This is an electronic version of an article published in: S. Jung, J. Ghaboussi and S.-D. Kwon (2004), Estimation of aeroelastic parameters of bridge decks using neural networks, ASCE Journal of Engineering Mechanics, v 130, n 11, p 1356-1364 http://dx.doi.org/10.1061/(ASCE)0733-9399(2004)130:11(1356)
ESTIMATION OF AEROELASTIC PARAMETERS OF BRIDGE DECKS USING NEURAL NETWORKS By Sungmoon Jung1, Student Member, ASCE, Jamshid Ghaboussi2, Member, ASCE, and Soon-Duck Kwon3
ABSTRACT: A new method of estimating flutter derivatives using artificial neural networks is proposed. Unlike other CFD based numerical analyses, the proposed method estimates flutter derivatives utilizing previously measured experimental data. One of the advantages of the neural networks approach is that they can approximate a function of many dimensions. An efficient method has been developed to quantify the geometry of deck sections for neural network input. The output of the neural network is flutter derivatives. The flutter derivatives estimation network, which has been trained by the proposed methodology, is tested both for training sets and novel testing sets. The network shows reasonable performance for the novel sets, as well as outstanding performance for the training sets. Two variations of the proposed network are also presented, along with their estimation capability. The paper shows the potential of applying neural networks to wind force approximations. 1
Graduate Research Assistant, Dept. of Civil Engineering, Univ. of Illinois at Urbana-Champaign, Urbana, IL,
email:
[email protected] 2
Professor, Dept. of Civil Eng., Univ. of Illinois at Urbana-Champaign, Urbana, IL, email:
[email protected]
3
Assistant Professor, Dept. of Civil Eng., Chonbuk National Univ., Chonbuk, Korea, email:
[email protected]
KEYWORDS: Aeroelasticity; Bridges; Bridge decks; Flutter; Numerical analysis; Neural networks;
INTRODUCTION Flutter is dynamic instability due to the interactions between structural motion and aerodynamic forces. In case of a bridge deck, for example, wind moves the deck and the movement changes the wind flow, which in turn will affect the movement of the deck. When these interactions cause a diverging oscillation, the phenomenon is called flutter. Several forms of aeroelastic models and normalization methods are available for analyzing flutter. The equation proposed by Scanlan (Simiu and Scanlan 1996) is employed in this paper. The equation consists of structural dynamic properties and aeroelastic parameters. In the aerodynamic design of bridges, the structural dynamic properties are known with a reasonable degree of accuracy, whereas the aeroelastic parameters are unknown. This paper discusses application of artificial neural networks in estimating aeroelastic parameters. The parameters are often called flutter derivatives in wind engineering practice. For bluff bodies that are common in bridge decks, it has not yet been possible to develop analytic expressions for the flutter derivatives. In the case of thin airfoils, the behavior is relatively simple and analytic equations have been proposed as early as 1935 (Theodorsen 1935). However, for bluff bodies such as bridge deck sections, by far the only reliable way of obtaining flutter derivatives is to perform wind tunnel test. In recent years, Computational Fluid Dynamics (CFD) has been used as an alternative method or to complement the wind tunnel tests. CFD requires powerful computing environments; consequently, its application in wind force calculation is fairly recent.
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Recent research has produced several successful applications of CFD in wind-structure interaction. These applications generally model the wind flow domain surrounding elastically supported rigid body. Very long computational time is required even on the fastest super computers because the method requires long transient flow analysis until the aerodynamic instabilities occurs. Furthermore, the wind velocity should be refined to cover the required wind velocity range. Indirect CFD analysis is an alternative that greatly reduces computational time. Indirect method is similar to wind tunnel test procedure. The method extracts flutter derivatives from the aerodynamic forces acting on a rigid body under prescribed harmonic motion. Larsen and Walther (1997) proposed an effective computational scheme for the flutter derivatives of bridge deck sections based on discrete vortex method. Jeong and Kwon (2000), Shimada et. al (2001), and Shirai and Ueda (2001) used finite element method, finite difference method, and finite volume method respectively. Most of the research results show the feasibility of CFD as an alternative to wind tunnel test or a complement to it. Recently, they have been used in the preliminary design of some bridge decks of simple polygon shapes. Although CFD analysis has its potential, certain characters of bluff body flutter are not completely understood yet (for example, turbulence), and there still remain great challenges to model the phenomenon solely by numerical simulation. Wind tunnel test is essential to capture the uncertain part. The motivation for this paper is to utilize existing test data, which contain invaluable information about flutter mechanics. The authors propose a novel approach of estimating flutter derivatives using artificial neural networks. The proposed method estimates flutter derivatives using previously measured experimental results. It utilizes function approximation capability of multi-layer feed-forward networks. The method is fundamentally
3
different from CFD based methods in two aspects. First, it utilizes previously measured experimental results. Most of the newly designed bridge sections are not much different from the previously used sections. Therefore, approximations based on the cumulated existing data can be expected to give us acceptable results. Secondly, it is based on soft computing method rather than strict mathematical formulation. Imprecision tolerance of artificial neural networks is a beneficial characteristic when we solve wind engineering problems. Wind tunnel test data contain noise, and neural networks are not sensitive to them. In fact, they generally perform better when the data contains noise (Ghaboussi 2001). Averaging over the noisey data smoothes the target function and prevents the network from overfitting of the training data (Reed and Marks 1999). After a brief discussion of the function estimation capability of artificial neural networks, the proposed methodology is presented, followed by an illustrative example.
