Computing S.vswms in Engineering, Vol. 5, No. 4 6, pp. 455 467, 1994
Pergamon
0956-0521(94)00019-0
Copyright <(, 1995 ElsevierScienceLtd Printed in Great Britain. All rights reserved 0956-0521/94 $7.1)0+ 0.00
MASSIVELY P A R A L L E L S T R U C T U R A L D E S I G N U S I N G STOCHASTIC O P T I M I Z A T I O N A N D M I X E D NEURALNET/FINITE ELEMENT ANALYSIS METHODS RONG C. SHIEH MRJ, Inc., 10455 White Granite Drive, Oakton, VA 22124, U.S.A. Abstract The title study is performed on the massively parallel processing (MPP) environment of Connection Machine (CM) computers using truss structural sizing design problems as example design problems. In this design optimization procedure, only the displacement solution is replaced by that based on neural net technology (under a given set of cross-sectional size parameters (e.g., areas) in the MPP finite element structural reanalysis). This structural reanalysis procedure, together with a vastly improved and parallelized version of the integral global optimization (IGO) stochastic algorithm, IIGO, forms the present MPP structural design methodology. In addition, a procedure to correct the final optimal design for constraint violation or too-conservatively satisfied constraint condition caused by inaccuracy of the NN (neural network) analysis model is also formulated. Evaluation of the numerical performance of the developed computational algorithm set, capability, and strategy is made, primarily on the Connection Machine CM-2 model computer by performing three neural-network-based truss structural reanalysis/minimum weight design problems.
1. INTRODUCTION
The present study is the second part of a multiplepart study program aimed at developing efficient NN-assisted or NN-based, as well as FE (finite element) based structural reanalysis and design technology, including optimization algorithms, in an MPP environment. In the preceding study, ~2 a mixed N N / L S (least-square) computational model-based structural reanalysis procedure/module, which completely replaces the finite element (FE) structural reanalysis procedure/module, is formulated, assessed, and shown to be exceptionally efficient, even in thc MPP environment of an older generation Connection Machine computer, the CM-2 model. In the case of a truss structural weight minimization design problem, the NN for the cross-sectional area (,4)/nodal displacement (D) I/O pair is trained using neural computational techniques, while the area (A)/ structural weight (W) and displacement (D)/stress (S) relationships are obtained using linear leastsquare (LS) fitting techniques. Similarly, in an earlier study, 13 a vastly improved and massively parallelized version of the integral global optimization (IGO) algorithm, I I G O (or I G O / K O O S H ) , is formulated and applied to the FE analysis methodbased structural design optimization of truss structures. In the present study, an alternate structural reanalysis strategy based on a mixed N N / F E computational model is formulated and used in structural design. In this strategy, only the most time-consuming portion, i.e., the linear equilibrium equation solution portion, e.g., via preconditioned conjugate gradient (PCG)
Applications of neural net (NN) technology in the structural analysis and design area have received considerable attention in the past few years. 1 12 These studies, most of which have dealt with problems of relatively small structure and network sizes, have demonstrated the usefulness and viability of the approach in structural analysis and design. However, as partially demonstrated by the author in the structural reanalysis area, l: such technology can be made immensely more powerful for large or larger size problems if the required calculations are performed in a massively parallel processing (MPP) environment. In an MPP environment, the large number of structural response analyses (each with a different structural parameter or design variable set) required for training a set of relatively large-size N N s can be generated simultaneously in parallel. Furthermore, an N N itself is an inherently massively parallel+ interconnected network of computing elements (cf. Fig. 1) which are particularly suitable and, thus+ efficient to be computable in an M P P environment. Finally, a set of trained N N s can be used directly in approximate structural analysis and/or design (optimization) in an MPP environment by developing efficient M P P algorithms to again exploit the massively parallel computational power of an MPP computer. Thus, these NN-based structural analysis and design optimizations have the potential to perform in an MPP environment with utmost efficiency. 455
456
RONG C. SHIEH
1st HiddenLayer
/nput Layer
2nd Hidden Layer
(m=2)
O =X
(I) i
12
W'
tj
(2:
y.
(21121+(
Output Layer
(m--3) 2
23
(3)
W~
2
Y-
J
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(m--M) (3)
X-
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34
W.'
K,n
Xt
V -Y n --
(4) .
(1) YI
.131 . .
Fig. 1. Example four-layer neural network with two hidden layers. method-based displacement response solution (under a randomly chosen structural element cross-sectional size), is replaced by an NN model, while the remaining portions (mostly data recovery) of the structural reanalysis, such as stress and objective function calculations, are performed by the corresponding portions of the finite element structural analysis module. Formation of a general structural design optimization problem and an MPP computer-based methodology are given in Section 2. Numerical examples are presented in Section 3. It should be noted that a trained NN presents some distinct advantages over the direct FE solution approach. For a given set of input variables, it provides a nearly instant output solution, particularly in the presence of nonlinearities in the analysis/reanalysis. Such rapid solution capability not only greatly accelerates the redesign computations, but is also essential in many important applications, such as in the case of a retargeting exercise where targeting (or pointing) errors are rapidly calculated by a trained NN.
2. P R O B L E M F O R M U L A T I O N A N D C O M P U T A T I O N A L CAPABILITY
2.1. Statement of the structural design problem A class of problems chosen for the study is the structural member sizing problems in minimum weight design. Typically, such a problem involves a structure (composed of ArE number of members or elements) subjected to several (Np) sets of loadings, {P}j ( j = 1, 2 . . . . . Np), with the structural weight, W, as the objective function, F({X}), and the member cross-sectional size variables, such as areas {A}, moments of inertia {I}, etc., as the design variable vector {X}, subjected to certain behavioral and design constraints. In mathematical terms, this can be written as W = F({X}) ~ minimum
(la)
with respect to (NA × 1) design variables, {X}. The constraints for the weight minimization problems typically are:
XLI ~ Xi ~ Xui(i = 1, 2 . . . . . NA)
IS~l <
Smj(j = 1, 2 . . . . . N+)
IDk[ ~
(lb) (lc) (ld)
where Dk are the components of the nodal displacement vector {D}, Sj the components of the stress vector {S}, and subscripts m, L, and U stand for the prescribed "maximum" allowable and +qower" and "upper" bounds, respectively. For a truss design problem, the design variables are normally the member cross-sectional areas, i.e., {X} = {A }. 2.2. A mixed NN/FE structural reanalysis strategyJbr structural design There are a number of ways in which the computational efficiency of structural reanalysis and/or design can be significantly or drastically enhanced by neural net technology, particularly in an MPP environment. One such structural reanalysis procedure, i.e., a mixed NN/LS structural reanalysis procedure, was advanced elsewhere by the author. 12 Because training of a nonlinear NN is usually the most time-consuming part of an NN-based structural reanalysis method, it is desirable to limit the number of NNs needed to be trained to the essential ones and to use other less time-consuming means to calculate the other quantities of interest. In a truss design problem, the essential NN is the area-displacement ({A}-{D}). Other quantities, such as stresses ({S}) and structural weight (W) can be calculated from the linear {S}-{D} and W-{A } relationships by using a trained linear NN model, 9A2 least-square model, 12 or the least time-consuming (stress data recovery) portions of an FE computational module and direct computation of weight data.
