TECHNICAL REPORT 033965-1-F Annual Report on NSF: ECS 9628972 MRTD: New Time Domain Schemes Based on Multiresolution Analysis by Linda P.B. Katehi, The University of Michigan and Andreas Cangellaris The University of Arizona January 1997 33965-1 -F = RL-2467 January 30, 1997 1

1. PERIOD COVERED BY REPORT: January 1, 1996 -31 December 1996 2. TITLE OF PROPOSAL: MRTD: New Time Domain Schemes Based on Multiresolution Analysis 3. CONTRACT OR GRAND NUMBER: ECS 9628972 4. NAME OF INSTITUTION: University of Michigan 5. AUTHORS OF REPORT: Linda P.B. Katehi and Andreas Cangellaris 7. LIST OF MANUSCRIPTS SUBMITTED OR PUBLISHED UNDER ARO SPONSORSHIP DURING THIS REPORTING PERIOD, INCLUDING JOURNAL REFERENCES: 1) E.M. Tentzeris, R.R. Robertson, L.P. Katehi and A. Cangellaris, "Space- and Time- Adaptive Griding Using MRTD Technique," accepted for presentation in the 1997 IEEE MTT-S International Symposium. (Appendix B) 2) E.M. Tentzeris, A. Cangellaris and L.P.B. Katehi, "Space/Time Adaptive Meshing Using the Multiresolution Time Domain Method," accepted for presentation in the 1997 ACES (Advanced Computation Electromagnetics Society). (Appendix C) 3) E. Tentzeris, J. Harvey and L.P.B. Katehi, "Time Adaptive Time-Domain Techniques for the Design of Microwave Circuits," accepted for presentation in the 1997 IEEE AP-S International Symposium. (Appendix D) 4) R. Robertson, E.M. Tentzeris, L.P.B. Katehi, "Modeling of Membrane Patch Antennas Using MRTD Analysis," accepted for presentation in the 1997 IEEE AP-S International Symposium. (Appendix E) 4) K. Goverdhanam, L.P.B. Katehi, A. Cangellaris, "Applications of Multiresolution Based FDTD Multigrid," accepted for presentation in the 1997 IEEE MTT-S International Symposium. (Appendix F) 8. SCIENTIFIC PERSONNEL SUPPORTED BY THIS PROJECT AND DEGREES AWARDED DURING THIS REPORTING PERIOD: Graduate Students Faculty E.M. Tentzens Linda P.B. Katehi K. Goverdhanam * These two students were funded part time by this contract. January 30, 1997

Brief Summary of Performed Study 1.0 Abstract The application of multiresolution analysis directly to Maxwell's equations results in new time domain schemes with unparalleled properties. This time domain approach, MRTD (MultiResolution Time Domain Method). allows for the development of schemes which are based on scaling functions only or on a combination of scaling functions and wavelets for the development of a variable riding. The dispersion of the MRTD schemes compared to the conventional FDTD Yee's scheme shows an excellent capability to approach the exact solution with negligible error for sampling rates which approach the Nyquist limit. Furthermore, due to the weak-interaction properties of the wavelets, MRTD schemes allow for time/space-adaptive grids. These recent developments in time domain techniques at the University of Michigan have strongly indicated the potential of MRTDs in creating a major impact to the area of computational electromagnetics [10, 11]. MRTD is not a new methodology. It is a correct and accurate generalization of the conventional discretization approaches. It provides the correct mathematical frame for solving problems in time domain and allows for the understanding of important issues in time-domain computational electromagnetics. 2.0 Introduction to MRTD The finite-difference time-denomain method (FDTD) has proven to be a powerful numerical technique in electromagnetic field computations [1,2]. However, despite its simplicity and modeling versatility, the technique suffers from serious limitations due to the substantial computer resources required to model electromagnetic problems with medium or large computational volumes. In addition, the FDTD method cannot provide the accuracy required for computer simulations of time-dependent electromagnetic interactions in electrically long regions or in regions which contain non-linear materials. Such simulations are very important for integrated device modeling, especially in relation to the design of non-linear photonic devices. The above limitations have always made it a matter of great interest to improve the efficiency ofYee's FDTD scheme and have led researchers to the development of hybrid combinations of FDTD with other propagation methods [3,4] and higher order FDTD schemes based on Yee's grid [5]. The method of moments provides a mathematically correct approach for the discretization of integral and partial differential equations [6]. Its application to the discretization of Maxwell's partial differential equations has provided the field theoretical foundation for TLM [7,8]. In addition, it has been shown recently [9] that Yee's FDTD scheme can be derived from the same approach when using pulse functions for the expansion of the unknown fields. Since the method of moments allows for the use of any complete and orthonormal set, the choice of an appropriate expansion set may lead to new powerful time domain schemes. The application of the method of moments using scaling and wavelet functions, known as multiresolution analysis (MRA), has been applied to Maxwell's partial differential equations and has lead to novel and powerful time domain schemes [10] and [11]. January 30. 1997 3

In a MRTD scheme the electromagnetic fields are represented by a two-fold expansion in scaling and wavelet functions with respect to space. The expansion in terms of scaling functions allows for a correct modeling of smoothly-varying electromagnetic fields. In regions characterized by strong field variations or field singularities, additional field sampling points are introduced by incorporating wavelets in the field expansions. These additional points are introduced only at specific locations, thus, allowing for a variable grid capability. The use of different families of functions leads to various time domain schemes. The exclusive use of scaling functions provides a variety of conventional schemes including FDTD and TLM. The MRTDs which have been recently developed at Michigan have used pulse functions as expansion and testing functions in the time domain in order to obtain a two-step finite difference scheme with respect to time. MRTD schemes based on cubic spline Battle-Lemarie scaling and wavelet functions have been developed and applied to a variety of problems. An extensive discussion of these derivations is presented in [10]. This orthonormal wavelet expansion has already been applied successfully to the computation of electromagnetic-field problems in the frequency domain and results have been presented for both 2-D and 3-D problems [17,18]. The Battle-Lemarie scaling and wavelet functions do not have compact support, thus the MRTD schemes have to be truncated with respect to space (see Figures 1,2). However, this disadvantage is offset by the low-pass and band-pass characteristics in spectral domain, allowing for an a priori estimate of the number of resolution levels necessary for a correct field modeling. Furthermore, for this type of scaling and wavelet functions, the evaluation of the moment method integrals during the discretization of Maxwell's PDEs is simplified due to the existence of closed form expressions in spectral domain and simple representations in terms of cubic spline functions in space domain. The use of non-localized basis functions cannot accommodate localized boundary conditions and cannot allow for a localized modeling of material properties. To overcome this difficulty, the image principle is used to model perfect electric and magnetic boundary conditions. As for the description of material parameters, the constitutive relations are discretized accordingly and the relationships between the electric/magnetic flux and the electric/magnetic field are given by two matrix equations. FIGURE 1. Battle-Lemarie Scaling Function and its Fourier Transforms. Battle Lemarie Scaling Function Founer Transform c;f Batlle-Lemane Scaling Function 1.2...............................a... '..........'.,:,,,........ 0....... -- --------- 0.4 0 0.2 -0.2 -::-:.:; i I:: i i 0 -3.5 -1.9 -0.3 1.2 2.8 4.4 -0.2 | I ' ' ' Argument -8 -6 -4 -2 0 2 4 6 8 Radial Frequency January 30, 1997 4 January 30, 1997 4

FIGURE 2. Battle-Lemarie Mother Wavelet and its Fourier Transforms. 1.5 1 0.5 0 -0.5 -1 Battle Lemare Wav-Itet Function rF.-1 K - L: ' '............................... - - - ^ _4,,, __...,,............. _ -3.5 -1.9 -0.3 1.2 2.8 4.4 Argument > -.77 0.8 - 0.6 - 0.4 r-. 0.2 0 --0.2 r -15 i I - 15 11 E -10 -5 0 5 10 Radial Frequency Complete dispersion analyses of the MRTD schemes including applications to 3-D problems and comparisons to Yee's FDTD scheme are given in [10] and show the superiority of MRTDs to all other existing discretization techniques. Specifically, the results show the capability of the MRTD method to provide excellent accuracy with up two points per wavelength which is the Nyquist sampling limit. The use of Battle Lemarie scaling and wavelet functions has provided very efficient solutions to open and shielded circuit problems (see Appendix B). Figure 4 shows field calculations for the even mode excited in coupled strips operating in an open environment as shown in Figure 3. The open boundaries have been accounted for by incorporating PML regions within the MRTD technique. The use of MRTD allowed for the placement of the matching layer right at the planar lines while it provided very high accuracy and high computational efficiency. Figure 5 shows the even-mode electric field distributions for a similar coupled strip geometry operating in a shielded environment as shown in Figure 6 [22] (see Appendix B). As with Battle-Lemarie functions, the application of the Haar basis functions (see Figure 7) has led into the development of a multigrid FDTD technique [23, 24] and has demonstrated the capability for a spatially adaptive grid in 2-dimensional waveguide and transmission line problems. Figure 8 shows the field distribution in a two-dimensional shielded stripline. On the same figure the spatially adaptive grid utilized in this problem is shown. For more details on this development see Appendices A and F. FIGURE 3. Open Coupled Strip Line PML PML LI mI January 30, 1997 5 January 30, 1997 5

FIGURE 4. Even-Mode Field Distribution in the Structure of Figure 3 LLu:.I * 0 0) ---*^ — X L..........:.:?;,i = a.....,.....;''.~~. '::::Q:.":. ~.~i:?.i.i.X~' January 30, 1997 6 January 30, 1997 6