NEURAL NETWORKS AS A NONLINEAR MAPPING SYSTEM The fundamental idea of neural networks approach is to describe a complex system using combinations of many small units. Inspired by neuroscience, simplified structure of biological neurons is used in designing artificial neurons. Fig. 1 shows an artificial neuron. Outputs from other units are transmitted along incoming connections and a weighted sum is calculated. The weight sum is passed through a nonlinear function called the sigmoid function.
Oi = g ∑ wijξ j j
(1)
4
Where ξj is output from other units, Oi is output of the current unit, wij is connection weight, g is sigmoid function. Typical choice for the sigmoid function is hyperbolic tangent function (tanh), which is also used in this study. Multi-layer feed-forward networks are layers of artificial neurons. The networks consist of an input layer, one or more hidden layers, and an output layer. Each layer has sets of neurons. Signals travel from the input layer to the output layer. Multi-layer feed-forward networks have the capability of constructing a statistical model of the given data sets. Training is a procedure whereby the connection weights of the networks are determined using given input and output sets (training sets). The trained neural network is tested to examine its performance. When determining the connection weights, back-propagation algorithm is commonly used. Error of a training set is expressed as E (w ) =
1 [ζ i − Oi ]2 ∑ 2 i
(2)
Where i is output node index, ζ is target pattern, O is neural networks output. Error is reduced by updating the weights using the gradient descent method:
∆wij = −η
∂E ∂wij
(3)
Where η is learning rate. In back-propagation algorithm, first an input pattern is passed through the neural network, and the output error is back propagated and the connection weights are updated. The details of the algorithm can be found in Hertz et. al (1991). Since the standard back-propagation algorithm is very slow in error minimization, one of the faster converging variations, rprop (Riedmiller and Braun 1993), is used in this study.
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Function Approximation Using Neural Networks Polynomials are often the preferred choice for function approximation. They can also be used when the problem variables are vectors. For example, if the terms of the input vector are x1, x2, …, x5, and there is only one output variable y, the 3rd order polynomial for the mapping is: 5
5
5
5
5
5
y = a 0 + ∑ ai xi + ∑∑ aij xi x j + ∑∑∑ aijk xi x j xk i =1
i =1 j =1
(4)
i =1 j =1 k =1
Assuming that the 3rd order polynomial can reasonably approximate the input-output relationship, 156 coefficients have to be determined in the above equation. On the other hand, a 5–5–5–1 neural network can approximate the same relationship with only 55 connection weights. As the dimensionality of the input space grows, the difference will be more significant. The real power of using neural networks in function approximation appears when both the input space and the output space are multi-dimensional. Neural networks represent a nonlinear function of many variables in terms of superpositions of many nonlinear functions (sigmoid functions) of a single variable (weight). When we train neural networks, the weights adapt to the data set as part of the training process. As a result, the number of required weights grows as the complexity of the problem grows, not as the dimension of the input space grows. The downside is the nonlinear functions of adaptive parameters. The procedure for determining parameters is computationally intensive and there may be multiple minima in the error function (Bishop 1995). Another fundamental point is that the neural network function approximations are non-universal; they are only valid within the range of variables covered by the training data set. The results become less meaningful away from that range (Ghaboussi 2001). It has been proven that any continuous function can be uniformly approximated by a continuous neural network having only one hidden layer, with an arbitrary continuous sigmoidal 6
nonlinearity (Cybenko 1989). However, in solving real world problems, it is often essential to have more than one hidden layers. This is because for many problems an approximation with one hidden layer would require an impractically large number of hidden units, whereas an adequate solution can be obtained with a reasonable network size by using more than one hidden layer (Hecht-Nielsen 1989). It is commonly accepted that using two hidden layers often leads to satisfactory results for most real world problems. Function Approximation in the Proposed Method Unsteady aerodynamic forces acting on bridge decks are expressed as a combination of motion dependent terms. The most commonly used form for unsteady aerodynamic forces are (Simiu and Scanlan 1996),
Lh =
1 h& Bα& h ρU 2 B KH 1* + KH 2* + K 2 H 3*α + K 2 H 4* 2 U U B
(5)
1 h& Bα& h ρU 2 B 2 KA1* + KA2* + K 2 A3*α + K 2 A4* 2 U U B
(6)
Mα =
Where Lh the lifting force, Mα the moment, h the vertical displacement, α the angular rotation, ρ the air density, U the velocity of the wind, B the deck width, K = Bω/U is the reduced frequency, and ω is the circular frequency of oscillation. Hi* and Ai* (i = 1,2,3,4) are called flutter derivatives. As was mentioned earlier, for a special case of streamlined sections, flutter derivatives can be analytically calculated. Theodorsen had developed theoretical expressions for unsteady aerodynamic forces of a harmonically oscillating flat plate airfoil (Theodorsen 1935). If we express the forces in terms of the flutter derivatives (Simiu and Scanlan 1996),
