Nonlinear unsteady bridge aerodynamics: Reduced-order modeling based on deep LSTM networks

Journal of Wind Engineering & Industrial Aerodynamics 198 (2020) 104116

Contents lists available at ScienceDirect

Journal of Wind Engineering & Industrial Aerodynamics journal homepage: www.elsevier.com/locate/jweia

Nonlinear unsteady bridge aerodynamics: Reduced-order modeling based on deep LSTM networks Tao Li a, b, Teng Wu b, *, Zhao Liu a a b

School of Civil Engineering, Southeast University, Nanjing, Jiangsu, 210096, China Department of Civil, Structural and Environmental Engineering, University at Buffalo, Buffalo, NY, 14260, USA

A R T I C L E I N F O

A B S T R A C T

Keywords: Nonlinear aerodynamics Bridge LSTM Deep learning Reduced-order modeling Post-flutter

Rapid increase in the bridge spans and the attendant innovative bridge deck cross-sections have placed significant importance on effectively modeling of the nonlinear, unsteady bridge aerodynamics. To this end, the deep long short-term memory (LSTM) networks are utilized in this study to develop a reduced-order model of the windbridge interaction system, where the model inputs are bridge deck motions and model outputs are motioninduced aerodynamics forces. The deep LSTM networks are first trained using the high-fidelity input-output aerodynamics datasets (e.g., based on the full-order computational fluid dynamics simulations). With the trained LSTM networks, it has been demonstrated that the bridge motion-induced nonlinear unsteady aerodynamics forces can be accurately and efficiently predicted. Numerical examples involving both the linear and nonlinear aerodynamics are employed to explore the flutter and post-flutter behaviors of bridges with the reduced-order model based on deep LSTM networks.

1. Introduction The wind-induced effects on bridges are conventionally modeled according to the linear analysis framework (Davenport, 1962; Scanlan and Tomko, 1971). The semi-empirical, linear, unsteady model is actually a time-linearized method, where a steady-flow field (statically nonlinear) is first determined, and then a small perturbation (linear) is added on this base flow (Dowell and Hall, 2001). While the linear schemes have been applied extensively both in research and design to investigate aerodynamic and aeroelastic behaviors of bridge decks under winds, the flow separation and hence nonlinear unsteady aerodynamics is prevalent as the fluid motion around the bridge deck cannot negotiate sudden changes in the deck profile (resulting in severe adverse pressure gradients). The linear methodology does not always have a satisfactory representation of the full nonlinear equations which govern the wind-bridge interactions (Wu and Kareem, 2013a). Actually, there remain important phenomena that the current linear schemes are not able to address due to their inherent limitations, e.g., nonlinear post-flutter limit cycle oscillations (LCOs). Other nonlinear bridge aerodynamics phenomena typically observed in the wind tunnel tests include: (i) non-proportional relationship between amplitudes of the input and output, (ii) single-frequency input resulting in multiple-frequency output, (iii) amplitude dependence of aerodynamic transfer functions, and (iv) hysteretic behavior of aerodynamics forces versus angles of

attack (Wu and Kareem, 2013b). It is often believed that the nonlinear aerodynamics usually has favorable effects on the wind-induced structural response due to LCOs, however, Dowell and Tang (2002) highlighted that its unfavorable effects on bluff structures do exist. The situation may become more complicated due to potential interactions between bridge aerodynamics nonlinearities and dynamics nonlinearities (e.g., geometric nonlinearities due to large deformation, material nonlinearities or complex damping features). Furthermore, rapid increase in the bridge spans and the attendant innovative bridge deck cross-sections have placed significant importance on effectively modeling of the nonlinear, unsteady bridge aerodynamics (Kareem and Wu, 2015). With the rapid developments of central processing unit (CPU) and graphics processing unit (GPU) computing speeds, the computational fluid dynamics (CFD) simulations are widely employed to capture the nonlinear wind-induced effects on bluff structures (Wu and Kareem, 2015; Xu et al., 2016). Although the CFD schemes show a great promise of simulating nonlinear bluff-body aerodynamics, the computational effort is extremely high considering the three-dimensional nature of wakes and intensive flow separations from the structure. For example, the computation complexity can easily involve the order of 106 or more degrees of freedom. In light of the high computational efficiency and simulation fidelity, the reduced-order models have been developed in this context in the simulations of nonlinear bluff-body aerodynamics, such as the describing function, trajectory piece-wise linearization,

* Corresponding author. E-mail address: [email protected] (T. Wu). https://doi.org/10.1016/j.jweia.2020.104116 Received 26 August 2019; Received in revised form 28 January 2020; Accepted 29 January 2020 Available online xxxx 0167-6105/© 2020 Elsevier Ltd. All rights reserved.

