"I

AlAA 95-1964

Investigations on the Properties of a Finite-Volume, Time-Domain Method for Computational Electromagnetics

Y. S. Weber CFD Research Branch Ae rornec han ics Division Wright Laboratory Wright-Patterson AFB, OH

26th AlAA Plasma Dynamics and Lasers Conference

June 19-22,1995 / San Diego, CA For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 370 CEnfant Promenade, S.W., Washington, DE. 20024

'4 Investigations on the Properties of a Finite-Volume, Time-Domain Method for Computational Electromagnetics

Yvette S. Weber' Wright Laboratory

Wright-Patterson Air Force Base, OH - - - -

Q R S - - A b s t r a c t

- Characteristic-based, finite-volume algorithms st - have been evaluated to determine the level of comput- ing power necessary to accurately simulate electromag- netic phenomena. Both second and third-order tech- V -

lution, media boundaries and domain size on wave prop- agation, radiation, and electromagnetic scattering. For ~ - wave propagation in free space, the third-order scheme has minimum dissipation when the grid is refined and a Courant number near the stability limit is utilized. For wave propagation across media boundaries, a lo- cal reduction of the scheme to second order allows for

this also causes additional dissipation of the reflected

- - - - -

niques have been used to study the impact of grid reso- + , Y , Z = - -

- - - - - - -

P

F > % C = - - -

P oscillation-free transmission and reflection. However, U -

and transmitted wave. Characteristic-based absorbing T - boundary conditions are demonstrated to be highly ef- Q - fective when the grid and wave propagation are aligned.

- - - - w

When this alignment cannot be enforced, the placement of the outer boundary becomes important. For RCS calculations, the scattered field formulation was found to produce a more accurate solution than the total-field formulation for a given grid system since it eliminates dissipation error of the incident field.

Subscripts - - - - S - t - - 0 -

45433

Dependent variables Reflection coefficient RCS integral Contour of integration Transmission coefficient Time Volume Cartesian coordinates Impedance Electric permittivity Wavelength Magnetic permeability Courant number Transformed coordinates Charge density Electric conductivity, radar tion Period Look angle Radian frequency

Incident Scattered Total Free space

N o m e n c l a t u r e Introduction

cross sec-

Magnetic flux density Electric displacement density Electric field strength Flux components Magnetic field strength Current density Equivalent electric current Wave number Equivalent magnetic current

The need for advances in the efficiency and accu- racy of computational electromagnetics (CEAI) contin- ues to be driven by the design of low observable (LO) aircraft and missiles. Specifications for modern weapon systems often cite radar cross-section (RCS) character- istics in addition to the more conventional performance parameters such as speed, weight and range. The radar signature of an aircraft is based on the scattering of elec- tromagnetic energy which occurs when a radar beam in- duces currents on the aircraft surface. It is comouted by

considering the resultant distribution of energy a t large distances from the scattering object[l].

Traditional techniques for computing RCS can be classified into high and low frequency methods. In the former, the scattering mechanism is primarily optical and is treated as a h a l Phenomenon. The body is aP- proximated by independent scattering centers. the SD

?Aerospace Engineer, CFD Research Branch, Aeromechanics ''U Division, Member A I A A .

paper is declared a aork of the U.S. ~ ~ ~ ~ ~ ~ ~ ~ , t and is not subject to copyright protection in the United States.

1

r)

Lax-IVendroff time integration in a finite-volume for- mulation. It has been successfully applied to many different configurations [ l l ] . tic based, flux-splitting methods have also been used to solve Maxwell's equations [12,13] and have also been applied successfully to many configurations.

The application of difference methods to CEM has several drawbacks which inhibit their use at the present time for practical, large-scale, high-frequency calcula- tions. Grid resolution is one of the pacing issues. Pre- vious work with both FDTD [3,4] and FVTD [11,14] indicates that each wavelength must be supported by a minimum of ten elementary cells or nodes for ade- quate spatial resolution. As the ratio of characteristic body length to signal wavelength increases ( / / A >> 1) for a general three-dimensional body, the grid size can quickly number in the millionsof points. IVave propaga- tion through material layers also increases the resolution requirement since the wavelength is reduced to smaller values. The use of high-order accurate spatial integra- tion schemes potentially reduce resolution requirements. but these methods usually involve more computations. Thus, the overall gain in computational efficiency may not he very high. Further, since the Maxwell equations constitute a linear, hyperbolic set of equations the time step size must be directly proportional to the grid spac- ing in order to honor the physics. Hence, wit,h grid refinement, not only do the number of computations J per time step become large, but the number of time steps needed is also large for a self-consistent solution. in which the signal or wave propagates sufficiently long enough for all portions of the scatterer to be causall? connected [3,4].

