search Center Grant NAG-1-1561, Dr. J. Sobieski,. Project Monitor. References. 1] R. T. Haftka and J. Sobieski. ... John Wiley and Sons, New York, New.
Response Surface Approximations for Discipline Coordination in Multidisciplinary Design Optimization R.S. Sellar M.A. Stelmacky S.M. Batillz J.E. Renaudx Department of Aerospace and Mechanical Engineering University of Notre Dame Notre Dame, Indiana Wfuel Wmotor Wpayload xle N T 0
Abstract The practical application of optimization methods to non-hierarchic, coupled, multidisciplinary systems has been hampered by the costs associated with the use of complex discipline-speci c analysis procedures. When the cost of performing these individual analyses is high, it is impractical to apply many current optimization methods to most practical systems in order to identify improved designs. This paper details an extended formulation and application of the Concurrent Subspace Optimization (CSSO) approach to this class of system design problems. In this application neural network based response surface mappings are used to allow the discipline designer to account for discipline coupling as well as for system level design coordination. The application is the design of a \hovercraft" and includes external con guration, structures, performance and propulsion disciplines.
I. Introduction The e ective design of complex, highly integrated and interdependent systems requires the coordinated e orts of a large number of discipline experts striving to accomplish a common goal. Though their decisions are driven by the design requirements for the complete system, often many other practical considerations lead to a situation where the e orts of discipline experts with design responsibility are not well coordinated nor are there well-established, rational schemes to provide that coordination.The goals of Multidisciplinary Design Optimization (MDO) include accounting for the couplings between disciplines and providing a framework in which improved system designs can be obtained. Much of the current work in MDO is directed toward the e ective inclusion of the advanced simulation and analysis tools into the design process at the earliest time possible, and in such a manner that the designer can eciently exploit the information provided by these tools. It appears that the successful integration of MDO methodologies into the actual system design process will therefore require a more complete understanding of the design problem formulation, the role of the tools, and the design decision processes. The current paper is an attempt to add to this discussion. In its simplest sense the Multidisciplinary Design Optimization (MDO) problem can be cast as a traditional optimization problem of the form: Minimize: f = f (x;y) Subject to: hi (x;y) = 0:0 (1) gj (x; y) 0:0
Nomenclature
b c
%cam
f FT FQ g h lrod MAC r RPM t tskin
span chord camber merit function force normal to rotor rotation force in plane of rotor rotation inequality constraint equality constraint rod length moment about aerodynamic center rod radius motor speed rod thickness skin thickness
Graduate Research Assistant, Member y Graduate Research Assistant z Professor, Associate Fellow AIAA x Assistant Professor, Member AIAA
fuel weight motor weight payload weight rod leading edge attachment point structural de ection Von Mises stress (FT direction) Von Mises stress (FQ direction) rotor blade incidence angle (static)
AIAA
1 Copyright
c 1996 by Stephen M. Batill. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission.
1
where the system merit function, f , is an arbitrary function of the bounded system design variables (x) and system states (y) which is minimized subject to equality, hi , and inequality, gj , constraints. Basic to the optimization problem is the ability to determine the system characteristics for a given set of design variables. This process, often referred to as a system analysis, necessitates iterating between complex numerical procedures for practical engineering systems. Consequently, simply performing a single system analysis to obtain a consistent set of states for a given design vector can be a costly and lengthy process. Optimizing such a system using conventional optimization strategies is beyond current and immediately foreseeable computational capabilities. Reference [1] surveys a number of the recent developments in MDO methods and highlights many of the other current concerns in developing practical MDO methodologies. Reference [2] introduces a framework for formulating problems of this type using an approach referred to as CSSO-NN. This approach provides a framework in which designers at the discipline level have the ability to improve the system design through the use of the tools and analyses with which they possess expertise. By developing response surface approximations to necessary states and constraints, the discipline experts are able to coordinate their design activities and work to achieve a common improved design. This also allows the designers to archive critical design data. This information could be of great value, particularly if one then wishes to perform \what if" studies and consider alternative design requirements. This paper details an application of the CSSO-NN algorithm and demonstrates a variety of issues related to its implementation using a conceptual vehicle design problem. This problem contains aerodynamic, structural, propulsion and performance discipline interactions. The next section provides a brief overview of certain features of this framework and the demonstration problem addresses speci c implementation issues.
