An analysis procedure has been developed and incorporated into COSMIC/NASTRAN that permits large dynamic degree of freedom models to be processed accurately with little or no extra effort required by the user. The method employs existing capabilities without the need for approximate Guyan reduction techniques. Comparisons to existing solution procedures presently within NASTRAN are discussed

Citerley, R. L.

Woytowitz, P. J.

English

NASA Technical Reports Server

RLTZ PROCEDURE FOR COSMIC/NASTRAN 
R. L. C i t e r l e y  and Y. J. Woytowitz 
Anamet Laboratories ,  k c . ,  San Carlos, CA 94070 
An a n a l y s i s  procedure has been developed and incorporated i n t o  COSMIC/ 
NASTRAN t h a t  permits l a r g e  dynamic degree of freedom models t o  be processed 
accu ra t e ly  with l i t t l e  o r  no e x t r a  e f f o r t  required by the  user. The method 
employs e x i s t i n g  c a p a b i l i t i e s  without t he  need f o r  approximate Guyan reduct ion  
techniques. Comparisons t o  e x i s t i n g  s o l u t i o n  procedures p re sen t ly  wi th in  
NASTM a r e  discussed.  
INTRODUCTION 
The search f o r  an e f f e c t i v e  methe f o r  performing a dynamic a n a l y s i s  of  
complex s t r u c t u r e s  has  been under continuing study f o r  many decades. In  order  
t o  reduce t h e  number of governing equations,  the  usual  approach was t o  use  
modal coordinates.  These modal coordinates a r e  develcped by employing c l a s -  
s i c a l  eigenvalue ex t r ac t ion  procedures. With t h i s  approach as the  accepted 
a n a l y s i s  procedure, a concerted e f f o r t  start in^ in the 1950's was made in 
developing var ious  eigenvalue e x t r a c t i o n  methods. Several  of these  op t ions  
p re sen t ly  e x i s t  i n  NASTRAN t o  develop these  dynamic proper t ies .  
In pu r su i t  of e f f i c i e n t  and accura te  eigenvalue methods, Jennings ( ~ e f .  1 )  
g ives  a r a t h e r  b r i e f  but complete review of methods f o r  s c lv ing  dynamic equa- 
t i ons  to determine c h a r a c t e r i s t i c  responses. The Lanczos method was repor ted  
a s  one o f  the more e f f i c i e n t  methods f o r  the so lu t ion  of  s t r u c t u r a l  eigenvalue 
problems. Further  i n v e s t i g a t i o n  by Nour-Ornid e t  a l .  ( ~ e f .  2). i l l u s t r a t e d  
t h a t  the Lanczos method had tremendous advantages over o the r  c l a s s i c a l  
eigenvalue ex t r ac t ion  methods. With a l l  t he  various eigenvalue procedures 
ava i l ab le ,  the choice of  which numerical procedure should be used f o r  so lv ing  
a dynamic response problem depends upon the  problem c h a r a c t e r i s t i c s  and 
s o l u t i o n  requirements. 
The so lu t ion  process f o r  the dynamic ana lys i s  developed i n  NASTRAN i s  
separated i n t o  th ree  phases: assembly, so lu t ion ,  and response recovery. A s  
pointed out i n  Sect ion 4.0 of the  NASTRAN Theore t ica l  Manual, a s  problem s i z e  
increases ,  the  cos t  of the  f i r s t  and t h i r d  phases inc reases  l i n e a r l y ;  whereas 
the  second increases  cubica l ly  with the  number of degrees of freedom. The 
eigenvalue so lu t ion  phase is usua l ly  considered the  most cos t ly .  
https://ntrs.nasa.gov/search.jsp?R=19850017566 2020-03-22T19:17:47+00:00Z
Wilson (Ref. 3) demonstrated how the  eigenvalue procedure could be c i r -  
cumvented. He c l a s s i f i e d  i t  a s  t he  "Ri t? . "  method. Arnold e t  a l .  (Ref. 4 ) ,  
demonstrated how the  R i t z  procedure could be incorporated i n t o  MSC/NASTRAN and 
showed t h a t  i t  would be l e s s  c o s t l y  and more e f f i c i e n t  than previous techniques. 
Reference 4 a l s o  d i scus ses  the r e l a t i o n s h i p  between Lanczos and Ritz .  