In the present work a characteristic-based, finite- volume CEM solver [I31 is implemented to solve EM fields associated with wave-propagation, radiation and scattering. This effort examines the level of computa- tional power necessary for this approach to accurately simulate electromagnetic phenomena. For wave prop- agation in one-dimensional media. the cuiniilative dis- persive and dissipative errors are examined versus grid refinement. Next, the effect of the outer boundary place- ment on the field induced by the oscillating electric dipole is examined. The wave and the grid are aligned for this configuration and represent the best perfor- mance of the characteristic-based boundary condition. Finally, the scattered field generated by a square cylin- der is computed using both total- and scattered-field for- mulations. Accuracy in surface current density and RCS values are evaluated versus the required grid-density and outer boundary placement. All computations are per- formed using a data-parallel version of the algorithm which exploits various parallelization strategies neces- sary to meet the computational requirements of larger.

Since then, characteris-' >

v

lutions of which are superimposed to obtain t h e overall field. These techniques are quite useful for computing the signature of full-scale configurations. At t h c lower end of the spectrum, integral methods are used to solve Maxwell's equations in the frequency domain. The stan- dard solution technique is called the method of riioiiients (MOM) and is a numerical approximation to thr exact formulation. This procedure couples all parts of the body together through a multiple scattering process [2] and is very similar to panel methods used in the com- putational fluid dynamics (CFD) community. MOM is particularly useful for computing surface traveling, creeping and edge waves which typically occur in the lower frequency, resonant scattering regime. IIowever, the methods are valid over all scattering regimes. The primary drawback is that solutions require the inversion of a large, dense matrix which becomes unwieldy for high frequencies and complex configurations containing inhomogeneous material properties. Nonethe1ei.i. Mob1 has been used successfully for many scattering configu- rations.

Recently, some modern finite-difference and finitc- volume methods have been transitioned from the ClzD community to the CEM community to solve the time- domain (TD) hlaxwell equations. T D techniqiirs offer several advantags over MOM. First, these methods of- fer a broader application base. They are not limited only to scattering but can also be used for commercial ap- plications such as antenna design, micro-circuitry, and biomedical imaging. Second, T D methods can support very general incident fields: harmonic, single-frequency, continuous-wave (CW) fields as well as transient. broad- band, single-pulse fields. This permits a the colnputa- tion of the scattered field at various frequencies. ranging from the Rayleigh regime through the optical regime in a single calculation [3,4]. Third, these difference tech- niques allow for variable material properties and treat- ment of active surfaces. However, with T D nwthods, the monostatic radar cross-section is limited to a single look-angle. In contrast the integral approach can pro- vide the resultant cross-section at all look-angles. but is limited to a single frequency.

T h e first finite-difference, time-domain ( I D T D ) method for CEl I was developed by Yee and was based on central differencing in space and a leap-frog integra- tion scheme in time [SI. Yee's method has been applied to many problenis in CEM, and more recently has been generalized to body-oriented grids [6-81. Nore mod- ern finite-volume and finite-difference techniques uti- lize a characteristic-based approach which better sim- ulates the physical and mathematical character of the hyperbolic equation set. Shankar e t a(. were the first to implement modern CFD techniques in CEXI [9,10]. Their method us- an upwind Riemann solver and a

n

more realistic configurations. 4

Numerical Method Governing Equations

The solution for any electromagnetic problem is governed by the time-dependent, Maxwell equations given below [15-171:

- V x H - J

- - V x E

aD at

at

_ - aB - -

V . D = p (3) V . B = 0 (4)

D = €E ( 5 ) B = pH (6) J = UE (7)

along with the constitutive equations,

and the conservation of electric charge

aP at - + V . J = O

t/' For simplicity consider free space, for which no sources are present (i.e. J = 0, p = 0). On a Cartesian grid, these equations can be rewritten in conservation form as

aQ aF aG aH - + - + - + - = o at ax ay a:

with

(9)

These equations can be recast for a generalized, curvi- linear space defined by:

E = E ( X > Y > Z )

'I = r l ( " , Y , Z )

c = C ( + . Y > Z )

In this new coordinate system the equations can be made to retain the same strong conservation form

where

Q = {QITV @ = { F , F + ~ , G + E , H } ~ V G = { V ~ F + ' I , G + ~ ~ ~ H ] ~ V H = { c ~ F + c , G + s , H } ~ v

In three dimensions, hlaxwell's equations involve three components of electric field {Ez, E,, E a ] and three components of magnetic field {Hz, H,, Hz}. In t w p dimensions, Maxwell's equations can be decoupled into two distinct sets: the transverse magnetic (TM) involv- ing {E,,H,,H,} and the transverse electric (TE) wave equations which involve {H2,Ez,Ey}, where in both cases the plane of incidence is defined as z = 0.