SA
CA1 CA2
CA3
???@@Convergence @@?? System Level
'c c $ &c %
Coordination
B
B
Response Surface Approximations
BB N
b` "H
J ] J J
? ? ? ? SSO1 SSO2 SSO3 ? ? ?
6
Subspace Optimizations
-
SA
CA1
CA2 CA3
Process Flow Information Flow
Figure 1: CSSO-NN Framework typically iterative and involves a coupled set of contributing or discipline analyses. It may represent the most costly single step in the process. The contributing analyses, CA's, are the discipline speci c analyses which are used to evaluate states and constraints. The purpose of a system analysis is to provide a consistent set of states for a given design (i.e. the design variables and associated states satisfy all of the coupled system analyses). Other components of the framework are the response surface approximations (neural networks), subspace optimizations (SSO), and the system level coordination. The role of each and their relationship to each other are brie y described below. The CSSO-NN algorithm begins with the selection of a baseline set of designs. The reason for starting with multiple designs is that a system approximation is constructed before subspace optimization occurs. These approximations to various \non-local" states are used to help the discipline designer understand the in uence of local design decisions on the system level constraints and system merit function. It is important to note that designs considered at this point need not be feasible, they only need be consistent. The next component is the response surface approximations used to provide coordination in the process. There exist many methods by which a system response surface approximation can be constructed; however, in this study response surface approximations will be limited to feedforward, sigmoid activation neural networks [3, 4, 5]. One would anticipate that a
II. The Framework The process used to select an appropriate combination of design variables to produce an optimal, feasible design is limited by the ability of the designer to predict system behavior. These predictions are used to drive design decisions. The sequence in which prediction and decisions are performed and the allocation of responsibilities within that process constitutes the framework for the multidisciplinary design process. The CSSO-NN framework used in this research is shown schematically in Figure 1 and detailed in Reference [2]. The following highlights particular features of this framework. This schematic illustrates the four main processes which comprise the CSSO-NN formulation. The process proceeds in an iterative fashion, counterclockwise beginning with system analysis, SA, in the upper left-hand corner. The system analysis is
2
and payload. The rotor is comprised of two rectangular lifting surfaces located on opposite ends of a hollow, circular shaft. The imposition of a hover condition requires that the system operate at the motor speed (RPM) which provides a thrust-to-weight ratio of unity. A schematic of the autonomous hovercraft problem, or AHC, is shown in Figure 2.
variety of response surface approximation types may be employed in an actual system design process. Once the system is approximated the subspace designers are given the freedom to optimize the system design based on the most appropriate analyses and their experience. This requires the subspace designers to solve the system optimization problem based upon accurate information about a particular region of the design space and approximate information about the rest of the space. Neural networks provide non-local state approximations during each subspace optimization. As will be detailed later in the paper, the allocation of responsibility for design variables and the selection of the appropriate response surface approximations is a central issues in developing this approach. The last component is the system level coordination which is accomplished using only the design space approximation in the form of the neural networks. This step could produce either a single \optimal" design or a number of potential design candidates. The cost to perform this step is low since all of the design space is mapped and evaluation of the approximate system states for a given set of design variables is a quick process. The coordination procedure then provides the next current design/s, a system analysis is performed on each, and a decision made to continue the process. The methods used to make the design decisions at the discipline level may vary from discipline to discipline or within a discipline as the design evolves. Numerical optimization methods or other design heuristics can be used in either the subspace optimization step or the system level coordination depending upon the type of design variables and nature of the system. Gradient based search techniques or genetic algorithms could be invoked depending upon whether continuous or discrete design variables are included. The important feature of this process is that as it proceeds, design data is being archived in the response surface mappings. This allows the design team to adapt to changes in design requirements, and instead of providing just a point design a more valuable parametric description of the design space is developed. The following sections describe the application of this approach to a simpli ed system design problem, an autonomous \hovercraft". In this example the traditional disciplines of aerodynamics, structures (stress and vibration), propulsion and performance are integrated into a single MDO application. Emphasis is placed upon the adaptation of the problem to the CSSO-NN formulation and the selection of the appropriate approximations and their role in each discipline's design decisions.