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H 1* = −
2πF K
π
GK H = − 2 2 F − K 2 * 3
A1* =
πF 2K
π
KG A = F− 2 2K 4 * 3
π
4G 1+ + F 2K K π 4G H 4* = 2K
H 2* = −
π K
KF −G − 2K 4 4 πG A4* = − 2K
A2* = −
(7)
2
(8)
Where F and G are real and imaginary part of Theodorsen’s circulation function, which is calculated from Bessel functions of first and second kind. The above equations imply that flutter derivatives (and corresponding aerodynamic forces) of a thin airfoil at a certain wind velocity can be determined from the reduced frequency K and the width B: Fd [ thin airfoil ] = f 0 ( K , B )
(9)
Where Fd = { Hi*, Ai* } (i = 1,2,3,4). We note that though the flutter derivatives are functions of both K and B, the only actual unknown variable is the width B because the reduced velocity K for a certain wind velocity is calculated throughout the flutter analysis. On the other hand, in case of bluff bodies, it is not yet possible to calculate the flutter derivatives explicitly, although we know that they are functions of the reduced frequency K and the section properties Sp:
Fd [ bluff body ] = f1 ( K , S p )
(10)
The goal of this paper is to estimate flutter derivatives Fd = { Hi*, Ai* } using function approximation capability of artificial neural networks. The input to the neural network is K and section properties, and the output is flutter derivatives. Neural networks enable us to define section properties in various ways, which will be presented in the next section.
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DATA PREPERATION AND PROCESSING Table 1 summarizes the experimental data used in this study. The sections are closed polygons symmetric with respect to the center line and only half of the section is shown in the table. The experimental data used in the study can be roughly divided into 3 groups. The first group is rectangular sections. The shape is fixed as rectangular and only deck width to height ratio (B/D ratio) is varied from 5 to 20. The second group is rectangular sections with center barrier, and the third group is various remaining sections. One section from each group is used as novel cases in testing. Those novel cases were chosen to represent characteristics of each group, but no systematic criteria were used in selecting the novel cases due to limited number of experimental data. 1694 sets processed from 14 sections are used in training and testing, 363 sets from 3 sections that are not included in the training are used as novel cases in testing. Each set includes processed section geometry and nondimensional velocity as neural network input, and processed flutter derivatives as neural network output. Nondimensional velocity U/fB is used instead of reduced frequency K = Bω/U, to follow wind engineering practice where flutter derivatives are expressed with respect to nondimensional velocity. Description of Deck Geometry Only half of the section is used to reduce the number of input nodes. All sections in Table 1 are first scaled to have the same width, and then are placed on the rectangle of 3 to 2 ratio. Fig. 2 illustrates the positioning of the section within the rectangle. The deck geometry can be described in different ways. One approach tried in the study was pixel description. The rectangle explained above was divided into 45 by 30 small squares. A 9
square was set to on (input value of 1.0) if it was fully or partially occupied by the section, otherwise, it was set to off (input value of -1.0). This procedure required 1350 neural network input nodes, one per square. However, the neural network failed to converge to a reasonable solution, even when combined with a data compression technique which had been successfully used in another research (Ghaboussi and Lin 1998a). The difficulty can be explained using the concept of curse of dimensionality (Bellman 1961). If the dimension of input space increases linearly, the complexity of the approximation increases exponentially. Although neural networks effectively deal with high dimensions, having too high dimensional space is not desirable. However, if we want to reduce the dimension of the input space, we should be cautious not to lose critical information, which may lead to poor performance. Another critical defect of the pixel representation is that small difference of deck geometry is disregarded due to large number of input nodes. The proposed method measures height of top and bottom surfaces from the horizontal center line. This is a simple but very effective method of describing the deck sections. Fig. 2 summarizes the deck section processing in the proposed method. The height of 45 points along the top surface and 45 points along the bottom surface are measured. These 90 nodes and 10 nodes for the nondimensional velocity compose the 100 input nodes of the flutter derivatives estimation network. Compared to the pixel representation, this method greatly reduces the number of input nodes; therefore avoid the problem of curse of dimensionality. It also does not lose important characteristics of the given section, which is crucial in the success of the neural networks training.
