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1、1262IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 51, NO. 4, APRIL 2003 Investigation of Boundary Algorithms for Multiresolution Analysis Martin Peschke and Wolfgang Menzel, Fellow, IEEE AbstractAn investigation on the multiresolution time-domain (MRTD) method utilizing different wavele
2、t levels in one mesh is presented. Contrary to adaptive thresholding techniques, only a rigid addition of higher order wavelets in certain critical cells is considered. Their effect is discussed analytically and verified by simulations of plain and dielectrically filled cavities with Daubechies and
3、BattleLemarie orthogonal, as well as CohenDaubechiesFeauveau(CDF)biorthogonalwavelets, showing their insufficiency unless used as a full set of expansion. It is pointed out that improvements cannot be expected from these fixed mesh refinements. Furthermore, an advanced treatment concerning thin meta
4、llization layers in CDF algorithms is pre- sented, leading to a reduction in cell number by a factor of three per space dimension compared to conventional finite difference time domain (FDTD), but limited to very special structures with infinitely thin irises. All MRTD results are compared to those
5、of conventional FDTD approaches. Index TermsBoundary conditions, multiresolution analysis, time-domain methods, wavelets. I. INTRODUCTION T HE multiresolution time-domain (MRTD) method has been under examination by various publications in the past. This includes the wavelet Galerkin scheme based on
6、BattleLemarie 1, Haar 2, Daubechies orthogonal (3, slightly different algorithm) and CohenDaubechiesFeauveau (CDF) biorthogonal wavelets 4. In structures with mainly harmonic spatial field distribution, it was shown analytically and by several simulations that the new approach reduces the numerical
7、phase error drastically, allowing a reduced cell number by up to two orders of magnitude. Additionally, all prementioned authors claim that MRTD achieves a natural mesh refinement such as introducing denser discretization rates for field components with fast spatial variation by adding wavelets of h
8、igher order. This feature is expectedto further reduce thenumerical effortand to contribute to an exact localization of different boundary conditions 5. Thus far, the only publication dealing with an a priori mesh refinement is 6, but results are compared only qualitatively to those of finite differ
9、ence time domain (FDTD) with a very high discretization rate. If any, this papers analytical survey and computational validation indicates that an advantage of such Manuscript received September 9, 2002; revised October 31, 20002. M. Peschke is with the Department of Optoelectronics, University of U
10、lm, D-89069 Ulm, Germany (e-mail: peschke_). W. Menzel is with Microwave Techniques,University of Ulm, D-89069 Ulm, Germany (e-mail: wolfgang.menzelieee.org). Digital Object Identifier 10.1109/TMTT.2003.809666 approaches cannot be expected in general. At last, dynamical mesh adaption remains a possi
11、bility for enhancing MRTDs performance (3, 7). Another often applied approach is the FDTD-like treatment of dielectric boundaries by local sampling for Daubechies or CDFwaveletsinscaling functionsonly 8,neglecting theexact material operator that has to actually be deployed. It will be shown that, ev
12、en with one-dimensional cavities, this simplified algorithmyieldsworseresultscomparedtoconventionalFDTD. MRTDs speed can be improved by pre-calculating boundary field dependencies (Massachusetts Institute of Technology (MIT), Cambridge, technique, 9), but this approach does not affect accurateness a
13、nd is very memory expensive for large calculation areas and arbitrary structures. Finally,allbutconcaveedgesrepresentaproblemforwavelet schemesincorporatinganimageprincipletomodelperfectelec- tric boundaries, explaining their rare appearance in most publi- cations thus far. This paper presents a new
14、 treatment for thin metallic irises by CDF algorithms. II. MRTD FORMULATION The (biorthogonal) wavelet representation of propagating fields, for simplicity in one dimension only, but for an arbitrary order of expansion, is as follows: (1) (2) is a dual-wavelet function of orderdisplaced by units,and
15、isthezeroth-orderrectangularHaarwavelet shifted byunitsin time 4. For orthogonal wavelet bases liketheBattleLemarieandHaar family,equals. Extension to the three-dimensional case is straightforward, replacing coefficients,by, and all components shifted according to the Yee scheme like in 1, e.g., (3)
16、 The letters , andindicate the Yee cell number in three dimensions, , ,andtheaccordingwaveletlevel,respectively. 0018-9480/03$17.00 2003 IEEE PESCHKE AND MENZEL: INVESTIGATION OF BOUNDARY ALGORITHMS FOR MULTIRESOLUTION ANALYSIS1263 TABLE I CONNECTIONCOEFFICIENTS FORHAARWAVELETS TABLE II CONNECTIONCO
