Data-driven approach to solve vertical drain under time-dependent loading

Abstract

Currently, the vertical drain consolidation problem is solved by numerous analytical solutions, such as time-dependent solutions and linear or parabolic radial drainage in the smear zone, and no artificial intelligence (AI) approach has been applied. Thus, in this study, a new hybrid model based on deep neural networks (DNNs), particle swarm optimization (PSO), and genetic algorithms (GAs) is proposed to solve this problem. The DNN can effectively simulate any sophisticated equation, and the PSO and GA can optimize the selected DNN and improve the performance of the prediction model. In the present study, analytical solutions to vertical drains in the literature are incorporated into the DNN–PSO and DNN–GA prediction models with three different radial drainage patterns in the smear zone under time-dependent loading. The verification performed with analytical solutions and measurements from three full-scale embankment tests revealed promising applications of the proposed approach.

Graphical abstract

Keywords

vertical drain / artificial neural network / time-dependent loading / deep learning network / genetic algorithm / particle swarm optimization

1 Introduction

Artificial intelligence (AI) is being increasingly applied in research studies owing to its ability to learn complex problems and easy implementation. The AI technology can foster easy navigation of many daily activities, which results in more speed and convenience in various aspects such as transportation, agriculture, education, robotics, automotive, astronomy, and healthcare. An artificial neural network (ANN), a type of AI, is composed of numerous connected mathematical equations, similar to the human biological neural system. ANNs have self-studying or training abilities, which enable them to learn sophisticated engineering solutions [1–6]. In the geotechnical field, ANNs have extensive applications in the estimation of pile capacity [7,8] and soil characteristics such as the stress history of clay [9], undrained shear strength [10], soil compression coefficients [11], and soil classification [12]. ANNs are also used to predict the probabilities of natural hazards, liquefaction, and landslides [13,14], shear strength parameters of soils [15], and intact rock strength [16].Recently, deep neural networks (DNNs) with more hidden layers than ANNs have demonstrated better performance than shallow neural networks [1,5,6]. Therefore, DNNs have strong potential for applications in engineering.

A vertical drain with preloading is widely used for soft ground improvement. It is simply installed in the ground, and the excess pore pressure generated by the preloading leads to a referential flow running horizontally toward the vertical drain and then along it to a permeable drainage layer [17]. This technique can significantly reduce the time required for the consolidation process because it shortens the drainage path. The consolidation problem of vertical drains is an attractive topic for numerous researchers, which have applied analytical solutions or the finite element method (FEM) to solve it [18–21]. However, to date, DNNs have not been exploited. To the best of the author’s knowledge, this study is the first application of a DNN model to the consolidation problem of a vertical drain. In this study, the performance of the proposed model was based on analytical solutions and full-scale embankment tests. This study was inspired by robotic fields, in which robot arm movements are trained by AI rather than an exact function [22]. With AI technology, the robot arm can perform more complex movements without requiring a lengthy mathematical solution to achieve similar movements. AI has changed the production process of the robotic field and will be increasingly applied because the training process does not require mathematical equations, which consume considerable time for design. Similarly, the solutions in this study, conventionally derived from several sophisticated analytical solutions, can be obtained from training DNN. Based on these concepts, this study may stimulate the application of this technology in the field of civil engineering, particularly in the geotechnical field.

The analytical solution considers the radial permeability variations in the smear zone of a vertical drain with three patterns [23–26]. Pattern I is an instantaneous reduction, and patterns II and III correspond to linear and parabolic reductions of the permeability, respectively. Based on the patterns, three databases were derived, and the sample distributions were evaluated. Single-ramp loading [24] was also incorporated in this study because most vertical drain constructions occur under multi-stage loadings to minimize the plastic deformation due to the soft subsoil. To enhance the accuracy of the prediction model [27], the DNN topology was optimized using two algorithms: particle swarm optimization (PSO) and genetic algorithm (GA). PSO and GA are meta-heuristic optimization algorithms, which are robust and simple to program [28,29]. They can be combined with DNNs to effectively determine the optimal solutions [1,5]. Finally, for validation, the results obtained by the proposed approach were compared with those of the analytical solutions and three full-scale embankment tests at the Bangkok International Airport. The good agreement between the results demonstrates promising applications with only a suitable database of real-site measurements and soil properties. Within the scope of this study, analytical solutions were determined under single-ramp loading, uniform subsoil, and ideal elastic deformation.

The remainder of this paper is divided as follows. The analytical solutions are reviewed in Section 2. The implementation of the DNN–PSO or DNN–GA models is described in Section 3. Section 4 presents the results and discussion of the proposed approach, along with the analytical results. Finally, in Section 5, the results predicted by the DNN–PSO or DNN–GA models are compared with those obtained by real full-scale embankment tests at the Bangkok International Airport.

2 Revised analytical solutions

In this section, analytical solutions to vertical drains with several modifications are discussed as the basis for obtaining a DNN database. Figure 1 illustrates the basic concept of calculating a unit cell using a cylinder model of a vertical drain, with three variations in the lateral drainage in the smear zones [23–26]. In pattern I, the instantaneous change in the radial permeability when it moves from an undisturbed zone to the smear zone is considered. In pattern II, the radial permeability is assumed to decrease linearly. In pattern III, there is a parabolic decrease in the radial permeability when it passes the smear zone. The surcharge loading is the ramp loading on the top of the boundary, where it reaches the ultimate loading, {\sigma }_u, in time T_{\mathrm{1}}, as shown in Fig. 2.

The solution to a vertical drain is based on a partial differential equation of the radial drainage of the cylinder unit cell model.

