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JAERI-Data/Code--96-031 JP9702007
JAERI-Data/Code96-031
Japan Atomic Energy Research Institute
28 ^ 7 t
(T319-H
This report is issued irregularly.
Inquiries about availability of the reports should be addressed to Research Information
Division, Department of Intellectual Resources, Japan Atomic Energy Research Institute,
Tokai-mura, Naka-gun, Ibaraki-ken, 319-H, Japan.
© Japan Atomic Energy Research Institute, 1996
JAERI-Data/Code 96-031
F D V I ^ u—s 3
>?-
(1996^10/3 1
Program Multiple
1/—V 3 y&f
: xl53 m« - 2 - 5 4
i
JAERI-Data/Code 96-031
A Conceptual Design of Multidisciplinary-Integrated C. F. D. Simulation
on Parallel Computers
Ryoichi ONISHI, Takashi OHTA and Toshiya KIMURA
Center for Promotion of Computational Science and Engineering
Japan Atomic Energy Research InstituteNakameguro, Meguro-ku, Tokyo
(Received October 1, 1996)
A design of a parallel aeroelastic code for aircraft integrated simulations is
conducted. The method for integrating aerodynamics and structural dynamics software on
parallel computers is devised by using the Euler/Navier-Stokes equations coupled with
wing-box finite element structures.
A synthesis of modern aircraft requires the optimizations of aerodynamics, structures,
controls, operabilities, or other design disciplines, and the R&D efforts to implement
Multidisciplinary Design Optimization environments using high performance computers are
made especially among the U.S. aerospace industries.
This report describes a Multiple Program Multiple Data(MPMD) parallelization of
aerodynamics and structural dynamics codes with a dynamic deformation grid. A three-
dimensional computation of a flowfield with dynamic deformation caused by a structural
deformation is performed, and a pressure data calculated is used for a computation of
the structural deformation which is imput again to a fluid dynamics code. This process
is repeated exchanging the computed data of pressures and deformations between flowfield
grids and structural elements. It enables to simulate the structure movements which
take into account of the interaction of fluid and structure.
The conceptual design for achieving the aforementioned various functions is reported.
Also the future extensions to incorporate control systems, which enable to simulate a
realistic aircraft configuration to be a major tool for Aircraft Integrated Simulation,
are investigated.
JAERI-Data/Code 96-031
Keywords: CFD, Aerodynamics, Structural Dynamics, Coupled Simulation, Grid Generation,
FDM, FEM, MDO, Design Optimization, Aircraft Design, Parallel Computer,
Parallel Processing, Super Computing
JAERI-Data/Code 96-031
2.2
2.3
2. 4
2. 5
2. 6
2. 7
3.
1. I? §fc 1
1. 1 # H 1
1.2 W&jl^ 2
##*«* 32. atefttfrF £*&&£* 4
8
10
11
14
15
16
17
a i mm^rn^ 173.2 tf-jjt#£ 20
3.3 m\iv&k 223.4 MPMDmmz^z&mfe^K&vzteTBj&tfammvffl 25m^Xffi 29
4. m&Mtir 30
4. 2 It^^fl i 31
4. 3 &m~? b l)9X<Dftf& 32
4. 4 H8^ f t# f c< fc *#P& • 34
4. 5 £&#©igf f l 34
4. 6 ^^J!ra^(C^>SiiJtt{gT©#Jt 35
4. 7 &?ij#yi0itff l 36
##^S^ 38
5. ^fiJMWS/XxAOfll^ 39
5. 1 Mfa m&V&fiHffiT 39
5. 2 W^©|o|3W 405. 3 WMt^-m 415. 4 -r-*<£Si ( >^-7i-X) 43