457
Massively parallel structural design Table 1. Optimal design weights for the ten-bar truss using an NN for analysis NN model and description AI0-6-6-D2
Design wt. # t
Max. FE disp (in.)
Adjusted wt. #
100 training sets used in a range of +25% about optimum 400 training sets used in a range of +25% about optimum 100 training sets used in a range of 0.01 55 in.e; output scaled to reduce range of variation "Exact" (FE) analysis-based optimal design solution
4692.49 4666.71 5010.22
2.1867 2.1701 2.0423
5130.82 5064.80 5116.08
5063.8I
2.0000
5063.81
q Ref. 2 results. Thus, in the mixed N N / F E structural reanalysis procedure, the (approximate) {D}-vector is calculated via the {A }-{D} N N model under a preassigned set of [,4 } (usually randomly generated in design optimization). The corresponding stresses are then calculated by using {S}-{D} relationships in the data recovery subroutine of an F E structural analysis module, while structural weight is directly calculated from the element length/area/density data. In an M P P environment, the stresses and weight for each element can be calculated concurrently in parallel in a most efficient manner. This structural reanalysis strategy is used in the following development work.
These seemingly contradictory results are due to different degrees of displacement constraint violations caused by using N N analysis models. The corrected weight results (with accompanying correction on the design areas not shown herel now show that the second N N gives the best weight result, as it should be. The fact that the third N N gives a slightly better weight result than the first result appears to be accidental. The simplest procedure (which can be called a modified linear design scaling procedure) of correcting or improving an N N analysis-based final structural design consists of the following steps:
2,3. Constraint t:iolation correction model Jbr N N analysis-based final design
( 1) At the end of the design, FE method is used to calculate the relevant response quantities. This is automatically done in the C M - S T R A N D code (cf. Subsection 2.4 below) using the builtin FE analysis module. (2) For the design case involving displacement and stress behavior constraints only, the design scaling factor k is then calculated as
An N N model-based structural analysis can only provide approximate solution results. Therefore, the final design results based on an N N or a mixed N N / F E structural analysis model, although satisfying optimization constrains under the N N calculated quantities, will either violate the constraint requirements or yield conservative design results under accurately calculated (FE-based) quantities. Without taking into account or correcting these constraint violation or too-conservatively-satisfied constraint conditions in the final design, the design may not be on the safe side or may be overdesigned and, above all, it is impossible to critically evaluate the performance of a trained N N structural reanalysis model in structural design. For example, shown in Table 1 are the optimal design weight 2 and presently corrected weight results (according to the constraint violation correction procedure given below) of 10-bar truss using an A10-6-6-D2 (10 input area, two 6 hidden layer nodes, and 2 output displacements) N N structural analysis model. In this design problem, the truss weight is minimized with respect to truss member cross-sectional areas subjected to displacement constraint limit of 2in. and minimum area of 0.1 ~ without stress constraint. Without correcting the constraint violations, it is seen that the use of a trained N N with a large number (400) of training example patterns yields a worse design weight result than that with the smaller (100) training set case, while the optimum weight based on an N N with training sets randomly generated in a much wider area range (0.1--55 in.-' ) is better than that using much narrower range ( + 2 5 % about optimum) of areas from the F E analysis-based optimum areas.
k = max[(max DFE)/D m, (max SvE )/Sin]
(2)
where max D and m a x S are the maximum absolute displacement and stress, respectively, and subscript FE denote the quantity being calculated by a FE model. Accept the design as the final design if k is within an acceptable constraint error limit. Otherwise, go to Steps 3 o r 4.
(3) If k is greater than unity (which means some constraints are violated), all design variables (areas) and structural weight are multiplied by k to obtain the final adjusted design value(s). (4) If k is less than unity, which means that all constraints are satisfied too conservatively, one either accepts the N N analysis-based design or taking the following substeps: • Adjust the final optimal design results as in the preceding step (Step 3). • Set those design variables (areas) that are smaller than their lower bound value(s) to their lower bound value(s). • Perform a new FE anal,ysis based on the new (resulting) adjusted areas (design variables). • Find a new design scaling factor,/,-', as in Step 2. l f k is less or equal unity, accept the
458
RONG C. SHIEH above adjusted design results as the final design results without additional correction. Otherwise, repeat Step 3.