T. Li et al.

Journal of Wind Engineering & Industrial Aerodynamics 198 (2020) 104116

Volterra series, autoregressive moving average, and artificial neural network. Improvement in the efficiency and robustness of these reduced-order models is a topic of cutting-edge research in the aerodynamics community (Lucia et al., 2004; Amsallem and Farhat, 2008; Wu and Kareem 2011, 2013c; Amsallem et al., 2012; Wu et al., 2013). The quasi-steady (QS) theory is central to a large class of nonlinear reduced-order models for bridge aerodynamics, and was first introduced to model galloping phenomenon (e.g., Parkinson and Brooks, 1961). The high amplitude oscillations of the original Tacoma Narrows Bridge deck are similar to galloping in a torsional degree-of-freedom (usually referred to as torsional flutter). Accordingly, it is convenient to illustrate this behavior by the QS theory. However, an obvious shortcoming of the QS theory is that it cannot take into consideration the unsteady features inherent in fluid-structure interaction with attendant fluid memory effects. The fluid memory indicates that the flow around the structure not only depends on the current relative states (structural motions and flow fluctuations) but also on their time histories. To this end, the Volterra series, consisting of linear and higher-order convolutions, are utilized to represent the complex mapping rules (static linear/nonlinear relationships) and time lags (fluid memory effects) between the bridge aerodynamics inputs and outputs (Wu et al., 2013; Wu and Kareem, 2014). Although the Volterra series-based reduced-order model shows great promise for the simulation of nonlinear, unsteady bridge aerodynamics, the efficient and accurate identification of higher-order Volterra kernels is quite challenging. For example, the Volterra kernel terms to be identified increase exponentially with the nonlinear degree (Wu and Kareem, 2014). On the other hand, the Volterra series has been shown to be equivalent to the recurrent neural network (RNN), and hence the identification of Volterra kernels could be accomplished through the training process of networks using back propagation of errors (Bialasiewicz and Soloway, 1990; Wray and Green, 1994). It is shown that the unfolded RNNs in time can be treated as very deep feedforward networks where all the layers share the same weights (LeCun et al., 2015). Although the Kolmogorov Neural Network existence theorem offers mathematical foundation for applying multilayered neural networks to approximate arbitrary nonlinear systems with any precision (Huang and Lippmann, 1988; Hornik et al., 1989), there existed challenges in the efficient training algorithm for deep networks. To advance the standard back propagation scheme with large training set, Hinton and Salakhutdinov. (2006) developed a fast, greedy learning algorithm to learn a deep, densely connected belief network one layer at a time, and hence the training speed of neural networks has been significantly improved. Although a time dimension is introduced into the RNN, theoretical and empirical evidence shows that traditional gradient-based networks (e.g., using standard back propagation through time) cannot reliably use information that lies more than 10 time steps in the past (e.g., Bengio et al., 1994; Gers et al., 1999). To let the networks remember inputs for a long time, an explicit memory, the long short-term memory (LSTM), was introduced to augment the network (Hochreiter and Schmidhuber, 1997). The vanishing and exploding gradients that occur in RNNs could be avoided by replacing the conventional neuron in the hidden layer with the LSTM cell, which has a recurrently self-connected linear unit called the ‘Constant Error Carousel’ and hence copies its own real-valued state and accumulates the external signal (LeCun et al., 2015). To circumvent the situation that the cell states tend to grow linearly during the presentation of a time series, Gers et al. (1999) designed an adaptive ‘forget gate’ to learn to reset memory cells once their contents are out of date, and demonstrated that the extended LSTM, whose self-connection is multiplicatively gated by another unit, successfully resolved the standard LSTM’s issue of uncontrolled growth of the internal states. It is expected that the LSTM networks enhanced by the forget gates may provide an effective modeling of nonlinear bridge aerodynamics with fading memory. In this study, the extended LSTM networks, with the structural motions as the inputs and the motion-induced aerodynamics forces as outputs, will be used to develop a reduced-order model of the wind-bridge

Fig. 1. Schematic of a LSTM memory cell.

interaction system. The deep LSTM networks are trained using the highfidelity input-output aerodynamics datasets, where the amplitude and frequency ranges of the input signals are appropriately selected. The numerical examples involving both the linear and nonlinear aerodynamics demonstrate that the bridge motion-induced nonlinear unsteady aerodynamics forces and hence the nonlinear post-flutter behaviors can be accurately and efficiently predicted with the trained LSTM networks. 2. LSTM networks The use of artificial neural networks (ANNs) has been widely explored over the last few decades as a reliable nonlinear model in many fields of engineering. ANNs have also been applied to simulate the wind field and related processes (e.g., Gurley et al., 1996; Huang and Xu, 2013). To introduce the time dimension into the network structure, the recurrent neural network (RNN) was developed to pass information across time steps. To this end, RNNs are well suited for the time series prediction where nonlinear input-output relationships exist. However, Bengio et al. (1994) presented theoretical and experimental evidences demonstrating the standard gradient-based learning algorithms in RNN face an increasingly difficult problem as the duration of the dependencies to be captured increases. A number of studies have been carried out to augment RNNs with a memory module, for example, the Neural Turing Marching (Graves et al., 2014) and memory networks (Weston et al., 2014). In this study, the long short-term memory (LSTM) introduced by Hochreiter and Schmidhuber (1997) to advance RNNs is utilized to effectively simulate the nonlinear, unsteady bridge aerodynamics. 2.1. Forward pass of LSTM The memory cell shown in Fig. 1 acts like a gated leaky neuron in the LSTM (LeCun et al., 2015), where xt represents the input at time t; ht denotes the cell output; ct and ct are the cell state and internal hidden state, respectively; ft, it and ot are the forget gate, input gate, output gate, respectively; W~ and b~ indicate the corresponding weight matrices and biases, and their dimensions are hyper parameters that are usually determined with trial and error; σ(⋅) and tanh(⋅) are the activation functions described using the logistic sigmoid 1þe1ðÞ (with range of [0 1]) and hyperbolic tangent sigmoid

eðÞ eðÞ eðÞ þeðÞ

(with range of [-1 1]), respec-

tively. The forget gate essentially removes out-of-date information from the cell state by using the nonlinear activation function and the multiplication operation. It is noted that the range of [0 1] for the activation 2

T. Li et al.

Journal of Wind Engineering & Industrial Aerodynamics 198 (2020) 104116

Fig. 2. The structure of LSTM networks.

(ii) Set information in the input gate by Eq. (1b), where it decides how much information contained in the current input will be saved into the current cell state. (iii) Set values to the hidden cell state by Eq. (1c), and calculate the cell state ct by effectively combining the previous memory and the new input [Eq. (1d)]. (iv) Set information in the output gate by Eq. (1e), where ot is actually a controller to determine the long-term memory effects on the current output. (v) Pass output to the next cell by Eq. (1f).

function of forget gate indicates that the LSTM cell can fully forget or memorize the previous state. The input gate is used to add new information to the cell state together with the internal hidden state, and the output gate decides the output information from the cell state. Accordingly, the variables in Fig. 1 can be expressed as:
 ft ¼ σ netf ; t

(1a)

it ¼ σ ðneti; t Þ

(1b)

ct ¼ tanhðnetc; t Þ

(1c)

ct ¼ ft  ct1 þ it  ct

(1d)

ot ¼ σ ðneto;t Þ

(1e)

ht ¼ οt  tanhðct Þ

(1f)