tskin
t c
r
x lea
l
rod
Fuel Engine Payload
θ0 b
Figure 2: Schematic of Autonomous Hovercraft Problem There are eleven design variables used in this example. These variables describe the physical dimensions of the rod and lifting surfaces and the amount of fuel carried by the rotorcraft. Other input parameters, taken to be constant, include structural material properties, air density, the aerodynamic eciency of the rectangular lifting surfaces, the rod drag coecient, and the weight of the payload. In addition to the operating RPM, the AHC system analysis determines the required motor weight, aerodynamic loads, structural stresses and deformation, the rst natural frequencies of the system in bending and torsion, fuel ow rate, tip Mach number and the endurance of the craft. This system analysis is made up of four discipline-speci c contributing analyses (CA's); they are aerodynamics, structures, propulsion and performance, and structural dynamics. The aerodynamics contributing analysis computes the resultant aerodynamic forces on the lifting surfaces. These include the resultant forces normal and parallel to the plane of rotation and the moment about the aerodynamic center of the lifting surface. The induced velocity at the lifting surfaces is estimated as a function of the total thrust produced. In addition to a subset of the complete vector of design variables, the aerodynamic analysis requires as input the motor RPM and the torsional deformation of the shaft. The aerodynamic loads are required as input to the structures contributing analysis. The structural analysis computes the axial and shear stresses at the hub of the rotor. These values are used to determine
III. Autonomous Hovercraft System Description The problem considered in this study is the design of an autonomous hovercraft. This problem models a physical system consisting of an engine, rotor,
3
ues until the thrust produced and system weight are the same within a prescribed tolerance. When both the torsional deformation and the motor RPM have converged, a consistent system analysis has been obtained.
the Von Mises stresses at the hub which are output as system states. The deformations computed in the structures CA includes the torsional deformation of the shaft, as well as the de ections of the wing normal and parallel to the rotor plane. The performance discipline is also coupled to the aerodynamic analysis. Inputs to the performance contributing analysis include the aerodynamic force components on each lifting surface which are used to calculate the total thrust produced and the torque necessary to spin the rotor. Another computation in the performance CA is the motor power required. This quantity depends on the required torque and the motor RPM and is used to determine the motor weight. The motor weight is then considered along with the weights of the rotor and fuel in order to determine the system weight. The Mach number at the tips of the rotors and the endurance are also calculated. The fourth contributing analysis, structural dynamics, is not coupled with any of the other disciplines; its input consists solely of design variables and constant parameters. The structural dynamics CA calculates the rst natural frequencies of the rotorcraft structure in bending and torsion. The ow of information between the various contributing analyses results in several feed-forward and feedback couplings. A dependency diagram for the AHC problem illustrates these couplings and is shown in Figure 3. The circuit between the aerodynamics
IV. CSSO-NN Implementation Implementation of the autonomous hovercraft problem into the CSSO-NN framework requires the categorization of the states and contributing analyses based on the way in which information is exchanged between disciplines. Figure 3 indicates that there exist couplings between the aerodynamics and structures and between the aerodynamics and performance contributing analyses. Information needs to be exchanged between these disciplines in order to iterate and obtain a consistent solution. One way to provide coupling information to designers in each discipline is to construct approximations to each discipline. The designer could then use the approximations to other disciplines in order to obtain the non-local state information necessary to perform an analysis. The approximation of a discipline is a function of the design variables in that discipline and the non-local state information required to determine that discipline's states. The resultant approximations for the aerodynamics and structures disciplines could be considered as shown in Figure 4.
