Struct Multidisc Optim (2010) 41:699–709 DOI 10.1007/s00158-009-0448-3
RESEARCH PAPER
Gradientless shape optimization using artificial neural networks Krishna K. Pathak · D. K. Sehgal
Received: 3 January 2007 / Revised: 1 October 2009 / Accepted: 6 October 2009 / Published online: 12 November 2009 c Springer-Verlag 2009
Abstract In this paper a new zero order method of structural shape optimization, in which material shrinks or grows perpendicular to the design boundary, has been proposed in order to satisfy fully stressed design criteria. To avoid mesh distortion that results in undesirable shape, design element concept and for nodal movement and convergence checking, fuzzy set theory have been used. To accelerate the convergence, artificial neural networks are employed. The proposed approach, named as GSN technique, has been incorporated in a FORTRAN software GSOANN. Using this software shape optimization of four structures are carried out. It is demonstrated that proposed technique overcomes most of the shortcomings of mundane zero order methods. Keywords Shape optimization · Finite element · Neural network · Fuzzy set · Design element · Zero order method
K. K. Pathak (B) · D. K. Sehgal Advanced Materials and Processes Research Institute (CSIR), Bhopal, Madhya Pradesh, 462064, India e-mail:
[email protected] D. K. Sehgal Department of Applied Mechanics, Indian Institute of Technology, New Delhi, 110016, India
1 Introduction Most of the shape optimization problems reported in the literature are solved using gradient-based (Zienkiewicz and Campbell 1973; Queau and Trompette 1980; Bennett and Botkin 1985; Braibant and Fleury 1984; Pourazady and Fu 1996) or gradientless methods. In gradientless methods of structural shape optimization gradients of structural response like displacements, stresses, frequencies etc. are not required that is why they are sometimes called ‘zero order methods’. In these methods stress results are directly used for shape modifications (Jose 1990; Sibal 1992). Some important literature on direct zero order method, are as follows: The work of Mattheck and co-workers (Mattheck and Burkhardt 1990; Mattheck and Moldenhauer 1990; Mattheck et al. 1992) is based on the interesting observation that ‘living structures’ appear to be able to add material in regions of high stresses and to reduce material in regions of low stresses to bring about an optimal shape that produces a constant von Mises stress distribution on the free surface. To mimic this process for engineering structures, they used an iterative approach where a fictitious adhesive layer is added to the boundary and is allowed to swell or contract in proportion to previously determined boundary stresses for the case with no adhesive layer. Hasengawa (1992) proposed two zero order methods namely, boundary changing methods, in which co-ordinates were changed, and thickness changing methods, in which thickness was changed. Heller et al. (1999) used a direct method to interface with a finite element code and optimised several examples including a adhesively bonded lap joint using this technique. Zhixue Wu (2005) presented an efficient zero order shape optimization approach for minimizing stress concentration factor. The design boundary is modeled by using cubic
700
K.K. Pathak, D.K. Sehgal
splines, which are determined by a number of control points. It is observed from the literature that zero order approaches have two major drawbacks: 1. Since co-ordinates of boundary nodes are directly taken as design variables without any consideration of boundary smoothness, it may create sharp corners which in turn may result in erroneous shape. 2. Since manual mesh refinement is carried out after few iteration, the optimization process may take a lot of time. To avoid above drawbacks of zero order method a new integrated zero order approach has been developed in this study. In this, to obtain smooth profile design element concept has been used. An automatic mesh generator is developed to avoid manual mesh refinement. Fuzzy set theory is used for nodal movement and prediction of the convergence. It has been also observed that even proposed zero order shape optimization technique takes lots of iterations to converge. To overcome this, artificial neural networks and fuzzy set theory have been clubbed up with this method. This new approach is named as GSN (Gradientless Shape Optimization Using Neural Networks) technique. Based on this a FORTRAN code, GSOANN, has been developed for shape optimization of two-dimensional structures. Four structural problems are successfully optimized using this software.
2 Integrated zero order method In this study, an integrated zero order approach of structural shape optimization is developed which has following features1. Use of minimum numbers of design variables. 2. Convenient geometric representation of boundary. 3. Incorporation of automatic mesh generation, remeshing and refinement capabilities. 4. Efficient optimization procedure.
