Two related methods for improving the dependent (OMIT set) displacements after performing a Guyan reduction are presented. The theoretical bases for the methods are derived. The NASTRAN DMAP ALTERs used to implement the methods in a NASTRAN execution are described. Data are presented that verify the methods and the NASTRAN DMAP ALTERs

Allen, Tom

Cook, Greg

Walls, Bill

English

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IMPROVED OMIT SET DISPLACEMENT
RECOVERIES IN DYNAMICS ANALYSIS
Tom Allen
Greg Cook
Bill Walls
McDonnell Douglas Aerospace
Huntsville Division
N94-17829
ABSTRACT
Two related methods for improving the dependent (OMIT set) displa_ments after performing a Guyan
reduction am presented. The theoretical bases for the methods am derived. The NASTRAN DMAP
ALTERs used to implement the methods in a NASTRAN execution are described. Data are pmsente, d that
verify the methods and the NASTRAN DMAP ALTERs.
1.0 INTRODUCTION
A NASTRAN usea" is faced with two major challenges when solving a dynamic eigenvalue problem.
First, an eigenvalue solution is expensive to perform for most structural problems encountcr_ in
engineeringanalysis,and second,many more degreesoffreedom (DOF) arcrequiredtodefinea
structure'selasticpropertiesthan arerequiredtodef'mcitsinertialproperties.
A popularmethod formeeting thesechallengesistoreducetheproblem sizeusing Guyan reduction
(ReferenceI). Guyan reductionallowstheu_crtopreservetheelasticpropertiesof the problem setwhile
reducing theproblem sizetoone thatismore manageable fora dynamic eigenvalueanalysis.At the same
tirnc,themass propertiesare alsocondensed with some .penaltyassociatedwith the redistributionof mass
from the coordinates elimi.'nated clmi_ngthe Guyan _rod.ucuon_ T__c_t dbedv_e_Pnl_g _hes
that correct the inaccuracies causco oy me conaensauon o_ u,_ ,,_,_,_ ,,,,,
solution time.
The theoretical development of the improvement methods is provided in Section 2. Section 3 describes the
NASTRAN DMAP ALTERs used to implement the algorithms used for both methods. Verification of the
two methods, the second of which is a refinement of the first, is presented in Section 4. Conclusions and
recommendations are provided in Section 5.
2.0 THE IMPROVEMENT METHOD
We begin by deriving the Guyan reduction scheme.
The dynamic eigenvalue problem is given by the equation
([K] - X[M]){¢} = 0 (1)
where
K __
M=
X =
¢ =
the structural stiffness matrix
the structural mass matrix
the system eigcnvalue
the eigenvector or modal displacements.
https://ntrs.nasa.gov/search.jsp?R=19940013356 2020-06-16T18:10:59+00:00Z
Wecan partition Equation 1 into independent DOF, designated in NASTRAN as the analysis set, or A-set,
and dependent DOF, designatedas theOMIT set,or O-seL Afterperforming thisoperationEquation I
becomes
where the subscript "a" denotes A-set DOF and the subscript "o" denotes O-set DOF.
Looking at the lower partitionf Equation 2 we can say
(2)
T T
kM_oO." 0K_,0 a + KooO o - _.MooOo = (3)
The Guyan reduction method (Reference 1) makes the assumption that the inertial forces on the O-set
displacements are much less important than the elastic forces transmitted by the A-set displacements. A
constraint equation for Guyan reduction can be derived by ignoring the mass terms in Equation 3. The
resulting constraint equation is given by
(4)
where
-I T
G O = -KooK.o (5)
This relationshipconstitutesa Ritztransformationof theeigcnvalueproblem. The wansfommtion written
in terms of the fulldisplacement setis
{} I°- o ,0.l-IOolI0:{_}= #o (6)
Using this Ritz transformation, the reduced mass and stiffness matrices become
[M..] = [G]T[M][G] (7)
and
[K..]= [G]T[K][G] (8)
The mass of the system is redistributed based upon the elastic connections between the O-set DOF and the
A-set DOF as shown in Equation 7,
The reduced mass and stiffness matrices shown in Equations 7 and 8, are then used m compute the
eigenvalues and the A-set displacements of the reduced system. Once the A-set displacements have been
computed, the Guyan reduction transformation of Equation 4 is used to recover the O-set displacements.
This back transformation ignores the inertial terms of the O-set displacements.
An improved back u-ansformation for ¢o can be found using Equation 3 (sec Reference 2). For mode i,
this back transformation is given by
{ o}i=-[Koo-XiMj' tK +¢o" (9)
