N91-19306 Task Title A Finite Element Conjugate Gradient FFT Method for Scattering Investigators J.D. Collins, Dan Ross, J.M.-Jin, A. Chatterjee and J.L. Volakis Period Cover¢_l Sept. 1990 - Feb. 1991 2 https://ntrs.nasa.gov/search.jsp?R=19910009993 2018-07-17T07:27:09+00:00Z
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N91-19306
Task Title A Finite Element ConjugateGradient FFT Method forScattering
Investigators J.D. Collins, Dan Ross,J.M.-Jin, A. Chatterjee andJ.L. Volakis
The corresponding plot of the reflection coefficient as a function of incidence is given in
figure 7 along with scattering patterns based on the proposed termination model. As seen,
for the chosen fictitious absorber the reflection coefficient is less than one percent for 0 up
to 62 degrees and less than 2 percent for 0 up to 77 degrees. For the same error criteria,
the corresponding angles associated with the second order Pad6 ABC are 35 and 41
degrees, respectively. The fictitious ABC has, therefore, a substantially better performance
over the existing ABCs, and its effectiveness will be examined further in the next few
months.
The three dimensional finite dement meshes required in the analysis will be
generated by SDRC IDEAS and we have already began to develop the software for
transforming the output of this commercial package to the input files of our analysis codes.
Similar drivers were already developed for the two dimensional code which was developed
last year.
Finally, during this period we performed extensive testing of the two-dimensional
code and have in the progress developed several new pre- and post- processing algorithms
for this code. Two of the new geometries (see fig. 8 and 11) whose scattering was
computed with our 2D finite element - CGFFT code are displayed in figures 9, 11 and 12.
These represent airfoil configurations, one of which is coated with a dielectric material.
CONCLUSIONS
The project continues to evolve in accordance with our original plan and schedule.
Most importantly, so far, our expectation of the finite element CGFFT formulation have
been realized and we are, therefore, pleased with its performance for the intended
applications.
TRANSITIONS
All of our efforts in the next six months will be devoted to the development of the
3D finite element boundary integral code for arbitrary structures. In the immediate future
we will also pursue improvements for our existing codes primarily directed at speeding the
convergence of the CG or BiCG algorithm.
6
\i i
_,_ °_
_ab_j_
_._
-- N
_ut
°la
o_mq
_ O
_ c_P
0_
7
<I
I
I
>>I
f
×
J
<<</
. i
7%.
i
15.0
10.0
5.0
0.0
-5.0
-10.0
-15.0
Conducting Sphere (p=0.5_,)..... I .... I .... I ..... I ..... I
i i , , . I ..... I .... I ..... I ..... I .....
0.0 30.0 60.0 90.0 120.0 150.0 180.0
0s [deg] (_s----O,0i--0 )
Figure 1: TM and TE bistatic scattering
pattern from a perfectly conducting sphere ofradius for axial incidence.
FFJBE (TE)
e CICERO (TE)
.........FE/BE (TM)
CICERO (TM)
Z
a_
I0.0
0.0
-I0.0
-20.0
-30.0
-40.0
-50.0
p---0.1_., I-I.0Xpc cylinder(m=0"-2)
..... I ..... I ..... I " " "" '" I ..... ! .... 't
t
.... , I ..... [ ..... I ..... I . , , . . I ....
0.0 30.0 60.0 90,0 120.0 150,0
0, [deg] (@,=0,0_=90 )
tSO.O
I _r__E CI'E)
® CICERO CI_)
.........FE/BIZ('rM)
m CICERO ('rld)
w
0.0
-I0.0
-20.0
-30.0
-40.0
p=0.IX,I=I.0_.pc cylinder..... I ..... I ..... I ..... I ..... I .....
,y t" ' \' /
. I ..... I .... , I ..... I ..... I .....a n " "
0.0 30.0 60.0 90.0 120.0 150.0 lgO,O
e,[deg](#,=o,0_=o)
• CICERO crE)
......... F_E CrM)
® CICE,RO (TM)
Figure 2: TM and TE bistatic scattering pattern from a perfectly conducting circular
cylinder of length lZ, and radius 0.1k for axial incidence. (a) modes 0-2, (b) converged
9
-- Z
¢q
"6
0.0
-10.0
-20.0
-30.0
-40.0
-50.0
p=.0889,., 1=1.0_ pc Ogive• ' " ' ' i ' ' ' ' ' I ..... I " ,,.,l°. '''l''''
t
..... ,t_...i ..... i ..... i ..... i .....
