A9992471 3 AIAA-99-1345

BLISS/S: A NEW METHOD FOR TWO-LEVEL STRUCTURAL OPTIMIZATION

Jaroslaw Sobieszczanski-SobieskP and Srinivas Kodiyalam’

Abstract-

The paper describes a Iwo-level method for structural optimization for a minimum weight under the local strength and the displacement constraints. The method divides the optimization task into separate optimizations of the individual substructures (in the extrem-e, the individual components) coordinated by the assembled structure optimization. The substructure optimizations use local cross- sections as design variables and satisfy the highly non-linear local constraints of strength and buckling. The design variables in the assembled structure optimization govern the structure overall shape and handle the displacement cons&aints. The assembled structure objective function is the objective in each of the above optimizations. The substructure optimizations are linked to- the assembled structure optimization by the sensitivity derivatives. The method was derived from a previously reported two- level optimization method for engineering systems, e.g., aerospace vehicles, that comprise interacting modules to be optimized independently, coordination provided by a system-level optimization. This scheme was adapted to structural optimization by treating each substructure as a module in a system, and using the standard Finite element analysis as the system analysis. A numerical example, a hub structure framework, is provided to show the new method agreement with a standard, no-decomposition optimization. The new method advantage lies primarily in the autonomy of the individual

. -- * Manager, Computational AeroSciences and Multidisciplinary Research Coordinator, MS139, NASA Langley Research Center, Hampton. VA 23681; j.sobieski@larc.nasa.gov; AIAA Fellow + Chief Technology Officer, Engineous Software Inc., Morrisville, NC 27560;srinivas@~engineous,com: AIAA Sr. Member Copyright 0 1998 by the Ainerican Institute of Aeronautics and Astronautics, Inc. No copyright is asserted in the United States under Title 17. U.S. Code. The U.S. Government has a royalty- free license to exercise all rights under the copyright claimed herein for Governmental Purposes, Al other rights are reserved by the copyright owner.

substructure optimization that enables concurrency of execution to compress the overall task elapsed time. The advantage increases with the magnitude of that task.

Notation

BB - black box, a module, in the mathematical model of a generic system; in a structure it stands for analysis of a substructure. BBA(Y,,(Z,X,)) - analysis of BB, to compute Y, for given Z,, local X, and inputs from the the other parts of the system. BBSA(D(Y, ,(Z,X,)) - sensitivity analysis of BB, to compute its output derivatives w.r.t. Z, and X, BBOPT,(X, ,W,, G,) - optimization in BB,

BBOWK opt 7 Z, Y,) - analysis of BB optimum for sensitivity to parameters D(Vl,V2) - derivative Vl with respect to V2 where V2 is an independent variable and Vl is either a composite or implicit function. d(Vl,V2) - partial derivative of a composite or implicit function Vl with respect to V2 FEA - Finite Element Analysis 0 - system objective function G, - vector of the system-level constraint functions; e.g., G” - displacement constraint, gr,, local to BB, , g,, <= 0 is a satisfied constraint

P - vector of constant parameters, e.g., mechanical properties of the structural material, or external loads. Q - structural internal forces acting on the boundary of i-th substructure. RS - response surface to approximate a multivariable function SA((Z,X),Y) - computation that outputs Y for a system defined by P, Z, and X

SOPT - system optimization

SSA(D(Y,(Z,X)) - system sensitivity analysis to compute sensitivity of the system response Y w.r.t. Z and X

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substructure - a part of a structure. It may be composed of one or more finite elements.

u - vector of structural displacements

X - vector of the design variables x+i , length IQ&, these variables are local to BB,;

X without subscript- a vector of all concatenated X, , length NX = CNx,

Y, - vector of variables output from BB,

Z - vector of the system design variables zI,

p - penalty factor set by user

1. IntraductioQ

The new method described herein is an adaptation of the method known as Bi-Level Integrated Synthesis (BLISS), Sobieski et. al., 1998, to applications in structures, hence the acronym BLISS/S. BLISS defined in the above reference was intended for synthesis of generic, multidisciplinary engineering systems, e.g., an aircraft, whose design is governed by the disciplines of aerodynamics, structures, control, etc. Fig.1 illustrates an example of such a system composed of three modules (also called Black Boxes). The number of modules in actual applications is not limited to only three. The Black Boxes are coupled by exchanging information Y while the variables X and Z, local and system- level, govern their design. BLISS was demonstrated to be effective in preliminary testing and its further development and validation continues. A new finding in this development was that the BLISS general algorithm might be specialized to structural optimization. This paper expands on that finding.