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Flutter Derivatives Processing Flutter derivatives in this paper follow notation of equation (5) and (6). Sometimes modified form of the equation was used in experiment, for example, half deck width b instead of full deck width B was used in expressing lifting force and moment. First the various forms of flutter derivatives experimental data were reduced to a uniform set. Then using piecewise cubic Hermitian interpolation polynomials, sets of nondimensional velocity versus flutter derivatives were generated. Interpolation was necessary because number of velocity measurements in some experiments was too small compared to the others. Interpolation equalizes the contribution of each experiment to the neural network approximation. Since some of the derivatives were measured only up to around U/fB = 12, this estimation range was also used in this study. For two of the experiments, due to early divergence, the experiment stopped before U/fB = 12. In these cases, extrapolation was used to complete the remaining part of the estimation range. Nearly all data was processed using interpolation. Extrapolation was only used over small ranges. The interpolation interval was ∆U/fB = 0.1, so the number of generated sets are 121 for each section. For example, the nondimensional velocity input of the first data set is 0.0, the second data set is 0.1, and the third data set is 0.2, and so on. Section property input does not change for all data set of the same section. The output nodes are flutter derivatives of the corresponding wind velocity.
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THE NEURAL NETWORK MODEL FOR ESTIMATION To obtain a good statistical model of the given problem using multi-layer feed-forward networks, the most important thing is to properly define the input and output nodes so that they can adequately reflect the nature of the problem. The next important parameters are the number of hidden nodes and the number of training iterations. In implementing equation (10) using neural networks, section properties can be abstracted in various ways to be used as neural networks input. Several different forms of section representation as well as different number of hidden nodes have been trained and tested in this study. The following error measures are used to systematize the testing process. 1 2N d
[ζ µ − O µ ] ∑∑ µ
(11)
1 = 2N µ
[ζ µ − O µ ] ∑∑ µ
(12)
E deck =
E avg
Nd
Ni
2
i
=1 i =1
Nµ
i
Ni
=1 i =1
2
i
i
Where, µ training set index, i output node index, Nd number of training sets for one deck, Nµ number of training sets that we want to average, Ni number of output nodes, ζ target pattern (i.e., flutter derivatives from experiment), O neural networks output (i.e., flutter derivatives from neural networks approximation). One of the error measure graphs used in testing is shown in Fig. 3. In the shown case, the network at iteration 10000 gives the smallest Eavg for the novel set, and small enough Eavg for the training set. Then, the network at iteration 10000 is chosen to estimate the flutter derivatives. Similar error measure graphs are used for different architectures of neural networks, in order to determine the best neural network model for the given problem.
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The Flutter Derivatives Estimation Network The proposed flutter derivatives estimation network is shown in Fig. 4. The architecture is chosen after the performance evaluation procedure explained above. The input nodes are nondimensional velocity (10 nodes, U/fB) and deck geometry information (90 nodes), and the output nodes are flutter derivatives (6 nodes, Hi* and Ai*, i = 1,2,3). Same velocity is given to 10 nodes to emphasize the effect of velocity, so that the velocity information is not overshadowed by the section information of 90 nodes. Since the purpose of the example is demonstration of the methodology, no additional effort was made to find the optimal number of velocity nodes. The fact that only 6 out of 8 flutter derivatives were used was not due to the limitation of the methodology; part of the training set was from experiments where only 6 derivatives were measured. The proposed methodology can estimate any number of flutter derivatives when sufficient data is available for training the neural network. If we express the network architecture in the symbolic notation (Ghaboussi et. al 1998b), Fd = Fd NN (K, Sp; 100, 20, 20, 6)
(13)
Performance of the flutter derivatives estimation network is first tested using the data included in the training sets. This is the case when we want to retrieve the result of the previously done experiment, or when we want to estimate flutter derivatives of a new deck section that is very close to the deck sections included in the experimental results. Fig. 5 shows comparison of the flutter derivatives of neural network estimation and experimental data in the case of the rectangular section B/D = 5. Experimental results are plotted at the interpolated interval of U/fB = 2 for the clarity of the presentation. Including this case, estimation performance of the training sets is very good, part of which are shown at Fig. 6 and Fig. 7 for the center barrier case and the general section case respectively.
