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Automated system for structural design using design window search approach: its application to fusion first wall design
ELSEVIER
Advances in Engineering Sofrwm 28 (1997) 103-l 13 0 1997 Elsevier ScienceLimited Printed in Great Britain. All rights reserved 0965~9978/97/$17.00
PII:SO965-9978(96)00049-X
Automated system for structural design using design window search approach: its application to fusion first wall design Yosbihiko Mocbizuki,
Sbinobu Yosbimura*
& Genki Yagawa
Department of Quantum Engineering and Systems Science, University of Tokyo, 7-3-l Hongo, Bunkyo, Tokyo 113, Japan
(Received 5 July 1996; accepted 3 September 1996) Some practical structures are required to be operated in severe environments where various mechanical, thermal or electromagnetic loadings occur in a complicated manner and various failure phenomena such as yielding, fracturing and melting compete among others. In designing such structures, it seemsvery useful to determine the design window (DW) that schematically indicates an area of satisfactory solutions in a permissible design space. The DW may give us more meaningful information than one satisfactory or optimum solution. However very little researchhas been performed on this topic so far. This paper describesa novel automated system for structural design based on the DW concept. The present systemconsists of three main modules and one sub-engine,that is, (a) the Analyzer, (b) the DW Search Engine, (c) the Design Modifier as the main modules and (d) a multilayer neural network as the sub-engine of the DW Search Engine. The Analyzer takes care of computational mechanics simulations for various sets of design parameters. To maintain high flexibility and extensibility, the Analyzer is constructed using an object-oriented knowledge representation and a data-flow processing technique. Some DW search methods are incorporated into the DW Search Engine. Using neural network technology, DWs are searched very efficiently. The Design Modiier has the role of tinding one satisfactory solution, which is given to the DW Search Engine so that some search methods can start searching. To demonstrate the practical performance of the present system, it is applied to designone of the typical structures to beoperated in a severeenvironment, i.e. the first wall structure of ITER (International Thermonuclear Experimental Reactor). 0 1997 Elsevier Science Limited. All rights reserved.
1 INTRODUCTION
This paper describes a novel automated system for structural design using a DW search approach. The present system consists of three main modules and one sub-engine. The main modules are (a) the Analyzer, (b) the DW Search Engine and (c) the Design Modifier,
Due to the progress of computational mechanics technology, e.g. the finite-element analyses, the accuracy and reliability of analyses of a uni-phenomenon, such as structural deformation, heat conduction or fluid dynamics, have been dramatically improving. In the field of structural design problems, structural optimization can often be solved based on mathematical methods combined with the computational mechanics.’ In practical design problems, it is more useful to obtain the design window (DW) that schematically indicates an area of satisfactory solutions in a permissible design space. Several methods to obtain one satisfactory solution have already been proposed.2 However very little research on. DWs has been performed so far.
while the sub-engine of the DW Search Engine is a multilayer
neural
network.
The Analyzer
performs
coupled finite-element simulations for various sets of design parameters. To maintain high flexibility and extensibility, the Analyzer is constructed using an object-oriented knowledge representation3 and a dataflow processing technique.4 Some search methods are incorporated
into the DW Search Engine. Using the
neuro Analyzer, DWs are searched very efficiently. The Design Modifier has the role of finding one satisfactory solution, which is given to the DW Search Engine so that some DW search methods can start searching. To demonstrate the practical performance of the
*Author to whom correspondence should be addressed. 103
104
Y. Mochizuki, S. Yoshimura, G. Yagawa
present system, it is applied to design one of the typical structures to be operated in a severe environment, i.e. the first wall structure of ITER (International Thermonuclear Experimental Reactor).’
(b) generate a finite element mesh (c) attach boundary conditions. In a main-process: (d) perform the finite element analyses.
2 SYSTEM
In a post-process:
CONFIGURATION
Figure 1 shows a schematic configuration of the present system. In the following sections, fundamental functions of the three main modules and the sub-engine of the present system are described in detail. 2.1 Analyzer
In the present Analyzer, an object-oriented knowledge representation technique is adopted to store several knowledge modules related to analyses of structures subjected to various loadings. A data-flow processing technique is also utilized as an inference mechanism among the knowledge modules. 2.1.1 Object-oriented knowledge representation An analysis process in automatic structural design usually consists of many elemental procedures as follows. In a pre-process after design parameters are given: (a) define an analysis domain
(e) extract some necessary physical values from the output data evaluate sensitivities of an objective function with 0 respect to the design parameters. The elemental procedures above are iterated by changing the design parameters. Each elemental procedure can be regarded as an analysis module here. This characteristic seems very compatible with the objectoriented knowledge representation. Figure 2 shows the schematic view of mesh generation object, FW-mesh. Figure 3 shows the same object expressed by LISP. Here, KCL (Kyoto Common Lisp) is employed. As shown in the figures, each object in the Analyzer consists of input/output slots and a functioncall or a method slot. Each object plays the role of an expert for each elemental procedure. In the present Analyzer, a number of objects are stored as shown in Fig. 4. 2.1.2 Data-flow processing The data-flow processing technique is utilized as an
Knowledge Engineering Techniques - Object-oriented Knowledge Representation * Data-flow Processing
Method of Multi-dimensional * Whole-areaSearchMethod (WSM) - Boundary Swelling Method (BSM) . Boundary Tracing Method (BTM)
* Moment Method
Fig. 1. Schematic configuration of the present system.
Structural design using design window search
RN-mesh input f--zicinterchannel
1m A
A
file
Fig. 2. Schematicview of PW-meshobject. inference mechanism controlling a global data stream among objects. Taking Figs 2 and 3 as an example, let us explain its fundamental concept. As soon as all the input data are prepared on the input slots and a flag of ‘not ready’ turns to ‘ready’ in Fig. 2, the function-call slot sends a ready-command to the outer processors. In the figure, the outer processor is a two-dimensional finite-element mesh generation code. After a mesh is generated, the output data are prepared on the output slots. Then, the data on the output slots are immediately s#entto any objects that need the data. Owing to the two techniques described above, all the objects are kept to be independent of one another, and the Analyzer succeedsin maintaining high flexibility and extensibility.6)7 2.2 Design modifier The Design Moldifier finds one satisfactory solution.
W-mesh (input (inter-channel (data scalar) (type design-prm) 0 notready (input-group 1)) (data scalar) (type design-prm) (thick-RN 0 notready (Inputgroup 1)) (dlstJW (data Scalar) (type design-prm) 0 notready (inputqroup 1)) (Idata scalar) (type deslgngrm) (helghtSH () notrwdy (input-group 1)) (data scalar) (type deelgngrm) (length-CH () notready (Input-group 1))) (output (mesh6JW (“mesh8.FW” file) (type 8-nodeJW_mesh))) (func (VW-mesh” ( thick_FW inter-channel dist_FW height_CH length-CH ) command-mode)) (group-level 20))
Fig. 3. F\V’_meshobjectexpressedby LISP.