Other methods using t h e  bas i c  concept of Lanczos have been suggested by 
Gupta (Ref. 5 )  using a Lanczos-Householder algori thm, Newman and Flanagan 
(Ref. 6 )  using FEEX, and by Newmen and Mann ( ~ e f .  7) us ing  complex FEER 
methods f o r  COSKIC/NASTRAN. These methods d i f f e r  from the  present  case in the  
s e l e c t i o n  process of t he  s t a r t i n g  vectors .  Also, t he  method of  obta in ing  t h e  
a d d i t i o n a l  vec to r s  is d i f f e r e n t .  
TECHNICAL DISCUSSION 
Xlthouph the  use of R i t z  vec to r s  can be appl ied  t o  t he  dynamic a n a l y s i s  
ot' any l i n e a r  system, it is p a r t i c u l a r l y  w e l l  s u i t e d  t o  ap; l icat ion i n  t h e  
f i e l d  of dynamic s t r u c t u r a l  ana lys is .  The na ture  of c e r t a i n  dynamic loadings,  
o r  r a t h e r  the  h n a l y t i c a l  r ep re sen ta t i on  of the  loadings,  a l lows the  R i t z  
approach t o  almost assuredly  use fewer modes f o r  t he  same accuracy a s  a  conven- 
t i o n a l  a n a l y s i s  using n a t u r a l  f requencies  and mode shapes. The use of a s t a t i c  
load vec tor  i n  the dynamic e x c i t a t i o n  d j r e c t i o n  i n  de r iv ing  the  s t a r t i n g  R i t z  
vec tor  e l imina t e s  t h e  need f o r  a  s t a t i c  co r r ec t ion  due to  h igher  order  modes. 
A s  menticned e a r l i e r ,  the  s o l u t i o n  of the  eigenvalue problem f o r  l a r g e  
systems is the  major computational task involved i n  dynamic ana lys l s .  F3r 
very l a r g e  systems, t h i s  -;ask is of ten  not even p r a c t i c a l ,  and i n ~ t e a d  is 
supplemented with an a d d i t i o n a l  s t e p  of s t a t i c  condensation ar.l Guyarl reduc- 
t i o n ,  o r  by use of subs t ruc tu r ing  techciques. A l l  these  techniques a r e  
acceptable  and a r e  i n  s tandard p r a c t i c e  with COSMIC/NASTRAN today; however, 
a l l  involve the  so lu t ion  nf l a rge  eigenvalue prsblems even i f  the  t o t a l  
problem ?.as been reducttd i n t o  workable subs t ruc tures .  The R i t z  algori thm, t o  
be defined i n  d e t a i l  l i t e r ,  a l lows the  ana lys t  t o  bypass the  s o l u t i o n  of a  
l a rge  eigenvalue problem, and ins tead  so lve  a  smal le r  e igenvalae  problem 
involv ing  vnly t he  P i t z  modes of s ign i f i cance  t o  the ana lys i s .  Nour-Omid and 
Clough ( ~ e f .  3) have shorn how even t h i s  s t e p  could be el iminated.  Also 
presented here is a method of us ing  an e r r o r  norm t o  de f ine  t he  n~.mber of R i t z  
vec to r s  requi rad  t o  obta in  accura te  dynamic a n a l y s i s  r a s u l t s .  
Deri-vation of Ri tz  Vectors 
Given a phys ica l  system whose mass and s t i f f n e s s  p r o p e r t i e s  have been 
d i s c r e t i z e d  t o  N degrees of freedom given by the  s t r u c t u r e  mass matr ix,  F1, and 
s t r u c t u r e  q t i f f n e s s  matr ix,  K, t he  i n i t i a l  s t e p  in the  algori thm is t o  f a c t o r  
K such t h a t  
The s t a r t i n g  vec tor  x* c a r  then be obtained by so lv ing  1 
where f  is a a t a t i c  load vec tor  which r ep re sen t s  t he  s p a t i a l  d i s t r i b u t i o n  of 
t he  dynamic loading. 
Once the f i r s t  Ri tz  vec to r  x7 is obtained,  an i t e r a t i v e  process is begu~; 
f o r  the  s o l u t i o n  of a  f u l l  L s e t  of Ritz  vec tors .  This process ..an be divided 
i n t o  t h r e e  s t eps  which a r e  given below. 