(1)c_h\left({\displaystyle \frac{\mathrm{1}}{r}}{\displaystyle \frac{\partial u}{\partial r}}+{\displaystyle \frac{{\partial }^{\mathrm{2}}u}{\partial r^{\mathrm{2}}}}\right)+c_v{\displaystyle \frac{{\partial }^{\mathrm{2}}u}{\partial z^{\mathrm{2}}}}={\displaystyle \frac{\partial \sigma }{\partial t}},

where u is the excess pore water pressure, c_v is coefficient of consolidation in the vertical direction , c_h is coefficient of consolidation in horizontal direction, zis depth, and ris radius.

The horizontal permeability coefficient is assumed to vary (from k_{\mathrm{h}} (horizontal coefficient of permeability of undisturbed zone) to k_{\mathrm{s}} (horizontal coefficient of permeability of smear zone)) in the radial direction from the undisturbed zone to the smear zone. A function describing these radial variations is as in Refs. [23–26]:

(2)k_r(r)=k_{h}f(r).

Applying Eq. (2) to Eq. (1), with some modifications, yields:

(3){\displaystyle \frac{\mathrm{1}}{r}}{\displaystyle \frac{\partial }{\partial r}}\left[{\displaystyle \frac{k_r(r)}{{\gamma }_w}}r{\displaystyle \frac{\partial u}{\partial r}}\right]+{\displaystyle \frac{k_v}{{\gamma }_w}}{\displaystyle \frac{{\partial }^{\mathrm{2}}\bar{u}}{\partial z^{\mathrm{2}}}}=-{\displaystyle \frac{\partial {\epsilon }_v}{\partial t}},r_d\le r\le r_e,

where{\gamma }_wis unit weight of water, {\epsilon }_v is the vertical strain of the entire unit cell model of the soil and the drain, k_vis vertical coefficient of permeability, r_dis radius of the drain, r_eis equivalent radius of the unit cell, and \bar{u} is the average excess pore water pressure, which can be integrated from the excess pore pressure at a certain depth, as:

(4)\bar{u}={\displaystyle \frac{\mathrm{1}}{\pi ({{\displaystyle r}}_e^{\mathrm{2}}-{{\displaystyle r}}_d^{\mathrm{2}})}}{\displaystyle \underset{{{\displaystyle r}}_d}{\overset{{{\displaystyle r}}_e}{\int }}\mathrm{2}\pi rudr.}

The strain rate can be derived from stress increment and average pore pressure dissipation, as:

(5){\displaystyle \frac{\partial {\epsilon }_v}{\partial t}}=m_v{\displaystyle \frac{\partial }{\partial t}}\left[\sigma -\bar{u}\right],

where m_v is the volumetric modulus, t is the consolidation time, and \sigmais the applied surcharge load ,.

The inflow water (Q_{\mathrm{1}}) to vertical drain can be defined as:

(6)Q_{\mathrm{1}}={\left[\mathrm{2}\pi r{\displaystyle \frac{k_r(r)}{{\gamma }_w}}{\displaystyle \frac{\partial u}{\partial r}}\right]|}_{r=r_d}.

The outflow water exiting the vertical drain is defined as:

(7)Q_{\mathrm{2}}=-\pi r_d^{\mathrm{2}}{\displaystyle \frac{k_d}{{\gamma }_w}}{\displaystyle \frac{\partial u_d}{\partial z^{\mathrm{2}}}}.

where u_d is the excess pore water pressure inside the drain, and k_d is the permeability of the drain.

Assuming that the water flowing into and flowing out is equal yields:

(8){\left[\mathrm{2}\pi r{\displaystyle \frac{k_r(r)}{{\gamma }_w}}{\displaystyle \frac{\partial u}{\partial r}}\right]|}_{r=r_d}=-\pi r_d^{\mathrm{2}}{\displaystyle \frac{k_d}{{\gamma }_w}}{\displaystyle \frac{\partial u_d}{\partial z^{\mathrm{2}}}}.

The boundary conditions are proposed based on the unit cell model (see Fig. 1):

(9){{\displaystyle \frac{\partial u}{\partial r}}|}_{r=r_e}=\mathrm{0}(\mathrm{the\; outer\; boundary\; is\; impervious}),

(10){u|}_{r=r_d}={u_d|}_{r=r_d}(\mathrm{continuous\; at\; the\; drain\; interface}),

(11){u_d|}_{z=\mathrm{0}}=\mathrm{0}(\mathrm{pervious\; boundary\; at\; the\; top\; of\; the\; drain}),

(12){\overline{u}|}_{z=\mathrm{0}}=\mathrm{0}(\mathrm{pervious\; boundary\; at\; the\; top\; of\; the\; soil}),

(13){{\displaystyle \frac{\partial u_d}{\partial z}}|}_{z=H}=\mathrm{0}(\mathrm{impervious\; boundary\; at\; the\; bottom\; of\; the\; drain}),

(14){{\displaystyle \frac{\partial \overline{u}}{\partial z}}|}_{z=H}=\mathrm{0}(\mathrm{impervious\; boundary\; at\; the\; bottom\; of\; the\; soil}).

Based on the functions derived after several steps, Lu et al. [23] presented a partial differential equation as follows:

(15){{\displaystyle c}}_v{\displaystyle \frac{{\partial }^{\mathrm{4}}{{\displaystyle u}}_d}{\partial z^{\mathrm{4}}}}-{\displaystyle \frac{{\partial }^{\mathrm{3}}{{\displaystyle u}}_d}{\partial z^{\mathrm{2}}\partial t}}-{{\displaystyle c}}_h{\displaystyle \frac{\mathrm{2}}{{{\displaystyle r}}_e^{\mathrm{2}}\mu }}\left[\mathrm{1}+(n^{\mathrm{2}}-\mathrm{1}){\displaystyle \frac{{{\displaystyle k}}_v}{{{\displaystyle k}}_d}}\right]{\displaystyle \frac{{\partial }^{\mathrm{2}}{{\displaystyle u}}_d}{\partial z^{\mathrm{2}}}}+(n^{\mathrm{2}}-\mathrm{1}){\displaystyle \frac{\mathrm{2}}{\mu }}{\displaystyle \frac{{{\displaystyle k}}_h}{{{\displaystyle k}}_d}}{{\displaystyle \left({\displaystyle \frac{\mathrm{1}}{{{\displaystyle r}}_e}}\right)}}^{\mathrm{2}}\left[{\displaystyle \frac{\partial {{\displaystyle u}}_d}{\partial t}}-{\displaystyle \frac{\partial \sigma }{\partial t}}\right]=\mathrm{0},

where c_v={\displaystyle \frac{k_v}{m_v{\gamma }_w}},c_h={\displaystyle \frac{k_h}{m_v{\gamma }_w}},s={\displaystyle \frac{r_s}{r_d}},n={\displaystyle \frac{r_e}{r_d}}, and \mu is a factor of the effect decay pattern of the horizontal permeability. Details of the determination of \mu with the three decay patterns were derived from previous studies [26].

