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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.

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 [

As in this study our main aim is to employ neural networks to predict the optimal design and maximum deflection of the double layer grids, the next paragraph is devoted to review the literature about the optimal design of space structures by soft computing techniques.

Erbatur et al

Much more other applications of neural networks in the field of civil engineering can be found in the literature [

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 an optimization algorithm. Although, in the recent years many new structural optimization algorithms have been proposed by the researchers [

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:

In this study, besides cross-sectional areas (

It is obvious that the computational burden of the above optimization problem is very high due to the fact that

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 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 to be optimized. The basic unconstrained SPSA optimization algorithm is in the general recursive stochastic approximation (SA) form [

It is observed that each iteration of SPSA needs only two objective function measurements independent of

The following step-by-step summary shows how SPSA iteratively produces a sequence of estimates [

Set counter index

Generate by Monte Carlo an

Obtain two measurements of the objective function

Generate the simultaneous perturbation approximation to the unknown gradient

Use the standard SA to update

Return to Step

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.

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.

The use of RBF in the design of neural networks was first introduced by Wasserman in 1993 [

There are two common ways to calculate the measure of spread

Find the measure of spread from the set of all training patterns grouped with each cluster center

Find the measure of spread from among the centers (

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.

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, ^{2} is applied to the nodes of the top layer.

The smallest and biggest structures in the considered interval.

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.

A similar cross sectional area is initially assigned to all elements of the structure.

The structure is analyzed through FE and axial stresses of all members are obtained.

All tension members of the structure are put into 6 groups according to their stress states as follows:

All compressive members of top and bottom layer elements of structure are put into 6 deferent groups according to their stress values as follows:

All compressive members of middle layer elements of structure are also put into 6 deferent groups based on their stresses as follows:

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

In order to train neural networks, the generated data should be separated to

Training data for optimal design predictor networks:

Data generation process.

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:

configuration processing of the selected space structures employing Formian,

selection a list of available tube sections from the standard lists,

implementation member grouping,

generation of some structures, based on span and height, to produce training set,

static analysis of the structures,

designing for optimal weight by SPSA according to AISC-ASD code,

training and testing RBF and GR to predict optimal design and maximum deflection,

improving generalization of the neural networks if it is necessary.

The flowchart of the proposed methodology is shown in Figure

Flowchart of the proposed methodology.

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

Typical topology of a neural network model to predict the optimal design.

Typical topology of a neural network model to predict the maximum deflection.

To find the optimal spread in the RBF and GR networks the minimum distance between training set and test set errors are employed [

RBF errors in approximation of optimal cross-sectional areas.

The errors of RBF for predicting the maximum deflections are shown in Figure

Summary of errors of RBF and GRNN in approximation of optimal designs.

Cross-sectional area | RBF | GRNN | ||
---|---|---|---|---|

Max. error | Mean of errors | Max. error | Mean of errors | |

1 | 32.8501 | 3.9054 | 8.8182 | 1.6575 |

2 | 30.0348 | 8.7955 | 22.8185 | 5.2312 |

3 | 18.3493 | 4.1153 | 15.6306 | 2.8343 |

4 | 36.8009 | 5.5074 | 10.8515 | 2.7220 |

5 | 15.6441 | 3.5933 | 13.4325 | 2.4476 |

6 | 24.8060 | 2.4974 | 6.1103 | 0.9992 |

7 | 33.3249 | 9.2145 | 20.4911 | 4.9154 |

8 | 18.7265 | 6.1936 | 33.8892 | 4.7607 |

9 | 26.1423 | 5.6690 | 16.1963 | 3.6055 |

10 | 29.7808 | 3.8814 | 10.6282 | 2.1702 |

11 | 21.9747 | 3.5231 | 11.3200 | 2.3858 |

12 | 29.3597 | 2.6233 | 2.6899 | 0.4795 |

13 | 26.4572 | 6.4117 | 28.6768 | 4.1939 |

14 | 23.7474 | 6.6549 | 37.5173 | 6.1356 |

15 | 29.8091 | 5.7633 | 18.2707 | 3.5529 |

16 | 32.8732 | 5.3975 | 17.5221 | 2.7931 |

17 | 21.5389 | 3.7532 | 13.8589 | 2.4312 |

18 | 29.6637 | 2.7991 | 4.4832 | 0.8361 |

| ||||

Avr. | 26.7714 | 5.0166 | 16.2892 | 3.0185 |

Summary of errors of RBF and GRNN in approximation of maximum deflection.

RBF | GRNN | ||
---|---|---|---|

Max. error | Mean of errors | Max. error | Mean of errors |

13.1166 | 1.6675 | 3.0084 | 0.4641 |

Errors of RBF for predicting the maximum deflections.

GR errors in approximation of optimal cross-sectional areas.

Errors of GR for predicting the maximum deflections.

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.

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.

It is the first study based on employing the SPSA optimization algorithm to optimize double layer grids with variable geometry.

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.

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.