Determining the three-dimensional magnetostatic field in current-induced situations has usually involved vector potentials, which can lead to excessive computational times. How such magnetic fields may be determined using scalar potentials is reviewed. It is shown how the heat transfer capability of NASTRAN level 17 was modified to take advantage of the new method

Hurwitz, M. M.

Schroeder, E. A.

English

NASA Technical Reports Server

SOLVING MAGNETOSTATIC FIELD PROBLIZMS WITH NASTRAN 
Myles M. Hurwitz and Erwin A, Schroeder 
David W. Taylor Naval Ship Research and Development Center 
SUMMARY 
Determining the three-dimensianal magnetostatic field in current-induced 
s i tuat ions  has usually involved vector potentials, which can lead to exceesive 
computarianal times. A recent paper shows how such magnetic f i e l d s  may be  
determined using scalar potentials. The present paper shows how the heat 
transfer c a p a b i l i t y  of NABTRAN Level I7 has been modi f i ed  to take advantage of 
the new method. 
INTRODUCTION 
All classical electromagnetic phenomena are governed by the four Maxwell 
equations: 
where the vector quantities are def ined  as follows: 
H Magnetic field strength or in tens i ty  
B Magnetic induction or flux dens i ty  
3 Current density 
E E l e c t r i c  field strength 
ll Electric displacement 
and the scalar quantities are defined as follows: 
p Charge density 
t T i m e  
There is also a constitutive relation between B and H ,  given by 
https://ntrs.nasa.gov/search.jsp?R=19780024541 2020-03-22T01:54:51+00:00Z
where u i s  the magnetic permeabi l i ty .  
Electromngneric problems a r e  o f t e n  solved by in t roducing  and so lv ing  f o r  
tho magnetic vec tor  p o t e n t i a l  A ,  where 
Spreeuw and Reefman ( re f .  1 )  used t h i s  nrethod wi th  N A S T W  i n  so lv ing  f o r  
harmonically o s c i l l a t i n g  e lec t romagnet ic  f i e l d s  i n  t he  presence of conductors 
car ry ing  a l t e r n a t i n g  c u r r e n t s ,  However, in order  t o  use the  e x i s t i n g  
s t r u c t l i r a l  and heat  t r a n s f e r  c a p a b i l i t i e s  i n  NAS'l'RAN, s imp l i fy ing  assumptfons 
had t o  be made. In  p a r t i c u l a r ,  t h e  magnetic vector p o t e n t i a l  A and t h e  source  
cur ren t  d e n s i t i e s  J were allowed t o  have components i n  only one d i r e c t i o n ,  and 
tllose conlponents were invariant:  i n  t h a t  d i r e c t i o n .  These a s s u m p t i ~ ~ ~ o  
e f f e c t i v e l y  reduce the  problem t o  one of so lv ing  f o r  a scalar p o t e n t i a l ,  which 
can b e  handled by NASTKAN1s heat t r a n s f e r  ana lyses ,  
Yn t h e  same paper ,  Spreeuw and Reefmnn a l s o  considered a  problem i n  which 
A was no t  u n i d i r e c t i o n a l  and were a b l e  t o  u se  NASTPAN's s t r u c t u r a l  a n a l y s i s  
c a p a b i l i t y  only because t h e  governing equat ions  uncoupled f o r  t h e  components of 
A.  
Another problem w i t h  " t i 5  Eoi-inu1~ti.o~ i s  t h e  requireinsot t h a t  
where E i s  the e l e c t r i c  p e r m i t t i v i t y .  
Spreeuw and Reefmnn used a s e p a r a t e  post-processor t o  handle t h i s  condi t ion .  
Frye and Kasper (ref. 2 ) ,  i n  s o l v i n g  magnetos ta t ic  problems using t h e  vec tor  
p o t e n t i a l ,  uae a  Lagrange m u l t i p l i e r  method ( s i m i l a r  t o  mult i -point  c o n s t r a i n t s )  
t o  s a t i s f y  cons t r a in t  ( 7 ) .  They a l so  po in t  ou t  t h a t  s p e c i a l  boundary condi t ions  
a r e  requi red  a t  boundaries  where t h e  permeabi l i ty  p changes. 
