Application of Finite Element Method to Analyze Inflatable ... · Application of Finite Element Method to Analyze Inflatable Waveguide Structures _. D. Deshpa_de ViCYAN, l_c.,llamplo_,

NASA/CR-1998-206928

Application of Finite Element Method to

Analyze Inflatable Waveguide Structures

_. D. Deshpa_de

ViCYAN, l_c., llamplo_, Virsz_ia

February 1998

https://ntrs.nasa.gov/search.jsp?R=19980031732 2020-07-06T20:25:39+00:00Z

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NASA/CR-1998-206928

Application of Finite Element Method to

Analyze Inflatable Waveguide Structures

_. D. Deshpa_de

ViCYAN, l_c., llamplo_, Virsz_ia

National Aeronautics and

Space Administration

Langley Research Center

Hampton, Virginia 23681-2199

Prepared for Langley Research Centerunder Contract NAS1-19341

February 1998

Available from the following:

NASA Center for AeroSpace Information (CASI)

800 Elkridge Landing Road

Linthicum Heights, MD 21090-2934

(301) 621-0390

National Technical Information Service (NTIS)

5285 Port Royal Road

Springfield, VA 22161-2171

(703) 487-4650

.

2.

2.1

3.

3.1

3.2

3.2.1

3.2.2

3.2.3

3.2.4

4.

A.1

A.2

B.1

Contents

List of Figures

List of Symbols

Abstract

Introduction

Theory

Finite element E-field formulation

Numerical Results

Rectangular waveguide without wall distortion

Rectangular waveguide with wall distortion

Inclined walls in y-direction

Inclined walls in x-direction

Rectangular waveguide with curved walls

Rectangular waveguide with randomly distorted walls

Conclusion

Appendix ADerivation of nodal basis function

Derivation of vector edge basis function

Appendix B

Expressions for matrix elements

References

2

5

7

7

10

10

15

15

16

17

17

18

18

19

19

19

20

21

22

22

Figure1

Figure2

Figure3

Figure4(a)

Figure4(b)

Figure5

Figure6

Figure7

Figure8

Figure9

Figure10

Figure11

Figure12

Figure13

List of Figures

Plot of main beam direction as a function of (_o- 1 ) %.

Plot ofresonant slot conductance G O as afunction of (_o-1 / %.

Geometry of a few cross sections of deformed rectangular waveguide

Geometry of cross section of a rectangular waveguide with distorted walls

Geometry of cross section of a rectangular waveguide with distorted walls and

triangular mesh

Geometry of a triangular element

Geometry of inhomogeneous rectangular waveguide with ( _b = 0.45, anda

d- = 0.5)b

Geometry of L-band rectangular waveguide with inclined walls in y-direction

Plot of percentage change in dispersion characteristics of L-band rectangular

waveguide for various inclination 0 shown in Figure 7

Electric field plot in the cross section of distorted waveguide shown in Figure 7

Geometry of L-band rectangular waveguide with inclined walls in x-direction


waveguide for various inclination 0 shown in Figure 10


Geometry of L-band rectangular waveguide with distortion in x walls

2

Figure 14

Figure 15

Figure 16

Figure 17

Figure 18

Figure 19

Figure 20

Figure 21

Figure 22

Figure 23

Figure 24

Figure 25

Figure 26


waveguide with distortion as shown in Figure 13


(frenquency = 1.4 GHz)

Geometry of L-band rectangular waveguide with distortion in x-walls





Geometry of L-band rectangular waveguide with distortion in x-walls





Geometry of L-band rectangular waveguide with distortion in y-walls





Geometry of L-band rectangular waveguide with distortion in y-walls



Figure27

Figure28

Figure29

Figure30

Figure31

Figure32

Figure33

Figure34

Figure35

Electricfield plot in thecrosssectionof distortedwaveguideshownin Figure25

(frenquency= 1.4GHz)

Geometryof L-bandrectangularwaveguidewith distortionin y-walls

Plotof percentagechangein dispersioncharacteristicsof L-bandrectangular

waveguidewith distortionasshownin Figure28

Electricfield plot in thecrosssectionof distortedwaveguideshownin Figure28

(frenquency= 1.4GHz)

Geometryof L-bandrectangularwaveguidewith dominantmodeelectricfield

for randomdistortionin all walls ( c_2 = 0.2 andtoleranceequalto+0.2 )


waveguidewith randomdistortionin all walls(50 runs)


waveguidewith meanvalue,lowerboundandupperboundcalculatedfrom

Figure32


waveguidewith randomdistortion( c_2 = 0.2 andtoleranceequalto +0.1 ) in

all walls( 50 runs )


waveguide with mean value, lower bound and upper bound calculated from

Figure 34

4

a

a i, b i, c i

aa, bb, cc

A

A

b

B

-->

E(x,y,z)

-->Et

EZ

e tm

f

gzn

Go

Hx, y,z )

J

ko

kc

h

S_z(m,m')

Rez (m, m')

Qd(m,m')

Pd (m, m')

Ud(m,m')

v l (m, m')

x l (m, m')

Yd(m,m')