In this procedure, only a finite element analysis computer code module is required. The other steps of computation can be performed by a desk calculator unless the design variables are excessive. The adjusted design weights given in Table 1 are calculated by using the above computational procedure (involving only Steps 1-3 because k values for all NN analysisbased design cases are greater than unity). 2.4. MPP computational capabilio' on the CM system (a) MPP CM-computer systems. There are several models of Connection Machines currently in use, such as the CM-2, CM-200, and CM-5. The CM-2 and its upgraded CM-200 model computers are data parallel, single-instruction-multiple-data (SIMD) architecture machines, while the CM-5 is a mixed S1MD/MIMD (multiple-instruction-multiple-data) architecture type computer, all of which are manufactured by Thinking Machines Corporation (TMC) in Cambridge, Massachusetts. The basic modular unit for the CM-2 has 8K (=8192; 1 K = 1024) 1-bit processors. These basic units can be configured to yield 8K, 16K, 32K, and 64K machines. Each processor can have dedicated RAM up to 12K bytes. These machines usually are equipped with the 32- and 64-bit Weitek co-processors to support single and double floating point calculations. Theoretical peak performance speeds for the CM-2, CM-200, and CM-5 are 20, 40, and 128 Giga-FLOPS, respectively. (b) NN training example generator Jbr structural reanalysis. An NN must be trained on an appropriately large number of known solution examples for the network l/O pair variables. For the class of truss design problems considered here, the I/O variables chosen are {A} and {D}, and each of the training example patterns represents solution of {D } under a randomly chosen area vector {A } within a preas~ In the present study, a multiple signed range of ~A j. concurrent FE structural analysis capability/module, CM-STRAN, of frame-type structures (also truss structures by inserting hinges at both ends of all beam elements), developed primarily on the Connection Machine (CM) CM-2 model computer, j2'j3 is adapted as the training example generator for the {A}-{D} network. The MPP algorithms used in the code capability are as follows. For each structural analysis under a given set of structural element cross-sectional parameters (areas, moments of inertia, etc.), the data computational/storage task for each of NE-structural elements (which may be substructures in the CM-5 environment case) is assigned to a processor. The equations of equilibrium are assembled implicitly in force vector form and solved iteratively by using the preconditioned conjugate gradient (PCG) method. To perform NA (=Np/NE) number of multiple con-
current structural analyses in parallel, each under a different set of cross-sectional parameters (e.g., areas), the Np-numbers of available processors (up to 64K 1-bit processor for the CM-2 and 1024 64-bit processors for the CM-5) are mobilized, with each group of N A processors performing an independent analysis. (c) Computational model for NN training. The NN computational capability used for the present study is an author's modified version, BACKPROP.RCS, of the CM-Fortran code, BACKPROP, originally written by Zhang. 14 It is based on the backpropagation algorithm and training example processor assignment strategy. Therefore, except for the serial operation of reading training examples via a front-end computer, the performance timing results of the code for NN training is virtually independent of the number of training examples as long as the number of processors is not smaller than eight times the number of training examples. The activation functions implemented in the code are of the sigmoid type ~5
g(z) = 1/[1 + e x p ( - z ) ]
(2)
and the linear type (added by the author). The code can train an NN up to two hidden (intermediate) layers and can also generate NN output data using a trained set of NN weight matrices and a given set of input variables. A convenient feature has been added by the author to perform a sub-network using a subset of the output variable set and a single, complete NN training example data file. (d) Structural optimizer using mixed NN/FE reanalysis strategy and IIGO optimization algorithm. The mixed N N / F E structural reanalysis strategy described above has been implemented into the FE structural analysis/design optimization code for space frame-type structures (CM-STRANOP) as CMSTRAND. In this code module, the PCG methodbased nodal displacement solution portion is replaced by an {A }-{D} NN-based displacement solution subroutine. The original CM-STRANOP code contains two modules: the FE structural analysis module, CM-STRAN, mentioned above and the optimization module, CM-OPTIM. The CM-OPTIM module is based on a vastly improved and parallelized version of the unconstrained, stochastic type integral global optimization (IGO) algorithm, ~6 IIGO, 13'17 and the linear design scaling type constraint treatment algorithm. The currently operational version of the CMSTRAND is limited to the design variable case of cross-sectional areas, although it can be easily (is currently being) extended to include moments of inertia (for frame design optimization problems). 3. N U M E R I C A L E X A M P L E S
3.1. CMNS CM-2 computer system The computer system used in the present numerical study is the CMNS CM-2 system at Cambridge,
459
Massively parallel structural design
3.2. Multiple concurrent structural analyses of 3-D
rectangular building frames
j
A class of 3-D building frame structures (Fig. 2) subjected to a set of static wind loads is used here to benchmark the numerical performance of the CMS T R A N code module in multiple concurrent structural response analyses on the C M N S CM-2 computer system for use as {A}-{D} N N training example patterns. The performance timing results are given in Table 2. In all these runs, the Sun-4 front-end machine utilization rates achieved were approximately 9 0 - 9 5 % . F r o m Table 2, one can observe the following:
'--~."--.
-g
><~,-
,--....
Fig. 2. 3-D building frame structure.
Massachusetts, supported by Thinking Machines Corporation (TMC) under a former D A R P A contract. The C M N S CM-2 system is a 32K processor machine controlled by four sequencers and equipped with 64-bit Weitek co-processors. Each C M processor has 64K bits (8K bytes) memory. One can run a job using one, two, or four sequencers (i.e., 8K, 16K, or 32K processors). The system has two frontend computers, Sun-4 and Sun-4C model workstations ( C M N S - S u n and C M N S - M o o n ) and is equipped with a data vault for mass data storage and to speed up writing and/or reading data to or from such a device. The front-end computers are shared by other users (i.e., non-dedicated ones) and the wallclock (WC) (i.e., CM-elapsed) time on the CM-2 (which is controlled by a front-end computer) is heavily influenced by the front-end machine utilization rate. Therefore, all CM-2 WC time results given throughout this paper are certainly larger than what one would obtain by using a fully dedicated CM-2 system (including the front-end computer).
o T h e speedup for the single and 128 multiple analysis cases of the BF1K frame on the 8K processor machine is seen to be 14.1. This number far exceeds 8, the number of multiple concurrent analyses that can be performed simultaneously in parallel on the 8K machine, because 15 repetitions of analyses require no additional reading of input data, and all required loading of data from the front-end to the CM processors is not needed in the repeated, multiple concurrent analysis case. This can be seen clearly for the two BF16K analysis cases on the 16K processor machine in which the speedup factor for the repeated single analysis case is 1.26 versus not more than l for the single analysis case. • Doubling and quadrupling of the CM processors from 8K to 16K and 32K in the BF1K multiple concurrent analysis cases resulted in speedup factors of 1.85 and 3.40, respectively. Similarly, doubling the CM processors from 16K to 32K for the B F I 6 K analysis case resulted in a speedup factor of 1.66. These speedup factors are less than the corresponding processor ratios because data processing in the front-end computer is serial. 3.3. Bar truss (T-72) desL~n problem under a single
loadin~ set The structural cross-sectional areas are to be sized for minimum weight under one (the first) or two loading set(s), stress and displacement constraints,
Table 2. CM2-STRAN code module single-run times for multiple concurrent structural analyses of rectangular building frames (Fig. 2) under different cross-sectional areas (PCG error nomr E r - l.e-6:Sun-4 front-end; 6 displacement component output) Str. id-# and No. elements
N, × N, x N_ = No. nodes
No. nonzero displacements
No. phys. processor
Nos. analys/ intern, loops
cm-elapsed timer /busy time (s)
Speedup factor
BFIK 095
9x8x6-432
360 x 6=2160
8K 8K 8K 16K 32K
1/1 8/1 128/16 128/8 128/4
87/68 66/48 792/735 427/384 233/199
1 10.5 14.1 26.1 47.8
9x8xS1-5832
5760 x 6 = 34560
16K 16K 32K
1/1 2/2 2/1
1342/691 2123/1462 1279/724
BF16K 15920
l 1.26 2.10
t Double precision run times; output data = structural weights and six displacement components of a nodal point at the top for all analysis sets. The Sun-4 front-end machine utilization rates were 85 92%.