FIGURE 5. Even-Mode Electric Field Distribution for the Structure of Figure 6 O -...I. ~ "'. "'""*I C... "'.........-.^ is.* m I..',::: ' 0 I C~i:: January 30, 1997 7 January 30, 1997 7

FIGURE 6. Shielded Coupled Strip Line.ii 3.0 Time/Space Adaptive MRTD Schemes The major advantage of the use of Multiresolution analysis to time domain is the capability to develop time and space adaptive schemes for an open as well as shielded space, and its application to linear and non-linear problems. The effectiveness of the technique is measured in terms of its accuracy and computational efficiency in comparison to conventional time domain techniques (FDTD, TLM). The development of a spatially and temporally adaptive MRTD method is based on the property of wavelet expansion functions to interact weakly and allow for a spatial sparsity that may vary with time as needed through a thresholding process [12-19]. The availability of such an adaptive method is extremely important for the accurate modeling of sharp field variations of the type encountered in beam focusing in nonlinear optics, wave propagation through narrow slits and apertures etc. A brief presentation of the fundamental steps required for such a adaptive grid is given in the following section. FIGURE 7. Haar Scaling Functions and Wavelets 0 11 2r X XX The use of multiresolution analysis for adaptive grid computations for PDEs has been suggested by Perrier and Basdevant [20] and Liantdrat and Tchamitchian [21]. To describe the basic ideas of such an adaptive scheme for Maxwell's hyperbolic system, let us cast Maxwell's equations in one spatial dimension in the form: January 30, 1997 S January 30, 1997 8

a 11 = Au at (I) where il = (E(z, t), H(z, t)), andA is the operator: 0 A = -I az -E(-) o (2) FIGURE 8. Field Distribution in a Printed Stripline Using MRTD Based on Haar Functions Januay 30,1997 January 30, 1997 9

The numerical solution of (1) subject to initial conditions and appropriate boundary conditions at the two boundary points is sought. Following appropriate derivations, the above equation can be written in the following form: M IL) = 0 (3) where M = hdt hD (4) -Z'hDz tZhd_ In equation 4, Zh, Th are half shift operators for the spatial and temporal coordinates z, t respectively and Zth, 7h, are their Hermitian conjugates. Furthermore, dr, D, are difference operators given by the following equations: dt= ^(Tth- Th (5) and 1i 8(i 8 D = ^ th a()-+ (i)Z- (6) = -9 i = -9 where Zt=Z-. (7) In equation 6, (a(i), a,,(i) are the coefficients associated with the scalar functions or wavelet expansion functions respectively. Since M is represented by block of band matrices, it can be shown that the domain where the field coefficients of u(n+) are non-negligible is at most equal to the corresponding domain of u(n) plus the width of the bands in matrix M that represent the operator A in the wavelet basis. If N(, NW are the total number of nonzero scaling and wavelet field coefficients (grid points), the number of operations required to compute M is of the order of 0( NI + N ). As it has been shown in [10,11], the total number of scaling grid points can be as low as two per wavelength. January 30, 1997 10

However, the resolution required in the wavelet grid points is determined by the nature of the boundaries in the problem of interest. We are now ready to describe the method suggested in [20,21] to adapt in space and time the wavelet grid and thus follow the sharp features of the waves as they develop and/or move on the grid. At each time step we keep both the wavelet field values that are larger than a given threshold as well as the adjacent values. An adjacent wavelet field value is defined on the basis of the wavelet resolution level(s) incorporated in the solution. The development of an appropriate definition can be considered. Let G(n) be the wavelet field values (grid points) which are kept and represent the approximate solution at the nth time step. From these let us call fundamental the wavelet coefficients that are greater than the threshold and adjacent the ones as defined above. From equation 3 we compute the wavelet field coefficients of u(n+l) corresponding to the fundamental and adjacent coefficients that constitute grid G(). We then adjust G(n) by changing into "fundamental" those field coefficients that are greater than the threshold and changing into adjacent their adjacent ones. This process creates the new grid G(n+ )* We project u onto the space corresponding to G(n + ) and we are ready for the next step update. Clearly the basic assumption behind this algorithm is that during a time At, the domain of the fundamental field coefficients does no spread beyond its border of adjacent coefficients. This method has already been applied to a variety of circuit problems [25] and is described in detail in Appendices C, D and E. Figure 10 shows the adaptive mesh following the pulse exciting the parallel plate waveguide structure shown in Figure 9, with a dense dielectric slab placed perpendicular to the parallel plates. This figure clearly shows the capability of the mesh to discretize based on field intensity and not on geometry. (see Appendix) FIGURE 9. Parallel Plate Waveguide Structure - 8 mm S January 30, 1997 11I

FIGURE 10. Time Adaptive Meshing Time of Incidence After Inckience 1 0.8 0.6 0.4 U. 0.2 u: z -0.2 -0.4 -0.6 I -- 6x800 FDTD II I A - - 2x100 MRTD 0 100 200 300 400 500 60070.00 0010 -0.~ M 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 1000 Time-Step Normal E-field (Time-Domain). January 30. 1997 12 January 30, 1997 12

APPENDIXA January 30, 1997 15 January 30, 1997 15

Appendix A: An FDTD Multigrid Based on MRTD Kavita Goverdhanam. Linda P.B. Katehi Radiation Laboratory. Department of Electrical Engineering and Computer Science University of Michigan, Ann Arbor, MI 48109-2122 Andreas Cangellaris Department of Electrical Engineering University of Arizona, Tucson, AZ

AN FDTD MULTIGRID BASED ON MRTD: Introduction The use of wavelets in the method of moments for the solution of integral equations in frequency domain has been known since the 1993 Antennas and Propagation Symposium in Ann Arbor. MI [L]. Recent publications have demonstrated that the application of the method of moments directly to Maxwell's equations allows as well for the use of multiresolution analysis in the time domain [2], [3], [4]. In fact, multiresolution time domain (NIRTD) schemes based on Battle-Lemarie scaling functions have shown to exhibit highly linear dispersion characteristics. In comparison to conventional FDTD. savings in computation time and memory of one and two orders of magnitude have been reported in [2]. Depending on the choice of basis functions, several different schemes result, each one carrying the signature of the basis functions used in MIRA. It is also important to note that the design of an MNRTD scheme can be accomplished using one's own application-specific basis functions. The objective of this work is to develop an FDTD multigrid using the Haar wavelet basis. It will be shown that NIRTD technique using Haar scaling functions results in the FDTD technique. Motivation for this work stems from the theory of MRA which says that a function which is expanded in terms of scaling functions of a lower resolution level, ml, can be improved to a higher resolution level, m2, by using wavelets of the intermediate levels. In other words, expanding a function using scaling function of resolution level ml and wavelets up to resolution level m2 gives the same accuracy as expanding the function. using just the scaling functions of resolution m2. However, the use of wavelet expansions has major implications in memory savings due to the fact that the wavelet expansion coefficients are significant only in areas of rapid field variations. This allows for the capability to discard wavelet expansion coefficients where they are not significant thereby leading to significant economy in memory. Different resolutions of wavelets can be combined so as to locally improve the accuracy of the approximation of the unknown function. This, combined with the fact that wavelet coefficients are significant only at abrupt field variations and discontinuities allows MIRTD to lend itself very naturally to a Multigrid capability. As mentioned above, the FDTD equations can be derived by applying the method of moments to Maxwell's equations using pulses as basis functions. Since pulse functions are the scaling functions in the Haar system, the FDTD technique can be considered as a specific MIRTD scheme. Based on these ideas, the multigrid for the first resolution level is derived in the following sections. This is then followed by the performance of dispersion analysis to demonstrate the improvement by adding the multigrid to the regular FDTD. The 2D MIRTD scheme based on Haar basis functions is then developed and applied to solve for the Electromagnetic fields in a waveguide and a shielded stripline. The results obtained are compared with those computed using conventional FDTD technique. It will be shown that the wavelet coefficients are significant only at locations with abrupt field variations. This facilitates in obtaining accurate solutions by combining the wavelet and scaling coefficients only in regions where the wavelet coefficients are significant (discontinuities).

Derivation of the FDTD Multigrid Using the Haar System As it is known in the literature, the Haar system is generated by a scaling function t(x) and a mother wavelet 4(x) given below: 1 for 0 < x < >(x)= - - - 0 elsewhere and 1 for0 < x < P(x) = -1 for < x < 1 0 otherwise Fig.1 shows these two generating functions. In the regular FDTD scheme. the electric and magnetic fields are expanded in terms of pulse functions as shown in the equation below: E(x. t) = vkm kEm hk(t) hm(x) where kEm is an unknown constant and hk(t) and hm(x) are the pulse functions centered at tk and xm. Applying this expansion to the one dimensional wave equation and using the method of moments (Galerkin's technique) we obtain the following discretized equation: Ax t(k+lE -k- Em) =k Em+l -k Ermwhich can also be denoted as: DtE = Em+l -k Emiwhere Dt is the differential operator in the time domain. This is the regular FDTD discretization scheme with the E and corresponding H components located at the same nodes. To apply multiresolution analysis, the scaling functions and wavelets of the Haar system are both used as shown below: E(x, t) = Skmhk(t)[(kErnm(x) +k Emrn(x)] Using Galerkin's method, two discretized equations are obtained for the one dimensional wave equation. Next, a dispersion analysis is performed in order to compare the characteristics of new scheme with that of the traditional FDTD technique in terms of the dispersion errors. A similar approach can be adopted for the 2D and 3D cases to perform the discretization. Dispersion Analysis The dispersion relation of the FDTD and the IMRTD scheme is calculated from the solution of the eigenvalue problem after transforming the equations in the frequency domain [5]

X - o 3 0 2 LL 4-.i 1 O0 C) I..L 0 1 2 3 X x - 3C) 0D (a 3 a) m 1 X 2 3 Figure 1: Scaling and Wavelet Functions.