Fig. 1 Design elements
should be updated for new boundary shape and loading conditions. This suggests that design variables which control the optimization model should also control the finite element mesh. For this design element concept (Imam 1982), along with isoparametric mapping (Krishnamoorthy 1994; Zienkiewicz and Taylor 1991) is used. The region of the structure to be modified during optimization process is defined by design element which contains a part of the mesh. Each design element is defined by a set of key nodes called master nodes which control the geometry of a design element. Boundaries of the design element are defined by using two dimensional isoparametric interpolation functions. Master nodes lying on the boundary, to be optimized, are considered as design nodes and their co-ordinates as design variables. For example, In Fig. 1, coordinates of the nodes 1 to 5 are design variables. Based on these five nodes finite element mesh is generated (Fig. 2). Each design element has eight master nodes and coordinates of generated finite element nodes are obtained with help of them using isoparametric mapping. If X and Y are the coordinates of generated nodes then: X= Y=
Nm i=1
Nm i=1
Ni (ξ , η) · Xi Ni (ξ , η) · Yi
In sections to follow, important features of the proposed integrated zero order approach, are described. 2.1 Finite element model generation The main concern in shape optimization is the change of shape in each iteration. It is obvious that older mesh is no longer valid for stress analysis. The finite element mesh
Fig. 2 FE mesh
(1)
Gradientless shape optimization using artificial neural networks
701
where Nm is the number of master nodes of a design element, Ni is the shape function, ξ and η are natural coordinates corresponding to the point (X, Y) and Xi and Yi are the coordinates of the master node i. It is obvious that change in the geometry is reflected from the modified coordinates of these master nodes. After each iteration the new coordinates of these master nodes are transferred from the optimizer to the mesh generator to create the new mesh. In this way smoothness of the boundary is maintained during optimization. Using these master nodes any degree of refinement can be carried out. In problems of stress concentration lowest order elements should not be used because of their erratic performance. Quadratic displacement function elements are a good compromise between accuracy of results and simplicity of usage. In this study nine noded Lagrangian element has been used for FE modeling. 2.2 Move strategy In zero order or gradientless method of shape optimization stress values at boundary nodes play important role in geometry modification for the next iteration. In fully stressed design criterion of shape optimization, in ideal case, stresses at all design nodes should be equal to the target stress. But it is rare to achieve this state of stresses which are more or less in fuzzy form. In proposed technique nodal movement of the geometry has been carried out with help of fuzzy set theory. The objective is to find the shape in which stresses are closest to the target value. To achieve this, the concept of fuzzy membership (Zimmermann 1996) is utilized. Two most prominent types of membership functions are, (a) triangular shape and (b) bell shape functions. If stress is equal to target stress σt , the membership value μ is equal to 1, otherwise it varies linearly or non-linearly according to the function chosen. The membership variation for both functions are as follows:
Fig. 3 Triangular function
(b) Bell shape function This membership function vary in non-linear fashion (Fig. 4). One of such curve is mathematically represented by1 μ (σ) = 1+ (σ − σt )4
Material addition and deletion is directly related to the nodal stress values. If stress is more than the target stress material should be added and if less material should be deleted. This concept can be easily incorporated through fuzzy membership concept in the sense that nodal movement of ith node is assumed to be proportional to move factor (MF) given by: If σi < σt
MF = 1 − μ (σi )
If σi > σt
MF = μ (σi ) − 1
μ (σ) =
if σ < σt
1
if σ = σt
2 − σ/σt 0
if σt < σ < 2σt
(2)
if σ > 2σt
where σt is the target stress.
(4)
The move factor vs stress plot for both types of membership functions are shown in Figs. 5 and 6.
