed. This Green's function was generalized to multilayered structures during the second quarter of the contract and was ported into the original EIGER code to create FSS-EIGER. The remaining effort during the second quarter was devoted to testing and code validation and this continued into the third quarter. In our second quarter report we showed some validations for stand-alone multilayered dipole FSS (see Appendices 2 and 3). During this quarter * we continued the validation of FSS-EIGER for slot arrays and for dipole antennas on multilayered FSS. Some results are shown in Figures 3 and 4. The results in Figure 4 refer to a dipole array on a dual layer dipole FSS and will be validated soon using FSS-BRICK and FSS-PRISM. It should be noted that these results refer to configurations that have not been considered in the literature due to their generality in terms of geometry and layered structure. Also note that the slot element in Figure 3 is identical to that used in Figure 2 for validating FSS-BRICK. The same slot element will be used later for validating FSS-PRISM. * Developed a geometry Driver for the FSS-EIGER using dipoles, slots, crossed slots and crossed-dipoles as primitives. The description of this Driver will be given as an independent report from the Univ. of Houston. * FSS-EIGER can model resistive cards and horizontal resistive loads which were already present in EIGER. However, testing of these capabilities has not yet been completed. The emphasis over the next quarter will be devoted to the non-commensurate arrays and FSS. FSS SLOT ARRAY -u TM INCIDENCE: 0 = 1~ =- 0~ 1 " 6U nitCell:: 0 I% \ *= 0e 2 - - - - - - j- - - - - - - - -- - - ------ 02 0. ---—... --- —- - a.; 5 10 15 20 25 FREQUENCY (GHz) Figure 3. Validation of the FSS-EIGER for slot arrays. 9

1 A n - s - - T - - - g - - g - - - |............Dipole / <20 ^ /! 0 / / Antenna h,= 0.40cm ~ (e"ji)(2010) Dipole-I /* * (0.25xO.5cm2) FSS-lI h= 0.25cm: |L(e,=(2.0,10) Dipole-2 '' - ' (0.25xO.6cm2) FSS-2 2. 5 h3= 0.15cm (3 ( l 0y= I cm L= 0.95 cm 0.8 --------- 0'8 -0.7 -------- 0.6 -------------------------- 0.5 ------------ 0.4 0.3 -------------— ^ --- —------------- 0.3 ------- ------- 0.2 0.1 -- - --- -- -- - n. - '.......... 0 I In rm 0 z m v 0 10 20 30 40 50 60 70 80 90 SCAN ANGLE (DEGREES) Figure 4. Broadside E-plane active reflection coefficient for a dipole array on a dual dipole FSS. Aleko 2.3 Development and Validation of FSS-PRISM FSS-PRISM refers to the hybrid FEM code. This code is the centerpiece of the proposed development and will combine the geometrical adaptability and generality of the FEM for modeling materials and inhomogeneities with the rigor of the moment method for mesh truncation. This code will therefore be the most general and capable among the all codes developed under this project. The code employs distorted prisms and will evolve to the doubly curved array code during the second year of this development. FSS-PRISM took shape and form during this quarter and was developed by Dr. Eibert who joined the project team in March and will be the responsible for the development of the doubly curved array code during the second year. The development of FSS-PRISM began with the existing code FEMA-PRISM which was a single element code for the analysis of conformal antennas on doubly curved surfaces. FEMA-PRISM was supported by Air Force contracts (Rome and Wright Laboratories) and was mentioned in the delivered proposal. During the quarter the quarter the Dr. Eibert modified FEMA-PRISM as followsfollows (see also attached report in section 3 and the viewgraphs in the Appendices): 10

* Incorporated periodic boundary conditions into the FEM domain to simulate the periodic nature of the FSS. * Replaced the free space Green's function in FEMA-PRISM with the faster Ewald Periodic green's function developed by the Univ. of Houston. * Validated the code for several trivial geometry for robustness * Validated the code for slot and dipole arrays on commensurate FSS. Figure 5 shows a validation of FSS-PRISM for a slot FSS. This figure displays results from all codes and it is pleasing to see that all codes are in excellent agreement. In our subsequent report we will demonstrate applications for dipoles and more complex configurations. FSS Slot Array Unit Cell:, 1 cm ~ Power Reflection Coefficient E o 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 e~r 4 O. lci,:) m Frequency, GHz Figure 5. Comparison of FSS-PRISM, FSS-EIGER and FSS-BRICK for computing the reflection coefficient of a slot FSS on a dielectric layer. In the following months, our goal will be * generalize the current version of FSS-PRISM to non-commensurate arrays * develop a geometry driver similar to that in FSS-BRICK * compare FSS-PRISM and FSS-EIGER in terms of CPU and memory requirements for modeling multilayered FSS. 11

3.1 Theory and Description of FSS-PRISM for Commensurate FSS: Upgrading FEMA-PRISM to Deal with Infinite Periodic Planar Structures 1 Introduction FEMA-PRISM-BI is a finite element (FE)/boundary element (BE)-hybrid code that works with prismatic elements in the volumetric FE-part and triangular elements in the BE-part. The prismatic elements can be right angled as well as distorted. So, the code vertically surface meshes at each layer with all geometrical adaptability whereas the volumetric FE-mesh can be easily generated by growing the mesh along the normals to the adjacent layer. The original version of FEMA-PRISM was implemented as a pure FE-code with artifical absorbers used to terminate the interface to free space. Because of this, no Green's function was needed to truncate the mesh, and as a result doubly curved finite surfaces could be modeled. During the first quarter of this project FEMA-PRISM was upgraded to a FE/BEhybrid code with the BE-part implemented for planar surfaces based on the free space Green's function. In this stage FEMA-PRISM with BI can deal with planar cavity backed antenna configurations as illustrated in Fig. 1. During the third quarter of this project FEMA-PRISM-BI was upgraded to a periodic FE/BE —hybrid code (PPRISM), capable of dealing with infinite periodic Figure 1: Cavity backed planar antenna configuration 12

