reaches a stasis. It is important to note here that, the strategy used to handle the coupling variables in this research is a signi cant departure from existing.
COEVOLUTIONARY GENETIC ADAPTATION : A NEW PARADIGM FOR DISTRIBUTED MULTIDISCIPLINARY DESIGN OPTIMIZATION Prasanth B. Nair and Andy J. Keaney University of Southampton, High eld, Southampton SO17 1BJ, U.K
Abstract
ploiting the synergistic e ects of the coupling in the problem physics at every stage of the design process. One advantage o ered by this methodology is the achievement of calendar time compression via concurrency in the product design and development cycle. It also allows for tradeo studies between performance considerations, manufacturing, supportability, economics, and life cycle issues to be conducted at all stages of the design process. Hence, the adoption of MDO methodology is not only expected to lead to better designs, but also the establishment of a more physically meaningful design practice as compared to the traditional sequential approach. A recent review of the state of the art in MDO has been presented by Sobieszczanski-Sobieski and Haftka1 . It was noted in this review that \organizational challenges" and \computational cost" are generally perceived as the major obstacles to the application of MDO methodology in industrial settings. A good deal of earlier research work in MDO has focused on system optimization approaches, also known as multidisciplinary feasible methods (see, for example, reference2). These approaches require a suite of highly integrated disciplinary analysis tools connected to a single optimizer. The major disadvantages of this line of approach include : (1) The high costs involved in software integration and maintainance of the integrated design system, (2) The disciplinary groups do not have any decision making power in the design process, and (3) The increase in computational cost and potential convergence problems due to the use of a single optimizer to handle all the design variables, and the requirement of iterations between the disciplines to arrive at a multidisciplinary feasible solution at each function evaluation. These drawbacks of system optimization approaches coupled with the ever increasing requirement of addressing organizational and computational challenges in MDO have motivated the development of distributed optimization architectures. The multilevel optimization architectures on which research has recently been pursued with particular vigor are Con-
This paper introduces a new paradigm for distributed multidisciplinary design optimization (MDO) of complex coupled systems. It is inspired by the phenomena of coevolutionary adaptation occurring in ecological systems. The focus of the paper is to develop
exible design architectures for addressing the organizational and computational challenges involved in application of formal optimization techniques to large-scale multidisciplinary systems. The distinguishing advantages o ered by the proposed MDO architectures include retainment of disciplinary autonomy and decentralized decision making, massive parallelism, reduced software integration and interdisciplinary communication overheads, accommodation of variable-complexity design parameterization with a mix of discrete and continuous variables, and robustness in nonconvex design spaces. The niche for coevolutionary adaptation based search strategies in the spectrum of MDO methodologies is discussed. Problems from the domain of structural optimization via substructuring are used to illustrate MDO methods based on the coevolutionary genetic adaptation paradigm.
Introduction
Multidisciplinary design optimization (MDO) is an enabling methodology for the design of complex systems, the physics of which involve couplings between various interacting disciplines/phenomena. The underlying focus of MDO methodology is to develop formal procedures for both exploring as well as exPhD Student, Computational Engineering and Design Center, Department of Mechanical Engineering, Member AIAA. y Head of Mechanical Engineering Department, Director of Computational Engineering and Design Center. c 1999 by Prasanth B. Nair. Copyright Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
1
models. In fact, in reference7, a feedforward neural network was rst constructed to map the design space before the optimization studies. Hence, when the dif culties of high computational cost are coupled with nonconvex design spaces, standard EA approaches are impractical and more advanced strategies are required. The development of computational frameworks for construction and management of approximation models to accelerate EAs has been the focus of some recent research (see, for example, references8 9). It is expected that the computational cost of using EAs could be overcome by using such advanced DSS strategies. As a consequence of the high computational cost involved in using GAs for DSS, its application to multilevel optimization becomes computationally prohibitive. An approach based on predator-prey coadaptation was applied to bilevel optimization problems by Venter and Haftka10 . It was shown that this approach could lead to savings in the computational cost. To the authors' best knowledge, this is the rst reported study on the application of coevolutionary algorithms to design optimization. An excellent description of the theoretical