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Starting from the solution, the DW Search Engine searchesa DW. Finding a satisfactory solution is called a feasible design. The DW Search Engine basically employs the ‘generate and test’ strategy. Here, design parameters have to be somehow modified within a design space when a former design candidate does not satisfy any of the design criteria. As one of the efficient techniques for design modification, much attention has been paid to mathematical approaches based on numerical sensitivity analyses. However, application of such approaches has been limited only to simple problems, such as shape design of simple structures subjected to mechanical loading. This is because the mathematical design modification is often trapped at one of the locally optimum points in complicated design problems. Thus, the present authors proposed a design modification approach based on experts’ empirical knowledge related to structural design in the previous studies.6-g One of the examples of experts’ empirical knowledge is that “if the maximum temperature occurred in a structural wall exceeds a permissible value, the wall thickness should be thinner”. These kinds of IF-THEN rules, which have been derived from experts’ qualitative inference, instruct how to modify a former design, considering the reason why the former design candidate violates design criteria. The empirical design modification approach combined with the fuzzy control* is incorporated into the Design Modifier described in this paper. 2.3 DW search engine and neuro analyzer 2.3.1 D W search engine As shown in Fig. 5, several large or small DWs may exist in a design space of practical structures. Some of them may be doughnut-shaped. If possible, one wishes to find all the DWs. In the present study, the original design spaceis divided into the following two subspaces,i.e. the permissible design space and the impossible design space.Here, the permissible design spaceis the subspace where all geometrical constraints are satisfied, while some of the constraints are violated in the impossible design space. The DWs are searched only within the permissible design space. The following three search methods are considered here: (a) the Whole-area Search Method (WSM), (b) the Boundary Swelling Method (BSM) and (c) the Boundary Tracing Method (BTM). These search methods are implemented in the DW Search Engine with the C language. The WSM does not need to know an initial satisfactory solution a priori, while the BSM and the BTM do. The principal characteristics of the three methods are summarized in Table 1. In the present paper, only the WSM is explained becausethe method can search plural multi-dimensional DWs even if any of the DWs to be searchedare plural or doughnut-shaped, and also because quasi-optimum
106
Y. Mochizuki,
S. Yoshimura, G. Yagawa
heal_FEM-Finas
:
stress-FEM-Flnas
:
heat-FEM-Post-Proc
:
srress-FEM-Post-Proc
:
heat_ifile-Fi%as
:
stress~ifi1e~Fina.s
:
FW-mesh
:
mat-dbaseprmor-CFC
:
mat-dbase-F W-SUS316 :
an object that performs 2D heat conduction analysis using the FEM code, i.e. FINAS an object Ihatperforms20 stressanalysis with rhermal effects using Ue FEM code,Le. FINAS an object that extracts temperature valuesfrom the output fJe made by heat_FEM_Finas object an object tbatfind the maximum equivaknt stress using the outputjile made by stress-FEM-Finas object an object that creates an input data fire for heaLFEM_Finns object an object that creates an input &ta j?k for stress-FEM-Finas object an objectteal generates20 FEM mesh, i.e. eight-noded isoparametrk mesh an object that manages the database of mater&l properties of the armor an object that manages the databaseof material
properliesof the mothermaterial :
designgrm Object
an object Ihat produces shape designparameters and boundary conditions
Dam Stream
Fig. 4. Objectnetwork and function list. solutions are obtained simultaneously. The algorithm of the WSM is as follows. At first, a multi-dimensional lattice that is empirically determined by engineers is generated in the design space as shown in Fig. 6. It is then examined one by one at all the intersection points of the lattice whether the points are inside DWs or not. During searching DWs a distribution of objective function over the permissible design space is also obtained if necessary. Though the WSM has several advantages as listed in Table 1, the number of points to be examined tends to become extremely huge. To apply the WSM to practical design problems, one has to solve such an intrinsic problem of the method. For this purpose, we utilize a neural network technology.’ Before explaining it in detail, we explain a process to separate Permissible Design Space
plural DWs from a set of satisfactory design points searched by the WSM in the following. In the case of two- or three-dimensional DWs, the DWs can be easily separated through visualization. When the number of design parameters is more than four, the DWs can not be visualized. The cluster analysis that is one of the multivariate statistical methods is employed here to separate plural DWs. Among several clustering methods, the hierarchical method,” i.e. the most neighboring method, is employed in the present study. Its analysis flow is as follows:
(4 Transform the coordinates of the searched satis@I (4 04
@>
factory design points so that all the unit steps for searching become 1.O. Calculate all the Euclidean distances between one satisfactory design point and any other ones. Assign each satisfactory point a group that consists of itself. Union two groups if the minimum distance between a point belonging to one group and a point belonging to the other group is less than Ji;, where n is the number of design parameters. Iterate the procedure (d) until all the points are checked.
2.3.2 Neuro analyzer
In this section, a multilayer neural network utilized as the Neuro Analyzer is first explained, and then the principle of the automatic DW search method using the neural network is described. Design Parameter 1 Fig. 5. Schematicview of DWs in two-dimensionaldesign parameterspace.
Network architecture and learning algorithm of mtdtilayer neural network. Figure 7 shows a unit of a
neural network consisting of multiple input slots and a
107
Structural design using design window search Table 1. Main features of three methods of searching DW In the case that an initial satisfactory In the case that an solution is known initial satisfactory solution is unknown Boundary Tracing Boundary Swelling Whole-area Method Method Search Method @TM) @SW W’SW Dimension of DW that can be searched
L2
22
2
possible
possible
impossible
possible
sometimes possible
impossible
Determination of quasi-optimum solutions
possible
possible
impossible
Number of
too large
large
small
Searching of doughnut-shaped DW Searching of plural DWs
searchedpoints
single output slot. The relation between the input and output data is formulated as follows: Oj =f(Uj)
= l/(1
Uj = k
WjiIi - 0j
(1)
+eXp(-2Uj/Uo)}
(2)
i=l
where Oj is the real-valued output of the jth unit, Uj the total input to the Jth unit, f( ) the activation function, i.e. the sigmoid function here, Us the temperature constant of the sigmoid function, IVii the connection weight between the ith and the jth units, Ii the input from the ith to thefih units, and 0, the bias value of theflh unit, respectively. Figure 8 illustrates an ordinary three-layer network.
2D DesignWindow
\v: satisfactorysolution 0 unsatisfactorysolution Fig. 6. Illustration of Whole-Area Search Method for design window.
All the units are formed into multiple
layers, i.e. an
input layer, intermediate (hidden) layer(s) and an output layer, with only feedforward connection between successive layers. The basic idea for training the neural network, i.e. a supervised learning algorithm is as follows. At first the following training error E is defined:
(3) k=l
A
where Ep is the square error for the pth learning pattern, Tpk the teacher signal to the kth unit in the output layer for the pth learning pattern, o,k the output signal from the kth unit in the output layer for the pth learning pattern, and n the number of output units, respectively. In the training process, the connection weights Wji and the basis 0, are modified repeatedly based on the gradient descent method in order to minimize the above square error. Since this modification proceeds downwards in Fig. 8, the training algorithm is called the back propagation. In the present study, the backpropogation algorithm combined with the momentum method is employed to attain stable convergence in
I, \Wil I* j* . 13w3. Qjf .. 91, ’ jr
Oj
Fig. 7. Schematic view of neuron unit.
108
Y. Mochizuki,
S. Yoshimura, G. Yagawa
Trslning Patterns : ( I pt -Ipn)vs(Tpt-Tpt);p=l-r Tpt ) TM 1 * Tw+ Teacher Signal l
Fig. 8. Three-layerneural network. training process.9*1’Through such an iterative training process, the network attains an ability of outputting the similar signal to the teaching one. It is theoretically proven that the three-layer neural network can approximate any continuous mapping. l1,12 However, the network has some limitations in reality due to poor convergence in training processeswhen the numbers of hidden layers and units increase. In the present study, an appropriate size of the network is determined through trial and error as in many practical applications, although several interesting studies on automatic determination of network size have been performed.131’4 The present authors have already applied the m~tilayer neural network to several inverse analyses successfully.15-2o iMain features of multilayer neural networks. The attractive features of the multilayer neural network can be summarized as follows:
because both thermal and stress analyses have to be executed. Thus, the neural network aided search method is proposed in the present study. As shown in Fig. 9, this method consists of three phases.At first, finite-element analysesare performed to prepare a number of learning data sets and unlearning ones by parametrically varying design parameters. The unlea~ng data sets are utilized to prevent the neural network from ‘overlearning’, which is such a phenomenon that estimation error for unlearning data setsstarts increasing even though the learning process advances. Next, a back-propogation neural network is trained using a number of learning data sets prepared in the previous phase. Here design parameters are given to the input units of the network, while physical values, such as the maximum temperature and equivalent stress, are given to its output units as teacher signal. After a sufficient number of training iterations, the network is expected to imitate a response of the original finiteelement analyzer. In the final phase, multi-~mensional DWs are searched quickly using the trained network. In the present search method, the finite-element analyses are performed only in phase 1. The number of the finite-element analysesrequired is much smaller in this method than in the search method without using the neural network. As the well-trained network calculates physical values very quickly, even the WSM can be executed in a reasonable processing time. 2.4 System en~o~ent The main portions of Analyzer, Design Modifier and DW Search Engine including Neuro Analyzer are
{a) One can automatically construct a nonlinear mapping function from multiple input data to multiple output data within the network through a learning process of some or many learning patterns. (b) The network has a feature of so-called ‘generalization’, i.e. a kind of interpolation, so that the well-trained neural network estimates appropriate output data even for unlearned patterns. (c) The trained network operates quickly in an power application process. Computational required for operating the trained network may be equivalent to only that of a personal computer. Principle of automatic D W Search using a neural network. In searching DWs, whether a searchedpoint is
is checked through a satisfactory solution computational mechanics simulations, e.g. the finiteelement analyses.However, such detailed calculations at every searchedpoint are very time-consuming, especially in designing high-temperature structural components
Fig. 9. Procedureof DW searchusing neural network.