V
JAERI-Data/Code 96-031
5. 5 ^H®5j*jHb 44
##£&* 45
6. frJflH'>XxA£©3ffi£&& 46
m^JCffi 497. 3i£0ii*W 50
m n 50APPENDIX A 51
APPENDIX B 53
VI
JAERI-Data/Code 96-031
Contents
1. Introduction 1
1. 1 Background 1
1. 2 Overview of the Research • 2
References 3
2. Fluid Grid and Structural Element 4
2. 1 Overview 4
2. 2 Geometric Data Generation 5
2. 3 Fluid Grid Generation 8
2. 4 Structural Element Generation 10
2. 5 Data Exchange between Grid and Element 11
2. 6 Dynamic Deformation of Fluid Grid 14
2. 7 Parallelization of Grid and Element Generation 15
References 16
3. Fluid Dynamics Analysis 17
3. 1 Basic Equations 17
3.2 Numerical Methods 20
3. 3 Parallelization 22
3. 4 Example of MPMD Parallel Calculation in a Moving Grid 25
References 29
4. Structural Dynamics Analysis 30
4. 1 Overview 31
4. 2 Process of a Computation 31
4. 3 Coefficient Matrix Generation 32
4. 4 Solution of Direct Integration 34
4. 5 Application of Composite Materials 34
4. 6 Stiffness Decrease by Aerodynamic Heating 35
4. 7 Application of Parallel Processing 36
References 38
5. Design of the Coupled Simulation System 39
5. 1 Coupled Analysis of Fluid-structural Dynamics 39
5. 2 Synchronization of Time 405. 3 Parallelization Method •••. 41
5. 4 Data Transfer of Coupled System 43
vii
JAERI-Data/Code 96-031
5. 5 Visualization 44
References 45
6. Concept on an Integration with Control System 46
References 49
7. Developmental Works 50
Acknowledgements 50
Appendix A 51
Appendix B 53
viii
JAERI-Data/Code 96-031
l. J? 6RB
i\L (Multidisciplinary Design Optimization)
Wkt-'Jj^-ifi&ffi-ZWLttMLMikOffiyu&fT^s ASTROS(Automated STRuctural Optimization
System) *fti#), &m<DQJiW&^-KcDW^^^«:;fToTV^63)4)5)o ASTROS^,
—lc NASAO USSAERO t MSC/NASTRAN
Paragon^IBM
tco Computational Aero Science Project ^ HiSAIR(High Speed
Airframe Integration Research) ^ n i ^ x ^ hfi, Zr&Wxft&Oftgtftlteyv^^^ hX\ Qt}
4Crayt£H<DfimNr
1
JAERI-Data/Code 96-031
1.2
t, y-
i^~
(2) m
(3)
- 2 -
JAERI-Data/Code 96-031
(4)
(5) mmz
ft is,
1) Sobieszczanski-Sobieski,J. and Chopra,L, "Multidisciplinary Optimization of Aeronautical
Systems" ,J. of Aircraft,Vol. 27,NO. 12,1990,pp. 977-978
2) Sobieszczanski-Sobieski,J., "Sensitivity Analysis and Multidisciplinary Optimization for Air-
craft Design:Recent Advances and Results", J. of Aircraft,Vol. 27,NO.12,1990,pp. 993-1001
3) Neil,D.J.,Johnson,E.H., and Canfield,R., "ASTROS-A Multidisciplinary Automated Struc-
tural Design Tool", J. of Aircraft,Vol. 27,NO. 12,1990,pp.l021-1027
4) Dodd,A.J.,Kadrinka,K.E.,Loikkanen,M.J.,Rommel,B.A.,Sikes,G.D.,Strong,R.C.,and
Tzong,T.J.,"Aeroelastic Design Optimization Program" ,J. of Aircraft,Vol. 27, NO. 12,1990,
pp. 1028-1036
5) Miura,H. and Neil,D.J.,"Applications to Fixed-Wing Aircraft and Spacecraft/Structural
Optimization: Status and Promise" ,Progress in Astronautics and Aeronautics, Vol. 150,
AIAA,1992, pp. 705-742
6) Coen,P.G.,Sobieszczanski-Sobieski,J.,and
Dollyhigh,S.M., "Preliminary Results from the High-Speed Airframe Integration Research
Project", AIAA-92-1004,Irvine,California,February 1992.
7)
o
JAERI-Data/Code 96-03 I
Nomenclature
DuDn
Li, L2, L>3, L4
LUL2,L3,LA
n
iVJ
P
51,52,53,54
Coon's
2.