S t ep  1: Solu t ion  f o r  x* is made using the  f ac to red  s t i f f n e s s  matrix from 
Equation (1 ) i 
Step  2: i = i + 1 , i f  i < L go to  1 ,  otherwise proceed 
Step 3: Orthogonalize xy with r e spec t  to  prevj.ous i - 1 v e c t o r s  
and normalize with r e spec t  t o  M t o  f ind  xi 
The r e s u l t  of t h i s  i t e r ~ . t i v e  process is an N by L matr ix of vec to r s ,  X, which 
i s  mass orthogonal ,  but st i l l  must be orthogonalized with r e s p s c t  t o  t h e  
s t i f f n e s s  matrix. The f i n a l  s e t  of Ri tz  vec to r s  muat be mass and s t i f f n e s s  
orthogonal to uncouple t he  equat ions of motion, and t h i s  or thogonal iza  t i o n  can 
be acccmp-ished by so lv ing  the fol lowing eigenvalue problem f o r  zi, 
where 
The f i n a l  matr ix of Ri tz  vec tors ,  OX, is then obtained using the  matri: of zi 
vec tors ,  2, a s  a t ransformation f o r  X,  
ox = xz (I c!) 
Error  Norm Def in i t i on  
The de f i r i i t i on  of a? e r r 3 r  norm is a key element in a p p l i c a t i o n  of t h e  
R i t z  algorithm. Wilsoii o r i g i n a l l y  devised an e r r o r  norm defined a s  
where 
T 
'j = OX .f = part ic=pat j- , i l  f a c t o r  J 
and the vectiors Ox. a r e  the  f i n a l  mass and s t i f f n e s s  or thogonal  Ritz  vec to r s  
obtained from E!qua$ion (I  0j. This e r r c r  norm v a r i e s  from e = 1.0 i f  no vec- 
t o r s  a r e  used, to  e = 0.0 i f  a l l  vec tors  a r e  used. Arnold, C i t e r l e y ,  e t  a i . ,  
used t h i s  same d e f i n i t i o n  of e r r o r ,  but appl ied  it using the  non-s t i f fnsss -  
or thogonalj  zed R i t z  vec to r s  x r a t h e r  than ox . .  This d e f i n i t i o n  then permit? j' 
e r r o r  e s t ima t ion  pr!.or to  eigenvalue extraot iot l  performed on Equation (7). I n  
add i t i on ,  they e s t ak l i shed  another  e r r o r  norm t o  quan t i fy  the  i n f luence  of the  
input  e x c i t a t i o n  frequency content  on the  adequacy of t h e  s e l ec t ed  B i t z  
vec tors .  Examination of t h i s  type of e r r o r  norm w i l l  be requi red  f o r  high 
frequency inpu t  exc i t a t i on .  
IMPLEMENTATION INTO NASTRAN AND NUMERICAL RESULTS 
The R?.tz procedure :;as implemented i n t o  NASTRAN using the DMAP capa- 
b i l i t i e s .  A flow cha r t  of  a t y p i c a l  MAP a l t e r  is shown in R g u r e  1. The 
procedure has been incorporated i n t o  SOL 3, 11, and 12. Although d a t a i l o  o f  
t h e  MAP a l t e r  a r e  s l i g h t l y  d i f f e r e n t  f o r  each r i g i d  format, the  flow cha r t  of  
Figura 1 o u t l i n e s  the  general  procedure. 
Block 1 of  Figure 1 processes the s t a t i c  loads. A s  many subcases a s  
necessary a r e  input  by the  user  t o  genera te  t he  s t a t i c  loads which a r e  t o  
represent  the s p a t i e l  d i s t r i b u t i o n  of the dynamic loads. Modules SSGI, SSG2, 
and SSG3 a r e  used f o r  the  processing and sllbsequent s e t  r e d u c t i o ~ ~ .  
Block 2 of  Figure 1 uses the  love r  t r i ~ n g u l a r  f a c t o r  of the s t i f f n e s s  
matrix (L of Equation (1 )) t o  so lve  f o r  t he  f i r s t  s e t  of  vectors. Block 3 is  
a loop, which c a l c u l a t e s  a d d i t i o n a l  vec tors  described previously. 