109
Structural design using design window search implemented on one of the popular engineering workstations, Sun SPARCstation ELC. The finite-element analyses are executed on one of the Graphic workstations, Titan STlOOO,which is connected to the ELC through the Ethernet LAN of the University of Tokyo (practical speed is 10kBPS). As for the finite-element analyses, one of the general purpose finite elements codes, FINAS” is employed. 3 DESIGN
MODELS
AND DESIGN
2
Cl-plasma zr hEt”5Gv 2 A Free Slip Condition
Q-blanket b--Jw
CRITERIA
The ITER (International Thermonuclear Experimental Reactor)’ is being designed through a co-operative work among Japan, EU, USA and the former USSR. Its Conceptual Design Activities (CDA) were completed in December 1990. Since then, ITER Engineering Design Activities have been performed in order to collect information necessary for detailed design, R&D and construction of ITER. The ITER first wall currently has two candidate designs regarding a cooling channel, i.e. rectangular and circular channels. Detailed explanation of the ITER project can be found elsewhere.5The present system is applied to search DWs for both models. The design models are shown in Fig. lO(a,b). The base material of the wall is subjected to membrane tensile loading, F, which might be caused by electromagnetic loading and pressure from breeder blanket, and to surface heat loading, Q-blanket, on the blanketside surface, and to volumetric heat loading, Q,-SUS316 (= 20.0 MW/m3). Magnitudes of F and Q-blanket will be given later. The armor material of the wall is subjected to surface heat loading, Qglasma (= 0.15 MW/m*), on the plasma-side surface, and to volumetric heat loading, Q,-armor (= 6.0 MW/m3). The temperature and pressure of cooling water are 100~0°C and 1.5MPa, respectively. The heat-transfer coefficient between the armor and the base materials is 1000.0W/ m*‘C, and that between the cooling water and the base material is 20,000*0W/m*“C. The type 316 stainless steel and CX-2002U are utilized as the base material and the armor material, respectively.The wall accompaniedby the armor is modeled with eight-noded isoparametric elements. A static thermal conduction analysis is performed. In an elastic thermal stressanalysis, only the basematerial is analyzed under the generalized plane strain condition. The finite element analyses are performed using the object-oriented Analyzer described previously. Both geometrical constraints and failure criteria are employed here as design criteria. The geometrical constraints are as follows: For the rectangular cooling channel model: D2a L>H+P WZD+B+y
Membrane Force F
(4)
I
(a) Rectangularcooling channelmodel. Membrane Force F QglasmaT
hrf Q-blanket W
-I
(b) Circular cooling channelmodel. Fig. 10. Designmodelsof ITER f&t wall. For the circular cooling channel model: D>a: LLR+P W>D+2R+y
(5) where D, L, W, H, B and R are the geometrical design parameters as shown in Fig. lO(a,b), while QI,,f3and y are the design margins that are empirically decided by engineers. The failure criteria employed are as follows: (a) TInax-Armor < ~OLinllor
(ha)
@)
(6b)
~max-SUS316
cc) ~max-SUS316
<
TO-SUS316
< gOo_SUS316
(k)
is the maximum temperature in the Gax-~or armor material portion, TO-Armor a permissible temperature for the armor material, Tm-SUS316 the maximum temperature in the base material portion, T0-sos3i6 a permissible temperature for the base material, omaxsus3i6 the maximum equivalent stress in the base material portion, and cOo_SUS316 a permissible equivalent where
110
Y. Mochizuki,
S. Yoshimura, G. Yagawa
stress for the base material, respectively. For the purpose
of simp~ck
TO&hor~
TO&SUS316 and
gOo_SUS316
are taken to be 1000~0,4OO*O”Cand a yielding stress of type 316 stainless steel, respectively.
4 RESULTS
I
Unlearned Patterns
AND DISCUSSIONS
4.1 Preliminary examination
Fundamental performances of the present DW search method are examined through searching (L - W) DWs of the rectangular cooling channel model. Here, D, H and B are fixed to be 2.0, 35 and 5.0 mm, respectively. Design margins for the geometrical constraints ,0 and y are taken to be 0.5. As for boundary conditions, F and Q-blanket are assumed to be 49*ON and 0.0 MW/m’, respectively. The multilayer neural network employed is of the three-layered type. The network has 2 input units, 40 hidden units and 1 output unit. The two input units correspond to the two geometrical design parameters L and W. One output unit is prepared to output the It should be maximum equivalent stress,i.e. oma-SUS316. noted here that the output units for Tmaxedor and Tmax-sus316are not prepared in this case because the temperature criteria of eqn (6a) and (6b) are always satisfied under the given boundary conditions. The number of 40 for hidden units seems rather large compared with the numbers of input and output units. This number is selectedon purpose to examine effectsof ‘overlearning’. Figure 11 illustrates 16 sets of the input data of learning patterns and 9 sets of the input data of unlearning patterns in the L-W design space. All the input data and output data are normalized to a unit range from 0.01 to 0.99. Figure 12 shows the history of learning process in the case that a constant of the Learning Patterns
Unlearning Patterns
18.0 17.0 16.0 15.0 Ts
14.0
3
11.0 10.0 9.0 8.0 3.5 4.0 4.5 5.0
5.5
6.0 6.5 7.0 7.5
8.0
L (x lo9 m)
Fig. 11. Learning and unlearning patterns in L-W design
parameterspace.
Iteration Number of Learning
Fig. 12. Training historiesof learnedand unlearnedpatterns.
sigmoid function U. is taken to be 0.6. Here the following mean error of estimation is employed for both learning and unlearning patterns: Mean-Error = f 2
)Tp - 0,)
(7)
p=l
where r
Tp 0,
is the number of learning or unlearning patterns is the correct solution of pth learning or unlearning pattern is the output data of pth learning or unlearning pattern
Overlearning is clearly observed in the figure. To examine the influence of overlearning to DWs quantitatively, DWs are searched using the following two kinds of trained neural networks: A:
B:
The network which adopts the connecting weights and the bias values at 226 learning iterations when the mean error of estimation for the unlearning patterns reaches the minimum value. The network that adopts the connecting weights at 5000 learning iterations when overlearning is fairly advanced.
Figure 13 shows the DWs obtained using the two trained neural networks above. The regions surrounded by chained open squares indicate the correct DWs that are searched using the Boundary Tracing Method (BTM) without the neural network, and the regions filled with plus-symbols show the DWs that are searched using the present Whole-area Search Method (WSM) with the neural network. Here the unit steps for searching AL and A W are taken to be 0.075 and 0*25mm, respectively. It can be easily seen from the figures that the DW obtained with the network A agrees well with the correct one, while that of the network B is rather distorted. Such less accuracy in the latter case may be caused due to overlearning. This comparison
Structural design using design window search
; ... -.. ..- . . ..
0 Minimum dma E::::.... !!&m Solution 1, ............... -a ........................ ........................... ............................... II y;:::::::::::::::::::::::::::::::.f’zgg ...................................... wlli ......................................... ......................................... .. ....................................... Ia . ............................................ 0.. ........................................ mm .............................................. .............................................. .a .............................................. .m ............................................... ,I.. .......................................... ........................................... mm .,*. . ........................................ m .................................... II I ..................................... I.. .............................*- . ......................... .................... ..I .............. I iize--=
16.0 15.0. g
14.0.
2
13.0
a 3
12.0.
clearly suggests that it is very important to utilize the network that stops learning when the mean error of estimation for unlearning patterns reachesthe minimum value. Open circles in Fig. 13(a, b) indicate the quasiminimum values of omasUS316. By using the WSM, the distribution of objective function over the searched DW and the quasi-optimum solution are simultaneously obtained.
D FEM + Nauro
17.0.
11.0. 10.0. 9.0 6.OLL
3.5
4.0
4.5
5.0
6.0
5.5
6.5
7.0
111
.
’
7.5
6.0
4.2 Comparison betweenrectangular and circular cooling channel models based on DWs
L(xlO%)
(a) L-W DW searchedby neurotraineduntil 226 learning iterations whenmean-errorof estimationfor unlearnedpatternsis minimized.
IfJ.O(---
0 FEM
I
+ Neuro 0 Minimum dmax
17.0. 16.0 15.0 p14.0. 0
13.0.