Aerodynamics
lrod c
Structures
b
Propulsion/ Performance
0 t=c
Structural Dynamics
=c 0
Figure 3: Coupling for Autonomous Hovercraft Problem
cc c cc c cc cc cc cc t
"" bb
fT F
fQ F
g M AC
r
lrod
xle
FT 0
FQ 0
"" bb
~
f N fT
MAC 0
Aerodynamics
Structures Figure 4: Candidate Response Surface Mapping Formulations
and structures CA's represents the traditional static aeroelastic problem; the deformation of the shaft is a function of the aerodynamic loads, while the loads themselves are dependent upon the torsional deformation. The other circuit, between the aerodynamics and performance CA's, involves the determination of the required motor size and operating \trim" speed. At the outset of each system analysis, an initial estimate of motor RPM is made. The aerodynamics CA computes the aerodynamic forces at this RPM. These values are fed forward into the performance CA and used to determine the required motor power and resultant motor weight. The total thrust and system weight, which must be equal for the hover condition to be satis ed, are then considered in determining the RPM value at the next iteration. This process contin-
Structural designers could use the aerodynamics approximation to estimate the applied loads and then determine the de ection with their analysis tool. However, an estimate of the de ection is needed to obtain the necessary loads. The process of obtaining a consistent solution is still iterative, but, the cost of performing this iteration is decreased because half of the iteration is approximate. A potential concern in employing this approach lies in the fact that iteration now occurs between an analysis and a response surface mapping of another discipline. Considering that the purpose of performing this approximation is so that structural designers can explore new designs without
4
The second feedback circuit contained in the autonomous hovercraft problem involves the exchange of information which is not explicitly determined. This information exchange occurs between the aerodynamics and performance disciplines. In order to calculate the aerodynamic forces produced by the spinning rotor, the rate at which the motor is turning is required. Given the motor RPM, a set of loads can be determined. These loads dictate the speed at which the motor can turn, setting up an iterative process. The di erence between this coupling and the aeroelastic coupling is that in this case the motor RPM is not explicitly determined. This state takes on a value which forces the system analysis to satisfy the hover condition which is imposed as part of the system analysis. Consequently, the analysis proceeds by assuming a motor speed, determining the aerodynamic forces, and adjusting the RPM based on the current thrust and total weight values. For this discussion the motor RPM will be described as an implicit system state, a state which is determined such that the system analysis results in a design which meets constraints imposed in the system analysis. This is di erent than the state information exchanged between structures and aerodynamics which was used to arrive at a consistent solution. Due to the fact that this implicit system state is not determined explicitly and will be a ected by any change in the system, this state is approximated by the full system design vector. Again the issue of the size of the approximation becomes a consideration. In typical feedback loops physical states are exchanged between disciplines. In dealing with implicit system states the responsibility for identifying system convergence and determining updated values of the implicit state lies within a speci c discipline or disciplines. By decomposing the system analysis through the use of response surface mapping the discipline(s) which formerly controlled the iteration now become reliant upon the response surface approximation. For the autonomous hovercraft problem this means that instead of an updated RPM being calculated in the performance contributing analysis, the states determined in this CA are a function of an approximation to RPM. Additionally, the disciplines to which this subspace are coupled, in this instance aerodynamics, also depend on this approximation. Including this coupling into the aerodynamics discipline results in an approximation of the form shown in Figure 6, where the
having to wait for aerodynamics results, then in general the accuracy of the approximation will decrease as the design moves away from the baseline point for the approximation. The result of the decreasing response surface accuracy is that there is no guarantee the iteration between structural analysis and approximate aerodynamics will converge. A similar situation arises in the aerodynamic analysis. In order to eliminate this diculty, an alternative formulation of the discipline approximations is considered for the current study. Examining the relationship between the structures and aerodynamics disciplines it can be seen that at the most basic level the states computed in each are simply functions of the local and non-local design variables. Looking at the structural analysis: = (t; r; lrod ; xle ; FT ; FQ; MAC ) (2) ( F ) (F ) T T
FQ = FQ (lrod ; c; b; 0 ; t=c; %cam) (3) MAC MAC Since is a function of FT , FQ, and MAC , it is im-
plicitly a function of the aerodynamic design variables. So, combining Equation 2 and Equation 3: = (t; r; lrod ; c;b; 0 ; t=c; xle; %cam) (4) The implication of Equation 4 is that for disciplines which share di erent information in two directions (feed forward and feed back) the states of both disciplines can be approximated as a function of the union of the subspace design vectors associated with each discipline. If the structures discipline design vector is denoted as xS = (t; r; lrod ; xlea) and the aerodynamics design vector as xA = (lrod ; c; b; 0 ; t=c; %cam) the state approximations for each discipline take the form shown in Figure 5. Analysis in the structural xA
[ xS
c c c c c c " b
fT F
fQ xA [ xS F g M AC
" b
~
f N fT
Aerodynamics
Structures Figure 5: Form of Approximation for Aerodynamics and Structures Disciplines
discipline can now be performed using the aerodynamics approximation without iteration. This is accomplished since for a given structural con guration the design variables which implicitly impact the loads are incorporated into the aerodynamics approximation. The trade-o in this formulation is that the number of variables used to construct the approximation can become large, having implications on the number of analyses required to construct approximations. Other researchers have begun to address the problem of large response surface mappings (see for example Reference [6]).
