Recent developments in multilevel optimization are briefly reviewed. The general nature of the multilevel design task, the use of approximations to develop and solve the analysis design task, the structure of the formal multidiscipline optimization problem, a simple cantilevered beam which demonstrates the concepts of multilevel design and the basic mathematical details of the optimization task and the system level are among the topics discussed

Kim, D.-S.

Vanderplaats, Garret N.

English

NASA Technical Reports Server

N89- 25203 . -  
RECENT DEVELOPMENTS IN MULTILEVEL OPTIMIZATION 
G. N. Vanderplaats 
and 
D-S Kim 
Department of Mechanical 6 Environmental Engineering 
The University of California 
Santa Barbara, California 93111 
PRECEBlNG PAGE BLANK NOT FPLMED 
1039 
https://ntrs.nasa.gov/search.jsp?R=19890015832 2020-03-20T01:49:38+00:00Z
Figure 1 identifies the general nature of the multidiscipline design task. The 
key point is that there are relatively few system level design variables, while 
there may be a great many subsystem design variables. For example, the overall 
length and diameter of the fuselage, the thickness, aspect ratio, sweep, etc. of the 
wing, and the maximum thrust may generally define the design of an aircraft as a 
system. On the other hand, the design of a subsystem such as a wing, consists of 
hundreds or even thousands of variables defining the aerodynamic shape, skin thick- 
ness distribution, spars, webs, etc. Also, this subsystem may be considered to be 
itself a collection of subsystems, including aerodynamics, structures, controls, 
hydraulics and others. There is seldom a clear mathematical structure to the over- 
all design task which would make it amenable to efficient solution techniques such 
as are available for many structural subsystem design problems. Also, the analysis 
tools for the various components range from purely experimental to empirical to for- 
mal solution of the governing equations by finite element or finite difference 
methods. In view of these complexities, it must be said at the outset that formal 
multidiscipline optimization is a technology that is still in its infancy. 
FEATURES OF THE MULTIDISCIPLINE PROBLEM 
RELATIVELY FEW SYSTEM DESIGN VARIABLES 
OFTEN COMPLEX/EXPENSIVE ANALYSIS 
ANALYTIC GRADIENTS ARE SELDOM AVAILABLE 
THERE IS NO CLEAR MATHEMATICAL STRUCTURE 
OPTIMIZATION IS TYPICALLY SEEN AS A "BLACK BOX" 
FIGURE 1 
, 
1040 
A key element in engineering design is the use of approximations to develop and 
solve the analysis/design task. These approximations may be very simple, such as 
empirical estimates of component weights based on historical data or they may be 
quite sophisticated such as the formal solution of the Navier Stokes equations. The 
motivation is usually to provide the efficiency necessary to the real design en- 
vironment. Figure 2 lists some of the motivations for making approximations. It is 
noteworthy that in the relatively well developed subsystem field of structural op- 
timization, the technology was pursued for over fifteen years before a formal ap- 
proach to creating high quality approximations was developed. In other areas such 
as aerodynamic or propulsion system optimization, this has yet to be pursued to a 
significant extent. 
APPROXIMATIONS 
AT THE SUBSYSTEM LEVEL 
PROVIDE NECESSARY EFFICIENCY 
IN STRUCTURAL OPTIMIZATION 
LINEARIZATION WITH RESPECT TO SOME 
INTERMEDIATE VARIABLES 
AT THE SYSTEM LEVEL 
DEAL WITH SUBSYSTEM RESPONSES 
ACCOUNT FOR INTERACTIONS AMONG SUBSYSTEMS 
SEND SYSTEM LEVEL INFORMATION TO SUBSYSTEMS 
SEND INTERACTION INFORMATION TO SUBSYSTEMS 
FIGURE 2 
1041 
Figure 3 depicts the structure of the formal multidiscipline optimization 
problem. It is noteworthy that this is a tree structure similar to the general 
design environment where the system may be thought of as the chief designer and the 
subsystems as engineering departments. Whenever the system variables are changed, 
the effects on the subsystems must be accounted for. Similarly, when the subsystem 
variables are changed, the effect of the overall system must be considered. Subsys- 
tems may be defined along discipline lines or by other criteria. For example, the 
design of a wing may be considered as a subsystem including aerodynamic, structural 
and other considerations. Alternatively, aerodynamics and structures may be con- 
sidered to be separate subsystems or lower level subsystems within the general 
category of wing design. It is clear that aerodynamics and structures play inter- 
acting and competing roles in the overall design and so these interactions must be 
properly accounted for via the system design control. Ideally, aerodynamic and 
structural design must be done simultaneously. However, this is counter to the 
