Prediction of Optimal Design and Deflection of Space Structures Using Neural Networks

The main aim of the present work is to determine the optimal design and maximum deflection of double layer grids spending low computational cost using neural networks. The design variables of the optimization problem are cross-sectional area of the elements as well as the length of the span and height of the structures. In this paper, a number of double layer grids with various random values of length and height are selected and optimized by simultaneous perturbation stochastic approximation algorithm. Then, radial basis function RBF and generalized regression GR neural networks are trained to predict the optimal design and maximum deflection of the structures. The numerical results demonstrate the efficiency of the proposed methodology.


Introduction
The history of the applications of Artificial Intelligence to civil and structural engineering is simultaneously brief and long.It is brief if compared to the history of civil and structural engineering, whose definition as a discipline can be fixed a very long time ago.It makes sense to consider civil and structural engineering as the most ancient applicative discipline, being founded in preclassical world by Egyptians and Babylonians.It is long, instead, if compared to the history of Artificial Intelligence, whose name first appeared in science at the end of the sixties of the twentieth century.The earliest applications to civil and structural engineering are very likely 1 , where authors review tools and techniques for knowledge-based expert system for engineering design.An even earlier paper whose scope was indeed wider, but introduced some fundamental themes, is 2 .We can definitely settle a start date in 1986 when saved.In the third stage, radial basis function RBF 27 and generalized regression GR 27 neural networks are trained to predict the optimal design and maximum deflection of the double layer grids.To design neural networks MATLAB 28 is employed.

Formulation of Optimization Problem
In optimal design problem of space trusses the aim is to minimize the weight of the truss under constraints on stresses and displacements.This optimization problem can be expressed as follows: minimize: w x 1 , . . ., x n , . . ., x ng ng n 1 x n nm m 1 γ m l m , 2.1 subject to: σ i ≤ σ all,i , i 1, 2, . . ., ne, δ j ≤ δ all,j, j 1, 2, . . ., nj,

2.2
where x n , γ m , and l m are cross-sectional area of members belonging to group n, weight density, and length of mth element in this group, respectively; ng and nm are the total number of groups in the structure and the number of members in group n, respectively; ne and nj are the total number of the elements and nodes in truss, respectively; σ i and δ j are stress in the ith element and displacement of the jth node, respectively.Also, σ all,i and δ all,j are allowable stress in the ith member and allowable deflection of the jth node, respectively.In this study, besides cross-sectional areas x n the geometry dependent parameters of the double layer grid, L and h, are also variables.In other words, the aim is to find optimal cross-sectional areas for each set of L and h.Thus, 2.1 can be reexpressed as follows: For each set of L and h minimize w x 1 , . . ., x n , . . ., x ng ng n 1 x n nm m 1 γ m l m .

2.3
It is obvious that the computational burden of the above optimization problem is very high due to the fact that L and h are variables.Employing the neural network technique can substantially reduce the computational costs.
As the SPSA requires less number of function evaluations structural analyses than the other type of gradient-based methods, it is selected as the optimizer in this study.The basic concepts of the SPSA are explained in the next section.

SPSA Optimization Algorithm
SPSA has recently attracted considerable international attention in areas such as statistical parameter estimation, feedback control, simulation-based optimization, signal and image processing, and experimental design.The essential feature of SPSA is the underlying gradient approximation that requires only two measurements of the objective function regardless of the dimension of the optimization problem.This feature allows for a significant reduction in computational effort of optimization, especially in problems with a large number of variables Mathematical Problems in Engineering to be optimized.The basic unconstrained SPSA optimization algorithm is in the general recursive stochastic approximation SA form 26 : where X k represents the estimate of X at kth iteration, a k > 0 represent a scalar gain coefficient, and G X k represent an approximate gradient at X k .Under appropriate condition, 3.1 will converge to optimum design X * in some stochastic sense.The essential part of 3.1 is the gradient approximation that is obtained using the simultaneous perturbation SP method.Let w • denote a measurement of objective function at a design level represented by the dot and let c k be some positive number.The SP approximation has all elements of X k randomly perturbed together to obtain two measurements of w • , but each component is formed from a ratio involving the individual components in the perturbation vector and the difference in the two corresponding measurement.For two sided simultaneous perturbation, we have where the distribution of the user-specified n v dimensional random perturbation vector Δ k {Δ k1 , Δ k2 , . . ., Δ kn v } T satisfies condition discussed in 26 .
It is observed that each iteration of SPSA needs only two objective function measurements independent of n v because the numerator is the same in all n v components.This circumstance provides the potential for SPSA to achieve a large savings in the total number of measurements required to estimate X * when n v is large.