aspects of CGAs and results for some application areas can be found in the dissertation of Potter11 . CGAs can be viewed as a generalization of the predator-prey model used in reference10. This paper introduces the paradigm of coevolutionary adaptation in the context of distributed MDO of complex coupled systems. Inspired by this paradigm, it is shown that the design process can be modeled as the process of coadaptation between sympatric species in an ecosystem (i.e., species which live in the same place, rather than being geographically isolated). Each discipline is modeled as a species which collaborates with the other species to improve the discipline speci c objectives and satisfy local constraints. In order to allow applicability to systems with arbitrary coupling bandwidth, the coupling variables are not explicitly represented as design variables. The strategy used here allows for satisfaction of interdisciplinary compatibility constraints at the optima under some mild assumptions. Two architectures based on whether the design variables are decomposed into disjoint sets or the disciplines are allowed to share design variables are developed here. In the former case, the disciplinary species are responsible for evolving independent sets of design variables. In contrast, the latter MDO architecture allows the disciplinary species to control the disciplinary as well as multidisciplinary design vari-
current Subspace Optimization (CSSO)3 4 and Collaborative Optimization (CO)5 . These architectures enable the solution of large-scale MDO problems in a distributed fashion and retain the advantages of division of labor. Multilevel optimization approaches to MDO in general involve the use of a system level optimizer which guides a number of disciplinary optimizers to improve the system performance, satisfy the disciplinary constraints, and ensure multidisciplinary feasibility (MF) of the converged solution. Since MF need not be enforced at each iteration, this approach circumvents the problem of \disciplinary sequencing", in which some of the disciplines have to wait for data from the other disciplines before carrying out their design studies. Hence, distributed optimization allows for the possibility of tackling MDO problems in a concurrent fashion (calendar-time compression) and also retains the autonomy of disciplinary specialists in the design process. The latter advantage is widely acknowledged to be a crucial factor in the acceptance of formal MDO methods by industry. Hence, distributed MDO methods are well suited to the hierarchical organizational structure and heterogeneous computing platforms found in many large industries. In summary, distributed multilevel optimization architectures combined with approximation techniques show a good deal of promise in addressing many of the challenges in MDO practice. On the lines of the discussion presented in Alexandrov and Kodiyalam6, MDO methods can be broadly classi ed on the basis of the design architecture or formulation, the optimization algorithms used for design space search (DSS), methods used for constructing approximation models to accelerate DSS, and the framework employed for managing variable- delity analysis models. The area of MDO architectures is closely coupled with the elds of system decomposition and optimization. The research work reported here focuses on the rst two points, i.e., the MDO architecture as well as the optimization algorithm. Stochastic nongradient optimization techniques such as Evolutionary Algorithms (EAs) show promise for tackling nonconvex MDO problems; see, for example, Lee and Hajela7. Even though, this class of optimization techniques o er many advantages compared to gradient search, the number of function evaluations required to converge to an optimal solution is considered prohibitive for many problems. This bottleneck of high computational cost has led to very few applications of EAs to MDO using high- delity analysis ;
;
2
n variables. Potter and Dejong12 used an approach in which the problem is decomposed into n species - one species for each variable. Hence, each species contain a population of alternative values for each variable. Collaboration between the species involves selection of representative values of each variable from all the species and combining them into a vector which is then used to evaluate f(x). An individual is hence rewarded based on how well it maximizes the function within the context of the variable values selected from the other species. This procedure is analogous to the univariate optimization algorithm wherein 1-D search is carried out using only one free variable. It was shown that a CGA approach to function optimization could lead to faster convergence (compared to a conventional GA) for low to moderate values of variable interdependencies (also referred to as epistasis in GA literature). This can be primarily attributed to the attendant reduction in the e ective search space due to coevolution of each variable concurrently, e.g., for the function optimization problem described earlier the original search space of size (2 ) has been reduced to a series of n constrained search spaces of size 2 ; where k is the number of bits used to represent each variable. Note here that it is also possible to decompose the original problem variables in to several blocks, with each species representing a block rather