--)T (x, y, z)

--->

Tt

List of Symbolsx-dimension of rectangular waveguide

constants

constants

a vector

area of triangular element

y-dimension of rectangular waveguide

a vector

electric field vector

transverse electric field vector

z-component of electric field

amplitudes of edge basis function

scalar function

amplitude of nodal basis function

Resonant conductance of shunt slot on a rectangular waveguide

magnetic field vector

free-space wave number

cut off wave number

unit normal vector drawn outwards

element matrix for single triangular element

element of coefficient






matrix for single triangular element






element of coefficient matrix for single triangular element

vector testing function

transverse component of testing function

TZ

m_

Wit/,/

x, y, z

O_n

Eo

go

Er

]Ar

_distorted

_undistorted

F

Vt

0

g

)_o

z-component of testing function

vector basis function associated with triangular element

Cartesian Coordinate system

unit vectors along x-, y-, and z-axis, respectively

scalar basis function associated with a node

permittivity of free-space

permeability of free- space

relative permittivity of medium in region II

relative permeability of medium in region II

propagation constant

propagation constant for deformed waveguide cross section

propagation constant for undistorted waveguide cross section

curve enclosing rectangular cross section

= 2 + y_--_

Angle of inclination in degrees with respect to x-axis

main beam direction in degrees

guide wavelength for dominant mode

free space wave length

6

AbstractA Finite Element Method (FEM) is presented to determine propagation

characteristics of deformed inflatable rectangular waveguide. Various deformations that

might be present in an inflatable waveguide are analyzed using the FEM. The FEM procedure

and the code developed here is so general that it can be used for any other deformations that

are not considered in this report. The code is validated by applying the present code to

rectangular waveguide without any deformations and comparing the numerical results with

earlier published results. The effect of the deformation in an inflatable waveguide on the

radiation pattern of linear rectangular slot array is also studied.

1.0 IntroductionRecently there has been considerable interest in the development of inflatable antenna

structures [1-3] for space applications. In inflatable antenna technology, the antenna structure is

packaged in a small volume during its launch phase and inflated or stretched to its full length after

reaching desired orbit. One such structure under development at NASA Langley Research Center

is an inflatable slotted rectangular waveguide antenna to be used in soil moisture measuring

radiometer. After full deployment of such structure in space, the waveguide surface may have

wrinkles, curved walls depending upon the pressure used to inflate the structure, and other

unaccounted forces acting on the structure. For successful operation of these antennas, it is

desirable to study and estimate adverse effects of these deformations in waveguide walls on the

antenna performance. Since these deformations cannot be completely eliminated, study of their

effects on antenna performance may lead to determine an allowable level of deformations in these

structures reducing high constraint on mechanical design.

An antenna array performance is usually specified by its radiation pattern, input impedance,

polarization, etc. For a linear slot array antenna consisting of the shunt slot elements on the

broadwall of arectangularwaveguide,themainbeamdirectionis givenby [4]

cos(_)) = n + -_g + nn )_o/ (2rid) where d is the physical spacing between the elements,

)_g and )_o are the guide and free space wavelengths, and n = 0, +2, +4 .... Usually for the broad

side radiation at a given frequency of operation the distance d is selected as )_gO/2, where )_gO is

the guide wave length of undistorted waveguide. For n = -2, the expression for beam direction

/becomes cos (_)) = ( )_g - 1 )_o/)_gO = - 1 _5o/k o where [_o and [_ are the dominant

mode propagation constants of undistorted and distorted rectangular waveguides, respectively,

and k o is the free space wave number. From these above expressions it is clear that if [_ = [_o

then main beam is in the broad side direction. However for [_, different from [_o, the main beam

shift from the broad side direction as shown in Figure 1. In order to relate various antenna

deformations to shift in mean beam direction, it is important to estimate the effects of various

deformation in waveguides on the propagation constant [_.

In the design of shunt slot array antennas, one of the most important expression designers

use is the resonant slot conductance [5] Go = 2.09( k° a ] m (rc_°_l]2 (_ob) LCOS_ ggo_o _). By selecting

proper slot displacement, an amplitude distribution for required radiation pattern is achieved.

However for [_, different from [_o, which is the case for deformed waveguide, the resonant slot

conductance will change and hence the amplitude distribution. Quantitatively the dependance of

G O on the propagation constant is shown in Figure 2. It is therefore essential to know the

propagation constant variation due to deformation in inflatable waveguides.

Thepurposeof this reportis to present ananalytical methodto determinethe

electromagneticfieldsandpropagationconstantin arectangularwaveguidewith deformedcross

sections.A few examplesof deformedcrosssectionsthat maybepresentin aninflatable

waveguideareshownin Figure3. Theanalysisof waveguidewith canonicalshapessuchas

rectangularor elliptical ( includingthecircularasa specialcase) crosssection is usuallycarried

outby solvingthe scalarHelmholtzequationsubjectedto Dirichlet andNeumannconditions.

