An efficient stochastic dynamic analysis of soil media using radial basis function artificial neural network

Abstract

Since a lot of engineering problems are along with uncertain parameters, stochastic methods are of great importance for incorporating random nature of a system property or random nature of a system input. In this study, the stochastic dynamic analysis of soil mass is performed by finite element method in the frequency domain. Two methods are used for stochastic analysis of soil media which are spectral decomposition and Monte Carlo methods. Shear modulus of soil is considered as a random field and the seismic excitation is also imposed as a random process. In this research, artificial neural network is proposed and added to Monte Carlo method for sake of reducing computational effort of the random analysis. Then, the effects of the proposed artificial neural network are illustrated on decreasing computational time of Monte Carlo simulations in comparison with standard Monte Carlo and spectral decomposition methods. Numerical verifications are provided to indicate capabilities, accuracy and efficiency of the proposed strategy compared to the other techniques.

Keywords

stochastic analysis / random seismic excitation / finite element method / artificial neural network / frequency domain analysis / Monte Carlo simulation

Introduction

Due to lack of knowledge and inherent properties of an earthquake, the generated ground motion of an earthquake is usually assumed as a random process. The assessment of earthquake response of soil medium is of great importance due to soil-induced amplifications and instability. Furthermore, soil properties always contains uncertainties resulting from natural structure of the soil, errors of measurements, and lack of suitable data [1–3].

Recent developments of the stochastic finite element method have provided a rational tool for probabilistic evaluation of uncertain engineering systems under random excitation [2]. This terminology is commonly utilized for the finite element analyses of uncertain systems, whether or not the inputs are random. Therefore, many researchers implemented these methods for numerous problems of geotechnical engineering and earthquake engineering [4–13]. Furthermore, uncertainty quantification has successfully been extended to other engineering and multidisciplinary subjects like nanotechnology and composite structures [14–16].

On the other hand, artificial neural network (ANN) is very successful for performing rapid analyses of complicated problems in which conventional solvers implemented in those problems require tedious and time-consuming computations. They are also very valuable for various engineering problems like pattern recognition [17], image and signal processing [18], optimization [19,20] and so on. Incorporating ANN in several research areas is in a progressive state of development such as computational and experimental mechanics [21], earthquake engineering [22], structural control [23], among others. The main merit of a well-defined ANN in comparison with the corresponding numerical analysis is that the ANN performs the analysis in a few clock cycles, which needs less computing demand than the corresponding numerical process, some applications were accomplished in Refs [20,24–26].

In this article, the ANN is utilized in Monte Carlo procedure as an alternative to stochastic finite element analysis. The ANN is trained to reduce computational time of probabilistic finite element analysis within the process wherein a soil medium is analyzed in the frequency domain. The random fluctuation of the shear modulus is defined through a random field modeling of this problem. Three stochastic analyses including Monte Carlo simulation, Monte Carlo simulation via the ANN, and stochastic finite element method are considered for the analysis of earthquake response of soil media. Illustrative numerical results are solved through these method for the demonstration.

This paper has been organized as follows: section 2 is about some descriptions and definitions of random seismic input in frequency domain, section 3 describes two methods of stochastic analysis which are used in this study, an introduction to the ANN is briefly discussed in section 4, the proposed algorithm for Monte Carlo simulation of random soil media using the ANN is explained in section 5, section 6 presents the numerical examples and outcomes.

Random seismic excitation

In this study, the earthquake ground motion is assumed to be a stationary random process with having constant frequency content, for simplicity. Here, the power spectral density function defined by Kanai and Tajimi [27] is utilized to characterize the motion as follows:

(1)S_{{\ddot{u}}_g}={\displaystyle \frac{S_{\mathrm{0}}(\mathrm{1}+\mathrm{4}{\xi }_g^{\mathrm{2}}{(\omega /{\omega }_g)}^{\mathrm{2}})}{{(\mathrm{1}-{(\omega /{\omega }_g)}^{\mathrm{2}})}^{\mathrm{2}}+\mathrm{4}{\xi }_g^{\mathrm{2}}{(\omega /{\omega }_g)}^{\mathrm{2}}}}

in which \omega, {\omega }_g, S₀, {\xi }_g are component frequency, the equivalent natural frequency, initial spectral density function or a scale factor, and damping ratio of the ground motion, respectively. The characterized ground motion indicates a stationary random process attained through filtering an initial spectral density function like white noise. In order to employ the non-stationary condition existing in real ground motions, a few corrections should be made. A procedure proposed by Hou [28] is followed herein for this correction. Three segment envelopes are implemented which consist of buildup, constant intensity and decay segments, defining the intensify function as follows

