Applied Mechanics and Materials Vols. 170-173 (2012) pp 3588-3593 Online available since 2012/May/14 at www.scientific.net © (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.170-173.3588
Prediction of dynamic impedances functions using an Artificial Neural Network (ANN) SBARTAI Badreddine1,a, GOUDJIL Kamel1,b 1
LMGHU laboratory, University of 20 Août 1955-Skikda, Algeria b
a
[email protected],
[email protected]
Keywords: Artificial Neural Networks, Foundation, Impedances Functions, Soil-Structure Interaction.
Abstract. Artificial Neural Networks (ANN) has seen an explosion of interest over the last few years. Indeed, anywhere that there are problems of prediction, classification or control, neural networks are being introduced. Hence, the main objective of this paper is to develop a model to predict the response of the soil-structure interaction system without using the calculate code based on sophisticate numerical methods by the employment of a statistical approach based on an Artificial Neural Network model (ANN). In this study, a data base which relates the impedance functions to the geometrics characteristic of the foundation and the dynamic properties of the soil is implemented. This leads to develop a neural network model to predict impedances functions (all modes) of a rectangular surface foundation. Then the results are compared with unused data to check the ANN model’s validity. Introduction The major problem that must be addressed during the construction of particular types of structures, such as conventional and nuclear power plants, chemical factories, liquid natural gas tanks, etc., concerns the rigorous safety conditions that must be established to avoid damage caused by various types of excitations to which these structures can be subjected. The strong interest in this problem is not only because of the constant increase in the quality and stability requirements of the constructions in the vicinity of these structures but also because of the need to protect sensitive material. The analysis of the response of these structures and that of the neighbouring soil is a wave-propagation problem that leads to consideration of the soil-structure interaction. To face these challenges, research in this field has been oriented toward numerical methods because the classical analytical methods, which are generally based on restrictive assumptions about the geometry of the foundation and the elastic properties of the soil, are not adapted to treat the problems of such great complexity. Also, the study of soil-structure interaction remains important and justifies the particular interest that many researchers have shown in it up to the present day. There are two main methods dealing with soil-structure interaction analysis: Direct Method, and Substructure Method. In the direct method, the response of the soil and structure is determined simultaneously by analysing the idealized soil–structure system in a single step. In substructure method the SSI problem is divided into sets of simpler problems, which are solved independently, and the results are then superposed to obtain the response of the structure. The basic step in the substructure approach is to determine the force–displacement characteristics of the soil. This relationship may be in the form of an impedance (stiffness) function, or, inversely, a compliance (flexibility) function. These impedances functions (Kij) are often represented in the form of a sum of two terms depending of the frequency. The first being the real part (kij) and the second the imaginary part (cij). The aim of this work is to avoid the calculation of the impedances functions with the numerical methods (FEM, BEM, and DEM) because of theirs complexities. Using the Artificial Neural Networks and the gathered data from Wong tables [1], necessary information can be reached with acceptable engineering approximation. In recent years, different computational solutions using neural networks have been studied, examples of which can be found in the literature. [2] Studied the dynamic soil-structure interaction of building by ANN. All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP, www.ttp.net. (ID: 41.96.30.23-08/11/12,18:23:30)
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In this study, a database which relates the impedance functions (dynamic stiffness) to the geometrics characteristic of the foundation and the dynamic properties of the soil is implemented. This leads to develop a neural network model to predict impedances functions (all modes) of a rectangular surface foundation. Then the results are compared with unused data to check the ANN model’s validity. Modeling by (ANN) Artificial Neural Network (ANN) gives an alternative for mathematical modellig and makes a part of the statistical nonparametric and nonlinear models capable to answering to the problematic of identification, of assistance to decision, of diagnostic, of prediction, etc. The application of this type of model only east appeared at the beginning of 1990 and their advantage resides in their capacity of generalization. Generally, the Artificial Neural Networks are often confused with the genetic algorithms, the cybernetics and the artificial intelligence. They constitute actually a current very precise of the artificial intelligence, of which developmental model is based on idea to copy of the algorithms of training on a simplified brain model. The Artificial Neural Networks are inspired of the biology and represent a mathematical model of the functioning of the biological neuron [3]. Idea is to present to ANN the input data and output, and to make him learn the relation between two by training process. This last consists in minimizing error by adjustment of the parameters model. A neural network, in general, has three layers namely, the input, output and hidden layers. The neurons in the input layer receive input from the external environment. Hidden layer, which receives inputs from the input layer, performs computation and provides the outputs to output layer. Out put layer consists of neurons that communicate the output of system to the user or external environment. The structure of a three-layer neural network is shown in (figure 1). At first we created our database starting with the tables of impedances functions (all modes) of a rigid rectangular foundation resting on a homogeneous visco-elastic semi-infinite soil of [1]. These impedances functions were gotten numerically by the code of computation (Continuum Linear Analysis of Soil-Structure Interaction) of [1]. This database is made of 24 tables which incorporate the values of the impedances according to the frequency ao, the relative length B/C (B and C are the sizes of the foundation), the coefficient of Poisson’s ratios ν and the material damping β.