25 12.0'
3
11.0' 10.0. 9.0 . s.oL 3.5
a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I” . . . . . . . . . . . . . . . . . . . . . . . . . *.. 1,., . . . . . . . . . . . . . . . . . . . I . . . . . . . . *. . . . . I,..... *. . . . -e......, I*.... -1:::’ mDlmQG‘I””
. 4 0 4.5
5.0
5.5
L(x 10
6.0
6.5
m mu7
7.0
7.5
1 6.0
%l)
(b) L-W DW searchedby tbe over-trainedneurountil 5000learning iterations.
Fig. 13. Effects of overlearning of neural network on DWs.
In this section, let us compare the rectangular and the circular cooling channel models in Fig. 10, taking three dimensional DWs of D, L and W. Figure 14 shows the three-dimensional DWs obtained for both models. The unit steps for searching AD, AL and A W are taken to be O-15,0.15 and 0.5 mm, respectively. Surface areas of both cooling channels are chosen to be identical. That is, the surface area of the rectangular model A,,,,,ti,, (= 2H + B = 2 x 3.5 + 5.0mm) and that of the circular model Acircular(= rR = 7r x 3.82mm) are taken to be 12.0mm. As to boundary conditions, F and Q-blanket are 9.8 N and 0.15 MW/m2, respectively. Q, /I and y are taken to be l.Omm. Both DWs are searched using the WSM combined with the neural network approach. Ordinary three-layer neural networks are employed. The network to design the rectangular channel model has 5 input units, 20 hidden units and 2 output units, while that of the circular channel model has 4 input units, 20 hidden units and 2 output units. D, L, W, H and B are the input data for the rectangular channel model, while D, L, Wand R are those for the circular one. The two output units output the maximum equivalent stress and the maximum temperature of the base material. The output unit for TmaxArmor is not considered here
Bracketedcoordinates denote(D, L, W). The unit is [mm].
‘thenumber of searchedpoints in DesignWindow is 356. Ihe minimum vdue of Chax is 174.05[Mpal at D=1,45[mm], L=4.55[mm] and W=8.5[mm].
The number of searchedpoints in Design Widow is 44. The minimum value of Omanis 166.35[Mpa] at D=l.O[mm], L=4.85[mm] and W=lO.O[mm].
Fig. 14.Comparison of three-dimensional DWs between rectangular cooling channel model and circular one. (a) DW of rectangular channel model. (b) DW of circular channel model.
because the temperature criterion for the armor material, i.e. eqn (Sa), is always satisfied under the given bounda~ conditions. Next, let us describe the learning and unlearning patterns employed. In the case of the rectangular channel model, 460 learning patterns and 120 unleaming ones are adopted, while 168 learning patterns and 56 unlearning ones are utilized in the case of the circular channel model. As for the rectangular channel model, the input data sets of learning patterns are prepared as follows: (a) Divide the interval between the maximum and the minimum values of each design parameter into three subsections as follows:
Here, r>-, Lmax, Wma, H,,, and Bmaxare 4.0, 10.0, 150, 95 and lo-Omm, respectively, while Dkn, Lh,,, Wtint &-,,A,, and &in are 1~0,35, 8.0, l-5 and 3*Omm, respectively. (b) Consider all possiblecombinations of D, L, W, H, B. (c) Choose the combinations that satisfy the geometrical constraints. These are utilized as the input data sets of learning patterns. On the other hand, the unlearning patterns are prepared as follows: (d) Make the following list of each design parameter using the learning data above: D
L
W
PI
+ D2)/2
CL1 + J52>/2
( w1+
Pi!
+ D3)P
G2 + L3>/2
cry,
f
CD3 + f)4)/2
CL3 + 9541P
(u;
+ W4hQ
H
3
(Hi
f Hz),‘2
@1+
W2)/2 %>/2
purpose is to prevent the neural network from extrapolating physical values. In the caseof the circular channel model, the learning and unlearning patterns are generated in the same manner. Dmax,Lax, W,, and R,, are 4.0, 10.0, 15.0 and 5aOmm, respectively, while L&,, Lmin, We, and Rhn are 1.0, 3.5, 8.0 and l-5mm, respectively. A constant of the sigmoid function U, is taken to be 0.6 for both models. The neural network is trained until the mean error of estimation for unlearned patterns reaches the ~nimum value. If these DWs are searched using the WSM without the neural network, 10651 and 6233 finite-element analyses have to be performed in the rectangular and the circular channel model cases, respectively. On the other hand, when using the WSM with the neural network approach, only 580 and 224 finite-element analyses are executed in both cases, respectively. The number of finite-element analyses denotes the sum of the number of the analyses required to prepare learning and unlearning patterns. Although a training process of the neural network takes some amount of time, its searching process is extremely short. Let us examine quasi-minimum cr-~us316 solutions. As for the rectangular channel model, the ~~~~us316 takes the mi~mum value of 174MPa at D = 1.45mm, L = 4.55 mm and W = %Smm, while it is 166MPa at D = 1.0mm, L = 4.85 mm and W = 10.0mm in the case of the circular channel model. The circular channel model is regarded as a slightly better design with respect to amax-StJS316*
Next, let us compare the sizes of the DWs of both models. The number of searched points within the DW of the rectangular channel model is 356, while that of the circular channel model is 44. Thus, the rectangular channel model has a larger DW. As seen in Fig. 14, sa~sfacto~ solutions exist in the caseof the r~~n~lar channel model, even if D is large. This may be because the cooling capability of the rectangular channel is higher than that of the circular one, and also becausethe stress concentration around the comers of the rectangular channel is not so severe. From the viewpoint of manufacturability and structural integrity, the rectangular channel model seemsmore suitable.
B2>/2
fH2 + ff31/‘2
G32 + B3)/2
W3 + H4h’2
(B3 + B4)/2
(e) Consider all combinations of D, L, W, H, B based on the list. (f) Choose the combinations that satisfy the geometrical constr~nts. These are utilized as the input data sets of unlearning patterns. In making these patterns, margins for the geometrical constraints CX,fl and 7 are taken to be 0+5mm which is less than the l+Omm used for searching DWs. This
5 CONCLUSIONS This paper proposes a novel automated structural design system using the DW search method. Three search methods of DWs are proposed, and incorporated into the present system.A multilayer neural network is applied to speedup searching DWs. Thanks to these techniques, the present systemcan searchmulti-dimensional DWs in a reasonably short processing time. Performances of the present system are clearly d~o~~a~ through the structural design of the ITER first wall.
113
Structural design using design window search
Learning representation by back-propogation errors.
ACKNOWLEDGEMENTS
This work was performed with financial support from the Grant-in-Aid of the Ministry of Education, Science and Culture of Japan. The present authors wish to thank Messrs H. Takatsu and M. Akiba of the Japan Atomic Energy Research Institute, the CRC Research Institute Ltd., and ASAHI CHEMICAL INDUSTRY for their kind help.
REFERENCES 1. Bennett, J. A. & Botkin, M. E. (eds.), The Optimum Shape (Automated Structural Design).
PlenumPress,New York,
1986. 2. Nakagiri, S. & Suzuki, K., A note on finite element synthesis of structures (Part 1): shift synthesis of vibration eigenpairs. Se&m-Kenkyu, J. Inst. Znd. Sci., 1987, Univ. of Tokyo, 39: 460-463. 3. Smith, R. G., STROBE: supported or structural object knowledge representation. Proc. 8th Znt. Joint Conf. Arti$cial Intelligence, ~~855-858, 1983. 4. Dennis, J. B., First version of a data flow procedure language. In Lecture Notes in Computer Science. Springer, Amsterdam, Vol. 19, pp. 362-376, 1974. 5. IAEA, ITER Conceptual Design Report. ITER Documentation Series 18, 1991. 6. Yoshimura, S., Yagawa, G. & Mochizuki, Y., Automation of thermal and structural design using AI techniques. Eng. Anal. Boundary Elements, 1990, 7, 73-77. I. Yoshimura, S., Yagawa, G. & Mochiiuki, Y., An artificial intelligence approach to efficient fusion first wall design. Lecture Notes in Computer Science (Computer-Aided Cooperative Product Development), Springer, Amsterdam,
pp. 502-521, 11390. 8. Yoshimura, S., Yagawa, G. & Mochizuki, Y., Automated structural design based on knowledge engineering and fuzzy control. Eng. Comput., 1995, 12, 593-608. 9. Rumelhart, D. E., Hinton, G. E. & Williams, G. E.,
Nature, 1986, 323, 533-536. 10. Manly, B. F. J., Multivariate
Statistical
Methods.
