Aircraft dynamic analyses are demanding of computer simulation capabilities. The modeling complexities of semi-monocoque construction, irregular geometry, high-performance materials, and high-accuracy analysis are present. At issue are the safety of the passengers and the integrity of the structure for a wide variety of flight-operating and emergency conditions. The technology which supports engineering of aircraft structures using computer simulation is examined. Available computer support is briefly described and improvement of accuracy and efficiency are recommended. Improved accuracy of simulation will lead to a more economical structure. Improved efficiency will result in lowering development time and expense

Chargin, Mladen

Melosh, R. J.

English

NASA Technical Reports Server

N89- 2 4 6 5 5  
IMPROVING TRANSIENT ANALYSIS T E C H N O L O G Y  
F O R  A I R C R A F T  STRUCTURES 
R .  J .  M e l o s h  
Duke  U n i v e r s i t y ,  D u r h a m ,  N C  
M l a d e n  C h a r g i n  
NASA A m e s  R e s e a r c h  C e n t e r ,  M o f € e t t  F i e l d ,  C A  
1. INTRODUCTION 
A i r c r a f t  d y n a m i c  a n a l y s e s  a r e  d e m a n d i n g  o f  c o m p u t e r  
s i m u l a t i o n  c a p a b i l i t i e s .  The m o d e l i n g  c o m p l e x i t i e s  o f  s e m i -  
m o n o c o q u e  c o n s t r u c t i o n ,  i r r e g u l a r  g e o m e t r y ,  h i g h - p e r f o r m a n c e  
m a t e r i a l s ,  a n d  h i g h - a c c u r a c y  a n a l y s i s  a r e  p r e s e n t .  A t  i s s u e  
a r e  t h e  s a f e t y  o f  p a s s e n g e r s  a n d  t h e  i n t e g r i t y  o f  t h e  
s t r u c t u r e  f o r  a w i d e  v a r i e t y  o f  f l i g h t - o p e r a t i n g  a n d  
e m e r g e n c y  c o n d i t i o n s .  
F i g u r e  1 i s  a s k e t c h  o f  a t y p i c a l  s t r u c t u r e .  I t  
d i p i c t s  o n e  o f  NASA A m e s  d e s i g n s  o f  a n  o b l i q u e  w i n g .  T h e  
w i n g  c h o r d  v a r i e s  f r o m  1 8 . 3 6  i n c h e s  a t  t h e  r o o t  t o  37.8 
i n c h e s  a t  i t s  254.4 i n c h  s p a n .  T h e  s k i n s  a r e  f o r m e d  o f  a 
0°/*45"/900 76%/14%/10% g r a p h i t e / e p o x y  c o m p o s i t e .  T h e  
s k i n  v a r i e s  i n  t h i c k n e s s  f r o m  .625 i n c h  a t  t h e  r o o t  t o  
.184 i n c h  a t  t h e  t i p .  T h e  s k i n s  a r e  s u p p o r t e d  by 5 
v e r t i c a l l y  s t i f f e n e d  s p a r s  a n d  14 s t i f f e n e d  r i b s .  A l l  t h e  
s u p p o r t  s t r u c t u r e  i s  d e s i g n e d  i n  a l u m i n u m .  T h e  w i n g  m u s t  
b e  p r o o f e d  a g a i n s t  l a n d i n g ,  l i f t  a n d  d r a g ,  g u s t ,  b u f f e t ,  
v i b r a t i o n ,  a n d  o s c i l l a t i n g  a e r o d y n a m i c  l o a d i n g .  
T h e  f i g u r e s  a n d  t e x t  t h a t  f o l l o w  examine t h e  t e c h n o l o g y  
w h i c h  s u p p o r t s  e n g i n e e r i n g  o f  a i r c r a f t  s t r u c t u r e s  u s i n g  
c o m p u t e r  s i m u l a t i o n .  T h e y  b r i e f l y  d e s c r i b e  a v a i l a b l e  
c o m p u t e r  s u p p o r t  a n d  r e c o m m e n d  i m p r o v i n g  a c c u r a c y  a n d  
e f f t c i e n c y .  I m p r o v e d  a c c u r a c y  o f  s i m u l a t i o n  w i l l  l e a d  t o  
more e c o n o m i c a l  s t r u c t u r e .  I m p r o v e d  e f f i c i e n c y  will r e s u l t  
i n  l o w e r i n g  d e v e l o p m e n t  t i m e  a n d  e x p e n s e .  
2. SIMULATION SUPPORT 
F i g u r e  2 l i s t s  t h e  d y n a m i c i s t s '  t a s k s  f o r  c o m p u t e r  
s i m u l a t i o n  of t r a n s i e n t  a n a l y s i s .  D y n a m i c i s t s  d e f i n e  t h e  