(2)I(t)=E\left[{\ddot{u}}_{g\hspace{0.17em}\max }\right]{\mathrm{\Delta }}_{{\ddot{u}}_g}{F}_{\mathrm{0}}{\psi }_{\mathrm{0}}

where E\left[{\ddot{u}}_{g\hspace{0.17em}\max }\right]is the expected value of maximum acceleration, which is obtained from the earthquake magnitude, M_e, and the epicentral distance, R in (km) according to the following relation:

(3)E\left[{\ddot{u}}_{g\hspace{0.17em}\max }\right]=\mathrm{1230}{e}_{\mathrm{0.8}{M}_e}{(R+\mathrm{25})}^{-\mathrm{2}}

Unit of acceleration is cm/s² here; {\mathrm{\Delta }}_{{\ddot{u}}_g}=\mathrm{2.05} is the correlation factor. The non-stationary behavior of the motion is imposed by a shape function like {\psi }_{\mathrm{0}}={\psi }_{\mathrm{0}}(t) as illustrated in [10]. In order to consider statistics of peak value, the non-stationary process is defined in terms of an equivalent stationary one with duration smaller than the real duration of the undertaken earthquake. The ratio of the standard deviation of acceleration amplitude {\sigma }_{{\ddot{u}}_g} to the expected highest peak of the process is known as the scale factor F₀, during the earthquake duration s. Thus, it is attained as follows [28]:

(4)\begin{array}{l}{F}_0=\frac{{\sigma }_{{\ddot{u}}_g}}{E[{\ddot{u}}_{g\hspace{0.17em}\max }]\mathrm{\Delta }{\ddot{u}}_g}={[2\ln ((2c_{0}s)/T_a)]}^{-0.5}\\ c_0=0.25+0.75t_l/s\end{array}

in which T_a is the expected value of mean period of the earthquake excitation, which is selected as 0.2246 s. The t_l is the duration of strong motion. From the aforementioned equations and considering that {\psi }_{\mathrm{0}}(t)=\mathrm{1} for the strong motion part of the seismic record, one can obtain the excitation variance as given by

(5){\sigma }_{{\ddot{u}}_g}^{\mathrm{2}}={\displaystyle \frac{E^{\mathrm{2}}\left[{\ddot{u}}_{g\hspace{0.17em}\max }\right]{\Delta }_{{\ddot{u}}_g}^{\mathrm{2}}}{\mathrm{2}\ln (\mathrm{2}{c}_{\mathrm{0}}s/T_a)}}

Furthermore, the scale factor S₀ is calculated as:

(6)S_{\mathrm{0}}={\displaystyle \frac{\mathrm{4}{\xi }_g{\sigma }_{{\ddot{u}}_g}^{\mathrm{2}}}{{\omega }_g(\mathrm{1}+\mathrm{4}{\xi }_g^{\mathrm{2}})}}

The natural frequency of the seismic ground motion varies between 2.1<w_g<21, and also the damping ratio is taken as{\xi }_g=\mathrm{0.64}. The mean and standard deviation of {\omega }_g and {\xi }_g for various site-specific situations may be found. By using S₀, {\omega }_g and {\xi }_g, in Eq. (1), the spectral density is computed.

Stochastic dynamic analysis of soil medium

Seismic response of soils

Earthquake response of a layered soil medium under shear wave propagation is governed by the following wave equation

(7)\begin{array}{l}\rho {\displaystyle \frac{{\partial }^{\mathrm{2}}u}{\partial t^{\mathrm{2}}}}=G^{*}{\displaystyle \frac{{\partial }^{\mathrm{2}}u}{\partial z^{\mathrm{2}}}},\\ G^{*}=G(\mathrm{1}+\mathrm{2}i\beta )\end{array}

where u,\rho, G*, G and \beta are horizontal displacement, soil density, complex shear modulus, shear modulus and damping ratio originating from the frictional loss of energy, respectively. Also, G,\betaand G* are frequency independent parameters due to experimental observations [10].