Though Equation 9 will yield improved results, the first term on the right hand side must be inverted for
each mode calculated during the eigenvalue analysis, a computationaUy inefficient process. Clearly, a
more direct substitution would make the O-set displacement recovery more efficient.
Recasting Equation 3 for all the computed modes, we get
r ML#, _ Moo#o _ = 0 (I0)K.o0 a+ Koo0 o -
where _.isa squarematrixwith thesystem cigenvaluesalong thediagonal. Solving forthe
displac-erncntshatarcnot multipliedby X,we get
-I z -I (II)
Go_. + KooM_od_. _ + KooMoo#o _ = 0o
From Equation 11 we can see that a closed form solution for _ does not exist. It is possible, however, to
use Equation 11 to obtain an improved approximation to _.
A first approximation to ¢o can be determined by using the O-set displacements recovered by Equation 4,
or
00) Go¢, (12)
O
Substituting these O-set displacements into Equation 11 yields
-I T _-I M cb(I)_=¢(o2) (13)Go* . + X_M_o0, _ +--oo- -oo-o
where ¢_ arc the corrected O-set displacements. Thcs_corrccted displacements can be substituted back
into Equation 13 for ¢_) and a better approximation, ¢o, can bc computed. This process can be repeated
until the displacements at the (i +1) iteration arc the same as the displacements at the i th iteration. These
"super" improved displacements will bc identical to those computed using Equation 9, and can be
determined without the computational penalty associated with inverting an O-set by O-set sized matrix for
each mode.
To summarize, three methods for rex.ovcring the O-set displacements after performing the Guyan reduction
and the reduced eigenvalue analysis have been presented. These three methods arc:
1) Standard Guyan reduction recovery using Equation 4, henceforth designated as Guyan
displacements.
2) Improved O-set displacement recovery using Equations 12 and 13, henceforth designated as
improved displacements.
3) Successively iterated improved O-set displacements using Equation 13, henceforth designated as
t! " v!
super tmproved displacements.
10
The reader will note that the A-set displacements are identical for all three methods described above. It is
assumed that the eigenvalues and the A-set displacements computed dm'ing the eigenvalue analysis are
"accurate". In other words, the accuracy (or inaccuracy) of the Guyan reduction itself is not in question.
Thus far, we have discussed improvements only in the O-set displacements. More importantly, any
quantity computed using these O-set displacements, such as element forces or element suesses, will also
be improved by methods 2 and 3.
The theory and methodology for improving the O-set displacements has been provided. The following
section describes the implementation of the improved displacement recoveries in NASTRAN.
3.0 IMPLEMENTATION IN NASTRAN
With themethodology inhand, theimplementationinNASTRAN becomes an exerciseindefiningthedam
blocksand theNASTRAN DMAP modules requiredtoperform thedesiredoperations.The DMAP
ALTER sequences used to recover the improved displacements are provided in Figure 1. The first ALTER
placestheUPARTN module followingthe SldPI module while the second ALTER placestheDMAP
modules used torecoverthe improved displacementsafterthe SDRI module. The usercontrolsthe
recoverymethod with theparamctersdefinedintheDMAP ALTERs. The allowableparameter valuesand
theresultingactiontaken areprovided inTable I. Note thatifno A-set isdefined,theO-set recovery
sectionissldppcd.
$
$ DMAP Alter to obtaln required matrices for improvement. Place after the SNF2 Module.
ALTER il $ where il - DMAP statement number of Module SMP2
UPARTN USET, MFF/, MAOT,, MOO/*F*/*A"/*O* $
$
$ DMAP Alter to perform O-set displacement improvement. Place after tl_e SDRI Module.
ALTER JJ $ where JJ - DMAP statement number of Module SDRI
COND SKIPIM, OMIT $
$
$ This PARAM defines whether Guyan recovery or improvement
$ recovery is to be performed (NOIMP < O, Guyan recovery)
PARAH II*NOP*INOIMP - -1 $
COND SKI PIM, NOIMP $
S
$ This PARAM defines what recovery improvement will be performed
$ If NREPT = 0, improve once, NREPT • 0, iterate NREPT times
PARAH II*NOP'INREPT =I0 $
$
$ MATGEN creates a square matrix from the LAMA table
HATGEN LAHA/HLAHA/3 / 2 $
MPYAD GO, PHIA,/PHIO/0/1/0/ $
F_ LOO, ,MAOT/C1/1/1 $
SMPYAD CI,PHIA, MLAMA,,,PHIO/A/3///1 $
F_ LOO,, MOOIB/I/I $