f i n i t e - e l e m e n t  r e p r e s e n t a t i o n  o f  t h e i r  s t r u c t u r e  a n d  i t s  
b o u n d a r y  c o n d i t i o n s .  T h e y  s e l e c t  t h e  p r o c e d u r e s  t o  u s e  i n  
i n t e g r a t i n g  t h e  e q u a t i o n s  o f  m o t i o n  o v e r  t i m e ,  a n d  d e f i n e  
t h e  m o d e l s  a n d  e x t e n t  o f  s t r e s s  e v a l u a t i o n .  T h e y  i n t e r p r e t  
a n a l y s i s  r e s u l t s  w i t h  r e s p e c t  t o  t h e  r e a l  s y s t e m ,  d r a w i n g  
343 
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upon their knowledge of the models, algorithms, and the 
computer configuration which implements the simulation. 
Figure 3 identifies the computer capabilities which 
support implementation of the tasks of Figure 2. Existing 
finite-element models provide for both Rayleigh-Ritz and 
heuristic models. Three methods o f  reducing the vector 
basis, four classes of numerical quadrature, and at least 
three processes for evaluating stresses are available. 
Interpretation software facilitates plotting and tabulating 
data. 
3 .  ACCURACY ASSESSMENT 
Figure 4 is typical of the type o f  data that would be 
useful to the dynamicist in assessing analysis accuracy. 
The continuous folded line on this figure plots the actual 
spatial discretization error for the first two resonant 
frequencies. The dashed folded line portrays the error 
predicted using accuracy qualifying logic. 
Figure 5 shows similar data qualifying the prediction 
of transient response with respect to spatial discretization 
error. The fact that this error can accumulate during the 
history emphasizes the need for continuous monitoring of 
this error source. 
Figure 6 notes the principal sources of inaccuracy in 
each of the simulation tasks. The sources include spatial 
discretization, time discretization, process, round-off, 
idealization, and human errors. These sources induce 
accuracy loss in each task which can accumulate from task 
to task and obliterate accuracy. 
Figure 7 is a bar chart of the comprehensiveness o f  
support o f  each error source in contemporary simulations. 
No known production computer code is complete with respect 
to any source. Most codes provide partial protection 
against process and roundoff error only. Consequently, we 
cannot regard transient analysis results as reliable. For 
some of these sources, new technology is needed to deter- 
mine accuracy; for others, suggested techniques require 
evaluation; and for the rest, only implementation in 
production codes is necessary. 
4 .  ANALYSIS EFFICIENCY 
Figure 8 cites the sources of inefficiency in 
simulation tasks with respect to technology and software. 
These sources involve use of non-optimum models, inappro- 
priate integration algorithms, and unsuitable space and 
time grids. Lack of efficiency measures in computer codes 
344 
inhibits experimental improvement of simulation efficiency 
in practice. 
Figure 9 illustrates the inefficiency of available beam 
models for predicting modal frequencies. This figure shows 
the logarithmic relation between the number of modal 
frequencies and the equivalent number of elements and nodal 
variables. The first is a measure of the computer resources 
needed for equation coefficient; the second, those needed 
for equation integration. The data show that the efficiency 
of  the Bernouilli-Euler beam model is less than 50 
percent of that of the ideal model. 
Figure 10 focuses on the efficiency of nodal siting 
€ o r  the beam. The abscissa of the graph measures the num- 
ber of calculations. The ordinate indicates the number of 
accurate modes. These curves illustrate the existence of 