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Next, the network is tested using novel test data that are not included in the training sets. Fig. 8 represents estimation of the rectangular section B/D = 12.5. For this novel case the estimation performance is good. The network estimates all flutter derivatives fairly well. For some derivatives, there is small offset near zero velocity, which is also observed in other tests. This does not affect flutter analysis too much because in flutter analysis we are interested in flutter derivatives at rather high wind velocities. However, non-negligible errors are observed in the center barrier case and the general section case. Fig. 9 shows testing of the novel center barrier case. H2* and A3* show poor estimation, but estimation of other flutter derivatives are fairly good. According to the study by Matsumoto et. al (1996), A2* tends to become negative if a section is aerodynamically more stable. It is also suggested in the author’s work that center barrier generally helps the aerodynamic stability. Fig. 9 shows that the network has captured those important characteristics well, although failed to capture the characteristic of less important H2* and A3*. In the testing of the novel general section (Fig. 10), A2* is also estimated fairly well but H2* is significantly overestimated. The cause of the error of the estimation may be because of inaccuracy of the experimental data itself, but it is more likely because of limited number of training sets. Considering complex mechanics of bluff body flutter, 14 training sets with limited diversity seem to be too small. Had the network been trained with more cases, the quality of estimation could be much better. Quantified errors of the estimations using equation (11) are shown in Table 2. Estimation with Different Neural Network Architectures In addition to the flutter derivatives estimation network explained above, two more different neural networks have also been trained and tested. The first variation has 1 velocity input rather than 10 velocity inputs (Fig. 11). In symbolic notation, 14
Fd = Fd NN (K, Sp; 91, 20, 20, 6)
(14)
Ideally, this network will perform as good as the original network with 10 velocity input nodes, but there is a possibility that importance of velocity is weakened by the large number of section information nodes. The second variation utilizes data compression technique to ease the training (Fig. 12). If a network has a large number of input nodes, the network is sometimes very hard to train, mainly because of the curse of dimensionality. Moreover, in some cases, the large number of input nodes is combination of information, and then the network has to learn several different things at the same time. In those cases, dividing the training process into a couple of parts and using different networks for them may improve the performance. For example, the flutter derivatives estimation problem may be better formulated by separating the training process into two parts; first, extraction of the key features from the deck section, second, estimation of the flutter derivatives using the extracted features. This two step procedure is shown in Fig. 12 as upper and lower diagram. The upper network compresses section properties, and the lower network estimates flutter derivatives using the compressed section properties. The data compression network has same input and output nodes, and three hidden layers. After training, the output of the network will be the same as the input. The network is called replicator neural network or auto-associative neural network in this aspect. The left half of the network compresses data, and the right half of the network decompresses data. The middle hidden layer is compressed vector of the input layer. In this example, 90 nodes of section properties are compressed into 6 nodes: Scp = Scp NN (Sp; 90, 22, 6)
(15)
15
Where Scp is compressed section properties. Then, flutter derivatives are estimated using 1 velocity input node and 6 compressed section properties nodes: Fd = Fd NN (K, Scp; 7, 7, 7, 6)
(16)
In some applications, this kind of two step procedure using data compression is essential to the success of the function approximation. For example, Ghaboussi and Lin (1998a) successfully utilized data compression network in the feature extraction of accelerograms, which contain too many points to be used as neural network input. However, in other applications, it is also possible that the data compression step makes the subsequent training step more difficult. If the function mapping between the compressed input and the output is too complex, the training involves decompression of the input to reduce the complexity. Another difficulty is possible loss of key information during the data compression. In estimating flutter derivatives, small difference in the input can cause large difference in the output, and the loss of key information can greatly degrade the quality of the estimation. To prevent this problem, all 17 sections were used as training sets in the compression step, so every compressed data could recover the desired section shape. In the second step of flutter derivatives estimation, 14 sections were used in training just as other cases. We can get around the loss of section information by this procedure, but the defect is that we have to repeat the whole training whenever flutter derivatives of a new section is to be estimated. After training the network, the performance was evaluated using the training sets and the novel testing sets. For both networks, estimation performance of the training sets was very good, but estimation of the novel sets was relatively poorer. Fig. 13 and Fig. 14 show estimation of the general section novel case. In this testing set and in other testing sets as well, the estimations were poorer than the original flutter derivatives estimation network. However, it was hard to tell
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whether the first variation network or the second variation network performed better. Analysis results by far indicate that the original network performs the best, but the second variation network may perform better when there are large number of training sets, because the original network may have difficulty as the number of training sets grow. In any case, the first variation network seems to be less desirable. Table 3 shows comparison of the estimation error of each network.
CONCLUDING REMARKS A new methodology is proposed to estimate the flutter derivatives using neural networks. The methodology includes simple but efficient way of describing generic deck sections of closed polygon shape for neural networks input. The proposed network has been trained and tested using data from 14 experiments, and also tested for novel cases using 3 experiments that are not included in the training sets. The network performs very well for the training sets, and it also performs reasonably well for the novel cases. Estimation of flutter derivatives of a novel rectangular section is remarkably good, and that of a novel general section is reasonable but shows more error. The main cause of error seems to be lack of sufficient training sets. Two variations of the proposed network are also studied. One uses less input nodes for velocity, and the other uses data compression technique to reduce the dimensionality of the section properties vector. Although their performance is acceptable, the original network exceeds the variations in performance.