Chapman and Hall, New York, 1986. 11. Kung, S. Y., Digital Neural Networks. PTR PrenticeHall, Englewood Cliffs, NJ, 1993. 12. Funahashi, K., On the approximation realization of continuous mapping by neural networks. Znt. J. Neural Networks, 1989, 2, 183-192. 13. Bello, M. G., Enhanced training algorithms and integrated training/architecture selection for multilayer perceptron networks. IEEE Trans. Neural Networks, 1992,3,864-875. 14. Maniezzo, V. Genetic evolution of the topology and weight distribution of neural networks. IEEE Trans. Neural Networks, 1994, 5, 39-53. 15. Yagawa, G., Yoshimura, S., Mochizuki, Y. & Oishi, T., Identification of crack shape hidden in solid by means of neural network and computational mechanics. Proc. ZUTAM Symp. Inverse Problems in Engineering Mechanics,
pp. 105-108, 1992. 16. Yoshimura, S., Hishida, H. & Yagawa, G., Parameter optimization of viscoplastic constitutive equation using hierarchical neural network. PYOC. VZZ Znt. Congress Experimental Mechanics, Vol.1, pp. 296-301, 1992. 17. Yoshimura, S., Matsuda, A. & Yagawa, G., New regularization by transformation for neural network based inverse analyses and its application to structure identification. Znt. J. Numer. Meth. Engng. (in press). 18. Yoshimura, S., Saito, Y. & Yagawa, G. Identification of two dissimilar surface cracks hidden in solid using neural networks and computational mechanics. Comput. Modeling Simulation Engng., 1996, 1, 477-491. 19. Yoshimura, S. & Yagawa, G., Inverse analysis by means of neural network and computational mechanics: its application to structural identification of vibrating plate. Znverse Problems (Specialized Monograph of ICES ‘93), Atlanta Technology Publications, pp. 184-193, 1993. 20. Qishi, A., Yamada, K., Yoshimura, S. & Yagawa, G. Quantitative nondestructive examination with ultrasonic method using neural networks and computational mechanics. Comput. Mech., 1995, 15, 521-533. 21. Power Reactor & Nucl. Fuel Develop. Corp. FZNAS User Manual, Version 11.0, 1989.
Advances in Engineering Sofrwm 28 (1997) 103-l 13 0 1997 Elsevier ScienceLimited Printed in Great Britain. All rights reserved 0965~9978/97/$17.00
PII:SO965-9978(96)00049-X
Automated system for structural design using design window search approach: its application to fusion first wall design Yosbihiko Mocbizuki,
Sbinobu Yosbimura*
& Genki Yagawa
Department of Quantum Engineering and Systems Science, University of Tokyo, 7-3-l Hongo, Bunkyo, Tokyo 113, Japan
(Received 5 July 1996; accepted 3 September 1996) Some practical structures are required to be operated in severe environments where various mechanical, thermal or electromagnetic loadings occur in a complicated manner and various failure phenomena such as yielding, fracturing and melting compete among others. In designing such structures, it seemsvery useful to determine the design window (DW) that schematically indicates an area of satisfactory solutions in a permissible design space. The DW may give us more meaningful information than one satisfactory or optimum solution. However very little researchhas been performed on this topic so far. This paper describesa novel automated system for structural design based on the DW concept. The present systemconsists of three main modules and one sub-engine,that is, (a) the Analyzer, (b) the DW Search Engine, (c) the Design Modifier as the main modules and (d) a multilayer neural network as the sub-engine of the DW Search Engine. The Analyzer takes care of computational mechanics simulations for various sets of design parameters. To maintain high flexibility and extensibility, the Analyzer is constructed using an object-oriented knowledge representation and a data-flow processing technique. Some DW search methods are incorporated into the DW Search Engine. Using neural network technology, DWs are searched very efficiently. The Design Modiier has the role of tinding one satisfactory solution, which is given to the DW Search Engine so that some search methods can start searching. To demonstrate the practical performance of the present system, it is applied to designone of the typical structures to beoperated in a severeenvironment, i.e. the first wall structure of ITER (International Thermonuclear Experimental Reactor). 0 1997 Elsevier Science Limited. All rights reserved.
1 INTRODUCTION
This paper describes a novel automated system for structural design using a DW search approach. The present system consists of three main modules and one sub-engine. The main modules are (a) the Analyzer, (b) the DW Search Engine and (c) the Design Modifier,
Due to the progress of computational mechanics technology, e.g. the finite-element analyses, the accuracy and reliability of analyses of a uni-phenomenon, such as structural deformation, heat conduction or fluid dynamics, have been dramatically improving. In the field of structural design problems, structural optimization can often be solved based on mathematical methods combined with the computational mechanics.’ In practical design problems, it is more useful to obtain the design window (DW) that schematically indicates an area of satisfactory solutions in a permissible design space. Several methods to obtain one satisfactory solution have already been proposed.2 However very little research on. DWs has been performed so far.
while the sub-engine of the DW Search Engine is a multilayer
neural
network.
The Analyzer
performs
coupled finite-element simulations for various sets of design parameters. To maintain high flexibility and extensibility, the Analyzer is constructed using an object-oriented knowledge representation3 and a dataflow processing technique.4 Some search methods are incorporated
into the DW Search Engine. Using the
neuro Analyzer, DWs are searched very efficiently. The Design Modifier has the role of finding one satisfactory solution, which is given to the DW Search Engine so that some DW search methods can start searching. To demonstrate the practical performance of the
*Author to whom correspondence should be addressed. 103
104
Y. Mochizuki, S. Yoshimura, G. Yagawa
present system, it is applied to design one of the typical structures to be operated in a severe environment, i.e. the first wall structure of ITER (International Thermonuclear Experimental Reactor).’
(b) generate a finite element mesh (c) attach boundary conditions. In a main-process: (d) perform the finite element analyses.
2 SYSTEM
In a post-process:
CONFIGURATION
Figure 1 shows a schematic configuration of the present system. In the following sections, fundamental functions of the three main modules and the sub-engine of the present system are described in detail. 2.1 Analyzer
In the present Analyzer, an object-oriented knowledge representation technique is adopted to store several knowledge modules related to analyses of structures subjected to various loadings. A data-flow processing technique is also utilized as an inference mechanism among the knowledge modules. 2.1.1 Object-oriented knowledge representation An analysis process in automatic structural design usually consists of many elemental procedures as follows. In a pre-process after design parameters are given: (a) define an analysis domain
(e) extract some necessary physical values from the output data evaluate sensitivities of an objective function with 0 respect to the design parameters. The elemental procedures above are iterated by changing the design parameters. Each elemental procedure can be regarded as an analysis module here. This characteristic seems very compatible with the objectoriented knowledge representation. Figure 2 shows the schematic view of mesh generation object, FW-mesh. Figure 3 shows the same object expressed by LISP. Here, KCL (Kyoto Common Lisp) is employed. As shown in the figures, each object in the Analyzer consists of input/output slots and a functioncall or a method slot. Each object plays the role of an expert for each elemental procedure. In the present Analyzer, a number of objects are stored as shown in Fig. 4. 2.1.2 Data-flow processing The data-flow processing technique is utilized as an
Knowledge Engineering Techniques - Object-oriented Knowledge Representation * Data-flow Processing
Method of Multi-dimensional * Whole-areaSearchMethod (WSM) - Boundary Swelling Method (BSM) . Boundary Tracing Method (BTM)
* Moment Method
Fig. 1. Schematic configuration of the present system.
Structural design using design window search
RN-mesh input f--zicinterchannel
1m A
A
file
Fig. 2. Schematicview of PW-meshobject. inference mechanism controlling a global data stream among objects. Taking Figs 2 and 3 as an example, let us explain its fundamental concept. As soon as all the input data are prepared on the input slots and a flag of ‘not ready’ turns to ‘ready’ in Fig. 2, the function-call slot sends a ready-command to the outer processors. In the figure, the outer processor is a two-dimensional finite-element mesh generation code. After a mesh is generated, the output data are prepared on the output slots. Then, the data on the output slots are immediately s#entto any objects that need the data. Owing to the two techniques described above, all the objects are kept to be independent of one another, and the Analyzer succeedsin maintaining high flexibility and extensibility.6)7 2.2 Design modifier The Design Moldifier finds one satisfactory solution.
W-mesh (input (inter-channel (data scalar) (type design-prm) 0 notready (input-group 1)) (data scalar) (type design-prm) (thick-RN 0 notready (Inputgroup 1)) (dlstJW (data Scalar) (type design-prm) 0 notready (inputqroup 1)) (Idata scalar) (type deslgngrm) (helghtSH () notrwdy (input-group 1)) (data scalar) (type deelgngrm) (length-CH () notready (Input-group 1))) (output (mesh6JW (“mesh8.FW” file) (type 8-nodeJW_mesh))) (func (VW-mesh” ( thick_FW inter-channel dist_FW height_CH length-CH ) command-mode)) (group-level 20))
Fig. 3. F\V’_meshobjectexpressedby LISP.