I n  re ference  3 ,  Zienlciewicz, Lyness, and Owen have developed a method f o r  
solving gene ra l ,  three-dimensional magnetos ta t ic  problems ustng s c a l a r  
p o t e n t i a l s , ,  s o  that: s tandard  hea t  t r a n s f e r  ana lyses  may be used and c o n s t r a i n t  
equat ion (7)  i s  no t  r equ i r ed .  They a l s o  i n d i c a t e  that  s p e c i a l  boundary condi- 
t i o n s ,  such as those mentioned i n  r e f e rence  2 ,  a r e  not  needed. 
The present  paper shows how t h i s  new method has been implemented i n  
NASTRAN Level 17. 
BASIC EQUATIONS AND ASSUMPTIONS 
The problem t o  be so'l.ved is  t h e  de te rmina t ion  o f  the magnetos ta t ic  f i e l d  
due t o  a body placed i n  an  e x i s t i n g  magnetic f i e l d  produced, f o r  example, by 
d i r e c t  cur ren t -car rv ing  loops.  The m a t e r i a l s  rre assumed to bc l i n e a r ,  bu t  
may be an i so t rop ic .  The governing equat ions  are: 
V x H  = J (8) 
Zienkiewicz separates H i n t o  two p a r t s ,  
H, i s  the f i e l d  i n  o homogeneous regl.on due t o  current J, s a t i s f i e s  
and i s  computed using the Biot-Savart low. H, i s  the unknown magnetic field 
s t r eng th  and satisfies 
v p v g  - t - v  * p H c =  0 (14 a) and 
where 4 i s  t h e  s c a l a r  p o t e n t i a l .  Zienkiewicz then uses s tandard  v a r i a t i o n a l  
p r i n c i p l e s ,  wit11  equat ions (9) and (lo), t a  arr ive a t  t h e  s tandard  f i n i t k  
element form 
where K i s  t h e  "stiffness" matr ix ,  
F is  the "load" v e c t o r ,  and 
where 
Ni is  the  f i n i t e  element shape func t ion  f o r  the iLh g r i d  p o i n t ,  and 
V is  the  volume of the f i n i t e  element.  
The formulat ion (16) f o r  k i j  i s  exac t ly  t h a t  of t h e  s tandard  h e a t  t ransEer  
conduct iv i ty  matr ix wi th  magnetic permeabi l i ty  p playing  the  r o l e  of thermal 
conduct iv i ty .  The EormuLation (17) f o r  Ei, however, is no t  a standard heat 
transfer quantity and must b e  computed e i t h e r  i n  a separate program and i n p u t  
t o  NASTRAN o r  in a new NASTRAN c a p a b i l i t y .  Also, note t h a t  fi i s  element- 
dependent, as evidenced by the shape function grad ien t  i n  equat ion  ( 1 7 ) .  
Equa t ion  (15) i s  solved fcr  4 s u b j e c t  t o  s t a n d a r d  n a t u r a l  o r  f o r c e d  
boundary c o n d i t i o n s ,  H, can t h e n  b e  camputed from e q u a t i o n  (14) ,  and t h e  f i n a l  
r e s u l t s  can be o b t a i n e d  u s i n g  equations (11) and ( 9 ) .  
NASTRAN IbPLENENThTION 
To s o l v e  n lagne tos ta t i c  problems w i t h  NASTMN L e v e l  1 7  u s i n g  t h e  met l~ods  of 
t h e  p rev ious  s e c t i o n ,  we select r i g i d  fo rmat  I, HEAT approach .  However, w e  
have modi f i ed  t h e  program t o  
1) compute t h e  Hc f i e l d  due t o  c i r c u l a r ,  d i r e c c  c u r r e n r - c a r r y i n g  l o o p s ;  
2) a c c e p t  a specified H, f i e l d ;  
3) compute f i  ( e q u a t i o n  ( 1 7 ) )  f o r  t h e  axisymmetr ic  s o l i d  r i n g  e l e m e n t s  
T W R G  and TRIARG; 
4) perform the a d d i t i o n  s p e c i f i e d  i n  e q u a t i o n  (11), where M, i s  a  
t r a n s f o r n l a t i o n  of the " temperature"  g r a d i e n t s ;  and 
5) o u t p u t  B ( e q u a t i o n  ( 9 ) )  f o r  subsequen t  NASTRAN p l o t t i n g .  