than a single variable. In CGA-based search, the term generation is used to refer to a cycle involving selection, recombination and tness evaluation for a single species, and the term ecosystem generation refers to an evolutionary cycle through all the species being coevolved. The reader is to referred to Potter11 for a detailed overview of CGA literature. From this brief overview of CGAs, it is clear that there exists the possibility of using coevolutionary genetic search techniques for decomposition and solution of large-scale nonconvex optimization problems. However, no research work has been reported in the literature which focuses on exploiting this potential of CGAs for design optimization problems. To illustrate the design architectures developed here, consider a typical MDO problem for aircraft wing design (adapted from the example in reference13) involving the disciplines of aerodynamics and structures shown in Figure 1. The variables in the owchart are described in Table 1. For this problem, x denotes the vector of multidisciplinary design variables; x1 and x2 are the vectors of disciplinary design variables corresponding to the aerodynamics and
ables. A Coevolutionary Genetic Algorithm (CGA) is used to model the patterns of interaction between the various species. Results are presented for structural optimization problems, wherein the concept of substructuring can be used to reformulate the optimization problem in terms of coupled subsystems. It is shown that the rst coevolutionary MDO (CMDO) architecture successfully converges to an optimal solution satisfying the disciplinary as well as interdisciplinary compatibility constraints. The e ect of coupling bandwidth and increase in problem size on the performance of the coevolutionary MDO architectures are also examined.
The Present Approach
The research work reported here continues the pursuit of more exible MDO architectures, the advantages of which include, (1) disciplinary autonomy in both analysis as well as optimization, and (2) applicability to systems with arbitrary coupling bandwidth. The features of a typical CGA are rst described in order to delineate the fundamentals underpinning the approaches developed. A CGA models an ecosystem consisting of two or more sympatric species having an ecological relationship of mutualism. The steps involved in a typical CGA algorithm are summarized below : Step 1 : Initialize a population of individuals for each species randomly. Step 2 : Evaluate the tness of the members of each species. Fitness evaluation in a CGA involves the use of the following algorithm : Choose representatives from all other species. FOR each individual i in a species being evaluated
k n
k
DO Form collaboration between i and representatives
from other species Evaluate the tness of collaboration by applying it to the target problem Assign tness of collaboration to i
ENDDO
Step 3 : If the termination criteria is not met, then apply a canonical GA involving the operators of reproduction and genetic recombination to arrive at a new population for each species. Go to Step 2. For a better understanding of the application of CGAs to function optimization, consider the problem involving maximization of f(x), which is a function of 3
structures disciplines, respectively. Note here that ar- In order to eliminate the couplings between the disriving at a MF solution of the coupled system would ciplinary analysis modules, the original system can be involve iterating between both the disciplines. decomposed as shown in Figure 2. Multidisciplinary feasibility is ensured via the use of coupling compatibility constraints. This decomposition of the original x, x1 Aerodynamics coupled system makes it possible to concurrently analyze a design. g1, f1 A1
Compatibility ||y12−y12*||=0 ||y21−y21*||=0
y21
y12
x, x2
Structures
g2, f2
A2
x, x2, y12
x, x1, y21 f1, g1, y12*
Figure 1 : Data Flow Diagram for a Typical MDO
Aerodynamics
Problem
f2, g2, y21* Structures
The state equations for the two disciplines can be Figure 2 : Distributed Multidisciplinary Analysis mathematically expressed as : The issues which are to be addressed in order to apply a CGA to this MDO problem are discussed A1(x; x1 ; u1 ; y21 (u2 )) = 0 next. [A] Problem Decomposition : and This issue is mainly concerned with how the coupled system should be decomposed into various A2(x; x2 ; u2 ; y12 (u1 )) = 0 species. In general, for MDO problems it would be where u1 and u2 are the state variables for the aerody- appropriate to decompose the design variables into namics and structures disciplines, respectively. Typ- groups on disciplinary lines. The variables which are ically, u1 is the vector of aerodynamic ow velocities, common to both the disciplines are referred to as muland u2 is the vector of structural nodal displacements. tidisciplinary design variables. The chromosome of each species generally consists of the variables it is allowed to control. In general, it may be preferable Table 1: Description of Variables in Figure 1 to decompose the design variable vector x into disjoint sets to circumvent the diculty of arriving at Var. Description a consensus on the independently evolved multidiscix Wing surface geometry plinary variables. For the case when the disciplinary x1 Airfoil geometry outside wing box species are allowed to search independently for the x2 Wing box structural dimensions, optimum values of the multidisciplinary variables, the skin thicknesses, etc. chromosome of each species would consist of both the y12 Aerodynamic loading disciplinary and multidisciplinary design variables. y21 Deformed surface geometry [B] Choice of Representative Individuals : g1 A/D Pressure Distributions, AOA limits, etc. As mentioned earlier in the description of CGAs, g2 Static and dynamic response constraints the species interact with each other via the use of repf1 Aerodynamic Drag resentative individuals. The evolution of each species f2 Structural Weight is constantly driven by evolutionary changes in the 4
