The development of efficient techniques for calculating sensitivity derivatives is studied. The objective is to present a computational procedure for calculating sensitivity derivatives as part of performing structural reanalysis for large-scale problems. The scope is limited to framed type structures. Both linear static analysis and free-vibration eigenvalue problems are considered

Noor, Ahmed K.

Whitworth, Sandra L.

English

NASA Technical Reports Server

-N87-18878
SENSITIVITY ANALYSIS FOR LARGE-SCALE PROBLEMS
Ahmed K. Noor and Sandra L. Whitworth
George Washington University Center
NASA Langley Research Center
Hampton, Virginia
INTRODUCTION
Due to the wide variety of uses of sensitivity derivatives, the development of
efficient computational procedures for the calculation of these derivatives has at-
tracted considerable attention in recent years. The calculation of sensitivity
derivatives forms the backbone of many optimization procedures and is the major
contributor to the cost and time of optimization of large systems. In addition,
sensitivity derivatives have several other applications in structural mechanics in-
cluding approximate analysis (and reanalysis) techniques, analytical model improve-
ment, and assessment of design trends. A review of the state of the art in
sensitivity calculations is contained in a survey paper (Ref. I), a monograph
(Ref. 2), as well as in some papers in these proceedings. Despite all the recent
advances made, the calculation of sensitivity derivatives for large structural
systems (with large number of degrees of freedom and design variables), is quite
expensive even on present-day large computers.
The present study focuses on the development of efficient techniques for cal-
culating sensitivity derivatives. Specifically, the objective and scope of the
present paper are listed in Fig. I. The objective is to present a computational
procedure for calculating sensitivity derivatives as part of performing structural
reanalysis for large-scale problems. The scope of the paper is limited to framed
type structures. Both linear static analysis and free-vibration eigenvalue problems
are considered.
Objective
To present a computational procedure for calculating
sensitivity derivatives as part of performing structural
reanalysis for large-scale problems
Scope
• Frame-type structures
• Linear static analysis
• Eigenvalue problems
Figure 1
357
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BASIC IDEA AND KEY ELEMENTS OF THE PROPOSED PROCEDURE
The basic idea and the three key elements of the proposed procedure are listed
in Fig. 2. The basic idea is to generate the solution of the modified structure
using large perturbations from that of the original structure. The three key ele-
ments are: a) lumping of the large number of design variables into one (or a small
number of) tracing parameters; b) application of operator splitting/reduction
technique; and c) for very large problems, use of single-level or multilevel sub-
structuring. Only the first two key elements are discussed in this paper. The
application of operator splitting/reduction technique proved to be effective in
reducing the computational effort in a number of structural mechanics problems (see,
for example, Refs. 3 to 6).
Basic idea
• Solution for modified structure is obtained using large
perturbations from that of original structure
Keyelements
• Lumpingof design variables into tracing parameter(s)
• Application of operator splitting/reduction technique
• Useof multilevel substructuring (for very large problems)
Figure 2
358
APPLICATION TO LINEAR STATIC ANALYSIS
In Fig. 3 the application of the proposed procedure to linear static analysis
is outlined. The governing finite element equations of the original and modified
structures are shown. The global stiffness matrices, load vectors, and responses
of the original and modified structures are designated by [K]o , [K]; {P}o' {P}; and
{X}o, {X}, respectively. The original and modified structure characteristics cor-
respond to the values of d? and d. of the design variables, respectively.
i i
The operator splitting technique is now applied, and the equations of the
modified structure are expressed in terms of the original structure equations plus
correction terms. A tracing parameter X is introduced and is attached to the cor-
rection terms. The tracing parameter is dimensionless and identifies _ the design
modifications. The original structure equations correspond to X=0, and the modified
structure equations correspond to X=I.
Operator splitting
Original structure
[K]otXt° =tPto:[K]o
Modified structure
K]o + _, K]- [K]oIX}= {Plo+ _,{PI- {P}o
_, = 0 _ Original structure
}, = 1 --.- Modified structure
Figure 3
359
REDUCTION METHOD FOR STATIC PROBLEMS
The response of the modified structure, {X}, is now expressed as a linear com-
bination of a few preselected global approximation vectors (or modes). This is
expressed by the transformation shown in Fig. 4. The columns of the matrix IF] are
the global approximation vectors, and the elements of the vector {4} are the ampli-
tudes of the approximation vectors which are, as yet, unknowns. Note that the
number of global approximation vectors, r, is considerably smaller than the total
number of degrees of freedom, n.
A Rayleigh-Ritz technique is now used to approximate the governing equations of
the modified structure by a much smaller system of equations in the unknowns {4}.