Equation (15) is rewritten in a simpler form as:

(16){{\displaystyle c}}_v{\displaystyle \frac{{\partial }^{\mathrm{4}}{{\displaystyle u}}_d}{\partial z^{\mathrm{4}}}}-{\displaystyle \frac{{\partial }^{\mathrm{3}}{{\displaystyle u}}_d}{\partial z^{\mathrm{2}}\partial t}}-\left[{\displaystyle \frac{\mathrm{2}{{\displaystyle c}}_h}{{{\displaystyle r}}_e^{\mathrm{2}}\mu }}+{{\displaystyle c}}_v\theta \right]{\displaystyle \frac{{\partial }^{\mathrm{2}}{{\displaystyle u}}_d}{\partial z^{\mathrm{2}}}}+\theta \left[{\displaystyle \frac{\partial {{\displaystyle u}}_d}{\partial t}}-{\displaystyle \frac{\partial \sigma }{\partial t}}\right]=\mathrm{0},

where \theta =\left({\displaystyle \frac{n^{\mathrm{2}}-\mathrm{1}}{n^{\mathrm{2}}}}\right){\displaystyle \frac{\mathrm{2}\pi c_h{m}_v{\gamma }_w}{\mu q_w}} and q_w is the discharge capacity..

The relationship between the average pore pressure and pore pressure on the drain was also derived from the solution of [23–26] as:

(17)\bar{u}=u_d-{\displaystyle \frac{\mathrm{1}}{\theta }}{\displaystyle \frac{{\partial }^{\mathrm{2}}{u}_d}{\partial z^{\mathrm{2}}}}.

The combination of Eqs. (19) and (20) under single-ramp loading (Fig. 2) can be solved by an approximated Fourier series [23–26] or by the Laplace transform technique. The solutions to the average degree of consolidation (DOC) were derived and summarized as:

(18)U(T_h)=\{\begin{array}{l}{\displaystyle \frac{T_h}{T_{\mathrm{1}}}}-{\displaystyle \frac{\mathrm{1}}{T_{\mathrm{1}}}}{\displaystyle \sum _{k=\mathrm{0}}^{\infty }{\displaystyle \frac{\mathrm{2}}{{\alpha }_m{M}^{\mathrm{2}}}}(\mathrm{1}-e^{-{\alpha }_m{T}_h})T_h\le T_{\mathrm{1}}}\\ \mathrm{1}-{\displaystyle \frac{\mathrm{1}}{T_{\mathrm{1}}}}{\displaystyle \sum _{k=\mathrm{0}}^{\infty }{\displaystyle \frac{\mathrm{2}}{{\alpha }_m{M}^{\mathrm{2}}}}(e^{-{\alpha }_m(T_{\mathrm{1}}-T_h)}-e^{-{\alpha }_m{T}_h})T_h>T_{\mathrm{1}}}\end{array},

where

{\alpha }_m=c_v{\displaystyle \frac{\mathrm{2}}{r_e^{\mathrm{2}}}}{\displaystyle \frac{\mathrm{1}}{\mu +{\displaystyle \frac{\mathrm{8}}{M^{\mathrm{2}}}}{\displaystyle \frac{n^{\mathrm{2}}-\mathrm{1}}{n^{\mathrm{2}}}}{\displaystyle \frac{k_h}{k_d}}{\left({\displaystyle \frac{H}{\mathrm{2}{r}_d}}\right)}^{\mathrm{2}}}}+c_h{\displaystyle \frac{M^{\mathrm{2}}}{H^{\mathrm{2}}}},M={\displaystyle \frac{\mathrm{2}m+\mathrm{1}}{\mathrm{2}}}\pi ,m=\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{....}\infty ,T_h={\displaystyle \frac{c_{h}t}{{(\mathrm{2}{r}_e)}_{}^{\mathrm{2}}}},

where His the total depth of soil, and T_{\mathrm{1}}={\displaystyle \frac{c_h{t}_{\mathrm{1}}}{{(\mathrm{2}{r}_e)}_{}^{\mathrm{2}}}}is time factor at the time t_{\mathrm{1}} of reaching maximum surcharge load.

Equation (18) was used to generate the database for the subsequent training of the DNN model. Figure 2 illustrates the single-ramp loading and DOC derived from Eq. (18). The following section introduces the application of a DNN to predict the average DOC based on an analytical solution.