of their respective representative individuals. This strategy reduces the extent of violation of the interdisciplinary compatibility constraints as coevolution reaches a stasis. It is important to note here that, the strategy used to handle the coupling variables in this research is a signi cant departure from existing multilevel MDO architectures. In contrast to other distributed MDO formulations, the present approach can tackle problems with arbitrary coupling bandwidth without an attendant increase in the problem size. The elements of the two coevolutionary MDO architectures which appear promising at this stage of the ongoing research are presented next.
species it interacts with. In genetics literature, this is referred to as the Red Queen hypothesis, wherein each species must constantly adapt just to remain in parity with the others. Hence, the issue of how to choose representative individuals from each species is of crucial importance in coevolutionary computation. It is generally preferable to use more than one representative individual in the coevolution procedure to reduce noise in evaluating the tness of collaborations. However, the computational cost in tness evaluation grows with the increase in number of representatives. In reference11 , two strategies were suggested for choosing the representative individual. The rst strategy was to select the individual with the highest tness, and the second strategy was to choose the second representative randomly and use both the representatives for tness evaluations. In the latter strategy, the maximum tness of the two collaborations is assigned to the individual under consideration. It was found via numerical experiments that the rst strategy is inherently greedy in nature and may not be robust when variable interdependencies are high. In contrast, the second strategy is more robust at the expense of slower convergence. In a general CGA, it is also not necessary for the species to interact at every ecosystem generation. This is an important characteristic which can further reduce interprocessor communication costs.
Coevolutionary MDO Architecture A
In this architecture, the design variables are decomposed into disjoint sets, and hence no variables are shared between the disciplines. The chromosome of the disciplinary species are hence composed of independent sets of variables. As mentioned earlier the coupling variables are not explicitly tackled as design variables in the design process. The details of the tness evaluation scheme is similar to that presented next for architecture B.
Coevolutionary MDO Architecture B
This architecture allows for sharing of the multidisciplinary design variables among the disciplines. In contrast to architecture A, an additional constraint is imposed so as to ensure consensus on the multidisciplinary design variables controlled independently by the disciplinary species. The chromosome representation and tness evaluation details for both the species are summarized below :
[C] Fitness Evaluation :
The tness of the disciplinary species is de ned as a function of the respective disciplinary objectives and constraints. When the disciplines are allowed to share design variables, the disciplinary species are also provided a tness incentive for ensuring that it arrives at a consensus on the value of the multidisciplinary variables with the other disciplinary species. Disciplinary Species - Aerodynamics Genotype : [x1 , x1 ] New MDO Architectures Fitness Evaluation : The value of y21 obtained Various MDO architectures based on di erent consid- from system analysis of a baseline design is used to erations for tackling the interdisciplinary compatibil- compute f1 and g1 in the rst few generations. In the ity constraints and problem decomposition character- subsequent ecosystem generations, the value of y21 istics can be developed within the framework of the evaluated in the structures discipline for their reprecoevolutionary genetic adaptation paradigm. How- sentative individual gives the new value of the couever, to ensure applicability to systems with broad pling variables. This discipline is also entrusted with coupling bandwidth, the coupling variables are not the job of matching the local copy of the multidisexplicitly considered as design variables. The cou- ciplinary variables with the structures species. This pling variables are initialized to values computed constraint is computed with respect to a representafrom system analysis of a baseline design. In the tive individual from the structures species. subsequent stages of the coadaptation procedure, the disciplinary species exchange values of the coupling variables computed during the disciplinary analysis 5
Disciplinary Species - Structures Genotype : [x2 , x2 ]
optimization.