It is noted that, due to the forget gate, the LSTM cell employed in this study can learn to reset memory blocks once their contents are out of date and hence useless (Gers et al., 1999). 2.2. Back pass of LSTM The back pass of LSTM utilized here is a slightly modified, truncated version of the standard back propagation through time (BPTT) (Williams and Peng, 1990). To efficiently determine the proper weight matrices and biases in the networks of Fig. 2, the gradient of each variable in the network is determined by mathematical expressions (Goodfellow et al., 2016). As shown in Fig. 2, the typical deep LSTM networks consist of multiple LSTM layers and single output layer. The flow of the input data [x] to the output [y] can be summarized as:

where the symbol  denotes the dot product operation. The inputs for the current gate in Eqs. 1(a) to (1f) can be calculated as: netf ; t ¼ Wf ½ht1 ; xt  þ bf ¼ Wfh ½ht1  þ Wfx ½xt  þ bf

(2a)

neti; t ¼ Wi ½ht1 ; xt  þ bi ¼ Wih ½ht1  þ Wix ½xt  þ bi

(2b)

netc; t ¼ Wc ½ht1 ; xt  þ bc ¼ Wch ½ht1  þ Wcx ½xt  þ bc

(2c)

neto; t ¼ Wo ½ht1 ; xt  þ bo ¼ Woh ½ht1  þ Wox ½xt  þ bo

(2d)

(i) For the first LSTM layer at time k, Eq. (1) is adopted to calculate the cell state and output with the input hk1 and xk . It is noted that the cell state c0 and output h0 at the initial time are set to be zero. Similar calculations based on Eq. (1) are executed for other time instants. (ii) The cell outputs hk in the first LSTM layer is set to be the input of the second LSTM layer, and then the same process in the first LSTM layer is executed with different weights and bias. (iii) The cell output ht at the last LSTM layer is multiplied by the corresponding weight matrix (and adding the bias), and then mapped to the output of the networks with the activation function. Usually, the linear activation function is adopted in this layer.

The above Eqs (1) and (2) basically describe the forward pass of LSTM networks, which can be summarized as: (i) Set information in the forget gate by Eq. (1a), where ft determines how much information contained in the previous ct-1 to be kept in the current cell state.

3

T. Li et al.

Journal of Wind Engineering & Industrial Aerodynamics 198 (2020) 104116

The BPTT of LSTM is an inverse process of the above (i)-(iii) steps. The stochastic gradient descent (SGD) is used to minimize the mean square error (MSE) of the networks, which is defined as (Goodfellow et al., 2016): E¼

N N
2 1 X 1 X by  yj Ej ¼ N j¼1 2N j¼1 j

δnc;t ¼

∂Ej ∂Ej ∂cnt ∂cnt ¼ ∂netnc;t ∂cnt ∂cnt ∂netnc;t

¼ε  n t

(3)

δno;t ¼

∂Ej ∂Ej ∂hnt ∂ont ¼ ∂netno;t ∂hnt ∂ont ∂netno;t

 
 0 ¼ δnt  tanh cnt  σ net no;t

where Ej and E represent the jth and total MSE of the training samples, respectively; N is the number of training samples; yj means the exact values; b y j is the predicted value in the output layer and can be calculated by b y j ¼ Wy hnt þ by in the case that a linear activate function is adopted,

δnk ¼

For the sake of brevity, the gradients of hlk and clk at time k in the lth layer, ∂Ej =∂hlk and ∂Ej =∂clk , are defined as δlk and εlk , respectively. Simi-

¼

larly, the gradients of net lf ;k , net li;k , net lc;k and net lo;k at time k in the lth layer, ∂Ej =∂net lf ;k , ∂Ej =∂net li;k , ∂Ej =∂net lc;k and ∂Ej =∂net lo;k , are defined as δlf ;k , δli;k , δlc;k and δlo;k , respectively. Since the BPTT is a reverse process of the forward algorithm, the back pass shall be illustrated from time t to 1 in the time scale and from nth to 1st layer in the spatial scale. At time t in the nth layer, δnt and εnt can be acquired based on the chain rule:  ∂Ej
¼ by j  yj Wy ∂hnt

εnt ¼

∂Ej ∂Ej ∂ ¼ ∂cnt ∂hnt ∂

hnt cnt

 0
¼ δnt  ont  tan h cnt

þ

∂

∂

∂Ej

¼

net nf;t

∂Ej ∂cnf;t ∂f nt ∂cnf;t ∂f nt ∂netnf;t

  0 ¼ εnt  cnt1  σ net nf;t δni;t ¼

∂

∂Ej netni;t

¼ε  n t

¼

cnt

∂Ej ∂cnt ∂int ∂cnt ∂int ∂netni;t

   σ net ni;t

∂Ej net no;kþ1

εnk ¼

(4a) ¼

∂Ej ∂hnk ∂

∂Ej

net nf;kþ1

∂netnf;kþ1 ∂Ej ∂netni;kþ1 ∂Ej ∂netnc;kþ1 þ þ n n n ∂hk ∂neti;kþ1 ∂hk ∂netnc;kþ1 ∂hnk

∂netno;kþ1 0 0 0 0 ¼ Wfh δf ;kþ1 þ Wih δi;kþ1 þ Wch δc;kþ1 þ Woh δo;kþ1 ∂hnk

∂Ej ∂Ej ∂hnk ∂Ej ∂cnkþ1 ¼ þ ∂cnk ∂hnk ∂cnk ∂cnkþ1 ∂cnk δnk




  tan h cnk þ εnkþ1 f nk

(6a)

(6b)

0

onk

With the same algorithms in Eqs. (5a)-(5d), δnf;k , δni;k , δnc;k and δno;k can be calculated once δnk and εnk are known. and εn1 are acquired as: At time t in the (n-1)th layer, δn1 t t

(4b)

Accordingly, δnf;t , δni;t , δnc;t and δno;t can be obtained as: δnf;t ¼

(5d)

At time k (k ¼ t-1–1) in the nth layer, δnk and εnk are recursively determined as:

where Wy and by represent the weight matrix and bias in this layer.