3 Deep neural network combined with particle swarm optimization and genetic algorithm

As shown in Fig. 2, the DOC is characterized by a curve, which is also considered as a focus of this study. Moreover, the curve is a function of the DOC with respect to time. Recurrent networks such as long short-term memory (LSTM) and gated recurrent units (GRU) can also simulate time-dependent problems. However, these techniques require a relevant database of input and target with the same series, such as the time interval (for input data) and temperature (for target data). In the current problem, the volume of input data are considerably smaller than that of the target data; thus, the conventional techniques of LSTM and GRU are not applicable for this study. The image classification technique of a DNN model is based on dividing a digital image into numerous small sections and layers, each of which includes a group of digits corresponding to their colors. Based on these features, the corresponding DNN model can be trained to recognize or classify objects. Motivated by this technique, this study provides a simple solution for the target problem by dividing the curve of the DOC into numerous discrete nodes with respect to time (Fig. 3). Furthermore, more nodes are provided in the early stage rather than in the later stage of the consolidation, such that the curve attains a better approximation. Specifically, the DOC line has more curvature in the early stage of the consolidation than in the later stage; therefore, the time spacing in the early stage should be shorter than that in the later stage. From this perspective, a simple equation for the increase in time spacing is provided by:

(19)T_{h,i}=T_{h,i-\mathrm{1}}+{\mathrm{\Delta }}_i,

where T_{h,i} is the time factor at node i and {\mathrm{\Delta }}_i is the increase in the time spacing.

The architecture of a multi-layer perceptron (MLP) of an ANN model is shown in Fig. 4. An MLP is a neural network that includes at least three layers: input, hidden, and output layers [30]. This is the most widely used network in the geotechnical field, with extensive applications in predicting the pile capacity, stress history of clay, undrained shear strength, and soil classification [7–16]. The input layer includes eight fundamental parameters that reflect the vertical drain distributions/properties and soil properties: n={\displaystyle \frac{r_e}{r_d}},s={\displaystyle \frac{r_s}{r_d}},R_s={\displaystyle \frac{k_h}{k_s}},H,c_v={\displaystyle \frac{k_v}{m_v{\gamma }_w}},c_h={\displaystyle \frac{k_h}{m_v{\gamma }_w}},{\displaystyle \frac{k_d}{k_h}}, and T_{\mathrm{1}}. As mentioned above, the output layer includes several DOCs (U_n) at a certain time of the increase. This study applied 25 intervals from T_h=\mathrm{0}to\mathrm{30} to a vector of 25 output data points fromU_{\mathrm{1}}\hspace{0.17em}\mathrm{to}{U}_{\mathrm{25}}. One hidden layer with numerous hidden neurons was placed in the middle of the input and output layers. This strategy has been widely applied by other scholars such as Kurup and Dudani [9], and Moghaddasi and Noorian-Bidgoli [31]. Although one hidden layer network is conventionally employed in the engineering field, recent research has demonstrated that it has lower performance than a DNN [1,5]. Therefore, to explore the effects of a DNN with different hidden layers on the performance of the neuron network, a DNN was applied in this study with one to six hidden layers. Two typical activation functions were applied: a tan-sigmoid function and a linear transfer function (Fig. 3). The combination of these two activation functions can fit any finite input–output problem. The tan-sigmoid function used between the input and hidden layers is defined as:

(20)f_{\mathrm{1}}(x)={\displaystyle \frac{\mathrm{2}}{\mathrm{1}+\exp (-\mathrm{2}x)}}-\mathrm{1},

whereas the linear transfer function applied between the hidden and output layers is defined as:

(21)f_{\mathrm{2}}(x)=x.

As previously mentioned, the ANN model is formed by an interconnected network of the above simple functions with respect to probabilistic weights and bias, as:

(22)x_i={\displaystyle \sum _{j=\mathrm{1}}^{j=n}{w}_{i,j}{y}_j+b_i},

where x_i is the input at the i\mathrm{th} neuron,

w_{i,j}

is the weight of the connection joining the i\mathrm{th} neuron with the j\mathrm{th} neuron in the previous layer, b_i is the bias at the i\mathrm{th} neuron, and n is the number of neurons in the previous layer.

Equations (20)–(22) are utilized through the network to obtain the predicted DOC of U_{p,i} in the output layer. The mean square error (MSE) reflects the second-order error between the target DOC of U_{t,i} and the predicted DOC of U_{p,i} and is expressed as follows:

(23)E(\widehat{w},\widehat{b})={\displaystyle \frac{\mathrm{1}}{N}}{\displaystyle \sum _{i=\mathrm{1}}^{i=N}{(U_{t,i}-U_{p,i})}^{\mathrm{2}}},

where E(\widehat{w},\widehat{b}) is the MSE, which is a function of the weight vector (\widehat{w}) and bias vector (\widehat{b}), and N is the number of pair targets and predicted values.

The performance of the DNN model also depends on the MSE value, which in turn depends on two vectors: weight (\widehat{w}) and bias (\widehat{b}). Therefore, the purpose of training the DNN model is to find a certain combination of the two vectors of weight (\widehat{w}) and bias (\widehat{b}) that minimizes the MSE. From this perspective, numerous training algorithms have been developed during the last decade, including the traditional backpropagation (BP), Polak–Ribiére conjugate gradient, Broyden–Fletcher–Goldfarb–Shanno quasi-Newton, scaled conjugate gradient, resilient BP, Fletcher–Powell conjugate gradient, one-step secant, Levenberg–Marquardt, and variable learning rate BP algorithms. The Levenberg–Marquardt algorithm modified by Hagan and Menhaj [32] is a fast-trained algorithm, and was employed owing to its speed and reliability. This algorithm is also one of the fastest methods for training moderate-sized, feed-forward neural networks [33]. To avoid overfitting problems, an early stop technique by Prechelt [34] was also applied. The technique is related to the convergence rate of both the training and validating convergence graphs. The convergence graph only shows a reduction in MSE during the training process, whereas it shows an increase or remains constant during the validation, indicating overfitting.

Three databases, one for each pattern, with 1000 samples each, were generated from Eq. (18). Each sample included a vector of eight parameters for the input data and a vector for the output data as the target with 25 DOCs (U_{t,i}), at each time increase (Eq. (19)). Table 1 lists the lower and upper limits of the three databases. These values were obtained from the practical points of the ground treatment method by applying a vertical drain, such as the thickness of the treatment layer (H) ranging from 5 to 30 m deep, the prefabricated vertical drain (PVD) spacing ranging from 0.5 to 2 m, which corresponds to a ratio (n) ranging from 15 to 40. The random method was employed to generate these databases with the detailed statistical values of average, min, max, range, and median (Tables 2, 3, and 4). Furthermore, to analyze the distributions of several features in each data set, the histograms of n,R_s,H,T_{h\mathrm{1}} for each pattern are presented in Figs. 5, 6, and 7, respectively. It can be clearly seen that each feature is randomly distributed among its limits. These data sets were then divided into 80% for training and 20% for validation.

