A Threshold Model of Tailings Sand Liquefaction Based on PSO-SVM

Abstract

The liquefaction of tailings sand during seismic activity poses significant risks to both property and the safety of individuals. This study focused on tailings sand and conducted liquefaction tests using a self-designed rigid test-box. Various factors were considered such as the dynamic factor, static factor, and drainage factor, which were all characterized and obtained. To determine the most influential factors, dimension reduction analysis was performed using SPSS. Additionally, a threshold model for predicting liquefaction was proposed by improving the parameters based on particle swarm optimization (PSO) and incorporating them into the support vector machine (SVM). Comparisons with two other established algorithms revealed that the improved algorithm not only exhibited a high accuracy rate of 92.7%, but also demonstrated faster performance. This model can serve as a crucial foundation for preventing earthquake disasters in tailings ponds.

1. Introduction

Recently, extreme climate change anomalies and frequent earthquakes have resulted in various natural disasters, such as landslides introduced by slope instability [1,2,3]. The stability problem of tailings dams as artificial slopes has always been a global concern, mainly in terms of preventing property damage and keeping the environment from being threatened [4,5,6]. At present, the upstream method is generally used in the construction of tailings dams [7], which directly leads to an excessively high flooding rate, even when the tailings sand is completely saturated. The tailings sand is very susceptible to liquefaction and deformation because of its sensitivity to ground movement, which can lead to instability and collapse. According to statistical analysis, since the beginning of the 20th century, there have been more than 200 tailings dam accidents caused by earthquakes in China [5,6,7]. The main reason is that the saturated tailings sand is very easy to liquefy and deform under the action of earthquakes, resulting in local dam failures [8,9,10]. Therefore, the crucial study in the area of tailings dam disaster is establishing the liquefaction parameters, liquefaction degree, and failure prediction of saturated tailings sand.

The possibility for sand liquefaction has been assessed using a variety of techniques, up to this point. However, most of these methods use some state to separate the non-liquefaction region from the liquefaction region to establish evaluation criteria [11,12,13,14,15,16]. However, it is challenging to adopt suitable empirical equations for regression analysis due to the unpredictability, complexity, and uncertainty of a soil mass [17,18,19,20,21]. As a result, many professionals and academics work to construct ways to identify the liquefaction of sandy soil and create simpler, more useful scientific analytical models [22].

In recent years, many scholars have tried to develop a simpler and more practical scientific analysis models in order to predict the possibility of liquefaction, and they have also established many methods for judging sand liquefaction [23]. Based on the GS search method, Zhang [24] et al. searched for the best parameters of the SVM, describing the training set of sand liquefaction with the help of the Libsvm program and then establishing a sand liquefaction prediction model on this basis. Liu [25] et al. selected eight influencing factors of sand liquefaction, including intensity, as predictive factors of the FDA model. Then the FDA discriminant model was established based on Fiser’s discriminant theory. Gao [26] et al. established a BP neural network and Elman network, and obtained a prediction model with faster operation speed and higher accuracy, with the help of Matlab scientific computing software. Yu [27] et al. proposed a gray weighted prediction method with an accuracy rate of up to 90% based on the gray system theory. In order to predict sand liquefaction, Pan [28] et al. applied a regression model and analyzed the sensitivity of different indicators under liquefaction based on the collected 173 sets of sample data. In the multi-index comprehensive discriminant method of sand liquefaction, Zhao [29,30] et al. extracted the initial samples based on KPCA and combined them with GPC to construct the KPCA-GPC model of sand liquefaction. Kurnaz [31] et al. suggested a novel classification-based integrated group data handling (EGMDH) model.

In summary, experts and scholars have conducted a lot of research on sand liquefaction prediction under dynamic load, but tailings sand is neither cohesive soil nor non-cohesive soil, and there are few studies on the liquefaction phenomenon after the complex coupling of solid–liquid–gas chemistry. Therefore, for the liquefaction of ore tailings, there is an urgent need to propose a comprehensive consideration of various factors, and the convergence rate is relatively fast to determine the liquefaction threshold. This article examines the primary liquefaction-causing causes using proto-type tailings sand as the material, and uses SPSS software for factor dimension reduction analysis to determine the most suitable for factor analysis of influencing factors. It then uses the particle swarm method of parameter optimization (PSO) to generate into the support vector machine (SVM) for liquefaction condition prediction. The established threshold model technology provides an important basis for earthquake prevention of tailings ponds, and can provide a reference for engineering applications.