(a) Triangular shape function The membership function value, μ(σ), varies linearly in this curve (Fig. 3). Mathematically it can be represented byσ/σt
(3)
Fig. 4 Bell shape function
702
K.K. Pathak, D.K. Sehgal
Fig. 5 Move factor—triangular shape
In Fig. 7, move directions of the design nodes are shown. Node i is a design node and j is direction node. The middle nodes of the Lagrangian elements are chosen as direction nodes. The minimum distance between all design and direction nodes is denoted by Lmin . Mathematically: Nd
Lmin = min (Li )
(5)
i=1
Li =
xi − xj
2
+ yi − yj
2
And Nd is the numbers of design nodes. For success of shape optimization process, nodal movements should not be very large as it may distort the geometry leading to undesirable shape. For this reason maximum nodal movement in any iteration has been restricted to 0.25 Lmin . Now movements of the design nodes are given by: mv (i) = 0.25Lmin .MF
Let xi , yi be new coordinate of (xi , yi ) and (xj , yj ) be coordinate of direction node then xi = xi + xj − xi .mv (i) / Li yi = yi + yj − yi . mv (i) / Li
where,
Fig. 7 Direction node
(6)
Based on the nodal stress values, move calculated by (6), will result in addition or deletion of the material. The co-ordinate of the design node i is changed in following manner-
(7)
After getting new coordinates of design or master nodes, this information is sent to mesh generator to create new mesh. Stress analysis is further carried out for this new mesh and above process is continuously repeated until convergence criterion is satisfied. 2.3 Convergence criterion In ideal condition of optimal shape, the membership values at all nodes should be equal to 1. This may be defined as crisp optimal design. But due to fuzzy nature of the stresses, it is difficult, if not impossible, to achieve this condition. A convergence criterion based on fuzzy membership value have been used in this study. For this, fuzzy intersection, also named minimum membership function (MMF), has been applied. This value should be the maximum in the optimum shape. The program execution is stopped when the MMF value starts decreasing during optimization. Mathematically it can be written as: Nd μ (σiter ) = min ∩ μ (σi )
(8)
i=1
If μ (σiter ) < μ (σiter + 1 ) μ (σiter ) > μ (σiter + 1 )
Fig. 6 Move factor—bell shape
continue optimization convergence is reached.
Due to local minima, μ(σiter ) or MMF value may decrease in some iteration and shoot up again. The convergence is
Gradientless shape optimization using artificial neural networks
declared to be reached when the diminishing trend continues at least for three iterations. 2.4 Selection of membership functions The selection of membership functions, i.e. triangular or bell shape, is a crucial decision for the success of optimization process. From Figs. 5 and 6, it can be visualized that move factor (MF) for bell shape function is higher than that of triangular function except near the optimum. Nodal movement for triangular function is linear whereas for bell shape function is non-linear. Convergence may be slow for triangular function whereas undesirable profile may be obtained for bell function because slow movement is required as the optimum shape reaches. A mixed approach may be a better choice in which initial few iterations are carried out with bell function and then switch over to triangular function. In this study triangular shape membership function has been used because of its simplicity and reliable results.
3 Gradientless shape optimization using artificial neural networks (GSN technique) In gradientless methods of structural shape optimization, geometrical modification of the boundary is carried out simply on the basis of boundary stresses in that particular iteration. The sensitivity of the stresses are not used in this process hence optimization process becomes very slow. Since past several years artificial neural networks (Hertz and Krogh 1991; Rumelhart et al. 1986), have been successfully employed in problems of structural engineering (Rehak et al. 1989; Cao et al. 1998; Mukherjee and
Fig. 8 Step jumps
703
Deshpande 1995; Topping et al. 1997; Mukherjee et al. 1996; Flood and Kartam 1994; Garrett et al. 1992; Berke and Hajela 1990; Berrais 1996; Waszczyszyn 1996; Hajela and Berke 1991; Berke et al. 1993; Papadrakakis et al. 1999). Lopez et al. (2008) investigated the feasibility of neural networks for solving shape optimization problems. Neural network predictions were found to be in close match with the analytical solutions. Yildiz et al. (2003) successfully employed neural network based image processing techniques for topology and shape optimization problems. Yu-Cheng Tang et al. (2008) developed a neural network based response surface model for preform shape tool optimization. To achieve fast convergence and at the same time employ stress based shape modifications, artificial neural networks along with above developed integrated zero order approach have been used in this study. In this, first few iterations of shape optimization are carried out using integrated zero order approach. Stress and move differences of these iterations are stored to train artificial neural networks. Let r0 and σ0 be the nodal positions and stresses at design nodes before the start of shape optimization. After one, two, three, four, five and six iterations nodal positions and stresses are r0 , r1 , r2 , r3 , r4 , r5 , r6 and σ1 , σ2 , σ3 , σ4 , σ5 , σ6 respectively. To speed up the optimization process one, two or three step jumps in these data may be considered for the training of the neural networks. There are ten data sets (five forward and five backward moves) in one step jump, eight data sets (four forward and four backward moves) in two step jump and six data sets (three forward and three backward moves) in three step jump respectively (Fig. 8). After training the neural network using these data, the next target stress difference at the design nodes is presented to the neural network to predict the corresponding move. From this move, geometry is modified for the next iteration. Again the
704
FEA is carried out and the optimization process is repeated until desired convergence is reached. One of the problems of neural network based approaches of shape optimization is that sometimes the network predicts oscillatory MMF (minimum membership function) value during optimization. In ideal situation the value of MMF should continuously increase till the final convergence is reached. To overcome this problem, a predefined value of MMF is kept as the limit. On reaching this value of the convergence the neural network prediction of the moves is switched to the integrated zero order approach. This new proposed technique of shape optimization is named as GSN technique.