- f. For the ID wave equation discussed above. the FDTD dispersion relation is show;-n below: A_.r Q n c t '2 '2 where = __At and \ = Axk, This equation is also derived when using Yee's cell. In the equations above. Ax and At are the space and time steps respectively and k, is the magnitude of the wave vector. The dispersion relation of the first order resolution MNRTD mesh (FDTD multigrid) for the ID wave is given by the following equation: Ax t y.s5n( ) =2sin( ) cAt 2 4 Fig.2 shows plots of the normalized wave vector component y as a function of the normalized frequency Q for the ideal case, the FDTD technique and MRTD scheme. From the figure it can be seen that the dispersion curve of the NIRTD scheme is much more linear and closer to the ideal than the FDTD scheme. Similar analysis has been performed for a 2D problem where the wavelets were applied in one direction and only scaling functions in the other direction. As expected, the dispersion relation is the same as that of the FDTD scheme in the direction in which no wavelets are applied while the dispersion curve substantially improves in the direction in which the wavelets are used. Figs. 3 and 4 show plots of y as a function of Q for different directions and validate the claims above. The 2D-MRTD scheme Consider the following 2-D scalar equation obtained from Maxwell's H-curl equation: OEr OH, e t = y + j3H (1) This equation can be rewritten in a differential operator form as shown below: L,(f(x, y, t)) + L(f2(x, y, t)) = g (2) where L1 and Lo are the operators and fl(x,y,t) and f2(x.y,t) represent the electric/magnetic fields. We now expand the fields using a Haar based MRA with scaling functions d and wavelet functions u The field expansion can be represented as follows: f(x. y, t) = [A][t(x)o(y)] + [B][o([:x),(y)] +[C][C-(x)(,(y)] + [D][\,[(x),(y)] (3)

>,FDTD N. N j Ideal 0.5 1 1.5 2 2.5 Figure 2: Dispersion Plots for 1D.

% N> FDTD 'S, N N \ N deal 0 0.5 1 1.5 2 2.5 Figure 3: Dispersion Plots for 2D in (1,0,0) direction.

MRTD D / / I / / Ideal 0 0.5 1 1.5 2 2.5 3 3.5 Figure 4: Dispersion Plots for 2D in (1,1,0) direction.

Cwhere o( )o( ). o( X)(V)] y [ ( x)o(y)] and [ ( x') ( 1 represent matrices Awhose elements are the corresponding basis functions in the computation domain of interest and [A]. [Bi. J(1'. iD] represent the matrices of the unknown coefficients which give information about the fields and their derivatives. Application of Galerkin's technique leads to 4 schemes which can be represented as follows: < [oo], L[( f) + L,(f2) >=< [oo].g >: oo5cheme (4) < [ot']. L(fl) + L^(f ) >=< [o].g >: oLScheme (5) < [uo].LI(fi) + L9(f2) >=< [,O],g >: coScheme (6) < [UIj]. L(fl) + L^(f) >=< [1 i].g >: uScheme (7) From this system, we obtain a set of simultaneous discretized equations. For the first resolution level of Haar wavelets. the above four schemes decouple and coupling can be achieved only through the excitation term and the boundaries. The shielded structures analyzed here are terminated at Perfect Electric Conductors (PEC) and the boundary conditions are obtained by applying the natural boundary condition for the electric field on a PEC as shown below: E%4~(x)o0(y) + E~' o(x )t(y) + Ej r(x)O(y) +. +E' U,(x)(y) = O....AtPEC. (8) where Et, Et', Et'~ and E" are the scaling and wavelet coefficients of the tangential electric field at the boundary nodes. The above equations are discretized by the use of Galerkin's method which results in a set of matrix equations of order N = M+1 where M is the order of the considered wavelet resolutions. These equations are solved simultaneously with the discretized 'Maxwell's equations to numerically apply the correct boundary conditions. Applications of 2D FDTD Multigrid and Results The 2-D MRTD scheme derived above has been applied to analyze the Electromagnetic fields in a waveguide and a shielded stripline. (a) Waveguide: An empty waveguide with cross-section of 12.7 x 25.4 mm is chosen. A coarse 5 x 8 mesh is used to discretize this mesh and 2D MRTD technique was applied to analyze the fields in this geometry. Fig. 5 shows the amplitudes of the wavelet and scaling coefficients of the electric field obtained by using NRTD technique. From this figure it can be seen that only the (o and ok coefficients make a significant contribution to the field and that the contribution of de and wy is negligible. From the computed coefficients, the total field is reconstructed using an appropriate combination of the scaling and significant wavelet coefficients. For the waveguide chosen here, elimination of the wavelet coefficients that have no significant contribution leads to 480 unknowns. The reconstructed field obtained by this mesh has the same accuracy as that of a 10 x 16 FDTD mesh with 960 unknowns which is in agreement with the theory of IRA. Fig. 6 shows the results of this comparison and demonstrates that the use of multigrid scheme provides a 50% economy in memory.

;& -I I I I I v (n c 'i C3D Ch o co Q o 0 c( M C 2 1.5 / / 1 0.5 / I - I - - - - - -" \. A.,, II -0 0.005 0.01 0.015 0.02 Distance along the cavity side in m 0.025 Figure 5: Amplitudes of Scaling and Wavelet Coefficients in a Waveguide.

I I i I --, 'Al?, o - I Ea 0 -L) -N CI o Z. 0.9 0.8 0.7 O.e 0.5 0.4 0.3 -: Analytical *: FDTD (960 unknowns) +: MRTD (480 unknowns) I-t 0.2 0.1 ys U-M - - - -.- - -. - - -.- - - Irk - -0 0.005 0.01 0.015 0.02 0.025 Distance along the cavity side in m 0.03 Figure 6: Comparison of MRTD, FDTD and Analytical Fields in a Waveguide.

(b) Shielded Stripline: Next. a striptine of width 1.27mm is considered. It is enclosed in a cavitv of area 12.7 x 12.7 mnm so that the side walls are sufficiently far away to ntot affect the propagation. The strip is placed 12.7rnm from the ground. A 40 x 40 mesh is used to analyze the.fields in this geometry with the 2D NIRTD technique. Fig. 7 shows the derived scaling and wavelet coefficients of the fields just below the strip. Fron the figure. it can be seen that among the wavelet coefficients, only t'o makes a significant contribution close to the vicinity of the strip where the field variation is rather abrupt. Fig. S shows the comparision of the total reconstructed field in the 40 x 40 MNRTD mesh with that of a -40 x 40 and SO x 0O FDTD mesh. From the figure it is clear that the field computed by -40 x 40 NIRTD mesh using only the significant wavelet coefficients follows the results of the finer SOxSO mesh very closely, demonstrating once again the significant economy in memory as illustrated in Table 1. Fig. 9 shows the Normal Electric field plot of the strip and the variable mesh resulting from MRTD. Table 1: Comparison of the memory requirements in FDTD and MRTD techniques Technique Unknown Coeff. 40x40 FDTD 9600 40x40 MRTD 11328 80x80 FDTD 38400 Conclusion: For the first time in literature, a mathematically correct approach for a FDTD multigrid has been given. Since FDTD is based on the expansion of the unknown fields in pulse functions, the principles of multiresoltion analysis allows for a consistent additional field expansion in terms of Haar wavelets. Due to the additional wavelets, the resulting NIRTD scheme exhibits dispersion characteristics with much less dispersion error than the traditional FDTD scheme. The Haar wavelet based 2D MRTD scheme that has been developed here has been applied to analyse the fields in a waveguide and a shielded stripline. The wavelet coefficients obtained are significant only in regions of rapid field variations. Thus the FDTD multigrid capability using NIRTD technique has demonstrated significant economy in memory.

(n c 5 -a, I' - a) U) U) o a) -o 0) 0 2 4 6 8 10 12 14 Distance along the direction of strip (mm) Figure 7: Amplitudes of Scaling and Wavelet Coefficients of a Shielded Stripline.

0 2 4 6 8 10 12 14 Distance along the direction of strip (mm) Figure 8: Comparison of Normal Electric Field under a stripline using MRTD and FDTD techniques.

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Figure 9: FDTD Multigrid and Field Plot of the Stripline.

References [1] B.Z. Steinberg. Y. Leviatan. "On the use of wavelet expansions in the method of moments". IEEE Trans. Antennas Propagat.. Vol. 41, pp.610-619. May 1993. [2] N1. Krumpholz. P. Russer. "On the Dispersion in TLM and FDTD". IEEE Trans. Microwave Theory Tech.. Vol. 42. No. 7, pp. 1275-1279. July 1994. [3] M.Krumpholz. L.P.B.EKatehi. "New Prospects for Time Domain Analysis", IEEE Microwave and Guided Wave Letters. pp. 382-384. November 1995. [4] MI.Krumpholz, L.P.B.Katehi. "M RTD: New Time Domain Schemes Based on Mlultiresolution Analysis"'. IEEE Transactions on Microwave Theory and Techniques, pp. 38;5 -391. April 1996. [5] L. N. Trefethen, — Group Velocity in Finite Difference Schemes", SIAM Review, V'ol. 24, No. 2, April 1982. [6] D. R. Lynch, K. D. Paulsen, "Origin of Vector Parasitics in Numerical Maxwell Solution". IEEE Trans. Microwave Theory Tech., Vol. 39, No. 3. pp. 383-394, March. 1991.