xsys
c c c cc gM RP
xA
[ xS
gM RP
" b
fT F
fQ F
g M AC
Aerodynamics
Figure 6: Form of Multi-level Approximation for AHC Problem 5
denotes approximate quantities. It is important to notice that disciplines which utilize these implicit system states (RPM in this case), will be relying on a \multilevel" approximation, an approximation based on approximate information. A concern in dealing with such approximations is the accuracy with which the nal states can be determined. An alternative approach to the multilevel approximation scheme described above is to approximate the disciplines involved with the system feasibility state by the full system design vector. This allows for implicit inclusion of variations in the implicit system state in discipline response surface mappings. Two reasons for not using such an approach exist. The rst reason again deals with the dimensionality of response surface mappings of the a ected disciplines. The second reason against such an approach is that aerodynamic designers and performance analysts need a value of RPM to perform their analyses. By not including the ability to determine RPM and consequently approximating the a ected disciplines using the full system design vector, designers in these disciplines are no longer included in the system optimization process. The power of the discipline experts to in uence system design is reduced. This is in direct opposition to one of the goals of the current CSSO-NN formulation. The form of the implicit system state approximation impacts not only the response surface mappings used in the disciplines directly involved with this quantity, but also any other disciplines coupled to the implicit system state iteration loop. For the AHC problem this occurs in the approximation to the structures contributing analysis. Recalling that the response surface mappings to the structures and aerodynamics disciplines are formulated based on the same input information, the aerodynamics approximation is dependant on RPM approximation and the structures response surface mapping must also take into account this information. This creates larger response surface mappings in every discipline which is coupled to the implicit system state iteration. Correspondingly, the aerodynamics, structures, and performance approximations all have inputs of the form of the aerodynamics imputs shown in Figure 6. A third type of contributing analysis exists in the autonomous hovercraft problem. This contributing analysis, the structural dynamics analysis, uses only information about design variables. It is not coupled in any way to the other analyses. In this case the discipline may be approximated simply as a function of the design variables used in the analysis. The response surface mapping of disciplines within the context of the current formulation of the CSSO algorithm has been generalized in Table 1. This table illustrates the form of the response surface mapping input vector for each class of contributing analysis which has currently been identi ed. The subscripts L and NL refer to local and non-local information (from the current discipline or from another discipline), respectively. For
CA Type Uncoupled Provides state information Needs state information Involved in system feasibility iteration Involved in other iteration
xL
RSM Input Vector
xL [ xNL
X X
xL [ yg NL
X X
X
Table 1: Input Vector for Di erent Classes of Contributing Analyses
contributing analyses which are members of more than one class, the desired input vector for the response surface mapping is the union of the prescribed input vectors for each coupling. The method of approximating response surfaces described above is utilized in analysis at the discipline level. The decreased time and e ort required to perform analyses at the discipline level allows designers to consider the impact of their design decisions on other designers and on the system. This creates an environment in which all designers can solve the same system optimization problem. The information used in the solution of the system optimization problem within each discipline is di erent since each designer has the ability and expertise to produce di erent data. For the AHC problem the system optimization problem takes the form: Minimize: Wempty = Wwing + Wrod + Wfuel +Wmotor Subject to:
g1 g2 g3 g4 g5 g6
= 1 ? xcle 0
= N ?1 0 allowable ?1 0 = T allowable !b
= kRPM ? 1 0 !t ?10 = kRPM
(5)
= MtipMallowable ? 1 0 tip
?10 = E E required Optimization of this problem at the discipline level requires the approximation of any non-local state information. Examining this problem from the perspective of a structural designer yields: 0 Minimize: Wempty = Wwing + Wrod + Wfuel +Wg motor g7
6
g1
=1?
g2
?1 0 = N allowable
g3 g4 g5 g6 g7
c 0 0
150
= T ?1 0 allowable 0
!eb ? 1 0 (6) g k RPM !et ?1 0 = g kRPM M tip allowable = ?1 0 Mtip 0 0 = E E ?10 required =
200
Iterations
Subject to:
xle
100
50
0 0.0
where the represents information obtained from a response surface mapping and the 0 denotes information which is calculated using approximate information. Designers in the structures discipline have the tools and ability to determine de ections and stresses based on approximations to the aerodynamic loads which are in turn based on an approximated RPM. Additionally, changes in the design vector made by structural designers will have an impact on each of the other constraints and the system merit function. Response surface mappings are used to evaluate these quantities so that structural designers can assess the impact of their design decisions on the system.