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In conclusion, this paper has shown the possibility of using neural networks in wind force approximation, especially in flutter derivatives estimation. Given a large number of training sets, the network is expected to perform much better. However, in order to improve the result with limited number of experimental data, the method needs to be improved in the future research. A possible approach is to improve the structure of the neural networks themselves (Ghaboussi and Sidarta 1998c) other than two variations studied in this paper. Another alternative is a hybrid method (Ghaboussi et. al 1998b) that combines CFD and neural networks to extract more information from experimental data. The proposed method is not intended to substitute wind tunnel test. However, with further improvements it can be effectively used in preliminary design.
ACKNOWLEDGEMENTS Most of the experimental data were provided by Professor Matsumoto of Kyoto University. This assistance is gratefully acknowledged.
APPENDIX. REFERENCES Bellman, R. (1961), Adaptive Control Process: A Guided Tour, Princeton University Press, Princeton, New Jersey Bishop, C. M. (1995), Neural Networks for Pattern Recognition, Oxford University Press, New York
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Cybenko, G. (1989), “Approximation by Superpositions of a Sigmoidal Function”, Mathematics of Control, Signals, and Systems, 2, 303-314 Ghaboussi, J. (2001), “Biologically Inspired Soft Computing Methods in Structural Mechanics and Engineering”, Structural Engineering and Mechanics, 11(5), 485-502 Ghaboussi, J., and Lin, C.-C. J. (1998a), “New Method of Generating Spectrum Compatible Accelerograms Using Neural Networks”, Earthquake Engineering and Structural Dynamics, 27, 377-396 Ghaboussi, J., Pecknold, D. A., Zhang, M., and Haj-Ali, R. M. (1998b), “Autoprogressive Training of Neural Network Constitutive Models”, Int. J. for Numerical Methods in Engineering, 42(1), 105-126 Ghaboussi, J., and Sidarta, D. E. (1998c), “New Nested Adaptive Neural Networks (NANN) for Constitutive Modeling”, Computers & Geotechnics, 22(1), 29-52 Hecht-Nielsen, R. (1989), “Theory of the Back-Propagation Neural Network”, Proceedings of the International Joint Conference on Neural Networks, 1, 593-605 Hertz, J., Krogh, A., and Palmer, R. G. (1991), Introduction to the Theory of Neural Computation, Perseus Books, Cambridge, Massachusetts Jeong, U.-Y., and Kwon, S.-D. (2000), “Prediction of Flutter Velocity Using Computational Fluid Dynamics”, 3rd International Symposium on Computational Wind Engineering, Birmingham, UK Larsen, A., and Walther, J. H. (1997), “Aeroelastic Analysis of Bridge Girder Sections Based on Discrete Vortex Simulations”, J. of Wind Engineering & Industrial Aerodynamics, 67&68, 253-265
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Matsumoto, M., Kobayashi, Y., and Shirato, H. (1996), “The Influence of Aerodynamic Derivatives on Flutter”, J. of Wind Engineering & Industrial Aerodynamics, 60, 227-239 Matsumoto, M., Yoshizumi, F., Yabutani, T., Abe, K., and Nakajima, N. (1999), “Flutter Stabilization and Heaving-Branch Flutter”, J. of Wind Engineering & Industrial Aerodynamics, 83, 289-299 Reed, R. D., and Marks, R. J. (1999), Neural Smithing, The MIT Press, Cambridge, Massachusetts Riedmiller, M., and Braun, H. (1993), “A Direct Adaptive Method for Faster Backpropagation Learning: The RPROP Algorithm”, 1993 IEEE International Conference on Neural Networks, 1, 586-591 Scanlan, R. H., and Tomko, J. J. (1971), “Airfoil and Bridge Deck Flutter Derivatives”, J. of Engineering Mechanics, ASCE, 97(6), 1717-1737 Shimada, K., Wakahara, T., and Satoh, H. (2001), “Prediction of Unsteady Wind Force Acting on the Separated Box Girder Cross-Section”, The Fifth Asia-Pacific Conference on Wind Engineering, Kyoto, Japan Shirai, S., and Ueda, T. (2001), “Aerodynamic Simulation by CFD on Flat Box Girder of SuperLong Span Suspension Bridge”, The Fifth Asia-Pacific Conference on Wind Engineering, Kyoto, Japan Simiu, E., and Scanlan, R. H. (1996), Wind Effects on Structures, 3rd Edition, John Wiley & Sons, New York Theodorsen, T. (1935), General Theory of Aerodynamic Instability and the Mechanism of Flutter, NACA Report No. 496