105
Starting from the solution, the DW Search Engine searchesa DW. Finding a satisfactory solution is called a feasible design. The DW Search Engine basically employs the ‘generate and test’ strategy. Here, design parameters have to be somehow modified within a design space when a former design candidate does not satisfy any of the design criteria. As one of the efficient techniques for design modification, much attention has been paid to mathematical approaches based on numerical sensitivity analyses. However, application of such approaches has been limited only to simple problems, such as shape design of simple structures subjected to mechanical loading. This is because the mathematical design modification is often trapped at one of the locally optimum points in complicated design problems. Thus, the present authors proposed a design modification approach based on experts’ empirical knowledge related to structural design in the previous studies.6-g One of the examples of experts’ empirical knowledge is that “if the maximum temperature occurred in a structural wall exceeds a permissible value, the wall thickness should be thinner”. These kinds of IF-THEN rules, which have been derived from experts’ qualitative inference, instruct how to modify a former design, considering the reason why the former design candidate violates design criteria. The empirical design modification approach combined with the fuzzy control* is incorporated into the Design Modifier described in this paper. 2.3 DW search engine and neuro analyzer 2.3.1 D W search engine As shown in Fig. 5, several large or small DWs may exist in a design space of practical structures. Some of them may be doughnut-shaped. If possible, one wishes to find all the DWs. In the present study, the original design spaceis divided into the following two subspaces,i.e. the permissible design space and the impossible design space.Here, the permissible design spaceis the subspace where all geometrical constraints are satisfied, while some of the constraints are violated in the impossible design space. The DWs are searched only within the permissible design space. The following three search methods are considered here: (a) the Whole-area Search Method (WSM), (b) the Boundary Swelling Method (BSM) and (c) the Boundary Tracing Method (BTM). These search methods are implemented in the DW Search Engine with the C language. The WSM does not need to know an initial satisfactory solution a priori, while the BSM and the BTM do. The principal characteristics of the three methods are summarized in Table 1. In the present paper, only the WSM is explained becausethe method can search plural multi-dimensional DWs even if any of the DWs to be searchedare plural or doughnut-shaped, and also because quasi-optimum
106
Y. Mochizuki,
S. Yoshimura, G. Yagawa
heal_FEM-Finas
:
stress-FEM-Flnas
:
heat-FEM-Post-Proc
:
srress-FEM-Post-Proc
:
heat_ifile-Fi%as
:
stress~ifi1e~Fina.s
:
FW-mesh
:
mat-dbaseprmor-CFC
:
mat-dbase-F W-SUS316 :
an object that performs 2D heat conduction analysis using the FEM code, i.e. FINAS an object Ihatperforms20 stressanalysis with rhermal effects using Ue FEM code,Le. FINAS an object that extracts temperature valuesfrom the output fJe made by heat_FEM_Finas object an object tbatfind the maximum equivaknt stress using the outputjile made by stress-FEM-Finas object an object that creates an input data fire for heaLFEM_Finns object an object that creates an input &ta j?k for stress-FEM-Finas object an objectteal generates20 FEM mesh, i.e. eight-noded isoparametrk mesh an object that manages the database of mater&l properties of the armor an object that manages the databaseof material
properliesof the mothermaterial :
designgrm Object
an object Ihat produces shape designparameters and boundary conditions
Dam Stream
Fig. 4. Objectnetwork and function list. solutions are obtained simultaneously. The algorithm of the WSM is as follows. At first, a multi-dimensional lattice that is empirically determined by engineers is generated in the design space as shown in Fig. 6. It is then examined one by one at all the intersection points of the lattice whether the points are inside DWs or not. During searching DWs a distribution of objective function over the permissible design space is also obtained if necessary. Though the WSM has several advantages as listed in Table 1, the number of points to be examined tends to become extremely huge. To apply the WSM to practical design problems, one has to solve such an intrinsic problem of the method. For this purpose, we utilize a neural network technology.’ Before explaining it in detail, we explain a process to separate Permissible Design Space
plural DWs from a set of satisfactory design points searched by the WSM in the following. In the case of two- or three-dimensional DWs, the DWs can be easily separated through visualization. When the number of design parameters is more than four, the DWs can not be visualized. The cluster analysis that is one of the multivariate statistical methods is employed here to separate plural DWs. Among several clustering methods, the hierarchical method,” i.e. the most neighboring method, is employed in the present study. Its analysis flow is as follows:
(4 Transform the coordinates of the searched satis@I (4 04
@>
factory design points so that all the unit steps for searching become 1.O. Calculate all the Euclidean distances between one satisfactory design point and any other ones. Assign each satisfactory point a group that consists of itself. Union two groups if the minimum distance between a point belonging to one group and a point belonging to the other group is less than Ji;, where n is the number of design parameters. Iterate the procedure (d) until all the points are checked.
2.3.2 Neuro analyzer
In this section, a multilayer neural network utilized as the Neuro Analyzer is first explained, and then the principle of the automatic DW search method using the neural network is described. Design Parameter 1 Fig. 5. Schematicview of DWs in two-dimensionaldesign parameterspace.
Network architecture and learning algorithm of mtdtilayer neural network. Figure 7 shows a unit of a
neural network consisting of multiple input slots and a
107
Structural design using design window search Table 1. Main features of three methods of searching DW In the case that an initial satisfactory In the case that an solution is known initial satisfactory solution is unknown Boundary Tracing Boundary Swelling Whole-area Method Method Search Method @TM) @SW W’SW Dimension of DW that can be searched
L2
22
2
possible
possible
impossible
possible
sometimes possible
impossible
Determination of quasi-optimum solutions
possible
possible
impossible
Number of
too large
large
small
Searching of doughnut-shaped DW Searching of plural DWs
searchedpoints
single output slot. The relation between the input and output data is formulated as follows: Oj =f(Uj)
= l/(1
Uj = k
WjiIi - 0j
(1)
+eXp(-2Uj/Uo)}
(2)
i=l
where Oj is the real-valued output of the jth unit, Uj the total input to the Jth unit, f( ) the activation function, i.e. the sigmoid function here, Us the temperature constant of the sigmoid function, IVii the connection weight between the ith and the jth units, Ii the input from the ith to thefih units, and 0, the bias value of theflh unit, respectively. Figure 8 illustrates an ordinary three-layer network.
2D DesignWindow
\v: satisfactorysolution 0 unsatisfactorysolution Fig. 6. Illustration of Whole-Area Search Method for design window.
All the units are formed into multiple
layers, i.e. an
input layer, intermediate (hidden) layer(s) and an output layer, with only feedforward connection between successive layers. The basic idea for training the neural network, i.e. a supervised learning algorithm is as follows. At first the following training error E is defined:
(3) k=l
A
where Ep is the square error for the pth learning pattern, Tpk the teacher signal to the kth unit in the output layer for the pth learning pattern, o,k the output signal from the kth unit in the output layer for the pth learning pattern, and n the number of output units, respectively. In the training process, the connection weights Wji and the basis 0, are modified repeatedly based on the gradient descent method in order to minimize the above square error. Since this modification proceeds downwards in Fig. 8, the training algorithm is called the back propagation. In the present study, the backpropogation algorithm combined with the momentum method is employed to attain stable convergence in
I, \Wil I* j* . 13w3. Qjf .. 91, ’ jr
Oj
Fig. 7. Schematic view of neuron unit.