Integration Method

The integration method for the time-domain Maxwell equations follows the work of Sbang e l a/. [12-141. The approach consists of an explicit, cell-centered, characteristic-based, finite-volume scheme based on the Steger-Warming splitting approach :18]. With this method the flux vectors are split along a di- rection aligned with the surface area vectors of each cell and balanced a t each cell center. The flux vectors, F. G, and H are split according to the signs of the eigenvalues of the coefficient matrix in each spatial direction.

pi++ = F+(Q;+:) + F-(Q:_+) ( 1 2 )

G+lj - - G+(Q;++)+G-(QP_~) (13)

H ~ + ~ = H+(Q;++) + H-(Q:-~) (14)

The solution of the hyperbolic solution is thus recon- structed from piecewise continuous da ta by a procedure which honors the direction of signal propagation, xyhich is consistent for hyperbolic systems. High-order accu- racy is achieved through the MUSCL approach[l9]. The dependent variables are computed a t each cell interface using the ti-scheme [19,20], written below as follow:

Q:+h = Qi + ;[(I- l i ) V + ( I + 6)AIQi m

(15)

with VQi = Qi - Qi-1 and AQi = Q i t l - Qi. By ad- justing the values of 4 and IC, the scheme can be made

first-, second- or third-order accurate. The time iritc,ga- tion is accomplished via a two-stage Runge-Kuttn tech- nique, which is formally second-order accurate. How- ever, more stages can be added to this proct.~lurc~ to increase the temporal accuracy of the solution.

Total and S c a t t e r e d F ie ld Formulation The electric and magnetic fields appearing in

Maxwell's equations are total fields. The linearity of Maxwell equations allows these total fields to be ex- pressed as the sum of the incident and scattered fields.

Et = E; + E, Hf = H; + H,

('7) (18)

The incident field must satisfy Maxwell's eqnations in free space. The scattered field, hoxever, arises on and within the scatterer in response to t,his incident field. Substituting the above expressions for the total fir:Id into Maxwell's equations yields 14)

+ J, = V x [Hi + H,] (19) a [ c o ~ i + ~ E , I at

The equation for the scattered field is then obtained by subtraction of the incident field

= + J , = V x H H , at

and for the scattered field outside the scattering body ( b f J , = O ) , this simply degenerates to:

a m at

at

- = V x H ,

- = - V X E , aBs

which is of the same form as the total-field formula- tion. Thus, the same solution technique can be used for either formulation. For scattering calculations, use of the total-field formulation requires that the incident field be propagated through the computational domain In the scattered-field formulation, the incident field ap- pears analytically in the boundary conditions.

B o u n d a r y Conditions For the present effort, two types of boundary con-

ditions are considered: the physical surface boundary condition for a perfect electrical conductor (PEC). and the far-field boundary condition for truncation of the computational domain.

The total fields at a PEC surface must satisfy i t s following conditions[l5]

(25 I W n x E f = 0 n . H t = 0 (2C!

For a TM or TE wave scattering from a two-dimensional surface the above boundary conditions provide va lue for two of the three electromagnetic components. 13 principle the third component can be found by consid- eration of the electric current density a t the surfac?. This, however, is an unknown in the calculation. Fcr the present work, this remaining coniponent is appro?% mated by a second-order, windward extrapolation. Fcr three-dimensional scatterers, the system of boundary conditions is closed with extrapolated numerical boun.'. ary conditions for the finite j ump properties at the SUI-

facc[21]. When the scattered-field formulation is iz- voked, the incident field must be included at the srr- face, since the boundary conditions apply to the totel electromagnetic field.

The truncation of the computational domain ne- cessitates an artificial boundary condition in the fzr- field. Ideally this boundary condition would simply a!- low the waves to propagate out of the domain with :.,> ivave reflection. In practice this is not easily achieve: and wave reflection from this boundary can iuterferi x i t h the distribution of energy generated by the sca:. terer, leading to large errors in surface currents and RCS values. Several finite-difference techniques utilize at- sorbing boundary conditions. The characteristic fern:.:. lation suggests an alternate approach where the incor;.. ing flux at the boundary is set to a null value as

F-(Ea, 0 , o = 0 (Z

This boundary condition is exact when the grid and I!.? Rave direction are aligned, and is an approximation f i r all other cases. The characteristic-based approach is pi- tentially much more computationally efficient than t t. absorbing boundary condition.