The key concept in the BLISS method was a decomposition of the design task into subtasks performed independently in each of the modules, see Fig. 1, and a system-level or coordination task giving rise to a two-level optimization. In general, decomposition was motivated by the obvious need to distribute work over many people and computers to compress the task calendar time. Equally important benefit from the decomposition is granting an autonomy to the groups of engineers responsible for each particular subtask in choosing their methods and tools for the subtask execution. As an additional advantage, the concurrent execution of the subtasks fits well the technology of massively concurrent processing that is now becoming available. The above motivation and benefits apply also in large-

scale structural optimization, especially for structures assembled of many dissimilar components or substructures. Applicability of two-level optimization to structures stems from the observation that, in general, a structure is defined by variables of two categories: the cross-sectional variables X, and the overall shape geometry variables Z. In optimization it is useful to distinguish between X and Z because:

The X variables are associated with individual components and, therefore, they tend to be clustered. Also, the constraints they govern directly, e.g., the stringer buckling in built-up, thin-walled structures typical of aerospace vehicles, tend to be highly nonlinear. The total number of the X variables in a typical airframe is in thousands but their number in an individual substructure is likely to be quite small.

The number of Z variables is much smaller than the total number of X variables.

Nonlinearity of the overall behavior constraints, such as displacements, with respect to X and Z tends to be much weaker than that of the local strength constraints.

Both Z and X influence entire structure, but the Z influence tends to be much stronger than that of X because it is exerted through the control of the structure overall shape while the X influence outside of the component they are associated with is governed by the degree of redundancy (that influence is zero in a statically determinate structure).

Accordingly, one may divide structural optimization procedure into two subtasks that alternate until convergence: 1. Separate, concurrently executed optimizations in

the X-subspaces, each subspace corresponding to the Xs associated with a component and dominated by the local, highly nonlinear constraints.

2. A single optimization in the Z -space in which only the displacement or frequency constraints of mild nonlinearity are present so that efficiency of Linear Programming may be exploited.

Because of the approximations involved at both levels, the optimizations in the X and Z spaces have to alternate iteratively until convergence.

The above decomposition is desirable not only for the reasons articulated in the foregoing but also

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because structural optimization performed in a conventional, all-in-one manner for a structure with a large number of design variables usually requires the use of approximations as surrogates of the full analysis in order to reduce the computational cost. However, if highly nonlinear constraints are present, the approximation error control requires imposition of narrow move limits in each stage based on the approximate analysis. That increases the number of stages required for convergence and may ultimately offset the intended benefit of the use of approximation as a cost control measure. A few different approaches to implementation of two-level optimization have become a part of optimization literature, e.g., a review in Balling and Sobieszczanski-Sobieski, 1996. These approaches apply to both generic, multidisciplinary engineering systems as well as to structural systems and they differ primarily in the way the Z-level and X-level optimizations are coordinated. A partial list of the techniques used in the past to solve the coordination problem in two-level optimization includes:

l The optimal X are tied to Z by means of the derivatives of the optimal X with respect to Z treated as parameters in the X-level optimization (e.g. Sobieszczanski-Sobieski, 1993);

l The optimal X are made dependent on Z by constructing response surfaces (or training Neural Nets) in the Z space built on the results of optimization performed at points judiciously placed that space using the Design of Experiments techniques (e.g., Sellar et. al., 1996, Stelmack and Batill, 1998).

= The BLISS algorithm, Sobieszczanski-Sobieski et. al., 1998, differs from the cited approaches by the following features in its solution to the coordination problem:

. the linearized system objective is the objective function in each X-space optimization.

. the influence of Z on the system objective is accounted for via the derivatives of the objective optima from each X-subspace with respect to Z. These derivatives are combined in a chain-differentiation manner with those from the system sensitivity analysis. The derivatives of the optimum objective with respect to the optimization parameters are computed in two different ways:

l Version A that uses Linear Programing to repeat optimization for perturbed

parameters followed by computing derivatives by a standard finite difference technique (Vanderplaats and Cai, 1986).

l Version B based on the concept of the Lagrange multipliers as shadow prices of the objective (Barthelcmy and Sobieszczanski- Sobieski, 1983).

The BLISS/S procedure adapts the above approach to structural optimization by treating the structure as a system whose modules are substructures, in the extreme a substructure may be a single element, e.g., a beam in the framework. Therefore, introduction of BLISS/S begins with a synopsis of the original BLISS procedure and a definition of the correspondence between the concepts and numerical operations in analysis and optimization of a generic system and a structure. Next, that correspondence is used to transform the generic BLISS procedure into BLISS/S.

A numerical example of a framework configured as a hub structure follows to demonstrate the BLISS/S procedure application. The paper concludes with a discussion of the BLISS/S performance that was found to agree in terms of the numerical results with optimization without decomposition. In regard to BLISS/S efficiency the results indicated that one should expect the efficiency advantage over the optimization without decomposition to increase with the scale of the optimization task.