The implementa t ion t h u s  f a r  h a s  been l i m i t e d  t o  s o l i d  axfsymmetr lc  
problems u s i n g  TRAPRG and TRIARG f i n i t e  e l e m e n t s  and i s  runn ing  on the DTNSRDC 
CDG 6000 computers.  
N U 4  BULK DATA CARDS 
Two new b u l k  d a t a  cards have been i n t r o d u c e d  i n t o  t h e  program f o r  
computing Hc EieLds. They a r e  CEMLOOP, for computing the H, f i e l d s  due t o  
c i r c u l a r  c u r r e n t  loops, and SPCP:JI, f o r  s p e c i f y i n g  Kc a t  s e l e c t e d  grid p o i n t s .  
(See figures 1 and 2 f o r  d e t a i l e d  d e s c r i p t i o n s  of t h e s e  cards.) 
MODIFIED NASTRAN ROUTINES 
Nineteen e x i s t i n g  N A S T M  r o u t i n e s  have been modi f i ed  t o  accommodate the  
new c a p a b i l i t y .  The r o u t i n e s  and t h e  n a t u r e  of t h e  m o d i f i c a t i o n s  are as 
f o l l o w s  : 
R o u t i n e  Reason f o r  Change 
IFP,IPS4P,IFXlBD-IFX7BD Recognize and check new bu lk  d a t a  c a r d s  
CElifLOOP and SPCFLD . 
Restart: with CEHLOOP and SPCFLD in Static Heat 
T r a n s f e r  Analysis. 
Recognize CEMLOCIP and SPCFLD as " h e a t  t r a n s f e r "  
l o a d  s p e c i f i c a t i o n s  and place on HSLT (Neat 
S t a t i c  Load T a b l e ) .  
SBSUD,  .QU'LUD 
SSESu.4 
OFP l A  
Reasoil f o r  Cllnllgc 
Retrieve CEELOOP nnrl SPCBLD s p a c i f i c n t i o n s  Eronr 
1 CEPiLOOP c a r d s  nre copied d i r e c t l y  t o  
d a t a  bloclc NEWSLT f o r  l a t e r  p r o c e s s i n g .  A l l  
SPCFLD s p e c i f i c a t i o n s  a r c  combined i n t o  one 
v e c t o r  g i v i n g  t o t a l  !Ic a t  a l l  grid p a i n t s .  
T h i s  v e c t o r  i s  placed at1 NUIJSLT. 
C a l l  nwt r o u t i n e  BANDM, which retrieves CBblLOOP 
ncd SPCFLII s p e c i f i c a t i o n s  frour NElJSLT and 
conlputes E as given i n  e q u a t i o n  (17) .  
S p e c i f y  t n b l c  updutes  s o  t h a t  NASTRAN w i l l  
r c c o g i ~ i z c  new Eunct io t ln l  modulo GPLPLD . 
C a l l  new f u n c t i o n a l  ~ n o d u l a  G b P L D .  
S p c c i f y  new l l ead i~ lgs  t a  output:  new d a t a  bloclc 
HOEM1 g i v i n g  ningne t i c  f i e l d  s t r e n g t h  and 
i t lduc t ion .  
C u l l  f o r  new hcadiaga from OPPlA wl~en WOEHL i s  
recogn ized .  
NEW ROUTINES 
Tlzree new r a u t i n r . ~  have been  developed.  S u b r o u t i n e  EhNDEI rends InLa open 
core CEbfLOOll and SPCFLII d a t a  fron~ dam block NEIJSL'II, r e a d s  infonnal:ion f o r  an  
e1eatr;nt Erotit the  Eleltlent Summary T a b l e  (EST) , and calls a n  e l e n ~ e n  t-dependent 
r o u t i n e  LO compute t h e  load as given  in e q u a t i o n  (17). 
Subroittitle EPllIENG i s  cal lec l  by GANDPI and computes t h e  load due t o  CEFILOOP 
and SPCBLU specifications for solid nxisymttletric t r o p c z o i d n l  (TRAPRG) nnd 
t r i a n g u l a r  (TRIAlIG) r i n g s .  iJa assume t h a t  H, is  c o n s t a n t  o v e r  an e l e m e r ~ t .  