0.0 30.0 60.0 90.0 120.0 150.0 180.0
0 s [deg] (_,=0, 0i=0 )FE/BE El'E)
® CICERO (TIE)
......... FE/BE (TM)
m CICERO (TM)
Figure 3: TM and TE bistatic scattering pattern from a perfectly
conducting ogive length 13. and maximum radius of 0.0883. for axial incidence.
10
FSS ARRAY SCAT-I"ERING
Y
12 x 12 FSS Array
f=24 GHz
_inc = 45 °
28,084
0inc = 75 °
Unknowns
I II I
I II I1.0cm
_X
f
40.
10.
-20.
_0 il90. 60.
¢=45 ° O (degrees) ¢=2250
90.
50.
m 20.
_ -10.
-40.
90.
'.i , _ i " _':,,'_ " ," i _.,
60. 30. 0. 30. 60. 90.
¢=45 ° 0 (degrees) ¢=225 °
Figure 4: Scattering by a 12 x 12 FSS array; comparison of the exact solution(solid line) with an approximate result obtained by truncating the infinite FSS.
qq
FSS ARRAY SCATTERING
24 x 24 FSS Array
0o
0o
20.
w
O.
90. 60.
_=0 ° 0 (degrees) _=180 °
f=24 GHz
oinc = 0 0
123,504
0inc = 45 °
Unknowns
60. 90.
30.
20.
10.
olJl
i'i
I, :.",
! \ "
• • em_.oe Q " oeoeww o teoB._o=.oooe.oo
-- TMincidence JTE iae_e
O. I I I i I I
0. 10. 20. 30. 40. 50. 60. 70.
Number of Iterations
Figure 5: Scattering by a 24 x 24 array; comparison of the exact solution(solid line) with an approximate result obtained by truncating the infinite FSS.
t2
LARGE PLATE SCATTERING
100.
Principal Plane Cut
10Lx 10Xplate, TM incidence, B iCG-FFF
,wml
O
E
Z
80.
60.
40.
20.
0_
1Q ®
®
100xl00 cells
120x120 cells
[ I I t I
10. 20. 30. 40. 50. 60. 70. 80.
0 (degrees)
90.
60.
10kxl0X plate, TM incidence, BiCG-FFF
50.
40.
30.
20.
10.
*
I J
100xl00 cells
o 120x120 cells
I I I I ....
0. 10. 20. 30. 40. 50. 60. 70. 80. 90.
0 (degrees)
Figure 6: Principal plane TM Scattering by a 10_. x 10_. rectangular plate.
13
Y
___FictitiousAbsorber
O IL X
(a)
IRI1.00
0.80
0.60
0.40
0.20
0.00
0.
! !
i
I0. 20. 30. 40. 50. 60. 70. 80. 90.
e (deles)
2.00
1.50
l._
O.50
0._
I
......... \.
O. 30. 60. 90. 120. 150. 180.
n-, (degrees)
(b)
0.75
0.50
0.25
0._
O. 30. 60. 90. 120. 150. 180.
x_ (degrees)
(c) (d)
Figure 7: Evaluation of the new fictitious ABC (a) geometry (b) reflection coefficient
plot (c) Hz field on a PEC Circular Cylinder (d) Ez field at a distance _10 from thesurface of a PEC Circular Cylinder.
"14
case 2: Coated Trailing Edge
Yin =t_+14 1-(x/2.5) 2
_+0.8232 A(x)
-2.5<x<0
0__x<2.5
A(x) - 5/(1-(x/2.7182)2) - 0.3926
Yout = +0.8116 B(x) 0. l_<_x<3.0
Yout=Yin elsewhere
B(x) - 5/ 1-[(x-. 1)/3.1416] 2 -.3846
13r=2- j 1 between Yin and Yout
FEM Mesh
Figure 8: Geometry and finite element mesh of the illustrated coated trailing edge
"15
¢..)