Implementation of SPSA
The following step-by-step summary shows how SPSA iteratively produces a sequence of estimates 26 .
Step 1 initialization and coefficient selection .Set counter index k 0. Pick initial guess and nonnegative coefficients a, c, A, α, and γ in the SPSA gain sequences a k a/ A k 1 α and c k c/ k 1 γ .The choice of gain sequences a k and c k is critical to the performance of SPSA.Spall provides some guidance on picking these coefficients in a practically manner.
Step 2 generation of the simultaneous perturbation vector .Generate by Monte Carlo an n v -dimensional random perturbation vector Δ k , where each of the n v components of Δ k is independently generated from a zero mean probability distribution satisfying some conditions.A simple choice for each component of Δ k is to use a Bernoulli ±1 distribution with probability of 1/2 for each ±1 outcome.Note that uniform and normal random variables are not allowed for the elements of Δ k by the SPSA regularity conditions.
Step 3 objective function evaluations .Obtain two measurements of the objective function w • based on simultaneous perturbation around the current X k : w X k c k Δ k and w X k − c k Δ k with the c k and Δ k from Steps 1 and 2. Step 4 gradient approximation .Generate the simultaneous perturbation approximation to the unknown gradient G X k : where Δ ki is the ith component of Δ k vector.
Step 5 updating X estimate .Use the standard SA to update X k to new value X k 1 .
Step 6 iteration or termination .Return to Step 2 with k 1 replacing k.Terminate the algorithm if there is little change in several successive iterates or the maximum allowable number of iterations has been reached.Figure 1 shows the flowchart of the SPSA.
In the present work, we suppose that the length and height of the double layer grids are varied in specific ranges.Our aim is to optimize all of the possible structures defined in the ranges.Therefore it can be observed that the additional difficulty is the huge computational burden of the optimization process.In order to mitigate the difficulty, RBF and GR neural networks are employed to predict the optimal design of the double layer grids with various length and height.

Neural Networks
In the recent years, neural networks are considered as more appropriate techniques for simplification of complex and time consuming problems.The interest shown to neural networks is mainly due to their ability to process and map external data and information based on past experiences.Neural networks are not programmed to solve specific problems.Indeed, neural networks never use rules or physic equations related to the specific problem in which they are employed.Neural networks use the knowledge gained from past experiences to adapt themselves to solve the new problems.

Radial Basis Function
The use of RBF in the design of neural networks was first introduced by Wasserman in 1993 27 .The RBF network basically involves three entirely different layers: an input layer, a hidden layer of high enough dimension, and an output layer.The transformation from the hidden unit to the output space is linear.Each output node is the weighted sums of the outputs of the hidden layer.However, the transformation from the input layer to the hidden layer is nonlinear.Each neuron or node in the hidden layer forming a linear combination of the basis or kernel functions which produces a localized response with respect to the input signals.This is to say that RBF produce a significant nonzero response only when the input falls within a small localized region of the input space.The most common basis of the RBF is a Gaussian kernel function of the form: where ϕ l is the output of the lth node in hidden layer; Z is the input pattern; C l is the weight vector for the lth node in hidden layer, that is, the center of the Gaussian for node l; σ l is the normalization parameter the measure of spread for the lth node; and q is the number of nodes in the hidden layer.The outputs are in the range from zero to one so that the closer the input is to the center of the Gaussian, the larger the response of the node is.The name RBF comes from the fact that these Gaussian kernels are radially symmetric; that is, each node produces an identical output for inputs that lie a fixed radial distance from the center of the kernel C l .The network outputs are given by where y i is the output of the ith node, Q i is the weight vector for this node, and M is the number of nodes in the output layer.
There are two common ways to calculate the measure of spread σ l .
1 Find the measure of spread from the set of all training patterns grouped with each cluster center C l ; that is, set them equal to the average distance between the cluster centers and the training patterns: where N l is the number of patterns that belong to the lth cluster and k is the index number of a pattern that belongs to the lth cluster.
2 Find the measure of spread from among the centers p-nearest neighbor heuristic :

Generalized Regression
Generalized regression network GR subsumes the basis function methods.This network does not require iterative training.The structure of GR is designated such that transpose of input matrix and transpose of desired output target matrix are chosen as first layer and second layer weight matrices, respectively.GR algorithm is based on nonlinear regression theory, a well established statistical technique for function estimation.Except the approach of adjusting of second layer weights, the other aspects of GR are identical to RBF neural networks.

Double Layer Grid Model
In this section dimensions of considered double layer grid structure and its corresponding model are described.The model considered here is a double layer grid with bar elements connected by pin joints.The length of the spans, L, is varied between 25 and 75 m with step of 5 m.The height is varied between 0.035 and 0.095 L with steps of 0.2 m.The smallest and biggest structures in this interval are shown in Figure 2. The sum of dead and live loads equal to 250 kg/m 2 is applied to the nodes of the top layer.
In order to satisfy practical demands, in the optimization of large-scaled structure such as space structures, the structural elements should be divided into some groups.In this study the elements are put into 18 different groups.For this purpose a step-by-step summary defined bellow is employed.
Step 1.A similar cross sectional area is initially assigned to all elements of the structure.