A'ear to Far Field T r a n s f o r m a t i o n Using the time-domain method described ahoy?.

the electromagnetic field is obtained at all points in the compiit~ational domain (near field) which usually es- tends no more than a few wavelengths from the bod:. RCS, however, is evaluated by consideration of the ?E-

ergy distribution at far distances from the scatterir; object and is computed in the frequency domain. Tt.? time-varying currents are transformed to the frequency domain using Fourier transforms. The bistatic RCS is t h e n obtained through a near to far-field transformaticn based on a Greens function [22] . For a TM wave sca:. tering from a two-dimensional object, the complex, RCS

4

integral can be evaluated for a specific look-angle, rY, as

lwtY cos rk)le-J~(=’,,,(Ytv’sin(Y)ds’ (29)

where the “equivalent” currents are

J = n x H , (29) M = - n x E , (30)

The radar cross-section, u is then obtained from

Wave Propagation and Radiation Uniform Field

The first set of tests is designed to assess the prop- erties of the numerical algorithm for one-dimensional wave propagation through uniform media. The errors associated with numerically simulating the wave equa- tion can be categorized as dissipative and dispersive. Dissipation causes amplitude attenuation while disper- sion causes incorrect wave speeds which lead to phase

These errors are cumulative and can lead to nonphysical solutions. Insight into the numerical errors generated by an algorithm can be gained by consider- ing the modified equation which describes the difference between the original PDE and the numerical approxinia- tion of the PDE [23]. After much algebra, the model. one-dimensional wave equation discretized by the algo- rithm referenced in this work, has the following modified equation:

t -’ errors.

6u 6u au au 6t 6 x - at ax - + a - - -+a-

- [ a h 3 (g + -)] ( K - 1) uzzzz

8

written in terms of E , Ax, the xave speed, a, and the Courant number, w . For brevity, only the leading terms of the truncation error have been retained. The nonzero coefficients multiplying the third derivative (uzrs) leads to dispersive error. Those multiplying the fourth deriva- tive (uzzzz) yield the dissipation error. The scheme then is second order in dispersive error with larger er- ror associated with the second-order (x = -1) scheme. The dissipation term may be used to determine the sta- bility limits. Warming and Hyett[24] have shown that a necessary, but not sufficient, condition for stability

W

may be obtained by satisfying the balance of dissipa- tive terms. This.occurs a t w = 0.87 for the third-order scheme (x = 1/3). Note tha t this stability limit can also be obtained from a separate van Neumann stabil- ity analysis [25]. In Figure 1 the amplification factor modulus, ]GI, is plotted versus wave number, p , for var- ious values of Courant number. Curves falling outside the unit circle, IGI = 1, are unstable. In Figure 2, the area near p = 0 is enlarged t o graphically illustrate the stability limit. It is clear from this figure that Courant numbers of 0.9 and greater are unstable while a Courant number Of 0.87 remains inside (left) of the unit circle. The results of the Von Neumann stability analysis for the second order scheme show the stability limit to be a t w=0.5 .

The effects of these errors on electromagnetic wave propagation are illustrated in the computation of a con- tinuous, single-frequency wave propagating through a computational domain 25X in length. Measurement sta- tions are located at distances of l X , 5X, l O X , and 20X from the inflow plane. Three values of grid resolution are considered: 10, 20 and 40 pts/X . Figure 3, displays the attenuation of the wave resulting from the application of the third-order scheme. Two values of Courant number. w = 0.5 and w = 0.8, are shown. First, it is clear that as the Courant number approaches the stability limit, there is notably less dissipation of the wave. In the limit of v = 0.97, it h a s been shown that the wave does not undergo significant dissipation[25]. However as the Courant number is decreased, there is greater attenua- tion of the wave. This disparity grows larger as the mesh is coarsened. Also, for a given location and Courant number, the dissipation error roughly follows the ( A z ) ~ variation as predicted by Eqn. 32. For a propagation length and Courant number of 5X and 0.8 respectively. the dissipation errors are approximately .66% on the finest mesh and 18.2% on the coarsest mesh. In Fig- ure 4, similar results are plotted for the second-order scheme. As expected, the dissipation is greater and fol- lows second-order behavior. At a distance of S A , a grid resolution of 10 pts/X yields errors of 5 i % while an in- creased resolution to 40 pts/X reduces the error to 2.8%. Note that for the second-order scheme the dissipation is less sensitive to the Courant number even for coarser meshes.

The phase errors for both the second- and third- order schemes were evaluated by comparing the time the waves reached the measurement stations to the theoret- ical values based on the phase velocity. Both the second and third-order schemes exhibited leading phase error. The third-order scheme shows less phase error, as pre- dicted by Eqn. 32. For a continuous wave, phase error is less crucial than dissipation because of the masking ef- fect of the Fourier transform tha t applied to the dataset

Figure 1: Amplification factor modulus for comhinrd 2- stage Runge-Kutta with Flux Vector Splitting, x = 1/3, d = l

"=1.0

Y = 0.9 I G l = I

~ Y = 0.87

0.2650(3

0.26480

0.26460

0.26440

0.26420

0.264QO 0.96420 0.96430 0.96440 0.96450 0.96460 0 96470

Figure 2: Enlargement of amplification modulus show- ing stability limit, u = 0.87, for K = 1/3, 4 = 1

40 ptsh

- - - .....--- 20 ptsh 0.8 ......