BI/PGF / i -- I - I PBC PBC Metal Figure 2: Metal backed periodic configuration BI/PGF PBC PBC BI/PGF Figure 3: Open periodic configuration planar antenna configurations as well as frequency selective surface (FSS) configurations with an arbitrary number of FSS layers (metallic patches or slots) and combinations of both. The code has the option to deal with metal backed configurations as illustrated in Fig. 2 and with configurations that are open at the top as well as the bottom surface of the FE-mesh (see Fig. 3). In this case the BE-method (or BI-method (boundary integral)) is used to terminate the top surface and the bottom surface. Due to the application of the BE-method on both surfaces the computational effort is larger and most often doubled. In the following, it is shown how the periodicity condition for the fields in the infinite periodic array can be used to model the infinite array problem using only with one cell of the array. This means that within the FE-model of this unit cell, the periodic boundary condtion (PBC) has to be enforced on the vertical walls of the mesh and on the boundary edges of the BE surface. Also in the BE surface the appropriate periodic Green's function (PGF) must be used. For this application, the PGF developed by Prof. D. Wilton and Prof. D. Jackson will be used. After a brief presentation of the formulation, numerical results are given to validate 13

the code. 2 Formulation of the Infinite Periodic Problem The infinite periodic problem is illustrated in Fig. 4. For the electromagnetic DYP cot Y Y x DYP LD DXP Figure 4: Infinite periodic configuration analysis of this configuration, it is sufficient to consider one single cell (unit cell) of the infinite periodic array because the fields in the array fulfill a periodicity condition given by E(x + m DXP + n DYP cot(-), y + n DYP) = E(x, y) e-jf(m DXP+n DYP cot(y)) e-jifynDYP (1) H(x + m DXP + n DYP cot(7), y + n DYP) = H(x, y) e-jz(m DXP+n DYP cot(y)) e-joynDYP (2) DXP and DYP are the distances between adjacent cells in the array in x- and y-directions, respectively. Also, 7y is the skewness angle of the lattice measured from the x-axis as illustrated in Fig. 4; fx and 3y are given by /x = ko sin Oocos Oo 3=y = ko sin o sin o, (3) (4) in which Oo and 4o are the angles describing the propagation direction of an incident plane wave or the scan direction of the array for antenna radiation, and ko is 14

Aci 5/12 4 21 3 19 20 / 22 Figure 5: FE-mesh of unit cell the free space wave number. The periodicity condition (1) has to be enforced on the vertical walls of the FEmesh and on the boundary edges of the surface meshes for the BE-method. Additionally, in the BE mesh the appropriate Green's function for the infinite periodic case must be employed. For the implementation of the periodic boundary condition into the FE volume let us consider the FE-mesh of a unit cell in Fig. 5. First, the FE-matrix is generated without considering the boundary condition at all, including all edges of the unit cell. Afterwards the right side wall is wrapped onto the left side wall and the upper side wall is wrapped onto the lower side wall via a matrix transformation. In this matrix transformation, all matrix entries involving unknowns associated with edges on the right and the upper vertical walls are eliminated and are replaced by new matrix entries involving the unknowns for the edges on the left and lower vertical walls. Consider for example edge #13 on the right boundary of the unit cell in Fig. 5. It can be seen that the corresponding periodic edge on the left boundary is edge #1. Therefore in all matrix entries the unknowns for edge #13 are replaced by Si,13 - Si, e-j I,DXP S13,i - Sl,i ej DXP, (5) where i is an arbitrary edge number (the testing edge) that is also not on the right or the upper boundary wall (periphery) of the cell. In the latter case, the matrix entries have to be transformed with respect to both edge numbers. For instance the self-term for edge #13 is replaced as S13,13 ~ S1,1. (6) Edges that are located on the top vertical periphery of the cell have to be wrapped onto the corresponding edges on the lower periphery with the phase term (ix3 DYP cot(y)+ 15

0 2 1 8 9 10 7 15 12 // 14 6 6 5 1 '3 4\ 17 14 45, 71718 3 19 20 Figure 6: Configuration for BE-calculations 3y DYP) and edges that are located at corners of the periodic cell (for instance edge #21) must be phase modified with respect to the x- and y-directions. The strategy for the calculation of the BE-matrices is similar to that for the calculation of the FE-matrix. First the matrix is calculated for all edges of the unit cell no matter whether they are located on the boundary of the unit cell or not. Then the matrix entries for the edges on the right and upper periphery section of the boundary are wrapped onto the associated matrix entries for the corresponding edges on the left and lower portion of the periphery with the help of the appropriate phase terms (see Fig. 6). Additionally, for the calculation of the matrix elements, the GCreen's function for an infinite periodic array must be used. In the given case, matrix elements of the form Zmn =-2koJJJ G(rr') (Wr x n) ds'' (Wm x ) ds Ti Tj +2J JG(r, r') V'. (Wn x z) ds'V. (Wm x z) ds (7) Ti Tj must be evaluated. As usual, W are the edge element expansion functions, Ti and Tj are the test and source triangles, respectively, and G is the periodic Green's function. In the original version of the PRISM-BI code, the free space Green's function is employed, 1 e-jkO-r' l Go= --- (8) 4w Ir-ri( where r is the observation point and r' is the source point. In the general case these coupling integrals are evaluated with a numerical quadrature technique, but for 16

near-coupling and especially self-coupling terms, the integration of the singularity of the Green's functions must be carefully performed to obtain accurate results. In the infinite periodic case, the proper Green's function can be obtained as an infinite sum over terms of the form Go with the coordinates of the source points of each term adjusted to the locations in the different cells in the lattice and with considerations for the phase terms due to the scan direction of the array. For practical implementations, this space domain summation is not applicable because of its very slow convergence of the series. For implementation into the FSS-PRISM code, the Green's function supplied by Prof. D. Wilton and Prof. D. Jackson, the University of Houston, was used. As noted in earlier reports, the EWALD acceleration scheme for the evaluation of doubly infinite series was employed. Using the EWALD summation scheme, one part of the infinite series is evaluated in the space domain whereas the other part is evaluated in the spectral domain. With a proper adjustment of the space and spectral domain parts of the series, a very fast convergence can be obtained. At the moment, the infinite array Green's function is available only for a lattice skew angle y = 90~ (right-angled lattice). If both the source and the test triangles are close to be boundaries of the unit cell, it can be observed (see Fig. 6) that additional singular terms of the Green's function series can be situated close to the integration domain. These situations have to be recognized for the evaluation of the integrals in eq. 7 and the additional singularities have to be dealt with carefully. 17

3 Validation of the Code 3.1 Example 1: As a first validation example, the trivial configuration in Fig. 7 was considered. The air layer was discretized with finite elements and a boundary integral with the periodic Green's function was applied on the top and bottom layers. The structure was excited by a plane wave propagating in the z-direction, and from Fig. 8 it is seen that we recover the expected unity transmission coefficient. Air I I PBC ~- Air I I Air BI/PGF /'__: PBC I I BI/PGF Figure 7: Infinite air layer U) Q).(1 0 C, 0.cn cn E C) - 1 0.8 0.6 0.4 0.2 0 r --- -— _ — _-__ -— _ — _ -- - _ _ _ _ _ _ _ - - - - - - - _ - -_ _ - - _ _- - l-_ _ _ _ _ _ _^ _- _ _ _^ _ - - - _. _ _ _ _ _ - _ _ _ _ _ i i I - - - - - - - - - --- - — I -- - - - -i - - - - - - - - - - - 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 Frequency (GHz) Figure 8: Power transmission coefficient for configuration in Fig. 7 18

