0015-7949387 53.00 + o.a, Pergamoo Joutn~ls Ltd. MULTILEVEL SUBSTRUCTURING SENSITIVITY ANALYSIS Dvc T. NGUYEN Department of Civil Engineering, Old Dominion University, Norfolk, VA 23508, U.S.A. (Rewired 20 Februury 1986) Abstract-Solution techniques for handling large scale engineering optimization problems are reviewed. Potentials for practical applications as well as their limited capabilities are discussed. A new solution algorithm for design sensitivity is proposed. The algorithm is based upon the multileve1 substru~tu~ng concept to be coupled with the adjoint method of sensitivity analysis. There are no approximations involved in the present algorithm except the usual approximations introduced due to the discretization of the finite element model. It is a well-known fact that substructuring concept can be exploited in order to reduce computer core memory requirement and the total solution time executed. These advantages have been realized not only for statics analysis, but also for design sensitivity analysis. For truly large scale structures, however, it is necessary to use multilevel substructuring. The present paper has formulated a multilevel algorithm for design sensitivity analysis. A two-level substructuring example is used to verify the proposed algorithm. The sensitivity vector obtained by the multilevel algorithm is exactly identical to the one obtained by a more conventional approach. 1. ~NTRODU~ON Existing computer hardwares and available solution techniques for non-linear mathematical pro- gramming problems allow the engineers today to optimally design more complex and larger structures than has been done in the past. Efforts have been concentrated in the recent years to develop solution methods in both areas of sensitivity analysis [l-3] and optimization [Z-41 for efficient design of truly large scale systems. The use of substNctu~ng[S] and multilevel sub- stNctu~ng techniques in structural analysis has been reported extensively in the literature [6,7J. Approxi- mate reanalysis with substructures has also been presented in [7]. In the field of optimization schemes for large scale structures using substructures, the interactions be- tween substructures are accounted for by using the model coordination and the goal coordination techniques [3]. The techniques suggested in [3] have limited success and further work is needed to facili- tate its usefulness. A multilevel approach to design aircraft structures has also been reported in [S]. An- other promising approach to treat large scale en- gineering design problems has recently been intro- duced in[9] and [lo] where each substructure is optimized independently. The coupling effects be- tween substructures are included by using informa- tion based upon sensitivity of optimum solutions [ 111. A number of successful applications have been re- ported and the method described in [9] and [lo] is still being developed toward a state of maturity required for industrial applications. The field of sensitivity analysis is emerging as one of the more fruitful areas of engin~~ng research. The reason for this is the recognition of the many prac- tical uses for design sensitivity information. Beyond the historical use of sensitivity information in con- nection with forma1 mathematical optimization tech- niques, recent work has been reported in using sensi- tivity derivatives in approximate analysis, assessing design trends, analytical model improvement, and determining the effects of parameter uncertainties [I]. Sensitivity analysis using adjoint variable tech- nique in conjunction with substructures has been developed 12, 121.Further applications of this method to the design of fail-safe structures (optimal design of structures under various damaged conditions) and other complex structures (mixed finite elements, multilayer composite materials) have also been suc- cessfully reported [12] by the author. For truly large scale structure, however, dividing the original structure into many substructures may not always be beneficial since the overall number of boundary degrees of freedom (DOF) may also in- crease substantially and thus defeat substructuring purpose. One obvious solution to cope with this situation is to use multilevel substructures via the adjoint variable technique for ~nsitivity analysis. In Section 2, general ideas about bottom-up and top-down process for multilevel substructuring are briefly introduced. The statics and design sensi- tivity analysis with substructures is summarized in Section 3 to facilitate the discussion of the multilevel sensitivity algorithm in Section 4. A very simple example where solutions can be obtained with a hand calculator is presented in Section 5 to study the proposed algorithm. Finally, conclusions are drawn in Section 6. C.A.S. 2~:t-c
Transcript
0015-7949387 53.00 + o.a, Pergamoo Joutn~ls Ltd.