.. m D

a 0.4

~ ~ . . . v = o s - v = 0.8

Figure 3: Dissipation error for various grid densities for K = 1/3, 4 = 1

Distance (A)

Figure 4: Dissipation error for various grid densities for K = -1, 4 = 1

prior to the RCS calculation. It should be noted, hov- ever. that phase error can become problematic when the scheme is used with transient, broad frequency spectrum pulses as input. There, the dispersion error effectively reshapes the incident pulse eliciting a different response from the scattering object.

Media Interfaces and PEC Boundaries L.'

In the second set of tests, the reflection and trans- mission coefficients are computed at five levels of grid density for the case of wave propagation through a me- dia interface. Theoretically these coefficients are func- tions of the impedance, Z , of the media, and are given as [MI:

(331

(34;

where 21 = and Z, = m. In these equa- lioiis, t,he double subscript,s on Z refer t.0 t,he reflection and transmission of E and H waves respectively. In this work a single sinusoidal wave pulse is propagated a total of 1OX1. The media interface is located at 5X: and separates two regions with €1 = p~ = 1 (free space) and € 2 = 4 , pz = 1 respectively. The wave amplitude is measured 2x1 from either side of the interface, and is normalized by the computed amplitude of an equiva- lent or control wave traveling 5x1 through free space. In this way the results are corrected at each value of grid density to account for dissipation due to simple wave propagation. The grid spacing is adjusted so that in both regions 1 and 2 , the resolution of the wave remains

6

constant. Since the material properties are discontinu- ous on either side of the interface a discontinuous be- havior of the solution vector, Q, is expected. Due to the flux-split nature of the algorithm, the interfaces are differenced in the direction of signal propagation, thus the interface requires no specialized formulation.

In Figures 5 and 6 the error of the computed trans- mission and reflection coefficients referenced to the the- oretical values is plotted versus the grid spacing per wax-elength. In each figure the dotted line represents a fully upwind discretization locally a t the media in- terface ( K = -1) while the solid line represents the third-order, upwind-biased discretization a t the inter- face ( K = 1/3). T h e algorithm used to compute these results is third-order accurate away from the interface. Figure 5 indicates that the upwind-biased interpolation produces more accurate values of the transmission coef- ficient. The higher error incurred by the fully upwind scheme may be due to additional dissipation introduced by the local reduction of the scheme to second order. The advantage of the fully upwind scheme is that the fluxes are fully one-sided, and no information is em- ployed in the reconstruction which is outside the wave’s domain of dependence. In the upwind-biased method, there is a small downwind contribution to the inter- face fluxes which causes a slight, spurious reflection of energy. As the relative permittivity of region 2 is in- creased, a sharper discontinuity of the wave across the interface is expected and the spurious oscillation intr- duced by the third-order scheme a t the interface may become more significant.

Figure 6 also illustrates tha t using the third-order scheme a t the interface produces the more accurate re- sult for the reflection Coefficient. I t should be noted, horT-ever, that the reflection coefficient is less sensitive to the order of the scheme a t the interface. In addi- tion. the relative errors are generally lower for the re- flection coefficient. The increased errors with respect to the transmission coefficient may in part be caused by the increased electrical distance the wave must travel to the measurement point. Recall tha t the measurement point is 2X1 on either side of the interface. In region 2, where the transmission coefficient is measured, this translates into 4x2, and thus the wave has had a longer distance over which to dissipate than the reflected wave. Both Figures 5 and 6 clearly show a reduction in error wi th grid resolution.

Media interface calculations were also conducted using the second-order accurate scheme. As expected, the relative errors were greater for both the transmission and reflection coefficients. For example, a t a grid spac- ingof 20 pts/X, the errors in the reflection and transmis- sion coefficients were approximately 18% and 26% for the globally second-order accurate scheme, compared to

Figure 5: Error in the transmission coefficient versus grid density, global K = 1/3, q5 = 1

Figure 6: Error in the reflection coefficient versus grid density, global K = 1/3,q5 = 1

t 1

Figure 7: Error in the reflection coefficient versus grid density for P E C boundary

6% and 10% from the third-order results illustrated in Figures 5 and 6. Note that the relative errors reported in this work were based on a fixed measurement, station and a Courant number of v = 0.2. Since the resiilts are normalized by the amplitude of the computed solution at the interface, the coefficients contained addit,ional dis- sipation error for wave propagation to the measiirement station. In the previous section it was concludcd that dissipation increased with lower Courant numbers and increasing propagation lengths. Rence, having thc mea- suring station closer to the interface and using a higher value of v would both decrease the dissipation of the wave, and would lead to reflection and transmission co- efficients which exhibit less error.