3.2 Example 2: In the second validation example illustrated in Fig. 9, the air layer was replaced by a dielectric layer with a permittity of 6r = 5. In this case, the reflection and transmission coefficients for plane waves can be calculated analytically by recursively applying the reflection and transmission coefficients at the layer interfaces. A comparision between the analytical and FSS-PRISM values for the transmission coefficients of a normally incident plane wave is given in Fig. 10. BI/PGF Air I I PBCI I I ~r= 5 I I I I ~m Air BI/PGF Figure 9: Infinite dielectric layer 0.7.9_ 0.65 0 C) o 0.6 0 (-. 0 0.55 (/) 0) 0.5 EO.5 I- 0.45 I ',-.Periodic PRISM -...! --------- - -- --,, ~...Analytical._ — _ l- -- '^ ^ -- — '- - -1 - - -- - - -— ' - — t — ' — _ - - - - - - - - - - - - - - - - - - - - ' '. - - -. - - -- - - - - ' 'I- - - - - - - - - - - - - i - - - - - - t - - r - - + - 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 Frequency (GHz) Figure 10: Power Tansmission coefficient for configuration in Fig. 9 19

3.3 Example 3: The geometry of the third example is given in Fig. 11. A rectangular periodic slot is backed by a dielectric layer with a permittivity of er = 4. The configuration is excited by a plane wave propagating along the z-direction with the electric field vector oriented parallel to the long side of the slot. In Fig. 12 the numerical results of FSS-PRISM: are compared to those generated by FSS-EIGER, and the results from the literature (Mittra et.al., Proc. IEEE, 1989). 1 cm ' --- — 0.75 cm 1 cm 0.1 cm i 'r -4 * r 0.2 cm i i Figure 11: Slot-FSS on dielectric layer 20

1 4-IC a) a) 0 0 0 C) a) a) fr 0.8 0.6 0.4 0.2 0 "-^ -,- -------- Periodic PRISM --------- - ----- -..EIGER ----— T-..... EIGER - - -----— ^ —...... Mittra I I I. --- —- ----- ---------------- I i * I4 I I - - - - I. - - - - ------ ---- -- - - - -- - - - - - - - - - - - - - - - - - - ------ ------ \ --- — ----—,, --------- I l I l& I I I A I I ~... - - - - - -T - r. ---. --- 1 1,- - -- -- ---- --- - -.-.-.-. -. —J-l-.- - ______ * * N____ 9 10 11 12 13 14 15 16 17 Frequency (GHz) Figure 12: Power reflection coefficient for configuration in Fig. 11 21

APPENDICIES APPENDIX 1: Project Goals The goal of the SERAT project at the University of Michigan (with subcontract to Univ. Of Houston) is to develop a suite of software for the analysis of strip and slot dipoles on multilayered substrates backed by a frequency selective surface. The dipoles are equipped with photonic switches permitting variable electrical dipole lengths for broadband performance and the FSS is suitably designed to simulate a variable substrate thickness for optimal operation. A general view of the geometry is given in Figure 1. The UM/UH team proposed to construct a code which combines various computational modules interfaced with appropriate pre-processors and post-processors. The computational modules include: * Stand-alone moment method simulation of the FSS with up to 10 layers with commensurate and non-commensurate periodicities. * Simple moment method simulation of the antenna elements on the FSS panels * Hybrid FEM simulation modules for small arrays, planar periodic arrays and curved arrays on FSS panels. Various options for modeling the FSS and for mesh truncation were proposed to provide a compromise between speed and accuracy. These are outlined in the proposal and summarized in the attached milestone chart (repeated from the proposal). As called in the milestone chart, we are proceeding in accordance with the schedule in our proposal. Specifically: 1. The commensurate moment method FSS and moment method periodic array code was completed and tested by Houston and delivered to UM. This code is referred to as FSS-EIGER 2. A geometry driver is also being developed for FSS-EIGER to handle the dipole, crossed-dipole, slot and crossed-slot elements. 3. UM completed ahead of schedule the small array FSS code and has tested it for slot and dipole array. 4. An automatic geometry generation Driver was developed for FSS-BRICK applicable to commensurate FSS panels 5. A first version of the centerpiece of this Code-Suite, the hybrid FEM code, is completed for commensurate arrays and combines the powerful FEM for material modeling with the robust boundary integral method. A Driver is now under development. 22

6. Several code validations have been given for slot/dipole periodic arrays on FSS structures. 23

SERAT panel + single strip crossed on different slots laminates FSS elements antenna panels spacer/dielectrics / N Curved SERAT Figure 1: General SERAT configuration. 24

Milestone Chart for EM Model Development (Tasks 1 and 2) Ouarterlv Progress Mesh generator for FSS elements Single Element and Small Array _ Pliartand Crve Planar-FEM/Moment Method Curved-FEM for antenna and FSS 4th 5th 6th 7th 8th Q. Q. Q. Q. Q p — No. b-b p — B- -__ _ -------- -- ".- - - - — I~ _ Planar Periodic Array ='-1 it __. ' a.L 11' /" _ Approximate Doubly Curved Doubly Curved with fast integral algorithms for mesh truncations Software Integration and I/O Software Support 25

APPENDIX 2: Presentation Given on June 25, 1997 at Nashua, N.H. 26

NAWC/Sanders SERAT Program -, - -., - - - ---- -- - - - -- ---- f Hybrid FEM Software for SERAT Simulation Project Team T. Eibert, Y. Erdemli and J. L. Volakis Radiation Laboratory Dept. of Electrical Engin. and Computer Science University of Michigan Ann Arbor, MI 48109-2122 D.R. Wilton and D.R. Jackson Dept. of Electrical and Computer Engineering University of Houston Univ. of Michigan/Univ. of Houston SERAT Code Development Team

_NAWC '/Sq~ndP~rQ S BRA PANE S17.R AT Pro onrrn. an wrltna;.4 %.... % *2Cfllent% 00 Univ. of M\ichigan/Univ. of Houston SERAT Code Developmaent Team

NAWC/Sanders SERAT Proeram Overall Project Goals * Develop a Suite of Codes - Combining Moment Methods, Finite Element and Hybrid Formulations for SERAT Modeling * Univ of Michigan - Develop Hybrid FEM Software and Integrate Univ. of Houston Codes into a single SERAT Simulation Package Addressing Geometrical Adaptability, Speed and Accuracy. * Univ of Houston - Develop Moment Method FSS Simulators and Modules for the Hybrid FEM Software Package Univ. of Michigan/Univ. of Houston SERAT Code Development Team

NAWC/ anders SERA Prog~ram -.- -,. -. - ---- -- - - -.- -- -- - - - I Y'ear 1 Year 1'Year1 and Y ear2 Univ. of M4ichigan/Univ. of Houston SERAT Code Development Teamn 0