MULTILEVEL SUBSTRUCTURING SENSITIVITY ANALYSIS
Dvc T. NGUYEN
Department of Civil Engineering, Old Dominion University, Norfolk, VA 23508, U.S.A.
(Rewired 20 Februury 1986)
Abstract-Solution techniques for handling large scale engineering optimization problems are reviewed. Potentials for practical applications as well as their limited capabilities are discussed. A new solution algorithm for design sensitivity is proposed. The algorithm is based upon the multileve1 substru~tu~ng concept to be coupled with the adjoint method of sensitivity analysis. There are no approximations involved in the present algorithm except the usual approximations introduced due to the discretization of the finite element model.
It is a well-known fact that substructuring concept can be exploited in order to reduce computer core memory requirement and the total solution time executed. These advantages have been realized not only for statics analysis, but also for design sensitivity analysis.
For truly large scale structures, however, it is necessary to use multilevel substructuring. The present paper has formulated a multilevel algorithm for design sensitivity analysis. A two-level substructuring example is used to verify the proposed algorithm. The sensitivity vector obtained by the multilevel algorithm is exactly identical to the one obtained by a more conventional approach.
1. ~NTRODU~ON
Existing computer hardwares and available solution techniques for non-linear mathematical pro- gramming problems allow the engineers today to optimally design more complex and larger structures than has been done in the past. Efforts have been concentrated in the recent years to develop solution methods in both areas of sensitivity analysis [l-3] and optimization [Z-41 for efficient design of truly large scale systems.
The use of substNctu~ng[S] and multilevel sub- stNctu~ng techniques in structural analysis has been reported extensively in the literature [6,7J. Approxi- mate reanalysis with substructures has also been presented in [7].
In the field of optimization schemes for large scale structures using substructures, the interactions be- tween substructures are accounted for by using the model coordination and the goal coordination techniques [3]. The techniques suggested in [3] have limited success and further work is needed to facili- tate its usefulness. A multilevel approach to design aircraft structures has also been reported in [S]. An- other promising approach to treat large scale en- gineering design problems has recently been intro- duced in[9] and [lo] where each substructure is optimized independently. The coupling effects be- tween substructures are included by using informa- tion based upon sensitivity of optimum solutions [ 111. A number of successful applications have been re- ported and the method described in [9] and [lo] is still being developed toward a state of maturity required for industrial applications.
The field of sensitivity analysis is emerging as one
of the more fruitful areas of engin~~ng research. The reason for this is the recognition of the many prac- tical uses for design sensitivity information. Beyond the historical use of sensitivity information in con- nection with forma1 mathematical optimization tech- niques, recent work has been reported in using sensi- tivity derivatives in approximate analysis, assessing design trends, analytical model improvement, and determining the effects of parameter uncertainties [I].
Sensitivity analysis using adjoint variable tech- nique in conjunction with substructures has been developed 12, 121. Further applications of this method to the design of fail-safe structures (optimal design of structures under various damaged conditions) and other complex structures (mixed finite elements, multilayer composite materials) have also been suc- cessfully reported [12] by the author.
For truly large scale structure, however, dividing the original structure into many substructures may not always be beneficial since the overall number of boundary degrees of freedom (DOF) may also in- crease substantially and thus defeat substructuring purpose. One obvious solution to cope with this situation is to use multilevel substructures via the adjoint variable technique for ~nsitivity analysis.
In Section 2, general ideas about bottom-up and top-down process for multilevel substructuring are briefly introduced. The statics and design sensi- tivity analysis with substructures is summarized in Section 3 to facilitate the discussion of the multilevel sensitivity algorithm in Section 4. A very simple example where solutions can be obtained with a hand calculator is presented in Section 5 to study the proposed algorithm. Finally, conclusions are drawn in Section 6.