In a third set of tests the reflection coeflicient is again measured at five values of grid density for t,he re- flection of the wave from a perfect electrically conduct- ing (PEC) boundary located at 5 X from the inflow platie. The incident field used to normalize the reflection coeff- cient is again the computed incident field at 5A for each value of the grid density. T h e reflected wave amplitude is measured at 2A from the boundary. In Figure 7 the relative amplitude error is plotted versus grid density for both second and third-order accurate schemes. As expected, the error decreases with increasing grid den- sity, and the error is minimized with the third- order scheme. At a grid density of 20 pts/A the error i n the reflection coefficient is 5% for the third-order sclteine versus 15% for the second-order method. These errors are approximately the same as for the reflection cocff- cients obtained in the study of media interfaces above, and suggest that the errors reported here are doniinated by dissipation rather than special treatments of the in- terface boundaries or PEC walls.

Radiation and Outer Boundary Conditions

The above discussion focussed on wave propaga- tion through uniform and layered media as w l l a the PEC boundary condition. In these next set of trsts an oscillating electric dipole is used as a model prohlem to examine the boundary conditions in the far field. For these calculations an alternating current elemetit is lo- cated at the origin of a spherical coordinate system as illustrated in Figure 8. The dipole is chosen becalm the wave propagation is radial, and hence the outer bound- ary condition given in Eqn. 27 becomes exact for the spherical grid system. Two grid systems are considered (25,24,45) and (94,24,48) where the first index refers to the resolution in the radial direction, and is chosen so that for outer boundaries, Ra = 1.OX and Rb = 4.0X, the grid spacing remains constant. The grid was initialized with the theoretical solotion(l6] at the start of the cal- culation. The Courant number was set to v = l , which mas stable because of the radially expanding coordinate

l 2 W

J X

5 ,

Figure 8: Spherical coordinate system for electric dip&

,d J

06 0.8 1 0

Figure 9: Effect of outer boundary condition place on the Z-component of the electric field generated by an oscillating dipole

Figure 10: Scattered electric field about a rectangular cylinder for ka = 1 -

8

system. The theoretical field solution for the oscillating

dipole was compared with both a computed solution a t an elapsed time corresponding to 4.5 periods of motion, t = 4 . 5 ~ . During this time any reflection of the elec- tromagnetic waves back into the computational domain has time to contaminate the .wlution space located be- tween 0 < T < 1.OX. In Figure 9, the Z-component of the electric field is plotted in a small region between 0.4X and 1.OX. For purely hyperbolic propagation with no wave reflection, the solutions for both grid systems should be exact. Figure 9 illustrates that while some slight differences exist, for t he most part the solution does represent pure wave propagation. Also note that an increase in the placement of the boundary from 1.OX to 4.0X represents only a very slight improvement in the solution. Hence, for outer boundaries which are aligned with the direction of wave propagation, the condition of Eqn. 27 represents a very good approximation to the physics.

4

Scattered Fields

In the previous section the dissipative and dis- persive characteristics of the algorithm, along with the treatment of interfaces, walls and outer boundaries were discussed. These are all necessary elements to consider when computing electromagnetic scattering. In this sec- tion the electromagnetic scattering resulting from a TM- polarized plane wave impinging on an infinite cylinder with a square cross section is considered. This is a canonical problem which is well documented [Z, 26-29] For all cases, time-domain calculations are carried out until the field is statistically stationary. Fourier trans- forms are used to obtain the complex field in the fre- quency domain once the field has reached the station- ary value. From this information the surface currents are extracted, and the bistatic RCS is computed using the near- to far-field transformation. All solutions in this section were generated on a uniform Cartesian grid using the third-order spatial scheme. Due to the geo- metrical singularities, the Courant number was set to

‘d

- ^._

where B is the incident angle and c = l/&F. In this case the field is aligned with the x-axis, and thus 0 = 0. For a ka = 1. the wavelength is greater than the linear dimension of the cylinder by a factor of 7, and thus the grid spacing is chosen to provide resolution of the scattering body rather than the wavelength. Contours of the scattered electric field are shown in Figure 10.

In the first series of tests the grid density is var- ied from 20 pts/2a to 60 pts/2a. The computational domain extends 0.5X from the surface of the cylinder in each direction. Using the total field formulation, the surface currents are computed around the upper half of the cylinder. Figure I1 displays the comparison of the surface currents with results from a conventional method of moments (MOM) calculation [%I. Note that the surface currents become infinite a t 90 deg corners: and thus the solution becomes discontinuous in this re- gion. Computational data is shown within two cells of the corners. From Figure 11 it is evident that all three values of grid density produce excellent agreement with the method of moments computation in the front (ab) and upper surfaces (bc) of the cylinder. Increasing the grid resolution, as expected, yields more accurate re- sults. It is interesting that in the shadow region (cd) all three levels of grid density converge to a value which is lower than predicted by the method of moments.