a distinct optimum grid for each mode. Analysis using the 
optimum grid requires only one-third the calculations of 
the average grid. 
Figure 11 gives the conventional wisdom for selecting 
the time integration process of transient analysis. This 
table pertains to linear dynamic analyses. Considering the 
number of calculations, the data indicates that a different 
algorithm is advisable depending upon whether the frequency 
content of response is high or low and whether the 
integration time is brief or extended compared with the 
period o f  the fundamental mode. Comparing the best to the 
worst choice of algorithm we find an advantage of a factor 
which is a function of the order and band o f  the integration 
operator matrix. 
Figure 1 2  provides data for comparing the efficiency of 
integration algorithms for a highly nonlinear transient 
analysis of a cylinder. These data indicate that explicit 
(central differences) and explicit (Newmark Beta) are 
competitive but modal synthesis is not. Choosing the better 
algorithm may reduce the number of calculations to 1 / 1 0 0  o f  
those of modal synthesis. 
Figure 1 3  summarizes the potential for improving 
simulation efficiency by improving both models and 
algorithms. It indicates the opportunity for reducing the 
number of calculations by three orders o f  magnitude. 
5. CONCLUSIONS 
Now, computer implementation of transient analysis of 
aircraft structures provides for accurate response 
predictions. The dynamicist can hope to determine the 
accuracy of his particular simulation only by "heroic" 
345 
efforts. Steps he may make to satisfy his desire f o r  
efficient analysis are heuristic. 
Thus, desirable new technology includes a validated 
comprehensive set o f  simulation accuracy and relative 
efficiency measures. Using these measures to identify 
research opportunities will lead naturally to better models 
and data processing algorithms. 
The ultimate benefit o f  accuracy measures will be that 
dynamicists will have the data they need to more fully 
understand and interpret the computer's time histories. The 
ultimate benefit of efficiency measures will be exploitation 
of the potential to reduce the number of  calculations o f  
transient analysis by one to three orders o f  magnitude. (Fig. 1 4 ) .  
346 
. A c c u r a c y  
. Efficiency 
F i g u r e  1 .  T y p i c a l  s t r u c t u r e :  N A S A  Ames d e s i g n  of o b l i q u e  
w i n g .  
FORM EQUATIONS OF NOTION 
. SELECT ELEMENT MODELS . GENERATE ELEMENT MATRICES 
. DESIGN, REDESIGN MESHES . ASSENBLE EQUATION COEFFICIENTS 
INTEGRATE EQUP.TIO#S OVER TINE 
. SELECT BASIC VECTORS . SELECT INTEGRATION RULES 
. REDUCE EQUATIONS . EVOLVE TIME HISTORIES 
EVALUATE RESPONSE DATA 
. FIND P E A K  D I S P L A C E M E N T S  
. SELECT STRESS FORMULAS 
. EVALUATE STRESSES 
. FIND PEAK STRESSES 
INTERPRET SINULP.TIOH RESULTS 
.VALIDATE MODELS, PROCESSES . QUALIFY ANALYSIS RESULTS 
.VERIFY PRODUCTION ANALYSIS . RELATE TO THE PHYSICAL 
F i g u r e  2 .  Tasks  of s i m u l a t i o n .  
347 
FORM EQUATIONS OF MOTION 
.LIBRARY OF ELEMENT MODELS .DATA GENERATORS 
.MODELING EXPERIENCE 'GRID GENERATORS AND SMOOTHERS 
.ASSEMBLY P R O C E S S  . GEOMETRY PLOTTERS 
.ELEMENT MODEL GENERATORS 
.MODAL, GUYAN, AND RITZ VECTOR TECHNOLOGY 
.SPARSE EATRIX OPERATIONS 
.EXPERIENCE IN TRANSIENT ANALYSIS 
. DUHAMEL, EXPLICIT, IYPLICIT, AND QUASI-STATIC INTEGRATORS 
EVALUATE RESPONSE DATA 
. INTERPOLATION L GIC 
. DISPLACEMENT DIFFERENTIATION, D'ALEMBERT, INERTIA RELIEF 
' P E A K  S T R E S S  L O G I C  
METHODS 
INTERPRET SI  MULAT 1 ON RESULTS 
' ELEMENT VALIDATION T E S T S  . SENSITIVITY ANALYSIS 
' P R O C E S S  VALIDATION TESTS . TABULATIONS AND RESPONSE P L O T S  