C.A.S. 2~:t-c
192 Due T. NGLXS
2. ,MULTILEVEL SUBSTRUCTURES: BOTTOM-UP AND TOP-DOWN PROCESS
A common and efficient way to solve a very large task (or structure) is to partition it into hierarchical levels with a number of smaller subtasks (or sub- structures) at each level. This method is inherently compatible with the way engineers cooperate in a design organization and with distributed computing capabilities provided by modern computer tech- nology. To simplify the discussions, however, Fig. 1 only shows a structure with three levels of sub- structuring. The first level is sometimes referred to as the top level, and the third level in this particular example can be referred to as the bottom level.
2. I Bottom -up process
Assembling of the total stiffness matrix (of the original structure) can be done by the bottom-up process as shown in Fig. 1. In this process, stiffness matrices of various substructures at the bottom level (or level 3) are first calculated. Each of these sub- structure stiffness matrices can be imagined as a stiffness matrix of a (super) finite element. Thus, the stiffness matrices of the next higher level (or level 2) can be readily obtained by assembling the stiffness contributions from each of the (super) elements in the lower level (or level 3). This assembling process is repeated until the top level (or level I) is reached. At this time, with proper boundary conditions intro- duced, the effective boundary stiffness matrix at the top level will be nonsingular and hence, boundary displacements at the top level can be solved from the equilibrium equations in statics.
2.2 Top-down process
Once the boundary displacements at the top level are known, the interior displacements and element stresses at the lower level can be found since all information about various substructures at all levels has already been calculated and stored during the bottom-up process. This concludes the statics analysis by multilevel substructures.
3. SUBSTRUCTURING FORMULATION FOR STATICS AND DESIGN
SENSITIVITY
For convenience, only a brief summary is given in this section; more details can be found in [2] and [ 121.
3.1 Stntics
Very briefly, the equilibrium equation from the finite element model is given by:
[;I 2;] (2) = {;}. (3.1)
where K is the stiffness matrix, Z is the nodal displacements vector and S is the equivalent nodal
loads. Subscripts B and I indicate quantities related to the Boundary and Interior DOF, respectively.
The iiterior displacements are first eliminated from the second half of eqn (3.1). the results are then substituted back into the first half of eqn (3.1) to obtain
Z, = G’(S, - &BZB) (3.2)
KJs= Fm (3.3)
where
KB= K,, - K,,K,’ Ku (3.4)
FB=SB- K,,K,‘S,. (3.5)
It should be noted that the effective boundary stiffness matrix KB and the effective boundary force FB can be obtained from the contributions of all substructures. Interior displacements for each sub- structure can be calculated from eqn (3.2) and ele- ment stresses can be obtained by a familiar finite element procedure.
3.2 Design sensitivity analysis
Very briefly, let JI (b, Zg, Z,) be a general function that may represent the cost of any constraint func- tion. In this notation, b is the design variables vector. The first order change in the function $ due to small changes in the design variable vector db is given as
68) =$Sb+-g62,+ g az,. (3.6) B I
By taking the first variation of the equilibrium equa- tion (3.1), eqn (3.6) can be transformed into the form
where G is called the sensitivity vector and is given as
G = $ + C:A, + CT& (3.8)
and
j _K-‘*T .I- I/
az, (3.9)
a a C, = - =$ (49BZB) -z Kd,) (3.10)
a a C, = - z VG,Zd - ;57; W,J,)
C=C,+Q’C,
Q = -K,‘K,,
(3.11)
(3.12)
(3.13)
(3.14)
(3.15)
4. MULTILEVEL SURSTRUCTURING FORMULATION FOR DESIGN
SENSITIVITY ANALYSIS
Statics analysis using multilevel substructures is omitted in here since it has been well documented in the literature. Based upon the discussions in the previous sections, a multilevel substructuring for design sensitivity analysis can be formulated accord- ing to the following (major) step-by-step algorithm. For simplicity, the discussion is restricted to a partic- ular type of constraint, say a displacement constraint.