The topology of a DNN can significantly affect the accuracy of the entire system based on the number of hidden layers and hidden neurons [35]. According to Kanellopoulos and Wilkinson [36], a DNN has an optimum number of hidden neurons that can be used for image classification. In this study, PSO and GA were used to search for the optimum number of hidden neurons to enhance the performance of the predicted model. The PSO is a robust and effective algorithm, which reflects the natural behavior of a flock of birds moving and finding food [28]. GA was derived from Darwin’s theory of natural selection, that is, that fitter individuals pass their traits through generations [29]. Each new offspring starts from a strong pair parent who survives through a natural selection characterized by a fitness function. Furthermore, a certain mutation ratio can increase the searching capability, which can effectively help overcome the local optimal to reach the final global optimal position. In this study, a mutation ratio of 20% was employed for the GA, and the ratio of crossover were the remaining 80%.

In the PSO method, each particle or bird has its own position and velocity, which are continuously updated based on the distance to the best location. The general equation of PSO is:

(24)v_{i+\mathrm{1}}=w{v}_i+c_{\mathrm{1}}{r}_{a\mathrm{1}}(p_{best}-x_i)+c_{\mathrm{2}}{r}_{a\mathrm{2}}(g_{best}-x_i),

(25)x_{i+\mathrm{1}}=x_i+v_{i+\mathrm{1}},

where r_{a\mathrm{1}} and r_{a\mathrm{2}} are random numbers between 0 and 1, x_i is the position of a particle, w is the weight, v_i is the velocity of a particle, p_{best} is the particle’s best value, g_{best} is the global best value, and c_{\mathrm{1}} and c_{\mathrm{2}} are the acceleration coefficients.

The maximum number of hidden neurons was 20, the number of populations was 50, and the number of interactions was 150. This optimization algorithm first randomly generated locations with different hidden neurons. Using the movement mechanism described above (Eqs. (24) and (25)), the search process was conducted across the solution space. Another problem with current searching algorithms by GA and PSO is that the constraint on integer values due to the number of neurons is an integer value. Both the PSO and GA search for new positions or new generations with real values, while constraining integer values may result in the optimization of the non-convex problem that may arise from the process of updating locations. In other words, the values of movements (PSO) and values of mixed generation (GA) are real values, whereas constraining these values into integer values may result in loss of accuracy of the correct positions and lead to a non-convex problem. This study provides a simple real and image position technique to address this problem. Figure 8 describes the concept of real and image positions of the four-step movement of a particle. The real position is the coordinate with real values for each individual in a population, whereas the image position is coordinated with the integer values that are closest to the real position by simply rounding off the real values. The image position can be used to determine the number of neurons in each hidden layer, whereas the real position can be employed for further movements or generations. This procedure can help to preserve the searching capability of PSO and GA and is also applicable for determining the natural values for the number of neurons.

In comparison with GA, PSO is simpler; however, it is difficult to overcome the local optimal position, whereas other techniques such as GA with a certain mutation ratio can effectively search the global optimal position. A proper tuning parameter for PSO can considerably help to overcome this limitation. Several trial-and-error procedures are also presented to select parameters for PSO in a simple hidden layer network with a data set of pattern I (Table 5). The best set of parameters among the 10 random cases were w ranging from 1.0 to 0.8, c_{\mathrm{1}}=\mathrm{1} and c_{\mathrm{2}}=\mathrm{2}with the results of the training process of MSE=\mathrm{1.72}E-\mathrm{5}and 11 optimum neurons (case 3). This set of parameters was selected for further analysis by PSO.

4 Results and discussions

Figure 9 illustrates the convergence of the training process with numbers of hidden layers varying from one to six. GA and PSO were employed to optimize the topology of the DNN for three data sets of patterns I, II, and III. Figure 9 clearly shows that DNNs with one and six hidden layers have lower performance (with large MSE) than the other networks for both optimization methods (PSO and GA). The shallow network of one hidden layer demonstrated lower accuracy, which is in good agreement with results obtained by other researchers [1,5]. However, in this study, the larger network of six hidden layers also showed a large error in the training process. The data sets may cause this issue because the convergence rate is different for different types of data sets (based on the characteristics of the feature and the complexity of the data set). The best performance was observed for the networks with two to five hidden layers. It can be concluded that for the data set of vertical drain, the networks with one and six hidden layers have worse performance in the training process than the networks with two to five hidden layers.

Tables 6, 7, and 8 summarize the analysis results for data set patterns I, II, and III, respectively. From these tables, there was an excellent agreement between the optimum number of neurons of the single hidden layer network (11 neurons) for both optimization methods (PSO and GA). The MSE variable may come from the initial weights, but the optimization of the initial weights is beyond the scope of this study. Another aspect from the tables is that the DNN–GA exhibited better performance than the DNN–PSO for both training and validating. During the training process, the MSE values of DNN–PSO were 7.51E-6, 4.15E-6, and 2.79E-6, which were larger than those of DNN–GA of 5.51E-6, 2.6E-6, and 2.5E-6, for patterns I, II, and III, respectively. Regarding the validating process, DNN–PSO exhibited the best MSE values of 2.55E-5, 2.25E-5, and 1.02E-5, which were larger than the values for DNN–GA of 1.42E-5, 1.07E-5, and 8.27E-6 for patterns I, II, and III, respectively. The better performances of DNN–GA may result from the basic optimization algorithm, in which PSO may still have many difficulties to overcome the local optimal positions. The GA with the mutation process can easily reach the global solution. Three training networks were selected based on the performance of validation: DNN–GA with four hidden layers (5,4,7,15) for pattern I, DNN–GA with four hidden layers (4,4,8,16) for pattern II, and DNN–GA with three hidden layers (4,7,12) for pattern II. The networks were used for further verification by analytical results and real full-scale embankment tests.