In the next series of tests the position of the outer boundary was varied for a fixed grid density of 20 pts/2a. Figure 12 displays the surface currents com- puted on three domain sizes where the outer or far-field (FF) boundary is placed a t distances of 0.5X, l . O X , and 2.0X from the cylinder surface. As noted with the grid density study, the surface currents converge to a lox value in the shadow regions for boundaries less than 1.OX from the cylinder surface. Figure 12 shows excel- lent agreement in the shadow region for boundaries of 1.OX and 2.0X. However, since the incident wave must propagate in from the boundary, the error in the shadow region which comes from having the grid in close must be balanced by the dissipation error of the wave as it travels over longer distances to reach the the surface of the scatterer.

In Figure 13 the bistatic RCS is plotted for a grid density of 40 pts/2a with an outer boundary located at 0.5/X. Note that over the full range of look angles the

2o pts/2a the maximumerror was found to be j,4y,, For all other cases the error was minimized by the additional

u = U.U. In the first set Of tests a total-fie1d is

the cylinder ( I = h = 2a). T h e incident field is continn- ous and represented by [29]:

implemented for a ka = 1, where a is the half height of maximumerror is 4.8%. F~~ the lower grid resolution of

grid density and outer boundary placement. In a third set of tests the surface current is com- E: = E,cosk(zcos8+ys inO-c t ) (35)

(36)

(37)

puted for an incident field which is a t 45 deg to the scat- terer. The resulting electric field is shown in Figure 14. Since all computations were performed on a Cartesian grid the incident field is also oblique to the grid. In Fig-

9

4 ,

'solid line - MOM

Position on Cylinder Surface

Figure 11: Surface current variation with grid density for ka =1. total field formulation

4 . . a k a = l d

Position on Cylinder Surface

Figure 12: Effect of outer boundary placement on sur- face current density for the square cylinder

2.0 . . , r , . , . I I

Max Error 4.8%

0 20 40 60 80 1W 120 140 160 180 Y (DEG)

Figure 13: Bistatic RCS for PEC cylinder, grid reso- lution: 40 pts/2a, outer boundary location, 0.SX from cylinder surface

lire 15 the surface current is plotted for outer boundarie located at 0.5X and 1.OX. Here again, the currents are more accurately predicted when the boundary is placed L O X away from the scat,tering surface. Surface current.: were computed using the scattered-field formulation for the same configuration, and the results of the two for- mulations were compared. Both formulations enforce the same surface and far-field boundary conditions, and thus the differences between the two results amounts to the effect of the incident field dissipation in the total field computation. In Figure 16 results from a scattered and a total field computation are displayed for a grid density of 40 pts/2a and an outer boundary of 1.OX. As expected, the scattered formulation produces the more accurate of the two results since it is free of the disi- pation error in the incident field. A separate analysis was also performed for far-field boundary placement for the scattered-field formulation, and it also was found 10

approximate the currents better when the outer bound- ary was 1.OX from the cylinder surface. In this case :he placement of the boundary further from the cylinder r e duces reflection error due to the nonalignment between the grid and wave direction, while not incurring addi- tional dissipation error on the incident wave.

Lastly, in a fourth series of tests the total-field for- mulation was used to compute the scattered field for an increased electrical size. ka = 3 . As in Drevious studie

d

4 both the impact of grid density and far-field boundary placement on the solution was assessed. In Figure 1; the surface current is plotted for three values of grid density: 20, 40 and 60 pts/2a. The far-field boundary is located 1.OX from the cylinder surface. Once again, the computed solution is compared to conventional XIOJL results. In this case the currents in the frontal porticn of the cylinder (ab) show substantial error for a gri3 density of 20 pts/2a (12.5 pts/X). Figures 18 and 19 show the surface currents computed with a constant grid spacing of 60 pts/2a (37.5 pts/X) for outer boundaries of l . O X , 2.0X and 4.0X in the front and shadow regions re- spectively. In the frontal region it is clear that. increasinz the distance to the outer boundary increases the dissi- pation error. However. for the shadow region there is no clear improvement with distance to the outer boundary.

Summary and Conclusions

In the present work the properties of a characteristic-based, finite-volume, CEM algorithm. which implements the 'MUSCL K-scheme for spatial d i i cretization and a 2-stage Runge-Kutta time integration scheme are evaluated. Electromagnetic fields associated with wave propagation, radiation, and scatiering are computed.