Block 4 uses a du module, MODB, t o  orthonormallze the  generated vec- 
t o r s  ( see  Bquntions ( 4 3 ) .  Block 5 then r l c u l a t e r  the g e n e r a l i e d  mass and 
s t i f f n e s s  matr ices of Equations (8) and (9). Block 6 uses the  REX9 module t o  
perform the eigenvalue/eigenvect.r ana lys i s  of Equation (7)  and form the Ri tz  
vec tors  using Equation (10). F ina l ly ,  Block 7 equivalences the Ri tz  vec tors  
t o  the regular  A s e i  eigenvectors  (PHIA) f o r  subsequent use in the  MAP r i g i d  
format. 
A dummy module, MOD9, was wr i t t en  to  perform seve ra l  tasks  de?ending 
upon the  input  parameters. It is used in Block 3 t o  append the  newly 
ca lcula ted  vec tors  t o  the previous vectors .  Addit ional lg,  although not shown, 
i t  was found necessary t o  n o m a l i ~ e  the  newly corcputed vectc''s i n  Block 3 
a f t e r  each pass through the  loop i n  order  to prevent numerical round-off 
problems. This was a l s o  perfarmed by the  dumn~y module. The dumnpr modu: .'s 
f i n a l  job was t o  orthonormalize the vec tors  I n  Block 4, using the  Gram-Schnidt 
orthonormalizst ion procedure. 
The Ri tz  procedure was p~*eviously implemented i n t o  the MacNeal-Schwendler 
version of NAETRAB. Comparisons of tine Ritz procedure t o  MSC/NASTRAN's Gener- 
a l i zed  Cynamic Procedure have been reported (Ref. 3). The Ritz  procedure was 
a l s o  compared to  COSMIC/NASTRAl's FEW method. Table 1 presents  frequency 
comp~r i sons  and t o t a l  CPU time using the FEER method and the Ri tz  method or a 
r e l a t  -?ly small  problem. A s  can be seen, both nethods y i e ld  s i m i l a r  r e s u l t s ,  
the Rita method being s l i g h t l y  f a s t e r .  
However, i t  is not proper t o  dLrec t ly  compare the FEEFi method with Ri tz ,  
s ince  they a r e  intended t o  serve d i f f e r e n t  purposes. The FEM (or any ocher 
eigen-extract ion) rou t ine  is designed to  f ind  a l l  the  e igenva l~ le s  and a ~ s o c i -  
a ted  eigenvectora below a c e r t a i n  user  s p e c i f i d  number. These modes may o r  
may not be good r ep resen ta t ions  f o r  use i n  a forced dynamic response. Often 
i t  is found t h a t  only c e r t a i n  modes p a r t i c i p a t e  due to  t he  na ture  of t h e  
forc ing  function.  The Ritz  method uses s t a t i c  r ep resen ta t ions  of the dynamic 
forces applied as a bas is  fo r  its "eigenvectors". The only "modes" a l c u l s t e d  
correspond to those excited by the forcing furction.  Therefore, fewer un- 
necessary modes a r e  used; addi t ional ly ,  a s  pointed out previously, s t a t i c  
correc1;ions (such as Modal ~ c c e l e r a t i o n )  tech- iques a r e  not necessary, s i n c e  
the W t z  vectors necessari ly accurately represent the t o t a l  force  applied t o  
the s t ructure .  
The R i t ~  procedure f o r  dynamic analys is  has been preaented. The procedure 
is re l s t ed  to other popular methods of eigenvalue/eige~.vector analjrsia; how- 
ever, it does not httempt to  obtain a l l  the modes of the system. Instead, 
only l tose  "modes" which a r e  excited bg the dynamic loading are obtainea. 
A flow char t  of the IMAP impleme2tation has been presented. RASTRAH's 
DUAP capabi l i ty ,  a l o a  d t h  +\le use of a dummy module, made t h i s  irplementation 
f a i r l y  simple. The method was compared to the PEER method f o r  one problem. 
The predicted frequencies compared ue l l ,  while the cost  of the Ritz method was 
less.  The same conclusions were shown by Arnold, Citerley,  et al .  with more 
dramatic cos t  savings fo r  very l a rge  analys is  problems. These r e s u l t s  c l e a r l y  
demonstrate the impact tha t  the Ritz algorithm can have on reducing ana lys i s  
cost. 
The benef i t s  of the Ritz algorithm in dynamic ana'qsis can therefore be 
summerized as: 
1. Order of magnitude ;,duction i n  deriving eigenvalues and eigenvectors f o r  
mode superposition analysis .  