APPENDIX B January 30, 1997 16 January 30, 1997 16

SPACE- AND TIME- ADAPTIVE GRIDDING USING MRTD TECHNIQUE Emmanouil M. Tentzeris, Robert R. Robertson, Linda P.B. Katehi Radiation Laboratory, Department of Electrical Engineering and Computer Science University of Michigan, Ann Arbor, MI 48109-2122 Andreas Cangellaris Department of Electrical and Computer Engineering University of Arizona, Tucson, AZ Abstract- The MRTD scheme is applied to the analysis of waveguide problems. Specifically, the field pattern and the S-parameters of a dielectric-loaded parallel-plate waveguide are calculated. The use of wavelets enables the implementation of a space- and time-adaptive gridding technique. The results are compared to those obtained by use of the conventional FDTD scheme to indicate considerable savings in memory and computational time. I Introduction Recently a new technique has been successfully applied [1-4] to a variety of microwave problems and has demonstrated unparalleled properties. This technique is derived by the use of multiresolution analysis for the discretization of the time-domain Maxwell's equations. The multiresolution time domain technique (MRTD) based on Battle Lemarie functions has been applied to linear as well as nonlinear propagation problems. The PML absorbing boundary condition has been generalized in order to analyze open planar structures. MRTD has demonstrated savings in time and memory of two orders of magnitude. In addition, the most important advantage of this new technique is its capability to provide space and time adaptive gridding without the problems that the conventional FDTD is encountering. This is due to the use of two separate sets of basis functions, the scal ing and wavelets and the capability to threshold the field coefficients due to the excellent conditioning of the formulated mathematical problem. In this paper, a space/time adaptive gridding algorithm based on the MRTD scheme is proposed and applied to the waveguide problems. As an example, the propagation of a Gabor pulse in a partiallyfilled parallel-plate waveguide is simulated and the S-parameters are evaluated. Wavelets are placed only at locations where the EM fields have significant values, creating a space- and time- adaptive dense mesh in regions of strong field variations, while maintaining a much coarser mesh elsewhere. II The 2D-MRTD scheme For simplicity the 2D-MRTD scheme for the TM, modes will be used herein. To derive the 2D-MRTD scheme, the field components are represented by a series of cubic spline Battle-Lemarie [5] scaling and wavelet functions to the longitudinal direction in space and pulse functions in time. After inserting the field expansions in Maxwell's equations, we sample them using pulse functions in time and scaling/wavelet functions in space domain. As an example, sampling aD,/9t,= - aHy/Oz in space and time, the following difference equation is obtained (i+1 - kD 2) t 1(k + D 1/2,m l+1/2,m 1

m+ml - ( E a(i)k/2H+li l/2 m=m-mmr+ms: + E b(i)*+l/2 H'y, ) ' (1) 1=m-ma 1(k+1Dz kD+ 1) - t l+l1/2,nm - 1+1/2,m = 1 m+m Ay( b(i)+l/2 1+1/2,i+l/2 -=m-m4 m+ms + E c(i)O+l/2 H+Y/2,+l/2), (2) s=m —me where kDlm and kHtY with ~=O (scaling),+ (wavelets) are the coefficients for the electric and magnetic field expansions. The indices 1, m and k are the discrete space and time indices, which are related to the space and time coordinates via z = IAz,z = mAz and t = kAt, where Az,Az are the space discretization intervals in x- and z-direction and At is the time discretization interval. The coefficients a(i), b(i), c(i) are derived and given in [2]. For an accuracy of 0.1% the values ml = m5 = 8,m2 = m3 = m4 = me = 9 have been used. For open structures, the perfectly matched layer (PML) technique can be applied by assuming that the conductivity is given in terms of scaling and wavelet functions instead of pulse functions with respect to space [4]. The spatial distribution of the conductivity for the absorbing layers is modelled by assuming that the amplitudes of the scaling functions have a parabolic distribution. The MRTD mesh is terminated by a perfect electric conductor (PEC) at the end of the PML region. Usually, 8-16 cells of PML medium with o, E=0.4S/m provide reflection coefficients smaller than -90 dB. In order to use a pulse excitation at z = mAz with respect to space and to obtain an excitation identical to an FDTD excitation, we decompose the pulse in terms of scaling and wavelet functions kE 'IsC EF(O, kAt) +4 +4 (ZE C(i)+, + E cO(i)iM+) (3) i=-4 i=-4 where the coefficients cO(i), cp(i) are given in Table 1 for i > 0. For i < 0 it is c,(-i) = c+(i) and C+(i) = c,(l - i). EF(0, kAt) is the time dependence of the excitation. For Iil < 4, the above excitation components are superimposed to the field values obtained by the MRTD algorithm. For example, the total E m+i will be given by Em tt = EF(O, kAt) c+(i) + Em+i (1- c(i)) Due to the nature of the Battle-Lemarie expansion functions, the total field is a summation of the contributions from the non-localized scaling and wavelet functions. For example, the total electric field E,(z, zo, to) with (k - 1/2) At < to < (k + 1/2) At is calculated in the same way with [2, 3] by II /i Ez(xo, zo, to) - +Zot) = Z kE1+1/2,mn' 1'+11/2(Xo) Om'(Zo) 1',m' = —,, +E Z kEO.iixm" 0+1/2(z0) Ormn(Zo) where rm (z) = m( - m) and Oi,m(z) = (w - m) represent the Battle-Lemarie scaling and i-resolution wavelet function respectively. For an accuracy of 0.1% the values 11 = 12,i = 4 have been used. There are many different ways to take advantage of the capability of the MRTD technique to provide space and time adaptive gridding. In DSP, thresholding of the wavelet coefficients over a specific timeand space- window (5-10 points) contribute significant memory economy, but increase the implementation complexity and the execution time. The simplest way is to threshold the wavelet components to a fraction (usually < 0.1%) of the scaling function at the same cell for each time-step. All components below this threshold are eliminated from the subsequent calculations. This is the simplest thresholding algorithm. It doesn't add any significant overhead in execution time, but it offers only a moderate (pessimistic) economy in menory (factor close to 2). Also, this algorithm allows for the dynamic memory allocation in its programming implementation. III Applications of 2D-MRTD The 2D-MRTD scheme is applied to the analysis of the partially-loaded parallel-plate waveguide of (Fig.l) for the frequency range 0-30GHz. For the 2

analysis based on Yee's FDTD scheme, a 16 x 800 mesh is used resulting in a total number of 14400 grid points. When the structure is analyzed with the 2D-MRTD scheme, a mesh 2 x 100 (200 grid points) is chosen (dz = 0.24Ao, dz = 0.4OA for f = 30GHz). This size is based on the number of the scaling functions, since the wavelets are used only when and where necessary. The time discretization interval is selected to be identical for both schemes and equal to the 1/10 of the 2D-MRTD maximum At. For the analysis we use 8,000 time-steps. The waveguide is excited with a Gabor function 0-30GHz along a vertical line for the FDTD simulation and for a rectangular region for the MRTD simulations. In all cases, the front and back open planes are terminated with a PML region of 16 cells and af =0.4S/m. The longitudinal distance between the excitation and the dielectric interface is chosen such that no reflections would appear before the Gabor function is complete. The capability of the MRTD technique to provide space and time adaptive gridding is verified by thresholding the wavelet components to the 0.1% of the value of the scaling function at the same cell for each time-step. It has been observed that the accuracy by using only a small number of wavelets is equal to what would be achieved if wavelets were used everywhere. Though this number is varying in time, its maximum value is 22 out of a total of 100 to the z-direction (economy in memory by a factor of 28-30). In addition, execution time is reduced by a factor 4-5. For larger thresholds, the ringing effect due to the elimination of the wavelets deteriorates the performance of the algorithm. For example, using a threshold of 1% (6 out of a 100 wavelets to the z-direction) increases the error by a factor of 2.5. The normal electric field is probed at a distance 10 cells away from the source and is plotted in (Fig.2) in time-domain. Comparable accuracy can be observed for the FDTD and the MRTD meshes. In addition, the reflection coefficient S11 is calculated by separating the incident and the reflected part of the probed field and taking the Fourier transform of their ratio (Fig.3). The results for 5 GHz (TEM propagation) are validated by comparison to the theoreti cal value obtained applying ideal transmission line theory [6] and are plotted at Table 2. The timeand space-adaptive character of the gridding is exploited in (Figs.4,5) which show that the wavelets follow the propagating pulses before and after the incidence to the dielectric interfaces and have negligible values elsewhere. The location and the number of the wavelet coefficients with significant values are different for each time-step, something that creates a dense mesh in regions of strong field variations, while maintaining a much coarser mesh for the other cells. IV Conclusion A space- and time- adaptive gridding algorithm based on a multiresolution time-domain scheme in two dimensions has been proposed and has been applied to the numerical analysis of a waveguide problem. The field pattern and the reflection coefficient have been calculated and verified by comparison to reference data. In comparison to Yee's conventional FDTD scheme, the proposed scheme offers memory savings by a factor of 5-6 per dimension maintaining a similar accuracy. The above algorithm can be effectively extended to three-dimension problems. V Acknowledgments This work has been funded by NSF. References [1] M.Krumpholz, L.P.B.Katehi, "New Prospects for Time Domain Analysis", IEEE Microwave and Guided Wave Letters, pp. 382-384, November 1995. [2] M.Krumpholz, L.P.B.Katehi, "MRTD: New Time Domain Schemes Based on Multiresolution Analysis", IEEE Transactions on Microwave Theory and Techniques, pp. 555-572, April 1996. [3] E.Tentzeris, M.Krumpholz and L.P.B. Katehi, "Application of MRTD to Printed Transmission Lines", Proc. MTT-S 1996, pp. 573-576. [4] E.Tentzeris, R.Robertson, M.Krumpholz and L.P.B. Katehi, "Application of the PML Absorber to the MRTD Technique", Proc. AP-S 1996, pp. 634-637. [5] I.Daubechies, "Ten Lectures on Wavelets", Philadelphia, PA: Society for Ind. and App. Math., 1992. [6] D.M.Pozar, "Microwave Engineering", pp. 94-96, Addison-Wesley, 1990. 3