Therefore, f o r  TRIARG e lements ,  kI, is  coi\\puted at: t h e  c e n t r o i d  due  t o  all 
CEPLOOP' s using a n  e l l i p t i c  i n  t e g r o l  f ormu1ati.011. If SPCBLD ' s a r e  g i v e n ,  Hc i s  
cor~iputcd co b e  t11e averago o f  t h e  HCis nt t h e  r l ~ r e c  v e r t i c e s .  Each TRAPRG 
cleniei~t ,ts ciivicled i n t o  f o u r  o v e r l a p p i n g  t , r i a n g l e s ,  and each  t r i a n g l e  i s  c r e a t e d  
a s  n TRIARG e le lnent .  Once 1.1, h a s  been canlputed f o r  a t r i a n g l e ,  cquatiolz (17)  
is used t o  coniputo t h e  l a n d  a t  each  g r i d  p o i n t  forming t h e  L r i a n g l e .  S u b r o u t i n e  
GPlRING a l s o  outpucs t o  Fortran f i l e  11 c e r t a i n  clement i n f o r m a t i o n ,  i n c l u d i n g  
I-I,, f o r  l a t e r  p r o c e s s i n g  by E~rnct io tznl  module EFIFLD. Tilis i s  n temporary 
method f o r  p a s s i n g  inforrt iat ion from EbRING to EPELD. ' the nortila1 a e r i ~ o d ,  of 
course, i s  t o  use d a t a  b locks .  llowever, s u b r o u t i ~ ~ e  EPlRING r e s i d e s  i n  f u n c t i o n a l  
niodule SSG1, and adding a new o u t p u t  d a t a  block t o  t h a t  e x i s t i n g  module would 
require a  chatlgc t o  evary r i g i d  Eornlar i n  t h e  progrtlru. These changes vi11 be 
rr~ade a t  n l a t e r  tinre. 
Subroucille EbELD, which i s  also a new E u n c t i o n n l  module, coinputes t h e  
magne t i c  f i e l d  s t r e r ~ g t h  and i t ~ d u c t i o n ,  a c c o r d i n g  t o  e q u n t l a n s  (11) and ( 9 j ,  for 
each f i n i t e  e l ement  i n  the model. ENFLU r e t r i e v e s  Ero~n data block HOEPl t h e  
" t e ~ l ~ p e r n t u r e "  g r a d i e n t  f o r  an e lement .  S i n c e  t h o  g r a d i e n t  i1,, was computed i n  
a n  element c o o r d i n a t e  systeur,  EbPLD t r a n s f o r m s  i t  i n t o  bas ic  c o o r d i n a t e s .  Then 
F o r t r a n  f i l e  11 i s  sea rched  t o  nletch t h e  cle!nent i d e n t i f l c o c i o n  number, and ,  on 
a nlntch, llc f o r  the  element: is r e t r i e v e d ,  added t o  t h e  t ransfarnled H I ,  and put  
o u t  t o  da ta  b l o c k  BOEH1. Also  cotnputed and output t o  HOER1 is  the magne t i c  
inducticln D. I i O E H l  i s  l a t e r  output: u s i n g  normal Output F i l e  P r o c e s s o r  (OPP) 
e x e c u t i o n .  EWLD a l s o  conlputes and o u t p u t s  o t h e r  i n f o r m t i o n  f o r  p l o t t i n g  
purposes ,  as exp la ined  i n  t h e  next: s e c t i o n .  