108
Y. Mochizuki,
S. Yoshimura, G. Yagawa
Trslning Patterns : ( I pt -Ipn)vs(Tpt-Tpt);p=l-r Tpt ) TM 1 * Tw+ Teacher Signal l
Fig. 8. Three-layerneural network. training process.9*1’Through such an iterative training process, the network attains an ability of outputting the similar signal to the teaching one. It is theoretically proven that the three-layer neural network can approximate any continuous mapping. l1,12 However, the network has some limitations in reality due to poor convergence in training processeswhen the numbers of hidden layers and units increase. In the present study, an appropriate size of the network is determined through trial and error as in many practical applications, although several interesting studies on automatic determination of network size have been performed.131’4 The present authors have already applied the m~tilayer neural network to several inverse analyses successfully.15-2o iMain features of multilayer neural networks. The attractive features of the multilayer neural network can be summarized as follows:
because both thermal and stress analyses have to be executed. Thus, the neural network aided search method is proposed in the present study. As shown in Fig. 9, this method consists of three phases.At first, finite-element analysesare performed to prepare a number of learning data sets and unlearning ones by parametrically varying design parameters. The unlea~ng data sets are utilized to prevent the neural network from ‘overlearning’, which is such a phenomenon that estimation error for unlearning data setsstarts increasing even though the learning process advances. Next, a back-propogation neural network is trained using a number of learning data sets prepared in the previous phase. Here design parameters are given to the input units of the network, while physical values, such as the maximum temperature and equivalent stress, are given to its output units as teacher signal. After a sufficient number of training iterations, the network is expected to imitate a response of the original finiteelement analyzer. In the final phase, multi-~mensional DWs are searched quickly using the trained network. In the present search method, the finite-element analyses are performed only in phase 1. The number of the finite-element analysesrequired is much smaller in this method than in the search method without using the neural network. As the well-trained network calculates physical values very quickly, even the WSM can be executed in a reasonable processing time. 2.4 System en~o~ent The main portions of Analyzer, Design Modifier and DW Search Engine including Neuro Analyzer are
{a) One can automatically construct a nonlinear mapping function from multiple input data to multiple output data within the network through a learning process of some or many learning patterns. (b) The network has a feature of so-called ‘generalization’, i.e. a kind of interpolation, so that the well-trained neural network estimates appropriate output data even for unlearned patterns. (c) The trained network operates quickly in an power application process. Computational required for operating the trained network may be equivalent to only that of a personal computer. Principle of automatic D W Search using a neural network. In searching DWs, whether a searchedpoint is
is checked through a satisfactory solution computational mechanics simulations, e.g. the finiteelement analyses.However, such detailed calculations at every searchedpoint are very time-consuming, especially in designing high-temperature structural components
Fig. 9. Procedureof DW searchusing neural network.
109
Structural design using design window search implemented on one of the popular engineering workstations, Sun SPARCstation ELC. The finite-element analyses are executed on one of the Graphic workstations, Titan STlOOO,which is connected to the ELC through the Ethernet LAN of the University of Tokyo (practical speed is 10kBPS). As for the finite-element analyses, one of the general purpose finite elements codes, FINAS” is employed. 3 DESIGN
MODELS
AND DESIGN
2
Cl-plasma zr hEt”5Gv 2 A Free Slip Condition
Q-blanket b--Jw
CRITERIA
The ITER (International Thermonuclear Experimental Reactor)’ is being designed through a co-operative work among Japan, EU, USA and the former USSR. Its Conceptual Design Activities (CDA) were completed in December 1990. Since then, ITER Engineering Design Activities have been performed in order to collect information necessary for detailed design, R&D and construction of ITER. The ITER first wall currently has two candidate designs regarding a cooling channel, i.e. rectangular and circular channels. Detailed explanation of the ITER project can be found elsewhere.5The present system is applied to search DWs for both models. The design models are shown in Fig. lO(a,b). The base material of the wall is subjected to membrane tensile loading, F, which might be caused by electromagnetic loading and pressure from breeder blanket, and to surface heat loading, Q-blanket, on the blanketside surface, and to volumetric heat loading, Q,-SUS316 (= 20.0 MW/m3). Magnitudes of F and Q-blanket will be given later. The armor material of the wall is subjected to surface heat loading, Qglasma (= 0.15 MW/m*), on the plasma-side surface, and to volumetric heat loading, Q,-armor (= 6.0 MW/m3). The temperature and pressure of cooling water are 100~0°C and 1.5MPa, respectively. The heat-transfer coefficient between the armor and the base materials is 1000.0W/ m*‘C, and that between the cooling water and the base material is 20,000*0W/m*“C. The type 316 stainless steel and CX-2002U are utilized as the base material and the armor material, respectively.The wall accompaniedby the armor is modeled with eight-noded isoparametric elements. A static thermal conduction analysis is performed. In an elastic thermal stressanalysis, only the basematerial is analyzed under the generalized plane strain condition. The finite element analyses are performed using the object-oriented Analyzer described previously. Both geometrical constraints and failure criteria are employed here as design criteria. The geometrical constraints are as follows: For the rectangular cooling channel model: D2a L>H+P WZD+B+y
Membrane Force F
(4)
I
(a) Rectangularcooling channelmodel. Membrane Force F QglasmaT
hrf Q-blanket W
-I
(b) Circular cooling channelmodel. Fig. 10. Designmodelsof ITER f&t wall. For the circular cooling channel model: D>a: LLR+P W>D+2R+y
(5) where D, L, W, H, B and R are the geometrical design parameters as shown in Fig. lO(a,b), while QI,,f3and y are the design margins that are empirically decided by engineers. The failure criteria employed are as follows: (a) TInax-Armor < ~OLinllor
(ha)
@)
(6b)
~max-SUS316
cc) ~max-SUS316
<
TO-SUS316
< gOo_SUS316
(k)
is the maximum temperature in the Gax-~or armor material portion, TO-Armor a permissible temperature for the armor material, Tm-SUS316 the maximum temperature in the base material portion, T0-sos3i6 a permissible temperature for the base material, omaxsus3i6 the maximum equivalent stress in the base material portion, and cOo_SUS316 a permissible equivalent where
110
Y. Mochizuki,
S. Yoshimura, G. Yagawa
stress for the base material, respectively. For the purpose
of simp~ck
TO&hor~
TO&SUS316 and
gOo_SUS316
are taken to be 1000~0,4OO*O”Cand a yielding stress of type 316 stainless steel, respectively.
4 RESULTS
I
Unlearned Patterns
AND DISCUSSIONS
4.1 Preliminary examination
Fundamental performances of the present DW search method are examined through searching (L - W) DWs of the rectangular cooling channel model. Here, D, H and B are fixed to be 2.0, 35 and 5.0 mm, respectively. Design margins for the geometrical constraints ,0 and y are taken to be 0.5. As for boundary conditions, F and Q-blanket are assumed to be 49*ON and 0.0 MW/m’, respectively. The multilayer neural network employed is of the three-layered type. The network has 2 input units, 40 hidden units and 1 output unit. The two input units correspond to the two geometrical design parameters L and W. One output unit is prepared to output the It should be maximum equivalent stress,i.e. oma-SUS316. noted here that the output units for Tmaxedor and Tmax-sus316are not prepared in this case because the temperature criteria of eqn (6a) and (6b) are always satisfied under the given boundary conditions. The number of 40 for hidden units seems rather large compared with the numbers of input and output units. This number is selectedon purpose to examine effectsof ‘overlearning’. Figure 11 illustrates 16 sets of the input data of learning patterns and 9 sets of the input data of unlearning patterns in the L-W design space. All the input data and output data are normalized to a unit range from 0.01 to 0.99. Figure 12 shows the history of learning process in the case that a constant of the Learning Patterns
Unlearning Patterns
18.0 17.0 16.0 15.0 Ts
14.0
3
11.0 10.0 9.0 8.0 3.5 4.0 4.5 5.0
5.5
6.0 6.5 7.0 7.5
8.0
L (x lo9 m)
Fig. 11. Learning and unlearning patterns in L-W design
parameterspace.
Iteration Number of Learning
Fig. 12. Training historiesof learnedand unlearnedpatterns.
sigmoid function U. is taken to be 0.6. Here the following mean error of estimation is employed for both learning and unlearning patterns: Mean-Error = f 2
)Tp - 0,)
(7)
p=l
where r
Tp 0,
is the number of learning or unlearning patterns is the correct solution of pth learning or unlearning pattern is the output data of pth learning or unlearning pattern
Overlearning is clearly observed in the figure. To examine the influence of overlearning to DWs quantitatively, DWs are searched using the following two kinds of trained neural networks: A:
B:
The network which adopts the connecting weights and the bias values at 226 learning iterations when the mean error of estimation for the unlearning patterns reaches the minimum value. The network that adopts the connecting weights at 5000 learning iterations when overlearning is fairly advanced.
Figure 13 shows the DWs obtained using the two trained neural networks above. The regions surrounded by chained open squares indicate the correct DWs that are searched using the Boundary Tracing Method (BTM) without the neural network, and the regions filled with plus-symbols show the DWs that are searched using the present Whole-area Search Method (WSM) with the neural network. Here the unit steps for searching AL and A W are taken to be 0.075 and 0*25mm, respectively. It can be easily seen from the figures that the DW obtained with the network A agrees well with the correct one, while that of the network B is rather distorted. Such less accuracy in the latter case may be caused due to overlearning. This comparison
Structural design using design window search
; ... -.. ..- . . ..