K.K. Pathak, D.K. Sehgal
Fig. 9 Cantilever beam
3.1 Computational steps Following steps are followed in the proposed GSN technique1. Define design nodes and generate FE mesh from them. 2. Carry out FEA for given loading conditions and calculate membership function values at design nodes. 3. Optimize the boundary shapes using integrated zero order methods. Change the coordinates of design variable. Repeat step 1 to 3 for a prescribed number of iterations and store the stress and geometry differences for the selected step jump. 4. Train artificial neural networks for stress differences as inputs and move differences as outputs for one, two and three step jumps. 5. Obtain the geometry from the neural network for given stress differences till the minimum membership function (MMF) value increases in each subsequent iteration, otherwise convergence is declared to be reached and terminate the network based shape optimization process. 6. Carry out integrated zero order based shape optimization with small moves till the final convergence is reached. Based on this algorithm a FORTRAN code GSOANN is written.
of top and bottom profiles. The beam is loaded at its tip by a 1,000 N concentrated load. The target maximum shear stress for the optimum shape is 10 N/mm2 . The Young’s modulus of elasticity and Poisson’s ratio are taken as 2 × 105 N/mm2 and 0.3 respectively. The beam is discretised into eight, nine node plane stress Lagrangian elements making a total of 45 nodes. There are two design elements and ten design nodes. Because of symmetry in geometrical and loading conditions, only top five design nodes are taken into the optimization process. The beam was first optimized using integrated zero order approach, proposed in Section 2, up to six iterations. From these results, artificial neural networks were trained for stress differences as inputs and move differences as outputs for one, two and three iteration jumps. There were total ten training patterns (five forward and five backward) for one step jump, eight training patterns for two step jump and six training sets for three step jump. Onwards, shape optimization, was carried out by neural network. Iteration number and minimum membership function values corresponding to that were recorded during optimization. Feed-forward back propagation neural networks of the architecture 5–6–5 were selected for training the data sets. The networks were trained for mean square error of values 0.001, 0.003 and 0.004 for the described three cases respectively. This increase in the error tolerance is to avoid over training of the neural networks as the numbers of data
Table 1 Maximum shear stresses in cantilever
4 Numerical examples Design node number
Using software GSOANN, four academic problems are optimized below. 4.1 Cantilever beam A cantilever beam of 120 mm span, 40 mm width and 10 mm thickness (Fig. 9) is chosen for shape optimization
Max. shear stress (N/mm2 ) Initial shape
Optimum shape
1
25.53
10.06
2
16.94
10.03
3
11.23
10.02
4
5.73
10.01
5
1.58
10.00
Max Shear Stress (N/mm2)
Gradientless shape optimization using artificial neural networks
705
30
Table 2 Maximum shear stresses in circular plate
25
Design node number
Max. shear stress (N/mm2 ) Initial shape
Optimum shape
1
49.39
29.52
2
23.40
29.52
3
18.09
29.30
20 15 10 5 0 1
2
3 Design Node
4
5
Initial Shape"
The optimized shape is shown in Fig. 9. The maximum shear stresses at design nodes for initial and optimum shape are given and shown in Table 1 and Fig. 10 respectively.
Optimal Shape
4.2 Circular plate with square hole Fig. 10 Stress distribution
set decrease. After getting the considerable high value of membership function, the iterative optimization process was again resorted to integrated zero order approach. Minimum membership function (MMF) value for integrated zero order and neural network based methods are recorded in each iteration. It is observed that using the single step method the MMF value equal to 0.84 is obtained after 13 iterations which, by using integrated zero order method is obtained after 23 iterations. Total numbers of iterations to reach the optimum profile were 67 and 45 for integrated zero order and single step jump method respectively. Using two step jump, the minimum membership function value obtained after 16 iterations is 0.84 which could be obtained by zero order approach after 23 iterations. Total number of iterations required to reach optimum shape is 51 for this case. Using three iteration jump, the minimum membership function value is obtained as 0.818 after 11 iterations which could be obtained after 22 iterations using zero order approach. It took 48 iterations to reach the optimum shape.