δnt ¼

(5c)

   tan h net nc;t 0

int

¼ δn1 t ¼

(5a)

∂ Ej ∂hn1 t

∂Ej ∂netnf;t ∂Ej ∂netni;t ∂Ej ∂netnc;t ∂Ej ∂netno;t n n n n1 þ n1 þ n1 þ ∂netf ;t ∂ht ∂neti;t ∂ht ∂netc;t ∂ht ∂netno;t ∂hn1 t

(7a)

¼ W nfx δnf;t þ W nix δni;t þ W ncx δnc;t þ W nox δno;t

εn1 ¼ t (5b) ¼

0

∂Ej ∂Ej ∂hn1 t ¼ n1 n1 n1 ∂ct ∂ht ∂ct

δn1 t



on1 t

 tan h

0




cn1 t



(7b)

At time k (k ¼ t-1–1) in the (n-1)th layer, δn1 and εn1 are recursively k k determined as:

δn1 ¼ k ¼

∂ Ej ∂Ej ∂hn1 ∂Ej ∂hnk ¼ n1 kþ1 þ n n1 ∂hk ∂hk1 ∂hn1 ∂ hkþ1 ∂hn1 k kþ1 ∂Ej

∂

netn1 f ;kþ1

∂netfn1 ∂Ej ∂netn1 ∂Ej ∂netn1 ∂Ej ∂netn1 ;kþ1 i;kþ1 c;kþ1 o;kþ1 þ n1 þ n1 n1 þ n1 n1 þ ∂neti;kþ1 ∂hk ∂netc;kþ1 ∂hk ∂netn1 ∂hk ∂hn1 o;kþ1 k (8a)

∂Ej ∂netnf;k ∂Ej ∂netni;k ∂Ej ∂netno;k ∂Ej ∂netnc;k n n n n1 þ n1 þ n1 þ ∂netf ;k ∂hk ∂neti;k ∂hk ∂neto;k ∂hk ∂netnc;k ∂hn1 k n1 n1 n1 n1 n1 n1 n1 ¼ W n1 fh δf ;kþ1 þ W ih δi;kþ1 þ W ch δc;kþ1 þ W ox δo;kþ1 þ

W nfx δnf;k þ W nix δni;k þ W ncx δnc;k þ W nox δno;k

4

T. Li et al.

Journal of Wind Engineering & Industrial Aerodynamics 198 (2020) 104116

both the velocity and acceleration can be obtained by weighting the displacement time series in the networks. Hence, only the bridge deck displacements are utilized here to be the inputs of the LSTM networks, as shown in Fig. 3 for the two-dimensional (2D) nonlinear, unsteady bridge aerodynamics simulations. The vt and αt in Fig. 3 represent the vertical and torsional displacements of the bridge deck at time t, respectively, and Ft is the calculated aerodynamics forces at time t based on LSTM networks. Other parameters such as the batch size and learn rate can be determined based on the general principles of deep learning (e.g., Goodfellow et al., 2016). To fully characterize a linear wind-bridge interaction system, the input signals for training the LSTM networks should cover the frequency range of interest. For the nonlinear bridge aerodynamics, the information of input amplitude range of interest should be nested in the training datasets as well. Hence, the harmonic signal is adopted here as the selected inputs during the training process of the LSTM networks:

Fig. 3. The mapping relationship between input and output data.

∂ Ej ∂Ej ∂hn1 ∂Ej ∂cn1 k ¼ n1 ¼ n1 n1 þ n1 kþ1 ∂c k ∂ckþ1 ∂cn1 ∂hk ∂ck k

ε

n1 k

¼

δn1 k



on1 k

0

 tan h




cn1 k



þε

n  X
 ~ j sin ω ~jt x t ¼ A

(8b)

~ j and ω ~ j are jth amplitude and frequency where x(t) denotes v(t) or α(t); A values randomly generated within the selected ranges. Compared to the widely-used pseudo-random binary sequence (PRBS) signal in system identification (e.g., Winter and Breitsamter, 2016), the CFD outputs based on the harmonic signals present better convergence features due to the relatively slow change of the flow field (Wu and Kareem, 2015). In ~ j and ω ~ j can be appropriately selected using the addition, the ranges of A structural dynamics properties of the bridge under investigation. The pre-determined amplitude and frequency ranges can significantly reduce the CFD costs and the LSTM training efforts. In the nonlinear, unsteady aerodynamics system, the wind-induced loads at the current time depend on not only the current moving state of the deck but also its moving history due to the fluid memory effects (Wu et al., 2013). Furthermore, the fluid memory effects on bridge aerodynamics present fading feature, which means that the wind-induced loads on structures are not dependent on the inputs of infinite past (Wu and Kareem, 2014). Therefore, one can describe the implicit relationship between the aerodynamics inputs (deck motions) and outputs (motion-induced aerodynamics forces) as:

n1 n1 kþ1 f k

The gradients in the lth layer at each time can be recursively obtained as:

∂ Ej ∂W eh;k l

∂Ej ∂W ex;k l

¼

¼

∂Ej

∂netle;k

∂nete;k ∂W leh;k l

∂Ej

∂netle;k

∂nete;k ∂W lex;k l

¼ δle;k hle;k1 ; k ¼ 2et

(9a)

¼ δle;k hl1 k ; k ¼ 1et

(9b)

l ∂ Ej ∂Ej ∂nete;k ¼ ¼ δle;k ; k ¼ 1et l l l ∂be;k ∂nete;k ∂be;k

(9c)


 dWy ¼ by j  yj hnt

(9d)

dby ¼ by  y

(9e)



The final gradients over all the training examples can be calculated as the sum of the gradients at all time instants: N X t X ∂E ∂Ej ¼ ∂We j¼1 k¼1 ∂We;k

FðtÞ ¼ <

∂E ∂We

vtiþ1 ; vtiþ2 ;   ; vt1 ; vt αtiþ1 ; αtiþ2 ;   ; αt1 ; αt

 (13)

where < is an unknown nonlinear function that will be captured using the LSTM networks; the parameter i indicates the duration of the fluid memory effects on the bridge aerodynamics system that could be predetermined during the training of the LSTM networks. Wu and Kareem (2014) used the Küssner function to demonstrate that the fluid memory effects become insignificant as the nondimensional time s (Ut=b b where U

(10)

Here W~ represents the weight matrices or the biases. Based on the obtained gradients of the LSTM, the SGD can then be executed by: We ¼ We  lr

(12)

j¼1

is wind speed and b b is half of deck width) is at 20. In present work, the parameter i is determined by a fluid memory length of 25 nondimensional time to accurately and efficiently simulate the nonlinear, unsteady bridge aerodynamics systems using the LSTM networks.