Figures 10, 11, and 12 compare the predicted results with the results obtained by Tang and Onitsuka [24] and Xie et al. [26] corresponding to patterns I, II, and III, respectively. In these cases, the same input parameters ofn=\mathrm{20},s=\mathrm{2},R_s=\mathrm{5},H=\mathrm{10.56},c_v=c_h,\mathrm{and}{k}_d/k_h=\mathrm{5000} were applied. Six different single-ramp loading scenarios were considered, corresponding to different T_{\mathrm{1}} values (T_{\mathrm{1}}=\mathrm{0},\mathrm{0.25},\hspace{0.17em}\mathrm{0.5},\hspace{0.17em}\mathrm{1},\mathrm{2},\mathrm{and}\mathrm{4}). For T_{\mathrm{1}}=\mathrm{0}, the scenario exhibited instantaneous loading. In general, except for the instantaneous loading, for which there were slight differences between the predicted and analytical results, in other loading scenarios, there was excellent agreement between these results.

5 Validation

The performances of the three full-scale embankment tests with a vertical drain at the Bangkok International Airport were reported by Bergado et al. [17] and subsequently simulated using three-dimensional FEM by Lin and Chang [37]. Numerous studies have also used these databases, including those of Lam et al. [38],. Therefore, they can be used to verify the proposed DNN model. The three embankments (TS1, TS2, and TS3) with three different PVD spacings of 1.5, 1.2, and 1.0 m, respectively, were distributed in a square pattern. The soil profile included soft to very soft clay with a depth of approximately 12 m, followed by medium stiff clay, stiff clay, and finally dense sand . The PVDs installed at a depth of 12 m below the ground surface were expected to improve the entire soft to very soft clay layers .The PVD installation plans also spread to the toe of the slope of the test embankment with a maximum surcharge height of 4.2 m. The loading sequences can be simplified as a single-ramp loading, as shown in Fig. 13, with t_{\mathrm{1}}=\mathrm{220}days considered as the time to reach the maximum surcharge loading. This simplification has several limitations, as it may cause underestimation during the early stage owing to the lower loading rate in the simplified scenario than in the real loading. However, the rate of the simplified scenario was faster than that of the real loading in the later stage (Fig. 13). These limitations can lead to a smooth curve when the DOC is predicted by the DNN model, although fluctuations are observed in the measured data before the surcharge height reaches a maximum. However, in the following waiting period, the predicted and measured DOCs should be similar. From this perspective, the input parameters of the proposed model, such as the smear ratio (R_s), horizontal coefficient of consolidation (c_h), vertical coefficient of consolidation (c_v), and permeability ratio (k_d/k_h), were carefully calibrated based on previous research findings and a trial-and-error method.

The smear ratio R_s=\mathrm{8} was selected, in agreement with Lam et al. [38], who simulated the ground improvement of the Bangkok International Airport using the PVD preloading method. .Furthermore, their research suggested that c_h=\mathrm{2}\mathrm{to\; 3}{c}_v was appropriate for the analysis. c_h in the field was back-analyzed by Bergado et al. [17] with values of approximately 3–8 (m²/year).In the present study, several parameters were selected based on previous findings, and are summarized in Table 9. The fact that only the permeability ratio (k_d/k_h) is directly related to the well resistance or discharge capacity of a vertical drain is controversial.

Bergado et al. [39] determined that the required discharge capacity was approximately 26–30 (m³/year) forn=\mathrm{20}\hspace{0.17em}\mathrm{to\; 30}.. Moreover, Deng et al. [40] proposed an analytical solution, based on which they back-analyzed the settlement of a treated embankment with an initial discharge capacity of 500 (m³/year) that exponentially decreased to 0.59 (m³/year) after 300 days. These disagreements may arise from the difficulty in obtaining the in situ measurements of the discharge capacity or well resistance of the PVD to confirm the laboratory test results. Based on the proposed DNN model, the permeability ratios were calibrated using a trial-and-error method to determine the most approximate DOC using field measurements. The final selected ratios are listed in Table 9, with k_d/k_h= 80000, 50000, and 35000 for embankments TS1, TS2, and TS3, respectively.

Figure 14 compares the DOCs predicted by the proposed DNN model in pattern I with the field data. The predicted and measured DOCs are in excellent agreement, except for several fluctuations in the early stage (before t_{\mathrm{1}}=\mathrm{220} days). Embankment TS1 (PVD spacing of 1.5 m) exhibited the largest permeability ratio k_d/k_h=80000, followed by embankment TS2 (PVD spacing of 1.2 m) with k_d/k_h=50000, and finally embankment TS3 (PVD spacing of 1.0 m), with k_d/k_h=35000. These results demonstrate that the well resistance increases when the spacing in the vertical drain decreases. This may indicate that the more settlement embankments, the better the resistance (influence of clogging and deformation). Therefore, the above permeability ratios should decrease when the spacing of the vertical drain is reduced. The predicted values from the DNN model in radial drainage pattern I are in accordance with the field data.