Demonstration Examples, Results and Discussion
Fitness Evaluation : The value of y12 computed via system analysis of a baseline design is used to compute f2 and g2 in the initial generations. In the subsequent ecosystem generations, the value of y12 evaluated in the aerodynamics discipline for their representative individual is used. Similar to the aerodynamics discipline, a constraint is added to the original disciplinary optimization problem to enforce matching of local values of multidisciplinary variables with the aerodynamics species.
This section presents the results of experimental studies conducted to generate computational data on the performance of the MDO architectures. The objective is to gain insights into the convergence characteristics and to make comparison studies with a traditional system optimization approach. The test problems considered here are taken from the domain of structural optimization via substructuring. Consider the equations for static equilibrium of a general structural system which is partitioned into two substructures as Substructure 1 K11 d1 = F1 ? K12 d2 Substructure 2 K22 d2 = F2 ? K21 d1 where d1 and d2 are the displacements corresponding to the nodes of each substructure; F1 and F2 are the external forces acting at the substructure degrees of freedom. This problem can be posed as a simulated MDO problem with two coupled subsystems. In the decomposition strategy chosen here, each subsystem handles the design variables corresponding to its substructure. The coupling variables for this problem are K12 ; d2 , K21 ; d1 and some components of the elements of K11 and K22 . Analysis of the rst subsystem involves solution of the equations corresponding to substructure 1. The displacement data thus obtained is used along with the value of the coupling displacement variables (d2 ) to compute the stresses in the substructure elements. Similarly, the second subsystem analysis involves solution of the equilibrium equations corresponding to substructure 2. This substructure displacement vector is used in conjunction with the coupling displacement vector (d1 ) to compute the stresses in this substructure. The objective of each subsystem is to minimize the substructure weight subject to stress constraints for the substructure elements. For all the experiments conducted here, a standard elitist GA with tournament selection, uniform crossover, bit, and creep mutation was used. The probability of crossover, bit and creep mutation were kept constant 0.5, 0.01, and 0.3, respectively. A high creep mutation is used here to prevent loss of diversity, and was found to give im-
Data Coordination, Surrogate Modeling, Decomposition, and Other Issues
A common blackboard for the coupling as well as the shared variables can be used for transferring data between the disciplines. The disciplinary species post the values of the coupling variables corresponding to the representative individuals on this blackboard during the coadaptation procedure. A local database for each discipline is also maintained, which could be used to construct simulation surrogate models to accelerate disciplinary DSS. See, reference9 for an approach to management of surrogate models in GAbased search. It is to be noted here that, since GAs constitute a global search paradigm, high quality space lling experimental data can potentially be obtained as a by-product of the coevolution procedure. Currently, investigations are underway to look at the possibility of using the information generated during the coevolutionary adaptation procedure to make decisions on system decomposition issues. This would allow for the possibility of the optimal problem decomposition structure to emerge rather than be xed a priori by the designer. An alternative is to decompose the multidisciplinary variables into disjoint sets by computing their main e ects for the disciplinary objectives via orthogonal array-based experimental designs. Since, the various species interact via a single representative individual, the optimization algorithms which can be used within the proposed MDO architectures are not restricted to evolutionary algorithms alone. In fact, there exists the possibility of using other optimization formulations/algorithms within the disciplines, e.g., the aerodynamics discipline which may deal with predominantly continuous variables may choose to use a Simultaneous Analysis and Design (SAND) formulation for the disciplinary 6
to this problem. The only interaction between the disciplinary species occur via exchange of coupling variables corresponding to the representative individuals. The best design in the population is chosen as the representative individual for each species. A population size of 25 was used for each species and the termination criteria was kept xed at 200 ecosystem generations, i.e., 5000 function evaluations for each species. The initial data for the coupling variables were generated assuming all design variables to be equal to 10.0 units.
proved results for a range of test problems. A binary string of 10 bits is used to represent all the continuous variables for the examples taken here. The stress constraints are incorporated into the tness functions via an exterior extended penalty function formulation. Note that for all the problems under consideration, the optima lies on the constraint boundary. In the rst ecosystem generation, the representative individuals were chosen randomly. In the subsequent generations, the best design vector of a species is chosen as its representative individual. The results obtained using the MDO architecture is also compared to the optimal solution obtained using a standard GA approach applied to the coupled/original structural model (which is equivalent to the Nested Analysis and Design (NAND) approach to MDO). Ten runs were carried out for each case to compute the averaged convergence trends and other statistics.