The shear modulus is here selected as a random field decomposed into a mean \bar{G} and fluctuation {\alpha }_G parts as below:

(8)G(z)=\bar{G}(z)+{\alpha }_G(z)

in which \bar{G}(z) and {\alpha }_G(z) are the mean value of shear modulus at z, and {\alpha }_G(z) is a zero mean fluctuation term, respectively. The following autocorrelation function with exponential distribution is considered:

(9)R_{\alpha \alpha }(z_{\mathrm{1}},z_{\mathrm{2}})={\sigma }_G^{\mathrm{2}}{e}^{-\left|z_{\mathrm{1}}-z_{\mathrm{2}}\right|/b}

in which {\sigma }_G and b show the standard deviation of the random process and correlation length (one-half of the fluctuation scale), respectively. Also, z₁ and z₂ are two arbitrary points of a domain.

The equation of motion for a soil medium under seismic excitation, is obtained by:

(10)M\ddot{u}+Ku=-m_I{\ddot{u}}_g

where uand \ddot{u} denote displacement and acceleration vectors, respectively. m_I=M\delta is an influence vector for imposing direction of excitation , and hence for horizontal excitation, {\delta }_i=\mathrm{1} when the degree of freedom corresponds to horizontal direction; otherwise {\delta }_i=\mathrm{0}. The global mass matrix and stiffness matrices are formed by assembling N element matrices over the entire domain as below

(11)M={\displaystyle \sum _{e=\mathrm{1}}^N{M}_e},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}K={\displaystyle \sum _{e=\mathrm{1}}^N{K}_e}

Moreover, stiffness matrix of an element is

(12)K_e={\displaystyle \frac{G_e^{*}}{l_e}}\left[\begin{array}{ll}\mathrm{1}& -\mathrm{1}\\ -\mathrm{1}& \mathrm{1}\end{array}\right]

and mass matrix of an element is attained from

(13)M_e={\displaystyle \frac{{\rho }_e{l}_e}{\mathrm{6}}}\left[\begin{array}{ll}\mathrm{2}& \mathrm{1}\\ \mathrm{1}& \mathrm{2}\end{array}\right]

while

G_e^{*}

and {\rho }_e and l_e are shear modulus, mass density and length of a soil element.

In order to solve Eq. (10) in the frequency domain, displacement of the domain under a ground acceleration with unit amplitude, {\ddot{u}}_g=\exp (i\omega t), should be found as follows

(14)u=H_u(\omega )e^{i\omega t}

and the acceleration is

(15)\ddot{u}=-{\omega }^{\mathrm{2}}{H}_u(\omega )e^{i\omega t}

By replacing the Eqs. (14) and (15) into the Eq. (10), the following relation is obtained

(16)[-{\omega }^{\mathrm{2}}M+K]H_u=-m_I

The transfer function of absolute acceleration is written as

(17)H_{\ddot{u}}=-{\omega }^{\mathrm{2}}{H}_u+\delta

Shear stress and shear strain of an element are expressed in terms of H_{ue}^{(\mathrm{1})}and H_{ue}^{(\mathrm{2})}, and thus a few arrays of solution vector H_u is the contribution of eth element. The transfer function of strain is explained as

(18)H_{\gamma e}={\displaystyle \frac{H_{ue}^{(\mathrm{1})}-H_{ue}^{(\mathrm{2})}}{l_e}}

The stress-strain relationship expresses the stress transfer function of an element as follows

(19)H_{\tau e}=G_e^{*}{H}_{\gamma e}

The response spectral density function is calculated based upon the transfer function as follows

(20)S_X(\omega )=diag[H_X(\omega )H_X^{*T}(\omega )]S_{{\ddot{u}}_g}(\omega )

where X is a target random function such as u,\hspace{0.17em}\ddot{u},\hspace{0.17em}\gamma ,\hspace{0.17em}or\hspace{0.17em}\tau .

More information about frequency domain analysis can be found in Refs [27,29].

SFEM: Monte Carlo simulation

Here, Stochastic Finite Element Method (SFEM) via Monte Carlo simulation is briefly discussed. The Monte Carlo Simulation method includes generating numerous realizations of a random field like\alpha (z) appearing in the Eq. (8) and computing the response with each realizations. After performing the simulations, the statistics of these responses are computed in terms of mean, standard deviation values and other statistical moments [10].

SFEM: spectral decomposition

Stochastic finite element analysis using spectral decomposition was proposed by Ref. [30]. This method employs Karhunen-Loeve and polynomial chaos expansions for imposing input and output variability, respectively. In this method, the random field {\alpha }_G(z)is illustrated by Karhunen-Loeve expansion as mentioned below

(21)\begin{array}{l}{\alpha }_G(z,\theta )={\displaystyle \sum _{k=\mathrm{1}}^m{\xi }_k(\theta )}\sqrt{{\lambda }_k}{\phi }_k(z),\\ {\xi }_k={\displaystyle \frac{\mathrm{1}}{\sqrt{{\lambda }_k}}}{\displaystyle \underset{l}{\int }{\alpha }_G{\phi }_k}\mathrm{d}z\end{array}

where {\xi }_k are random coefficients and {\lambda }_k are the eigenvalues and the {\phi }_k are the eigenfunctions of the autocorrelation function R_{\alpha \alpha }(z)

(22){\displaystyle \underset{l}{\int }{R}_{\alpha \alpha }}(z){\phi }_k(z_{\mathrm{2}})\mathrm{d}{z}_{\mathrm{2}}={\lambda }_k{\phi }_k(z_{\mathrm{1}})

Therefore, Eqs. (21) and (8) gives

(23)G(z,\theta )=\bar{G}(z)+{\displaystyle \sum _{k=\mathrm{1}}^m{\xi }_k(\theta )}\sqrt{{\lambda }_k}{\phi }_k(z)

In the above equation, \theta denotes randomness of the variable.

In order to decompose the random field and implement the autocorrelation function given by the Eq. (9), the integral equation in Eq. (22) becomes [30]:

(24){\sigma }_G^{\mathrm{2}}{\displaystyle {\int }_{-a}^a{e}^{-\frac{\left|z_{\mathrm{1}}-z_{\mathrm{2}}\right|}{b}}}{\phi }_k(z_{\mathrm{2}})\mathrm{d}{z}_{\mathrm{2}}={\lambda }_k{\phi }_k(z_{\mathrm{1}})

in which b is the correction length and a=l/\mathrm{2}.

Analytical solution of Eq. (24) is expressed by

(25){\phi }_k(z)={\displaystyle \frac{\cos \left({\overline{\omega }}_{k}z\right)}{\sqrt{a+\sin \left(\mathrm{2}{\overline{\omega }}_{k}a\right)/\mathrm{2}{\overline{\omega }}_k}}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(for\hspace{0.17em}odd\hspace{0.17em}k)

(26){\phi }_k(z)={\displaystyle \frac{\cos \left({\overline{\omega }}_{k}z\right)}{\sqrt{a+\sin \left(\mathrm{2}{\overline{\omega }}_k^{*}a\right)/\mathrm{2}{\overline{\omega }}_k}}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(for\hspace{0.17em}odd\hspace{0.17em}k)

The corresponding eigenvalues are determined by:

(27){\lambda }_k={\displaystyle \frac{\mathrm{2}c}{{\bar{\omega }}_k^{\mathrm{2}}+c^{\mathrm{2}}}}{\sigma }_G^{\mathrm{2}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}k=\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{...}

Hence, stochastic expansion of element stiffness matrix leads to

(28)\begin{array}{l}{K}_e={\bar{K}}_e+{\displaystyle \sum _{k=\mathrm{1}}^m{\xi }_k}{K}_{ek}\\ {\bar{K}}_e={\displaystyle \frac{\bar{G}(\mathrm{1}+\mathrm{2}i\beta )}{l_e}}\left[\begin{array}{ll}\mathrm{1}& -\mathrm{1}\\ -\mathrm{1}& \mathrm{1}\end{array}\right]\\ {\bar{K}}_{ek}={\displaystyle \frac{\mathrm{1}+\mathrm{2}i\beta }{l_e}}\sqrt{{\lambda }_k{\phi }_k}\left[\begin{array}{ll}\mathrm{1}& -\mathrm{1}\\ -\mathrm{1}& \mathrm{1}\end{array}\right]\end{array}

and the global stiffness matrix is formed by

(29)K={\displaystyle \sum _{e=\mathrm{1}}^N{K}_e}={\displaystyle \sum _{e=\mathrm{1}}^N{\bar{K}}_e}+{\displaystyle \sum _{e=\mathrm{1}}^N{\displaystyle \sum _{k=\mathrm{1}}^m{\xi }_k}{K}_{ek}}=\bar{K}+{\displaystyle \sum _{k=\mathrm{1}}^m{\xi }_k}{K}_k

in which

(30)K_k={\displaystyle \sum _{e=\mathrm{1}}^N{K}_{ek}}

Now, the Eq. (16) is rewritten as

(31)\left[-{\omega }^{\mathrm{2}}M+\bar{K}+{\displaystyle \sum _{k=\mathrm{1}}^m{\xi }_k{K}_k}\right]H_u=-m_I

Let

(32)D=-{\omega }^{\mathrm{2}}M+\bar{K}

After simplifying the above equation, we have

(33)\begin{array}{l}\left[I+{\displaystyle \sum _{k=\mathrm{1}}^m{\xi }_k{Q}_k}\right]H_u=f\\ Q_k=D^{-\mathrm{1}}{K}_k\\ f=D^{-\mathrm{1}}(-m_I)\end{array}

The solution of equation is

(34)H_u={\left[I+{\displaystyle \sum _{k=\mathrm{1}}^m{\xi }_k{Q}_k}\right]}^{-\mathrm{1}}f

The equation of motion is expressed as

(35)\left(-{\omega }^{\mathrm{2}}M+\bar{K}+{\displaystyle \sum _{k=\mathrm{1}}^m{\xi }_k{K}_k}\right)H_u=-m_I

With K_{\mathrm{0}}=-{\omega }^{\mathrm{2}}M+\bar{K},\hspace{0.17em}{\xi }_{\mathrm{0}}=\mathrm{1}, the above can be written as:

(36)\left({\displaystyle \sum _{k=\mathrm{0}}^m{\xi }_k{K}_k}\right)H_u=-m_I

Expanding H_u by polynomial chaos, the jth element of H_u is formed as:

(37)H_{uj}=a_{j\mathrm{0}}{\Gamma }_{\mathrm{0}}+{\displaystyle \sum _{i_{\mathrm{1}}=\mathrm{1}}^{\infty }{a}_{j{i}_{\mathrm{1}}}{\Gamma }_{\mathrm{1}}}[{\xi }_{i\mathrm{1}}]+{\displaystyle \sum _{i_{\mathrm{1}}=\mathrm{1}}^{\infty }{\displaystyle \sum _{i_{\mathrm{2}}=\mathrm{1}}^{i_{\mathrm{1}}}{a}_{j{i}_{\mathrm{1}}}{\Gamma }_{\mathrm{2}}}}[{\xi }_{i_{\mathrm{1}}}{\xi }_{i_{\mathrm{2}}}]+{\displaystyle \sum _{i_{\mathrm{1}}=\mathrm{1}}^{\infty }{\displaystyle \sum _{i_{\mathrm{2}}=\mathrm{1}}^{i_{\mathrm{1}}}{\displaystyle \sum _{i_{\mathrm{3}}=\mathrm{1}}^{i_{\mathrm{2}}}{a}_{i_{\mathrm{1}}{i}_{\mathrm{2}}{i}_{\mathrm{3}}}{\Gamma }_{\mathrm{3}}}}}[{\xi }_{i_{\mathrm{1}}}{\xi }_{i_{\mathrm{2}}}{\xi }_{i_{\mathrm{3}}}]+\mathrm{...}

in which {\Gamma }_p[]is pth order polynomial chaos, that is:

(38){\mathrm{\Gamma }}_p[{\xi }_{i_{\mathrm{1}}}\hspace{0.17em}{\xi }_{i_{\mathrm{2}}}\hspace{0.17em}\mathrm{...}{\xi }_{i_n}]={\displaystyle \frac{{(-\mathrm{1})}^n}{e^{(-\mathrm{1}/\mathrm{2})\xi {\xi }^{\mathrm{T}}}}}\left[{\displaystyle \frac{{\partial }_n}{\partial {\xi }_{i_{\mathrm{1}}}\partial {\xi }_{i_{\mathrm{2}}}\mathrm{...}\partial {\xi }_{i_n}}}{e}^{(-\mathrm{1}/\mathrm{2})\xi {\xi }^{\mathrm{T}}}\right]

where P vector of random variables. After expanding the summation in the Eq. (42), the H_ni can also be formed as

(39)H_{ni}={\displaystyle \sum _{i=\mathrm{0}}^P{c}_{ni}{r}_i}

in which P is the number of polynomial chaos, excluding the zero order term. By combining the Eqs. (41) and (44), we arrive at

(40)\left({\displaystyle \sum _{k=\mathrm{0}}^m{\xi }_k{K}_k}\right)\left({\displaystyle \sum _{i=\mathrm{0}}^P{c}_{ni}{r}_i}\right)=-m_I

After multiplying the both sides by r_i, i=0,…,P, and taking mathematical expectation, we have

(41)\begin{array}{l}KC=F,\\ K_{ij}={\displaystyle \sum _{m=\mathrm{0}}^M{K}_m}\langle r_i{r}_j{\xi }_m\rangle , i,j=\mathrm{0},\mathrm{...},P\\ F_i=\langle -m_I{r}_i\rangle , i=\mathrm{0},\mathrm{...},P\\ F_i=-m_I for\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=\mathrm{0}\\ F_i=\mathrm{0} for\hspace{0.17em}\hspace{0.17em}i\ne \mathrm{0}\end{array}

Finally, we will have [10]:

(42)\begin{array}{l}{S}_{un}=(H_{un}\times H_{un}^{*T})S_{{\ddot{u}}_g}=A_{un}{S}_{{\ddot{u}}_g}\\ \langle A_{un}\rangle ={\displaystyle \sum _{i=\mathrm{0}}^P{c}_{ni}^{\mathrm{2}}}\times \langle r_i^{\mathrm{2}}\rangle \end{array}

Radial basis function neural networks

One of the well-known types of ANNs is radial basis function (RBF) network which needs more number of neurons than standard feed-forward back-propagation networks. However, they are able to be trained, swiftly [31]. The design of a neural network is manifested as a curve-fitting problem by searching a best fitness function over the training data space. The RBF design of neural networks was firstly proposed by Lowe and Broomhead [32]. The RBF networks have two layer feedforward networks so that the first layer contains RBF neurons with Gaussian activation functions.

There are several suggestions for designing a RBF network. One of the model adopted in MATLAB software and its architecture is shown in Fig. 1. For each training pattern n=1,2,…,N, the input of the first layer is constructed by the values x_nl, l=1,2,…,L, where L is the number of input. This information is transferred to the hidden layer (i.e., the second layer) without modifications due to weights, at a difference with respect to the multilayer perceptron. For each input vector x_n=[x_n1, x_n2,…,x_nl], the activation of each hidden neuron m, m=1,2,...,M is handled by the Euclidean distance which separates a center {\bar{c}}_{ml} from the input vector defining each hidden neuron:

(43)d_{nm}={\left[{\displaystyle \sum _{l=\mathrm{1}}^L{(x_{nl}-{\bar{c}}_{ml})}^{\mathrm{2}}}\right]}^{\mathrm{1}/\mathrm{2}}

This center is randomly defined as another input vector. The most significant part of the RBF is an activation function of the hidden neurons with a locally active support of radial pattern. One of the most popular RBF is Gaussian one, specified as:

(44)f(\bar{s})=e^{-{\bar{s}}^{\mathrm{2}}}

Here, the output of each hidden neuron is obtained by the Euclidean distance multiplied by a bias \bar{b} defining the sensitivity of the receptive field. In other words, short bias is desirable for regression analysis, whereas a large one is more sufficient for an accurate reproduction of the training outputs in detail.

The matching of the hidden neuron output to the training outputs y_nk, k=1,2,…,K is attained by a simple linear transformation as follows

(45)y_{nk}={\displaystyle \sum _{m=\mathrm{1}}^M{\bar{a}}_{nm}{w}_{mk}}+{\bar{b}}_k

in which the weights of the second layer connections w_{mk} and their corresponding biases {\bar{b}}_k are the unknowns of Eq. (45). The standard least square procedures are employed for finding these parameters. The numerical results many engineering applications have illustrated that the RBF networks are very powerful interpolators with fast training stage.

Monte Carlo simulation of random soil medium using ANN

In this paper, computational effort of Monte Carlo adapted stochastic finite element analysis of soil medium is reduced by ANN. Since Monte Carlo simulation is a realization of desired random space in each step (as a sample), this procedure continues iteratively until reaching the number of selected simulations or samples. Here, realization of random shear modulus of soil elements is utilized as input data of ANN. Thus, input matrix consists of a number of vectors involving shear modulus of soil elements. For training of ANN, spectral density of absolute acceleration of ground surface is considered as a target of corresponding input vector.

The following procedure is suggested for Monte Carlo method via ANN:

1) A number of samples is selected for training of ANN. Here, 500 samples are considered.

2) The Monte Carlo simulation is accomplished with 500 samples and then the generated shear modulus and the corresponding results consisting of spectral density of acceleration of ground surface as input and target matrices are stored, respectively.

3) In this step, the trained neural network is utilized in order to perform Monte Carlo simulation, and hence instead of deterministic finite element and frequency domain analyses, this trained neural network is used as a solver.

4) The Monte Carlo based stochastic finite element analysis is carried out via ANN considering the number of samples which is selected. Here, 5000 samples are selected. Therefore, incorporating ANN Monte Carlo simulation significantly reduces the computational time.

It should be noticed that 500 input vectors make the input matrix and 500 target vectors make the target matrix considering 500 samples. Each vector has n arrays. n is taken as the number of soil elements’ nodes.

Numerical results

In this section, the three approaches of stochastic finite element, which one of them (Monte Carlo via ANN) are proposed here, are applied to the seismic analysis of the soil medium at Kawagishi-Cho during the Niigata earthquake of 16 June 1964 [10]. This benchmark problem is studied here to validate and demonstrate the capabilities of the proposed Monte Carlo ANN approach in comparison with standard Monte Carlo and stochastic finite element approaches.

Seismic excitation parameters

Niigata earthquake had magnitude of 7.3 at an epicentral distance about 56 km (35 miles). The earthquake’s duration and the duration of the strong motion part of the accelerogram were 26 sec and 8 sec, respectively.

The spectral density function of Kanai-Tajimi (see Fig. 2) is considered to characterize the input ground acceleration with having the ground natural frequency of{\omega }_g=\mathrm{15.56}\hspace{0.17em}\hspace{0.17em}rad/s, and the damping ratio of {\xi }_g=\mathrm{0.64}. The scale factor is attained by using Eq. (46) as follows

(46)S_{\mathrm{0}}={\displaystyle \frac{\mathrm{4}\xi {\sigma }_{{\ddot{u}}_g}^{\mathrm{2}}}{\mathrm{\pi }{\omega }_g(\mathrm{1}+\mathrm{4}{\xi }_g^{\mathrm{2}})}}={\displaystyle \frac{\mathrm{4}\times \mathrm{0.64}\times {\mathrm{1.41}}^{\mathrm{2}}}{\mathrm{\pi }\times \mathrm{15.56}\times (\mathrm{1}+\mathrm{4}\times {\mathrm{0.64}}^{\mathrm{2}})}}=\mathrm{0.039}\hspace{0.17em}\hspace{0.17em}\mathrm{f}{\mathrm{t}}^{\mathrm{2}}/{\mathrm{s}}^{\mathrm{3}}

Properties of soil

The soil deposit at Niigata, had the total depth of 200 ft. The upper layer with thickness of 100 ft was sand, and the lower one with thickness of 100 ft was either stiff clay or clayey sand. Since the shear moduli of sand and stiff clays are almost constant, the deposit can practically be considered as sand throughout the total depth.