. ,> 
PLOTTING tfiGNETOSTATTC RESULTS 
Normal N A S T W  p l o t  p r o c e s s i n g  a l l o w s  f o r  de fo rn~ed  p l o t s  based on g r i d  
p o i n t  d i s p l a c e m e n t s  o r  c o n t o u r  p l o t s  of  s t r e s s e s .  I n  t h e  p r e s e n t  a n a l y s i s ,  
however, the "displnccments"  ( t h e  s c a l a r  p o t e n t i a l s )  a r e  of l i t t l e  use  by 
the i t~sc lves .  The " s t r e s s e s "  (N o r  B f i e l d s  coming Prom EEWLD) a r e  more u s e f u l ,  
b u t  what w e  would r e a l l y  l i k e  t o  see f o r  "nice1'  p l o t s  are t h e  l i n e s  of 
i n d u c t i o n .  The l i n e s  of i n d u c t i o n  i n d i c a t e  t h e  d i r e c t i o n  of  t h e  magneric 
i n d u c t i o n  B, and t h e  aumbcr of l i n e s  p e r  u n i t  area i n d i c a t e s  tl-re magnitude of 
B .  While w e  d o  not  p r e s e n t l y  p l o t  t h e s e  l i n e s  of i n d u c t i o n ,  w e  do plot :  t h e  
actual i n d u c t i o n ,  magnitude and d i r e c t l i o n ,  f o r  each  e lement .  T h e r e f o r e ,  
f u n c t i o n a l  module EMl?LD o u t p u t s  o t h e r  q u a n t i t i e s  as f o l l o w s .  For each  e l e m e n t ,  
two c o i n c i d e n t  g r i d  p o i n t s  a r e  c r e a t e d  a t  the c e n t r o i d  of t h e  e lement ,  and the 
cor respond ing  G R I D  c a r d s  are punched. Also punched is  a PLOTEL card connecting 
t h e  two g r i d  p o i n t s .  (The l e n g t h  of tlte PLOTEL e l e n ~ e ~ ~ t  is z e r o . )  T11en n 
"displacement"  v e c t o r  is  c r e a t e d  by assigning z e r o  v a l u e s  t o  each  of tlze 
original g r i d  p o i n r s  i n  t h e  msdel and a s s i g n i n g  t h e  magnet ic  i n d u c t i o n  v a l u e  
f o r  an  e lement  t.o each of t h e  two c o i n c i d e n t  g r i d  p o i n t s  c r e a t e d  f o r  the 
element .  (The 'Ldisplacement l l  v e c t o r  u s e s  s i x  degrees of freedom p e r  g r i d  p o i n t  
s i n c e  B is  a v e c t o r ,  n o t  a s c a l a r . )  T h i s  v e c t o r  i s  packed in EMFLD and o u t p u t  
i n  DIW u s i n g  module OUTPUTT. On a subsequen t  NASTRAN r u n ,  tile new GRID and 
PLOTEL c a r d s  arc merged w i t h  t h e  o r i g i n a l  d a t a ,  t h e  "displaceizent"  v e c t o r  is 
r e t r i e v e d  u s i n g  DMAP module INPUTT1 a s  d a t a  b l o c k  UGV,  and a defortued p l o t  i s  
r e q u e s t e d  wit11 t h e  VECTOR B o p t i o n .  T h i s  NAS'IIRAN r u n  i s  performed w i t h  r i g i d  
format  I, DISP approach,  and AL'rERs are used t o  e x e c u t e  o n l y  t h o s e  modules 
r e q u i r e d  f o r  deformed p l o t s .  The p l o t s  show t h e  o r i g t n a l  s t r u c t u r e  a s  an 
under lay ,  and a v e c t o r  is drawn a t  each element  c e n t r o i d  i n d i c a t i n g  the  magni- 
tude  and d i r e c t i o n  of B i n  t h a t  e l ement .  
SAMPLE PROBLEMS 
A t  t h e  time t h a t  t h i s  paper  was b e i n g  p r e p a r e d ,  t h e  on ly  a x i s y m e t r i c  
problems r u n  w i t h  NASTRAN f o r  which a n a l y t i c a l  r e s u l t s  were r e a d i l y  a v a i l a b l e  
were p r o b l c ~ n s  w i t h  uniform p e r m e a b i l i t y .  The conrparison between t h e  NASTRAN 
and a n a l y t i c a l  r e s u l t s  was very good. Although problems w i t h  nonuniform 
p e r m e a b i l i t y  have been r u n  w i t h  NrlSTRAN and "reasonable- lookingt '  r e s u l t s  have  
been o b t a i n e d ,  a n a l y t i c a l  r e s u l t s ,  r e q u i r e d  f o r  v e r i f i c a t i o n ,  are s t i l l  
for thcoming.  
1. Sproeuw, E.; and Roofmnn, R.J.U.: Rigid Portllat Alter Packets  f o r  t h e  
A n a l y s i s  of BlectromagnoCic F i e l d  P r o b l e t i ~ s .  Four th  NhSTlUN Users' 
Colloquiuru, NAS'L'lUN: Users' Lsperiences, NASA TEI X-3278, Septetlrlcr 3975, 
p p .  557-570. 