(11)

where lr is the learn rate usually in range of [0.01 0.25]. With the BPTT algorithm, one can acquire the gradients of the networks and accordingly update the weights using Eq. (11). The training process stops as the MSE is smaller than the pre-determined threshold.

3. Numerical example 3.1. Governing equation of bridge deck motion

2.3. Training data design

A unit-length section of a bridge deck subjected to the uniform wind flow is considered here as a two-degree-of-freedom system with vertical and rotational motions. Accordingly, the governing equations of motion are written as:

The selection of proper input variables in the training of the LSTM networks for nonlinear, unsteady bridge aerodynamics is a critical issue. In conventional semi-empirical linear or quasi-steady nonlinear models (Scanlan and Tomko, 1971; Diana et al., 1999), both the displacement and velocity of the bridge deck moving states are considered. The deck acceleration is also occasionally included in the inputs to consider the added mass effects on bridge aerodynamics. However, it is noted that


 m v€ þ 2ωv ξv v_ þ ω2v v ¼ FL ðtÞ

5

(14a)

T. Li et al.

Journal of Wind Engineering & Industrial Aerodynamics 198 (2020) 104116

Table 1 The frequency-independent coefficient values for flat plate. Cvv 2

Cv1α

Cv3α

Cvv 1

Value

0.0197

1.5693

0.6458

2.8083

1.1564

1.0402

0.4953

Coefficient for torsional moment Value

Cα1v 0.0007

Cα2v 0.3921

Cα3v 0.1614

Cαα 1 0.7000

Cαα 2 0.1079

Cαα 3 0.2547

rα 0.5225


 € þ 2ωα ξα α_ þ ω2α α ¼ MT ðtÞ I α

Cvv 3

Cv2α

Coefficient for lift force

rv

frequency hybrid expressions in Eqs. (15a) and (15b) can be transformed into pure time domain with Roger’s approximation (Roger, 1977):

(14b)

where m and I are the effective mass and moment inertia of the bridge deck section, respectively; ωv and ωα are the heaving and torsional circular frequencies; ξv and ξα are the heaving and torsional mechanical damping ratio; v€, v_ and v are the vertical acceleration, velocity, and €, α_ and α are the torsional acceleration, velocity, and displacement; α displacement; FL(t) and MT(t) denote the motion-induced lift force and torsional moment, respectively.

2 FL ¼ ρU

vv B C vv 1 vðtÞ þ C 2

24

3

Z

U

_ þ C vv vðtÞ 3

t

þ ρU

B4C v1α

B ðtÞ þ C v2α

α

U

_ ðtÞ þ C v3α

α

_ τÞe vð

0

2 2

rv U B ðτtÞ

Z 0

dτ5 3

t

rv U B ðτtÞ

α_ ðτÞe

dτ5

3 Z t rα U αv αv B αv ð τ tÞ 4 _ þ C3 vð _ τÞe B d τ5 MT ¼ ρU B C 1 vðtÞ þ C 2 vðtÞ U 0 2 3 Z t rα U ð 2 2 4 αα αα B αα τ tÞ α_ ðτÞe B dτ5 þ ρU B C1 αðtÞ þ C 2 α_ ðtÞ þ C 3 U 0

(16a)

2

3.2. Linear aerodynamics case

2

The lift force and torsional moment per unit span of the bridge deck based on the semi-empirical linear analysis framework can be expressed as (Scanlan and Tomko, 1971):  v_ 1 Bα_ v FL ¼ ρU 2 ð2BÞ KH *1 þ KH *2 þ K 2 H *3 α þ K 2 H *4 U 2 B U

(15a)


 v_ 1 Bα_ v MT ¼ ρU 2 2B2 KA*1 þ KA*2 þ K 2 A*3 α þ K 2 A*4 U 2 B U

(15b)

(16b)

pq pq where the frequency-independent coefficients Cpq 1 , C2 , C 3 and rm (p, q and m denoting ‘v’ or ‘α’) can be identified using nonlinear optimization in a least-squares sense to approximate the experimentally obtained flutter derivatives.

where ρ is the air density; B represents the deck width; Hi* and Ai* (i ¼ 1, …, 4) are the flutter derivatives estimated at the reduced frequency K¼Bω/U; ω is the circular frequency of bridge deck vibration. The time-

3.2.1. Theoretical solution of linear unsteady aerodynamics For the sake of simplicity, the frequency-independent coefficients are

Fig. 4. The theoretical and fitted flutter derivatives of the flat plate.

Fig. 5. Selected input signal of the flat plate. 6

T. Li et al.

Journal of Wind Engineering & Industrial Aerodynamics 198 (2020) 104116

Fig. 6. Validation of the trained LSTM networks for linear, unsteady aerodynamics at U ¼ 100 m/s.

determined based on the flutter derivatives of a flat plate, as listed in Table 1. The fitted flutter derivatives based on the identified coefficients pq pq (Cpq 1 , C2 , C 3 and rm ) are compared with the theoretical solution of the flat plate (e.g., Theodorsen, 1935) in Fig. 4 [where f (Hz) is the oscillation frequency of bridge deck], and the excellent agreement indicates the numerical approximations in Table 1 provide satisfactory results.

frequency-independent coefficient values in Table 1. Totally, 13,000 input-output datasets under the wind speed of 100 m/s have been generated to train and validate the reduced-order modeling of the linear, unsteady aerodynamics system based on the deep LSTM networks. Accordingly, the LSTM networks are built on Tensorflow (GitHub, 2018) with single LSTM layer and 25 cells are employed based on trial and error approach. In the training process, the first 12,000 datasets are used. After 1500 learning cycles, the LSTM networks have actually been appropriately trained. With the trained LSTM, the validations are conducted using the left 1000 datasets at the wind speed U ¼ 100 m/s. As shown in Fig. 6, the trained LSTM can accurately predict the new validation data.