usual division along discipline lines and so little emphasis has been directed 
toward the combined design process, even at the research level. 
FORMAL MULTIDISCIPLINE OPTIMIZATION 
ADDITIONAL LEVELS OF SUBSYSTEMS MAY EXIST 
FIGURE 3 
1042  
Figure 4 presents a simple cantilevered beam which demonstrates the concepts of 
multilevel design. The objective is to minimize the material volume subject to 
limits on the deflection at the beam junction and at the tip, and on the maximum 
bending stresses and height to width ratios of the members. The design variables of 
interest are the width, Bi, and height, Hi, of each element, and the length, L1 (Lz 
= L - L1). Clearly, for such a simple problem, this would be solved directly. 
However, for demonstration purposes, i t  is possible to formulate i t  as a multilevel 
problem with a system level and two subsystems. 
The system level problem may be stated as, find the beam length, L1, and dimen- 
sions B 1 ,  H , B2 and H to minimize the volume subject to constraints on the deflec- 
tions. Additionally, $n the present method, subsystem constraints will be imposed, 
in linearized form, on the stresses and the height to width ratio on the members. 
Each member can be taken as a subsystem and, during subsystem optimization, the mem- 
ber volume will be minimized subject to constraints on the member stresses and 
height to width ratio. At this level, the purely system level design variable, L1, 
will be held fixed, but the system level constraints (deflections) will be included 
in linearized form. 
Note that, at the system level, all design variables are included. At the sub- 
system, the design variables that are important to the subsystem are considered, but 
the strictly system level variable, L1, is held fixed. 
Because of the interdependence between the system and subsystem variables, each 
level will affect the other. The key issue is how to account for these interactions 
and how to account for competition between subsystems. 
I 
MULTILEVEL DESIGN OF A CANTILEVER BEAM 
P 
'I I 
FIGURE 4 
1043 
A variety of methods have been proposed to deal with the multidiscipline design 
task i n  a formal way. Figure 5 presents a recent method developed in an effort to 
simplify the overall process while maintaining the traditional separation of d i s -  
linearized. Each of the subsystem optimizations is then performed, presumably in 
parallel. At the optimum for the subsystem, its constraints (or a critical and near 
critical subset) are linearized and returned to the system. The system level op- 
timization is then performed, including these linearized subsystem constraints. 
Also, at this point, the subsystem design variables are included along with the 
strictly system level variables. The process is repeated until it has converged to 
an optimum. As with any linearization technique, move limits must be imposed at 
each level and these are reduced as the optimization proceeds. 
I ciplines [ l ] .  Initially, all system level functions (objective and constraints) are 
MULTILEVEL OPTIMIZATION PROCESS 
EVALUATE SYSTEM LEVEL FUNCTIONS 
CREATE LINEAR APPROXIMATION TO SYSTEM LEVEL FUNCTIONS 
SOLVE EACH SUBSYSTEM PROBLEM, 
INCLUDING LINEARIZED SYSTEM LEVEL CONSTRAINTS 
CREATE LINEAR APPROXIMATION TO ALL SUBSYSTEM CONSTRAINTS 
SOLVE SYSTEM LEVEL PROBLEM 
INCLUDING LINEARIZED SUBSYSTEM CONSTRAINTS 
REPEAT TO CONVERGENCE 
MOVE L I M I T S  ARE USED AT EACH LEVEL 
F I G U R E  5 
Figure 6 provides the basic mathematical details of the optimization task at 
the system level. Here, capital letters indicate strictly system level design vari- 
ables, objective and constraints, and lower case letters indicate subsystem level 
variables and constraints. Here, the subsystem variables are included along with 
the system variables. The subsystem constraints are included in their linearized 
form. Note that the number of design variables, as well as the number of con- 
straints to be considered here is greatly increased from the number of strictly sys- 
tem level variables and constraints. However, the subsystem constraints are 
linearized and so are relatively easily dealt with. This is a departure from previ- 
ous methods which used a cumulative constraint for each subsystem as well as a set 
of "optimum sensitivities" from the subsystems. The tradeoff is that the functions 
here are linearized at the expense of an increase in the number of design variables 
and constraints. However, the need to deal with nonlinear inequality constraints at 
the subsystem, as well as the need to calculate sensitivities of the optimized sub- 
systems is avoided. 
PRESENT METHOD 
AT THE SYSTEM LEVEL 
N DESIGN VARIABLES, X, xl, x2, ... x 
OBJECTIVE, F(X,xl,x2, . . .  xN) 