A circular plate of 120 mm diameter and 1 mm thickness is undertaken for shape optimization of the central square hole of 20 mm side width. The plate is applied with uniformly distributed pressure of 5 N/mm2 at the outer boundary. Because of symmetry along x and y axes only one quarter of the plate is considered. The finite element mesh, loading and boundary conditions are shown in Fig. 11. The plate is discretized into eight nine node Lagrangian plane stress elements and there are total 45 nodes. The plate is divided into two design elements and all 5 master nodes of the design elements lying on the inner boundary are considered as design nodes. The target maximum shear stress for shape optimization is considered equal to 30 N/mm2 . The Young’s modulus of elasticity and Poisson’s ratio are taken as 2 × 105 N/mm2 and 0.3 respectively. The problem was first solved by integrated zero order approach and then by neural network based approaches for one, two and three step jumps. Because of stress concentration at the corner of the square hole, first five iterations of zero order approach were
Max Shear stress (N/mm2)
50
40
30
20
10 1
2
3
4
Design Node Initial Shape Optimal Shape
Fig. 11 Circular plate with square hole
Fig. 12 Stress distribution
5
706
K.K. Pathak, D.K. Sehgal
Max Shear Stress (N/mm2)
80
Fig. 13 Fixed ends beams
60
40
20
0 1
2
3
4
5
Design Node
not considered for training the network. Iteration numbers 7 to 12 were used for training. Due to symmetry in loading and geometry, three out of five deign nodes were considered in optimization process and remaining two were modified by changing their x and y co-ordinates. Feed-forward back propagation neural networks of the architecture 3–5–3 were selected for training the data sets. The networks were trained for mean square error of values 0.001, 0.003 and 0.004 for the described three cases respectively. The integrated zero order approach took 80 iterations to reach the optimum profile for a given minimum membership function (MMF) value equal to 0.976 which was 0.357 for initial shape. The one step jump predicts the same rate of convergence as zero order approach. Two step jump is found to convergence very fast for this case. Here MMF value after 28 iterations is 0.996 which could not be obtained even after 80 iterations using zero order approach. The convergence rate for three step jump is also parallel to the zero order method. The optimised shape is shown in Fig. 11. The initial and final maximum shear stress values at design nodes are given and shown in Table 2 and Fig. 12 respectively.
Initial Shape Optimal Shape
Fig. 14 Stress distribution
design elements which are further discretized into 16 nine node Lagrangian elements and 85 nodes. The Young’s modulus and Poisson’s ratio values are taken as 2 × 105 N/mm2 and 0.3. There are total ten design variables but due to various symmetries only three master nodes on top half profile is considered as design variables. This problem like all above problems is also optimized by integrated zero order and one, two and three step jump approaches. Feed-forward back propagation neural networks of the architecture 3–4–3 were selected for training the data sets. The networks were trained for mean square error of values 0.001, 0.003 and 0.004 for the described three cases respectively. The integrated zero order approach took 35 iterations to reach maximum MMF value equal to 0.946 whereas the initial MMF value was 0.0221.
4.3 Fixed ends beam A both ends fixed beam of 240 mm span, 40 mm depth and 10 mm thickness, loaded by a concentrated load of 10 KN magnitude at the center, is shape optimized for top and bottom profile. The target maximum shear stress is 60 N/mm2 . The finite element mesh, loading and boundary conditions are shown in Fig. 13. The beam is discretized into two
Table 3 Maximum shear stresses in fixed ends beam Design node number
Max. shear stress (N/mm2 ) Initial shape
Optimum shape
1
66.22
59.76
2
4.68
58.94
3
47.56
59.81 Fig. 15 Column
Gradientless shape optimization using artificial neural networks
707
Fig. 16 Optimized shape Max Shear Stress (N/mm2)
70 60 50 40 30 20 10 0 0
1
2
3
4
Design Node Initial Shape Optimal Shape
Fig. 17 Stress distribution
The one step jump approach reached maximum MMF equal to 0.925 after 14 iterations for which zero order approach took 31 iterations. The two step jump approach reached a maximum MMF value equal to 0.853 after 13 iterations which could be reached by zero order approach only after 21 iterations. The three step jump approach reached a maximum MMF value equal to 0.657 after 13 iterations for which zero order approach took 16 iterations. The optimised shape is shown in Fig. 13. Maximum shear stress values in initial and final shapes are given and shown in Table 3 and Fig. 14 respectively. 4.4 Column The last example considered for shape optimization is a column. The dimensions and the loading conditions for this are shown in Fig. 15. It was modeled by one design element which was further discretised into 16 nine node plane stress elements and 81 nodes. The bottom and the side faces of the column were considered for shape optimization.