Basis reduction
tXt n, 1 = [Fin, r t¢/t r, 1 ; r < < n
where {$} = amplitudes of global approximation vectors
Reduced system of equations
Rayleigh-Ritz technique used to approximate the equations of
the m_ified structure
K ° + _ K - K o = {Pt o -)' PI-IP}o
where [K]o = [r]t JK]o [r]
['KI= [rlt[Kllrl
° = lr-It {Pt0
N
{P} - lrl tIP}
Figure 4
360
SELECTIONANDEVALUATIONOF THEGLOBALAPPROXIMATION
The effectiveness of the proposed procedure depends, to a great extent, on the
proper choice of the global approximation vectors. In the present study the global
approximation vectors are selected to be the response of the original structure,
{X}o_ and its various-order derivatives with respect to the parameter X. The re-
cursmon relations for evaluating the approximation vectors are obtained by successive
differentiation of the original finite element equations. Note that the matrix on
the left hand sides of these equations, [K]o , is the same(see Fig. 5).
Ixl= Jr]t_}
_X _2X
where[r]=[txtoI_l0I--I0 .]
_2 " "
[K]o {X}o = {P}o
_([.].[.]o) X o+tPt- P,[K]° _-fo
_2X _X["]o,_}o:-_(["]- e"]o){_}o
0
Same left hand side•
Figure 5
361
COMPUTATIONAL PROCEDURE
The computational procedure for generating the solution of the modified struc-
ture and the sensitivity derivatives is outlined in Fig. 6.
The first step is to generate the global approximation vectors at %=0 through
decomposition of the stiffness matrix of the original structure. The derivatives
with respect to % provide information about the sensitivity of the response to
the design modifications. Because % is dimensionless, the derivatives with respect
to X have the same dimensions as the original response quantities, and consequently
an assessment of the effect of desig_ modifications on the response can be easily
made.
The second step is to generate the reduced equations and solve them for the
amplitudes of the global approximation vectors.
• Evaluate global approximation vectors at _ = 0
• . Derivatives with respect to ;_ represent sensitivity of
the response to design modifications
• Generate reduced equations
Solve reduced equations and find amplitudes of
global approximation vectors
Figure 6
362
RELATIONSHIPBETWEENTHEPRECONDITIONEDCONJUGATEGRADIENT(PCG)
TECHNIQUEANDTHEPROPOSEDCOMPUTATIONALPROCEDURE
If the proposed computational procedure is contrasted with the preconditioned
conjugate gradient (PCG)technique in which the preconditioning matrix is selected
to be the global stiffness matrix of the original structure, [K] , the relationshipsoshownin Fig. 7 can be identified. These relations express the preconditioned re-
siduals {Y}o' {Y}I' ... of the PCGtechnique in terms of the global approximation
aX {82X}
vectors of the foregoing technique, {_}o' a%2 o' ....
Equivalence
)] +,(,P,]0+_([_][_]0txt=tPto
[K] o = preconditioning matrix
PCG Proposed procedure
XIo ""-"
Preconditioned
residuals
{Y}o
{Y}I
{Y}i
{X}o
0
c_X + 2 {_
_(1-C o) _-_ )
i+I Cj ,{c_Jxt
0
Figure 7
363
IMPLICATIONSOFTHESIMILARITIESBETWEENTHEPROPOSED
PROCEDUREANDTHEPCGTECHNIQUE
The implications of the similarities between the proposed computational pro-
cedure and the PCGtechnique are listed in Fig. 8.
For the PCGtechnique, the similarities can be exploited to provide a rational
approach for selecting the preconditioning matrix (as the global stiffness matrix
of the original structure), and a physical meaning for the preconditioned residual
vectors (in terms of sensitivity derivatives).
For the proposed procedure, someof the work done on parallelizing the PCGon
multiprocessor computers can be exploited.
PCG
• Rationalchoiceforpreconditioningmatrix
• Physical meaning for preconditioned residuals
(in terms of sensitivity derivatives)
Proposed procedure
• Exploiting work done on parallelizing PCG on
multiprocessor computers
Figure 8
364
APPLICATIONTOEIGENVALUEPROBLEMS
The application of the proposed computational procedure to free vibration
(eigenvalue) problems is outlined in Fig. 9. The governing equations of the origin-
al and modified structures are shown. Again, the operator splitting technique is
applied, and the stiffness and massmatrices of the modified structure are written
as the sumof the corresponding matrices of the original structure plus correction
terms. The correction terms are identified by the tracing parameter X. WhenX=0,
the original structure equations are recovered, and when X=I the modified structure
equations are obtained.
Original structure
Operator splitting
[[q0 0]I×10-0
Modified structure
or,
[[K]- n[M]]Ix_:o
[(N0 0 a
where
;_ = 0 --,,-- original structure
_, = I _ modified structure
[K]a = [K]-[K]o
[M]a = [M]-[M]o
=0
Figure 9
365
REDUCTION METHOD FOR EIGENVALUE PROBLEMS
The application of the reduction method to the free vibration problem is outlined
in Fig. i0. As in the static analysis, the eigenvectors of the modified structure,
{X}, are approximated by a linear combination of a few global approximation vectors.