NAWC/Sanders SERAT Program First Year Goals(Summary)-Planar * Develop accelerated (simple) Moment Method FSS codes * Integrate Moment Method FSS code with antenna elements * Develop small array Hybrid FEM code as a testbed for the Hybrid FSS code * Integrate FSS Green's function with FEM for a general purpose planar SERAT array hybrid code Second Year Goals(Summary)-Curved * Generalize hybrid FEM codes for SERAT arrays on doubly curved platforms * Introduce fast algorithms and new solvers to speed-up computations for planar and curved SERAT arrays Univ. of Michigan/Univ. of Houston SERAT Code Development Team I

NAWC/Sanders SERAT Prograi Code Development Status * FSS_BRICK (UM) Small array code - Capable of large finite array analysis using fast Bl solvers - Fast but more restricted geometry due to brick modeling - Will be upgraded to periodic arrays * FSS_EIGER (UH) Simple moment code - Upgraded from EIGER to handle arrays on FSS - Includes fast evaluation/Ewald of periodic Green's function - Validated for commensurate slot and dipole arrays on FSS - Delivered to UM and is being tested * FSS_PRISM (UM) Planar SERAT hybrid FEM code - Our most capable code —combines FEM and MoM capabilities - Implemented with Periodic Boundary Conditions and Eiger free space Green's function. - Can handle FSS and antenna elements at any layer location. - Validated with FSS EIGER for slot and dipole elements - Slower than FSS_BRICK but full geometrical adaptability Univ. of Michigan/Univ. of Houston SERAT Code Development Team Ti ts) toj

NAWC/S anders SERAT PrograI FSS-BRICK (Small Array Code) (original UM version of the Air-Force XBRICK) - Rectangular brick elements for modeling SERAT volume - Includes capabilities for lumped loads, vertical wires, resistive sheets and feeds of arbitrary orientation - Boundary integral for mesh truncation and the FFT for the fastest known matrix compression —THIS IS THE GIVES A MAJOR SPEED ADVANTAGE FOR BRICK - BRICK was demonstrated for slot and strip FSS elements, dipoles on multilayered FSS. - Geometry Driver was developed for the SERAT commensurate panels (antenna and multilayered FSS) This code was scheduled for completion end of first year. It is ready, except for various upgrades in Geometry interface, lumped elements, more validation. Goal is to extend BRICK to a Periodic Code and compare to the Periodic PRISM in terms of speed and capability. Univ. of Michigan/Univ. of Houston SERAT Code Development Team 3fl (k> wJ

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NAWC/Sanders FSSBRICK Validation SERAT Program - `- - ` ` - - - ---- - - - -- I ' Single Layer FSS: Slot Element z X Slot (0.2x0.75cm2) Power Reflection Coefficient 1 FSS _ h=1 cm Unit Cell (lxlcm2) 0.9 0.8 0.7 0.6 I52 0.5 FSS-Brick Code: TM incidence,Oi =1~, i =00 FEM sample size: dx = 0.1, dy = 0.05 cm f,= 13 GHz 2x2 Array 3x3 Array No. of unknowns: 6795 15600 FFT pad size: 64x128 64x128 No. of iterations: 98 202 CPU time (sec): 139 247 0.4 0.3 0.2 0.1 0 8 10 12 14 16 18 20 Frequency, GHz Univ. of Michigan/Univ. of Houston SERAT Code Development Team

NAWC/Sanders Double L er FSS with Antenna SERAT Program km' z,/_x //: h,=0.40cm (e,, -) FSS-1 h2= 0.25cm ( | )= ~- Dipole Antenna - Dipole-1, (0.25x0.5cm2) / J l = 0-90~, ~= 0o fo= 17.25 GHz FEM sample size = 0.05 cm FSS with 1-cell: No. of unknowns = 16702 FFT pad size = 64x64 No. of iterations = 323 CPU time = 333.25 sec.0) Dipole-2,(0.25x0.6cm2) FSS-2 h3= 0.15cm / (Er,.r.) = (2.0,1.0) y=1 cmGi yGain Lx= 0.95 cm 1. 1 0. I 1-1........ /,= 13.50 GH Zi= 22.6-j26.3 oh 270 - = 17.25 GHz Zin= 118.8+j23.2 ohrm --- I I I A a __j Univ. of Michigan/Univ. of Houston SERAT Code Development Team

NAWC/Sanders SERAT Progra] * FSS Green's function and Code - Efforts during the first quarter concentrated on accelarating the periodic Green's function using Ewald's method - Achieved substantial speed-ups using the Ewald method as demonstrated in the 1st Quarterly report - Speed-up is usually 2 orders of magnitude - Validated Green's function by comparison with printed dipole and slot FSS - Ewald free space periodic Green's function was delivered to UM for integration with Periodic-PRISM code. This task was to be completed by the end of first quarter and was done on schedule Univ. of Michigan/Univ. of Houston SERAT Code Development Team Enl 00

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NAWC/Sanders SERAT Pro ram II COMPARISON OF CALCULATION TIMES 2 Ewald calculation time 2X(2(NEwald +I)) x1 (Asue evautoofefz)itetiemoeepnietaevlaonfth 1 Lb caellne: 1 Singh calculation time,2 2x (2Nsih +1 I ^ Singh 2 (Assumes evaluation of erfc(z) is ten times more expensive than evaluation of the free-space Green's function.) sample calculations: * Assume a required error of 0.1% * * 1 =1.11 * Assume a required error of 10-7 T=O.0028 Univ. of Michigan/Univ. of Houston SERAT Code Development Team

NAWC/Sanders FSS EIGER SERAT Progra * Simple MoM code Using New FSS Green's functio * Began with EIGER, a general-purpose code framework that allows flexible subdomain modeling with a wide variety of element types (triangles, rectangles, wires, bricks, etc.). * Developed and imported Periodic Green's Function with EWALD acceleration. * Added pre-processor (user interface) to generate SERAT geometries (dipoles, slots, crosses) and excitation data. * Added post-processor to calculate figures of merit for SERAT anlaysis and design (reflection, transmission coeffs., impedance). * Models any number of FSS and dielectric layers with lumped or distributed loads, arbitrary element shapes, linear or higher-order modeling, combined metallics/dielectrics. Commensurate FSS/MoM code was completed end of 2nd quarter as scheduled Non-commensurate FSS/MoM code and Drivers are in Progress Univ. of Michigan/Univ. of Houston SERAT Code Development Team m in

FS EIGE COD VALIDATIO SEATPgr NAWC/Sr anders -&j11 or rII THRE-AYEFSINFEESACE REFLECTION LOSS OF (0,0) MODE 0 LI) 0 -J 0 a1) a1) -5 -10 -15 -20 -25 ' — 8.5 9 9.5 10 10.5 11 FREQUENCY (GHz) k) Univ. of M~ichigan/Univ. of Houston SERAT Code Developm-ent Team