2.1
<o Fig.2.1
JAERI-Data/Code 96-031
ir MPMD(Multiple Program Multiple Data)
I I
Fig.2.1
2.2 Wfitk"r-
Fig.2.2
Fig.2.4
(Fig.2.3)
10-20
100
(Fig.2.4)
(2)
V
(3)
(Fig.2.5)
c
JAERI-Data/Code 96-031
# WING DESIGN PARAMETERS.# Canard Subsonic Ultra High Capacity Aircraft DR-1## Aircraft type# Section profiles at root, mean, and tip.# Root, Mid, & tip chord.(ft)# Whole , inboard,& outboard span.(ft)# Fuselage width.(ft) , tip twist angle (degree)# Root stagger, leading edge sweep, & dihedral angle.(degree)#uhcalsc20714 sc20714 64a01050.0 32.0 13.0105.0 40.0 65.022.0 -2.02.0 35.0 6.0
Fig.2.3
Fig.2.4
7 -
JAERI-Data/Code 96-031
2.3
Fig.2.5
< HifttH*« t
Fig.2.6
JAERI-Data/Code 96-031
(2)
(3)
(5)
- 9
JAERI-Data/Code 96-031
(2.1)
/4 =
Dn —
d\(Xn—X\)
dn(Xn —
...,n +1)
2.4
51t*>mM#*%i£*V,
Tabiei
20 g
Z-, Tablel
1 0 -
JAERI-Data/Code 96-031
Table 1
STRUCTUREPROPlPROP2
rm©
Nl N2 N3 N4
Nl N2
X Y Z #J3OD
2.5
t) (7)^e— p« 1/ V
Fig.2.8
*), ^ t l i 9 afl
(2.3)
Tablel (D LOAD
- 11 -
JAERI-Data/Code 96-031
L2
Fig.2.8
\d\d2d3d4 — dsd&didft)
\g\d% — g^djjd^d^ — \g2d5 —- gzdijd^d^j_,2 = = •—• •• — —-—— •—— }f
P
Lx = »Xi((V4 - Vi) x (V2 ~ V!))/|(V4 - Vi) x (V2 -
£3 - ™£3((V4 - Va) x (V2 - V8))/|(V4 - Vs) x (V2 - Va)|
L4 = -L 4 ( (Vi~"V 4 )x(V3"V 4 ) ) / | (V 1™V 4 )x(V 3 -V 4 ) |
(2-2)
(2.3)
12
JAERI-Data/Code 96-03 i
(2) mti Fig.2.9
Fig.2.9
- 13
JAERI-Data/Code 96-031
2.6
t
____ (
111
- ^ ^ ^
> -H
- .L — 1
i
I
t i
i 1i — — a
> —
» S: 46.x (x.n v.n 7 . n \
*J J J
K.n-\ v.n-l z.n-l \
m
- -e-
Fig.2.10
w; =
-n - l
.n - l
4-Ax?n ^ ^ _
\NJNJ
i)zl\
NJn(NJ~j
At \NJ-l)Ay" /7VJ--A
At VivJ-iyA^ /JVJ^-jX
At \NJ-lJ
(2.4)
1 4 -
JAERI-Data/Code 96-031
2.7
(2)
fc^, PE
(3)
(4)
(5)
w\m- 3 fcmm^m^ x
15
JAERI-Data/Code 96-031
1) Liu, M. and Gorman, D.G., "Symmetrization of Equations of Motion for Coupled Systems
Subject to Fluid-Structural Interactions", Communications in Numerical Methods in Engi-
neering, Vol.11,831-838 (1995)
2) Belanger, F. et.al., "Dynamics of Coaxial Cylinders in Laminar Annular Flow By Simul-
taneous Integration of the Navier-Stokes and Structural Equations", Journal of Fluids and
Structures, Vol.8, 747-770 (1994)
3) Guruswamy, G.P. and Byun, C,
"Fluid-Structural Interactions Using Navier-Stokes Flow Equations Coupled with Shell Finite
Element Structures" ,AIAA-93-3087,Orlando,Florida, July 1993.
4) Byun,C. and Guruswamy,G.P. ,
"Wing-Body Aeroelasticity Using Finite-Difference Fluid/Finite Element Structural Equa-
tions on Parallel Computers", AIAA-94-1487-CP (1994)
5) wmnAs&Mnttfe^mft&wjmwMms,^v~t-#.,m44
6) Hoffmann,K.A. and Chiang,S.T., "Computational Fluid Dynamics for Engineers-Volumel"
,Engineering Education System,1993.
7) Coen,P.G.,Sobieszczanski-Sobieski,J.,and Dollyhigh,S.M.,
"Preliminary Results from the High-Speed Airframe Integration Research Project" ,AIAA-
92-1004,Irvine,California,February 1992.