SYSTEM CONSTRAINTS, GJ(X,xl,x2, ... xN) 
SUBSYSTEM CONSTRAINTS, gj(X,xl,x2, ... xN) 
WHERE 
n s s  
7 
I 
i=l  
FIGURE 6 
1045 
The basic mathematical details for the subsystem optimization are given in 
Figure 7. The inputs to the subsystem problem include the boundary conditions, sys- 
tem level variables and system level constraints, all in linearized form. The 
that these may be functions of the subsystem variables and are not assumed to be 
constant in the present method. For example, the forces in the members of a struc- 
ture may be functions of the local variables. Also,  if the strictly system vari- 
ables are functions of the subsystem variables, this must be accounted for. Then, 
when the approximate system level constraints are calculated, i t  is first necessary 
to calculate the approximate values of the system variables and subsystem boundary 
conditions since the system constraints are functions of these. While this appears 
to be a bit cumbersome, i t  must be remembered that these computations are relatively 
simple matrix operations and so are efficiently performed. A l s o ,  if sufficient in- 
formation is available to calculate these parameters precisely, this may be done to 
improve the overall efficiency. 
I reason that the system level variables and boundary conditions must be linearized is 
PRESENT METHOD 
AT EACH SUBSYSTEM 
DESIGN VARIABLES xi 
OBJECTIVE, f(BC,X,xi) 
SUBSYSTEM CONSTRAINTS g.(BC,X,xi) 
J 
SYSTEM CONSTRAINTS, GJ(BC,X,xl,x2, ... xN) 
WHERE 
0 0 BC = BC + V BC.(X - x ) -X 
0 x = xo + v X.(x - x ) 
-X 
0 0 0 
G .  = G + V G .(X - X ) + p G . ( E  - ) + gBCGj.(BC - -  - BC ) x j  -X j - - 3 j 
FIGURE 7 
I 1046 
Figure 8 presents the iteration history for the cantilevered beam shown in 
Figure 4. The problem was also solved by direct application of optimization and 
those results are shown also. The initial design violated constraints, so the 
direct method first increased the volume in order to overcome these constraint 
violations. While i t  appears from the figure that the multilevel method provided an 
equivalent convergence rate, it  must be remembered that one iteration of the multi- 
level method consists of optimization of all subsystems followed by a system level 
cient computationally. This is generally true for problems that can be solved 
directly. The value of the multilevel method is for design problems where i t  is 
necessary to separate the problem for other reasons. 
I optimization. Thus, for this simple example, the direct method is much more cffi- 
900  
8 0 0  
h 
cr) 7 0 0  2 
H 
W 
2 6 0 0  
3 
l-a 
0 
5 0 0  
3 9 3  
300 
ITERATION HISTORY FOR CANTILEVER BEAM 
S I N G L E  LEVEL 
. 1 I t 
0 1 2 3 4 5 6 7 8 
I T E R A T I O N  NUMBER 
FIGURE 8 
1047 
The portal frame shown in Figure 9 was designed using the proposed method. 
This is considered to be a standard test case, and the details of the loads, 
materials, system and subsystem constraint calculations are presented in Reference 
2. The system level constraints are displacement and rotation limits at the joint 
where the loads are applied. The subsystem constraints included stress, local buck- 
ling, and sizing limits. There are three subsystems, being the design of the in- 
dividual beam elements. The subsystem design variables are the six individual 
dimensions of the cross-section of each element. The objective function at both the 
system and subsystem levels is to minimize the volume of material. 
Two cases were considered. In the first, th6 initial design was well within 
the feasible region, while in the second, the initial design was quite infeasible. 
The iteration histories for the two cases are shown in Figures 10 and 11. The mul- 
tilevel approach did not produce as good an optimum in either case, but did produce 
a near optimum. There is no clear reason for the differences, although this struc- 
ture is known to have relative minima. 
MULTILEVEL OPTIMIZATION OF A PORTAL FRAME 
@- L- 
I C h  -1 
I - . m  
L-. 
T 
A 
.f 
L 
FIGURE 9 
I 1048 
ITERATION HISTORIES FOP.PORTAL FRAME CASE I 
60 
280 
2 4 0  
-- 200 
I 
0 4
X 
m 
160 
Ej 
0 
120 
80 
4 0  
I 
. I 
t 
I 
- SINGLE LEVEL 
---- KULTI LEVEL 
w 
0 2 4 6 8 10 12 14 16 18 20 
ITEPATION NUMBER 
140 1 
FIGURE 10 
- SINGLE LEVEL 
- - -- MULTI LEVEL 
II I 
II ! 
201 . . ' ' ' . . . *  
0 2 4 6 8 10 1 2  14 16 
ITERATION NUMBER 
FIGURE 11 
1049 
The two-bay frame shown in Figure 12 was designed for minimum material using 
the proposed method. The design variables and material properties for each beam and 
the subsystem constraints are the same as for the portal frame. The system level 
constraints are shown in the figure, as well as the loading conditions. Symmetry 
was used so the system is comprised of four subsystems, being the vertical members 