0.2
0.4
0.6
0.8
1.0
k
Figure 7: System Analysis Convergence because each iteration involves performing three of the four contributing analyses in the AHC problem. The contributing analyses that made up the AHC problem were short computer routines, so it is easy to rapidly perform many system analyses. This simple study demonstrates that the eciency with which they are performed is a key concern. Performing a system analysis for at each corner and at the center of the design space using k = 0:2 (a total of 2049 (211 + 1) points) showed that an average of 73 contributing analyses were required for each system analysis. This means that an average system analysis required 24 iterations of the aerodynamics, structures, and performance disciplines, plus one analysis in the dynamics discipline. Thus, if the contributing analyses were more complex, a single system analysis itself would become a very expensive task in terms of time and resources. Conventional all-at-once optimizations, using the generalized reduced gradient method [7], were performed on the AHC problem from fty di erent starting points. One of these points was the center of the design space; 22 additional points were obtained by letting one of the eleven design variables start from its upper and lower bounds while the remaining ten were halfway between their bounds. The other 27 starting points were speci ed by a 3-level orthogonal array populated by di erent combinations of design variables at their lower and upper bounds, and the midpoints between those bounds. Of the 50 system optimizations that were performed, 48 of them converged to a point at or near the global optimum shown below in Table 2. The notations (L) and (U) indicate variables at their lower and upper bounds, respectively. Two of the seven constraints were active at this point, the constraint on the stress in the rod at the root top (g2 ), and the en-
V. CSSO Application and Results As described previously, the AHC system analysis includes a hover condition which requires the motor to operate at an RPM such that the thrust produced equals total system weight. For a given set of design variables, this RPM value is calculated iteratively, with each estimate of RPM being dependent on the values of RPM, thrust, and total system weight at the previous iteration. The general form of this scheme is given by Equation 7:
Wtotal k i (7) RPMi+1 = RPMi Thrust i where the exponent k is greater than zero. It is noteworthy that the value of the exponent k in Equation 7
had a considerable e ect on the eciency of the system analysis iteration. Figure 7 shows the number of iterations required to obtain a consistent system analysis as a function of the exponent k. The system analyses performed to obtain this data used the same design vector (the center of the design space) and initial estimates of RPM and torsional deformation. The plot shows that a value of k = 0:5 resulted in the most ecient system analysis. This is an important result
7
Variable Global Optimum Local Optimum t inches 0.15 (L) 0.15 (L) r inches 1.2 (L) 1.2 (L) lrod inches 58.5 60.0 (U) c inches 6.0 (L) 6.0 (L) b inches 42.0 (U) 40.33 0 8:22 7:50 t=c 0.05 (L) 0.05 (L) xle inches 0.0 (L) 6.0 (U) %cam 2.77 5.00 (U) tskin inches 0.05 (L) 0.141 Wfuel pounds 15.9 17.5 Wempty pounds 61.5 74.8
Variable CSSO Solution Global Optimum t inches 0.15 (L) 0.15 (L) r inches 1.2 (L) 1.2 (L) lrod inches 47.8 58.5 c inches 6.0 (L) 6.0 (L) b inches 40.5 42.0 (U) 0 9:04 8:22 t=c 0.05 (L) 0.05 (L) xle inches 4.5 0.0 (L) %cam 1.55 2.77 tskin inches 0.05 (L) 0.05 (L) Wfuel pounds 17.8 15.9 Wempty pounds 64.6 61.5
Table 2: Comparison of Global and Local Optimal
Table 3: Comparison Between Average CSSO-NN
durance of the hovercraft (g7 ). The remaining two system optimizations converged to a locally optimal point shown in the table at which a third constraint, the rst natural bending frequency constraint (g4 ), was active. The importance of ecient analysis and design methods is even more pronounced by the results of these system optimizations. For the 50 conventional optimization trials, the average number of system analyses required for one optimization was 1155. Furthermore, due to the iterative nature of the system analysis, each conventional optimization required an average of 56,423 contributing analyses, 18422 each for aerodynamics, structures and performance, and 1155 for structural dynamics. The CSSO-NN algorithm was applied to the AHC problem as described earlier in an attempt to reduce the number of analyses performed at both the discipline and system levels. The results presented here are based on four trials beginning from the center of the design space. Initial neural network response surface mappings were constructed about this initial point using a 10% perturbation on each design variable independently, yielding 12 initial designs. The cost of analyzing these 12 designs is recovered immediately compared to determining nite di erence gradients in the full system optimization procedure. The most stringent requirement for assessing the validity of the CSSO-NN algorithm is its ability to locate the global optimal design. The average design identi ed using this method is compared with the global optimum in Table 3. The average empty weight determined by the CSSO-NN process is 64.6 pounds compared to the actual minimum empty weight of 61.5 pounds. This is a result of one trial arriving at a nal fuel weight of 23.7 pounds due to a local minimum in the approximate design space. The remaining three trials were able to locate solutions near the global optimum but all slightly infeasible with an average empty weight from these three trials of 60.99 pounds. The greatest variations as a percentage of the design variable range between the global optimum and the solutions identi ed by the CSSO-NN algorithm occur for design variables lrod , xle, and %cam (x3 , x8 , and x9 ).