Theelectromagneticfield in thesecrosssectionscanbewrittenin termsof sine,cosine,or Bessel

functionsbecauseof the separabilityof variables[6,7].However,for theirregularshapesshown

in Figure1,the simpleseparationof variablesmethodgivenin [6,7]becomesmoretediousand

hencenot preferred. In thisreportaversatileandpowerfulnumericaltechnique,namelythe

FiniteElementMethod,is usedto analyzethesedistortedstructures.

Theproblemof findingeigenvaluesandpropagationconstantof awaveguideof an

arbitrarily shapedcrosssectioncanbesolvedby invokingtheweakform of vectorwaveequation

[8,9]. By dividing thewaveguidecrosssectioninto triangularsubdomainsandexpressingthe

electricfield (for E-field formulation)or themagneticfield (for H-field formulation)into

appropriatevectorbasisfunction [9], theweakform of vectorwaveequationis reducedto a

matrixequation.Theresultingmatrix equationis thensolvedfor eigenvaluesandpropagation

constantusingstandardmathematicalsubroutines.Theremainderof thereportis organizedas

follows. Theformulationof theproblemin termsof weakform of vectorwaveequationandits

reductionto amatrixequationisdevelopedin section2. Thedetailstepsinvolvedin casting the

matrixequationintoaneigenvalueproblemis alsogivenin section2. Thequantitativeestimates

of effectsof waveguidecrosssectiondeformationonpropagationconstantof aL-band

rectangularwaveguidearegivenin section3. Theeffectof wall distortionon radiationpattern

of linearslotarrayson distortedrectangularwalls is alsonumericallystudiedin section3. The

reportis concludedin section4 with recommendationsbasedon thenumericalresultspresented

in section3 andfuturework to becompleted.

2.0 Theory2.1 Finite Element E-Field Formulation :

The waveguide cross sections to be analyzed are shown in Figure 3. To determine

effects of these irregularities on the cut-off frequency, propagation constant, and characteristic

impedance, the numerical technique such as Finite Element Method is developed in this section.

The electric field in the cross sections shown in Figure 4 satisfies the Maxwell's equations:

(1)

(2)

Substituting (1) in (2), the

--> ->

VxH = jo3eE

where g and e are the permeability and permittivity of the medium.

vector wave equation with electric field is obtained as

--> 2 -->

_r

Similar vector wave equation for the magnetic field can be obtained by substituting (2) in (1).

However, we will restrict here to the E-field vector wave equation. Assuming the waveguide to

be infinite in the z-direction, the electric field can be written as

E = E t + _E z (4)

....)

where E t = 2E x + _Ey, 2, Y, _ being the unit vectors along the x-, y-, and z-directions respec-

tively and [3 is the propagation constant in the z-direction. In the equation (4) it is assumed that

the wave is traveling from z = -_ to z = + _. Substituting (4) into (3) and carrying out sim-

ple mathematical operations, the following equation is obtained:

10

[Jr VxLte )--KoErLte

+ vxl(Vx_Eze4_Z)-k_er2Eze4_Z=O (5)_-[F \

Substituting the gradient operator in equation (5) as V = V _- 2 (j[}) and performing simple

mathematical manipulation, equation (5) can be written as

The equation (6) can be written in component form as

= 0 (6)

Vrr_. VrrE , j_VtEz-[3 ,)-koe r ,= 0 (7)

In order to make coefficients of field components real, equations (7) and (8) after the substitution

E = j_E are written as

The expressions

v_, v_e, +7, v,E+ ,)-ko_, ,= o (9)

• + koerE 0 (10)_, v, v,e+ =(9) and (10) are required equations to be solved either for the propagation con-

stant [_ for a given frequency or for the cut-off wave number k c = k o for [_ = 0. In either case

to solve equations (9) and (10) using the Galerkin's procedure, we select a testing function

--) --)T = Tt+_T.

Z

.-)

Multiply equations (9) and (10) with T: and T, respectively, and integrating over

the cross section we get

11

I Vdc VpcE t + VtE Z + E l -koerEt • Ttdxdy = 0cross section

S S(_,,-, +cross section

Using the vector identities

+ > > > (._xa)A • VtxB = VtxAt • B- V t •2

(11)

(12)

--) --> --)

fV t •A = +V t •fA -A•Vf

I I Vt•A dxdy = •hdlcross section F

where h is the outward drawn unit normal vector to the curve F enclosing the cross section.

equations (11) and (12) can be written as

s/ //f vA, ±v_L _ >• - koe ,.- _t •Tt dxdy_-L I.

cross section

and

I • Ttdxdy = - t • h × --V/cEtdFVtEz+ btr

cross section F(13)

_ _<_,Er d*dS J'v,r-1->S S(v,_.-v,. ) + S _,.,<,x<,ycross section cross section

1 -)

= [1TVtE•hclr+ [--LEt.hdF (14)_t,. r_t,.

where h is the outward drawn unit normal vector to the curve 17 enclosing the cross section. To

solve the weak forms of differential equations given in (13) and (14) numerically, the cross

section shown in Figure 4(a) is discretized into triangular domain as shown in figure 4(b). The

12

transverse and longitudinal components over a triangle (shown in 5) are then expressed as

3--) --)

E, = _ e. W,.,(x,y) (15)

m 1

3

E = _ gzo_n(x,y) (16)

n 1

where m =1,2,3 are the three edges of the triangle and n = 1,2,3 are the three nodes of the triangle.