During earthquake-induced vibrations, the moduli are taken as G=\mathrm{1}\times {\mathrm{10}}^{\mathrm{5}}\times {\sigma }_{\mathrm{0}}^{\mathrm{1}/\mathrm{3}}psf for very loose sands, or G=\mathrm{2}\times {\mathrm{10}}^{\mathrm{5}}\times {\sigma }_{\mathrm{0}}^{\mathrm{1}/\mathrm{3}}psf for very dense sand, wherein {\sigma }_{\mathrm{0}} is the overburden pressure. If the aforementioned values are applicable for sands with relative density ranging from 0 to 100 per cent, then suitable shear modulus for sand at every relative density can be calculated by interpolation. Here, 40 soil elements are employed and the soil medium is drawn in Fig. 3.

The relative density is about 45 per cent at the top 10 ft, while it drops to about 40 per cent at depth equal to 20 ft, and then it increases to a relative density of 80 per cent at depth of 60 ft.as an assumption, the relative density ranges from 80 to 90 per cent in the lower 130 ft of the deposit.

In highly damaged zone of Niigata, the water table is 3 ft below the ground surface. Supposing that the sand weight density is 110 lb/ft³ above the water table, and the buoyant weight is 50 lb/ft³ below the water table, the overburden pressures can easily be calculated. The determined distribution of shear modulus is plotted Fig. 4. Available experimental data shows that the damping ratio of \beta =\mathrm{0.04} can be taken for the sand. Randomness of the shear modulus of sand is supposed to be expressed as the standard deviation {\sigma }_G=\mathrm{0.1}G and the fluctuation scale, {\delta }_G=\mathrm{20}ft.

Numerical example

Three methods are incorporated to evaluate spectral density function of absolute acceleration of ground surface (site response), as already stated. First, stochastic finite element analysis incorporating Karhunen Loeve and Polynomial Chaos expansion is accomplished, and orders of these expansions are taken as 20 and 1, respectively. Fig. 5 and Fig. 6 show the mean value of acceleration spectral density function and its standard deviation versus frequency variation, respectively, that are obtained from this method. Spectral density functions which are attained by Monte Carlo simulation and Monte Carlo simulation using the ANN have been illustrated in Fig. 7 and Fig. 8. They represent accuracy of the ANN based Monte Carlo method in comparison with standard Monte Carlo method. Also, the calculated errors between outcomes of these two methods are visible in Fig. 9 and Fig. 10. Convergence curve of the ANN using 500 samples is shown in Fig. 11. Therefore, results and Table 1 demonstrate computational efficiency and suitable accuracy of the proposed ANN based Monte Carlo method for stochastic dynamic analysis of soil response. In Table 1 computational time of the three methods are compared. Total run time of the ANN based Monte Carlo method is about 717 seconds, whereas standard Monte Carlo method takes about 4343 seconds. Thus, ratio of these times is only 17 per cent. Slight difference between SFEM and Monte Carlo is visible; this is because of the selected order and main parameters such as correlation length. In spite of good performance of SFEM, Monte Carlo methods are still a robust tool for random analyses and hence employing the ANN along with Monte Carlo is an interesting combination.

Conclusions

Implementation of uncertainty in engineering computations leads to robust analysis and design of systems. This is more important for the materials that their parameters cannot be measured exactly due to their inhomogeneity and instrumental limitations such as soil parameters. In this study, the stochastic dynamic response of a soil medium is carried out in the frequency domain. Here, randomness is considered for the both seismic excitations and soil shear modulus. Two methods are used for stochastic analysis of soil media which are spectral decomposition and Monte Carlo method. The shear modulus of soil medium is considered as a random field model, and also the earthquake motion is implemented as a stochastic process. The ANN is utilized and added to Monte Carlo method for sake of reducing computational effort of the random analysis problem. The numerical example is solved and compared through three methods and results demonstrate desirable accuracy and performance of the proposed ANN based Monte Carlo method for stochastic dynamic analysis of soil media.

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