Step 1
Starting at the bottom level (level 3). obtain the matrices [A 2,,,], x yi0L3 and [C,.?lm x ND,,, according to eqns (3.15) and (3.12). The first subscript of these matrices indicates the level number and the second subscript indicates the substructure number in a particular level. Thus, matrix (C,.,] means the [C] matrix of the rth substructure in the third level. The dimensions shown for matrices [A 21 and [C] have the following meaning:
B3 = the number of boundary DOF of a particular substructure in level 3
ViOL3 = the number of (displacement) con- straint violations of a particukzsub- structure in level 3
NDV3 = the number of design variables of a particular substructure in level 3.
Step 2
Calculate [C~:,],-W X TJ [j.,,,,ln X y,0L13 as shown in eqn (3.8) where
ViOLI = the number of (interior displacement) constraint violations of a particular substructure in level 3.
This step completes the sensitivity analysis of all substructures at a bottom level.
Comments. To facilitate the discussion in the next step, assume that B3 = 7 and the boundary DOF of the rth substructure (in level 3) are 1, 2, 3, 5, 7, 8 and 9.
Step 3
At the next higher level (or level 2), it is possible that only DOF numbers 1, 3, 7, 8 and 9 still remain as boundary DOF, whereas DOF numbers 2 and 5 become interior DOF (with respect to the current level (level 2)). Thus, we can rearrange the matrix [A2,.,] according to the following order:
1 3
[A '&.,I = i A 25:; 9
__ _______
2 5 _ A 2:!!
(4.1)
An analogous equation of the form of eqn (3.15) can be used to calculate the following matrix:
1
IA2.,1B2 x Vi0.U = : A 47
8 9 I + [Q:,l,xn
2r 1
’ (4.2)
nxvxm
where
82 = the number of boundary DOF of a particular rth substructure in level 2
12 = the number of interior DOF of a particular rth substructure in level 2.
The last matrix of eqn (4.2) can be identified as the extended form of the bottom submatrix in eqn (4.1).
Comments. (1) The assembling process for the next (higher) level of matrix [C,.,] (as appeared in Step 1)
194 Due T. NGUYEN
can be done in the same ways as described in the previous steps via eqn (3.12). For example:
and
(4.3)
2 C\!t 5 x 4 1 0 0.. .o 1 . (4.4)
6 L 0 0.. .O 1 RxNDV3
Again, the last matrix of eqn (4.4) can be identified as the extended form of the bottom submatrix in eqn (4.3).
kO = (1000) ; :, ;
i 0 0 -1 0 0 0
(2) The above steps are repeated as one continues to go up to the next higher level of substructures.
Step 4 k =
0
Having completed the sensitivity calculation at the top level, one can solve for the adjoint vector 1, from eqn (3.14):
1, = [K&‘[A2], (4.5)
where KB has already been calculated during the statics analysis phase, [A21 and [C] have also been computed during the bottom-up sensitivity assemb- ling process (Steps l-3). Next, one can evaluate the product C’i., in order to obtain the sensitivity vec- tor(s) G as shown in eqn (3.8) since @bT/ab is zero (unless $ is the buckling constraint for example).
kO = (1000) _; I 0 1 0 0
; ;
-1 0
5. AN EXAMPLE
The multilevel sensitivity algorithm presented in the previous section has no restriction on the number of levels in a tree hierarchy. The following simple example is designed to study the algorithm and to show all intermediate calculations. Hopefully, it will clarify many of the steps presented earlier.
L
5.1 Problem description Fig. 3. Two substructures in level 1.
A two-dimensional truss with its loading and sup- port conditions is shown in Fig. 2. A two-level substructure is used for statics and design sensitivity analysis (refer to Figs 3 and 4). All relateddata are summarized in Table 1. Member cross-sectional areas are used as design variables. This problem has four design variables as defined in Table 1. The objective is to compute the sensitivity vector corresponding to Fig. 4. Four substructures in level 2.