16
JAERI-Data/Code 96-031
a
ffi
(CFDrComputational Fluid Dynamics)
3.1
Navier-Stokes(NS) ^
(Cartesian)
, 3
Q
E L dG dEv dFv dGv - 05a: 6*3/ 9^ 9a: dy dz
p y
pupv
pwe j
( *" )pu2 + p
puv
puw^ (e + p)u j
pvu
pv2 +ppvw
(e + p)v j
' pw i
pwupwv
pw2 -f p, (e + p)w J
(3-1)
Ev = Tyx
Tzx
TXXU + TyXV
+TZXW
,FV ~
'xy
TVV
TXyU + TyyV
,GV —TZz
TXZU + TyzV
+rzzw ~
(7 c - i
p .
- 17 -
JAERI-Data/Code 96-031
(body-fitted coordinate system) t> L <
^ TV (Jacobian)
(x,y,z) fab-
(generalized coordinate system)
(metrics) £/BVvr (3.1)
v _ (3.2)
ppapvpw
\ e )
IJ
pV
T}xp
pvV +pwV +
puW
+pwW
Ev
0
syTxy
syTzy
+
0
C T "4~ C T ~f~ C Tx xx sy xy • sz xz
Sx^"yx i sy^"yy "r* Cz^"yz
Cx'T'zx + CyTzy + CzTzz
^ CxO"x + Cy^y + Cz< z
+ TfotJy + f72cr2
\
J\t Jacobian t
- 18
JAERI-Data/Code 96-031
d(x,y,z)
]) y 9 (metrics) ~C „
£x iy iz
Wx Vy 'Hz
Cx sy Qz
, V - T)t-\-T)XU + 7}yV + TjzW, W =
, r\t
(u (v - yt)^y + (w -
V = (w - xt)f}x + (v- yt)iiy + (w -
_ 2Txx ~ ^
(w
dv dwdw\ __ 2 / dv dw du\ __ 2
du dv dv dwdyj'
dTTXXU
dw du dv" ^ ~~dx~~dy
( dw du'_ I
v + ryzw-\-kdy
- 1 9 -
JAERI-Data/Code 96-031
TZXU + rzyvdT
(3.2)
3
(3.2)
Diminishing)
, Chakravarthy and Osher6)(D 3
ft # ^ 2 Hk^tclt 4
TVD(Total Variation
Runge-Kutta&^ffi
, Chakravarthy and Osher (D TVD fen 2
(Fig.3.1)
- 20
JAERI-Data/Code 96-031
Fig.3.1 NACA0012 H[HI «9 <D 2
ffft 5*s, Fig.3.1
21
JAERI-Data/Code 96-031
3.3
STARl]
t=t+in=n+l
END
Fig.3.2
Program Multiple
SPMD(Single
22 -
JAERI-Data/Code 96-031
Fig.3.3
ncpul,
lot, ([4]ff)
parameter(nng=100,nne=100) [1]
parameter(ncpul=2,ncpu2=2,ncpu=ncpul*ncpu2) [2]
paxameter(ng=(nng+2)/ncpuH-l,ne=(nne+2)/ncpu2+l) [3]
no
JAERI-Data/Code 96-031
dimension q(ng,ne) ,qo(ng,ne) ,fo(ng,ne) [4]
iprocs( 1) =ncpu 1
iprocs(2)=ncpu2
bound(l) —.false.
bound(2) =.false.
call mpijnit(ierr) [5]
call mpi_comm_size(mpi_comm_world,nprocs,ierr) [6]
call mpi_comm_rank(mpi_comm_world,myrank,ierr) [7]
call mpi_cart_create(mpi_comm_world,2,iprocs,bound, .false.,icomm,ierr) [8]
call mpi_cart_coords(icomm,myrank,2,mycoord,ierr) [9]
call para_range(l,nng,iprocs(l),mycoord(l),ista,iend) [10]
call para_range(l,nne,iprocs(2),mycoord(2) jstajend) [11]
do 100 j=jstajend
jj=j-jsta+l
do 100 i=ista,iend
ii=i-ista+l
voliv=1.0d0/vol(iijj)
100 continue
c
[5], [6],
, jend \$&y y?^ t \Z. [6], [7]
24
JAERI-Data/Code 96-031
3.4
MPMD MPMD ^ < £ > - # J £ I t ,
MPMD (Multiple Program Multiple Data)
y •?• jt-e SPMD ti: ft 5
t »
25
JAERI-Data/Code 96-031
1
(time)
<*>!