Both fully upwind, second-order ( K = -1) and

10

Figure 14: Scattered electric field for ka = 1 , 8 = 45

A B

Position on Cylinder Surface

C

Position on Cylinder Surface

Figure 17: Variation of surface currents with grid den- sity for ka = 5

Ez 2.6 -

r- 2.4 - H, g

2 2.2 -

ka = 5

,"I: 0

0 0

MOM FF at 1 .OX FF at 2.0.2 FF at 4.0.2

Position on Cylinder Surface

Figure 18: Variation of surface currents around the front face of the cylinder with outer boundar: placement for ka = 5

Figure 15: Variation of surface currents with outer boundary placement for 8 = 45 deg

I

Position on Cylinder Surface S!O 0.5 1 .o 1.5

ka=5

0.12

H.

, "On0,- "oq61, 0.091," ' 1.6 ' 1.7 1.8 1.9

Position on Cylinder Surface

Figure 19: Variation of surface currents in the shado\v region of the cylinder with outer boundary for ka = 5

Figure 16: Comparison of surface currents from total and scattered field formulations with MOM for B = 45 deg 4

11

upivind-biased, third-order (K = 113) schemes were used to computed one-dimensional wave propagation In a iiniform medium, the third-order scheme demonstrated minimum dissipation error for increased grid density and Courant numbers close to the stability limit of u = 0.87. The second-order scheme displayed larger dissipat,ion er- ror which could be minimized only by grid refiiiement. These results suggest that when the third-order algo- rithm is applied to practical configurations, the grid geometry and boundary conditions should be carefully constructed so tha t no additional stability constraints are imposed which lower the Courant number below the stability limit thereby introducing larger dissipation er- rors. Both second and third-order spatial schemes dis- play second-order leading phase error associated with the second-order accurate time integration scheme.

The transmission and reflection of waves from an impedance boundary was also studied. For the tbirtl- order method, local reduction at the boiindary to pro- vide a fully-upwind dinerencing stencil ensures 110 spu- rious oscillations. However t,his technique docs t,rigger additional dissipation of the resulting waves. Similar results were also obtained for wave reflection from a PEC wall. For a relative permittivity, cI = 4, rctaiii- ing a fully third-order scheme through the boiindary produced more accurate results despite the small oscil- lation. As the relative permittivity is increased, errors introduced by the oscillation may become significant. For this case, a reduction to second order a t the bound- ar? coupled with local grid refinement may produce the most accurate results.

The electromagnetic field radiated by ail oscil- lating electric dipole was computed by the third-order method on a spherical grid. For this case, the grid and wave are aligned and the characteristic-based absorb- ing boundary condition is very accurate. When the grid and wave are skewed, the accuracy is expected to deteri- orate, however, both the dipole and the square cylinder show an outer boundary of 1X is sufficient.

The third-order scheme was applied to the scatter- ing of electromagnetic energy from an infinite cylinder with a square cross section. Because of the geometric singularities, a further restriction in stability led to a reduction of the Courant number to 0.25. Two electri- cal sizes, ka = 1, and k a = 5 were considered along with two incident angles, # = Odeg and 0 = 15deg. For ka = 1, resolution of the body leads to a dense grid in terms of wavelength resolution and thus 20 pts per side (60 pts/A) was sufficient to compute the sur- face current. An outer boundary placement 1.OA from the cylinder surface was found to be optimal. Similar results were found for 0 = 45deg. At the higher fre- quency, ka = 5, the same surface resolution yields 12.5 pts/X. This resolution produced significant error in I.he

currents along the front face of the cylinder. Doubling the grid resolution showed considerable improvement in the solution.

For the total-field formulation, increasing the di.- tance to the outer boundary typically enhanced the ac- curacy in the shadow region. However, since the inci- dent field must be propagated from the outer boundary to the surface of the scatterer, dissipation becomes sig- nificant. The scattered-field formulation utilizes an an- alytic representation of the incident field on the surface boundary and is free from this error.

In summary, the third-order spatial scheme in combination with the scattered field formulation is an accurate and efficient method for determining surface currents and radar cross section. However, grid res- olution, surface and far-field boundary conditions s t i l l remain pacing issues. Hence, future work must focus on higher-order algorithms. boundary conditions, and high performance computing in order for time domain techniques to be used for realistic configurations.

J

Acknowledgments The author wishes to thank Joseph Shang, Datta

Gaitonde and Ken Moran for many instructional conver- sations with regard to algorithmic issues and numerical methods, and Jeff Young and Kueichien Hill for many helpful conversations in electromagnetic theory. Com- putational resources for this work were supported by the ‘ J DOD HPC Distributed Resource Centers a t the Army High Performance Computational Resource Center and the Center for Computational Sciences at the Naval Re- search Laboratory.

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