Fig. 2. A 2-D truss structure.
a nodal displacement constraint at node 2 (see Fig. 2) by the multilevel approach and the results are com- pared to a more conventional approach (without using substructures).
5.2 Numerical solutions
Element stiffness matrices in the global reference are given as
and the DOFs associated with this matrix are 1,2 and 4 where 1 and 2 are boundary DOF and 4 is the interior DOF. Thus the submatrices K,,, K,,, K,, and K,, can be identified directly from the matrix shown in eqn (5.19).
Applying eqns (3.10) and (3.1 I), one has
a [K”’ “1 -___ = ab, *Z=(-5)
or
0
- 153.13708
-40
(5.20)
Using eqn (3.13), one obtains
[Q@=‘)lr= -j&&j = -(> { -:}*[&] or
(5.26)
A2=0 (5.27)
IQ'.."lT=(-&){-;} c = c, =
-a[K(‘-‘)]*z ab
2
(5.21) = -(?) [A :I*{-15P,,,,8}
From eqn (3.12), one has
C=C,+QTC2
or
c={-i;;}+(S) (-;~*[3.55E-O5]
c= - 800 - 9.27273E - 06
800 + 9.27273E - 06
where
e, E 9.27273E - 06.
From eqn (3.9), one obtains
(5.22)
A,= K,’ f&=[-&--*[;I
1, = - 0.0369398. (5.23)
As part of the sensitivity vector(s) G, see eqn (3.8) one calculates
/I(‘=” E C’i., = [3.55E - 051 [ -0.03693981
= - 1.31136E - 06. (5.24)
Using eqn (3.15) one has
or
. (5.25)
Substructure 2: In this substructure, one has
c= 0 iI 0 . (5.28)
Substructure 3: In this substructure, one has
A2=0
C, iI -a[K'"')]*Z
c, = ab,
(5.29)
(5.30)
198 Due 1. NGUYEN
Qr= --K&y1 Using eqns (5.29) and (5.34). one has
= -(=e) i~~*[~] Qr= (3) {:} (5.32) Using eqns (5.30) and (5.35), one obtains
A pkvel?) = 0 0 0 .
Using eqns (5.33) and (5.36). one has
(5.33) At the next higher level for level 1)
Substructure 4: In this substructure, one has
/$@-4) s c;j., = 0
A2=0
c=c,= -,[K(‘“4J1*Z
(3b4
(5.40)
(5.41)
(5.42)
Assembling the sensitivity information from the lower level (level 2):
In this level one obtains
(5.34) Using eqns (5.37) and
(5.35) - 1.31
= - (~) [:, :] {- 153.13708j Also,
c= 0 {I 0 *
(5.36)
5.40), one has
36E - 06 0
0 (5.43)
0
A 2(1sk=11) = A 2(/,kVC!:, + A 2(//hel2)~
Using eqns (5.38) and (5.41), one obtains Assembling sensitivity info~ation ~substructures 1 and 2):
Substituting eqns (5.24) and 5.26) into the above operation, one has Using eqns (5.39) and (5.42) one has
As part of eqn (3.8), one calculates For truly large scale structures, however, it is
nECT&= [ ;;)--; *:O+;]
necessary to use multilevel subst~ctu~ng. The present paper has formulated a multilevel algorithm for design sensitivity analysis. A simple example is used in the paper to clarify many intermediate steps of the proposed algorithm. The sensitivity vector obtained by the multilevel algorithm is exactly identi- cal to the one obtained by a more conventional
- 2.49998E - 05 approach.
Acknowledgemenr-The financial support for the author through the NASA-ASEE Summer 1984 Program is ac-
(5.47) knowledged.