Table 1: Excitation Decomposition Coeffs r i 0 1 2 3 4 c (i) 0.915 0.038 0.010 -0.009 0.005 c (i) -0.103 -0.103 0.121 -0.030 0.015 Table 2: Sll calculated by 2D-MRTD I, I. - I I I Analyt. Value [6] 16x800 FDTD 2x100 MRTD r S11 (12) 0.4298 0.4283 0.4360 -4 Relative error 0.0% -0.3% +1.4% 3 I I i er 1 IV 4.8 mm Figure 3: S11 values (Frequency-Domain). * --- 8 mm -- Figure 1: Dielectric-loaded Waveguide. Tme of Incdence Ater Incidence 0. z Figure 4: Adaptive Grid Demonstration. 000 2000 3000 4000 5000 6000 7000 8000 9000 1 Time-Step Figure 2: Normal E-field (Time-Domain). 4

APPENDIX C January 30, 1997 17 January 30, 1997 17

Space/Time Adaptive Meshing and Multiresolution Time Domain Method (MRTD) Emmanouil Tentzeris1, Andreas Cangellaris2, Linda P.B. Katehi' 'Radiation Laboratory, EECS Department, University of Michigan, Ann Arbor, MI 48109-2122, USA 2University of Arizona, Tucson, AZ, USA I Introduction Recently the principles of the Multiresolution Analysis have been successfully applied [1, 2] to the time-domain numerical techniques used for the analysis of a variety of microwave problems. New techniques have been derived by the use of scaling and wavelet functions for the discretization of the time-domain Maxwell's equations. The multiresolution time domain technique (MRTD) based on Battle Lemarie functions has been used for the simulations of planar circuits and resonating structures. The conventional FDTD absorbers (e.g. PMIL) have been generalized in order to analyze open planar structures. MRTD has demonstrated unparalleled savings in execution time and memory requirements (2 orders of magnitude for 3D problems). In addition to time and memory, MRTD technique can provide space- and time- adaptive meshing without the problems that the conventional FDTD variable grids are encountering (e.g. reflections between dense-coarse regions). This unique feature stems from the use of two separate sets of basis functions, the scaling and wavelets. Due to the excellent conditioning of the formulated mathematical problem, MRTD offers the capability to threshold the wavelet field coefficients. This advantage of the MRTD Technique is demonstrated herein by performing a space-/timeadaptive meshing. In this paper, a space-/time- adaptive meshing algorithm based on the MRTD scheme is proposed and validated for a specific waveguide problem. Wavelets up to the second resolution are placed only at locations where the EM fields have significant values. These locations are changing with the time as the pulse is propagating inside the waveguide and with the space as the pulse is approaching regions of discontinuities. The proposed algorithm offers the opportunity of a space-/time- adaptive mesh with variable resolution of the field representation. In this way, significant memory and execution time savings can be achieved in comparison to the conventional variable-mesh FDTD algorithms.

II MRTD Formulation Without loss of generality, the 2D-MRTD scheme for the TiMV modes will be described herein. To derive the scheme equations, the field components are represented by a series of cubic spline Battle-Lemarie scaling and 1-order wavelet functions along the z-direction, while pulses are used for the time representation. Wavelets of higher-order can be included in a similar way. After inserting these series expansions in Maxwell's equations and sampling them with pulse functions in time and scaling/wavelet functions in space domain, we derive the following equations for the electric field: 1 1 m+mi m+m3 t't(+l D+./2m - tkDl+/lm)= ( a(i)k+1/2 H 1/2+1/2 + b(i)k+l/2 H /212) AtVk+~ 1+1/2 /=z 2) t =m7-m2 i=m-m4 1 +m3 m+ms -(k /2,m - 1/2) = ( b(i)k+1/2 /2i+1/2 + C(i)k+/2 Hl+ /2.+l/1/2) At I+/2=m-m4 t=m-m6 i --- —12 1 1 +13 At(k+ l,m+l/2 - l,m+1/2) A= ( E a(i)1+,/, l +2.,/2m+1/2) =1-14 At(k+Dm+1/2 Dlm+1/2) = a( c(i) k+l /2,1+1/2) where kD, kEIr and kH(Y with r=O (scaling),, (wavelets) are the coefficients for the electric flux, electric and magnetic field expansions. The indices 1, m and k are the discrete space and time indices, which are related to the space and time coordinates via x = lAx,z = mAz and t = kAt, where Ax,Az are the space discretization intervals in x- and z-direction and At is the time discretization interval. The coefficients a(i), b(i), c(i) are derived and given in [1]. For an accuracy of 0.1% the values ml = ms = 8, m2 = m3 = m4 = m6 = 9 have been used. The indices,i have to take similar values to achieve tha same accuracy in the summations. The use of non-localized basis functions in the 2D-MRTD scheme causes significant effects. Localized boundary conditions are impossible to be implemented, so the perfect electric boundary conditions are modelled by use of the image principle in a generic way. The implementation of the image theory is performed automatically for any number of PEC, PMC boundaries. The material discontinuities are represented in terms of scaling and wavelet functions resulting into a linear matrix equation as explained in [1, 3] where this technique was used in the modeling of anisotropic dielectric media. In addition, the total value of a field component at a specific point of the mesh is a summation of the contributions from the neighbooring non-localized scaling and wavelet functions. The fiold values at the neighbooring cells can be combined appropriately by adjusting the scaling and wavelet function values and by applying the image principle. The demand for the simulation of open structures led to the generalization of the perfectly matched layer (PML) technique [4], so as it can be used in the MRTD simulations. The conductivity is expanded in terms of scaling functions instead of pulse functions with respect to

space. The amplitudes of the expansion scaling functions follow the PL spatial conductiity distribution. In our simulations, the parabolic distribution was used, though the realization of other distributions (linear, cubic....) is straightforward. For example, if we assume that the PML absorbing material (E, /,u gE) extends to the z-direction, substituting D(i)xZ(, t) = (i:, t)(e-it/z, (1) and H(i)Y(, z, t) = (i)Y(x. z, t)e-')(t/ (2) for i=O, i, leads to the following equation: ADx a<HY - - -. (3) Following a procedure similar to the one used for the derivation of the non-PML region equations, we get for Dr components k+Di1/2m - e- (a ^) t/ A m+mi m+m3 - e(m 5O.S E C a(i)+l/2 H+/ / + E b(i)k+l/2 Hl +) 1k+1D+ /2,m e +l/2,m At., 0 m+m3 m+ms - - () t/( E 6(i)k+l/2 H /2, /2+ E C(i)k+1/2 /2,1/2) a i=m-m4 tim-m6 The finite-difference equations for D(1k,)z and H(0")^Y are similar. For all simulations, a parabolic distribution of the conductivity o is used in the PML region (N cells): E,H _E,Hrn f O (m =) = a,( )2 for m=0,,..,N, (4) with aoJH the maximum conductivity at the end of the absorbing layer. As in [5], the "magnetic" conductivity aH is given by: ~E H (ML) (mA ) for m=O,1,..,N, (5) and the MiRTD mesh is terminated by a perfect electric conductor (PEC) at the end of the PML region. This PEC is modelled by applying the image theory.

III Space/Time Adaptive Meshing The wavelet components' amplitudes have negligible values away from the discontinuities or at regions where the excitation pulse has not propagated yet. There are numerous ways of taking advantage of the above feature. The simplest one is to threshold the wavelet components to a fraction (usually < 0.1%) of the scaling component at the same cell (space adaptivity) for each time-step. All components below this threshold are eliminated from the subsequent calculations for the same time-step (time adaptivity). This procedure offers only a moderate economy in memory (factor close to 2). Also, this algorithm allows for the dynamic memory allocation in its programming implementation, while maintaining a low complexity. The above space-/time- adaptive meshing scheme is applied to the analysis of the partiallyloaded parallel-plate waveguide of (Fig.l) for the frequency range 0-22.5GHz. The waveguide is half-filled with air and half-filled with dielectric with e, = 2.56. An FDTD 16 x 640 (10240 cells) mesh and an MRTD 2 x 80 (160 cells) mesh (160 grid points with dx = 0.18A0, dz = 0.3A0 - close to the Nyquist Limit for f = 22.5GHz) are used for the Time-Domain simulations (3,000 time-steps). The 160 grid points of the MRTD mesh express the number of the used scaling functions. The number of the wavelets is varying with time and depends on the predefined threshold. For consistency, the time step for both schemes is chosen to be equal to the 1/8 of the FDTD maximum At. The waveguide is excited with a Gabor function 0-22.5GHz along a vertical line for the FDTD simulation and for a rectangular region of 12 cells to the longitudinal direction (due to the non-localized character of the Battle-Lemarie scaling and wavelet functions) for the MRTD simulations. Other excitations (e.g.Gaussian) can be applied i a straightforward way. For both cases, a PML region of 16 cells and oa, =0.4S/m absorbs the waves in the front and back open planes. The capability of the MRTD technique to provide space- and time- adaptive gridding is verified by thresholding the wavelet components to the 0.1% of the value of the scaling function at the same cell for each time-step. The accuracy achieved by using only the wavelets with values above the threshold is equal to what would be if wavelets were used everywhere. Though this number is varying in time, its maximum value is 36 out of a total of 160 to the z-direction (economy in memory by a factor of 52 instead of 32). In addition, execution time is reduced by a factor 4-5. For larger thresholds, the ringing effect due to the elimination of the wavelets deteriorates the performance of the algorithm. For example, using a threshold of 1% (13 out of a 160 wavelets to the z-direction) increases the error by a factor of 2.1. The results for the Reflection Coefficient for 10 GHz are validated by comparison to the theoretical value IRf = 0.231 (=(V2.56-1.0)/(v2.56+1.0)). MRTD gives the value 0.2296 and FDTD gives 0.2304 (similar accuracy). The normal electric field is probed at a distance 10 cells away from the source and is plotted in (Fig.2) in time-domain. Similar accuracy can be observed for