E R u o a  MEASURES . DATA RETRIEVAL LOGIC 
Figure 3. S u p p o r t i n g  s i m u l a t i o n  technology. 
30 
25 
20 
15 
10 
5 
0 
t 
actual error - - - - - estimated 
- 1' 
1 2  4 8 
Number of finite elements 
16 
F i g u r e  4 .  Spatial discretization errors in eigenvalues. 
348 
C 
3 
c 
m 
- a c t u a l  error  9 t - - - es t imated  I 
c .rl 
k 
0 k
k 
al 
al 
> rl 
U 
4 
al 
m 
* t  
7 / 
4 e lements  
227 
1 1 6  e l e m e n t s ,  
0 
. 5 ?  .02 . C3 .03 .C5 .06 . :7 .58 . ? ?  .10 
Time a f t e r  impact  s e c o n d s  
F i g u r e  5 .  T r a n s i e n t  a n a l y s i s  d i s c r e t i z a t i o n  e r r o r s .  
FORM EQUATIONS OF NOTION 
. HUMAN E R R O R  
' lDEALlZATlON OF GEOMETRY, MATERIALS, B.C.  ' SPATIAL DISCRETIZATION 
INTEGRATE EQUATIONS OVER TIRE 
' SPATIAL DISCRETIZATION 
' TIME DISCRETlZATlDN 
' INTEGRATION PROCESS 
. ROUND-OFF 
EVALUATE RESPONSE DATA 
' S P A T I A L  DISCRETIZATION 
' T I M E  DISCRETlZATlON 
'STRESS EVALUATION PROCESS 
. ROUND-OFF 
INTERPPET SIMULATION RESULTS 
. Hur,AN ERROR 
' TECHNOLOGY OMISSIONS 
F i g u r e  6 .  S o u r c e s  of i n a c c u r a c y  i n  t r a n s i e n t  a n a l y s i s .  
349 
1 00 
90 
80 
70 
60 
50 
4 0  
30 
2 0  
1 0  
0 
Error source 
F i g u r e  7 .  C o n t r o l  of i n a c c u r a c i e s .  
FORM EQUATIONS OF NOTION 
' INEFFICIENT LEMENT MODELS 
. NON-OPTIMUM GRIDS 
. NON-OPTIMUM MESHES 
INTEGRATE EQUATIONS OVER TlFlE 
. IRRELEVANT GENERALIZED COORDINATES 
. EXCESSIVE TIME STEPS 
. IRREGULAR TIME STEPS 
EVALUATE RESPONSE DATA 
. EXCESSIVE SAMPLING IN TIME 
. EXCESSIVE SAMPLING IN SPACE 
INTERPRET SI M U L A 1  I ON RESULTS 
' INADEQUATE AND INEFFICIENT DOCUMENTATION 
. INSUFFICIENT VALIDATION 
. INEXPERIENCED ANALYSTS 
F i g u r e  8. S o u r c e s  of inefficiency. 
350 
RATIONAL - . . . . . .  
0 4 E 12 16 20 24 28 3: 
EQUIVALENT NUMEER OF ELEMENTS C0/40 
3 0 E 16 24 32 40 48 56 6 4  
I 
EOUIVALENT NUMBER OF EOUATONS C,/4 
F i g u r e  9 .  E f f i c i e n c y  of  beam e lement  m o d e l s .  
v a
a 
3 
V 
V a 
Ol  1 I I I I 
0 2 4 ' 6 .  8 IO 
EQUIVALENT NUMBER OF FINITE ELEMENTS 
F i g u r e  10 .  E f f i c i e n c y  of g r i d  d e s i g n s .  
351 
ANALY S I S 
CLASS (1) 
H . B .  
L.B. 
H.E. 
L.E.  
BEST INTEGRATION N O .  OF 
ALGOR1 THM CALCS . ( 2 )  
C e n t r a l  or Newmark Nb2 
Wilson  R i t z  16Nb 
t 
Newmark 4 N b 9  
A t i  
Modal S y n t h e s i s  8 N k  
A t i  
N 
b 
/b 
N 
(1) 
H = h i g h  f r e q u e n c y  r e s p o n s e  i m p o r t a n t :  L = l o w  
B = b r i e f  p e r i o d  of i n t e g r a t i o n ;  E = e x t e n s i v e  p e r i o d  
( 2 )  
N = no .  o f  e q u a t i o n s  of m o t i o n  
b = semi -band  w i d t h  o f  i n t e g r a t i o n  o p e r a t o r  m a t r i x  
tp = p e r i o d  o f  i n t e g r a t i o n  
A t i  = t i m e  s t e p  r e q u i r e d  
( 3 )  
Comparing C e n t r a l  D i f f e r e n c e s ,  Newmark Beta, Modal 
S y n t h e s i s ,  a n d  W i l s o n ' s  R i t z  Vectors me thods .  
Figure 11 .  Efficient time integration-linear. 
Intrusions Goal '9 - Accelerations Goal -
M C 
INTEGRATION APPROACH 
F i gure 12. Calculations f o r  nonlinear analysis. 
352 
SIMULATION APPROACH IMPROVEMENT 
WITH TIME 
F i g u r e  1 3 .  I m p r o v e m e n t  of  s i m u l a t i o n  e f f i c i e n c y .  
STATE-OF-THE-ART 
CAPABILITY FOR ACCURACY EXISTS 
ACCURACY NOT WELL QUANTIFIED 
EFFICIENCY EMPIRICAL 
PROSPECTIVE IMP R O V E M E N T S  
COMPREHENSIVE SET OF ACCURACY SENSORS 
A SET OF EFFICIENCY SENSORS 
BENEFITS 
SELF-QUALIFIED TRANSIENT ANALYSIS 
MUCH IMPROVED EFF I c I ENCY 
F i g u r e  1 4 .  Conclusions. 
353 


Improving transient analysis technology for aircraft structures

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