3.2.2. Reduced-order modeling of linear unsteady aerodynamics To obtain the linear, unsteady aerodynamics datasets for training/ validating the LSTM networks, the vertical and torsional motions described in Eq. (12) are utilized to be the input signals as shown in Fig. 5 (where the frequency range is selected as [0.75 rad/s 12.5 rad/s] to cover the structural vibration frequencies in Sect. 3.2.3) and the simulation time interval is set to be 0.05s. The linear, unsteady relationship between the flat plate motion inputs and the motion-induced aerodynamics outputs are fully characterized by Eqs. (16a) and (16b) together with the

3.2.3. LSTM application of linear unsteady aerodynamics Based on the trained LSTM, Eqs. (14a) and (14b) can be iteratively solved with an arbitrarily selected set of dynamics parameters of the flat plate as: m ¼ 10000 kg, I ¼ 2800000 kg m2, ωv ¼ 2.2 rad/s, ωα ¼ 3.77

Fig. 7. The simulated and theoretical torsional response time history. 7

T. Li et al.

Journal of Wind Engineering & Industrial Aerodynamics 198 (2020) 104116

Fig. 8. CFD approach of nonlinear, unsteady bridge aerodynamics.

rad/s, ξv ¼ ξα ¼ 0.005, and the given initial state of α_ ¼ 0:5 and α ¼ v ¼ v_ ¼ 0. The structural motion input vector x(t) ¼   vtiþ1 ; vtiþ2 ⋯vt1 ; vt is first generated, and then the aerodynamics αtiþ1 ; αti ⋯αt1 ; αt force output vector F(t) ¼ [FL(t); MT(t)] can be predicted using the trained LSTM. It is noted that the input x(t) for t < 0 can be set to be zero. The predicted F(t) is utilized to solve Eqs. (14a) and (14b) using the New-

precludes the formulation of an analytical expression of the bridge aerodynamics. In this study, the CFD approach is treated as a full-order modeling of the nonlinear, unsteady bridge aerodynamics. Accordingly, the high-fidelity input-output aerodynamics datasets obtained from CFD scheme are employed to train and validate the LSTM networks. 3.3.1. Computational fluid dynamics of nonlinear unsteady aerodynamics The Eulerian description is favored in fluid dynamics, while a Lagrangian description is preferred in structural dynamics. Since a moving interface between the wind and the rigid bridge deck exists in the current simulations, an arbitrary Lagrangian-Eulerian (ALE) description for the fluid domain is employed. Accordingly, the fluid near the bridge deck is described utilizing Lagrangian framework while the traditional Eulerian description of fluid far from the deck remains. Accordingly, the governing Reynolds averaged Navier-Stokes (RANS) equations for 2D, incompressible, viscous flow in the ALE form is expressed as:

_

mark-β algorithm. If the newly obtained F ðtÞ [based on the solutions of _

Eqs. (14a) and (14b)] satisfies jF ðtÞ  FðtÞj  104 , the simulation moves _

to the next time step. Otherwise, set F(t) equal to F ðtÞ for the iteration process. The simulation results of the convergent, constant and divergent oscillations corresponding to the wind speed U ¼ 100 m/s, U ¼ 125 m/s and U ¼ 135 m/s are obtained and the torsional responses of the bridge deck (with flat plate aerodynamics) are presented in Fig. 7. Compared to the exact solution, the simulation results indicate that the trained LSTM networks can be used to effectively and accurately solve the linear, unsteady bridge aerodynamics.










∂ρui ∂ρ uj  us ui ∂ρui ui ∂p ∂ ∂ui þ þ ¼ þ μ þ Si ∂t ∂zj ∂zi ∂zi ∂zj eff ∂zj

(17)

with mass conservation:


3.3. Nonlinear aerodynamics case



∂ui ∂ uj  us þ ¼0 ∂zi ∂zj

For the bridge deck with a bluff cross-section, the nonlinear, unsteady characteristics resulting from the flow separation, reattachment around the deck and three-dimensional (3D) nature of the ensuring wake

(18)

where p is the pressure field; ui and uj are the velocity components; us is 8

T. Li et al.

Journal of Wind Engineering & Industrial Aerodynamics 198 (2020) 104116

Fig. 9. Selected input signal of the bridge deck.

Fig. 10. Validation of the trained LSTM for nonlinear, unsteady aerodynamics at U ¼ 13 m/s.

the velocity of the moving boundary; zi and zj are the Cartesian spatial coordinates; Si indicates possible additional momentum source contribution; μeff is the effective viscosity including laminar and turbulent

contributions. In this study, the CFD is carried out using the AcuSolve commercial software, where the spatial domain is discretized with the finite element method (FEM). The Galerkin least-squares method is

Fig. 11. The comparation between the LSTM-based predictions and CFD simulations at U ¼ 9 m/s. 9

T. Li et al.

Journal of Wind Engineering & Industrial Aerodynamics 198 (2020) 104116

Fig. 12. The comparation between the LSTM-based predictions and CFD simulations at U ¼ 11 m/s.

the time interval in the transient simulations is set to be 0.005s. The grid and time-step independent studies have been carried out to ensure the computationally efficient spatial and temporal discretization strategies were used in the simulations. Although the accuracy of the CFD scheme employed here needs further improvement for general applications of bluff-body aerodynamics, it seems to offer acceptable accuracy in the simulation of motion-induced aerodynamics forces. This observation has been widely demonstrated in the literature of bridge aerodynamics (e.g., de Miranda et al., 2014). In addition to the verification using the

utilized with the Spalart-Allmaras turbulence model to solve the discretized governing equations. The steel box cross-section of Taohuayu self-anchored suspension bridge in China (main span length of 406 m) is employed here as a case study. The computational domain and boundary conditions are schematically shown in Fig. 8(a), the dimension of the deck cross-section with a geometrical scale of 1:50 is presented in Fig. 8(b), and a close-up of the mesh generated around the deck section with an initial attack angle of 5 is given in Fig. 8(c). The distance of the first grid layer to the bridge deck surface is 0.0001m to ensure yþ<4, and

Fig. 13. The comparation between the LSTM-based predictions and CFD simulations at U ¼ 13 m/s. 10

T. Li et al.

Journal of Wind Engineering & Industrial Aerodynamics 198 (2020) 104116

Fig. 14. The comparison of LSTM-based and CFD-based LCO amplitudes.

flat-plate case (comparing the CFD response with theoretical aerodynamics), the CFD results of the bridge-deck case (i.e., post-flutter behaviors in terms of LCO amplitudes) have been also well compared with the numerical simulations from Ying et al. (2017). All these observations indicate that the adopted CFD approach presents acceptable accuracy in the simulation of motion-induced effects on bridge deck.