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NAWC/ anders FS IGR D~IPLIFS SEAT Prog..rami BROADSIDE-MATCHED ACTIVE REFLECTION COEFFICIENT E-PLANE SCAN h1= 0.4Ocm FSS-i1 h2= 0.25cm FSS-2 -h3= 0. 15cm PDipole Antenna r —Dipole- I (0.25x0.5CM2) p-Dipole-2 _(0.25x0.6cm2) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 n ------— r — ----- -- I e*jy)= (2.0, 1.0) Ly!Icm p ) ) 0 10 20 30 40 50 60 70 80 90 SCAN ANGLE (DEGREES) "% L = 0.95 cm W Univ. of M~ichigan/Univ. of Houston SERAT Code Developmaent Team

NAWC/S anders Orignal Version of RS as of O ctober FEERAlk- 1*>:*""'IA AkTwor I Li 4B SEAT Programa FEMA.-PRS wvas Developed under Air Force & N avy Sonsorship ]Vlesh G~erierati cn ID is tcmrted prism I4,e iidllic te rimi-go Uo.Aborb~ ig IIaxeria ------ 11 (Perspectw-ve vierw) (CrC'- SS- SeC-; Features.A.rialyz~es aritoerria s cri d <ubl1y c ur v ed p1 atfc<rrrias.Arii ctncDpic! rri aterial oc~catirigs/siibstrate-s are acmocx-da)ted CE3-rcv~, s vcDlurne! mesh using di stcrte~d pri srri S. (3-i ye~n the su~rf ace mnesh. vc~lurrle rresh,grc~xur alc~rig surfaces ri~rrrial s lB Iui 1 t-i ri surf ace me-=sh geraeratcDr f<Dr ci auld ar ar-id rectarigul ar patch ariterira s IAesh turic~ate~d u Si rig i scDtnc'pi clarii'sc~tncpi a artifiial A abscorber s Univ. of M4ichigan/Univ. of Houston SERAT Code Development Team

NAWC/Sanders.qFR AT Prnarrn n * FEMA-PRISM was upgraded during 1st quarter for accurate boundary integral truncation and future 11 interface with FSS Green's function Artificial Ab '- j sorber Sparse Matrix x0 Antpna [A] {E} Dense i nt \ - Boundary Integral Truncation ={ffnc} i E''1 f int Ebound iLE J [ r Aint A A rOSSi E Abound E b A IF c A Tcross I Uo j Lo,int 1 ext Univ. of Michigan/Univ. of Houston SERAT Code Development Team

NAWC/Sanders SERAT Program FSS-PRISM(Planar SERAT Code: FEM+Moment Method) * Development of FSS-PRISM started during second quarter * Implemented Periodic Boundary Conditions(PBCs) along vertical walls of the periodic element * Modified code for FSS (open boundary at bottom surface) and antenna analysis * Wrote interface and ported UH Periodic Green's function with Ewald Acceleration. * Began testing code for simple FSS and antenna elements. * Even with the Ewald speed-up, the periodic Green's function reduced speed by at least an order of magnitude Univ. of Michigan/Univ. of Houston SERAT Code Development Team

FSS NAWC/S anders PRISM SERAT Program -. -.- - --- - -- - ---- -- -- - - Periodic Boundary Condition (FE) Unit Cell: Infinite Periodic Array DYP cot Y 22 Periodic Free-Space Green's Function (BI) Unit Cell with Image Sources: x 4 II 8 9 1 2 12 1 62 1 3 1 3 19 20 DXP Univ. of Michigan/Univ. of Houston SERAT Code Developnient Tean

NAWC/Sanders SERAT Program -.- -.. - - - ---- -- - - - - - - -- - - - - - - -I FSSPRISM Current Capabilities * Periodic Boundary Condition (PBC) for FE-part * Periodic Green's Function (PGF) for BI-part * Metallic patches in all layers possible (Antenna, FSS) * Probe current feeds and plane wave excitation * Lumped impedances / resistive sheets (to be done; already available in FSS-BRICK code) Metallic backing (Antenna) BI/FPGF /l i - PBC i PBC Metal BI on top and bottom surfaces (Transmission, FSS) BI/ PGF / I - i PBC *' PBC BI /PGF Univ. of Michigan/Univ. of Houston SERAT Code Development Team 0o

NAWC/Sanders SERAT Program -. -. - - --- - - - -- - - FSS PRISM Validation FSS Slot Array Unit Cell: t<.1 cm I "kP Power Reflection Coefficient A tO,-.q IF12 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 A O.lcm v 0 8 10 12 14 16 18 20 Frequency, GHz Univ. of Michigan/Univ. of Houston SERAT Code Development Team

NAWC/Sanders SERAT Progra] Periodic-PRISM...next few months Planar hybrid FEM code is scheduled for delivery on the 5th quarter We already have a working version at this point. Goals for next few months: - develop Geometry Driver for commensurate and noncommensurate elements - validate and compare (speed and accuracy) with Houston's Periodic-Eiger code - optimize code - port multilayer FSS Green's function - Enhance modeling capabilities (loads, feeds, sheets, etc) - implement fast algorithms for BI speed-up rn t\ Univ. of Michigan/Univ. of Houston SERAT Code Development Team

NAWC/Sanders SERAT Program POSSIBLE APPROACH FOR NON-COMMENSURATE MODELING IN FSS PRISM Vertical PBCs I x PEC Horizontal PBCs Univ. of Michigan/Univ. of Houston SERAT Code Development Team (.A

NAWC/Sanders SERAT Proeram -. --- -- - - - - - - -- "..n TASK SCHEDULE FOR FSS_EIGER (1997) I June July August Sept. I 1 2 3 -- p — -44m - F ~ba- -E 1 Further code validation, improvements, documentation, driver development, loaded elements, calculation of additional figures of merit for antenna and FSS structures. 2 Numerical Green's function and integration with FSS_PRISM 3 Non-commensurate periodic structures. Univ. of Michigan/Univ. of Houston SERAT Code Development Team

NAWC/Sanders General Assessment SERAT Prograi At this point we have available: - 1. Periodic FEM codes(FSS-BRICK and FSS-PRISM) for SERAT modeling with geometry Drivers for commensurate periods PERIODIC FEM CODES ARE AHEAD OF SCHEDULE - 2. Periodic MoM code (FSS-EIGER) for commensurate FSS periods THIS TASK IS ON SCHEDULE - These codes are currently more capable than other Periodic codes available at the start of the project. * In principle, both codes have the capability of modeling arbitrary antenna and FSS elements * The FEM code has the potential for modeling any non-planar details such as feeds and loads or material inhomogeneities Univ. of Michigan/Univ. of Houston SERAT Code Development Team 'l