For this three master nodes at bottom surface were considered as design nodes. The target maximum shear stress was 60 N/mm2 . The Young’s modulus and Poisson’s ratio are taken as 2.0 × 105 N/mm2 and 0.3. Column like all other examples, was optimized by zero order, one, two and three step jump approaches. Feed-forward back propagation neural networks of the architecture 3–5–3 were selected for training the data sets. The networks were trained for mean square error of values 0.001, 0.003 and 0.004 for the described three cases respectively. The initial maximum MMF value was 0.019 which improved to the optimum 0.18 in 60 iterations by zero order approach. Single step jump approach proved to be very fast and optimum was reached after 20 iterations. In two step jump approach maximum MMF obtained is 0.083 after 14 iterations whereas using three step jump approach maximum MMF value was obtained as 0.1045 after ten iterations. The optimized shape is shown in Fig. 16. The maximum shear stress values at design nodes in initial and optimum shape are given and shown in Table 4 and Fig. 17 respectively.
Table 5 Iterations required by different approaches Example
Table 4 Maximum shear stresses in column Design node number
zero order jump
Max. shear stress (N/mm2 ) Initial shape
Integrated One step Two step Three step Max. time
Optimum shape
Cantilever
jump
jump
saving (%)
67
45
51
48
32.83
Circular plate 80
76
28
78
65.00
Fixed ends
35
18
27
31
48.57
60
20
42
24
66.67
1
21.13
59.99
2
1.15
5.20
beam
3
21.13
59.99
Column
708
K.K. Pathak, D.K. Sehgal
5 Discussion of results
References
From the results of above examples it is obvious that neural network based shape optimization reduces the number of iterations drastically. The number of iterations taken to converge to the final shape depends on the stress values at design nodes, not on their numbers. The number of iterations taken by zero order, one, two and three step jump approach to reach optimum profile for all the four examples are given in Table 5. It can be seen that application of neural network can reduce the computational time up to 66.67% as compared to the integrated zero order method discussed in Section 2. In three out of four examples, one step jump is the fastest whereas in circular plate example two iteration jump takes the lead. The three step jump is slower in convergence than the first two. It can be concluded that one step jump can be used for all problems except stress concentration related problems where two step jump may be more suitable. It is also observed in these problems that network training takes very less time. This may be due to good quality data obtained from zero order approach. The proposed approach can be easily extended to three dimensional real world problem using brick elements. This major task for such problems will be of defining the design elements and design nodes. Once these are done, GSN technique can be successfully applied.
Bennett JA, Botkin ME (1985) Structural shape optimization with geometric description and adaptive mesh refinement. AIAA J 23:458–464 Berke L, Hajela P (1990) Applications of artificial neural networks in structural mechanics. NASA 331–348, TM-102420 Berke L, Patnaik SN, Murthy PLN (1993) Optimum design of aerospace structural components using artificial neural networks. Int J Compt Struct 48:1001–1010 Berrais A (1996) Neural networks in civil engineering: state of the art. In: Toping BHV (ed) Advances in computational structures technology. CIVIL-COMP, Edinburgh, pp 93–101 Braibant V, Fleury C (1984) Shape optimal design using B-splines. Comp Meth Appl Meth Engg 44:247–267 Cao X et al. (1998) Application of artificial neural networks to load identification. Int J Comp Struct 69:63–78 Flood I, Kartam N (1994) Neural networks in civil engineering I—principles and understanding. J Comput Civ Eng ASCE 8: 131–148 Garrett JH et al. (1992) Neural network and their application to civil engineering problems. In: Allen RH (ed) Expert systems for civil engineers: knowledge representation. ASCE, Reston Hajela P, Berke L (1991) Neurobiological computational model in structural analysis and design. Int J Comp Struct 41:657–667 Hasengawa A (1992) Shape optimization of two dimensional bodies by boundary changing method and thickness changing method. Int J Numer Methods Eng 34:889–892 Heller M, Kaye R, Rose LRF (1999) A