the FDTD and the MIRTD meshes. Fig.3 demonstrates the space- and time-adaptive character of the meshing algorithm. It is clearly shownw that the wavelets follow the propagating exciation pulse before and after the incidence to the dielectric interface and can be omitted elsewhere. The location and the number of the xwavelet coefficients with values above the threshold ("effective wavelets'") are different for each time-step, something that creates a mesh with high resolution ("dense') in regions of strong field variations, while maintaining a much lower resolution ("coarse") for the rest cells. IV Conclusion A simple space- and time- adaptive meshing algorithm based on an MRTD scheme has been proposed and has been validated for a parallel-plate waveguide problem. The electric field value and the reflection coefficient have been calculated and verified by comparison to reference data. The proposed scheme exhibits memory savings by a factor of.52 in 2D, as well as execution time savings by a factor of 4-5, while maintaining a similar accuracy with Yee's conventional FDTD scheme. In addition, this algorithm doesn't increase the programming complexity and can be effectively extended to 3D problems. V Acknowledgments This work has been partially funded by ARO and by NSF. References [1] M.Krumpholz, L.P.B.Katehi, "MRTD: New Time Domain Schemes Based on Multiresolution Analysis", IEEE Trans. Microwave Theory and Techniques, vol. 44, no. 4, pp. 555-561, April 1996. [2] E.Tentzeris, M.Krumpholz and L.P.B. Katehi, "Application of MRTD to Printed Transmission Lines", Proc. MTT-S 1996, pp. 573-576. [3] R. Robertson, E. Tentzeris, M. Krumpholz, L.P.B. Katehi, "Application of MRTD Analysis to Dielectric Cavity Structures", Proc. MTT-S 1996, pp.. [4] E.Tentzeris, R.Robertson, M.Krumpholz and L.P.B. Katehi, "Application of the PML Absorber to the MRTD Technique", Proc. AP-S 1996, pp. 634-637. [5] J.-P. Berenger, "A Perfectly Matched Layer for the Absorption of Electromagnetic Waves", J. Computational Physics, vol. 114, pp. 185-200, 1994.

Figure 1: Dielectric-loaded Waveguide. LL 7000 TInm-Stp Figure 2: Normal E-ffeld Time Evolution. I I 1 1' I I I I. I.: ~~ I ~ I II t I ' ~mmi~jmIi~ I I I 0 I I t=0 t before incidence t after incidence Figure 3: Space-/Time- Adaptive Meshing Demonstration.

APPENDIX D January 30, 1997 18 January 30, 1997 18

Revised Version of Paper Previously Submitted to the IEEE AP-S 1997 Time Adaptive Time-Domain Techniques for the Design of Microwave Circuits Emmanouil Tentzeris1, James Harvey2, Linda P.B. Katehi1 'Radiation Laboratory, EECS Department, University of Michigan 2Army Research Office Abstract The recently developed MRTD schemes are used for the development of a time adaptive timedomain technique for circuit design. The new technique exhibits considerable savings in memory and computational times in comparison to the conventional FDTD scheme. I Introduction Significant attention is being devoted now-a-days to the analysis and design of various types of microwave circuits. The finite-difference-time-domain (FDTD) scheme is one of the most powerful numerical techniques used for numerical simulations. However, despite its simplicity and modeling versatility, the FDTD scheme suffers from serious limitations due to the substantial computer resources required to model electromagnetic problems with medium or large computational volumes. In addition, the FDTD scheme cannot provide the accuracy required for computer simulations of time-dependent electromagnetic interactions in electrically long regions or in regions dbh;^h -^nt?)n. non-linear materials. Such simulations are very important for integrated device modelling, especially in relation to the design of non-linear photonic devices. To alleviate these problems hybrid combinations of FDTD with other numerical techniques and higher order FDTD schemes based on Yee's grid have been proposed. MRTD (MultiResolution Time Domain Method) [1, 2] has shown unparalled properties in comparison to Yee's FDTD. MRTD is not a new methodology. It is a correct and accurate generalization of the conventional discretization approaches. It provides the correct mathematical frame for solving problems in time domain and allows for the development of time/space adaptive grids. II Introduction to MRTD It is well known that the method of moments provides a mathematically correct approach for the discretization of integral and partial differential equations. Since it allows for the use of any complete and orthonormal set, the choice of an appropriate expansion set may lead to different time domain schemes. For example, the expansion of the unknown fields using pulse

functions leads to Yee's FDTD scheme. In a MIRTD scheme the fields are represented by a twofold expansion in scaling and wavelet functions with respect to time/space. Scaling functions guarantee a correct modelling of smoothly-varying fields. In regions characterized by strong field variations or field singularities, higher resolution is enhanced by incorporating wavelets in the field expansions. Wavelets are introduced only at specific locations, allowing for a time/space adaptive grid capability. MRTD schemes based on cubic spline Battle-Lemarie scaling and wavelet functions (Figh.) have been successfully applied to the simulation of 2D and 3D open and shielded problems [1, 2, 3, 4]. The functions of this family do not have compact support, thus the MRTD schemes have to be truncated with respect to space. Localized boundary conditions (PECs. PilCs etc.) and material properties are modelled by use of the image principle and of matrix equations respectively. However, this disadvantage is offset by the low-pass (scaling) and band-pass (wavelets) characteristics in spectral domain, allowing for an a priori estimate of the number of resolution levels necessary for a correct field modelling. In addition, the evaluation of the moment method integrals during the discretization of Maxwell's PDEs is simplified due to the existence of closed form expressions in spectral domain and simple representations in space domain. Dispersion analysis of this MRTD scheme shows the capability of excellent accuracy with up to 2 points/wavelength (Nvquist Limit). However, specific circuit problems may require the use of functions with compact support. For that reason, Haar basis functions have been utilized and have led to [5]. As an extension to this approach, intervalic wavelets of higher order may be incorporated into the solution of SPICE-type circuits. Results from that new technique will be shown at the Conference. III Time Adaptive MRTD Scheme The major advantage of the use of Mutiresolution analysis to time domain is the capability to develop time and space adaptive schemes. This is due to the property of the wavelet expansion functions to interact weakly and allow for a spatial sparsity that may vary with time through a thresholding process. The adaptive character of this technique is extremely important for the accurate modelling of sharp field variations of the type encountered in beam focusing in nonlinear optics, etc. The use of the principles of the multiresolution analysis for adaptive grid computations for PDEs has been suggested by Perrier and Basdevant [6]. To understand the fundamental steps of such an adaptive scheme for Maxwell's hyperbolic system, let's consider Maxwell's equations in 2D (1 for space and 1 for time): at = -,(z)- u=(E(z, t), H(z,t))T,(1) After manipulation, the above equation can be written as After manipulation, the above equation can be written as

MAjt D ] (2) where Zh, Th are half shift operators for space and time coordinates z,t and Zt, Tt are their Hermitian conjugates. Dt, Dz are difference operators given by: 1 8 9 = - E at(i)rT'+ E at(i)T-) D = ( E az(i)z-'+ E az(i)z-') (3) it =-9 i=-9 i=-9 i=-9 where a,, an, are the coefficients associated with the scalar and the wavelet functions respectively. At each time step we keep both the wavelet field values that are larger than a given threshold as well as the adjacent values. An adjacent wavelet field value is defined on the basis of the wavelet resolution level(s) incorporated in the solution. Recently, an efficient space/time adaptive meshing prosedure was proposed [7] for Battle-Lemarie expansion functions. In this paper, intervalic wavelets are used for the expansion of the fields (Fig.2). The adaptive mesh will be applied to a variety of circuit problems and results will be discussed during the presentation. IV Conclusion A Time Adaptive Time-Domain Technique based on intervalic wavelets has been proposed and applied to various types of circuits problems with lumped and distributed elements. This scheme exhibits significant savings in execution time and memory requirements while maintaining a similar accuracy with conventional circuit simulators. V Acknowledgments This work has been partially funded by NSF. References [1] M.Krumpholz, L.P.B.Katehi, "MRTD: New Time Domain Schemes Based on Multiresolution Analysis", IEEE Trans. Microwave Theory and Techniques, vol. 44, no. 4, pp. 555-561, April 1996. [2] E.Tentzeris, M.Krumpholz and L.P.B. Katehi, "Application of MRTD to Printed Transmission Lines", Proc. MTT-S 1996, pp. 573-576. [3] R. Robertson, E. Tentzeris, M. Krumpholz, L.P.B. Katehi, "Application of MRTD Analysis to Dielectric Cavity Structures", Proc. MTT-S 1996, pp.. [4] E.Tentzeris, R.Robertson, M.Krumpholz and L.P.B. Katehi, "Application of the PML Absorber to the MRTD Technique", Proc. AP-S 1996, pp. 634-637. [5] K.Goverdhanam, E.Tentzeris, M.Krumpholz and L.P.B. Katehi, "An FDTD Multigrid based on Multiresolution Analysis", Proc. AP-S 1996, pp.. [6] V.Perrier and C.Basdevant, "La decomposition en eondelettes periodiques: un outil pour 'analyse des champs inhomogenes. Theorie et algorithmes", La Recherche Aerospatiale, no.3, pp.53-67, 1989. [7] E.Tentzeris, R.Robertson. A.Cangellaris and L.P.B. Katehi, "Space- and Time- Adaptive Gridding Using MRTD", to be presented in the 1997 IEEE MTT-S, Denver, CO.