NAWC/Sanders SERAT Program Activity That Will Affect Next Year's Effort -New results on fast algorithms hold much promise for substantial speed-up above the present state-of-the-art. These algorithms handle the boundary integral (full) section of the hybrid system matrix -CVSS, a new sparse LU solver was examined for handling the large FEM matrix section of the hybrid matrix. The dramatic speed-up of this solver will allow full scale FEM implementations of the SERAT. -Compared several matrix compression schemes to improve speed of iterative solvers for hybrid FEM systems -Frequency extrapolation methods can reduce CPU time for broadband response evaluations -We made substantial advances in integrating design optimization methods with antenna codes. Univ. of Michigan/Univ. of Houston SERAT Code Development Team

N11AWC/Sanders SERAT Programa.A.'IA.AL.T %W.A.A I-IWIROA- -.F_ - -- L7 —2-%- - Slotz Fed DualtachAtenhDsg Substrate Slot length =?. >D~~esign ~1~arables Bottom P atch Substrate Slot Sura rte Performance at dl=O.85 mm, d2=O.55mm, slot length=4.O mm M. lR% r%, r FINAL DESIGN: VSWVR = 1.414 BWVV= 18% w K- 69 P4 4 —P R 9 so [ is I0 - Rin\ I1 II /I Rn 73 ~ 70 71.11 m...30 L7 L7..5 LB0 LBS5 LS L95 06 Frequency (GHz) Univ. of M\ichigan/Univ. of Houston SERAT Code Developmient Team

APPENDIX 3: Univ. of Houston (FSS-EIGER) Presentation on the May 30, 1997 review held in Ann Arbor. 58

NAWC/Sanders SERAT Program FSS CODE DEVELOPMENT * FSS analysis is accomplished by modifying existing nonperiodic multilayer code "EIGER" * EIGER has the capability to handle conductors of arbitrary shape and arbitrary element loading * Efforts are focused on accelerating the convergence of the periodic series to improve efficiency: Extraction of image terms from the spectral Green's function along with application of the "Ewald method" Univ. of Michigan/Univ. of Houston SERAT Code Development Team

NAWC/S anders EIGER USES A MIXED SERAT Program POTENTIAL INTEGRAL EQUATION FORMULATION =- j - VO —VxF E H =-jcoF -, 11 1 1 VxA du A=A GA J(r )dS S F=4EGF F-S S *M(r')dS' GA,GF dyadic potentials s S 9 (D ( 0 r rr dS'+ S z r9 rr 9 alit9 S s V M(r')K(r, r') dS'+ S z r I z r(r afd ' Univ. of Michigan/Univ. of Houston SERAT Code Development Team

NAWC/S anders SERAT Program EFFICIENT CALCULATION OF POTENTIALS REQUIRES A SOPHISTICATED SERIES ACCELERATION ALGORITHM GA = A - Ai (spectral form of layered media Green's function with asymptotic form removed) ~ +Y~[G9l S'6ASG 1s]] +pE[ G pq + r,s rO sO Gr,s (Ewald hybrid spectral - spatial summation method for terms representing periodic direct and quasi - static images) (self terms removed from Ewald series for handling singularities) Univ. of Michigan/Univ. of Houston SERAT Code Development Team

NAWC/Sanders SERAT Program EWALD ACCELERATION METHOD * The Ewald method is an efficient technique for calculating the periodic free-space Green's function * The periodic free-space Green' s function is the slowest converging part of the multilayered Green's function Ewald method converges much faster than conventional acceleration schemes Typically, the summation limits may be truncated at N = 1 Univ. of Michigan/Univ. of Houston SERAT Code Development Team

NAWC/Sanders SERAT Program OVERVIEW OF EWALD METHOD G(r,r') = 00 00 v e- jkt OOPmn m=-oo n=-oo e-jkRmn 4 irRmn elk^jo Q422 1) Use the following identity: e- jkRmn 4JzRmn 2- e 2 ~ 0 ds 2) Split the integral into two parts: Gl(r,r') 00 00 = 1- Xe- ktOO'Pmn m=-oo n=-co 2E 2 2j_ -R 2s2+ -- 0e 4s2 e oo k2 2 -Rm2~S + — 2~ E e 4s2 ds G2 r, 00 00 v e- JktOOPmn m=-oo n=-oo ds Univ. of Michigan/Univ. of Houston SERAT Code Development Team

NAWC/Sanders SERAT Program OVERVIEW OF EWALD METHOD 3) The second integral is cast into the form of the complimentary error function (which decays exponentially): k 2 4 e 2 2 ds 2 0 4s2 ds ~ Ee 1 2Rmn erfc RmnE - + ejkmnerfct 2E RmnE+ Hence: G2(r,r') 00 00 m=-oo n=-o e- ktOO'Pmn 1 e 8llRm 2Rmn L Jkmn erfc Rmn E -k 2E + e jkRmn erfc R m jk) RmnE + I 2E) I 4) The Poisson transformation is used to transform the first sum into one that also involves the complimentary error function. G,(r,r') = A A 1 ne4jk zm [ 2E 2E) 2E Univ. of Michigan/Univ. of Houston SERAT Code Development Team

NAWC/Sanders SERAT Program ACCELERATION RESULTS Results are presented for the following case: y 4 a =b = 0520 x'=y'=z'=O x,= y9= z)= 0 x = y = 0.25Ai z=O kxo = ky =0 (unit cell dimensions) (source is at the origin) (observation point is at the center of the unit cell) (observation point is in the source plane) (incident wavenumbers are zero) A A.i obs. point 0 b source point a Univ. of Michigan/Univ. of Houston SERAT Code Development Team

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NAWC/Sanders SERAT Program COMPARISON OF CALCULATION TIMES define: Ewald calculation time Singh calculation time 2x(2(NEwa +1))2 X10 2 x (2(ingh + 1)) (Assumes evaluation of erfc(z) is ten times more expensive than evaluation of the free-space Green's function.) Ratio T depends on the level of accuracy required (T is smaller as the accuracy increases). sample calculations: * Assume a required error of 0.1%: * Assume a required error of 10 - T- 1.11 %: T - 0.0028 Univ. of Michigan/Univ. of Houston SERAT Code Development Team

NAWC/Sanders SERAT Program CODE VALIDATION EIGER code with Ewald acceleration scheme has been validated for various cases: - arrays of dipole antennas in free space or over ground plane - arrays of dipole antennas printed on grounded substrate - FSS dipole arrays - FSS slot arrays Univ. of Michigan/Univ. of Houston SERAT Code Development Team