--+

The detail derivation of the vector edge basis function W.. and the scalar basis function

o_n (x, y) are given in Appendix A. Substituting (15) and (16) in (13) and (14) we get

_ et. ' VpcWt. . VpcWt. koe , Wt. • Wt. dxdym 1 triangle _r

+ 2(3n 1 triangle

)

Vto_n • W.. dxdy + 3 --)

_ e.. I I _"''W'''dxdy

m 1 triangle

3

E etm l

m 1 triangle

= - t " h × --VflcEtdFr _t

3

" W..dxdy +, I • koe ,.(o_o_,) dxdyn 1 triangle _r

(17)

3 3

I 1 11= + _ gzn r_°_nVt°_n" hdF + _ e,. uO_nW,., hdF (18)n 1 " m 1 F _ r

For the waveguide cross section enclosed by metallic boundaries, the line integrals appearing on

right hand sides of equations (17) and (18) are always zero. This is true because of the tangential

electric field being zero on the perfectly conducting boundaries. With these considerations, the

equations (17) and (18) can be written in a matrix form

Igel(m"m) _1 [etrrt = -_2IRel(m:'m) Qel(m"n)] Ietrrto L&d P,(n',n)JL&d

suitable for calculations of propagation constant for a given frequency.

(19)

13

For calculation of cut-off wave number when [_ = 0 the equation (17) and (18) can be written in

a matrix form

Vel(n',n) Lg_d 0

suitable for calculation of cut-off wave number.

in equations (19) and (20) are given by

Yd(n', n) lg_d

The elements of various submatrices appearing

Qd (m', n) (23)

s s( -> ' --> -> ->;)S d (m', m) = V#Wtt.' • --V#Wtm - koe r Wtm • Wtm' dxdy (21)triangle _tr

Rd(m,m) = 1 f (22)' • Wt,,'dxdy

_rtriangle"

:-1 .f fv,o_,,._,,,,<_,<dy_tr triangle"

"_,_,,,,,).r .r(v,o,,,_,v,o,,,_ )' = • - koe ,. (o_no_n,) dxdy (24)triangle _tr

U d (m', m) (25)= VdcW..' • VrxW.. dxdytriangle -_r

: .r .r(v,o<,<.-'v,o<,_<,<<,ytriangle _tr J

V d (n', n) (26)

rrt -> _)X d (m', m) = e Wt,. • Wf,.' dxdytriangle

c,(,',',,',): I I_,o<,,o<,,<'<<'ytriangle

The double integrations over the triangle

(27)

appearing in (21)-(28)

Details of the numerical integration are given in Appendix B.

3.0 Numerical Results

A FORTRAN code is written to solve the eigenvalue problems described in equations (19)

and (20). The matrix elements appearing in (19) and (20) are evaluated numerically (see

Appendix B ). To validate the code, the cutoff wave numbers for various modes in a rectangular

waveguide without wall distortion are first determined and compared with analytical results [7].

(28)

are numerically evaluated.

14

3.1 Rectangular Waveguide Without Wall Distortion:

The eigenvalues are then determined using standard mathematical subroutines. For

avalidation of the code, a rectangular waveguide with _) = 2 and without wall distortion is

selected as a first example. The cut-off wave numbers calculated using the present code are given

in Table 1 along with the results reported earlier [7]. It is found that the percentage error in the

calculated wave numbers using the present code is very small (less than 3 percent). From the

results shown in Table 1, it is also observed that the percentage error increases with the mode

order.

Table 1: Cut-off wave number of rectangular waveguide a/b =2

kcaModes

Reference [ 1] Present Method %Error

TE Modes

TElo 3.142 3.1397 0.007

TE2o 6.285 6.276 0.143

TEol 6.285 6.267 0.286

TEll 7.027 7.139 1.59

TE3o 9.428 9.376 0.552

TE21 8.889 9.115 2.54

TM Modes

TMll 7.027 7.026 0.001

TM21 8.889 8.9012 0.137

TM31 11.331 11.337 0.052

15

For thesecondexample,aninhomogenuosrectangularwaveguidewithout anywall

distortionasshownin Figure6 is considered. For this geometry,usingthe presentcode the

propagationconstantasafunction of frequencyis calculatedandgivenin Table2 alongwith

earlierpublisheddata. Thenumericalresultsobtainedby thepresentcodearewithin 5percentof

theanalyticalresults[7,9].

Table 2: Dispersion characteristic of lowest order in a rectangular waveguide

[3/k 0 For lowest order modeb/)_

Reference [1 } Present Method % Error

0.2 0.48 0.462 3.75

0.3 1.00 1.01 1.00

0.4 1.18 1.18 0.00

0.5 1.26 1.28 1.59

0.6 1.30 1.36 4.62

3.2 Rectangular waveguide with wall distortion:

In an inflatable rectangular waveguide, distortion in the walls may be of type shown in

Figure 3. In this section, effect of each type of distortion on the propagation constant is

numerically studied. It should be noted that while analyzing the effects of distortion, the

perimeter of distorted waveguide remains the same as that of undistorted waveguide. This is due

to inelastic characteristics of the material used for the inflatable waveguide. In the present code,

under the constant perimeter constrain effect of each type of distortion, the propagation constant

is numerically studied.

3.2.1 Inclined walls in y-direction:

A rectangular waveguide with inclined walls in y-direction is shown in Figure 7. The

16

dispersioncharacteristics_51isfo,.fe,tof anL-bandrectangularwaveguidewith dimension

16.5x 8.26 cm and walls in the y-direction inclined at 0 are calculated using the present code.

If _Sn,tisfo,.fe,t is the dispersion characteristics of undistorted L band rectangular waveguide, the

percentage change in the dispersion characteristics of distorted waveguide is given by

_5_isf°"f_ - _5"n_isf°"f_ 100 (29)percentage Change in I_ = _undistorted

The percentage change in the dispersion characteristics using (29) is then calculated and

presented in Figure 8 for various values of 0. From Figure 8 it may be concluded that there is not

a significant effect of the distortion shown in Figure 7 on the propagation characteristics. Figure 9

shows the electric field pattern in the cross section of the rectangular waveguide. The arrow

direction gives the direction of electric field and the length of arrows show the magnitude of the

electric field.

3.2.2 Inclined walls in x-direction:

A rectangular waveguide with inclined walls in the x-direction is shown in Figure 10. The

dispersion characteristics _distorted of an L-band rectangular waveguide with dimension

16.5 x 8.26 cm and walls in the x-direction inclined at 0 are calculated using the present code.

The percentage change in the dispersion characteristics using (29) is then calculated and pre-

sented in Figure 11 for various values of 0. Figure 12 shows the electric field pattern in the cross

section of the rectangular waveguide. The arrow direction gives the direction of electric field and

the length of arrows shows the magnitude of the electric field.

3.2.3 Rectangular waveguide with curved walls:

Rectangular waveguides with curved walls may take shapes as shown in Figures 13, 16,

19, 22, 25, and 28. These waveguide shapes are modelled using GEOSTAR, and the percentage

17

changein thedispersioncharacteristicscalculatedusingequation(29)is presentedin Figures14,

17,20,23,26,and29. Correspondingelectricfield plotsfor thesegeometriesareshownin

Figures15,18,21,24,27,and30. FromFigures14,17,and20 it maybeconcludedthat

distortionsof formsgivenin Figures16and19causemorechangesin propagationconstantthan

thedistortionshowninFigure13. Similarconclusionmaybedrawnfrom Figures23,26,and29.

Thedistortionof formsgivenin Figures25 and28causemorechangesin thepropagation

constantthanthedistortionshownin Figure23.

3.2.4Rectangularwaveguidewith randomly distorted walls:

A rectangular waveguide with distorted walls is shown in Figure 31. The randomly

distorted rectangular cross section shown in Figure 31 is obtained using the following procedure.

Random distortion in the walls is obtained by using a random number satisfying Gaussian

2distribution with varience c_ = 0.2 and zero mean value. Using the tolerance of _+0.2 and

2variance c_ = 0.2, random numbers satisfying the Gaussian distribution are generated. A

randomly distorted cross section of L-band rectangular waveguide as shown in Figure 31 is then

obtained by displacing the boundary nodes of undistorted L-band rectangular waveguide using

these random numbers. The percentage change in the dispersion characteristics using (29) is

2then calculated for c_ = 0.2 for the tolerance of 0.2. In order to determine the true statistical

2nature, 50 runs were performed for c_ = 0.2 and tolerance equal to 0.2 and the percentage

change in the dispersion characteristics for each case are presented in Figure 32. From these

results, mean and standard deviation values for the [_ are calculated and presented in Figure 33.

2Figures 34 and 35 show results of similar run for c_ = 0.2 and the tolerance equal to _+0.1.

18

4.0 Conclusion

Simple formulas are developed to show dependence of slot array performance on the

dominant mode propagation constant of the rectangular waveguide feeding the slot array. Using

the Finite Element Method it has been shown how various types of mechanical deformation can

alter the propagation constant and hence the array performance. The variety of deformation/

distortions that might be present in an inflatable rectangular waveguide are analyzed and their

effects on the dominant mode propagation constant are numerically studied. The study will help

in determining allowable dimensional tolerances in an inflatable rectangular waveguide to be

used in the space antennas.

Appendix A

A.1 Derivation of Nodal Basis Function:

Consider a triangle as shown in Figure 5 where ezl , ez2 , ez3

ponent of electric field at the three nodes reduplicative. Assuming linear variation

gle, E Z (x, y) can be written as

are the amplitudes of z-com-

over the trian-

E (x,y) = aa+bbx+ccy (30)

The constants aa, bb, cc can be determined from

[: il eli I]fa!] lXlylzbb = x2 Y2 (31)

c x 3 y

Substituting (31) into (30)

where

and rearranging the terms, (30) can be written as

3

E (x,y) = ___eio_ i(x,y) (32)

i 1

19

_i(x,Y)

1_i (x, y) = _-_ (a i + bix + ciY ) with i = 1, 2, 3

a i = XjYk--Xky j

bi = Yj-Yk

c i = Xk--X j

1X 1 YlA 1 x 2 Y2

1 x 3 Y3

given in (33) is the required nodal basis function.