LABEL IMPRV $
$MPYAD B, PHIO, MI2t_A,,,ICI31/II $
ADD A,C/PHIO/(1.0, 0.0) / (1.0,0.0) $
REPT IMP RV, NREPT $
UMERGE USET, PHIA, PHIO/PHIF/*F*/*A'/*O*
UHERGE USET,PHIF, IPHIN/*N'/*F*/*$* S
MPYAD GH, PHIN, IPHIMI011/0/ $
UMERGE USET, PHIN, PHIM/PHIG/*_ _/*N*/*M *
LABEL SKIPM $
Figure 1. O-set Displacement Improvement DMAP ALTERs
II
Table I.
Execution
No A-set
Guyan
Improved
Su_cr Improved
DMAP Parameter Settings
NOIMP
N/A
-l
0
0
I NREPT
NIA
N/A
0
#mp, ctitions
Once the O-set displacements have been recovered, the rest of the standard solution sequence is executed.
This allows the user to define all data recoveries using the familiar NASTRAN Case Control Deck
commands. Displacements, element forces, element stresses, or any other user requested data will be
printed and handled in the normal fashion. No special provisions are required to view the improved data.
4.0 METHOD VERIFICATION
Two sample problems were created to verify the method and the DMAP described in Section 3. The first
sample problem consists of a simple four story building. This problem was used to verify the
methodology and the DMAP ALTERs shown in Figure 1. The second problem consists of a 3600 DOF
substructured model. Element forces for this model were recovered from a transient response analysis
using the three O-set displacement recovery methods and compared to the benchmark element forces
obtained when no Guyan reduction was performed. These sample problems verify the improvement
methods and the DMAP ALTERs.
k i= 400.0
k== 800.0
_'-U l
_'2=U 2
_'3_U 3
K ==
m i= 2.0
ks= 1200.0
U 4
k(= 1600.0
\\\\\
m z=2.0
m s=2.0
m(= 2.0
400 -400 0
-400 1200 -800
0 -800 2000
0 0 -1200
I 2 0 0
0 2 0
M= 0 0 2
0 0 0
1/3 2/3Go = 0 317
fl = 1.278 Hz
A-set DOF = u z and u3
-1200
2800 -J
°100
2
Figure 2. Simplified Four Story Building
12
The four story building used for sample problem 1 is shown in Figt_ 2. This problem was selected
because it is easily represented with NASTRAN elements and may be solved using the NASTRAN
program. It may aLso easily be solved by hand so that the data produced by the DMAP ALTERs can be
verified. Data were recovered for the first mode only.
Table 2 pre_nts the O-set displacements for the three methods as well as the unreduoed benchmark
displacements. The data in Table 2 were recovered fzorn NASTRAN using the DMAP ALTERs described
in Section 3. The reader can easily verify that the Guyan results are identical to those recovered by hand
using Equation 4, the improved results are identical to those recovered by hand using Equation 11, and the
superimproved dataare identicaltothoserecoveredby hand usingEquation 9. These dataverifythe
DMAP sequence describedinSection3.
Disp.
u 1
Table 2.
Guyan
1.0000
Displacement Comparison
Improved
1.0000
Super •
Improved
1.0000
Benchmark(No A-set)
1.0000
u2 0.6015 0.6681 0.6764 0.6775
u3 0.4023 0.4023 0.4023 0.4069
u4 0.1724 0.1806 0.1810 0.1828
0.99995MAC
These data
0.999980.99730
were recovered using 10 iterations
N/A
The Medal Assurance Criterion(MAC) definedinReference4 isused tomeasure the accuracy of the
eigcnvectorsprovided inTable 2. MAC valueswillvary between zero,indicatingno correlationbetween
modes, tounity,indicatingperfectcorrelationbetween modes. Based on theMAC values,itisclearthat
both improvement methods produce better O-set displacements than the standard Guyan recovery method
produces alone.
The advantage of using the improved O-set recovery methods is clearer when element data, e.g. element
forces or stresses, are compared. The modal spring forces for all three O-set displacement recovery
methods are compared to the benchmark data in Table 3. From this it is clear that the improved
displacements produce spring forces that are vastly superior to those of Guyan reduction.
Based on this simple problem, the displacements can be dramatically improved by using the methods
described in Section 2. The next sample problem will show this more clearly.
The second sample problem uses the 3600 DOF Spacelab Pallet model shown in Figure 3. A transient
response analysis was performed with this model in two configurations, an unreduced configuration and a
Guyan reduced configuration. Transient element forces of all the bar elements were rex_vered using four
distinct PHASE3 executions, i.e. no A-set, Guyan, improved, and super improved.
13
Spring
Forces
F 1
F2
F3