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NAWC/Sanders SERAT Progran ANALYSIS OF FSS STRUCTURES WITH NONCOMMENSURATE PERIODS * Formulation has been developed for analysis of FSS structures consisting of multiple screens with noncommensurate periods * Analysis is based on a discrete spectral propagator method which cascades both propagating and evanescent parts of the spectrum between layers * Method allows for arbitrary periodicity in the different FSS screens that are cascaded 1 Univ. of Michigan/Univ. of Houston SERAT Code Development Team

NAWC/Sanders SERAT Prograrr ANALYSIS OF FSS WITH NON-COMMENSURATE PERIODS * The formulation of a systematic method for the general analysis of an FSS structure with non-commensurate periods has been developed. inc inc (): ^ k=n() k x, ky ) ky ( kxm kyn ) I- - 0 0 0 * Discrete Spectral Propagator method: Introduce a fixed set of wavenumbers that discretizes wavenumber space (independent of structure). Propagating and evanescent wavenumbers are included in the set. Univ. of Michigan/Univ. of Houston SERAT Code Development Team 1 00 tKi

NAWC/Sanders SERAT Program * Each screen is characterized by a generalized scattering matrix [S] in which all scattered wavenumbers are approximately recast into the fixed wavenumber set (kxm' kyn) - f Amn) ' F (m,n) lD[S i I I s W 21 I S22 12 / f 2 -521 1 522- S 2, * Each incident wavenumbers wavenumber (kxm,kyn) (kxp kyq ): k(mp kxm d produces an infinite set of k(n + 2zq yq - yn - * Therefore, to obtain the elements renormalization procedure is used to scattered waves into an approximate propagator set (kXm,ky). of the scattering matrix a recast the amplitudes of the representation in the discrete 00 C~J Univ. of Michigan/Univ. of Houston SERAT Code Development Team

NAWC/Sanders SERAT Program renormalization: q+ (m,n) _ E /+ (pq) W(pq) vim,n) p,q I The weighting coefficients W (p,q) are determined from linear interpolation in wavenumber space: (m,n) - Am(kxp)An(kyq) where, e.g., Am(kx) is a linear (rooftop) interpolating function, centered at k = k Am (kx) m-1 m m+1 - * X ' * kX kxm Univ. of Michigan/Univ. of Houston SERAT Code Development Team 00 4~

NAWC/Sanders SERAT Programr * The scattering matrices for different screens are cascaded to obtain the overall scattering matrix for the entire FSS structure. * The scattering matrix can be used to help optimize the FSS design, and will also be used to terminate the FEM mesh for the analysis of the combined antenna/ FSS structure. I Univ. of Michigan/Univ. of Houston SERAT Code Development Team

NAWC/Sanders SERAT Program NUMERICAL GREEN'S FUNCTION /' PEC / i.. ---...........," // n // m/ UNIT CELL Interaction needed between magnetic current j and testing function i: [Yn] i -[Pif ][Zmnr ]f [mj ] Univ. of Michigan/Univ. of Houston SERAT Code Development Team

NAWC/Sanders SERAT Program Matrix Element Definitions DEFINE: E(J,M) - electric field operator H(J,M) - magnetic field operator J = InA (r) - electric current on conductors n M = VjA" (r) - magnetic currents at interface j yn -(A,H(An, )) Zmn =-(Am, E(An,0)) / mj =(AmE(OA;)) in =(A H(An,0)) 00.,, Univ. of Michigan/Univ. of Houston SERAT Code Development Team

NAWC/Sanders SERAT Program Obtain System Matrix for an Aperture at the Interface * system matrix for an aperture problem: ] Ff' [Pm1] \ in ] [ i L~,niJ [v ]_ [,rcj [Zmn] in] [0] * Fill upper half space with PMC so that Yext -. * Write system matrix blocks to a data file and form [Yj] [in][Xnj where [Zmn] [xn]= [m6] x Univ. of Michigan/Univ. of Houston SERAT Code Development Team

NAWC/Sanders SERAT Progr; NEAR-TERM GOALS * Calculation of reactions to provide interface between FSS and FEM codes * Further code validation, improvements, documentation, driver development, loaded elements calculation of additional figures of merit for antenna and FSS structures. * Begin coding for FSS structures with noncommensurate periods Univ. of Michigan/Univ. of Houston SERAT Code Development Team am n

NAWC/Sanders TASK SCHEDULE ( June I July August SERAT Program 1997) Sept. 1 2 3 —.Nd dm-mmmmamma~gg8 8 -18 1 q r/^ [ Further code validation, improvements, documentation, driver development, loaded elements, calculation of additional figures of merit for antenna and FSS structures. > Numerical Green's function. i Non-commensurate periodic structures. Univ. of Michigan/Univ. of Houston SERAT Code Development Team O0 0

APPENDIX 4: FSS-PRISM presentation by T. Eibert given on the May 30, 1997 review held in Ann Arbor 91

PRISM-Development PRISM: Finite Element/Boundary Integral Code Based on RightAngled or Distorted Prismatic Finite Elements and Triangular Boundary Elements * Modeling Flexibility in FE-Part, (Arbitrary Materials, Arbitrary Geometries,...) * Geometrical Adaptability * Ease of Meshing Starting Point: PRISM-code for Cavity-Backed Planar Antennas * Metallic Boundaries * Antenna Elements in Top Surface 3* Probe Current Feeds j *. Vertically Grown Volume Mesh Starting from a Given Surface Mesh MEMEMIN * Boundary Integral Based on FreeSpace Green's Function 92

Periodic PRISM-Code * Periodic Boundary Condition (PBC) for FE-part * Periodic Green's Function (PGF) for BI-Part (from Houston) * Metallic Patches in all Layers Possible (Antennas, FSS,...) * Probe Current Feeds and Plane Wave Excitation * Lumped Impedances / Resisitive Sheets (to be done, already available in BRICK-Code) * Metallic Backing BI/PGF (Antenna) 7 r r i - -I PBC PBC Metal * BI on Top and Bottom Surface (Transmission, FSS) BI-/-PGF - i PBC PBC BI/PGF 93

Inf iniLte Periodic Arrayr DYP cot 7 -I DXP Periodicity Condition: E (x *znDXP *n DYP CO t7, yZ2 nDYP)= x E(X. Jr) e J PX (m DXP + 12 LYP CO ty) e-jpy 12 L)YP H (xy*m DXP.-n DYP CO tyI,v ynLYP) = H (x., x) eJP x (m DXP + 12 DYP CO t yk jP y1 n DY' e e 3x= ~kosin13b COS (P0 P ~k sin13b sin p la' O.,o: Scan Angle 94

Periodic Free-Space Green's Function (BI) ( from Houston ) Unit Cell with Image Sources: 8 9 10 2 1 7/// 6 4 1 15 I5 12. 5 16 "' 17 1 3 19 20 * Transformation of Matrix Elements analog to FE-Part * Special Care for the Singular Image Sources * Implementation of Plane-Wave Excitation * Extraction of Transmission Coefficients and Radiation Patterns 95

