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Finite-Volume Maxwellian Absorber
on
Unstructured Grid
Krishnaswamy Sankaran, Christophe Fumeaux and Rudiger Vahldieck
Laboratory for Electromagnetic Field Theory and Microwave Electronics IFH Swiss Federal Institute of Technology ETH Zurich Zurich, CH-8092, Switzerland, Email:
[email protected] Telephone: (+41) 44 632 66 71, Fax: (+41) 44 632 11 98 -
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Abstract- A novel finite-volume time-domain (FVTD) model for the Maxwellian absorber is presented to aid numerical simulations on unstructured grid. In the present approach all the electromagnetic (EM) field quantities are colocated in both space and time. Theoretical development of the co-located FVTD formulation and a practical application of the Maxwellian absorber as an unsplit perfectly matched layer (PML) for waveguide problems are presented. For variations of the angle of incidence from near normal to 50 degree, the reflection coefficient of the FVTD-Maxwellian absorber is lower than -40 dB. Index Terms- Finite-Volume Time-Domain-FVTD, Maxwellian Absorber, Perfectly Matched Layer-PML, Computational Electromagnetics-CEM. I. INTRODUCTION
A novel finite-volume based model for the Maxwellian absorber as an unsplit perfectly matched layer (PML) on unstructured grid is presented. Although, this time-domain model utilizes the modified Lorentz material response of lossy dielectric media as proposed in [1], [2], [3], the present FVTD approach naturally adapts the model for an unstructured grid formulation. Furthermore, in order to reduce the computational overhead due to increased number of field equations a modified update equation technique is used. In addition, there is no need for any recursive convolution in time as in [4], [5]. This model satisfies Maxwell equations both inside and outside the absorber region without any unphysical field-splitting as in the case of Berenger's split-PMLs [6], [7]. Previous models for unsplit-PML were mainly studied using the structured finite-difference time-domain (FDTD) method, which involves staggering in spatial and temporal quantities. For geometries with complex boundary structures, the standard FDTD stair-case approximation lacks flexibility and accuracy. This creates a strong motivation for the development of FVTD method on unstructured space-time grid, which additionally enables spatially and temporally co-located field storage. The structure of the present paper is as follows. In Sec. II, the FVTD approach is briefly summarized for the present problem. In Sec. III, a 2D FVTD model of modified lossy Lorentz material is developed for the numerical simulation on unstructured grid. In addition, the definitions of flux-functions are discussed to finally derive the FVTD field update equations. Sec. IV focuses on the numerical implementation and discusses the
0-7803-9542-5/06/$20.00 C2006 IEEE
differences between the previous FDTD-based Maxwellian absorber and the present FVTD-based approach. In Sec. V, a practical problem is defined and the procedure for calculating the reflection coefficient is discussed. The results are presented in Sec. VI together with a discussion on the performance of the FVTD-Maxwellian absorber. II. FVTD APPROACH
Numerical simulation of electromagnetic (EM) fields involves discretization of the two Maxwell curl-equations using finite space-time samples. In the context of the FVTD method each spatial-sample Qi is called the controlvolume or cell and the temporal-sample At corresponds to the update time-step. Depending on the method of storing and updating the field quantities, different FVTD schemes are possible. For further references on the FVTD method, readers are directed to [8], [9], [10], [11]. The model presented here provides simple and consistent co-located spatial and temporal field variations. In the following analysis a 2D transverse electric (TEj) model with the electric field along the z-axis (E ) and the magnetic field in the xy-plane (H, and Hy) is employed. Using the divergence theorem, the FVTD update equations inside each cell of the dielectric medium is cast in the form [10],
atU =-
ZE
k=l
Fuk nk Sk |)- a1tQ
(1)
where U = [H, Hy:, E ]T denotes the EM field-vector with the superscript 'T' representing matrix transpose. Each i-th polyhedral cell is made of f faces and has a control-volume Vi 1. Each k-th face has an area of Sk and a unit outward-normal nk. a takes the value of free space permeability (,uo) and permittivity (Eo) for the magnetic and electric field update-equations respectively. The vector Q = [Mr, My: P / 0]T represents the components of magnetization M and polarization P fields inside the dielectric medium. The factor 'FUk nk' is called as the flux-function in the FVTD nomenclature and plays a crucial role in information exchange between adjacent cells. III. FVTD - MODIFIED LORENTZ MATERIAL MODEL
The polarization P and magnetization M field-vectors of the modified lossy Lorentz media were previously
used to model FDTD-based absorbing boundary condition (ABC) [1], [2]. These auxillary field quantities, when included in the FVTD formulation, result in computational overhead due to additional update equations involving flux terms. In this paper, a modified approach which reduces the computational overhead in the FVTD method is employed to model Maxwellian absorber on a 2D unstructured triangular grid. The TEz model of the modified (timederivative) Lorentz material yields the magnetization and polarization equations as follows [3],
(t2 + rat +w02) Q
(wo2xaz
+
WOXB&t) U (2)
where the material resonance-frequency is denoted as wo and the resonance bandwidth is given by F. The factor xog relates P and M to E and H, respectively. Similarly, the term Xa couples P and M with time-histories of E or H respectively. In particular, the factor Xa plays a crucial role for numerical modeling and will be a subject of detailed analysis in the next section. For a perfectly matched interface, a uniaxial absorber along the x-axis is considered. The frequency-domain electric and magnetic susceptibilities of the modified Lorentz media x§, results then in the relative permittivity and permeability tensors given by 0 O (I + Xw)1 + x§( (3) 0
(
0
0
I +X)(
where the superscript 'm' denotes the modified Lorentz model due to the additional time-derivative term in (2). The frequency-domain electric and magnetic susceptibilities are directly obtained from (2) using Fourier transformation as follows, w
[x0
(w 'wo)xal
(4) wi bwi2 -iFw~ where the subscript 'w' represents the frequency-domain value, with w corresponding to the center frequency of the incident EM excitation. As proposed in [1], the modified Lorentz model behaves like a uniaxial absorber if it satisfies the following constraints, 1) w >» w0 and F 0 inside 0abs. This highlights an important advantage of Maxwellian absorber as compared to the split-PML [6].