AF I
&F 2
AF 3
AF 4
Table 3.
Guyan
159
159
276
276
30
-58
7
-17
Spring Force Comparison
Improved
133
213
266
289
4
-4
-3
4
129
219
266
29O
0
2
-3
-3
Benchmark
1
129
217
269
293
N/A
N/A
N/A
N/A
Figure 3. NASTRAN Model for Sample Problem 2
14
The maximum absolute values for all of the bar forces for the three recovery methods were compared to
the benchmark case. These comparisons are shown in Table 4. The data are arranged acccmiing to a
percentage difference range. For each of the recovery methods, the percentage of the forces falling within
this range as well as the maximum difference between the benchmark data and the data produced by the
three recovery methods within this delta percentage range are provided.
For example, in the range between two and five percent, 8.53 percent of the forces from the Guyan
recovery method fell within this range with the maximum difference between the Guyan recovered data
and the benchmark data being 397. For the improved recovery method, only 0.10 percent of the forces
fell into this range with a maximum difference between the benchmark and the improved data being 5. The
percentage of items failing in this range for the super improved method was 0.09, with a maximum delta
of 7.
Table 4. Bar Element Force Comparisons
for Sample Problem 2
A%
Range
0-2
2-5
5-10
10-25
25-50
>50
Percentage
in Range
89.05
G U_ ran
Maximum
IAI
Improved
* These
Percentage
in Range
1045 99.90
8.53 397 0.10
1.48 48 0.00
0.60 82 0.00
0.03 0 0.00
0.32 2281 0.00
lata were recovered using 10 iterations
1V'taxilnum
IAI
102
5 0.09
0 0.03
0
0
0
Percentage
in Range
99.76
0.00
0.00
0.13
Super Improved*
Maximum
lal
114
7
4
0
0
36
As was the case for the simplified model used for sample problem 1, the improved recovery methods
produce data that are superior to those data computed using Guyan reduction. The data appear to be the
most accurate for the simple improvement method. This is especially true when the computer CPU time
required to produce the data is considered. The improved displacement recoveries required 30 percent
more CPU time than the Guyan recovery, while the super improved displacement recoveries required 150
percent more CPU time than the Guyan recovery.
Because of the simplicity of this model, however, it would be premature to use these data to cast the super
improved method aside without first considering more complex models with equally complex loading.
5.0 CONCLUSIONS AND RECOMMENDATIONS
Two methods for improving the O-set displacements were provided. It was demonstrated that both
improvement methods produce O-set displacements that are vastly superior to those produced using the
standard Guyan recovery alone. In addition, the NASTRAN DMAP ALTERs required to perform these
operations were presented along with the supporting data used to verify them. It remains only to
determine whether the additional accuracy that may be obtainable through the iterative procedure of Method
15
3 is justified by the extra computational effort After all, a significant degree of approximation is already
guaranteed by the initial use of Guyan reduction to determine the A-set displacements.
Because this study did not provide enough information to determine which of the two improved recovery
methods was best suited for the problems encountered in most engineering applications, it is recorded
that additional studies be perfimned to compare improved displacements from a set of models with va_." g
complexity to the benchmark unreduced data. These additional cases can be used to definitively determine
which improvement method is better in terms of accuracy and computational efficiency. Finally, it would
be of great interest to compar_ the results of a multi-mode transient response analysis before and after
modal improvement to assess its practical siguificancc in tmns of the end result.
6.0
1. Guyan, R.J.,
2.
o
4.
REFERENCES
"Reduction of Stiffness and Mass Matrices," AIAA Journal, Volume 3, pg. 380, 1965.
Cook, R.D., Concepts and Applications of Finite Ele_, John Wiley & Sons Inc., New
York, Second Edition, 1981.
NASTRAN User's Manual, NASA SP-222, Volume I, June 1986.
Ewins, D.J., Modal Testing: Theory and Practice, John Wiley & Sons Inc., New York, June 1985.
16


Improved omit set displacement recoveries in dynamic analysis

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