2. Svnomis of BLISS

The optimization problem for a generic system shown in Fig.1 may be formally stated as

1) Find X and Z

Minimize Q, wa

Satisfy G(X,Z) <=O and g(X,Z) <= 0

The BLISS procedure solves the above problem by interleaving optimizations in subspaces X corresponding to the modules with a system-level optimization (the coordination problem). The procedure step-by-step recipe is defined in Sobieszczanski-Sobieski et al., 1998 as follows

1. Initialize X and Z variables;

2. Execute System Analysis (SA) to compute the behavior variables;

3. Examine termination criteria; stop or continue. 1276

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4.

5.

6.

7.

8.

9.

Execute sensitivity analyses of the modules to obtain derivatives of each module output to its input; the input includes data transmitted from other modules, variables Z, and the local variables X. Execute System Sensitivity Analysis (SSA) using data from #4, Execute autonomous optimizations in the modules using local X in each to minimize contribution of each module to the system objective. Execute optimum sensitivity analyses in each module to obtain derivatives of the optimum with respect to parameters. The parameters comprise Z and the inputs from other modules.

Execute the system-level optimization using the Z variables and approximating the influence of Z on the module optima by means of the derivatives from #4 and #7. Update X and Z and repeat from #2.

The above procedure developed for a general, multidisciplinary optimization is modified and adapted to structures in a way described below.

3.cialized for Structural Optimizatia

Assume X to stand for the detailed variables defining cross-sections of structural components. A component may be a substructure in the finite element model or a single element in that model. The Z-variables define the overall shape geometry for the structure. The optimization objective is, typically, the structure weight, and the constraints are divided into the local component-level constraints g = g(X,Z), and the overall behavior constraints G= G(Z,X). For example, in a thin-walled wing structure a local buckling constraint is an instance of a component-level constraint, while a limit on the wingtip deflection in bending is an overall behavior constraint. In this paper, the objective function @ (X,Z) is the structural weight, but any other objective, e.g., the cost, could be prescribed as long as it is computable as a function of X and Z.

The generic formulation in eq. 1 when interpreted with the above definitions poses an optimization problem for a structure that may be represented as a coupled system shown in Fig.2. The system components are labeled in Fig.2 as beams

only because the numerical example presented below is a framework, but there is no restriction on the types of structural elements that can be accommodated.

3.1 The BLISS/S I’rocedur e.

The BLISS/S procedure solves the above problem by decomposition. In optimization of i-th substructure performed for constant Z and Q, the local constraints depend on Z directly through the substructure geometry and, indirectly, through the influence of Z on Q. The X variables local to i-th substructure exert direct influence on the local constraints, and an indirect influence through the substructure stiffness coefficients that contribute to FEA and, therefore, to forces Q, not only the local ones but also to those acting on all substructures. Hence, the local X affects the optimization results in all substructures, not only the i-th substructure. The system-level displacement constraints are controlled by both Z and X that affect the substructure stiffness properties through its geometry, overall and cross- sectional. This web of influences is analogous to the one in a general modular system for which the original BLISS was developed.

Comparing to a general, multidisciplinary engineering system for which BLISS was originally developed, the structural system is degenerate in the sense that the intermodular data exchange is limited to the flow of the internal forces data from the Finite Element Analysis Black Box to the Black Boxes representing the substructures (the structural elements). Also, the entire system analysis and sensitivity analysis are both contained within the Finite Element Analysis. Therefore, BLISS may be adapted to structures by substituting the elements of a structural optimization problem solution in the BLISS step-by-step recipe according to the correspondences displayed in Table I.

The above correspondences provide a basis to convert the BLISS standard recipe into the a step-by- step prescription for BLISS/S written below ror k-th cycle

1. (//) Update X and Z to the new values generated in the previous cycle k-l (Initialize X and Z with the best guess if this is the first cycle).

2. (//) In BB, for i-th substructure calculate the stiffness properties needed to represent the substructure in the FEA of, the assembled structure. Repeat for all substructures.

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3.

4.

5.

6.

7.

9.

Execute SA (FEA of the assembled structure) to compute the behavior variables: the displacement (u), the structure internal forces (Q) acting on the boundary of each substructure, and structural weight.

(//) Execute BBA for i-th substructure to compute its strength constraints (typically: stress and buckling), and its structural weight W,. Repeat for all substructures. (optionally, the structural weight W, for the substructures may be computed in Step 3 instead).

Check the termination criteria and continue or stop.

Execute SSA (FE Sensitivity Analysis) to obtain: D(G”,z,), D(W,,z,), D(Q,z,J, D(G”,xi), D(Q,xj) , and for all elements in Z and X.