5.3
Preparing a neural network is achieved in three stages: data generating, training, and testing.In the first stage, a number of input and output pairs are provided and divided into training and testing sets.In the second stage, the training set is used and the modifiable parameters of the neural network are adjusted.In the last stage the performance generality of the trained neural network is examined through the testing set.
In order to provide the required data data generation , a number of double layer grids according to their L and h are randomly selected.All of the selected structures are optimized using SPSA.Optimal designs of the selected structures and their corresponding maximum deflections are saved.This process is shown in Figure 3.
In order to train neural networks, the generated data should be separated to n tr training data and n ts testing data n tr n ts n s as follows.
Training data for optimal design predictor networks: Training data : . . .
The optimization process is terminated

Design vector is updated
The SP vector is generated The unknown gradient is approximated  Testing data for optimal design predictor networks: Testing data : . . .

Main Steps in Training Neural Network
As a summary the main steps in training of RBF and GR NNs to predict optimal design and maximum deflection of the structure are as follows: 1 configuration processing of the selected space structures employing Formian, 2 selection a list of available tube sections from the standard lists, 3 implementation member grouping, 4 generation of some structures, based on span and height, to produce training set, 5 static analysis of the structures, 6 designing for optimal weight by SPSA according to AISC-ASD code,   7 training and testing RBF and GR to predict optimal design and maximum deflection, 8 improving generalization of the neural networks if it is necessary.

Flowchart of the Methodology
The flowchart of the proposed methodology is shown in Figure 4.This flowchart includes three main blocks: data generation, optimization, and NN training.The data generation block includes the optimization block.In these two blocks the data needed for neural network training is produced.The mentioned data are stated through 5.4 to 5.7 .

Numerical Results
Typical topology of the RBF and GR neural networks to predict the optimal design and maximum deflection of the double layer grids is shown in Figures 5 and 6, respectively.To find the optimal spread in the RBF and GR networks the minimum distance between training set and test set errors are employed 29 .The spread values in RBF networks trained to predict the optimal design and maximum deflection are 11.5 and 11.75 and for GR are 12.5 and 10.25, respectively.The results of RBF for predicting the optimal cross-sectional areas are shown in Figure 7.
The errors of RBF for predicting the maximum deflections are shown in Figure 8.The results of GR for predicting the optimal cross-sectional areas are shown in Figure 9.The errors of GR for predicting the maximum deflections are shown in Figure 10.Maximum and mean of errors of RBF and GRNN in approximation of optimal designs and maximum deflection are given in Tables 1 and 2, respectively.
The numerical results demonstrate that the generality of the GR is better than that of the RBF neural network in prediction of optimal design and maximum deflection of the double layer grids.

Conclusion
In this investigation, an innovative methodology is proposed to predict the optimal design and maximum deflection of the square-on-square double layer grids.This methodology consists of three stages.In the first stage, a number of the double layer grids with random spans and heights are generated.In the second stage the generated double layer grids are optimized by SPSA algorithm.Also, the maximum deflections of the optimal structures are saved.In the third stage, RBF and GR neural networks are trained to predict the optimal design and maximum deflection of the double layer grids.
By concerning the following points, it can be observed that the proposed methodology is novel and innovative.
1 It is the first study based on employing the SPSA optimization algorithm to optimize double layer grids with variable geometry.
2 Application of the RBF and GR neural networks to predict the optimal design and maximum deflection of the double layer is achieved for the first time in this study.
3 The main advantage of the proposed methodology is to predict the optimal design and maximum deflection of the double layer grids with high speed and trivial errors in comparison with the traditional methods.

Figure 2 :
Figure 2: The smallest and biggest structures in the considered interval.

Figure 4 :
Figure 4: Flowchart of the proposed methodology.

Figure 5 :Figure 6 :
Figure 5: Typical topology of a neural network model to predict the optimal design.

Figure 7 :
Figure 7: RBF errors in approximation of optimal cross-sectional areas.

Figure 8 :
Figure 8: Errors of RBF for predicting the maximum deflections.

Figure 9 :
Figure 9: GR errors in approximation of optimal cross-sectional areas.

Figure 10 :
Figure 10: Errors of GR for predicting the maximum deflections.

Table 1 :
Summary of errors of RBF and GRNN in approximation of optimal designs.

Table 2 :
Summary of errors of RBF and GRNN in approximation of maximum deflection.