2. Materials and Methods

2.1. Test Equipment

Direct field testing is exceedingly challenging because of how destructive and unpredictable earthquakes are. Therefore, based on the findings of the current research conduct, the liquefaction test was carried out through the self-designed rigid test-box [32]. The Liaoning University of Engineering Technology’s ANCO R-232H seismic simulator conducted the model test of tailings sand liquefaction. The major components of the data acquisition system employed in the experiment were as follows: A DH-3817K dynamic and static data acquisition equipment was used for the collection of pore pressure and earth pressure data. A DH-5981 dynamic data acquisition tool was used to conduct acceleration acquisition [33].

2.2. Test Material

Table 1. Physical and mechanical properties of the tailings sand.

Plasticity	Liquid Limit	Plasticity Index	Optimum Moisture Content	Void Ratio	Specific Gravity	Maximum Dry Density
ωp/%	ωl/%	Ip/%	ωop/%	e/%	G/g/cm3	Ρdmax/g/cm3
13.1%	20.5	7.4	14.2	0.892	2.84	1.92

displays the fundamental mechanical and physical indices of the ore tailings, and Figure 1 depicts the precise grading curve.

According to the research in [34], the dynamic strength of unsaturated tailings sand was higher than saturated tailings sand during the two phases of the tailings sand vibration process.

In addition, the preparation method of saturated sand was as follows:

(1)

Before preparing the model, the pore pressure gauge, earth pressure gauge, and accelerometer were arranged at the predetermined position of the test-box by flexible arrangement;

(2)

A certain amount of pure water was injected into the test-box;

(3)

The tailings sand layers were prepared in 10 layers, and the vibrating sand shaker evenly sprinkled the tailing sand from the sand outlet. If there was any unevenness, the surface was quickly leveled to ensure each tailing sand layer was consistent;

(4)

The height of the water surface was controlled at 5 cm above the sand layer in each operation to ensure that the prepared tailing sand was in a saturated state (maximum thickness 75 cm). However, the tailings sand between 75–90 cm from the platform height was not saturated, to guarantee that the tailings sand was in a half-soaked state, in order to more accurately imitate the saturation state of the tailings sand in the actual tailings dam;

(5)

The test was performed after the elements were still for 24 h.

2.3. Test Model Box

An enhanced new rigid test-box was presented after taking into account the benefits and drawbacks of rigid boxes, flexible boxes, and shearing test-boxes. To ensure that the test-box was applicable, tailings sand was used in the vibrating table model test. According to the experimental findings, the natural frequencies of the test-box before and after the test were 12.25 Hz and 2.36 Hz, respectively. The fundamental frequency of the empty test-box is five times that of the test-box containing tailings. The test-box has no effect on the vibration properties of the model tailings sand’s free region and does not cause self-vibration. The vibrating frequency satisfies the vibrating table’s test standards.

The boundary of the test-box was processed in accordance with the benefits and drawbacks of the rigid box, flexible box, and shear test-box:

(1)

Treatment of the friction bottom: First, the inner side of the bottom plate was dusted with gravel of various sizes to simulate a friction boundary in order to prevent relative sliding of the test tailings sand and the bottom of the test-box as much as possible and to ensure that they have a good bond.

(2)

Sliding boundary treatment: Lubricating oil was applied to the inner wall of the test-box as the sliding boundary treatment to minimize the frictional resistance and increase the stiffness of the tailings sand.

(3)

Flexible boundary treatment: The sidewall of the test-box was coated with a molded polystyrene foam board as the flexible boundary treatment.

In addition, four rectangular pipes were welded vertically along the four corners of the frame. Each side was welded into an “x” shape with rectangular pipes. The test-box can hardly be bent and deformed. It is difficult for the model floor and the bottom of the tailings to slip relative to each other, but due to the great restraint formed around it, the model tailings sand’s free shear deformation in the horizontal direction is inescapably constrained. The tailings sand shall not be twisted or distorted horizontally during the design process based on the rigid box.