(//) Execute BBSA for each BBi to obtain d(Wi,Xj)t d(g,ZJ and d(g,U d(g,QJ 8.1 (//) If Z exist, execute BBOPT for BBi,

Find Ax

Minimize W = Wk-’ + xi(d(Wi,X,) + (JXW, tQ>D(Qi &I>> A3

Satisfy g <= 0; and G” <= 0

In the above approximate G” = (G”)‘.’ + D(G”,Xi) AX; Use d(Wi,Xj) and d(g,Q) from #7, D(Q,X,) from #6, and D(Wi,Qi) from #9 in the previous cycle k- 1.

8.2 (//) If Z does not exist, execute BBOPT for BBiv

Find Ax

Minimize W = Wk.’ + x,(d(Wi,Xi) + (D(Wi,Q>D(Qi,X,))>Aq + p d(G”&)AX,

Satisfy g <= 0; and G” <= 0

In the above approximate G” = (G”)k-’ + D(G”,X,) AX; Use d(W,,X,), and d(g,Q) from #7; D(G”,X,) and D(Q,,Xi)from #6; and D(Wi,Q,) from #9 in the previous cycle k-l.

(//) Execute BBOSA to compute D(W,,Z,) and D(W,,Q,)using(as in #7, BLISS/B) the algorithm from Barthelemy and Sobieszczanski-Sobieski, 1983. The algorithm is summarized in Appendix.

10. Execute SOPT, IF Z present, BYPASS if Z absent

Find AZ ~~

Minimize w=cwi

Satisfy G” <= 0

In the above, approximate

Wi = (W,),, + (D(W,,Z) + (D(W,vQ> D(Q,,Z)))AZ using D(W,,Z), D(W,,Qi) from BBOSA in #9, and D(Q,Z) from SSA in #6

G” = (G”), + D(G”.Z) AZ using the derivatives from SSA in #6.

Begin the next cycle, k=k+l, from #I.

Notes:

In the first cycle, k=l, set D(W,,z,) = 0 and DWi,Q) = 0;

Concurrent, coarse-grained, processing opportunities are marked by (/I), They are the opportunities created by BLISS/S. The other opportunities that might be intrinsic in FEA and FE Sensitivity Analysis, e.g., concurrent processing of many right-hand side vectors in structural sensitivity analysis are not marked.

In #8.2, BBOPT is a sole means of satisfying G”, hence the penalty term appended to the objective. The penalty factor p should be set so that, initially, the penalty term magnitude is of the same order as the other terms in the expression.

In #8.1 and #8.2, any suitable search algorithm may be used. It does not have to be the same for all substructures.

In #lo, owing to the linearization of all the functions, one may use a Linear Programing technique to obtain AZ

As mentioned before, the key to effectiveness OF

the BLISS/S procedure is a judicious USC of the sensitivity information. In BBOPT, step #8, the direct influence of X, on Wi is captured by d(W,,X) and the indirect influence of X through the change of Q due to the change of X is represented by the term D(Wi,Q,) D(Qi,Xj). As mentioned before, this indirect influence is important in redundant structures, and in contrast to the term d(Wi,X) that is purely local, the term reflects the influence of X, on the entire structure.. Similarly, in SOPT, the term (D(W,,Z) + (D(W,,Q,) D(Q,Z)) plays an analogous role for W, and Z.

Although SOPT, step #lO does not explicitly address the local constraints g, satisfaction of these

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is protected by the use of D(W,,Z), and D(W,,Q,) obtained from BBOSA, step #9. It is so because the algorithm of Barthelemy and Sobieszczanski- Sobieski, 1983 (see also Appendix) generates these derivatives as constrained derivatives. In other words the algorithm treats Z and Q as the parameters of the optimization that was executed in BBOPT, and coordinates the changes aW,, aZ, and aQ, so as to preserve g =O. In absence of the highly nonlinear local constraints, the optimization in step 10 may be efficiently carried out by linearization of the displacement constraints in terms of Z and use of Linear Programing. This is the key benefit from the BLISS/S algorithm.

Finally, one should note that when there are no variables Z, the procedure becomes a special case of optimization by piece-wise linear approximations as SOPT in Step #10 is bypassed. However, even in this case the optimization remains decomposed because each substructure is optimized separately in Step #8.

As emphasized in the foregoing, the substructure optimizations are independent of each other and unrestricted in regard to the choice of the method. The use of a Response Surface (RS) Method to represent the BBA operation is an example.