Radial drainage pattern I is widely used because of its simplicity, whereas patterns II and III have limited research and application. However, patterns II and III were considered to be more realistic distributions owing to the PVD installation effect. The input parameters (based on the previous findings) previously presented for pattern I were reused for patterns II and III to simplify the calibration. Only the permeability ratios were calibrated using trial-and-error. These calibrations have a strong relevance to practical designers when they switch from the traditional solution of radial drainage pattern I to new patterns II and III. The tendencies of patterns II and III lead to simple and suitable solutions that are not very different from those of common methods. The proposed approach can easily bridge the applications from simplified solutions with pattern I to more realistic solutions with patterns II and III. Tables 10 and 11 list the selected input parameters of the proposed ANN model for patterns II and III, respectively. It can be seen that the permeability ratios are similar to that of pattern I. The permeability ratios decreased when the spacing of the vertical drain was reduced. However, pattern III with parabolic radial permeability distribution had lower values of permeability ratio than pattern II with a linear radial permeability distribution.

Figures 15 and 16 compare the DOCs predicted by the proposed DNN model with the measured DOCs. As previously mentioned, the simplified loading scenarios have several limitations, leading to differences between the predicted and measured DOCs in the early stage. However, in the later stages, both values exhibited good agreement. Despite limitations such as a simplified loading scenario, considering uniform subsoil and ideal elastic deformation, the proposed DNN model demonstrates an excellent ability to predict DOCs in comparison with analytical solutions and field measurements. The verification of the analytical results and field data indicates that the method using the DNN model has potential applications for design of vertical drains. Based on its good performance, it is reasonable to state that the solution to the consolidation of vertical drains by the DNN model can be even better than the concept of using predicting solutions such as analytical and numerical solutions, and FEM. Specifically, it can be used for sophisticated and real consolidation problems of vertical drains with only sufficient databases, including the input and output parameters.

6 Conclusions

In this study, a new approach was proposed for predicting the DOC of a vertical drain under time-dependent loading using a data-driven approach based on DNN–PSO and DNN–GA. The following conclusions were drawn.

1) The current approach combines DNN–PSO and DNN–GA to predict the vertical drain consolidation problem. The DOC curve is a function of the time. The discretization of this line into numerous nodes with a certain time factor was proposed. Three data sets of variations in the radial permeability in the smear zone with patterns I, II, and III were generated (1000 samples for each data set). Furthermore, statistical diagrams were plotted to show the distribution of the features. The data set was then split into 80% for training and 20% for validation.

2) Different DNNs with hidden layers varying from one to six were employed in this study to explore the effect of DNNs, and the topology of these DNNs was optimized by PSO and GA methods.

3) The networks with one and six hidden layers showed lower performance in training and validation for both DNN–GA and DNN–PSO, and the best networks were those with hidden layers from two to five. Both the DNN–GA and DNN–PSO showed good agreement regarding the number of optimum neurons of 11 in the shallow network (one hidden layer). The network with one hidden layer may easily lead to overfitting, even with a small number of iterations. Thus, networks with a larger number of hidden layers are recommended.

4) When comparing the best network for each data set, DNN–GA showed a stronger performance than DNN–PSO for both training and validation. This may be due to the GA technique mutation, which can effectively help to overcome the local optimal points and reach global points. The three best DNN–GAs were selected for validation with a full-scale embankment project.

5) Regarding the accuracy and reliability of the proposed method, we compared the present results with those of the analytical solutions and the three embankment tests. The corresponding measured results were in good agreement with the predicted results.

References

[1] Guo H, Zhuang X, Rabczuk T. A deep collocation method for the bending analysis of Kirchhoff plate. Computers, Materials & Continua, 2019, 59(2): 433–456

[2] Anitescu C, Atroshchenko E, Alajlan N, Rabczuk T. Artificial neural network methods for the solution of second order boundary value problems. Computers, Materials & Continua, 2019, 59(1): 345–359

[3] Samaniego E, Anitescu C, Goswami S, Nguyen-Thanh V M, Guo H, Hamdia K, Zhuang X, Rabczuk T. An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications. Computer Methods in Applied Mechanics and Engineering, 2020, 362: 112790

[4] Guo H, Zhuang X, Meng X, Rabczuk T. Integrated intelligent Jaya Runge-Kutta method for solving Falkner-Skan equations for various wedge angles. 2020, arXiv:2010.05682

[5] Hamdia K M, Zhuang X, Rabczuk T. An efficient optimization approach for designing machine learning models based on genetic algorithm. Neural Computing & Applications, 2021, 33(6): 1923–1933

[6] Zhuang X, Guo H, Alajlan N, Rabczuk T. Deep autoencoder based energy method for the bending, vibration, and buckling analysis of Kirchhoff plates. European Journal of Mechanics. A, Solids, 2020, 2010: 05698

[7] Ahmad I, Hesham El Naggar M, Khan A N. Artificial neural network application to estimate kinematic soil pile interaction response parameters. Soil Dynamics and Earthquake Engineering, 2007, 27(9): 892–905

[8] Momeni E, Nazir R, Jahed Armaghani D, Maizir H. Prediction of pile bearing capacity using a hybrid genetic algorithm-based ANN. Measurement, 2014, 57: 122–131

[9] Kurup P U, Dudani N K. Neural networks for profiling stress history of clays from PCPT data. Journal of Geotechnical and Geoenvironmental Engineering, 2002, 128(7): 569–579

[10] Lee S J, Lee S R, Kim Y S. An approach to estimate unsaturated shear strength using artificial neural network and hyperbolic formulation. Computers and Geotechnics, 2003, 30(6): 489–503

[11] ham B T, Nguyen M D, Dao D V, Prakash I, Ly H B, Le T T, Ho L S, Nguyen K T, Ngo T Q, Hoang V, Son L H, Ngo H T T, Tran H T, Do N M, Van Le H, Ho H L, Tien Bui D. Development of artificial intelligence models for the prediction of compression coefficient of soil: An application of Monte Carlo sensitivity analysis. Science of the Total Environment, 2019, 679: 172–184

[12] Beucher A, Møller A B, Greve M H. Artificial neural networks and decision tree classification for predicting soil drainage classes in Denmark. Geoderma, 2019, 352: 351–359

[13] Abbaszadeh Shahri A, Spross J, Johansson F, Larsson S. Landslide susceptibility hazard map in southwest Sweden using artificial neural network. Catena, 2019, 183: 104225