-0.0158576 0 J REFERENCES
1. H. M. Adelman, R. T. Haftka, C. J. Carnarda and
Finally, combining eqns (5.43) and (5.47), the sensi- J. L. Walsh, Structural sensitivity analysis: methods, applications and needs. NASA TM 85821 (1984). tivity vectort G of eqn (3.8) is given as
2. E. J. Haug and J. S. Arora, Applied Opr&al design: Mechanical and Structural Svsfems. Wiley. New York -. (1979).
3. U. Kirsch, Optimum Structural Design. McGraw-Hill, New York (1981).
4. G. N. Vanderplaats. Numerical Up~~~za~io~ Techniques for Engineering Design With Appiica~io~. McGraw- Hill, New York (1984).
5. J. S. Prxemieniecki, lkeory of Matrix Srrucrural Anai- ysis. McGraw-Hill, New York fl968).
6. R. H. Dodds Jr and L. A. Lopez, Substructuring in Linear and Nonlinear Analysis. inr. J. Numer. Merh.
Engn.g 15, 583-597 (1980).
(5.48) 7. A.-K Noor, H. A. Kamel and R. E. Fulton, Sub- structurinn techniaues-status and projections. Com- put. Srruc;. 8, 621632 (1978). _ -
8. L. A. S&nit Jr and M. Mehrinfar, Multilevel optimum
6. CONCLUSION
It is a well-known fact that subst~ctu~ng concept 9. can be exploited in order to reduce computer core memory requirement and the total solution time 1o
’ executed. These advantages have been realized not only for statics analysis, but also for design sensitivity analysis (2, 121.
11.
t The sensitivity vector obtained by the more con- ventional approach as shown in the appendix (the original structure is analyzed directly without using any sub- 12. structures) is exacriy iden~icaf to the multilevef sensitivity algorithm presented in &is paper.
design of structures with fiber-composite stiffened-panel components. AIAA paper 80-0723. Presented at the 2k.t AIhiIAShfE[ASCEpd% StNCtUm, StmCtUrai
Dynamics and Materials Conference, Seattle, Washing- ton, 12-14 May 1980. J. S. Sobitski, A linear d~orn~~tion method for large optimization problems-bluep~nt for development. NASA TM 83248 (1982). J. S. Sobieski, J. F. M. Barthelemy and G. L. Giles, Aerospace engineering design by systematic decom- position and multilevel optimization. Presented at 14th Congress of the International Council of the Aero- nautical Sciences (ICAS), Toulouse, France, IO-14 Sept. 1984. J. S. Sobieski, J. F. M. Bartbelemy and K. M. Riley, Sensitivity of optimum solutions to problem parame- ters. AIAA paper 81-0548R. AIAA Ji 20, 1291-1299 (1982). D. 1. Nguyen and J. S. Arora, Fail-safe optimal design of complex structures with substructures and composite materials. ASME Jnf Meek. Des, 1W (1982).
x0 Due T. NGUYES
APPESDIX CALCULATIONS OF SENSITIVITY YECTOR FOR DISPLACEIMEBT COSSTRAIST BY A
COSVENTIONAL APPROACH I. Stalks
Having eliminated all the DOFs associated with support conditions. the stiffness matrix [K] of the origmal structure can be obtained from the contributions of stiffness matrices in members 16.
VI =
+ 1000 + Oi+ 0 + 0
500 0 --
Jz 0 + 0
0 +z 4
+0+ 0
+o+o
0 + 0
+0+0
-- ;+ 0
0 -500 d
.+ 0 + 0
+ 0 + 0
0 + 0
+ 0 + 0
500
-3’O
+ 0 + o’+ 0 + 0
0 + oi-‘oo+ 0
___--J-c- 0 +z
/; 0 + 0 1 i 0 + *o- ( + 0 + 0 -i o+o A!?!?+ 0
fi
1000 + z! 3
0 + 0
+ o+o+o+o + 0+0+0+0
- (A.0
+ 0 + 0 1 + 0 + 10001
+ 0
+
0 1 500 f-+
fi 0 1
-500 -500
-500 -500
-500 ioooJ2+ 500 0 0 1000~+500 !
The DOF associated with this matrix are 1, 2, 4 and 8 (see Fig. 5). Nodal load vector is given by:
