A two-phase neuro-modal linear method for seismic analysis of structures
Introduction
Design of many structures and buildings in earthquake-prone regions is controlled by seismic loads [1,2]. Seismic codes and standards have been developed to define the regulations governing the design of structures in order to maximize the safety and welfare of occupants at the time of earthquake [3]. Seismic analysis requires the solution of a time-dependent system of second order differential equations. There are several direct and indirect numerical methods to solve such a problem, however, these methods are usually expensive; especially for systems with a large number of degrees of freedom and uncertain seismic loads [1,2]. Furthermore, because seismic analyses are usually conducted within design, optimization, and risk analysis procedures [4], one usually needs to perform numerous analyses to achieve a desired solution. In practice, seismic analyses are usually simplified to equivalent static solutions in order to avoid tremendous dynamic computations [1,2]. Therefore, developing efficient methods for seismic analysis of structures and mechanical systems aimed at reducing computations and time has always been of interest among researchers [5].
Artificial neural networks (AANs) have been extensively applied to different problems in civil engineering and structural mechanics [6]. These methods have been used to describe mechanical behavior of materials such as strength of concrete mixtures [7] and load capacity of structural members [8]. The first effort for tackling simulation problems using ANNs was a guideline developed to design and train an inexpensive ANN-based meta-model to simulate a computationally expensive structural analysis [9]. ANNs were also employed to solve the problems of probabilistic reliability analysis of structures [10], analysis of steel moment-resisting frames [11], and estimation of inter-story drift of structures in Monte Carlo simulations [12]. One limitation of the earlier ANN-based models is most of these models were developed to tackle specific problems rather than developing general frameworks to solve a class of problems [6]. Furthermore, network architectures used in these ANN-based models were not as powerful as state-of-the-art architectures, such as dynamic networks, developed in recent years in the field of deep learning.
ANNs have been limited to shallow architectures for several decades. In recent years, with advances in efficient methods for training deep networks as well as availability of high performance platforms, deep neural networks have dominated the field of machine learning. The applications of deep neural networks range from computer vision [13,14], speech recognition [15], and natural language processing [16] to bioinformatics [17], drug design [18], etc. Specifically, one of the applications of ANNs is in analysis of sequences and time-series data through recurrent neural networks (RNNs) [19]. In recent years, and in parallel with the advances in deep learning algorithms, architecture of RNNs has been advanced to more powerful architectures such as Long Short-Term Memories (LSTMs) [20]. LSTM networks can capture dynamics of sequences and time-series through cycles in the network and, as opposed to vanilla RNNs, does not suffer from vanishing gradients [20]. An LSTM layer can receive a full sequence of data from time 0 to t as input and return a corresponding full sequence of data as response. Therefore, structure of LSTM seems to be well compatible with seismic analysis wherein time-dependent loads from time 0 to t are applied to a mechanical system and corresponding responses (i.e., displacements) are calculated at each time step.
In this paper we have developed a two-phase neuro-modal solution for seismic analysis of multi degree-of-freedom (DOF) structures including high-rise skeletal buildings through integration of modal analysis with LSTM neural networks. To improve efficiency and accuracy of the training process, we first decomposed the coupled system of differential equations through diagonalization (i.e., modal analysis). The problem in the modal space was implemented into a Long Short-Term Memory neural network and solved as a time-series problem. The input and output to the network were full sequences of time-dependent seismic loads and the corresponding displacement field, respectively. Since a single training point contained full time-series of loads and displacements, the trained network was able to receive a random earthquake sequence at once and promptly return the entire displacement field of the structure. Since neural networks estimate the relation between inputs and outputs using closed-from solutions in the form of composite functions, the proposed method was efficient at the time of testing/application as it only needed to perform function evaluation by entering inputs to the functions and returning outputs. The fixed parameters (weights, biases) of these functions in the neural network model were defined using a one-time training process. The efficiency of the model was also compared against a 4th-order Runge-Kutta method, which is a popular method for numerical solution of ordinary differential equations. The developed model was validated through comparison between model's predictions and a set of unseen ground truth values obtained from simulated and real data.
Section snippets
Modal analysis of mechanical systems: decomposition of coupled differential equations
The equations of motion for a multi DOF mechanical system iswhere M is mass matrix, C is damping matrix, K is stiffness matrix, u is vector of displacements, and p(t) is the vector of time-dependent loads. Damping matrix is usually considered as a linear combination of M and K matrices, C = αK + βM.
Here the intention is to uncouple the equations of motion through diagonalization by making a linear change of variables as u(t) = Qz(t), where Q is a constant matrix
LSTM neural networks
Long Short-Term Memory (LSTM) is an artificial Recurrent Neural Network (RNN) architecture in the field of deep learning proposed by Hochreiter and Schmidhuber in late 90 s [19]. LSTM neural network is a powerful tool for processing sequences and has been extensively used in recent years in computer vision (e.g., image captioning, visual question answering), machine translation (e.g., sentence translation from French to English), and time-series analyses. Like vanilla RNNs, LSTM can capture
Seismic modeling and analysis using LSTM neural networks
In previous sections the formulation for seismic analysis of mechanical systems was derived and the architecture for LSTM neural networks was introduced. Here, the intension is to implement the seismic formulation into the LSTM network to be able to train a predictive dynamic model that receives seismic loads at each time step as input and returns response of the system (displacement field) at each time step as output for the single DOF differential equation in Eq. (7) (Fig. 3). To generate
5.1. Simulated ground motion analysis
Since the modal analysis and the transformation from modal to original space had closed-form mathematical solutions and were accurate, here we focused on modeling and evaluation of the performance of LSTM network. To generate inputs that resemble seismic loads, Fourier series were used: where amplitudes and frequencies were randomly generated. For amplitudes, they were chosen as a number generated through multiplication of a random integer between 5 and 15 and a
Conclusions
In this paper a two-phase neuro-modal solution for seismic analysis of skeletal structures was developed through integration of modal analysis with LSTM neural networks. The method allowed us to convert a time-dependent problem to a recurrent neural network with fixed architecture, functions, and parameters. The trained LSTM neural network was equivalent to a closed-form solution in the form of composite functions such that at the time testing/application one only needed to perform function
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2023, Chaos, Solitons and Fractals
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Iman Shojaei co-authored this article in his personal capacity. The views expressed are his own and do not necessarily represent the views of Align Technology, Inc.