Table 2 : Comparison of Results for Problem 1 Approach NAND Average Weight Standard Deviation Minimum Weight
Problem 1
The rst design problem considered is the well documented 10 bar planar truss structure with 2 bays. The structure is parameterized in terms of the sizing variables. The design objective is to minimize the weight subject to stress constraints.
1661
CMDO Architecture A 1617
36
28
1613
1595
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10
Interdisciplinary Compatibility Constraint Violation Stress Constraint Violation
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Figure 4 : Comparison of Averaged Convergence Characteristics of the Interdisciplinary Compatibility
Figure 3 : Averaged Convergence Characteristics of Constraints and Cumulative Percentage Stress ConSubstructure Weights for Problem 1
straint Violation for Problem 1
Two species which collaboratively coevolve the variables corresponding to each bay are set up to solve this simulated MDO problem. The set of design variables corresponding to each bay is de ned as the chromosome of the two species. The design variables for this problem are bounded between 0.1 and 11.0. Since, the design variables are partitioned into disjoint sets, architecture A can be directly applied
Figure 3 compares the averaged convergence characteristics (of the substructure weights) of the two species which coevolve to the optima of the original coupled system. It can be seen from the gure that, both the species have converged quite rapidly to the optimal solution. The statistics of the optimal solution obtained using the CMDO architecture is compared to a standard GA approach (obtained using 7
a termination criteria of 10,000 evaluations) in Table 1. It can be clearly seen that the CMDO architecture gives a better solution as compared to a standard GA approach. This can be attributed to the dynamic nature of the search space for the two subsystems. This could a key factor which enables the CMDO architecture to circumvent the problem of entrapment in a false optima. Figure 4 depicts the averaged convergence trends of the cumulative percentage violation of stress constraints and interdisciplinary compatibility constraints for the structure. It can be seen from the gure that, even though interdisciplinary compatibility constraints are not explicitly enforced in the formulation, it rapidly converges close to zero. It was also found during the numerical experiments that the coevolution process is relatively insensitive to the initialization values used for the coupling variables.
Since there are 6 interface degrees of freedom for the two bays, the coupling bandwidth is much higher as compared to the earlier problem. The coupling variables for the rst substructure are the interface forces and the sti ness of the members connecting nodes 1-3 from nodes 4-6. The coupling variables for the second substructure are the displacements at nodes 1-3 and the interface forces computed during analysis of substructure 1. The tness landscape of this simulated MDO problem is nonconvex and discontinuous. A population size of 50 was used for each species, and the termination criteria was kept xed at 200 ecosystem generations, i.e., 10,000 evaluations for each species. The convergence trends of the substructure weights averaged over 10 runs is shown in Figure 6. The statistics of the optimal solution is compared with a standard GA approach in Table 3. The standard GA solution was obtained using a population size of 100 over 200 generations. The convergence trends of the CMDO architecture show more complex dynamic behavior as compared to problem 1 due to the high coupling bandwidth between the two substructures. The best solution using the CMDO architecture is marginally better than that obtained using a standard GA approach. It is encouraging to note that the performance of the CMDO architecture show no signi cant degradation with increase in the coupling bandwidth.
Problem 2
The second problem involves weight minimization of a 2 bay 18 bar truss structure (see Figure 5) subject to stress constraints. The cross sectional areas of the 18 truss members are considered as discrete design chosen from the set 0:1i; i = 1; : : : ; 128. The coordinates of the joints are considered as continuous design variables to yield a total of 30 design variables and 18 constraints for this problem. The structure is decomposed into two subsystems corresponding to the two bays to simulate a MDO scenario.