0 Minimum dma E::::.... !!&m Solution 1, ............... -a ........................ ........................... ............................... II y;:::::::::::::::::::::::::::::::.f’zgg ...................................... wlli ......................................... ......................................... .. ....................................... Ia . ............................................ 0.. ........................................ mm .............................................. .............................................. .a .............................................. .m ............................................... ,I.. .......................................... ........................................... mm .,*. . ........................................ m .................................... II I ..................................... I.. .............................*- . ......................... .................... ..I .............. I iize--=
16.0 15.0. g
14.0.
2
13.0
a 3
12.0.
clearly suggests that it is very important to utilize the network that stops learning when the mean error of estimation for unlearning patterns reachesthe minimum value. Open circles in Fig. 13(a, b) indicate the quasiminimum values of omasUS316. By using the WSM, the distribution of objective function over the searched DW and the quasi-optimum solution are simultaneously obtained.
D FEM + Nauro
17.0.
11.0. 10.0. 9.0 6.OLL
3.5
4.0
4.5
5.0
6.0
5.5
6.5
7.0
111
.
’
7.5
6.0
4.2 Comparison betweenrectangular and circular cooling channel models based on DWs
L(xlO%)
(a) L-W DW searchedby neurotraineduntil 226 learning iterations whenmean-errorof estimationfor unlearnedpatternsis minimized.
IfJ.O(---
0 FEM
I
+ Neuro 0 Minimum dmax
17.0. 16.0 15.0 p14.0. 0
13.0.
25 12.0'
3
11.0' 10.0. 9.0 . s.oL 3.5
a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I” . . . . . . . . . . . . . . . . . . . . . . . . . *.. 1,., . . . . . . . . . . . . . . . . . . . I . . . . . . . . *. . . . . I,..... *. . . . -e......, I*.... -1:::’ mDlmQG‘I””
. 4 0 4.5
5.0
5.5
L(x 10
6.0
6.5
m mu7
7.0
7.5
1 6.0
%l)
(b) L-W DW searchedby tbe over-trainedneurountil 5000learning iterations.
Fig. 13. Effects of overlearning of neural network on DWs.
In this section, let us compare the rectangular and the circular cooling channel models in Fig. 10, taking three dimensional DWs of D, L and W. Figure 14 shows the three-dimensional DWs obtained for both models. The unit steps for searching AD, AL and A W are taken to be O-15,0.15 and 0.5 mm, respectively. Surface areas of both cooling channels are chosen to be identical. That is, the surface area of the rectangular model A,,,,,ti,, (= 2H + B = 2 x 3.5 + 5.0mm) and that of the circular model Acircular(= rR = 7r x 3.82mm) are taken to be 12.0mm. As to boundary conditions, F and Q-blanket are 9.8 N and 0.15 MW/m2, respectively. Q, /I and y are taken to be l.Omm. Both DWs are searched using the WSM combined with the neural network approach. Ordinary three-layer neural networks are employed. The network to design the rectangular channel model has 5 input units, 20 hidden units and 2 output units, while that of the circular channel model has 4 input units, 20 hidden units and 2 output units. D, L, W, H and B are the input data for the rectangular channel model, while D, L, Wand R are those for the circular one. The two output units output the maximum equivalent stress and the maximum temperature of the base material. The output unit for TmaxArmor is not considered here
Bracketedcoordinates denote(D, L, W). The unit is [mm].
‘thenumber of searchedpoints in DesignWindow is 356. Ihe minimum vdue of Chax is 174.05[Mpal at D=1,45[mm], L=4.55[mm] and W=8.5[mm].
The number of searchedpoints in Design Widow is 44. The minimum value of Omanis 166.35[Mpa] at D=l.O[mm], L=4.85[mm] and W=lO.O[mm].
Fig. 14.Comparison of three-dimensional DWs between rectangular cooling channel model and circular one. (a) DW of rectangular channel model. (b) DW of circular channel model.
because the temperature criterion for the armor material, i.e. eqn (Sa), is always satisfied under the given bounda~ conditions. Next, let us describe the learning and unlearning patterns employed. In the case of the rectangular channel model, 460 learning patterns and 120 unleaming ones are adopted, while 168 learning patterns and 56 unlearning ones are utilized in the case of the circular channel model. As for the rectangular channel model, the input data sets of learning patterns are prepared as follows: (a) Divide the interval between the maximum and the minimum values of each design parameter into three subsections as follows:
Here, r>-, Lmax, Wma, H,,, and Bmaxare 4.0, 10.0, 150, 95 and lo-Omm, respectively, while Dkn, Lh,,, Wtint &-,,A,, and &in are 1~0,35, 8.0, l-5 and 3*Omm, respectively. (b) Consider all possiblecombinations of D, L, W, H, B. (c) Choose the combinations that satisfy the geometrical constraints. These are utilized as the input data sets of learning patterns. On the other hand, the unlearning patterns are prepared as follows: (d) Make the following list of each design parameter using the learning data above: D
L
W
PI
+ D2)/2
CL1 + J52>/2
( w1+
Pi!
+ D3)P
G2 + L3>/2
cry,
f
CD3 + f)4)/2
CL3 + 9541P
(u;
+ W4hQ
H
3
(Hi
f Hz),‘2
@1+
W2)/2 %>/2
purpose is to prevent the neural network from extrapolating physical values. In the caseof the circular channel model, the learning and unlearning patterns are generated in the same manner. Dmax,Lax, W,, and R,, are 4.0, 10.0, 15.0 and 5aOmm, respectively, while L&,, Lmin, We, and Rhn are 1.0, 3.5, 8.0 and l-5mm, respectively. A constant of the sigmoid function U, is taken to be 0.6 for both models. The neural network is trained until the mean error of estimation for unlearned patterns reaches the ~nimum value. If these DWs are searched using the WSM without the neural network, 10651 and 6233 finite-element analyses have to be performed in the rectangular and the circular channel model cases, respectively. On the other hand, when using the WSM with the neural network approach, only 580 and 224 finite-element analyses are executed in both cases, respectively. The number of finite-element analyses denotes the sum of the number of the analyses required to prepare learning and unlearning patterns. Although a training process of the neural network takes some amount of time, its searching process is extremely short. Let us examine quasi-minimum cr-~us316 solutions. As for the rectangular channel model, the ~~~~us316 takes the mi~mum value of 174MPa at D = 1.45mm, L = 4.55 mm and W = %Smm, while it is 166MPa at D = 1.0mm, L = 4.85 mm and W = 10.0mm in the case of the circular channel model. The circular channel model is regarded as a slightly better design with respect to amax-StJS316*
Next, let us compare the sizes of the DWs of both models. The number of searched points within the DW of the rectangular channel model is 356, while that of the circular channel model is 44. Thus, the rectangular channel model has a larger DW. As seen in Fig. 14, sa~sfacto~ solutions exist in the caseof the r~~n~lar channel model, even if D is large. This may be because the cooling capability of the rectangular channel is higher than that of the circular one, and also becausethe stress concentration around the comers of the rectangular channel is not so severe. From the viewpoint of manufacturability and structural integrity, the rectangular channel model seemsmore suitable.
B2>/2
fH2 + ff31/‘2
G32 + B3)/2
W3 + H4h’2
(B3 + B4)/2
(e) Consider all combinations of D, L, W, H, B based on the list. (f) Choose the combinations that satisfy the geometrical constr~nts. These are utilized as the input data sets of unlearning patterns. In making these patterns, margins for the geometrical constraints CX,fl and 7 are taken to be 0+5mm which is less than the l+Omm used for searching DWs. This
5 CONCLUSIONS This paper proposes a novel automated structural design system using the DW search method. Three search methods of DWs are proposed, and incorporated into the present system.A multilayer neural network is applied to speedup searching DWs. Thanks to these techniques, the present systemcan searchmulti-dimensional DWs in a reasonably short processing time. Performances of the present system are clearly d~o~~a~ through the structural design of the ITER first wall.
113
Structural design using design window search
Learning representation by back-propogation errors.
ACKNOWLEDGEMENTS
This work was performed with financial support from the Grant-in-Aid of the Ministry of Education, Science and Culture of Japan. The present authors wish to thank Messrs H. Takatsu and M. Akiba of the Japan Atomic Energy Research Institute, the CRC Research Institute Ltd., and ASAHI CHEMICAL INDUSTRY for their kind help.
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