Case 2: Coated trailing edge, _'=2-jl), H-pol20.0
FEM
......... MoM
30.0 60.0 90.0 120.0 150.0 180.0
Angle [deg]
Case 2: Coated trailing edge, _'=2-jl), E-pol20.0
t-
FEM
......... MoM
30.0 60.0 90.0 120.0 150.0 180.0
Angle [deg]
Figure 9: E and H polarization scattering patterns for the configuration shown in figure8. Comparison of results between the FE-CGFFF and the moment method.
16
Perfectly conducting airfoil
All dimensions in wavelenlFhs. The _foil section is made by 5 _cs:
• OA : strLil_ht line
• AB : circle of radius _ = 7_o and of center 02
• BC : polynomia/parametric equation
• CD : polynomial parametric equation
• DO : circle of radius R1 = 9,Xo aad of center O1
Figure 10: Geometry of a PEC Airfoil whose scattering is given in figures l I & 12.
17
O
t_
¢9
20.0
10.0
0.0
-30.0
-40.0-180.0
E-POL FE-CGFFT
UNCOATED
........ I ........ I........ I ........ I ........ I,, i i ....
-120.0 -60.0 0.0 60.0 120.0 180.0
Angle [deg]
Figure 11: E-polarization echowidth for the airfoil given in figure I0.
"18
0
r_
(,,)
H-POL FE-CGFFT
20.0
10.0
0.0
-30.0
........ I ........ I ........ I ........ I ........ I ........
i
UNCOATED
........ I ........ I........ I ....... ,1, ,, ,,,, ,I ........
-120.0 -60.0 0.0 60.0 120.0 180.0
Angle [deg]
Figure 12: H-polarization echowidth for the airfoil given in figure 10.
19
Appendix A
Finite element formulation for tetrahedralelements and edge-based expansion basis
1 Derivation of finite element equations
Let us consider a three dimensional inhomogeneous body occupying the
volume V. In order to discretize the electric field E inside the body, we
subdivide the volume V into a number of small tetrahedra, each occupying
volume V_(e = 1,2,..., M) with M being the total number of tetrahedraI
elements. Within each tetrahedron, the electric field satisfies the vector
wave equation
iv =V x x E - ko2e,E 0 (1)#,
where p, is the permeability of the medium, e, is the medium permittivity
and ko is the free space wave number. The next step is to expand the
electric field within V, as
6
E = _ E;W; (2)j=l
where W_ are edge-based vector basis functions and E; denote the
expansion coefficients of the basis, all defined within the volume V,. W_ is
tangential to the jth edge of the eth tetrahedron with zero tangential
component along the other edges of the tetrahedral element. On
substituting (2) into (1), we obtain
Z, E; v x x w_ - ko_,w, = o (3)jml _r
In order to solve for the unknown expansion coefficients E_, we take the dot
product of (3) with W_ and then integrate the resulting equation over the
element volume V, (Galerkin's technique). The wave equation thus reducesto
2O
2:)E_ W, . V x × W_- koe,Ws dv = O (4)j=l #r
The first term in the integral of the above expression can be simplified by
using basic vector identities. Since
w_. v x x w; = v. (v x wj) x w_ + _(v x w_). (v x w;)P*
the divergence theorem can be readily applied to (4) resulting in the
following expression:
0 = _E; (V x WT). (V x W;)- koe,Wi.Wj dvj=l
(_)
where Se denotes the surface enclosing V_. Using vector identities , (5) can
be further simplified to yield the weak form of Maxwell's equation:
,., e 2 _ _ W_.(n × H)dss t (v × w_). (v x wj) - ko_W,.Wj dv = S_oj=l
(6)
where n × H is the tangential magnetic field on the exterior dielectric
surface. Equation (6) can be conveniently written in matrix form as
[A _][E "] = [B "] (7)
where
Ai_ = (V × W_). (V x W;)- koe,.Wi.W.i
B_ = 3W#O /s W_.(n x n)ds (9)e
On assembling all the M tetrahedral elements that make up the geometry,
we obtain a system of equations whose solution yields the field components
over the entire body. Therefore, summing over all M elements, we have
21
which gives
M M
[A e][E e] = _[B e] (10)e=l e=l
[A][E] = [B] (11)
where [A] is a N x N matrix with N being the total number of edges
resulting from the subdivision of the body and [El is a N x 1 column vector
denoting the edge fields. Due to the continuity of the tangential component
of the magnetic field at the interface between two dielectrics, an element
face lying inside the body does not contribute to [B] since the surface
integrals over the faces of adjacent tetrahedra cancel each other. As a result,
[B] is a column vector containing the tangential magnetic field only over
the exterior surface of the body. Equation (11) can therefore be written as
As,Ea+ AaiEi = Ha
AiaEs + AiiEi = 0 (12)
where the subscript s denotes the edges on the surface and i represents the
edges inside the body. It is thus readily seen that (11) relates the electric
field inside and on the surface of the body to the on-surface tangential
magnetic field.