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APPENDIX. NOTATION The following symbols are used in this paper. Ai*
=
flutter derivatives;
a
=
coefficient of a general polynomial;
B
=
deck width;
b
=
half deck width;
E
=
error measure;
Edeck
=
error measure for one deck;
Eavg
=
average error measure;
Fd
=
flutter derivatives Hi* and Ai*;
F
=
real part of the Theodorsen’s circulation function;
f0
=
function describing flutter derivatives of thin airfoil;
f1
=
function describing flutter derivatives of bluff body;
G
=
imaginary part of the Theodorsen’s circulation function;
g
=
sigmoid function;
Hi*
=
flutter derivatives;
h
=
vertical degree-of-freedom;
K
=
reduced frequency;
Lh
=
lifting force of wind;
Mα
=
moment of wind;
Nd
=
number of training sets for one deck;
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Nµ
=
number of training sets that we want to average;
Ni
=
number of output nodes;
Oi, Oiµ =
output of a neural network unit;
Scp
=
compressed section properties;
Sp
=
deck section properties;
U
=
wind velocity;
w
=
connection weights vector;
wij
=
connection weight;
x
=
variable of a general polynomial;
α
=
angular degree-of-freedom;
η
=
learning rate;
ρ
=
air density;
ω
=
angular frequency;
ξj
=
output of a neural network unit;
ζ i, ζ iµ =
target pattern;
22
Characteristic
Training Set
1
Ellipse (B/D = 3)
○
2
B/D = 5
○
3
B/D = 8
○
4
B/D = 10
○
5
B/D = 12.5
6
B/D = 15
○
7
B/D = 20
○
8
B/D = 20, h/D = 1.0
○
9
B/D = 20, h/D = 0.5
○
10
B/D = 10, h/D = 0.5
11
B/D = 5, h/D = 0.25
Deck Shape Index
12 13
Diamond (B/D = 3.53) Inverse Triangle (B/D = 6)
Source
Matsumoto et. al 1996
○ ○ ○
14
Generic Deck
15
Generic Deck
○
16
Generic Deck
○
17
Generic Deck
○
Matsumoto et. al 1999
Scanlan and Tomko 1971
TABLE 1. Half Deck Shapes From 17 Experiments (B= width, D= depth, h= height of center barrier)
Deck Index
Error E(deck)
1
0.000504
2
0.001775
3
0.002054
4
0.001643
5
0.002553
6
0.000908
7
0.001627
8
0.001664
9
0.002795
10
0.033902
11
0.000999
12
0.001113
13
0.000590
14
0.073562
15
0.004041
16
0.003933
17
0.001598
TABLE 2. Estimation Error of the Flutter Derivatives Estimation Network
Original Network
Variation A
Variation B
E(avg) (training set)
0.001803
0.001309
0.012145
E(avg) (novel set)
0.036673
0.061308
0.046627
TABLE 3. Comparison of the Estimation Error of Each Network
List of Figures
FIG. 1. Schematic Diagram of an Artificial Neuron FIG. 2. Description of the Deck Geometry in the Proposed Method FIG. 3. Performance Evaluation of the Trained Network FIG. 4. Flutter Derivatives Estimation Network FIG. 5. Test of the Flutter Derivatives Estimation Network – Part of the Training set (deck 2) FIG. 6. Test of the Flutter Derivatives Estimation Network – Part of the Training set (deck 9) FIG. 7. Test of the Flutter Derivatives Estimation Network – Part of the Training set (deck 13) FIG. 8. Test of the Flutter Derivatives Estimation Network – Novel Case (deck 5) FIG. 9. Test of the Flutter Derivatives Estimation Network – Novel Case (deck 10) FIG. 10. Test of the Flutter Derivatives Estimation Network – Novel Case (deck 14) FIG. 11. Variation A: Using 1 Node for Velocity FIG. 12. Variation B: Using Data Compression FIG. 13. Test of the “Variation A” Network – Novel Case (deck 14) FIG. 14. Test of the “Variation B” Network – Novel Case (deck 14)