However, the variation of empty weight with these variables is 1.7%, 0.02%, and 0.02% over the design variable range around the optimal solution. With the neural network training error of 3% used in this study, these variations are undetectable in the approximate design space and are of minimal signi cance in this problem. A measure used to assess the performance of the CSSO-NN algorithm is the number of system and contributing analyses required in order to nd the optimal solution. For the four trials conducted on the AHC problem, the CSSO-NN algorithm was able to identify the global optimal solution within 55 system analyses and (4351,2586,2861,309) analyses of the aerodynamics, structures, performance, and structural dynamics disciplines. This represents a 1-2 order of magnitude reduction in the number of system analyses and a 1 order of magnitude reduction in the number of discipline analyses over the all-at-once optimization procedure. It should be noted that although the number of analyses performed at the discipline level is decreased compared to all-at-once optimization, the ratio of the number of discipline analyses to system analyses is greater. Despite these signi cant savings, the number of discipline analyses required to nd the optimal solution may still be unattainable for realistic problems in a practical design environment. Work in the area of variable complexity modeling may aid in reducing the amount of computational expense associated with performing these subspace optimizations [8]. The number of system analyses performed during the MDO process is purely a function of the number of iterations required for the process to converge and the number of analyses required to construct initial response surface mappings. The speed with which the optimal design is reached is illustrated in Figure 8. This gure depicts the merit function (Wempty ) history at the system level (fully approximate) and within the structures discipline. Since the structural designers only control a portion of the design vector and subspace optimization preceeds system coordination in the design process it is expected that the actual designs obtained as a result of analyzing the neural net-
All-At-Once Optimization Solutions
and Global Optimal Solutions
8
150
130
Nonparametric Iteration 0 Iteration 5 Iteration 10
725 700
110 RPM
Empty Weight (pounds)
750
System Level, Actual System Level, Approximate Structures Subspace, Actual Structures Subspace, Approximate
90
675 650 625
70
600
50
0
1
2
3
4 5 6 Iteration
7
8
9
design point, iteration 10
10
10
15
20 25 30 35 40 Fuel Weight (pounds)
45
50
(a) Single Level Approximation
Figure 8: Merit Function Convergence
15000 von Mises Stress in Rod (psi)
work predicted optima from the system coordination would be better than those which result from subspace optimization at a given iteration. This is in fact the case. Consequently, the structural designers are limited in the extent to which they can a ect the system by the location of the baseline design which results from the previous system coordination. Another view of this is that the system coordination is e ective in integrating design information from the various disciplines in order to obtain an improved system design. A previously unexplored facet of the CSSO-NN approach to MDO is the way in which multi-level approximations impact the process. In particular, it is of interest to examine the error of multi-level approximations and the error in the approximation on which they are based. This is shown in Figure 9. Figure 9a illustrates the neural network representation of the motor RPM as a function of the fuel weight in the vicinity of the nal design point (Wfuel = 16:6 pounds) with all other design variables held xed at their nal values. The approximation at this point improves as the design evolves. By iteration 10 the response surface mapping is accurate, but only in the vicinity of the nal design as illustrated in Figure 9a. The impact of this approximation on the von Mises stress at the base of the rod on the side perpendicular to the plane of rotation is shown in Figure 9b. In this case the stress prediction follows the behavior of the RPM approximation in that it is fairly accurate near the design point and does not vary away from this point. To further assess the multilevel approximation impact consider Figure 9c. This gure represents the stress variation as a function of the actual motor RPM. It can be seen from this gure that the multilevel nature of the stress approximation shown in Figure 9b does little to a ect a straightforward approximation in this instance.