(33)

(34)

(35)

(36)

(37)

A.2 Derivation of Vector Edge Basis Function:

From the current basis functions given in [10]

between nodes 2 and 3 (see 5) can be written as

the vector edge function for the edge

---> LI^

W1 = _-_z× (._(x-x1) +Y(Y-Yl)) (38)

---)The vector edge function defined in (38) satisfies the condition Vf • W1 = 0 ; and if 71 is the unit

vector along the #1 edge, then ?1 • W1 = 1 . The edge vector functions in general can be written

as

L.

Wi = 2A 2 x (2 (x-xi) + y (y -Yi) ) (39)

..__)Wi given in (39) is the required vector edge basis function.

Appendix B

B.1 Expressions for Matrix Elements:

Using the basis function given in (33) and (39) and using expressions (21)-(28), the matrix

elements of matrix equations (19) and (20) can be written as

20

2

1 LmLm' ko8rSel(m',m) - f f((x-x,n)(x-x,n,)+(y-y,n)(y-ym,))dxdy (40)

_r A 4A 2 triangle ....

Re/(m,,m ) _ 1 f f ( (x-xm) (x-xm') + (Y-Ym) (Y-Ym') )dxdy (41)

Qe/(m', i) (42)

(43)

_r4A2triangle

- 4A 2 f f (ci(x-xm,)-bi(y-ym,) )dxdy_t r triangle

" f f(ai+bix+ciy)(ai'+bi'x+ci'y))dxdyPel (1, i) = 1 (bi,bi + cici,) 4A2triang le

1 LmLm'

_tr A

1Eel (i', i) - (bi, b i + cici, )

_tr4A

E

r 2 f f((x-x,n)(x-x,_,')+(Y-Y,n)(Y-Ym'))dxdy4A triangle

Ye/(i''i) _ e,. f f(ai+bix+ciY)(ai,+bi,x+ci,Y))dxdy

4A2triangle

Ue/(m', m) (44)

(45)

Xe/(m', m) - (46)

(47)

Using 13 point integration formulas given in [ 11], the integration over triangle appearing in

(40)-(47) are evaluated.

References[1] R. Freeland, et al., "Inflatable antenna technology with preliminary shuttle experiment

results and potential applications, "Eighteenth Annual Meeting & Symposium Antenna

Measurement Techniques Association, pp. 3-8, Sept. 30-Oct. 3, 1996, Seattle, Washington.

[2] R.E. Freeland, et al., " Development of flight hardware for large, inflatable-deployable

antenna experiment, "IAF Paper 95-1.5.01, presented at the 46th Congress of the

International Astronautical Federation, Oslo, Norway, October 2-6, 1995.

21

[3] R.E. Freeland, et al., " Validation of a unique concept for a low-cost, light weight space-

deployable antenna structure, "IAF Paper 93-I. 1.204, Presented at the 44th Congress of

the International Astronautical Federation, Graz, Austria, October 16, 1993.

[4] R.S. Elliott, "Antenna Theory and Design, "Prentice Hall, Inc., 1981.

[5] H. Jasik, "Antenna Engineering Handbook, "McGraw-Hill Book Co., Inc., 1961.

[6] R.E. Collins, "Field Theory of Guided Waves, "McGraw Hill Book Company, 1961.

[7] R.F. Harrington, "Time-Harmonic Electromagnetic Field, "McGraw-Hill Book Co., Inc.,

1961.

[8] J. Jin, "The finite element method in electromagnetics, " John Wiley & Sons, Inc., New

York, 1993.

[9] C.J. Reddy, et al., "Finite element method for eigenvalue problem in electromagnetics, "

NASA Technical Paper 3485, December 1994.

[10] S.M. Rao, D. R. Wilton, and A. W. Glisson, "Electromagnetic scattering by surfaces of

arbitrary shape, "IEEE Trans. on Antennas and Propagation, Vol. AP-30, pp. 409-418,

March 1982.

[11] J.N. Reddy, "An introduction to the finite element method, "McGraw-Hill Book Company,

1984.

22

_3(D

_0(D

-e-

90

85

80

75

x<..-.....

\ - .. --.'..

frequency = 1.12 GHz _" ..'"-." \1.16 GHz _'-- .'"-..'\

......... 1.20 GHz "_"...'-...\\1.24 GHz _ ".. "-. "

70 , i i i I i i i i I i i i i I

0 5 10 15

Figure 1 Plot of main beam direction _) as function of6o

23

frequency = 1.12 GHz

1.16 GHz1.20 GHz

1.24 GHz

0 i i i i I i i i i I i i i i I

0 5 10 15

Figure 2 Plot of resonant slot conductance G O as a function of [_o "

24

Y

Xy

(a) Undistorted Rectangularwaveguide

Y

_X

(b) Rectangular waveguide withx-wall inclined

(c) Rectangular waveguide withy-walls inclined

(d) Rectangular waveguide withcurved walls

Y

X

(e) Rectangular waveguide withwrinkles in walls

Figure 3 Geometry of few cross sections of deformed rectangular waveguide.