Inputs
Multiple input neurons
P1 W1, 1 P2 P3 PN
Σ
W1, N
n
f
b
a = f (WP + b) Figure 1. Neural Network structure
a
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The database used for the implementation of (ANN) is constituted of 504 samples of which 420 samples are used for the training of the network and 84 for the test (validation). The (ANN) used is constituted by an input layer consisting of four neurons, two hidden layers composed respectively of eight and sixteen neurons and an output layer consisting of two neurons (4 - 8 16 - 2), Fig 2. The activation function used for all neurons is a sigmoid type: f (x) = 1/ (1+ e−x)
(1)
First, we created our database from tables impedance functions (all modes) of a rigid rectangular foundation placed on the surface of a homogeneous soil viscoelastic half-space of Wong and Luco (1985). These impedances were obtained numerically by the computer code (Continuum Linear Analysis of Soil-Structure Interaction) by (Wong and Luco, 1985). This database can be summarized in 24 tables that contain the impedance values as a function of dimensionless frequency ao, the relative length B / C (B and C are the dimensions of the foundation), Poisson's ratio ν and amortization hysteresis β. The database is formed of the following parameters: • the adimensional frequency ao ranging from 0 to 10, • the adimensional width of the founding B/C ranging from 1 to 4, • Poisson's ratio ν varying from 0.1 to 0.5, • the hysteresis damping coefficient β ranging from 0.01 to 0.05. The output parameters of the (NNA), represents the complex impedance functions (Ki) of a rectangular foundation: • the real part of the impedance is (ki), • the imaginary part of impedance is (ci). (i): represents the type of mode of the vibrant foundation. biais
biais
biais
a0 β ν B/C
ki ci Output layer
Input layer First hidden layer
Second hidden layer
Figure 2. Configuration of the neural network After the phase of training, the validation (test) of RNA is used to judge the capability of generalization of the model [4]. This technique consists in testing the model on not used data for the adjustment of the weights. The data of the phase test represent generally 20% of the database. The programming of the different algorithms was carried out in the environment Matlab ®.
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Before choosing the type of network to make the training, one has shared the database in two: the twenty first tables for training and the last four tables for the test without forgetting to normalize input data and output. After chosen the type of network, we carried out the training. This last stop them if the epoch number were perfect or if the goal (error) east reaches (fig.3). Because of the large number of the training patterns and the complexity of the problem a two hidden layers was used and trained with the back propagation method. It is the best known training algorithm for neural networks, and still one of the most useful. This algorithm is based on the computation of the gradient of error, and aims to minimize error in output network.
Results The figure 4a illustrates the correlation between the true horizontal dynamic stiffness k11 of the foundation and predicted by the model for the training process. The correlation is excellent with an absolute error of prediction inferior to 1 %. On the twenty tables, note us that 99, 93 % of the examples were predicted. On the database of validation (testing), the same remarks were observed with however less of precision (fig. 4b). On the four tables that this database composes, we note that 99, 77 % of the examples were predicted. These results testify about the capabilities of generalization of adopted (ANN). On the figure 5a, we show the correlation between true horizontal damping C11 and predicted by the model based on the database of training. It is noticed clearly that the model predict damping with an excellent precision. Indeed, 99, 92 % of the data of the twenty four tables are predicted. On the database of test (figure 5b), note us that 99.45 % of the data of the four remaining tables were predict but with less precision.
Figure 3. Training process
To highlight the performance of our network, we compared the real and simulated values predicted by the (ANN). Due to space limitations, we present in Tables 1 and 2 only the first ten values.
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(a) Training
(b) Test
Figure 4. Correlation between actual dynamic stiffness and predicted by the ANN Model
(a) Training
(b) Test
Figure 5. Correlation between real damping and predicted of (ANN) Model Table 1. Comparison between simulated and real values (horizontal stiffness kxx) a0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Real (kxx) 5.92 5.95 5.91 5.86 5.81 5.76 5.71 5.62 5.53 5.42
Simulated 5.929 5.949 5.902 5.845 5.797 5.752 5.696 5.621 5.528 5.425
Error (%) 0.9 0.03 0.82 1.53 1.24 0.78 1.42 0.1 0.14 0.5
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Table 2. Comparison between simulated and real values (horizontal damping cxx) a0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Real ( cxx) 8.82 10.2 11.70 13.20 14.60 16.10 17.60 19.00 20.50 22.00
Simulated 8.84 10.25 11.67 13.12 14.60 16.07 17.56 19.03 20.50 21.97
Error (%) 2.08 5.3 2.73 7.2 0.1 2.3 4.1 3.3 0.6 2.3
Summary The purpose of this study is to develop a model able of predicting the analysis of a problem of interaction soil– structure problem without passing by sophisticated numerical methods established in the calculation codes (example: calculation of the impedances functions of a foundation), by using a statistical approach based on the concept of artificial neuronal networks (ANN). For this reason a database was constituted based on the tables of impedances functions by [1], linking up the input parameters (geometry of foundation, mechanical characteristics of the soil, frequency of excitation) with the dynamic impedance function of foundation. At the end of the training, the set up model was considered satisfactory. This model allows predicting the impedance (all modes) of a rigid rectangular foundation resting on the surface of a semi-infinite soil with an excellent precision (± 0.5 %). This model is valid only for a rigid surface foundation, but can be developed to predict the case of embedded foundations or the case of the piles. These cases will be the object of a next paper. References [1] H.L. Wong, J.E. Luco, Tables of impedance functions for square foundation on layered media, Journal of Soils Dynamics and Earthquake Engineering. 4(2) (1985) 64–81. [2] M. Pala, N. Caglar, M. Elmas , A. Cevik , M. Saribiyik, Dynamic soil–structure interaction analysis of buildings by neural networks, Construction and Building Materials. 22(3) (2006), 330-342. [3] J.F. Jodouin, Les réseaux de neurones, Hermès, Paris, 1994. [4] G. Dreyfus, J.M. Martinez, M. Samuelides, M.B. Gordon, F. Badran, S. Thiria and L. Hérault, Réseaux de neurones, Méthodologie et applications, Eyrolles, Paris, 2004.
Progress in Civil Engineering 10.4028/www.scientific.net/AMM.170-173
Prediction of Dynamic Impedances Functions Using an Artificial Neural Network (ANN) 10.4028/www.scientific.net/AMM.170-173.3588