In this application a polynomial response surface-based optimization employs an “adaptable” response surface model (RSM) in place of the substructure analysis. That model is implemented in iSIGHT software (Golovidov et al., 1998). In this approach, a minimum number of designs are used to construct an initial RS around the baseline design. Typically, a linear RS is constructed initially, although the user has an option to request a quadratic initial RS. For a linear RS, this number would be (N,,,+l), where Ninp is the number of inputs. After the best design is found using this RS within the specified design space bounds, the design is analyzed using the “exact analysis”, the data are included into the RS data set, and the RS is regenerated.

The validation results include test cases compared to the benchmark all-in-one (A-i-O) optimization for accuracy of the final results and the convergence characteristics. The validation test case is a hub

framework depicted in Fig. 3 that appears also in Balling and Sbbieszczanski-Sobieski, l994. Utility of the hub structure as an optimization test case stems from its ability to include as many members as desired without increasing the dimensionality of the load-deflection equations. These equations remain 3x3 for a 2D hub structure regardless of the number of members. While analytically simple, the hub structure design space is complex because the stress, displacement, and buckling constraints are rich in nonlinearities and couplings among the design variables.

In the hub structure herein, each beam has an I- shaped cross-section. The X variables are the dimensions of bl, h, b2, tl, t2, and t3 shown in Fig. 3. The top and bottom flanges of the I-beam are not of the same dimensions, hence the cross-section of each I-beam requires 6 design variables. The Z variables are the horizontal and vertical coordinates of the hub point where the beam members are rigidly connected. The change in the coordinates of the hub results in the change of the angles between the beams.

An example of g is the local buckling of the top flange in beam #2, and an example of G is the horizontal displacement of the hub. Appendix contains all the detail of the constraint formulation (these details may also be found in Balling, and Sobieszczanski-Sobieski, 1994).

The BLISS/S procedure was tested on the above structure in a two-beam and 20-beam versions, both considered under two loading conditions specified in the appendix. The two versions had, respectively,12 local design variables, 76 local constraints, and 120 local design variables,760 local constraints. The local constraints included the stress and local buckling constraints for both loading conditions. In both versions, the system-level constraints were imposed on the resultant translations and one rotation at the hub point for each of the two loading conditions.

The tests were organized in three cases beginning with the one in which Z and G are absent but X and g present, and ending with Z, G, X, and g all present. The first two cases were carried out for a two-member hub structure, while the third case included also a 20-member version of the structure. The remainder of this section shows the results in tables and figures.

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Each table is labeled withthe case description and shows the objective function (structural weight), and the maximum constraint values for the initial and optimal states for the benchmark A-i-O method, BLISS/S, and BLISS/SIRS.

BLISS not tested in this report is its amenability lo concurrent execution of the substructure optimizations and associated analyses.

The benchmark A-i-O method is a piece-wise approximate optimization in which there is no decomposition. The structure FEA includes computation of gradients by finite differences (one- step-forward) that are then used to form a linear extrapolation as- an approximate analysis for optimization within move limits. The optimizations within move limits were performed by the usable- feasible directions method. The same method was used at the substructure and system levels in BLISS/S. The gradients at both levels were also computed by finite differences for consistency of comparison with the A-i-O method, except the derivatives of the objective with respect to the optimization parameters that were computed by the algorithm described in the appendix. The BLISS/S procedure was implemented in the a software framework called iSIGHT (1998).

The case in Table 5 comprises two subcases labeled 10 % and 100 %. The percentage labels refer to the side constraints imposed on the horizontal and vertical location of the hub. For example, in the 10 % subcase the horizontal coordinate of the hub location, a Z variable, was restricted to be less or equal to 0.1 length of the horizontal member. Thus, the IO0 % subcase is special so that it allows an extreme reconfiguration of the structure such that the hub moves all the way to the root of the horizontal member. The expected result was that given that freedom, the procedure should eliminate the one of the members so that the load would be applied directly to the wall. The remaining member should then shrink to minimum gages resulting in a very light, degenerate structure. The results in Table 5 confirmed the above expectation, and Fig. 4 illustrates the corresponding reconfiguration of the structure.

As a measure of the numerical labor, Tables 2, 4, 5, and 7, show how many calls were issued to the assembled structure FEA and the total of such calls to the substructure analyses. The latter is not shown for the benchmark method. In that method the substructure analyses are a part of SA because there is no decomposition so that each substructure is analyzed once in each execution of SA.

The tables show that the difference between the BLISS/S objective minimum and the benchmark remains well under 1 %, except for BLISS/S/RS where it reaches 2.7 % for the two-member case. Regarding the comparison of the individual design variables, the volume of data is too large to show in full, therefore, only a typical sample is given in Table 3.