Figs. 11–13, together with the corresponding CFD simulations (free vibration scenarios). After the transient response stage (around the first 3s), it is noted that excellent agreement between the LSTM-based predictions and the CFD simulations is achieved. It is well known that there is a divergent response in the linear flutter analysis beyond the critical wind speed, however, the limit-cycle oscillations (LCOs) are obtained at U ¼ 11 m/s and U ¼ 13 m/s in the case study. This observation indicates a nonlinear post-flutter behavior of the employed bridge deck. Furthermore, obvious higher-order harmonics appearing in the aerodynamics lift force and torsional moment at U ¼ 11 m/s and U ¼ 13 m/s explicitly present the nonlinear features in the bridge aerodynamics system. Five LSTM-based LCO amplitude predictions corresponding to the wind speed in the range of 9m/s-13 m/s are given in Fig. 14 together with the CFD simulations, and good agreement is observed between the reduced-order and full-order modeling results. It will be more challenging for the LSTMbased reduced-order model to accurately capture the bridge aerodynamics resulting from more advanced CFD simulation (e.g., large eddy simulation), where more complicated nonlinearities may exist. In such a case, other techniques (e.g., adaptive data-driven model reduction) need to be used to further enhance the developed LSTM network (Peherstorfer and Willcox, 2016).

3.3.2. Reduced-order modeling of nonlinear unsteady aerodynamics To obtain the nonlinear, unsteady aerodynamics datasets for training/validating the LSTM networks, the vertical and torsional motions described in Eq. (12) are utilized to be the input signals as shown in Fig. 9 (where the frequency range is selected as [9.5 rad/s 31 rad/s] to cover the vibration frequencies of bridge deck in Sect. 3.3.3). The nonlinear, unsteady relationship between the bridge deck motion inputs and the motion-induced aerodynamics outputs are fully characterized by the CFD simulations discussed in Sect. 3.3.1. Totally, 14,880 input-output datasets have been generated using CFD simulations to train and validate the reduced-order modeling of the nonlinear, unsteady aerodynamics system based on the deep LSTM networks. Accordingly, the LSTM networks are built on Tensorflow (GitHub, 2018) and the structure of the network is set as three LSTM layers. Through trial and error, the weight matrix dimensions are set as 20, 10 and 5, respectively. Totally, 2500 learning cycles are performed on the first 13,000 datasets, and the validations are conducted using the left 1880 datasets at the inlet velocity U ¼ 13 m/s. As shown in Fig. 10, the excellent aggrement between the LSTM-based results and CFD simulations indicates that the trained LSTM can be used as an accurate reduced-order model of the nonlinear, unsteady bridge aerodynamcis system.

4. Concluding remarks In this study, the deep LSTM networks are utilized as a reduced-order modeling of the nonlinear, unsteady bridge aerodynamics, where the model inputs are bridge deck motions and model outputs are motioninduced aerodynamics forces. The numerical examples demonstrate that the LSTM networks, after trained well using the harmonic input signals with appropriately selected frequency and amplitude ranges, could efficiently (compared to the CFD simulations) and accurately (compared to the linear models) predict the nonlinear, unsteady bridge aerodynamics. The trained LSTM can be conveniently used to calculate the deck vibrations under the nonlinear, unsteady wind loads. The LSTMbased simulations of LCO amplitudes indicate it may be possible that the design wind speed for long-span bridges can be set to be higher than the linear flutter instability solution due to nonlinear post-flutter behaviors. The LSTM-based reduced-order model for the nonlinear, unsteady aerodynamics of the 2D bridge deck cross-section can be readily extended to a 3D full-bridge case, where the multiple LSTM networks along the bridge axis should be simultaneously employed. There are a number of bridge aerodynamics applications for the developed LSTM-based reduce-order model, for example, real-time control of wind-induced vibration and wind-induced response prediction of a bridge deck with various structural properties.

3.3.3. LSTM application of nonlinear unsteady aerodynamics Based on the trained LSTM and the procedure discussed in Sect. 3.2.3, Eqs. (14a) and (14b) can be iteratively solved with the dynamics parameters of the bridge deck selected as: m ¼ 12.003 kg, I ¼ 0.4375 kg m2, ωv ¼ 12.63 rad/s, ωα ¼ 23.79 rad/s, ξv ¼ ξα ¼ 0.005 (Ying et al., 2017). The initial states are set as α ¼ α_ ¼ v ¼ v_ ¼ 0. It should be noted that the aerodynamics forces on the bridge deck in the CFD simulations actually involve steady-state, signature turbulence-induced and self-excited components, and the first two provide the initial excitation to the dynamic system. Once the training phase is completed, the LSTM networks provide aerodynamics outputs of the wind-bridge interaction system through a simple arithmetic operation with deck motion inputs (within the ranges of amplitudes and frequencies used in the training phase), and hence circumvent the extreme computational cost of classical numerical methods (e.g., finite element scheme). In general, around 10 steps can be simulated within 1s (wall-clock) time using the single core CPU with the frequency 3.4 GHz based on the abovementioned procedures. On the other hand, more than 40s (wall-clock) time is generally needed for one simulation step using CFD approach. The LSTM-based prediction results (aerodynamics forces and responses) at U ¼ 9 m/s, U ¼ 11 m/s and U ¼ 13 m/s are shown in

Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence 11

T. Li et al.

Journal of Wind Engineering & Industrial Aerodynamics 198 (2020) 104116

the work reported in this paper.