2 .  Frye, 3.11. ; and Kasper ,  R.G. : Analysis of Magnetic Fields U s i n g  Vnrintiotlal 
P r i n c i p l e s  and CELAS2 Glearents . S i x t h  NASTIZAN Users ' C o l l o q u i u a ~ ,  NASA 
Conference Publicmtion 2018, October 1977, p p .  175-192. 
3 .  Zicnkicwicz  , 0. C .  ; L y n e s s  , John; nlld OIJC~I ,  D. R. J. : Three-DF~~lcnsionol. 
blagnetic F i e l d  Dctertuir~ntion Using tl ScuLar P o t e n t i a l  - A F i n i t e  Elen~ctlt 
Solution. IEEIS Trnrrsoc t i o n s  on Ilagne t i c s ,  Vol .  llAG-13, No. 5, 
September 1977, p p .  1649-165G. 
Input Data Card CEhCLOOP Circular Current Loop 
Description: Defines n circular  c u r r e n t  loop i n  magnetic Eield problems. 
Forn~o t and Example : 
Fie ld  Contents 
S I D  Load set  i d e n t i f i c a t i o n  number ( i n t e g e r  > 0). 
J Current through l o o p  (units of p o s i t i v e  charge tsec)  ( r e a l  2 0) . 
AX1 = 0 nonaxisynmletric problem (no t  yet implemented) 
= 1 axisymmetric problem; T W R G  and TRIhRG elements a r e  
implied (integer) . 
1 2 3 4 5 6 7 8 9 10 
X l , Y l , Z l  Coordinates of two points through which the  loop passes 
X2,Y2,Z2 (given in c o o r d i n a t e  system CID)(real). 
CEMLOOP 
+A 
CEMLOOP 
XC,YC,ZC Coordinates o f  c e n t e r  o f  loop (given i n  coordinate system 
C I U )  ( rea l ) .  
C ID Coordinate system identification number (integer 2 0) .  
SID 
22 
3 
Remarks : 
1. Load se ts  m u s r  b e  s e l e c t e d  i n  t h e  Case Control Deck (LONl=SID) to 
be used by NASTRAN. 
J 
XC 
2.5 
2. I f  AXI-1, Y 1  must be 0. or blank, and a l l  data f i e l d s  a f t e r  21 musr 
be  0. o r  blank. (Continuation card need not be present.) 
Figure 1. Bulk Data Description of CEkILOOP 
4 4  AX1 
YC 
1 
Y2 
--- 
XI 
ZC 
5 . 2  
Y1 21 XZ 
- 
C 
0. 2.25 
I n p u t  Dnto Card SPCFLD SpcciEied MogncLic Field 
Uescriptlon: Specifies nlagnctic f i e l d  a t  selected &rid poincs .  
I SPCFLD SID HCS SPCFLD 18 12.25 0. 
- 
Field Contents 
--- 
S ID Load set idcntificarioi~ number (integer 0). 
kICX,HCY,HCZ Components of s p e c i f i e d  Hc f i e l d  ( r ea l ) .  
Gi,GIDi ' Grid p o i n t  identification numbers (integer > 0). 
- 1 Implies the Hc f i e l d  a p p l i e s  a t  all grid points. 
SPCFLD 
SPCFLD 
- 
SPCFLD 
SPCFLD 
Remarks : 
I. Load sets tuust be selected i n  the Case C o n t r o l  Deck (LON)=SID) t o  
be used by NASTRAN. 
SID 
18 
2 .  A l l  g r i d  p o i n t s  referenced by SLDl THRU GID2 m u s t  e x j s t .  
SID 
18 
Figure 2. Bulk Data D e s c r i p t i o n  oE SPCFLD 
HGZ 
- 
62. 
lICX 
12,25 
)LC); 
-- 
1 2 . 2 5  
HCY 
0. 
GLDl 
9 
1lCZ 
62. 
-
HCY 
0 .  
-1 
-I 
THRU 
TI-IRU 
GI132 
27 


Solving magnetostatic field problems with NASTRAN

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