Power Transmission Through an Air Layer (sanity check) BI/PO F PBC I I I I I Air Air I I I I I I I -I, —PBC A' I I Air BI/PG F Transmission coefficient has to be 1.0 -- 0a) (C Cn E cn C E c1 1 0.8 0.6 0.4 0.2 0 I I I I I I I I I I I I I I I I i I I I I2 I I I I I I I I I I I I I I I I I I I I I I I _ I I I I L L I II I I I I i 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 Frequency (GHz) 96

Power Transmission Through a Dielectric Layer Air BI/PG F Air 0.7 -I I0 PBC 0.55 - - - - PBC r_5 i-__ -- -I Air BI/PGF 0.7 0.5 ------- Periodic PRISM I --- Analytical CD ) I I I 0 0.55 < I r cn cn C) - 0.45 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 Frequency (GHz) Frequency (GHz) 97

FSS Slot Array 1 cm 0.75 cm 1 cm s L l- 0.2 cm 4 -4 _ I r i _ J_______ _ _0.1 cm Power Reflection 1 a) 4-t C a) 0 4 -0 ca a) cc 0.8 0.6 0.4 0.2 0 I I I I I! I, —,, _ Periodic PRISM t --- —------- 4, _ _ I I I -_ -: S- -.-.. EIGER - 4 I, I I I ------- -..... Mittra + - - — r --- —I 4.1 I I I, - --- - - I- - — i- -- -- - - \ - - -1- - - - - - - - -- - - - - - - -- - - - - - - L - - - - -- I I I I I - \ - -,- - - -- - -.... - ~. - I I I I I I I - - - - l - - - -^ - A -, _ - - _,- _ -, ^ _ - -T - - - - - -r - - - - — ~ _ i I \ I I I I I I 4 I I i ~ I I --------- -- - - ---- —.-: --- —------- I 41 I * I I I I I.1 i I 4. I I I - I — I 4 % --- —-~ --- —-- --— T I I I, I. I ' - 4 I. - - - - - - - - - - - - - - - J - - - k - -, - ^ - - - - - - - ^- - - — I- — I ___***L L__ 9 10 11 12 13 14 15 16 17 Frequency (GHz) 98

APPENDIX 5: FSS-BRICK Presentation Given by Y. Erdemli on the May 30, 1997 review held in Ann Arbor 99

09 w *0 94 9z 0ll* 9 F* w 0 w cm 4 W) ONO 94 4 1 cn CA wo 100

FEATURES: * Driver: Geometry I/O tool * No need for external meshing package * Mesh: Rectangular brick elements * Driver requires the following inputs: * Computation type: * Bistatic scattering * Backscattering * Antenna radiation * Number of periodicity for commensurate FSS geometry (for both x- and y-direction) * Physical dimension of periodic FSS unit cell (all dimensions are in cm) * Sample size in x- and y-directions * Depth of cavity, and no. of layers along the cavity Fss-Brick Code's Mesh y 3 2 1 2 3..... N Aperture yL A-.x 1 - 1 I r 1 11) PEC 3 Nz --- 3z -------- -- PEC AZ -x -- Cavity z Y -7 x z / 7 D, / a Dz DX ] O

FEATURES (cont'd.): * Thickness and electrical parameters (er,Ur) for each layer along the depth * Operational frequencies in GHz * Polarization, incidence, and scanning angles * Type of antenna placed at the cavity aperture * Physical dimension of the center-fed antenna * Excitation information: Amplitude and phase of the probe-current (x- or y- directed) * Physical dimension of each FSS layer's elements * Geometry display of the antenna and FSS elements * Although they are not available in the driver as input options yet, the FSS-Brick code has also the following features: * z-directed probe feeds * Short-circuit pins * Lumped impedance loads * Resistive cards * Dielectric patches * Transmission coefficient or Gain is output on screen. * Similarly, a geometry driver for PRISM code will be developed.

FSS Element Types Available in BRICK Driver Strip Slot Crossed-Strip Crossed-Slot 1..L.JJ....... _II....L.. Li.J..ii..L I -.. i.! I! 1 1 1 I 1 1 1 I! Dipole Antenna Slot Antenna Crossed-Dipole Antenna [ 1 I

Single Layer FSS: Slot Element Power Reflection Coefficient 1 hi= 0.05 cm FSS h2=0.10 cmI h3=0.15cm yI 0.9 (0.2x0.75cm2) I Absorber Layer i/ = 1 cm Lx= 1 cm 0.8 0.7 0.6 IF12 0.5 0.4 0.3 0.2 0.1. Fss-B rick Code: TM incidence, i = 10, Oi= 0o fr = 12.25 GHz FEM sample size = 0.05 cm FSS with 1-cell: No. of unknowns = 5977 FFT pad size = 64x64 No. of iterations = 217 CPU time = 171.44 sec 8 9 10 11 12 13 14 15 16 17 18 19 Frequency, GHz

Single Layer FSS with Antenna hi= 0.40 cm FSS.1 cm h2= 0.1 cm (E V f Dipole Antenna Gain 90 o Dipole (0.25x0.5cm2) 7 LI= 1 cm 0 Er2'gr2) = (2.0,1.0) Lx= 0.95 cm A Fss-Brick Code: 0 = 0-90o, 0 = 0 f = 18 G Hz FEM sam ple size = 0.05 cm FSS with 1-cell: No. of unknowns = 10461 FFT pad size = 64x64 No. of iterations = 306 C PU tim e = 224.63 sec 270 f....... = 16.0 GHz Zin= 1.7-j45.3 ohm f,= 18.0 GHz Zin=135.3+j 1.3 ohm 8i -- --— ~~ — ~~ ---- ~ --- —--— ~ ~-~~~-~~~~ ~ ---~ ~~ ---- ~~-~~~~- ---- -~~-~ --- -— ~~~~- -- -~ -- -- ~~~~ ~~ ---- ~~ —~~ ~ —~ —~

00 Cl) Cu 0 Cu) 0 0lo 0c IIlt CCOD 5 7~Z Z II II I r 106

rA 0=0 9 44 4 0*:4 *0 *4 w 9z C 0 cn Cl) rzo w F* Z) cn 4 w 0 C W) 1 4 C 4 04 0 44 u -11 0 9 00 44 -4! w 04 m 0=4 A cn 0 00 0. 0 U I- - -I I "I-\ I I -------- -------- I A 1% I I A N / A I, '4, N / I, N ---- ---- A-4c-444 - - - - -- K N /... -. I. -... -- I I. -. I. - K N 4 N -— "7 v '111 N'. ------ 90 r= C*n ol 0 CI) 107