460
RONG C. SHIEH Z
1- Typical Bar Elements in Each Bay
n@6~
Load set 1 2 2 2 2
Node
X
Y
Z
4n+l 5000 5000 -5000 4n+ 1 0 0 -5000 4n+2 0 0 -5000 4n+3 0 0 -5000 4n+4 0 0 -5000
Design/material data
Value
Displacement limits Young modulus (AI) Specific mass Allowable stresses Minimum c.-s. area
-f- 0.25 in 104 ksi 0.1 Ibm/in3 + 25 ksi 0.1 in2
Fig. 3. 72-Bar truss design problem (n =4). Note: For clarity, not all elements are shown.
and symmetrical cross-sectional area conditions as shown in Fig. 3. Because of the symmetric structure condition, there are only 16 design variables (independent areas). This 3-D minimum weight design problem was previously studied by other authors TM using the FE method for structural reanalysis. The performance timing results for Single and multiple concurrent structural analyses are tabulated in Table 3a. Because of out.putting large amounts of data (areas, displacements, weights) to a front-end disk (done ,serially through the Sun-4C front-end computer), the speed,up factors as .a resalt of doubling or quadrupling processors from the 8K processors are
not seen to be as good as those for the frame analysis case (Table 2) in which only 6 displacement components per analysis are written to a front-end disk. Nevertheless, a speedup factor of 158 is seen when using 32K processors to generate 910 analysis sets, compared with using 8K processors to generate a single analysis set. To accelerate the output, one can use the data vault. After generation of a large number of displacement solution sets (each with 48 nonzero nodal displacement components) under the corresponding sets of randomly generated areas (each set with 16-independent areas) subjected to structure (area) symmetry condition, appropriate number of training example sets are prepared via the PRENET code model. The chosen NN architecture for this analysis problem is A16-27-27-D48 (16 input areas, two 27 node hidden layers, and 48 output displacements). This choice was partly based on the rule of thumb for determining numbers of hidden layer nodes (i.e., an appropriate total hidden layer node number is somewhere between the sum and average of input and output node numbers) l and partly based on the maximum hidden layer nodes acceptable to the CMNS CM-2 computer system. (The RAM for this CM-2 system is only 8K bytes per processor and can handle an A16 input and DI6 output NN with up to two 27 node hidden layers without causing memory overflow.) With this architecture and including bias nodes, the total unknown number of NN weight matrix elements is nx = (n~+ 1) x nnl + (nm + 1) x nn2 +(nn2+l)xno
(3)
(=2559), where nj ( j = I , O , H1, or H2) denotes number o f j t h layer with subscripts I, O, HI, and H2 standing for input, output, and first and second hidden layers, respectively. Theoretically, to have a one-to-one fitted result for the nx-number of unknowns, the same (nx-number) of training example patterns are needed. However, this condition is not a necessary condition for successfully training an NN. An NN is often under-trained (fitted) with respect to number (nT) of training example sets (i.e. nT < Nx). For example, only 100 patterns versus N x = 122 is used elsewhere 1"2to train an AI0-6-6-D2 NN, which, in turn, is used in optimal design of a 10-bar truss with good design results (cf. Table 1). Therefore, n v = l l 2 0 (approximately 10 machineful multiple analyses of the 8K processor CM-2) is first selected herein to represent an under-fitted NN, and it is later increased to 2600 example patterns to represent a slightly over-fitted case. The minimum weight design results using the NN/FE-IIGO methods are given in Table 3b as functions of network training errors (rms and maximum errors) using the NN configuration of A16-2727-D48.T1120 (with 1120 training (T) patterns). As
461
Massively parallel structural design Table 3a. CM2-STRAN code module run times for multiple concurrent structural analysis of 72-hinged beam element structure (truss) {PCG error norm E r = 1.e-4; Sun-4C front-end) NXx N, x N. = No. of nodes 2x2x
5=20
No. nonzero displacements
No. phys. processors
16x3-48
8K 8K 8K 16K 32K
Nos. analyses/ loops 1/I 113/1 904/8 908/4 910/2
cm-elapse time+ /busy time (s)
Speedup factor
12/9 20/14 131/98 93/56 69/35
I 68 83 117 158
+ Double precision run times; output data for each analysis set = 16 areas, the total structural weight, and 48 nonzero components of nodal displacements. The Sun-4C front-end computer utilization rates - 75 85%.
can be seen, the approximate minimum structural weight inches up toward that based on the purely finite element structural reanalysis/IIGO optimization method of the S T R A N O P code module ( ~ 370 lbs). Also shown in this table are the corresponding accurate (FE-predicted) maximum displacements (DFE) at "~optimum" at which the NN-predicted maximum displacements are all 0.25 in. { = the displacement constraint limit). Therefore, to accurately correct the inaccuracy of the NN-predicted maximum displacement results, the N N / F E - b a s e d design areas and weight for each design case must be linearly scaled up (adjusted) by the corresponding design scaling factor (k = DFE/0.25) given in Table 3b (cf. Subsection 2.3). The weight so adjusted are also listed in Table 3b as "adjusted". For the case with largest error pairs, 0.00494/0.1232, the design solution didn't converge at all, i.e., displacement and stress constraints couldn't be satisfied. This nonconvergent solution condition is believed to be caused by the corresponding trained N N model whose maximum error is 0.123. Therefore, an N N should be trained to such a degree that both rms (root-mean-square) and maximum errors are reasonably small, This fact seems to not have received
sufficient attention in the previous NN-assisted structural analysis and design studies, which are mostly concerned only with the rms error. The discrepancy between the F E and adjusted N N based structural weight results is seen to initially decrease as the N N training errors decrease up to certain error limits, but the discrepancy increases thereafter with further decrease in the rms error. It is unclear whether this phenomenon is caused by ac-
companying increase in the maximum error as the rms error further decreases. It should be noted, as demonstrated in Ref. 1, that an excessively (overly) accurately trained N N may actually predict design results (not contained in the N N training sets) less accurately. Shown in Table 3c are the new optimal design area and structural weight results based on the purely FE and mixed N N / F E structural reanalysis and I I G O optimization methods of C M - S T R A N D code. In the N N / F E - I I G O method-based design cases (Cases 2-4), the three trained N N s used (i.e., AI6-27-27D48.TN with n = 1120, 1140, and 2600 training (T) examples) are also given in the lower portion of Table 3c. So are the maximum FE-method predicted displacements at optimal design, for which NN-predicted maximum displacements are all equal to the displacement constraint limit value of 0.25 in. The total approximate optimal structural weight (349 lbs) for Case 2 (with 1120 training examples under randomly generated input areas) is seen to be 5.7% smaller than that of the finite element method-based ( ~ 370 lbs). To improve the accuracy of the N N / F E structural analysis method-based (approximate) optimal design results, it is good to use the approximate optimal areas and the corresponding F E displacements as a special additional training example pattern. Therefore, in the Case 3 design, 20 identical patterns of this special pattern are added to the 1120 training examples of Case 2 to form the 1140 example patterns case (Case 3) as a way of assigning a larger weight to this special example pattern. The corresponding minimum weight result of 379.64 lbs is seen to be only
Table 3b. Mixed NN/FE structural reanalysis method-based minimum weight design results as functions of NN training errors using the A16-27-27-D48.T1120 NN configuration NN No.