Fig.3.4
t+dt
- 26
JAERI-Data/Code 96-031
~>r-? (ii^mpLcomm.world)
(Fig.3.5#J§)o &TF\Z.S
mpLcomiiworld
icons cfd
PEi
1
-
-
1
1
r1—i
iconiR_grid
PE
_
i
1
Fig.3.5
call mpijnit(ierr) [Al]
call mpi_comm_size(mpi_comm_world,nprocs,ierr) [A2]
call mpi_comm_rank(mpi_comin_world,myrank,ierr) [A3]
icolor=l
key =myrank
call mpi_comm_split(mpi_comm_world,icolor,key,icomm_cfd,ierr) [A4]
call mpi_comm_size(icomm_cfd,newprocs,ierr) [A5]
call mpi_comm_rank(icomm_cfd,newrank,ierr) [A6]
call mpi_send(time,l,mpi_double_precision,nprocs-l,l5 mpi_comm_world,ierr) [A7]
call mpi_send(dt,l,mpi_double_precision,nprocs-l,l, mpi_comm_world,ierr) [A8]
call mpi_init(ierr) [Bl]
call mpi_comm_size(mpi_comm_world,nprocs,ierr) [B2]
call mpi_comm_rank(mpi_comm_world,myrank,ierr) [B3]
icolor=2
- 2 7 -
JAER1-Data/Code 96-03 1
key =myrank
call mpi_comm_split(mpi_comm_world,icolor,key, icomm_grid,ierr) [B4]
call mpi_comm_size(icomm_grid,newprocs,ierr) [B5]
call mpi_comm_rank(icomm^rid,newrank,ierr) [B6]
c
call mpi_recv(time,l,mpi_double_precision,0,l, mpi_comm_world,istatus,ierr) [B7]
call mpi_recv(dt,l,mpi_double_precision,0,l, mpi_comm_world,istatus,ierr) [B8]
[Al], [A2],
, [A6] X\ icomm_cfd
[A7]N [A8]T% B# iJ time t\%mM& dt
=1 ^ ^ ^ { t ^ # ? : ^ r p _ _ w
s-l) ^r#x.6o S t ^ / n ^ ^ ^ ^ T f i , icomm_cfd
l/fr newrank>
SPMD
tfs [B4]-[B6]-C,
, [B8]T%
28
JAERI-Data/Code 96-031
# # ^ Sfc
1) Cebeci,T. and Smith,A.M.O.,1974, in Analysis of Turbulent Boundary Layers, New York:
Academic Press
2) Baldwin,B. and Lomax,H.,1978, "Thin layer approximation and algebraic model for separated
turbulent flow",AIAA paper 78-0257
3) Launder,B.E. and Spalding,B.,1972, in Mathmatical Models of Turbulence, New
YorkrAcademic Press
4) Jones,W.,P. and Launder,B.,E.,1972, "The prediction of laminarization with a two-equation
model of turbulence",International Developments in Heat Transfer, 15,303-314
5) Peter,V.L.,Rodi,W., and Scheurer,G.,1985, "Turbulence models for near-wall and low-
Reynolds number flows: a review",AIAA Journal 23,1308-1319
6) Chakravarthy,S.,R. and Osher,S.,1985, "A new class of high accuracy TVD schemes for hy-
perbolic conservation laws",AIAA paper 85-0363
7) Gropp,W.,Lusk,E., and Skjellum,A.,1994, in Using MPLPortable Parallel Programming with
the Message-Passing Interface, The MIT Press
8) Geist,A.,Beguelin,A.,Dongarra,J.,Jiang,W.,Manchek,R.,
and Sunderam,V.,1994, in PVM:Parallel Virtual Machine - A Users' Guide and Turorial
for Networked Parallel Computing, The MIT Press
- 29
JAERI-Data/Code 96-031
Nomenclature
[B]
ID]
e
IF)
9
H\,
in
L
[M]
M[N]
IT]
t
u
Vfa,0
P
Pf
h V v
h V y
b V y
b y y
h }) y
i&ft-r h y
Rayleigh MM^ V * —
3 0 -
JAERI-Data/Code 96-031
4.1
mm
mfo Elastic Aircraft
[F] (4.1)
(Mode
Superposition)
~h*ila*, NASA <D ENSAERO1)
Newmark /3 & ^ ^
4.2
- 31
JAERI-Data/Code 96-031
Fig.4.1
4.3
Appendix (31 !B$$
m*^h y yM(O 30 - 40 %}
(Lumped Mass) * # J U %XZ>
32
JAERI-Data/Code 96-031
Appendix -A-N 4.5 ^ Appendix -B-, R& 4.6
Appendix -A-^ :
[m] =
[ft] =
[c] = a[m]
4 4
t=lj=l
[F]
Lix
Liy
Li2
0.0.