25

_J

J

./J

./

Figure 4 Geometry of cross section of a rectangular waveguide with distortedwalls.

26

ezl Node 2 (X 2, Y2) Edge #3

Edge #1 _ -_ et3/

ezz/__O.....1...._et2/ ( 1, Yl)

Node 3 Edge #2(x3, Y3)

Figure 5 Geometry of a triangular element.

b

Y

X

y

a

Figure 6 Geometry of inhomogeneous rectangular waveguide with

b_ = 0.45 and d_ = 0.5.a b

27

16.5 cm

Figure 7 Geometry of L-band rectangular waveguide with inclined walls in y-direction.

28

t9 = 5 Degrees

19 = 10

.......... 19 = 15

............. 19 = 20

19 = 25

"\\

\\

\\

iIi ii iIi iii Iii ii Iii ii Iii ii I iii iI

0.24 0.26 0.28 0.30 0.32 0.34 0.36

Figure 8 Plot of percentage change in dispersion characteristics of L-band rectangular

waveguide for various inclination, 19 , in y-direction.

29

Figure9 Electricfield in thecrosssectionof distortedL-bandrectangularwaveguideshownin Figure7.

30

4

16.5 cm

\,

0

-5 0 5

8.26

Figure 10 Geometry of L-band rectangular waveguide with inclined walls in x-direction.

31

m

4

3

o_,-_

(D

2(D

(D

(D

1

0

-1

0 = 5 Degrees

0 = 10

.......... 0=15

0 = 20

........... 0=25

"\

\

•°_

_°

°

.............................................................-_.-_._-._.-_.-_.-.-._.-_.-_._-._

iIi iii Iii ii Iii ii Iii iiI iii iIi ii iI

0.24 0.26 0.28 0.30 0.32 0.34 0.36

ko

Figure 11 Plot of percentage change in dispersion characteristics of L-band rectangularwaveguide for various inclination with respect to x-axis.

32

Figure 12 Electric field in the cross section of distorted L-band rectangular waveguideshown in Figure 8 (frequency = 1.4 GHz).

33

_T

,d 300cm

o

h = 16.5 cm

•9 15.4 cm

//

\

\J

/

/

\/

\

Figure 13 Geometry of L-band rectangular waveguide with distortion in x-walls.

34

20-

15

(D

Z=

(D

10(D

(D

\\

\\

d = 1.Ocm

d = 2.0cm

.......... d = 3.0cm

d = 4.0cm

\\

\\

\\

\

_°

°_o

r--r-_ i i i I i i i i I i i i i I i i i i I i i i i I i i i i I

0.240 0.250 0.260 0.270 0.280 0.290 0.300

k o

Figure 14 Plot of percentage change in dispersion characteristics of L-band rectangularwaveguide with distortion as shown in Figure 13.

35

\\

%

,\\\ll

, \ \ 1\

\ \

IIi

T

1

T

1 T t !

!!

I

II '1

II i/

lltl/

PII/_lilz,ilz,

! /

\ I

% !

l

/

7

Figure 15 Electric field in the cross section of distorted L-band rectangular waveguide

shown in Figure 13 (frequency = 1.4 GHz).

36

d=3.tm

8.26 u:m \

¢..-

15.4 cm


37

i

-5

_z-.10o_,-_

hi3

Z=

-15

¢3

-20

-25

-30

-35

-400.24

/

/I

/

I'

/

I"

/

/.

/,

f

/,

_ _ _ I111 I 11s

/ I"

/ i. t"

/I I*/ "

/ io I

/ / /.

x / f*

/ ,/"

/ .//

J z",/ 7

/

/ // /

/ // d = l.Ocm

//

/ / d = 2.0cm/ /'

d = 3.0cm/// ...........

/ d = 4.0cm/

i i i i Ii ii i I i i i i I i i ii Ii i i i I i i i i I

0.26 0.28 0.30 0.32 0.34 0.36

k o

Figure 17 Plot of percentage change in dispersion characteristics of L-band rectangularwaveguide with distortion as shown in Figure 16.

38

\

\

\

\

I

l!

\ \

i 11 l

Ji l

l/ l

/1 l/

\

\

j ll

Ii

1 1 1

JJJ1

I I

JJl

1, \

/

/ / "

l 1 "

l l '

1 l

1

\\

\x



39

8. 6c

15.4 cm


40

0

d = 1.Ocm

d = 2.0cm

.......... d = 3.0cm

............. d = 4.0cm

'\

'\

\

\

illlllllllllllllllllllllllllllll

0.24 0.26 0.28 0.30 0.32 0.34 0.36

k o


waveguide with distortion as shown in Figure 19.

41

' 1

1 \

1 1

\

l 1 1 '

1 11 l

!\ !