NN's rms/max errors
Design wt. (lb)
% wt. diff. vs. FE sol.
1 2 3 4 5 6
0.00494/0.1232 0.00307/0.0497 0.00289/0.0520 0.00275/0.0535 0.00259/0.0281 0.00240/0.0340
273.75 311.08 330.18 338.79 348.88 351.47
-- 26 - 16 -11.0 - 8.5 - 5.4 -5.0
Max. FE disp., DyE (in) 0.3504 0.3227 0.3100 0.3123 0.3165
Scaling factor DFE/0.25
Adjusted values wt. (lb)
Constraints badly violated 1.402 436.01 1.291 426.20 1.240 420.10 1.249 435.82 1.266 444.96
%wt. diff. 17.8 15.2 13.5 17.5 20.2
462
RONG C. SHIEH
Table 3c. Purely FE and mixed NN/FE structural reanalysis method-based weight minimization results of 72-bar truss under loading set 1 FE and 3 cases of mixed NN/FE reanalysis method-based optimal design areas (in.2)t plus the adjusted areas/weight for the last case Bay No.
1-4
I 2 3 4
Intra-bay truss element No. 5-12 13 16
17 18
1.876 1.813/2.198/1.943 1.233 1.293/1.180/1.298 0.5006 0.7046/0.8983/0.5027 0.1000 0.3860/0.2872/0.1000
0.5056 0.5310/0.5481/0.5234 0.4885 0.3255/0.3683/0.5052 0.5068 0.5333/0.6050/0.5155 0.5192 0.4622/0.5141/0.5529
0.1003 0.1034/0.1010/0.1018 0.1003 0.1043/0.1009/0.1021 0.1001 0.1017/0.1023/0.1009 0.4102 0.1013/0.1003/0.4688
0.1000 0.1037/0.1041/0.1018 0.1010 0.1040/0.1021/0.1020 0.1163 0.1005/0.1008/0.1022 0.5662 0.5691/0.6432/0.5753
Total weight (lb)
370.11 348.88/379.64/384.37
Adjusted weight:~ (lb)
370.11 436.80/435.82/370.61 II
Weight ratio
1.0/0.943/1.025/1.038
Adj. weight ratio
1.0/1.18/l.18/1.001
: Adjusted design areas for case 4rl
Bay Bay Bay Bay
# 1: 1.870, 0.5039, 0.1003, 0.1003 #2: 1.249, 0.4863, 0.1003, 0.1003 #3: 0.4839, 0.4962, 0.1003, 0.1003 #4: 0.1003, 0.5322, 0.4512, 0.5538
Cases 2~, speed factors vs. Case 1 (FE-IIGO): 1.44/1.79/4.26
t The first quantity (case 1) and the remaining 3 quantities (Cases 2-4) in each cell correspond to the FE and mixed NN/FE (Cases 2-4) structural analysis-based optimal design results, respectively. The trained NN models and the computational speedup factors in the optimal designs using the latter (NN) vs. the former (FE) analysis models are as follows: Case 2 3 4
NN configuration A16-27-27-D48.T1120 A16-27-27-D48.T11401[ AI6-27-27-D48.T2600
rms/max errors 0.00259/0.0281 0.00256/0.0339 0.00299/0.0495
kp = max DYE~0.25 0.313/0.25 = 1.252 0.287/0.25 = 1.148 0.240/0.25 = 0.960
:~Scaled approximate weights and areas obtained by mixed NN/FE models according to the max. displacement ratios in the above sub-table to actually (accurately) satisfy the displacement constraint of 0.25 inch and, for the Case 4 design, to limit the areas above the lower area bound of 0.1 in.2[[ § The 1140 training (t) examples in Case 3 are obtained by adding 20 identical examples (with input areas equal the Case 2 areas listed above and the output displacements corresponding to those obtained by FE analysis results) to the 1120 examples in Case 2. ]l The adjusted design values are obtained by first scaling down the unadjusted areas by a factor of 0.94 as in the other cases (cf. footnote §), setting those resulting areas smaller than the lower bound value of 0.I inf to 0.1 in. 2, performing a FE analysis based on the resulting area set to find the maximum displacement of 0.25068 inch and weight of 369.16 lbs, and finally scaling up the last area set and weight to obtain the adjusted areas and weight listed above. 2.5% greater than the FE-based optimal design result of 370 lbs. However, large discrepancies between the accurate (FE-based) and approximate (NN-based) design areas are seen for some elements (e.g., 13-16 elements of the fourth bay). To further improve the design results, particularly to get out from the lower bound area zone for elements 13-16 of the fourth truss bay toward the FE-analysis based design area value, 2600 example patterns, as explained above, are used in training the N N for the Case 4 design to slightly over-fit the 2559 unknowns to the training data sets. The design results (the last one in each cell of Table 3c) are seen to significantly improve, particularly for the design areas and maximum displacement, which now are much closer to (if not in excellent agreement with) those based on the F E method. The NN-based minimum design weight is only 3.8% greater than the FE-based result. However, as discussed in Subsection 2.3, this standard assessment method of the performance of an NN-based structural design contains a critical flaw, because it does not take into account actual constraint violations (if any) caused by inaccuracy in the
N N structural analysis model. Therefore, the design results are adjusted according to the procedure presented in Subsection 2.3, and some adjusted results are listed in Table 3c. The adjusted optimal weight results for N N analysis based designs (Cases 2-4) are seen to be heavier than the F E analysis-based result by 18%, 18%, and 3.8%, respectively. For the second and third design cases, the deviations can be reduced to 13.5% if the trained N N No. 4 instead of No. 5 in Table 3b and a similarly trained A16-27-27D48.1140 are respectively used. For the mostly satisfactory Case 4 design, all but one adjusted N N / F E analysis-based design areas deviate from the FEbased design area by less than 3.5%. Even this maverick area (i.e., the third Bay # 4 area) cited above deviates from its FE counterpart by less than 10%, which is considered to be very satisfactory. The computational speedup factors for obtaining the N N / F E - I I G O - b a s e d optimal results for Cases 2 - 4 versus the F E - I I G O - b a s e d optimization results are also shown in Table 3c as 1.44, 1.79, and 4.26. The speedup factor is seen to significantly or dramatically increase as the N N structural analysis model
463
Massively parallel structural design improves (as measured by its ability to yield better design results)i Therefore, it is important to create and train a structural analysis N N properly for use in structural design. It should be noted that these speedup factors can be made larger if the input areas in the N N models are unscaled rather than linearly scaled (mapped) into values between 0.1 and 0.9 as in the N N output displacement value case, which have to be scaled back (to their physical quantities) and forth during the .design optimization process. 3.4. 72-Bar truss (T-72) design problem under two loading sets This design problem differs from the preceding one in that the structure is to be optimized under the two nonsimultaneously applied loading sets given in Fig. 3. In addition to the AI6-27-27-D48.n.L1 N N s (n = 1120, 1140~ and 2600) (each of which now is a sub-NN) under the first loading set (LI) trained in the preceding example problem, an additional sub-NN~ A16-27-27-D48.565.L2 under the second loading set (k2) is also trained to the rms error of 0.00299 and maximum error of 0.0326. A smaller number of training examples of 565 was used for the second
loading set case because only 8 (vs. 28 in the first sub-NN case) among 48 non-zero displacement components are independent (having distinguished values) due to both structure and loading symmetry conditions. Five cases (sets) of optimal design results using the F E - G S D (gradient search direction) TM for Case 1, F E - I I G O for Case 2, and mixed N N / F E I I G O for the last three cases (Cases 3-5), together with the three associated trained N N sets, are given in Table 4. The minimum structural weight and area results obtained by using the F E - I I G O methods is seen to agree excellently (e.g. with weight discrepancy of less than 0.4%) with those using the F E - G S D method. The discrepancies between these two sets of optimal results can undoubtedly be made even smaller by improving the FE-I1GO results at additional computational costs. As in the preceding example case, the N N / F E IIGO-based design results are adjusted according to the procedure used in the preceding example, i.e., that formulated in Subsection 2.3. The adjusted weight results are also tabulated in Table 4. These adjusted design weight results show that the under-fitted N N s (using the training example patterns of I 120 and 1140
Table 4. 72-Bar truss design optimization results for the two-loading set case using the purely FE and mixed NN/FE structural analysis methods Five sets of optimal design areas (in.:)+ Member No. Bay No.