[T]T[K]e[T]
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
~ 33
JAERI-Data/Code 96-031
4.4
^ (4.
W i l s o n ^ Crank-Nicolson&U
i & & (Linear Acceleration Method),
iftfHr [H) f f
+ afc [ci + {W}n + 1
2 {«>n 2 {n ~ - M n ) (4.7)
4.5
(Aeroelastic Tayloring) NASA (7) X-29
- 3 4 -
JAERI-Data/Code 96-031
h V y?X% Appendix -B-Cl
Zn
El, E2, G12, v 12, v21
Z l
Fig.4.2
4.6
2.5-3.0
CFD
1E.G
[ D ]
^ Fig.4.3
[FJ [ K ]
Fig.4.3
or
JAERI-Data/Code 96-031
4.7
SPMD(Single Program Multiple Data) ^r^(C£ «9 ff 5
PEl PE2
PEl PE2
PEl
PEl - PE2
PEl PE2
PEl
1PE2
PEn
PEn
PEn
PEn
PEn
Fig.4.4
PE
36
JAERI-Data/Code 96-031
hVy? CPU xff b fc
(4)
mm
mk i t
fci?) PE
(6)
ti-f PE|HJtf>jIfi
- 37
JAERI-Data/Code 96-031
1) MacMurdy,D.E.,Guruswamy,G.P. and Kapania,R.K., "Static Aeroelastic Analysis of Wings
Using Euler/Navier-Stokes Equations Coupled with Improved Wing-Box Finite Element
Structures" ,AIAA-94-1587-CP,1993,pp. 2146-2158.
2) Byun,C. and Guruswamy,G.P.,
"Wing-Body Aeroelasticity Using Finite-Difference Fluid/Finite-Element Structural Equa-
tions on Parallel Computers" ,AIAA-94-1487-CP,1994,pp. 1356-1365.
3) Dodd,A.J.,Kadrinka,K.E.,Loikkanen,M.J.,Rommel,B.A.,Sikes,G.D.,Strong,R.C.,and
Tzong,T.J., "Aeroelastic Design Optim ization Program",J. of Aircraft,Vol. 27,
NO. 12,1990,pp. 1028-1036.
4) &)\\7cm,WftM,ttGmZ*mn,tir&tJ¥k CAEi/y-Xl,i£jiLflt,1994,pp. 171-179.
6) I.M. K^xm
7) Miura,H. and Neil,D.J.,"Structural Optimization: Status and Promise/Applications to
Fixed-Wing Aircraft and Spacecraft",Progress in Astronautics and Aeronautics, Vol. 150,
AIAA,1992,pp. 705-742.
8) Hoskin,B.C. and Baker,A.A./'Composite Materials for Aircraft Structures",
AIAA Education Series,1986.
9) Grande,D.L., "HSCT Materials Sz Structures Program Introduction" ,Boeing Commercial
Airplane Group, Seattle, Washington, June 20-23, 1994.
CAE */ v - x 7,
3 8 ~
JAERI-Data/Code 96-031
APPENDIX B
[D] =
A12 A22 A26 Bu B22 B26
Aie A26 A$e Bis B2G Bee
Bn B12 Bie Dn D\2 DIG
B\2 B22 B26 D12 D22 D26
B26 Bee Die ^26 ^66
By
Di
*J (6k) (zk+l ~k=in
011
022
016
(Ok)(0k)
026 (0k). 066 (0k) .
C2*2
CS3
c2s2
cs(c*
s2) -<
4 c V
-4c 2 s 2
4c2*2
- c 3 s 2cs(c2 -
-2cV c252 (c2 - s2)2
0i i (0)012(0)
022(0)
. 066 (0)
c = cos
011 (0)
012(0)
022(0)
066(0)
(1 -1/12^21)
(1 -1/12^21)
(1 -1/12^21)
5 3 -
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