6tGX
'~ Gx
IV. FVTD - MAXWELLIAN ABSORBER: NUMERICAL IMPLEMENTATION A. Spatial and Temporal Discretizations The central idea of spatial discretization in the FVTD approach lies in the computation of flux-function across
each face of the control volume. The flux-function is split into incoming and outgoing fluxes across each interface as shown on the right-hand side of Fig. 1. On the other hand, the FDTD approach depends on the spatially staggered field quantities as shown on the left-hand side of Fig. 1. The concept of flux-function used in the equations (Il)(12) and (14)-(15) enables the FVTD method to naturally adapt the Maxwellian absorber to an unstructured grid. Furthermore, to simultaneously update the field quantities, a co-located (Predictor-Corrector) Lax-Wendroff time stepping is used [10]. Structured FDTD
7nstructured L l FVTD
~~~ ~~~~~ ~~Flux
Fig. 1.
Comparison between FDTD and FVTD Maxwellian absorber
field orientation model.
B. Damping and Coupling Coefficient As mentioned before, the factor Xa controls the absorber material response based on the previous time-history of field quantities E and H. In other words, it couples the time-derivative values of E and H to the corresponding values of P and M inside the lossy dielectric media. In fact, the modified Lorentz material model forces the material response to follow previous time-step field values. This essentially sets an upper limit in choosing (, in other words chosing Xa once wo is fixed. In discretized form, the value of ( is gradually increased from zero at the free space-absorber interface to the maximum value at the external truncating boundary of the absorber. A parabolic increase of the material loss parameter inside the absorber is employed as in [6], [1]. This is an optimal choice to reduce reflections both from the free spaceMaxwellian absorber interface and the truncating boundary of the absorber. V. PRACTICAL PROBLEM DEFINITION In order to test the effectiveness of the FVTD Maxwellian absorber on an unstructured grid, a 2D parallel-plate waveguide problem is simulated. The waveguide is assumed to have infinite symmetry along the
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transverse z-axis. The wave propagation direction is towards +x-axis. A perfect electrically conducting (PEC) boundary condition (BC) is forced on the two sides of the waveguide. The proposed FVTD-Maxwellian absorber is used to truncate the waveguide perpendicularly to the x-axis. In order to verify the practical applicability of the FVTD-Maxwellian absorber under the constraint of limited computational resource, the thickness of the absorber in all the analyses is fixed to dabs lA, where A correponds to the wavelength at the center-frequency w of the EM excitation (pulse or harmonic). A triangular spatial discretization with linear cell dimensions of A/12 is used for all presented results. In order to quantify the performance, the numerical reflection coefficient from the FVTD-Maxwellian absorber is calculated for various angles of wave incidence. Considering the plane wave decomposition model of a waveguide mode, changing the width of the waveguide is equivalent to changing the angle of incidence with respect to the free space-absorber interface. For each angle of incidence two models are built namely the reference and test models. The reference model is divided into two parts, one (DA) with exactly the same domain cells as in the test model and the other (DB), an extension, which is truncated by a first-order Silver-Mueller ABC (SM-ABC) as shown in the Fig. 2. The numerical reflection is computed by subtracting the reference field values from those of the test model. -
PEC-,,
m
m m m
DA -
PEC
PEC-'
DB
m m m
..... fi..-
i
I
I
Absorber
"Sensors
Fig. 2. Models for calculating reflection coefficient. Top: Reference model. Bottom: Test model.
VI. RESULTS AND DISCUSSION
The model in Fig. 2 is used to study the reflection coefficient at different angles of incidence using a firstorder TE mode excitation. As an example, the numerical reflection coefficient computed for an angle of incidence approximately equal to 30 degree with respect to the free-space-absorber interface is shown in the Fig. 3. The material loss parameter is chosen as ( 2w, where w corresponds to the center frequency of the EM excitation. The analysis is extented to a range of incidence angles from near normal to 60 degree which is of practical interest. The numerical reflection coefficient is calculated for different material loss parameter ( and the results are
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