Histogram in Fig. 5 shows the objective function convergence for the benchtnark method and BLISS/S. It indicates that BLISS/S converges the objective to the benchmark value within 5 cycles, effectively recovering from the error in extrapolation based on the optimum sensitivity derivatives. A similar phenomenon was reported in Sobieszczanski- Sobieski, 1993. The histogram in Fig. 6 shows the most violated constraint convergence. In this regard, BLISS/S holds the most violated constraint satisfied from start to finish. In contrast, the benchmark method allows one of the initially feasible constraints to become violated and displays an irregular, oscillatory convergence of that constraint. This confirms the expectation that decomposition in BLISS/S makes satisfaction of highly nonlinear local constraints easier.

Examination of the data showed that, as expected, the discrepancies for the individual design variables are greater that those for the objective function. In terms of the numerical labor, BLISS/S shows significant reduction of the number of calls to the assembled structure analysis. To be fair one should note that the BLISS/S advantage in this regard is amplified by the use of finite difference gradients in both BLISS/S and in the benchmark method. That advantage would be less if analytical gradient calculation was used in both methods. One should emphasize that one potential advantage of

5. Summarv and Concludine Remarks

A two-level optimization method known from previous publication as BLISS for Bi-Level System Synthesis was adapted to structural optimization purposes and labeled BLISS/S for BLISS/Structures. The original method decomposes a modular system optimization into subtask optimization, that may be executed concurrently, and the system optimization that coordinates the former. Transformation of BLISS into BLISS/S was accomplished by treating lhe substructures (ultimately, the individual members) as

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modules in a generic system and by specifying the Finite Element Analysis as the equivalent of the system analysis. The resulting procedure separates the multitude of the cross-sectional variables from the overall structure geometry (shape) variables. Also, the highly non-linear local constraints, e.g., the local buckling, remain in the individual substructure optimizations and do not directly enter the assembled structure optimization performed under the system constraints, e.g., the displacement constraints.

The validation tests performed using a hub framework structure with up to 20 members (120 variables, 760 constraints) showed satisfactory agreement with the benchmark results obtained by optimization without decomposition, in terms of the minimum of the objective. They showed that the BLISS/S ability to satisfy the local constraints as better than that of the benchmark method. This advantage is expected to be amplified with the increase of the number of the substructures and the degree of non-linearity of the constraints. In terms of the numerical labor, the results showed that BLISS/S reduces that labor substantially with regards to the number of full FEA but requires additional substructure analyses for the substructure optimizations. Therefore, the degree of the pay-off from the use of BLISS/S instead of the A-i-O method will increase with the number of the design variables if the problem is large enough so that the cost of the FEA is dominant.

However, the BLISS/S additional advantage is its amenability to execute substructure optimizations concurrently and autonomously so that different optimization techniques may be used if these substructures are heterogeneous. Implementation of BLISS/S in a heterogenous computing environment to exploit the latter advantage is the future development of a potentially high pay-off.

6.Aknowledvemenl

The second author’s work was supported by the NASA Contract L68259D

7. References

Balling, R.J.; and Sobieszczanski-Sobieski, J.: Optimization of Coupled Systems: A Critical Overview of Approaches. AIAA J., Vol. 34, No. I, pp.6-17, Jan. 1996.

Balling, R.J.; and Sobieszczanski-Sobieski, J.: An algorithm for solving the system-level problem in multilevel optimization, NASA CR: 195015, December 1994.

Barthelemy, J.-F; and Sobieszczanski-Sobieski, J.: Optimum Sensitivity Derivatives of Objective Functions in Nonlinear Programming. AIAA Journal, Vol. 21, No. 6, pp. 913-915, June 1983.

Golovidov, O., Kodiyalam, S., Marineau, P. et al., “Flexible Implementation of Approximation Concepts in an MD0 Framework,” Proceedings, 7’ AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, AIAA, St. Louis, Missouri, September 1998. AIAA Paper No. 98-4959.

Haftka, R. T.; and Gurdal, Z.: Elements of Structural Optimization, Kluwer Academic Publ., 1992.

iSIGHT Designers and Developers Manual, version 4.0, Engineous Software Inc., Morrisville, North Carolina, 1998.

Sellar, R. S.; Batill, S. M.; Renaud, J. E.: Response Surface Based, Concurrent Subspace Optimization for Multidisciplinary System Design. 34* AIAA Aerospace Sciences Meeting, Reno, Nevada, AIA4 96-0714, January 1996.

Sobieszczanski-Sobieski, J.; Agte, J.; and Sandusky, Jr., R.:Bi-level Integrated System Synthesis (BLISS), AIAA 98-4916, 7’ AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, September 2-4, 1998, St. Louis, MO.

Sobieszczanski-Sobieski, J.: Optimization by Decomposition in Structural and Multidisciplinary Applications; in Optimization of Large Structural Systems, Vol.1; (G.I.N. Rozvany, ed.) NATO AS1 Series E Applied Sciences - Vol. 231. Kluwer Academic Publishers 1993.