1 2
j
wi1 i
wi2 wij
∑
summation: linear activation: nonlinear, differentiable
FIG. 1. Schematic Diagram of an Artificial Neuron
45 measurement points +1.0
0.0
-1.0
FIG. 2. Description of the Deck Geometry in the Proposed Method
0.044
0.007 training set novel set
avg
0.005
0.041
0.004
0.04 0.003
0.039 0.038
0.002
0.037
0.001
0.036 0
10000
20000
30000
40000
(training set)
E
0.006 avg
0.042
E
(novel set)
0.043
0 50000
training iterations
FIG. 3. Performance Evaluation of the Trained Network
U/fB (10 nodes)
flutter derivatives (6 nodes)
section properties (90 nodes)
hidden layers (20 nodes each)
FIG. 4. Flutter Derivatives Estimation Network
H1* (NN) H2* (NN) H3* (NN)
H1* (experiment) H2* (experiment) H3* (experiment)
A1* (NN) A2* (NN) A3* (NN)
20
A1* (experiment) A2* (experiment) A3* (experiment)
5 4
10
3 0 2 -10 1 -20
-30
0
0
2
4
6 U/fB
8
10
12
-1
0
2
4
6
8
10
12
U/fB
FIG. 5. Test of the Flutter Derivatives Estimation Network – Part of the Training set (deck 2)
H1* (NN) H2* (NN) H3* (NN)
H1* (experiment) H2* (experiment) H3* (experiment)
A1* (NN) A2* (NN) A3* (NN)
5
A1* (experiment) A2* (experiment) A3* (experiment)
6
4 0 2 -5 0 -10 -2
-15
0
2
4
6 U/fB
8
10
12
-4
0
2
4
6
8
10
12
U/fB
FIG. 6. Test of the Flutter Derivatives Estimation Network – Part of the Training set (deck 9)
H1* (NN) H2* (NN) H3* (NN)
H1* (experiment) H2* (experiment) H3* (experiment)
A1* (NN) A2* (NN) A3* (NN)
10
5
5
4
A1* (experiment) A2* (experiment) A3* (experiment)
3 0 2 -5 1 -10 0 -15 -20
-1
0
2
4
6 U/fB
8
10
12
-2
0
2
4
6
8
10
12
U/fB
FIG. 7. Test of the Flutter Derivatives Estimation Network – Part of the Training set (deck 13)
H1* (NN) H2* (NN) H3* (NN)
H1* (experiment) H2* (experiment) H3* (experiment)
A1* (NN) A2* (NN) A3* (NN)
5
4
0
3
-5
2
-10
1
-15
0
-20
0
2
4
6 U/fB
8
10
12
-1
0
2
4
A1* (experiment) A2* (experiment) A3* (experiment)
6
8
10
U/fB
FIG. 8. Test of the Flutter Derivatives Estimation Network – Novel Case (deck 5)
12
H1* (NN) H2* (NN) H3* (NN)
H1* (experiment) H2* (experiment) H3* (experiment)
A1* (NN) A2* (NN) A3* (NN)
5
A1* (experiment) A2* (experiment) A3* (experiment)
6 5
0
4 -5
3
-10
2 1
-15
0 -20 -25
-1 0
2
4
6 U/fB
8
10
12
-2
0
2
4
6
8
10
U/fB
FIG. 9. Test of the Flutter Derivatives Estimation Network – Novel Case (deck 10)
12
H1* (NN) H2* (NN) H3* (NN)
H1* (experiment) H2* (experiment) H3* (experiment)
A1* (NN) A2* (NN) A3* (NN)
20
A1* (experiment) A2* (experiment) A3* (experiment)
4
15 3 10 2
5 0
1
-5 0 -10 -15
0
2
4
6 U/fB
8
10
12
-1
0
2
4
6
8
10
U/fB
FIG. 10. Test of the Flutter Derivatives Estimation Network – Novel Case (deck 14)
12
U/fB (1 node)
flutter derivatives (6 nodes)
section properties (90 nodes)
hidden layers (20 nodes each)
FIG. 11. Variation A: Using 1 Node for Velocity
compression
decompression
same section properties as input (90 nodes)
section properties (90 nodes)
compressed section properties (6 nodes) U/fB (1 node) flutter derivatives (6 nodes)
compressed section properties (6 nodes)
hidden layers (7 nodes each)
FIG. 12. Variation B: Using Data Compression
H1* (NN) H2* (NN) H3* (NN)
H1* (experiment) H2* (experiment) H3* (experiment)
A1* (NN) A2* (NN) A3* (NN)
15
A1* (experiment) A2* (experiment) A3* (experiment)
6
10 4 5 2
0 -5
0
-10 -2 -15 -20
0
2
4
6 U/fB
8
10
12
-4
0
2
4
6
8
U/fB
FIG. 13. Test of the “Variation A” Network – Novel Case (deck 14)
10
12
H1* (NN) H2* (NN) H3* (NN)
H1* (experiment) H2* (experiment) H3* (experiment)
A1* (NN) A2* (NN) A3* (NN)
20
A1* (experiment) A2* (experiment) A3* (experiment)
4
15 3 10 5
2
0 1
-5 -10
0 -15 -20
0
2
4
6 U/fB
8
10
12
-1
0
2
4
6
8
U/fB
FIG. 14. Test of the “Variation B” Network – Novel Case (deck 14)
10
12