1M
5 12
13 16
I7 18
/1.897/(1.6701 [1.6241 11.9371 (2.000) /I.147/(1.3451 [1.245]{1.1821. ( 1.3881 /0.6660/(0.5225) [0.8759] 10.7458]. (0.53065 ,'0.1510/(0.15541 [0.39131 [0.3085[. (0.29275
/0.5158/(0.5148) [0.5375] .[0.5573'~. (0.5089) /0.5119/(0.5315) [0.3406] {0.3853 ~, (0.5141 ) /'0.5255/(0.5274) [0.55131 [0.58781 (0.50141 /0.5863/~0.56401 [0.43831 10.5293 ~, (0.6068)
/0.1000/(0.10051 [0.1003] {0.1001 } (0.10031 /0.1000/(0.100I ) [0.10041 10.10021 (0.10035 /0.1000,,' (0.10011 [0.10091 10.1004 ~, (0.10041 /0.4099/10.3887~ [0.1005110.1016~, (0.51361
,0.1000/(0.1011) [0.1012] [0.10011 (0.1(1245 ,'(1.1000/(0.10041 [0.1002] ',0.10011 (0.1001) ,0.1000/IO. 11461 [O. lOOl] 10.10121 (0. I 1385 /0.5690,' 10.55861 [0.6247] 10.81961 (0.6403)
Total weight (Ib) Weight ratio Adjusted weight and weight ratio:l: NN/FF/IIGO vs. FE/IIGO speedup factor
/379.66/(381.06) [350.71] [380.99} (401.935 /1.000/ I 1.0037) [0.9237] [ 1.0035~ ( 1.05875 /379.66/(381.06) [439.93] [427.09} (392.79) / 1.000/ ( 1.00371 [1.1587] I 1.12491 (1.0343) /7,' (I.001 [2.311 [3.821 (4.775
+ Shown here are the five sets of optimal design areas based on the FE-GSD (Gradient Search Direction; Case l), FE-IIGO (Case 2), and NN/FE-IIGO (Cases 3 5) methods using the following sub-NN models: Case NN configuration NN rms/max, errors k D - max. D~L/0.25 3 A16-27-27-D48.T1120.L1 0.002395/0.03404 0.3136/0.25 in. -- 1.254 A 16-27-27-D48.T565. L2 0.002994/0.03260 0.2802/0.25 in. - 1.121 4 A 16-27-27-D48.T 1140.L 1§ 0.002558/0.03387 A 16-27-27-D48.T565.L2 0.002994/0.03260 5 A 16-27-27-D48.T2600. L 1 0.002992/0.04952 0.2442/0.25 in. = 0.977 A 16-27-27-D48.T565.L2 0.002994/0.03260 where AI6, D48, Tn, L1, and L2 denote 16 input areas, 48 output displacements, n-number of training (T) examples, the first loading set, and the second loading set, respectively. ;~Adjusted weights and weight ratios using the ratios (last column) in the above sub-table as linear scaling (multiplication) factors for total weights (also areas) to actually/"exactly" satisfy the maximum displacement constraint of 0.25 in. Note that such a constraint is satisfied by the NN calculated (approximate) displacements, but not by the FE (accurate) displacement results using the corresponding areas listed. § See the last footnote in Table 3c.