Stelmack, M.; and Batill, S.: Neural Network Approximation of Mixed Continuous/Discrete Systems in Multidisciplinary Design., Univ. of Notre Dame, Notre Dame, IN. AIAA-98-0916

Vanderplaats, G-N., and Cai, H.D.: Alternative Methods for Calculating Sensitivity of Optimized Designs to Problem Parameters. NASA CP-2457, Proceedings of the Conference on Sensitivity

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Analysis in Engineering; NASA Langley Research member buckling. The following defines the Center, Hampton, VA, September 1986. constraints.

Appendix: Aborithm and I’&merical Example Details

Displacement Constraints at the hub: d/d, - 1 < 0, and q/q, - 1 < O-where d = resultant translational displacement, and q = rotational displacement

1. Algorithm for Optimum Sensitivity Analvsis.

The derivatives of the minimum of the objective function F = F(X, P), where P comprises the parameters that are kept constant in the optimization are denoted D(F,P) and may be calculated directly, without computing the derivatives of the optimal values of X, by means of an algorithm based on the well-known notion that the Lagrange multiplier of an active constraint C is the price for that constraint change measured in units of the objective. The algorithm whose derivation may be found in Barthelemy and Sobieszczanski-Sobieski, 1983, yields D(F,P) for a particular parameter P, as

~ Stress Constraints: Normal and shear stresses (s and t) are evaluated on the cross section at the top and bottom extreme fibers, at the centroid, and at the top and bottom of the web. This is done at both ends of the member except for the stresses at the centroid which are constant along the length of the member. The following stress constraints are imposed at each location: s&, - I < 0 % = von Mises-Huber equivalent stress = (s2 + 3t2)“2 s, = allowable stress = 25 kN/cm’ In-Plane Buckling Constraint for each beam: ND$, - 1 < 0

D(F,Pi) = d(F,P) + LT d(C,Pi)

N = axial force (compression positive) N, = 2.057c2 E&,/L2 E = modulus of elasticity = 20,000 kN/cm2

where L stands for the vector of the Lagrange multipliers associated with the vector of the active constraints C.

I,, = strong axis moment of inertia L = member length

In the application herein, for a particular substructure P contains Z that influence the substructure geometry and the substructure Q. However, Q depends on Z hence the chain derivative terms in the extrapolation of Wi in step#lO of BLISS/S.

Out-of-Plane I Lateral-Torsional Buckling Constraint (at each end): (N/N,,) + (M/M,,)‘.‘5 - 1 < 0 N = axial force (compression positive) M = magnitude of bending moment N, = 2.057c2-EL,&’ M, = ~(EI,,GI,,)“2/L

2. Numerical Example Details

The structure is a framework configured as shown in Fig. 3. AII the members lie in a common plane. Each member is rigidIy attached to a common hub and clamped at the other end. The hub has the freedom to rotate and translate in the plane formed by the members. The solution method is the standard, linear, displacement-based finite element method, using the beam element in a slender beam formulation that neglect the transverse shear stress deformations and preserves the Kirchoff’s assumption of the beam cross-sections remaining planar under the load. The structure is two-dimensional for analysis purposes, with the exception of out-of-plane

G = shear modulus of elasticity = E/2(l+v) v = Poissons ratio = 0.3 q= weak axis moment of inertia I, = torsional moment of inertia = b,ti3+b2t2j+(h- t,- t,N,‘v where h is the cross-section height

Local Flange and Web Buckling Constraints (at each end):

S/Scr + (t/t,,)* - 1 < 0; where s = normal stress (compression positive), and t = shear stress.

The critical stress values are computed by the following formulas

tL3 tL3 Top Range: Top Range: 0.55E(2t,/bJ2 0.55E(2t,/bJ2 0.4 1 E&-bJ2 0.4 1 E&-bJ2 Bottom Range: Bottom Range: 0.55E(2t,/b2)* 0.55E(2t2/b2)* 0.41E(2t,/bJ2 0.41E(2t,/bJ2 Web: Web: 4.80E(t,/(h-t,-t,))* 3.60E(t,/(h-t,-t2))2 4.80E(t,/(h-t,-t,))* 3.60E(t,/(h-t,-t2))2

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The side constraints on the cross-sectional dimensions:

lower upper bl 2.0 cm 6.0 cm

tl 0.1 cm I.0 cm

b2 2.0 cm 6.0 cm

t2 0.1 cm 1.0 cm

lower upper tj: 0.1 cm 1.0 cm h: 3.0 cm 8.0 cm A: 0.68 cm* 10.00 cm2 I: 1 .OO cm4 100.0 cm4

The two loading conditions are as follows:

1. 100 kN at the hub at 45 degrees to horizontal acting toward upper left in Fig.3.