17、EFFICIENTS FOR?WAVELETS TABLE III CONNECTIONCOEFFICIENTS FORBATTLELEMARIEWAVELETS IntroducingthefieldexpansionintoMaxwellsequations,the update instructions for the one-dimensional case are derived by testing with the nondual wavelet functions (4) (5) with the connection coefficient (6) Numericalvalu
18、esuptothefirstorderaregiveninTablesIIII for recently used wavelet families with the symmetry relation- ship, , and. Recently, 6 presented a way of calculating the connection co- efficients directly out of wavelets filter coefficients. III. ANALYTICALANALYSIS A. Homogeneous Formulations Analytical in
19、vestigations on MRTDs dispersion properties have been done by 4 and 10. However, only homogeneous expansions of zeroth or first order across the whole calculation areahavebeenconsidered,andspurioussolutionsareneglected. According to the formulation in 10, first-order fields are ar- ranged as follows
20、: (7) (8) (9) (10) denotes the numerical wavenumber in coeffi- cient space, which differs from the real casefor a fixed angle frequency. By introducing (7)(10) and after some mathematical manipulations, the update (4) and (5) yield an eigenvalue problem for the plane-wave amplitudesto after some mat
21、hematical operations as follows: (11) (12) (13) (14) (15) The matrix elementsare the connecting terms between the ordersand ,is the Courant number, is the ratio between real and numerical wavelength, is the number of Yee cells per wavelength, andis the free-space wave impedance. The dispersion relat
22、ionship is obtained fromas follows: (16) This is almost the same result as in 4, where the factor 2 in front of the forth term obviously has been forgotten. Fig. 1 compares the implicit equation (16) in the form of a wavelength error over discretization rate for wavelets with the zeroth-order soluti
23、on. Like for the BattleLemarie family, a spurious branch is obtained in addition to the improved curve. 1264IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 51, NO. 4, APRIL 2003 Fig. 1.Dispersion properties of complete zeroth- and first-order approaches using?wavelets. The Courant number
24、is? ? ? ?. The numerical wavelength error is plotted versus the number of cells per wavelength for wavelets of order 0 and 1. Fig.2.Dispersionpropertiesofthemultiresolutionapproachusing ?wavelets. The Courant number is? ? ? ?. As in Fig. 1, the plot shows the numerical wavelength error versus the nu
25、mber of cells per wavelength. The mixed approach shows the biggest wavelength error. B. Multiresolution Formulations The dispersion properties of a multiresolution approach will now be examined. Consider a one-dimensional dielectric res- onator that is filled with two different dielectrics. Accordin
26、g to Maxwells equations for inhomogeneous media, electric fields are expected to be differentiable smooth on the surface in the center of the cavity, while magnetic fields are not. This problem would lead to a multiresolution algorithm with low order only, but high order. Choosing, the following pre
27、-examinations are performed. With all, the system matrix of the eigenvalue problem can be written as (17) leading to the dispersion relationship (18) The mixed-order curves are expected to lie between the ho- mogeneous ones. However, in fact, the results are worse than those of zeroth order at any g
28、iven resolution, as indicated in Fig. 2. With this prior research, it cannot be anticipated that a mul- tiresolution formulation with higher orderin the whole cal- TABLE IV MATERIALOPERATOR? ?FORZEROTH-AND FIRST-ORDER?WAVELETS culation area or just at the surface between the two dielectrics will pro
29、vide better results than a simple zeroth-order one. Anal- ogous graphs are obtained for BattleLemarie families. IV.BOUNDARYALGORITHMS A. Dielectric Boundaries As the stencil size of CDF wavelets exceeds one, exact treat- ment of dielectric boundaries is only possible with the material operator deriv
30、ed by 1. For one-dimensional propagation and a dielectric surface located at, this operator reads as (19) (20) Forwavelets, these integrals have to be calculated numerically making use of the cascade algorithm provided by 11. For 20 iterations, the results are displayed in Table IV. Note thatrationa
31、lfractionsreplacedthenumericalvalues, being identical in all correct digits. Perhaps they will be shown to be exact by analytical investigations. These results are presented in 6 as well, but their exact quantity is omitted. Concerning the multiresolution approach, the following field representation
32、 was chosen. 1) As the electric field is continuous and differentiable smooth, only zeroth-order coefficients are applied. 2) Asthemagneticfieldiscontinuous,butnotdifferentiable, one first-order wavelet is arranged in the boundary plane to model the bump see Fig. 3(a). PESCHKE AND MENZEL: INVESTIGAT
33、ION OF BOUNDARY ALGORITHMS FOR MULTIRESOLUTION ANALYSIS1265 (a)(b) Fig. 3.Modeling of special boundary field curves (top) including zeroth- (thick lines) and first-order (thin lines) wavelets (bottom). (a) Modeling of?, (b) Modeling of?. The interface between the two dielectric media lies at ? ? ?(s