464
RONG C. SHIEH
that are fewer than its threshold number, nx, cf. Eq. (3) for the Cases 3 and 4 designs) are capable of yielding reasonably good results, with discrepancies of 16% and 12%, respectively, from the FE-GSDbased design weight result of Case 1. However, if a much better result is desired, the number of training example patterns must be increased, preferably to nx-number or larger as in Case 5 design. For the slightly over-fitted case (Case 5) of using 2600 training example patterns, the unadjusted and adjusted weight results are seen to be only 5.9% and 3.4% larger than that of the Case 1 design, i.e., both unadjusted and adjusted design results for Case 5 are very satisfactory. Also shown in Table 4 are the computational speedup factors for the N N / F E - I I G O method design cases (Cases 3-5) versus the FE-IIGO-based design case. These speedup factors for the 3-5 design cases are found to be 2.31, 3.82, and 4.77, respectively. As in the preceding example case, the speedup factor increases as the accuracy of NN increases. 3.5. Ten-bar truss (T-10) design problem The truss configuration and design parameters for this textbook 2-D truss design problem are given in Fig. 4.18'j9 The loaded structure is subjected to a maximum displacement limit of 2 in. The main purpose of restudying this textbook type problem is to assess numerical performance of the developed mixed N N / F E model in optimal design of this small-scale design problem with wide range of optimal design areas (ranging from 0.1 to 30.52in.2). The range of areas used in randomly generating N N training example patterns used in the present study (as well as previous ones L2) is even wider, i.e., 0.01-55 in 2 than the optimum value range. As in the preceding examples, in order to use the frame structural analysis code capability for analysis of truss structures in this paper, hinges are inserted at both ends of all beam elements. (This truss modeling technique is much less efficient that using rod elements directly.) The WC (CM-elapsed) time required for simultaneously generating in parallel 819 independent structural analyses (with output of 10-areas, 8-displacements, and weight per analysis) using an 8K-processor machine was found to be 23.04 s. The numerical performance results of training A10-9-9-D8 N N can be found in the author's preceding study. ~2 The FE-IIGO method-based optimal design results shown in Table 5 are seen to be virtually identical to the best previously obtained optimum results ts for both weight and areas. The N N / F E - I I G O methods-based optimal design area and weight design results using the two cases of trained A10-9-9-D8.Tn NNs (NN Nos 1 and 2) with training (T) sets of n = 200 and 800, respectively, are also tabulated in Table 5. So are the adjusted weight results for both cases and adjusted area results for the one based on N N No. 2 model. The adjusted final weights are seen to be respectively 25% and 7%
heavier than the best FE analysis-based optimal results. Thus, a much larger number (say, 800, as in the N N No. 2 case) of training sets than its threshold value of n x = 269 are seen to be required to obtain good optimal design results in this example problem because the optimal design areas are widely scattered between 0.1 and 30.52in. 2 Further increase in the number of training sets, as in the preceding 72-bar truss design case under two loading sets, appears to be required if one wishes to obtain a better set of optimal design results, particularly to have the design area for member No. 6 to move out of the lower area bound zone toward the FE-IIGO-based region design area of 0.55 in. 2 Thus, the number of N N training sets required to yield good optimal design results is a function of N N size as well as the range of optimal areas. The CM-BACKPROP code is particularly powerful in training these classes of problems that require a large number of training example sets to yield accurate structural analysis results. This is because an MPP computer, such as the CM-2 or the CM-5, has a large number of processors, and the performance time for NN training via the CMBACKPROP code is virtually independent of the number of training sets used. 3.6. Discussion In passing, it should be noted that, in addition to using the fully configured 64K processor CM-2, further significant or drastic improvements in the performance results can be made by (1) using rod elements (that involve only translational and rotational axial deformations and forces) to model the truss structures, (2) reading the input and writing the output data on the data vault instead of directly from or to the front-end computer, and/or (3) as noted above, using the CM-5 model computer, which is
J_ -
-
360""
~-
~
360"--------+--
2
7
360"
3
Material properties & constraints
Loading (kips)
Item
Value
Node
X
Y
Modulus of elasticity Material density Stress limits Min. cross-sect, area Max. allow, displacements
104 ksi 0.I Ib/in + + 25 ksi 0.1 in2 + 2.0 in
1 3
0 0
-100 -100
Fig. 4+ Ten-bar truss design problem.
Massively parallel structural design
465
Table 5. Purely FE and mixed NN/FE structural reanalysis method-based optimum design results for l0 bar truss Structural analysis-optimization method used and final area (in. e) Mixed NN/FE-IIGO (Present) Member No. I 2 3 4 5 6 7 8 9 10 Wt. (lbl Wt. ratio
FE-GSD'~ (Ref. 20)
FE-GSD (Refs 18, 19)
30.670 0.100 23.760 14.590 0. I00 0.100 8.578 21.070 20.960 0.100
30.52 0.100 23.20 15.22 0.100 0.55 7.46 21.04 21.53 0.100
5076.85 1.003
5060.85 1.000
FE-11GO (Present) 30.52 0.1000 23.20 15.22 0.1000 0.5513 7.457 21.04 2 I. 53 0.1000 5060.88 1.000
NN # 1+ +
NN #2:]:
33.68 14.97 20.37 6.37 0.127 1.029 10. 723 21.01 19.055 0.127
29.50 0.1156 23.38 11.64 0.1027 0.1009 11.90 21.34 15.04 0.1016
5347 (6327)11 1.057 ( 1.25)14
4786.7
Adj. NN #2§ 33.43 0.1310 26.92 13.19 0.1164 0.1143 13.48 24.18 17.04 0.I 151 5423.8
0.9458
1.072
+ GSD = Gradient search direction design optimization method. + Used the following trained network in the area range of 0.01 to 5 in. 2 I l) A10-9-9-D8.200 NN with rms error/max error/max Dv~ (FE disp.) of 0.0096/0.0592/2.3665 in. [2) AI0-9-9-D8.800 with rms error/max error/max DFE of 0.00517/0.0871/2.2662 (in). §The adjusted tscaled-up) areas and weight of the 2rid NN-based design areas and weight by a factor of 2.2662/2 = 1.1331 to satisfy exactly the D constraint limit of 2.0 in. [I The adjusted weight or weight ratio according to k - 2.36665/2.0 to satisfy accurately the displacement constraint limit of 2.0 in.
generally 6.4 times faster than the CM-2. This general speedup factor of 6.4 was confirmed by using the C M - S T R A N D code in a test run of the above F E - I I G O method-based 72-bar truss optimal design problem under two loading sets on the CM-5 at Naval Research Laboratory. Training a nonlinear N N using the sigmoid activation function is found to be the most time-consuming element of structural reanalysis and design optimization computations in the N N reanalysis method-based structural design. For example, by considering the structure and load symmetric condition of the present 72-bar truss problem under the first loading set (cf. Subsection 3.3), the numbers of independent cross-sectional areas and non-zero displacement components are 16 and 28, respectively. An appropriate N N architecture using the aforementioned rule of the sum for determining the numbers of two hidden layer nodes is AI6-22-22D28. II took 2 h:16 m CM-2 (approximately 22 min. CM-5) wall-clock time in 22,800 cycles of iteration to train this N N on 1000 example patterns to rms and maximum errors of 1% and 16.6%, respectively. 12 At least twice as much time is estimated to be needed to reduce these errors to their one-third values for use in obtaining reasonably accurate minimum weight design results. Therefore, it is important to formulate efficient methods or strategies to keep the number of N N s to be trained to a minimum (as in the present and preceding author's studies ~-') and/or to greatly reduce the time required for training the N N s so that the overall computational time required in the structural design can be kept as small as possible. For this,