2. 135 kNcm at the hub acting counterclockwise.

Table 1: Correspondence of operations between BLISS/S and BLISS

Elements of BLISS/S

Finite Element Analysis including direct or adjoint sensitivity Analysis

Substructure local optimization using local X to minimize the substructure contribution to the entire structure weight and to satisfy local constraints for constant Z and the internal forces Q

Elements of BLISS

SA, SSA in BLISS #2, #4, #5

BLISS #6

Optimum sensitivity analysis of a substructure to obtain derivatives of the optimum data with respect to Z and the internal forces Q

BLISS #7

Optimization of the assembled structure for the objective of minimum weight and satisfaction of constraints. If there are some constraints that cannot be satisfied at the structural element level, e.g. displacement constraints, this optimization is performed using Z at the system level.

BLISS #8

Table 2: Two Member Hub Frame Solutions for Cosc~ 1 Case Initial Initial Max Constraint

Design Violation Objective

A-I-O 1988.0 -0.162662

Final Final Max Constraint Number Number of Design Violation Of System Substructure Objective FEA Analyses

1045.5 o.OQO93 165 10 cycles

BLISS/S 1988.0 -0.162662 1045.5 0.00132 131 841 10 cycles

BLISS/S (with RSA in BBOFTj)

1988.0 -0.162662 1073.45 -0.0084

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i9 6 cycles

486

Table 3; Comparison of design variable values for Case 1

I A-I-0 Final 1 BLISS/S

Value (cm) Value (cm) Final (cm)

( ;i;a;S;;, RSA) 1

Ml-b1 5.0 4.47 5.04 4.66

Ml-b2 5.0 4.43 4.44 4.39

Ml-b3 0.4 0.26 0.21 0.12

Ml -tl I 0.4 1 0.28 IO.27 IO.34 I

Note: Ml and M2 are the member numbers

Table 4: Two Member Hub Frame Solutions for Case 2 Case Initial Initial Max Final Final Max Number Number of

Design Constraint Design Constraint Of System Substructure Objective Violation Objective Violation FEA Analyses

A-I-O 1988.0 -0.162662 1538.16 O.OQO82 106 6 cycles

BLISS/S 1988.0 -0.162662 1537.38 -cm365 73 503 6 cycles

BLISS/S 1988.0 -0.162662 1592.83 0.0022 79 268 (with RSA in 6 cycles BBOPTj)

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Table 5: Two Member Hub Frame Solutions for Case 3 Case Initial Initial Max Final

Design Constraint Design Objective Violation Objective

10% 1988.0 -0.162662 1447.97 A-I-O

Final Max Constraint Violation

0.0039

Number Of System FEX

242 I4 cycles

10% 1988.0 -0.162662 1470.89 0.0037 211 BLISS/S 14 cycles

100% 1988.0 -0.162662 8.55796 -0.140 A-I-O

497

100% 1988.0 -0.162662 8.33236 -0.0031 BLISS/S

Number of Substructure Analyses

1295

f

2219

Table 6: : Two member hubframe -Location variable comparison for Case 3 (100%)

Hub Initial A-I-O Final BLISS/S Location Location (cm) Location Final X 150.0 0.93 1.05 Y 70.0 199 19~8

Table 7: Twenfy Member Hub Solutions for Case 3 Case Initial - Initial Max Final

Design Constraint Design Objective Violation Objective

10% 20939.9 I. 13E-03 10224.35 A-I-O 10% 20939.9 l.l3E-03 10180.87 BLISS/S

Final Max Number Constraint Of System Violation FEA

O.OCOO4 5020 (39 cycles)

-0.00054 1354 (11 cycles)

Number of Substructure Analyses

17104

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2 - wing sweep angle. aspect ratio

X, -sheet metal thickness

- Y, 1 - strct. weight Z

4 2 - Per1

An,

Figure 1. Engineering system of three modules typical of aircraft.

(SA) & Sensitivity Analysis by FEA

XI x2

Evaluation of Evaluation of

Figure 2. Modular system adapted to represent a two-beam structure.

Two members

Hub structure Cross-section b-,

Twenty members (not all shown)

Figure 3. Hub structure: 2-beams and 20-beams versions, and a beam cross-section.

Hub Lot.)

Figure 4. Hub structure, two-beams configuration: optimal hub location for 10 % and 20 % side

constraint.

l A-1-0 a BLISS/S

Iteration Number

Figure 5. Histogram of the objective function.

1.5 -

+ A-I-01 . A-I-02 A BLISS/S

I I I I I IO 15 20 25 30

Iteration Number

Figure 6. Histogram of maximum constraint.

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