2 Basis functions
Vector fields within tetrahedral domains in three dimensional space can be
conveniently represented by expansion functions that are linear in the
spatial variables and have either zero divergence or zero curl. The basis
functions defined below are associated with the six edges of the tetrahedron
and have zero divergence and constant curt. Assuming the four nodes and
the six edges of a tetrahedron are numbered according to Table 1, the
vector basis functions associated with the (7 - i)th edge of the tetrahedron
are defined as
f7-i + gT-i × r, r in the tetrahedron (13)WT-i = O, otherwise
where i = 1,2,... ,6 and f and g are constant vectors. On direct
evaluation, it is readily seen that
22
V'Wi -- 0
V x Wi = 2gi
(14)
(15)
Since the complex scalar Ej in (2) is the projection of the electric field
onto the jth edge of the tetrahedral element,
Wi.ej [r on jth _ag. = 6ij (16)
where 6ij is the Kronecker delta. Solving (13) and (16) for the unknown
vectors yield[l]
br-i
fT-i -- 6vrq X ri2
b_bT-ieig7-i --
6V
(17)
(18)
where V is the volume of the tetrahedral dement, ei = (ri= - ril)/b_ is the
unit vector of the ith edge and bi = ]ri2 - ri, [ is the length of the ith edge.
All distances are measured with respect to the origin.
Since there are two numbering systems, local and global, a unique global
direction is defined (e.g., always pointing from the smaller node number to
the larger node number) to ensure the continuity of n × E across all edges.
This implies that (13) should be multiplied by (-1) if the local edge vector
(as defined in Table 1) does not have the same direction as the global edge
direction. Even though Wi forces no conditions on the normal component
of E, it has been shown[2] that the continuity of electric flux can be satisfied
within the degree of approximation with the above formulation. Finally,
since V- Wi = 0 the electric field obtained through (2) exactly satisfies the
divergence equation within the element, i.e.V. E = 0. Therefore, the finite
element solution is free from contamination of spurious solutions[2].
3 Mesh termination
Differential equation methods, such as finite elements, can only solve
boundary value problems. Since electromagnetic problems are open
boundary-infinite domain types, a means to truncate the solution domain
to lie within a finite boundary must be found. On this boundary, a
condition is enforced thus ensuring that the fields will obey the Sommerfeld
23
radiation condition at distancesasymptotically far from the object. Theseabsorbingboundary conditions (ABCs) have a significant advantageoverthe global methods of solving unboundedproblemsusingfinite elementsinthat they are local in nature. Due to this, the sparsematrix structure ofthe finite element formulation is retained. One disadvantage,however,isthat ABCs are approximate and do not model the exterior field exactly.The objective of absorbingboundary conditions is to truncate the finiteelementmeshwith boundary conditions that causeminimum reflectionsofan outgoing wave.TheseABCs shouldprovide small, acceptableerrorswhile minimising the distancefrom the object of interest to the outerboundary. This minimal distanceis required to reducethe number ofunknownsin the problem for computational efficiency.A three dimensionalvector boundary condition will be investigatedherefor terminating thefinite elementmeshof the body describedin section 1.1. We begin with theWilcox representation[3]of the electric field which hasan expansion
E(r) =-- e-Jkroo A.(e, ¢) (19)r n=0 rn
From (19), we get
{ I+D1} e-Jkr_-._nAn,V×E= jk_x-- E (20)r r 2 r n
n=l
where Ant = t x A,_ is the transverse component of An and, for a vector F,