[14] Chen W, Pourghasemi H R, Kornejady A, Zhang N. Landslide spatial modeling: Introducing new ensembles of ANN, MaxEnt, and SVM machine learning techniques. Geoderma, 2017, 305: 314–327

[15] Khanlari G R, Heidari M, Momeni A A, Abdilor Y. Prediction of shear strength parameters of soils using artificial neural networks and multivariate regression methods. Engineering Geology, 2012, 131–132: 11–18

[16] Tiryaki B. Predicting intact rock strength for mechanical excavation using multivariate statistics, artificial neural networks, and regression trees. Engineering Geology, 2008, 99(1–2): 51–60

[17] Bergado D T, Balasubramaniam A S, Fannin R J, Holtz R D. Prefabricated vertical drains (PVDs) in soft Bangkok clay: A case study of the new Bangkok International Airport project. Canadian Geotechnical Journal, 2002, 39(2): 304–315

[18] Bergado D T, Chaiyaput S, Artidteang S, Nguyen T N. Microstructures within and outside the smear zones for soft clay improvement using PVD only, Vacuum-PVD, Thermo-PVD and Thermo-Vacuum-PVD. Geotextiles and Geomembranes, 2020, 48(6): 828–843

[19] Nghia N T, Lam L G, Shukla S K. A new approach to solution for partially penetrated prefabricated vertical drains. International Journal of Geosynthetics and Ground Engineering, 2018, 4(2): 11–17

[20] Nghia-Nguyen T, Shukla S K, Nguyen D D C, Lam L G, H-Dang P, Nguyen P C. A new discrete method for solution to consolidation problem of ground with vertical drains subjected to surcharge and vacuum loadings. Engineering Computations, 2019, 37(4): 1213–1236

[21] Nguyen T N, Bergado D T, Kikumoto M, Dang H P, Chaiyaput S, Nguyen P C. A simple solution for prefabricated vertical drain with surcharge preloading combined with vacuum consolidation. Geotextiles and Geomembranes, 2021, 49(1): 304–322

[22] Shome R, Tang W N, Song C, Mitash C, Kourtev H, Yu J, Boularias A, Bekris K E. Towards robust product packing with a minimalistic end-effector. In: 2019 International Conference on Robotics and Automation (ICRA). IEEE, 2019, 9007–9013

[23] Lu M M, Xie K H, Wang S Y. Consolidation of vertical drain with depth-varying stress induced by multi-stage loading. Computers and Geotechnics, 2011, 38(8): 1096–1011

[24] Tang X W, Onitsuka K. Consolidation by vertical drains under time-dependent loading. International Journal for Numerical and Analytical Methods in Geomechanics, 2000, 24(9): 739–751

[25] Rujikiatkamjorn C, Indraratna B. Analytical solution for radial consolidation considering soil structure characteristics. Canadian Geotechnical Journal, 2015, 52(7): 947–960

[26] Xie K H, Lu M M, Liu G B. Equal strain consolidation for stone columns reinforced foundation. International Journal for Numerical and Analytical Methods in Geomechanics, 2009, 33(15): 1721–1735

[27] Dey N, Borra S, Ashour A Sand Shi F. Machine Learning in Bio-Signal Analysis and Diagnostic Imaging. Academic Press, 2018, 159–182

[28] Kennedy J, Eberhart R. Particle swarm optimization. In: Proceedings of the IEEE International Conference on Neural Networks. IEEE, 1995, 4: 1942–1948

[29] Golberg D E. Genetic Algorithms in Search, Optimization, and Machine Learning. Addison Wesley, 1989

[30] Van Der Malsburg C. Frank Rosenblatt: Principles of neurodynamics: Perceptrons and the Theory of Brain Mechanisms. In: Palm G, Aertsen A, eds. Brain Theory. Berlin: Springer, 1986, 245–248

[31] Moghaddasi M R, Noorian-Bidgoli M. ICA-ANN, ANN and multiple regression models for prediction of surface settlement caused by tunneling. Tunnelling and Underground Space Technology, 2018, 79: 197–209

[32] Hagan M T, Menhaj M B. Training feedforward networks with the Marquardt algorithm. IEEE Transactions on Neural Networks, 1994, 5(6): 989–993

[33] Rafiq M Y, Bugmann G, Easterbrook D J. Neural network design for engineering applications. Computers & Structures, 2001, 79(17): 1541–1552

[34] Prechelt L. Early stopping—But when, neural networks: Tricks of the trade. In: Montavon G, Orr G B, Müller K R, eds. Neural Networks: Tricks of the Trade. Springer, 1998, 55–69

[35] Huang S C, Huang Y F. Bounds on the number of hidden neurons in multilayer perceptrons. IEEE Transactions on Neural Networks, 1991, 2(1): 47–55

[36] Kanellopoulos I, Wilkinson G G. Strategies and best practice for neural network image classification. International Journal of Remote Sensing, 1997, 18(4): 711–725

[37] Lin D G, Chang K T. Three-dimensional numerical modelling of soft ground improved by prefabricated vertical drains. Geosynthetics International, 2009, 16(5): 339–353

[38] Lam L G, Bergado D T, Hino T. PVD improvement of soft Bangkok clay with and without vacuum preloading using analytical and numerical analyses. Geotextiles and Geomembranes, 2015, 43(6): 547–557

[39] Bergado D T, Manivannan R, Balasubramaniam A S. Proposed criteria for discharge capacity of prefabricated vertical drains. Geotextiles and Geomembranes, 1996, 14(9): 481–505

[40] Deng Y B, Liu G B, Lu M M, Xie K H. Consolidation behavior of soft deposits considering the variation of prefabricated vertical drain discharge capacity. Computers and Geotechnics, 2014, 62: 310–316

RIGHTS & PERMISSIONS

Higher Education Press