This is accomplished by the transformation shown. An efficient choice of the ap-
proximation vectors was found to be a few eigenvectors for the original structure
(corresponding to X=0) and their derivatives with respect to X, evaluated at X=0.
Then, the Rayleigh-Ritz technique is used to approximate the original large
eigenvalue problem by the reduced one shown in Fig. i0. The solution of the reduced
eigenvalue problem gives the amplitudes of the global approximation vectors.
Basis reduction
• Eigenvectors of modified structure, IX}, are approximated by:
tX}n, 1 = [F]n, r lOtr, 1 ; r <<n
where
2 X=O
Reduced system of equations
• Rayleigh-Ritz technique is used to approximate the original
eigenvalue problem by a reduced one
where
o- Q[M] 0) +;_
[_1o
[K!a
I_1o
I_1_
- _[M] Ie}= o
a
=lrltlKlolrl
=lrltlKla It]
=lrlt lMlolrl
=lrltlMlalrl
Figure I0
366
EVALUATIONOFGLOBALAPPROXIMATIONVECTORS
The recursion equations used in generating the eigenvectors at %=0and their
first derivatives with respect to %are shownin Fig. ii. Note that the left hand
sides of all these equations are the same. The expression of the first derivatives
of the eigenvalues with respect to %appearing on the right hand sides of the equ-
ations are given in Fig. ii.
Since the matrix on the left hand side of the recursion equations used in
evaluating these derivatives is singular, the solution of each set of equations can
be expressed as the sumof a homogeneous olution (multiple of the eigenvector) and
a particular solution, {Q}I The equations used in evaluating the particular solu-
tion {Q}I are given in Fig#'ll. The details of this procedure are given in Ref. 5.
Recursion formulas
0 0
IN [M]0 0
IXl = 0
+
=,o,,÷c,x,
Where I Q} 1 = particular solution.
Figure ii
367
COMPUTATIONALPROCEDUREFOREIGENVALUEPROBLEMS
The procedure for extracting the eigenvectors of the modified structure and for
generating the sensitivity of the eigenvectors to design modifications is outlined
in Fig. 12.
First: A few eigenvectors of the original structure (corresponding to %=0)are
generated.
Second: The derivatives of the eigenvectors with respect to %are generated at
%=0. In the process, derivatives of the eigenvalues are also computed. These de-
rivatives provide sensitivity information regarding the effect of all the design
modifications on the eigenvectors and eigenvalues. The reduced equations are
generated.
Third: The reduced eigenvalue problem is solved at %=1.
• Generateeigenvectors for original structure (_ = O)
Generateglobal approximation vectors (derivatives of
eigenvectors w.r.t. _) and reduced equations. In the
process, derivatives of Q w.r.t. _ are computed
• Solve reduced eigenvalue problem at _ = 1
Figure 12
368
CANTILEVEREDLATTICETRUSS
To assess the effectiveness of the proposed computational procedure, a number
of problems were solved by this procedure. Comparisonwas madewith the direct solu-
tion of the structure. Herein a typical problem of a five-bay cantilevered lattice
truss is considered (see Fig. 13). In the original structure all the longerons had
the samecross section, and all the battens and diagonals had the samecross section.
The design variables consisted of the cross sectional areas, momentsof inertia and
torsional constants. The characteristics of the original and modified structures
are given in Fig. 13.
Z
,v,_r st bay
I. '1
0.75m
16 Design variables (8 varied)
4 Cross-sectional areas
8 Moments of inertia
4 Torsional constants
rt_Pe: _ 109x 12 4!109xi 4 108xj ¢P ty 10 x A m" m 3 m m
0.3
0.15
0.075
0.04
6.0
O.65
1.5
0.15
6.0
0.65
1.5
0.15
Original design: Longerons - type 1
Battens and diagonals -- type 2
Modified design: First bay -- Longerons -- type 1
Battens and diagonals -- type 2
Other bays - Longerons -- type 3
Battens and diagonals -- type 4
1.2
0.13
0.3
O.03
Figure 13
369
STATICLOADING
The first problem considered is that of the static response due to a transverse
load in the z direction at the free end of the cantilever. Figure 14 showsa sum-
mary of the results. The transverse displacement w and rotation at the free end
(point a) of the original structure are given. The sensitivity of these quantities
_X {_-X} Also shownin Fig. 14 areto design modifications is provided by {_}oand 212
the corresponding w and _2 for the modified structure (which are considerably larger
than those for the original structure). The solution obtained using the proposed
procedure with four global approximation vectors was identical to the direct solution
of the modified structure to at least three significant digits.