12500 10000 Nonparametric Iteration 0 Iteration 5 Iteration 10
7500 5000 2500 0
design point, iteration 10 10
15
20 25 30 35 40 Fuel Weight (pounds)
45
50
(b) Multi-level Approximation
Von Mises Stress in Rod (psi)
14000 Nonparametric Single Level Approx., Iteration 10
13000
12000
design point 11000 660.0
680.0 700.0 720.0 Motor Speed (RPM)
740.0
(c) Single Level Approximation of Stress Figure 9: Error in Multi-level Approximation
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multiple \optimum" designs are obtained (currently one from each discipline) in each design iteration. It has been demonstrated that these designs can e ectively be integrated to obtain an improved system design. A new facet of the CSSO-NN algorithm implementation has been explored in this paper, that of the multi-level response surface mapping. Although the design space for which the response surfaces were constructed is rather benign, the initial implementation of the multi-level approximation to the design space illustrates the opportunity for application of this methodology to more complex problems. Finally, application of the CSSO-NN multidisciplinary design optimization strategy to a series of increasingly larger problems has shown a small increase in the number of analyses at the system and discipline levels required to obtain optimal system solutions. This bodes well for application of this methodology to more realistic design problems.
The current CSSO-NN implementation represents the third in a series of test problems. Two previous implementations are discussed in Reference [2]. Table 4 below summarizes the system analysis results of the test problems to date. Although other many other Design Contributing # System Problem Variables Analyses Analyses Analytic Problem 3 2 33 Aircraft Concept 6 3 35 AHC 11 4 55
Table 4: Trends in Number of System Analyses to Locate Optimal Solution
factors may in uence the number of system analyses required to identify the global optimal solution, for the example problems studied to date the e ect of doubling the size of the design space results in an increase of only 70% in the number of system analyses for identi cation of the global optimum.
Acknowledgements
VI. Conclusions
This work was supported in part by the National Aeronautics and Space Administration Langley Research Center Grant NAG-1-1561, Dr. J. Sobieski, Project Monitor.
Current practices in engineering design focus on discipline level design based on discipline measures of merit. This is illustrated by discipline analysis tools which contain as part of their functionality the ability to optimize based on discipline level performance objectives; e.g. nite element analyses which provide the ability to minimize weight. It is recognized that the optimal system is comprised not of optimal parts, but of subsystems which represent the best compromise between discipline goals. The systematization of response surface mapping within the CSSO-NN algorithm to allow decoupling of the subspace analyses facilitates the solution of a common problem in all disciplines. This decomposition has been shown to lead to the design of optimal systems. In addition to providing a means by which designers at the discipline level can determine the impact of their decisions on the system, the CSSO-NN algorithm has been demonstrated to reduce the cost and time associated with designing a system in comparison to traditional optimization of a system. Since traditional optimization of a system is currently an unattainable goal, the order of magnitude reduction in the number of system and discipline analyses is a step toward application of MDO methodologies to practical design problems. A fundamental premise of the CSSO-NN algorithm is that the discipline level designers can play an important role in the determination of optimal systems. In order to preserve this role the current method provides a means by which discipline designers can utilize their tools and expertise in an ecient manner to solve the system optimization problem. As a result,
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[6] I.P. Sobieski and I.M. Kroo. Aircraft Design Using Collaborative Optimization. 34th AIAA Aerospace Sciences Meeting and Exhibit, 96-0715, Reno, Nevada, January 1996. [7] G. V. Reklaitis, A. Ravindran, and K.M. Ragsdell. Engineering Optimization, Methods and Applications. John Wiley and Sons, New York, New York, 1983. [8] S.N. Gangadharan, R.T. Haftka, and Y.I. Fiocca. Variable complexity modeling structural optimization using response surface methodology. AIAA/ASME/ASCE/AHS/ASC 36th Structures, Structural Dynamics and Materials Conference, AIAA 95-1164, New Orleans, Louisiana, April 1995.
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