sub-NN concepts in conjunction with development and use of simultaneous multiple concurrent N N computational/training strategy/capability on an MPP computer, approximate linear reciprocal-area/ displacement N N models, and/or alternate neural computational algorithms, etc., can be used. Formulation/development of these concepts and capabilities in the MPP environment of the CM-5 are underway at M R J , Inc. under a N A S A Lewis Research Centersponsored, SBIR Phase II study. 4. C O N C L U D I N G
REMARKS
A computational strategy that replaces the most time-consuming portion, i.e., the conjugate gradient method-based linear equilibrium equation solver, of finite element structural reanalysis with an N N reanalysis model in structural design optimization was presented and numerically shown to be an efficient and viable approach in the MPP environment of the CM-2 computer. Extension of the study to the CM-5 computer platform is underway. Also formulated is a correction/improvement procedure for constraint violation or too-conservatively-satisfied constraint condition (arising from approximate nature of N N analysis) of N N analysis-based, final structural design results. The purpose of correcting the final design results is not only to obtain a sound or improved design, but also to critically evaluate the numerical performance of N N analysis-based design results. Through several example problems, it was demonstrated that both new F E - I I G O and N N / F E - I I G O
466
RONG C. SHIEH
algorithm sets implemented on the C M computers as the C M - S T R A N D code are capable of performing multiple concurrent structural analyses and/or optimal design excellently and efficiently under a properly configured and trained NN. As for the formulated correction/improvement model of inaccuracy in the final N N or N N / F E analysis-based optimal design, it was shown or pointed out that such a model not only provides a better or safe set of design results, but also enhances the basis for critically assessing the performance results of N N analysis-based optimal design. In any case, an N N analysis-based final optimal design set should be checked and adjusted to satisfy all constraint requirements. It was also found that in addition to the rms error, the maximum error measurement of a trained N N is also important in assessing the degree of adequacy of the trained N N . The number of training example sets is best kept at or above its data fitting threshold value, n x (Eq. (3)), in training an N N , and for a small size N N with wide range of optimal design areas, the number may need several times of the threshold value, as in the case of the 10-bar truss design problem. This conclusion is at least true for the present N N / F E analysis and stochastic type I I G O optimization method-based optimal design and needs further numerical experimentation to confirm this. It should be noted that the use of a large number of training example sets has only small impact on N N training/computational time using the C M - B A C K P R O P code on the CM-2 or the CM-5. (The impacted areas are mainly in reading the training sets serially into the front-end of a C M computer and added training effort/iteration cycle in bringing down the larger maximum error induced by having a larger number of training sets.) It is desirable in the follow-on study to develop (or further develop) and/or implement additional efficient, fully or partially NN-based structural reanalysis strategies for structural design (e.g., one based on mixed N N / L S (least-square) models12), N N computational models (e.g., an approximate linear reciprocal-area/displacement N N model), and N N training strategies (e.g., one based on the N N partitioning/multiple concurrent N N training concepts). Acknowledgements--This work was supported by NASA Lewis Research Center SBIR Phases I & II, Contract Nos NAS3-26842 and NAS3-27412. The NASA Project Engineer is Dr Laszlo Berke. The author also wishes to thank Dr George Wilson of MR J, Inc., for his CM programming support in implementing the neural network-based portion of structural reanalysis capability into the CM-2 STRANOP code. The CMNS CM-2 computer resources used in the study were provided by Thinking Machines Corporation (TMC). REFERENCES
1. L. Berke and P. Hajela, "Applications of artificial neuralnets in structural mechanics," NASA Technical Memorandum 10240, 1990.
2. P. Hajela and L. Berke, "Neurobiological computational models in structural analysis and design," Proceedings of the 31st AIAA/ASME/ASCE/AHS/ ASC Structures, Structural Dynamics and Material Conference, Long Beach, CA, pp. 345 355, April 1990. 3. P. Hajela and L. Berke, "'Neural network based decomposition in optimal structural synthesis," Computing Systems in Engineering 2, 473-481 (1991). 4. P. Hajela, B. Fu and L. Berke, "'Art networks in automated conceptual design of structural systems," NATO/AGARD Advanced Study Institute on Optimization of Large Structural Systems, Berchtesgaden, Germany, 23 Sept.-5 Oct. 1991. 5. D. A. Brown, P. L. N. Murthy and L. Berke, "Computational simulation of composite ply micromechanics using artificial neural networks," Microcomputers in Civil Engineering 6, 87-97 (1991). 6. R. A. Swift and S. M.Batill, "'Application of neural networks to preliminary structural design," Proceedings of the 32nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Material Conference, Baltimore, MD, pp. 335-343, April 199l. 7. W. C. Carpenter and J. Barthelemy, "A comparison of polynomial approximations and artificial neural sets as response surfaces," Proceedings of the 33rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Material Conference, Dallas, TX, pp. 2474-2482, April 1992. 8. R. A. Swift and S. M. Batill, "Simulated annealing utilizing neural networks for discrete variable optimization problems in structural design," Proceedings of the 33nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Material Conference, Dallas, TX, pp. 2536 2544, April 1992. 9. L. Berke, W. Hafez and Y.-H. Pao, "Neural networks for structural design: an integrated system implementation," Proceedings of the 4th AIAA/USAF/ NASA/OAf Symposium On Multidisciplinary Analysis and Optimization, pp. 915 923, 21 23 September 1992. 10. X. Zhang, M. McKenna, J. P. Mesirov and D. J. Waltz, "The backpropagation algorithm on grid and hypercube architectures," Parallel Computing 14, 317 327 (1990). 11. Z. Szewczyk and P. Hajela, "Neurocomputing strategies in decomposition based structural design," Proceedings of the 34th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Material Conference, La Jolla, CA, pp. 24585465, 19-22 April 1993. 12. R. C. Shieh, "Neural network-assisted large-scale structural analysis/reanalysis in a massively parallel environment," Proceedings of the 35th AIAA/ASME/ASCE/ AHS/ASC Structures, Structural Dynamics, and Materials Conference, Hilton Head, SC, pp. 1533 1545, 18 20 April. 13. R. C. Shieh, "Finite element method structural analysis and optimization of space frames and trusses in a massively parallel environment," Proceedings of the 4th AIAA/USAF/NASA Symposiums on Multidisciplinary Analysis and Optimization, Cleveland, OH, pp. 1069 1077, 21 23 September 1992. 14. X. Zhang, "'Backpropagation algorithm on the connection machine systems," Thinking Machines Corporation, Cambridge, MA, 1992. 15. J. Hertz, A. Krogh and R. G. Palmer, Introduction to the Theory o[" Neural Computation, Addison-Wesley, Redwood City, CA, 1991. 16. S. H. Chew and Q. Zheng, Integral Global Optimization, Springer-Verlag, New York, 1988. 17. R. C. Shieh and G. V. Wilson, "A massively parallel nonlinear optimization code capability and its application in structural optimization," Proceedings of
Massively parallel structural design the 32nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Baltimore, MD, pp. 636-643, 8-10 April 1991. 8. C. Fleury and L. A. Schmit, "'Dual methods and approximation concepts in structural synthesis," NASA CR-3226, December 1980.
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19. R. T. Haftka, Z. Gurdal and M. P. Kamat, Element oJ Structural Optimization, 2nd edn, Kluwer, Boston, MA, 1990. 20. L. A. Schmit and H. Miura, "'Approximation concepts for efficient structural synthesis," NASA Technical Report CR-2552, 1976.