gradientless finite element procedure for shape optimization. J Strain Anal 34:323–336 Hertz J, Krogh A (1991) Introduction to the theory of neural networks. Addison-Wesley, Redwood Imam MH (1982) Three-dimensional shape optimization. Int J Numer Methods Eng 18:661–673 Jose PC (1990) Strengthening and optimization of gravity dams. PhD Dissertation, Applied Mechanics Dept. IIT Delhi Krishnamoorthy CS (1994) Finite element analysis theory and programming. McGraw Hill, New Delhi Lopez R, Diego X, Flores R, Chiumenti M, Onate E (2008) Artificial neural networks for the solution of optimal shape design problems. In: 8th World conference on computational mechanics; June 30–July 5, Venice, Italy Mattheck C, Burkhardt S (1990) A new method of structural shape optimization based on biological growth. Int J Fatigue 12: 185–190 Mattheck C, Moldenhauer H (1990) An intelligent CAD method based on biological growth. Fatigue Fract Eng Mater Struct 13: 41–51 Mattheck C, Erb D, Bethge K, Begemann U (1992) Three dimensional shape optimization of a bar with a rectangular hole. Fatigue Fract Eng Mater Struct 15:347–351 Mukherjee A et al. (1996) Prediction of buckling load of columns using neural networks. J Struct Eng ASCE 122:1385–1387 Mukherjee A, Deshpande JM (1995) Application of artificial neural networks in structural design expert systems. Int J Comp Struct 54:367–375 Papadrakakis M et al. (1999) Innovative computational methods for structural optimization. In: Papadrakakis M, Toping BHV (eds) Innovative computational methods for structural optimization. Saxe-Coburg Publication, Saxe-Coburg, pp 195–222 Pourazady M, Fu Z (1996) An integrated approach to structural shape optimization. Int J Comp Struct 60:274–289 Queau JP, Trompette P (1980) Two dimensional shape optimal design by finite element method. Int J Numer Methods Eng 15:1603– 1612
6 Conclusions In this paper a new zero order shape optimization technique using artificial neural networks (GSN) has been proposed for structural shape optimization. This has been incorporated in a FORTRAN code named as GSOANN. Shape optimizations of four structures are successfully carried out using this software and results are compared with those obtained from the proposed integrated zero order method. Following are the salient findings of this study:
1. Application of neural networks helps reduce the computing time and improves the final convergence. 2. Application of triangular fuzzy membership function has been found very suitable for nodal movements and convergence prediction. 3. One step jump is most suitable for general problems except in stress concentration related problems. 4. First few iteration of zero order should not be used to train the neural network in stress concentration problems. Two step jump is most suitable in these cases.
Gradientless shape optimization using artificial neural networks Rehak DR et al. (1989) Neural network approaches in structural mechanics computations. In: Nelson JK Jr (ed) Computer utilization in structural engineering. ASCE Proceedings from Structural Congress Rumelhart DE et al. (1986) Learning representations of backpropagating errors. Nature 323:533–536 Sibal SK (1992) Optimal shape design of concrete gravity dams. M.Tech. Thesis, Applied Mechanics Dept., IIT, Delhi Tang Y-C, Zhou X-H, Chen J (2008) Preform tool shape optimization and redesign based on neural network response surface methodology. Finite Elem Anal Des 44(6):462–471 Topping BHV et al (1997) Parallel training of neural networks for finite element mesh decomposition. Int J Comp Struct 63:693– 707
709 Waszczyszyn Z (1996) Some recent and current problems in neurocomputing in civil and structural engineering. In: Toping BHV (ed) Advances in computational structures technology. CIVILCOMP, Edinburgh, pp 43–58 Yildiz AR, Ozturk N, Kaya N, Ozturk F (2003) Integrated optimal topology and shape optimization using neural networks. Struct Multidiscipl Optim 25:252–260 Zhixue Wu (2005) An efficient approach for shape optimization of components. Int J Mech Sci 47:1595–1610 Zienkiewicz OC, Campbell JS (1973) Shape optimization and sequential linear programming. In: Gallaghar RH, Zienkiewcz OC (eds) Optimum structural design. Wiley, New York Zienkiewicz OC, Taylor RL (1991) The finite element method, vol 1 & 2, 4th edn. McGraw Hill, London Zimmermann HJ (1996) Fuzzy set theory. Kluwer, Dordrecht