34、ee text for details). Fig. 4.Tangential electric field in the plane of a thin metallic iris extended up to? ? ?. To ensure zero tangential field on the iris, wavelet coefficients up to ?must vanish. 3) As the electric flux jumps, two first order functions are selected to the left and right of the di
35、electric interface, as shown in Fig. 3(b). Atlast,thesimplifiedlocalsamplingapproachfor wavelets will be observed in the simulations part. Similar to FDTD, this one only uses the main diagonal elements of the first Table IV. Note the equality to zeroth-order Daubechies wavelets with local sampling,
36、employing the same connection coefficients. B. ImprovedAlgorithm for Thin Perfect Electric Conductors (PECs) As for other MRTD formulations, PECs have to be modeled inalgorithms utilizing the image principle. This ap- plies well for enclosed concave cavities,but yields certain prob- lems when applie
37、d to convex edges or thin metallic irises. Espe- cially for the latter, it is easy to improve the performance with a simple change in geometry. ConsiderFig.4,whichshowsthe -componentoftheelectric fieldinaplaneparalleltothe-surfacethatishalfoccupied by a thin iris extended up to, but with no limits i
38、n the -direction. For this setup,lies tangential to the PEC and should vanish in the left half-space . This is usually achieved by adding uneven images in the columnsoccupied by the iris while processing the update equations. For,itisnotsufficienttoapplytheimageprinciple only to the negative columns
39、. As Fig. 4 indicates, this proce- dure leaves the wavelet coefficientuntouched, producing nonzero fields on the metallization. A simple extension of the iris by one cell can solve this problem, producing a linear rising (a)(b) Fig. 5.Two one-dimensional resonators under investigation. (a) Air fille
40、d. (b) With dielectric charge. Fig. 6.First-order simulation with spurious modes. Comparison of simulation results and analytically derived wavelength error graphs of Fig. 1. in front of the edge. Since this evaluation is not straight- forward for other field components or wavelet types, it is only
41、possible to talk about an effective aperture height in the MRTD domain, which is supposed to be one cell size larger than the actual geometry in the CDF-MRTD case. V. SIMULATIONS A. Simulation of Dielectric Boundaries At first, the air-filled resonator in Fig. 5(a) was analyzed in order to verify th
42、e results of Section III. The Courant number was chosen to befor FDTD andfor MRTD. The cells per real wavelengthwere calculated by resonant order and spatial discretization lengthfor each of the first three resonances, as well as the spurious modes at six and ten overall grid points. Errors in the r
43、esonant frequency compared to the trivial analytic case were recalculated to numerical wave- length errors. The results are displayed in Figs. 6 and 7 together with the dispersion graphs. A good match can be observed for both diagrams. Additional errors are addressed to nonideal boundary positions i
44、n the sim- ulation, which cannot be taken into account by the analytical investigation of free-space plane waves. These results do not encourage higher order attempts. Complete first-order expansions suffer under spurious solu- tions, what is not bearable for-parameter extraction. Mixed approaches a
45、re worse than one of a complete lower order even in this trivial case so they cannot be expected to be better in general. In 6, results are compared only qualitatively to FDTD with a very high resolution, which does not justify the emphasis of their superiority. 1266IEEE TRANSACTIONS ON MICROWAVE TH
46、EORY AND TECHNIQUES, VOL. 51, NO. 4, APRIL 2003 Fig. 7.Mixed-order simulations. Comparison of simulation results and analytically derived wavelength error graphs of Fig. 2. TABLE V NUMERICALRESULTS FOR THECHARGEDRESONATOR. THEANALYTIC RESONANTFREQUENCY OF THEDOMINANTMODE IS59.82 MHz The next study d
47、ealt with a dielectrically charged resonator utilizingdifferentboundaryalgorithmsforwavelets. Having maximum fields in the middle of the cavity close to the boundary, the basic resonant mode is the most interesting one. Simulationparameterswereasbefore,andthestepinthedielec- tric permittivity was ch
48、osen to bein order to obtain spatially strong varying fields at the interface. Results for the dominant mode are displayed in Table V. The most striking aspect are the poor results for the multires- olution column. Theyarefar worse than those of anyother tech- nique and improve only by adding wavele
49、ts of higher order to each field component in every cell. This strongly supports the idea of partially higher order approaches yielding worse results compared to uniform lower order ones because of the truncated field expansion that was put up in the analytic examination in Section III. Even the local sampling approach used in many publications so far (e.g., 3, 5 and 8) cannot hold against FDTD of the sameresolution.Notethatthealgorithmforwavelets in zero order wi