of each bay and the floor members of each bay. Each subsystem consists of s i x  
design variables for a total of twenty four independent design variables. 
The results for single level and multilevel optimization are shown in Figures 
13 and 14 for an initially feasible design and an initially infeasible design, 
respectively. 
MJLTILEVEL DESIGN OF A TWO-BAY FRAME 
16000 l b d  
7 
16000 
16000 
500 l b / i n  
L 6  5 1 .8  i n  
t 
120 in 
1 
1 
- 
120 in 
7- 
6 < 1.0 in - 
FIGURE 1 2  
ITERATION HISTOP.IES FOR A TWO-BAY F W  
2 
2 
21 
11 - 
II 
'0 11 
II- 1 1  
4 
X 
.. 
rl - 
w 1; 
L 
3 
9 1 c  
8 
6 
4 
2 
0 
SINGLE LEVEL 
MULTI LEVEL 
- 
---- 
0 2 4 6 8 1 0  1 2  1 4  1 6  1 8  * 
ITEPATION NUMBER 
FIGURE 13 
- SINGLE LEVEL t 
0 2 4 6 8 10 1 2  1 4  1 6  
ITERATION NUMBER 
FIGURE 14 
1051 
Figure 15 lists the advantages and disadvantages of the present approach. A 
key advantage is that the concept is relatively simple. It does not require the use 
of nonlinear equality constraints at the subsystem level or the calculation of sen- 
sitivities of the subsystem optimum with respect to the system variables as are re- 
quired in earlier methods. The principal disadvantages are that the number of sys- 
tem level variables is greatly increased and that the quality of the linearizations 
is important. Regarding this last issue, i t  should be remembered that the term 
"linearization" used here does not infer strict linearizations. For example 
linearizations in reciprocal space may be preferred for  structural design problems. 
The key idea is that the approximations used at each level are explicit. 
In summary, i t  is clear that much research remains to be done before decomposi- 
tion methods such as this reach the state of reliability that is available in stan- 
dard structural optimization today. However, for optimization to find widespread 
use in the multidiscipline environment, i t  is clear that methods must be developed 
that will interface with designers with a minimum of disruption to the traditional 
design environment. 
ADVANTAGES OF PRESENT APPROACH 
EACH SUBSYSTEM IS SOLVED AS THE ENGINEER CHOOSES 
ONLY A SIMPLE SET OF LINEAR CONSTRAINTS MUST BE ADDED 
THE SYSTEM AND SUBSYSTEM OBJECTIVES MAY BE DIFFERENT 
DEPENDING ON THE NEEDS/MOTIVATIONS OF THE INDIVIDUAL LEVEL 
THE SYSTEM LEVEL CONTROLS THE OVERALL OBJECTIVE 
THE CONCEPT IS SIMPLE 
DISADVANTAGES OF THE PRESENT APPROACH 
THE NUMBER OF SYSTEM LEVEL DESIGN VARIABLES IS GREATLY INCREASED 
THE QUALITY OF THE LINEARIZATIONS IS IMPORTANT 
MOVE LIMITS ARE IMPORTANT 
THE METHOD HAS THE SAME OVERALL ADVANTAGES AND 
DISADVANTAGES AS SEQUENTIAL LINEAR PROGRAMMING 
FIGURE 15 
1 1052  
A key issue in the development of "user friendly" multilevel and multidis- 
cipline optimization methods is the user interface. Figure 16 is a general diagram 
showing the essential components of such a system, and this is the subject of Cur- 
rent research at UC Santa Barbara. The control module directs the activities rela- 
tive to the system and subsystem tasks, as well as basic data management, All data 
transfer between modules is via a data management system which may be a general sys- 
tem or may be a specialized system for the multidiscipline optimization task. The 
important aspect of this approach is that the system and subsystems are provided 
with a specific form of their input and output which is general enough to accom- 
modate the need to operate either independently or within the multidiscipline en- 
vironment. This only requires a general degree of standardization and the in- 
dividual disciplines are otherwise free to operate as usual. Also, at the subsystem 
level, a similar standardization is required to allow the user to perform analysis 
alone, optimization without an interface to a controlling system, and optimization 
within the overall system. The purpose of the pilot code being developed is to 
create such an environment for testing on a variety of multilevel and multidis- 
cipline problems, This is expected to identify more clearly the strengths and 
weaknesses of the present method as well as identify future research needs. 
I 
DATA TRANSFER 
FIGURE 16 
1053 
REFERENCES 
1. Vanderplaats, G. N., Yang, Y. J .  and Kim, D. S., "An Efficient Multilevel Op- 
timization Method for Engineering Design, Proc. 1988 AIAA/ASME/ASCE/AHS Struc- 
tures, Structural Dynamics and Materials Conference, Williamsburg, Virginia. 
2. Sobieszczanski-Sobieski, J., " A  linear Decomposition Method for Large Op- 
timization Problems - Blueprint for  Development," NASA Technical Memorandum 
83,248, February 1982. 
I 1054 


Recent developments in multilevel optimization

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server