Or* % i0 0 L-4 13 I 1Li (I N PI * C. I I — w- - -- I.I I I I I I I p: I I 6 0 kA 0 iA 4 LA1 Al ' at N+ N C., cnl II I-:*i 6J 0 ~ 0 0 0 IP - P - I r F. - I I 9 I 9 t 11% I -. - I: I I I I I 11 i Battle Lemorns Moilhe( Wavelet 6 0 0 0 0 - ~in b........ ib UV t4 I. - - LA - 1 0 0 0 0 - iN 4 a in C

APPENDIX E January 30. 1997 19 January 30, 1997 19

Modeling of Membrane Patch Antennas Using MRTD Analysis R. Robertson, E.M. Tentzeris, Linda P.B. Katehi Radiation Laboratory, Department of Electrical Engineering and Computer Science University of Michigan, Ann Arbor, MI 48109-2122 Abstract The Multiresolution Time-Domain (MRTD) scheme with perfectly matched layer (PNML) absorbing boundaries is applied to the analysis of a membrane patch antenna. The results are compared to those obtained by use of the conventional FDTD technique and substantial reductions in memory requirements are observed. I Introduction Recently Multiresolution Time-Domain Analysis has been successfully applied to simulate a variety of microwave structures [1]. MRTD has performed complete analysis of both planar circuits [2] and resonating structures [3]. Additionally conventional FDTD absorbers, such as PML [4] have been generalized in order to analyze open planar structures [5]. In all cases, MRTD has demonstrated a high degree of savings in execution time and memory requirements with respect to FDTD. In this paper the techniques described are applied to the simulation of a membrane patch antenna, with a center frequency of 9 GHz. Full 3D MRTD analysis with PML along three coordinate directions is used to simulate the antenna. The MRTD scheme is applied to the calculation of S-parameters for the membrane antenna and is compared to conventional FDTD. II Application of PML to the 3D MRTD scheme r To derive the 3D MRTD scheme, the field components in Maxwell's E-curl and H-curl equations are represented by a series of cubic spline Battle-Lemarie scaling functions in space and pulse functions in time. These equations are sampled with pulse functions in time- and scaling functions in space-domain, as detailed in [1]. As an example, consider the discretization of: aE aH, aHy(1) &t ay a For a homogeneous medium with the permittivity c, expanding and sampling aE:/9t, alz/ay and aHy/zl with scaling and pulse functions in space and time gives ^ (n+ Ei+1/2,m,n - kEt+/2,mn) m+8 n+8 a(i) k+1/2H 2 - a(i) k+2 /2, 2 (2) ply E ().21+1/2,i+1/2,n Lz E ( ) k l/2 1+1/2,m,i+1/2 y=m-9 i=n-9 where kE,,n kHm, and kH*Yl are the coefficients for the electric and magnetic field expansions. The indices 1, m, n and k are the discrete space and time indices related to the space and time coordinates via x = lzx, y = mAy, z = nAz and t = kAt, where Az, my, Az are the space discretization intervals in x-, yand z-directions and At is the time discretization interval. The coefficients a(i) are given in [1]. To derive the perfectly matched layer (PML) technique [4] along one coordinate direction it is assumed that the conductivity is given in terms of scaling functions with respect to space. The spatial distribution of the 1

conductivity for the absorbing layers is simulated by assuming that the amplitudes of the scaling functions have a parabolic distribution [5]. For the PML absorbing material in the y-direction with c., A and conductivity aE the term o(y)E must he added to the left side of eq.(1). Then, substituting the following into eq.(1): E,(x, y, z, t) = Ei(x, y, z, t)e-),t/' (3) Hi(x, y, z, t) = i(x, y, z, t)ee-i(), (4) and assuming that the PML is only along the y-direction leads to the following equation: +lE/2,m,n = e(m )EkE E (/)/ I m+8 + m+8 + 1 E a(i)+1/2HI /,,m+l/22)5, i=m-9 For all simulations, a parabolic distribution of the conductivity a is used in the PML region (N cells): E,H E,H( 2 for m=0,1,..,N, (6) where aErH is the maximum conductivity at the end of the absorbing layer. As in [4], the magnetic conductivity o-H has to be chosen as: E 1H (y) = (mAy) for m=0,1,..,N, (7) E / for a perfect absorption of the outgoing waves. The MRTD mesh is terminated by a perfect electric conductor (PEC) at the end of the PML region, modelled by applying the image theory. While the above derivation is adequate for a structure which only needs to be terminated with PML along one direction, such as a shielded thru-line, structures such as patch antennas need PML termination in all three coordinate directions. In this case, the derivation discussed above needs to be extended to three dimensions. The procedure is straightforward and results in the following equation: k+1. - e(1B*OAt/Ee-( At/ e( - a At/c Elk E: + I+l 1/2,m,n e (la) e +1/2,m,n m+8 + E /) 1+1/2,i+1/2,n(8) =i=mm+s +ai (8) Az E. ( i) 1+1/2,..,+1/2) i=m — As in eq.(6) a parabolic distribution for the conductivity is applied in the x-, y- and z-directions. III Applications of the 3D-MRTD scheme The object of this paper is to apply the 3D MRTD scheme to the analysis of membrane patch antennas. However, in order to test the application of PML to the 3D MRTD scheme, a microwave thru-line is analyzed using MRTD and FDTD. The thru-line has a width of 0.4 mm and length of 10.0 cm and is placed in the center of a cavity with dimensions 1.6mm x 10.0cm x 1.6mm. A Gaussian pulse is used to excite the thru-line 2

with fma, = 50GHz [6] and is placed in the middle of the center conductor 9 mm from the PML layer along the y-axis. The FDTD analysis uses 16 x 100 x 16 mesh while the MRTD analysis uses a 8 x 20 x 8 mesh. Additionally six cells of PML are used along the y-direction at either end of the thru-line with a.a = 30. Therefore the total discretization of the thru line is 16 x 112 x 16 for FDTD and 8 x 32 x 8 for the MRTD scheme, resulting in a factor of 14.0 savings in memory. The time discretization interval for the MRTD scheme is At = 3.92. 10-14s, while the FDTD scheme has At = 6.335 10-14s. In both cases, the simulation is performed for 6000 time steps. A comparison plot of time vs. Ez-field amplitude is shown in Figure 1. Note that the amplitude of the Gaussian has been normalized and the time-steps multiplied by a constant factor in order to compare the two plots more easily. The initial Gaussian pulse has been completely absorbed by the PML layer along the y-direction. The membrane,patch antenna shown in Figure 2 is simulated using 3D MRTD and FDTD. A full description of the parameters of the antenna can be found in [7]. A PML layer of six cells is used along the ~x, ~y and +z directions, resulting in an FDTD mesh of 72 x 112 x 28 and a MRTD mesh of 42 x 62 x 12, a factor of 7.22 savings in memory. In the PML layers aE = Ey = = 3.0 for FDTD and MRTD. The time discretization interval used for the MRTD scheme is At = 1.6008 10-'3s while the FDTD time discretization interval is At = 1.3297 10-13s. In both cases the simulation is performed for 7000 time steps. The antenna feed line is 20 mm long and the Gaussian pulse is sent from a point y=4 mm from the edge of the PML layer in the FDTD and MRTD simulations. Figure 3 shows a plot of Ez field values vs. time for MRTD and FDTD. Measurement of the intial and reflected normalized Gaussian pulses occurred at y = 14 mm from the edge of the PML layer. Figure 4 shows a plot of the calculated S11 [7] for the membrane patch antenna. Note that excellent correlation is achieved between FDTD and MRTD results. IV Conclusion A membrane patch antenna is successfully simulated using the 3D MRTD scheme with PML absorber along the x-, y- and z-directions. With respect to calculated Sl results MRTD shows excellent correlation with FDTD while exhibiting a memory savings with a factor of 7.22. V Acknowledgements This work has been partially funded by NSF and ARL. References [l] M. Krumpholz and L.P.B. Katehi, "MRTD: New Time Domain Schemes Based on Multiresolution Analysis", IEEE Trans. Microwave Theory and Techniques, vol. 44,no. 4,pp. 555-561. [2] E. Tentzeris, M. Krumpholz and L.P.B. Katehi, "Application of MRTD to Printed Transmission Lines", Proc. MTT-S 1996, pp. 573-576. [3] R. Robertson, E. Tentzeris, M. Krumpholz and L.P.B. Katehi, "Application of MRTD Analysis to Dielectric Cavity Structures",Proc. MTT-S 1996, pp. 1840-1843. [4] J.P. Berenger, "A Perfectly Matched Layer for the Absorption of Elecromagnetic Waves", J.Comput. Physics, vol. 114, pp. 185-200, 1994. [5] E. Tentzeris, R. Robertson, M. Krumphoz and L.P.. Katehi, "Application of the PML Absorber to the MRTD Technique", Proc. AP-S 1996, pp. 634-637. [6] X. Zhang and K.K. Mei, "Time-Domain Finite Difference Approach to the Calculation of the Frequency-Dependent Characteristics of Microstrip Discontinuities", IEEE Trans. Microwave Theory and Techniques, vol. 36, no. 12, pp. 1775-1787. [7] D. Sheen, S. Ali, M. Abouzahra and J. Kong, "Application of the Three-Dimensional Finite-Difference Time-Domain Method to the Analysis of Planar Microstrip Circuits", IEEE Trans. Microwave Theory and Techniques, vol.:38, no. 7, pp. 849-856. 3

0 ( L. 3 0.794 mm 0o 100 2000 3000 4oo0 5ooo Tanw Sep (d) Figure 1: Time-Domain Analysis of a Microstrip Thru-Line. Figure 2: Membrane Patch Antenna. rew so (dr F gure 3: Time Domain AnalJyss o( a Membrane Patch Antenna. 02 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Frequency (Hz)e Path Figure 4: Simulated Sl1 results for the Patch Antenna.