Hochreiter, S., Schmidhuber, J., 1997. Long short-term memory. Neural Comput. 9 (8), 1735–1780. Hornik, K., Stinchcombe, M., White, H., 1989. Multilayer feedforward networks are universal approximators. Neural Network. 2 (5), 359–366. Huang, W.Y., Lippmann, R.P., 1988. Neural net and traditional classifiers. In: Neural Information Processing Systems, pp. 387–396. Huang, W.F., Xu, Y.L., 2013. Prediction of typhoon design wind speed and profile over complex terrain. Struct. Eng. Mech. 45 (1), 1–18. Kareem, A., Wu, T., 2015. Changing dynamic of bridge aerodynamics. Proc. Inst. Civil Eng. Struct. Build. 168 (2), 94–106. Lecun, Y., Bengio, Y., Hinton, G., 2015. Deep learning. Nature 521 (7553), 436. Lucia, D.J., Beran, P.S., Silva, W.A., 2004. Reduced-order modeling: new approaches for computational physics. Prog. Aeosp. Sci. 40 (1–2), 51–117. Parkinson, G., Brooks, N., 1961. On the aeroelastic instability of bluff cylinders. J. Appl. Mech. 28 (2), 252–258. Peherstorfer, B., Willcox, K., 2016. Dynamic data-driven model reduction: adapting reduced models from incomplete data. Adv. Model. Simulat. Eng. Sci. 3 (1), 11. Roger, K., 1977. Airplane Math Modeling Methods for Active Control Design. AGARD-CP288. Scanlan, R.H., Tomko, J., 1971. Airfoil and bridge deck flutter derivatives. J. Soil Mech. Found Div. 97 (EM6), 1717–1737. Theodorsen, T., 1935. General theory of aerodynamic instability and the mechanism of flutter. NACA Technical Report 496. In: Aeronautics, 20th Annual Report of the National Advisory Committee for Aeronautics. NACA. Weston, J., Chopra, S., Bordes, A., 2014. Memory Networks. https://arxiv.org/abs/ 1410.3916. Williams, R.J., Peng, J., 1990. An efficient gradient-based algorithm for on-line training of recurrent network trajectories. Neural Comput. 2 (4), 490–501. Winter, M., Breitsamter, C., 2016. Neurofuzzy-model-based unsteady aerodynamic computations across varying freestream conditions. AIAA J. 54 (9), 1–16. Wray, J., Green, G.G., 1994. Calculation of the Volterra kernels of non-linear dynamic systems using an artificial neural network. Biol. Cybern. 71 (3), 187–195. Wu, T., Kareem, A., 2011. Modeling hysteretic nonlinear behavior of bridge aerodynamics via cellular automata nested neural network. J. Wind Eng. Ind. Aerod. 99 (4), 378–388. Wu, T., Kareem, A., 2013a. Bridge aerodynamics and aeroelasticity: a comparison of modeling schemes. J. Fluid Struct. 43, 347–370. Wu, T., Kareem, A., 2013b. Aerodynamics and aeroelasticity of cable-supported bridges: identification of nonlinear features. J. Eng. Mech. 139 (12), 1886–1893. Wu, T., Kareem, A., 2013c. A nonlinear convolution scheme to simulate bridge aerodynamics. Comput. Struct. 128, 259–271. Wu, T., Kareem, A., 2014. Simulation of nonlinear bridge aerodynamics: a sparse thirdorder Volterra model. J. Sound Vib. 333 (1), 178–188. Wu, T., Kareem, A., 2015. A nonlinear analysis framework for bluff-body aerodynamics: a Volterra representation of the solution of Navier-Stokes equations. J. Fluid Struct. 54, 479–502. Wu, T., Kareem, A., Ge, Y., 2013. Linear and nonlinear aeroelastic analysis frameworks for cable-supported bridges. Nonlinear Dynam. 74 (3), 487–516. Xu, F., Wu, T., Ying, X., Kareem, A., 2016. Higher-order self-excited drag forces on bridge decks. Journal of Engineering Mechanics 142 (3), 06015007. Ying, X., Xu, F., Zhang, M., Zhang, Z., 2017. Numerical explorations of the limit cycle flutter characteristics of a bridge deck. J. Wind Eng. Ind. Aerod. 169, 30–38.

CRediT authorship contribution statement Tao Li: Writing - original draft, Methodology, Validation, Software. Teng Wu: Conceptualization, Methodology, Validation, Writing - review & editing, Supervision. Zhao Liu: Supervision. Acknowledgments The support for this project provided by the National Natural Science Foundation of China (No. 51678148 and No. 51778495) and the Institute of Bridge Engineering at the University at Buffalo is gratefully acknowledged. References Amsallem, D., Farhat, C., 2008. Interpolation method for adapting reduced-order models and application to aeroelasticity. AIAA J. 46 (7), 1803–1813. Amsallem, D., Zahr, M.J., Farhat, C., 2012. Nonlinear model order reduction based on local reduce-order bases. Int. J. Numer. Methods Eng. 92 (10), 891–916. Bengio, Y., Simard, P., Frasconi, P., 1994. Learning long-term dependencies with gradient descent is difficult. IEEE Trans. Neural Network. 5 (2), 157–166. Bialasiewicz, J.T., Soloway, D., 1990. Neural network modeling of dynamical systems. In: Proceedings of 5th IEEE International Symposium on Intelligent Control, pp. 500–505. Davenport, A.G., 1962. Buffeting of a suspension bridge by storm winds. J. Struct. Div. 88 (3), 233–270. de Miranda, S., Patruno, L., Ubertini, F., Vairo, G., 2014. On the identification of flutter derivatives of bridge decks via RANS turbulence models: benchmarking on rectangular prisms. Eng. Struct. 76, 359–370. Diana, G., Falco, M., Cheli, F., Cigada, A., 1999. Experience gained in the Messina bridge aeroelastic project. In: Long-Span Bridges and Aerodynamics. Springer, London. Dowell, E.H., Hall, K.C., 2001. Modeling of fluid-structure interaction. Annu. Rev. Fluid Mech. 33, 445–490. Dowell, E.H., Tang, D., 2002. Nonlinear aeroelasticity and unsteady aerodynamics. AIAA J. 40 (9), 1697–1707. Gers, F.A., Schmidhuber, J., Cummins, F., 1999. Continual prediction using LSTM with forget gates. In: Neural Nets WIRN Vietri-99. Springer, London. GitHub, 2018. An open Source Machine Learning Framework for Everyone. https://gith ub.com/tensorflow/tensorflow. December 5, 2018. Goodfellow, I., Bengio, Y., Courville, A., 2016. Deep Learning. MIT Press, Cambridge. Graves, A., Wayne, G., Danihelka, I., 2014. Neural Turing Machines. https://arxiv.org/ abs/1410.5401. Gurley, K.R., Kareem, A., Tognarelli, M.A., 1996. Simulation of a class of non-normal random processes. Int. J. Non-linear Mech. 31 (5), 601–617. Hinton, G.E., Salakhutdinov, R.R., 2006. Reducing the dimensionality of data with neural networks. Science 313 (5786), 504–507.

12