Original
structure
X-O
Modified
structure
;_- 1
Static analysis
Ixl
Ixl
(1_,21o
w at a ¢ 2at a
O. 102
O.0737
O. 110
O. 394
O. 394
-0.0397
-0. 0257
-0. 0388
-0.142
-0.143
Full system O. 394 -0. 143
.:_."-...
..:• ....
....,;........,,.
...L:_......-
Figure 14
370
FREEVIBRATIONS
The second problem considered is that of the free vibrations of the samelattice
structure. Figure 15 shows a summaryof the results. The first three eigenvalues
(squares of the vibration frequencies) and their first two derivatives with respect
to I are listed. The corresponding eigenvalues of the modified structure are also
listed. The frequencies predicted by the proposed procedure with eight approximation
vectors (four eigenvectors and their first derivatives with respect to I) and twelve
approximation vectors (four eigenvectors and their first two derivatives with respect
to X) are listed. The predictions of the eight-vector approximation are accurate for
the first two eigenvalues, but not the succeeding ones. On the other hand, the pre-
dictions of the twelve-vector approximation are accurate for the first three
eigenvalues.
Original
structure
_,=0
Modified
structure
X=I
lO-5x Q
O_
lO-5x a%
-5 c_2Q
10 x 2
-510 x
Q
r=8
r= 12
Full
system
Mode
O.324
-0.134
-0.0911
O.122
O.122
O.122
2
O.502
-0. 207
-0.141
0.191
0.191
0.191
3
1.815
-0.486
-0. 472
0.583
O.896
O.896
Figure 15
371
MODE SHAPES
The first three mode shapes of the modified structure are shown in Fig. 16.
Note that the first two vibration modes are bending modes and the third is a
torsional mode.
Free vibrations
._'',.,
-: ; ._.
."" •N""""-""
. _.,- .... -
•,., : .,; •
•, i .": l'. • • • ,.
• : ..," ...:." ..: "... • ".
: ,...:... - .......'..
..-., • ." . . .." .-. .
%.o . _. . .1o.. %.
,4 I I 14 e I I I . I, el %_ 41
• ° eo" ." .. t--
.": .._," : "" ......-. :.
: ," • _.'" "i""
:,-, .: • ...,.:.
• , °. ,. •
.°1 .._ .'..
:: ," ...
. . _: :
Q - 0.122K105 Q - 0.191x105 Q - 0.896 x 105
I 2 3
Figure 16
372
SU_fl_ARY
In summary, a computational procedure has been developed for calculating the
sensitivity derivatives of large structural systems as part of structural reanalysis
(see Fig. 17). The three key elements of the procedure are:
a) lumping of the large number of design variables into one (or small number
of) tracing parameter(s);
b) application of operator splitting/reduction technique; and,
c) for very large problems use of multilevel substructuring technique.
The proposed procedure can be considered as a general computational strategy for
generating the response of the modified structure using large perturbations from the
response of the original structure.
For static problems the similarities between the proposed procedure and precon-
ditioned conjugate gradient technique are identified and are exploited to provide a
rational procedure for selecting the preconditioning matrix and a physical meaning
for the preconditioned residual vectors.
Future work includes:
o extension to more complex structures and to shape design modifications
o generation of sensitivity information with respect to design variables.
• Computational procedure presented for calculating
sensitivity derivatives as part of performing
structural reanalysis for large-scale problems
• Lumping of design variables into tracing parameter(s)
• Application of operator splitting/reduction technique
• Use of multilevel substructuring
• Future work includes:
• Extension to more complex structures and to shape
design modifications
• Generation of sensitivity information w.r.t, design variables
Figure 17
373
I.
.
.
.
.
.
REFERENCES
Adelman, H. M. and Haftka, R. T.: Sensitivity Analysis of Discrete Structural
Systems. AIAA Journal, vol. 24, no. 5, May 1986, pp. 823-832.
Haug, E. J., Choi, K. K., and Komkov, V.: Design Sensitivity Analysis of
Structural Systems, Academic Press, New York, 1986.
Noor, A. K. and Peters, J. M.: Model Size Reduction Technique for the Analysis
of Symmetric Anisotropic Structures. Engineering Computations, vol. 2, no. 4,
December 1985, pp. 285-292.
Noor, A. K. and Peters, J. M.: Nonlinear Analysis of Anisotropic Panels. AIAA
Journal, vol. 24, no. 9, September 1986, pp. 1545-1553.
Noor, A. K. and Whitworth, S. L.: Model Size Reduction for the Buckling and
Vibration Analyses of Anisotropic Panels. Journal of the Engineering Mechanics
Division, ASCE, vol. 113, no. 2, February 1987, pp. 170-185.
Noor, A. K. and Whitworth,S. L.: Model Size Reduction for the Analysis of
Symmetric Structures with Unsymmetric Boundary Conditions. International
Journal for Numerical Methods in Engineering, 1987.
374


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