L
l
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A 3 7 16 A 6 @ ?@ ? ? ? 8@ 17 @ ? ? @ ? @ ? @ @ 6 ? ? 15 ? @ ? @ 9 @ @ ? 18 A ? 3 @ @ A? @ 5 @? 2 @ ? ? @ ? ? 12 4@ @ ? ? @ ? 13@ ? @ @ 11 ? ? 2 ? @ ? @ @ @ ? 14 A ? 5 @ @ @? @ 4 . A? 1 1 10
l
l l l l l l l l l l l l l 100
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Figure 6 : Averaged Convergence Characteristics of Substructure Weights for Problem 2 The averaged convergence trends of the interdis-
Figure 5 : 18 Bar Planar Truss Structure 8
The results obtained for the two test problems are quite encouraging and give a good idea of the convergence characteristics of MDO Architecture A. It appears that this architecture in general presents a useful tool for DSS of large-scale nonconvex MDO problems with a mix of discrete and continuous variables in a distributed fashion. The data generated during this process can subsequently be used to construct approximation models, which could then be used in conjunction with a local search technique to ne tune the design variables. On a di erent note, the formulation used for the example problems also suggests a computationally ecient way to solve large-scale structural optimization problems via substructuring, since the substructure analysis cost is signi cantly lower as compared to analysis of the original structural model. Work is currently underway to generate computational data on the performance of MDO Architecture B. This architecture is expected to introduce complex dynamics during the coevolutionary adaptation of the disciplinary species. The performance of this architecture and its ability to aid the disciplinary species to arrive at a consensus on the multidisciplinary design variables remains to be seen. It is expected that results from the area of computational immunology14 would be useful in this context. An ongoing research e ort is focused on integration of surrogate model management frameworks9 with the coevolutionary MDO architecture. This paragraph from Kau man et. al.15 appropriately sums up the motivations as well as the promises o ered by coevolutionary problem solving techniques for complex problem domains. \: : :Contrary to intuition, optimization of systems with many con icting constraints is often better achieved by partitioning the overall system into sel shly optimizing subsystems. Partitioning allows each subsystem to ignore constraints and avoids trapping on poor optima. Overall good performance arises as a collective emergent behavior of the interacting, coevolving subsystems. Here, in fact, is an invisible hand made visible. It is presumably no accident that organizations which have developed to solve hard problems with many con icting constraints are typically divided into departments, pro t centers, and other semi-independent sub-organizations. Indeed, abundant, if anecdotal, evidence suggests that monolithic hierarchical organizations in business, government, and elsewhere perform and adapt poorly. Calls for decentralization and \ attening" of organizations
ciplinary compatibility constraints and stress constraints for this problem are shown in Figure 8. It was observed that the maximum violation of the stress constraints for this problem across all the runs was of the order of 0.01%.
Table 3 : Comparison of Results for Problem 2 Approach Standard CMDO Analysis Architecture A Average 2497 2500 Weight Standard 44 97 Deviation Minimum 2419 2344 Weight 0
10
Constraint Violation Norm
Interdisciplinary Compatibility Constraint Violation Stress Constraint Violation −1
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10
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200
Figure 7 : Comparison of Averaged Convergence Characteristics of the Interdisciplinary Compatibility Constraints and Cumulative Percentage Stress Constraint Violation for Problem 2
Concluding Remarks
Distributed MDO architectures inspired by the phenomena of coevolutionary genetic adaptation in ecological systems is introduced in this paper. It is shown that this approach to MDO o ers a number of advantages which include, the retainment of disciplinary autonomy and decentralized decision making capabilities in product design, massive parallelism, and robustness in nonconvex design spaces. The other distinguishing advantage o ered by this approach is the applicability to systems with arbitrary coupling bandwidth without an attendant increase in the problem size. 9
are common. The results demonstrated here may be an important theoretical foundation for useful decentralization. Coevolutionary problem solving may be a very general means by which human agents collectively solve hard social and practical problems."
Acknowledgments
This work was supported by a grant from the Faculty of Engineering and Applied Sciences at the University of Southampton.
References
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