42

Arc length8.26cm

_c_ /\

\/\ //\/\\\

/\////////\\

\/_\\ \

16.5 cm

Figure 22 Geometry of L-band rectangular waveguide with distortion in y-walls.

43

2,0 -

1.5

1.0

et_

'05

z=C)(D

0.0(D

-0.5

d = 0.5cm

d = 1.Ocm

d = 1.5cm

d = 2.0cm

d = 2.5cm

.1.0 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

0.24 0.26 0.28 0.30 0.32 0.34 0.36

k o



44

4

L|



45

Arc Length = 8.26 cm1.5cm

1

\\\\\\\\/\\ //_

16.5 cm


46

20-

\

\

\

'\

15

\

(D

_10

(D

(D

(D

5

0

\

\

'\

\\

\

\\

\

d = 0.5cm

d = 1.Ocm

\

\\•

\\

\\,

\\

.......... d = 1.5cm

"\

\

\

.......... d = 2.0cm

d = 2.5cm

\,

\\

\\

\\

\

\\

illlllllllllllllllllllllllllllll

0.24 0.26 0.28 0.30 0.32 0.34 0.36

ko



47

l

1

/

' I I

l 1

l l!

1

L



48

Arc Length 8.26 cm

//\\//\\\\\_

\\\_\\)\//"7.7cm /

_// // ///_ 1.5cm/

16.5 cm


49

i

-2O

_-40

(D

Z=

(D

-60r_¢)

-8O

-100

/

/

//

d = 0.5cm

d = 1.0cm

d = 1.5cm

d = 2.0cm

d = 2.5cm

/i lllllllllllllllllllllllllllllll

0.24 0.26 0.28 0.30 0.32 0.34 0.36

k o



50

r l

, I I

T T l

T, t l l l T T l ,

T

T

T

T

T

T

T

l ,



51

I

L

1 11

1 1

r I l

l l

' I

J

J

Figure 31 Geometries of rectangular waveguides with random distortion in wall2

boundaries ( c_ = 0.2 and tolerance = +/-0.2 )(cont.).

52

' 1

J I

J

J

J L-"_

J ,\

J

j ,j L

L

Figure 31 Geometries of rectangular waveguides with random distortion in wall2

boundaries ( c_ = 0.2 and tolerance = +/-0.2 ) (completed).

53

-1

-2

-3

_-4

(D

-5z=c)(D

-6(D

7I-8 I,,,,I,,,, I,,,,I,,,,I,,,,I,,

0.24 0.26 0.28 0.30 0.32 0.34

k o

Figure 32 Plot of percentage change in dispersion characteristics of L-band rectangular2

waveguide for c_ = 0.2 and tolerance = +0.2.

54

0

-2

-4

o_,-_

C)

_8

-10

- -O ...... • ...... _ .....

J/

/

//

/

0

Mean Value

Mean + standard deviation

Mean - standard deviation

lilll lil ill Bill lUll ilill ilil

0.24 0.26 0.28 0.30 0.32 0.34

k o


waveguide, mean value, and standard deviation calculated fromFigure 31.

55

0

k o

Figure 34 Plot of percentage change in dispersion characteristics of L-band rectangular2

waveguide for c_ = 0.2 and tolerance = +0.1.

56

-2

= -4

eao

c)

eao

6

-8

-10

..... -O--..... O- O- • -0- O, O

Mean Value

- - O- - Mean + standard deviation

_3 Mean - standard deviation

illllllllllllllllllllllllllll

0.24 0.26 0.28 0.30 0.32 0.34

k o


waveguide, mean value, and standard deviation calculated fromFigure 34.

57

Form ApprovedREPORT DOCUMENTATION PAGE OMBNo. 0704-0188

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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE | 3. REPORT TYPE AND DATES COVERED

Februar_ 1998 I Contractor Report4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Application of Finite Element Method to Analyze Inflatable WaveguideStructures NAS 1-19341

6. AUTHOR(S)

M. D. Deshpande

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Langley Research Center

Hampton, VA 23681-2199

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

522-11-41-02

8. PERFORMING ORGANIZATIONREPORT NUMBER

10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

NASA/CR- 1998-206928

11. SUPPLEMENTARY NOTES

Langley Technical Monitor: M. C. Bailey

12a. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified-Unlimited

Subject Category 17 Distribution: Nonstandard

Availability: NASA CASI (301) 621-0390

12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

A Finite Element Method (FEM) is presented to determine propagation characteristics of deformed inflatable

rectangular waveguide. Various deformations that might be present in an inflatable waveguide are analyzed using

the FEM. The FEM procedure and the code developed here are so general that they can be used for any other

deformations that are not considered in this report. The code is validated by applying the present code to

rectangular waveguide without any deformations and comparing the numerical results with earlier published

results. The effect of the deformation in an inflatable waveguide on the radiation pattern of linear rectangular slot

array is also studied.

14. SUBJECT TERMS

Inflatable Waveguides, Finite Element Method, Waveguide Discontinuity

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OF REPORT

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OF ABSTRACT

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