APPENDIX F January 30, 1997 20 January 30, 1997 20

APPLICATIONS OF MULTIRESOLUTION BASED FDTD MULTIGRID Kavita Goverdhanam, Linda P.B. Katehi Radiation Laboratory, Department of Electrical Engineering and Computer Science University of Michigan, Ann Arbor, MI 48109-2122 Andreas Cangellaris Department of Electrical Engineering University of Arizona, Tucson, AZ Abstract- A Multigrid 2-D Finite Difference Time Domain (FDTD) technique based on Multiresolution analysis with Haar wavelets is used to analyze structures such as an empty waveguide and a shielded stripline. The results obtained are compared with those computed using a finer resolution regular FDTD mesh. This comparative study illustrates the benefits of using wavelets in FDTD analysis. I Introduction Multiresolution Time Domain (MRTD) Technique is a new approach to solving time domain problems. This technique uses Multiresolution Analysis (MRA) to Discretize Maxwell's equations in time domain and demonstrates excellent capability in solving Electromagnetics problems [1], [2]. Depending on the choice of basis functions, several different schemes result, each one carrying the signature of the basis functions used in MRA. It is also important to note that the design of an MRTD scheme can be accomplished using one's own application-specific basis functions. MRTD technique using Haar scaling functions results in the FDTD technique [3]. Recently, an FDTD multigrid using the Haar wavelet basis has been developed and it has demonstrated that such a scheme exhibits highly linear dispersion characteristics [3]. Motivation for this work stems from the theory of MRA which says that a function which is expanded in terms of scaling functions of a lower resolution level, ml, can be improved to a higher resolution level, m2, by using wavelets of the intermediate levels. In other words, expanding a function using scaling function of resolution level ml and wavelets upto resolution level m2 gives the same accuracy as expanding the function using just the scaling functions of resolution m2. However, the use of wavelet expansions has major implications in memory savings due to the fact that the wavelet expansion coefficients are significant only in areas of rapid field variations. This allows for the capability to discard wavelet expansion coefficients where they are not significant thereby leading to significant economy in memory. Different resolutions of wavelets can be combined so as to locally improve the accuracy of the approximation of the unknown function. This, combined with the fact that wavelet coefficients are significant only at abrupt field variations and discontinuities allows MRTD to lend itself very naturally to a Multigrid capability. In this paper, a 2D MRTD scheme based on Haar basis functions (first order resolution) is developed and applied to solve for the Electromagnetic fields in a waveguide and a shielded stripline. The results obtained are compared with those computed using conventional FDTD technique. It will be shown that the wavelet coefficients are significant only at locations with abrupt field variations. This facilitates in obtaining accurate solutions by combining the wavelet 1

and scaling coefficients only in regions where the wavelet coefficients are significant (discontinuities). II The 2D-MRTD scheme Consider the following 2-D scalar equation obtained from Maxwell's H-curl equation: 9Er _OH r- + 3H (1) This equation can be rewritten in a differential operator form as shown below: Ll(f(x, y, t)) + L2(f2(2(, y, t)) = g (2) where L1 and L2 are the operators and fl(x,y,t) and f2(x,y,t) represent the electric/magnetic fields. We now expand the fields using a Haar based MRA with scaling functions 0 and wavelet functions 0 [3]. The field expansion can be represented as follows: f(x, y, t) = [A][O(z)x(y)] + [B][O(z)x(y)] +[C][~(x)O(y)] + [D][O(xz)i(y)] (3) where [O(x)(y)] [x)(y)], [((x)+(y)] and [V(x)+(y)] represent matrices whose elements are the corresponding basis functions in the computation domain of interest and [A], [B], [C], [D] represent the matrices of the unknown coefficients which give information about the fields and their derivatives. Application of Galerkin's technique leads to 4 schemes which can be represented as follows: < [<$], L(fi) + L2(f2) >=< [<a], 9 >: t~Scheme < [0I], LI(fi) + L2(f2) >=< [fik], g >: OOqScheme < [V0], L~(fi)+ L2(f2) >=< [(0], g >: ~OScheme < [Oi/], Ll(fl) + L2(f2) >=< [ikk], g >: OIkScheme From this system, we obtain a set of simultaneous discretized equations. For the first resolution level of Haar wavelets, the above four schemes decouple and coupling can be achieved only through the excitation term and the boundaries. The shielded structures analyzed here are terminated at Perfect Electric Conductors (PEC) and the boundary conditions are obtained by applying the natural boundary condition for the electric field on a PEC as shown below: E~?O(x)o(y) + E"6(xr)k(y) + Et '(x, )(y) + +Et ' )(xz)(y) = O....At PEC. (8) where E>, E", E and E' are the scaling and wavelet coefficients of the tangential electric field at the boundary nodes. The above equations are discretized by the use of Galerkin's method which results in a set of matrix equations of order N = M+1 where M is the order of the considered wavelet resolutions. These equations are solved simultaneously with the discretized Maxwell's equations to numerically apply the correct boundary conditions. III Applications of 2D FDTD Multigrid and Results The 2-D MRTD scheme derived above has been applied to analyze the Electromagnetic fields in a waveguide and a shielded stripline. (a) Waveguide: An empty waveguide with crosssection of 12.7 x 25.4 mm is chosen. A coarse 5 x 8 mesh is used to discretize this mesh and 2D MRTD technique was applied to analyze the fields in this geometry. Fig. 1 shows the amplitudes of the wavelet and scaling coefficients of the electric field obtained by using MRTD technique. From this figure it can be seen that only the 0f and qi coef(4) ficients make a significant contribution to the field and that the contribution of i0 and kI/ is negligible. From the computed coefficients, the total field is re(6) constructed using an appropriate combination of the (7) scaling and significant wavelet coefficients. For the waveguide chosen here, elimination of the wavelet coefficients that have no significant contribution leads to 480 unknowns. The reconstructed field obtained by this mesh has the same accuracy as that of a 10 x 16 FDTD mesh with 960 unknowns which is in agreement with the theory of MRA. Fig. 2 shows the results of this comparison and demonstrates that the use of multigrid scheme provides a 50% economy in memory. 2

(b) Shielded Stripline: Next, a stripline of width 1.27mm is considered. It is enclosed in a cavity of area 12.7 x 12.7 mm so that the side walls are sufficiently far away to not affect the propagation. The strip is placed 12.7mm from the ground. A 40 x 40 mesh is used to analyze the fields in this geometry with the 2D MRTD technique. Fig. 3 shows the derived scaling and wavelet coefficients of the fields just below the strip. From the figure, it can be seen that among the wavelet coefficients, only ~0 makes a significant contribution close to the vicinity of the strip where the field variation is rather abrupt. Fig. 4 shows the comparision of the total reconstructed field in the 40 x 40 MRTD mesh with that of a 40 x 40 and 80 x 80 FDTD mesh. From the figure it is clear that the field computed by 40 x 40 MRTD mesh using only the significant wavelet coefficients follows the results of the finer 80x80 mesh very closely, demonstrating once again the significant economy in memory as illustrated in Table 1. Fig. 5 shows the Normal Electric field plot of the strip and the variable mesh resulting from MRTD. Table 1: Comparison of the memory requirements in FDTD and MRTD techniques References [1] M.Krumpholz. L.P.B.Katehi, 'New Prospects for Time Domain Analysis", IEEE Microwave and Guided Wave Letters, pp. 382-384, November 1995. [2] M.Krumpholz, L.P.B.Katehi, "MRTD: New Time Domain Schemes Based on Multiresolution Analysis', IEEE Transactions on Microwave Theory and Techniques. pp. 385-391, April 1996. [3] K.Goverdhanam, E.Tentzeris, M.K-rumpholz and L.P.B. Katehi, "An FDTD Multigrid based on Multiresolution Analysis", Proc. AP-S 1996, pp. 352-355. 2.51 1 S c 0 =" 21 S I Q 0 I I I I I/ II -- PIP \\;;;i 0r -- -- L -- --— 0 - - 0 ---. 0 0.005 0.01 0.015 0.02 0.025 Distance along the cavity side in m Figure 1: Amplitudes of Scaling and Wavelet Coefficients in a Waveguide. Technique Unknown Coeff. 40x40 FDTD 9600 40x40 MRTD 11328 80x80 FDTD 38400 IV Conclusion A Haar wavelet based 2D MRTD scheme was developed and applied to analyse the fields in a waveguide and a shielded stripline. The wavelet coefficients obtained are significant only in regions of rapid field variations. Thus the FDTD multigrid capability using MRTD technique has demonstrated significant economy in memory. V Acknowledgments The authors are greatful to NSF and ARO for their support. a a o z Distance along the cavity side in m Figure 2: Comparison of MRTD, FDTD and Analytical Fields in a Waveguide. 3

- - I - 1 -3) I - - '3 D-,..I J) i - o Y\Ijw i i,1 1.. '2 - I I I \ I \., II i,, 4 o 8 10 Distance along the direction of strip (mm) 12 14 Figure 5: FDTD Multigrid and Field Plot of the Stripline. Figure 3: Amplitudes of Scaling and Wavelet Coefficients of a Shielded Stripline. i.9 Z0.7 - i a,a '0.S o C.2r Z Z;2 3U j j 2 i 14 I I 4 6 8 10 12 Distance along the direction of strip (mm) Figure 4: Comparison of Normal Electric Field under a stripline using MRTD and FDTD techniques. 4