2. Better  accuracy f o r  fewer vectors. 
3. No s t a t i c  correction needed f o r  higher order modes. 
4. No s t a t i c  condensation o r  Guyan reduction required f o r  l s r g e  systems. 
5- Error estimation can be made to  determine the adequacy of se lec ted Ri le  
vectors. 
REFERENCES 
! Jenninga, A., "Eigenvalue Uethoda fo r  Vibration Analysis 11," Shock & 
Vibration Digest, Vol. 16, No. 1, Jan. 1984. 
2. IVour-0mid. B.. Par le t t .  B. R.. and Tarlor. R. L.. "Lanczos versus Subspace 
~ t e r a t i o n -  for-  solut ion-  of Eigenvalue h-obiems, " bt . J. f o r  Numerical - 
Methods in Enginezrin., Vol. 19, pp. 859-871, 1983. 
3. Wilson, E. L. ,Yuan, ?I., and Dicksns, J., 'Dynamic .Inaiysis by E r e c t  
Superposition of R i  t z  Vectors, " Earthquake !hgineering and St ruc tu ra l  
~ W ~ C S ,  Val. 10, 1902. 
4. Arnold, R. R., Citerley,  R. L., Chargin, M., and Gelanl., D., "Application 
of R i t x  Vectors f o r  Dynamic Analya3 of Large Structures, '  presented a t  
the S y a p o s i * ~  on Advances and Trends Sn Structures  and m i c s ,  sponsored 
by AASA, Washington, DC, Oct. 1984, to  he published in Computers and 
Structures.  
5. Gupta, V.. Cole, 3.. '-24 Mock, U., "A Cost-Effective Eigensolution 
Method 'or Zscrc Systems w i t k t  Rockuell HASTRAH," Tueli th NAS'TEUB L'seras 
Colloqui~um, ' 32. 
5. !ktxar,, 3. and fianagan, P. P., 'Rigenvalue B t t r a c t i m  in RlSTRAX by the 
Tric'iagonal 9 d u c t i o n  (PEER) Method - Real Eigenvalue Analysis,' RASA 
CR-277., Aug. 1976. 
7. Reurnen, 11. aad b n n  F., " Complex Eigenvalue Extraction in IcASTRAX by the 
Tridiagcnal Reduciiion (FESR) Method,' NASA CU-145258, Sep* . 1977. 
8. Neur-Cmid, 3. and Zlough, P. U., '3gnanic Analysis of St ructures  ' J s ' , ~  
Ganczos ~ o o r d i n a t e s ,  ' ~ a r t % ~ u a l m  -eering &d S t ruc tu ra l  Dynamics, - 
i'ol. 12, pp. 565-577, 1984. 
TABLE 1 
FREQUEWY AWD TINE RESULTS FOR A W L  ma, 
R I T Z  VERSUS FEER 
~ I C  FREQUENCY 
FEER (tIZ) 
2 -83647 1E+02 
5.01 5972EW 
7 -401 528EW 
8.7921 %EM2 
9 -876671 E+02 
1 ,18921 OEM3 
1 -691 335€+03 
2.1 65648EW3 
2.467984EHI3 
2.9061 39E+03 
270 
R I n  (HZ) 
- 
1 
2 
3 
4 
5 
6 
7 
8 
3 
I 10 
I 
Total Run Time 
(cpu-sec) 
2.836471 € 4 2  
5 -01 5972E+02 
7.41 8652E+02 
8.908411 E+02 
9.876880€+@2 
1 ,18921 3E+03 
1 .902981E+G3 
2.1 78580EN3 
2.852047E+O3 
3.239252Ei03 
218 
* 1  
Process static loads ( f )  using SSG1, 
5562, and 5563 arodules 
I Calculate f i r s t  vector I 
. 
1 
Orthonormal i ze vectors using nODB 
A 
Calculate 
I Perform e i  genval ueleigenvector 
extraction usinq READ MODULE 
\ 
t e 
Equivalence O X  to PHIA for use i n  
rest of DMAP 
L 
Figure 1 Flow Chart for Implementing Ritz Procedure into NASTRAN 
233 
3 
Loop to calculate additional vectors 
x? = K-I H xTe1 
1 
i + i + l  
. 
Y2s 
a I 1 


Ritz Procedure for COSMIC/NASTRAN

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