The 2-norm method formula is as follows:

$$\alpha =\frac{\Vert {\mathrm{x}}_{\mathrm{i}}-{\mathrm{x}}_{\mathrm{j}}\Vert }{\Vert {\mathrm{x}}_{\mathrm{i}}\Vert }=\sqrt{\frac{\mathsf{\Sigma }{\left({\mathrm{x}}_{\mathrm{i}}-{\mathrm{x}}_{\mathrm{j}}\right)}^2}{\mathsf{\Sigma }{\left({\mathrm{x}}_{\mathrm{i}}\right)}^2}}$$

(1)

where x_i is the reference point acceleration (central acceleration) and x_j is the comparing point acceleration (boundary point acceleration). Both are the maximal acceleration values at these points, and “i” and “j” both represent different time moments.

From the definition, if α is 0, the boundary effect may be entirely avoided because the two signals are equal.

Table 2. Norm index of the input seismic wave model boundary of the shaking table.

Working Condition	Seismic Wave	Peak Acceleration /m/s2	Reference Point Acceleration /m/s2	Comparing Point Acceleration /m/s2	Boundary Norm Index /%
1	II (El Centro)	0.2	0.246	0.250	1.868
2		0.4	0.457	0.465	1.965
3		0.6	0.665	0.670	0.755
4	III (Traft)	0.1	0.157	0.158	1.061
5		0.3	0.317	0.340	7.652
6		0.5	0.560	0.566	1.245
7	IV (Shanghai wave)	0.4	0.453	0.453	0.011
8		0.2	0.289	0.284	1.653
9		0.4	0.415	0.454	9.282
10		0.5	0.578	0.580	0.335

shows that the boundary norm index for various seismic wave types is less than 10%, which satisfies the test-box’s boundary design specifications. Peak acceleration is the one that applies to the box and is the actual acceleration that was collected. The center acceleration serves as the reference point acceleration. The acceleration at the border serves as the comparison point acceleration.

2.4. Sensor and Measuring Point Arrangement

Among them, the BS-1 piezometer has high sensitivity, good permeability and strong sealing; BX-1-type has a measuring range of 0.1MPa and an external dimension of Φ17 × 7 mm; On the one hand, CE0-type piezoelectric voltage (IEPE) acceleration sensors are processed to obtain accurate acceleration data from the front of the ground along the direction of the ground motion when it is buried. On the other hand, it must be sealed and waterproof. Therefore, the acceleration sensor is placed in a small box made of PVC board before the test and uses Special Kraft 304 silicone rubber for sealing and waterproofing; The range of GJBLS-1 tension sensor is 0-3kN. The sensor arrangement diagram is shown in Figure 2.

2.5. Test Loading Scheme

According to the geological survey report, with a preset seismic acceleration of 0.10 g and a characteristic period of 0.45 s, the corresponding shear wave velocity exceeds 284 m/s, indicating a moderately hard site. A Traft wave is an actual strong earthquake record with a characteristic period of 0.44 s, suitable for the medium hard site. Therefore, this test will explore the dynamic response characteristics of a Traft wave, El Centro seismic wave, and Shanghai seismic wave. The actual situation of the three seismic waves is shown in Figure 3.

A relevant summary of vibration liquefaction in research and soil dynamics is available. In this experiment PGA (0.1 g, 0.2 g and 0.3 g) was used to determine the magnitude. Different local waves (II, III, and IV) made up the waveform, while ground motion duration made up the frequency. Shear strain served as a defining feature of the stress–strain history in soil parameters. Pore pressure variations in the burial components were used to describe the drainage conditions, and buried depth was used to describe the overlying effective stresses.

Table 3. Model test loading conditions.

Site Type	Seismic Wave	Earthquake Intensity (PGA)	Working Condition
III	Traft	0.1 g	1
		0.2 g	2
		0.3 g	3
II	El Centro	0.1 g	4
		0.2 g	5
		0.3 g	6
IV	Shanghai wave	0.1 g	7
		0.2 g	8
		0.3 g	9

displays the precise model test loading circumstances.

3. Algorithm Design Based on PSO-SVM

3.1. Particle Swarm Optimization (PSO)

PSO typically uses actual coding and determines its new speed based on its prior speed, the distance between its present location, and both its own and its neighbor’s best historical positions, in the absence of cross-validation and variation. In order to prevent particles from wandering too far beyond the search area, the values of each component in ν_i can often be sandwiched by [ν_min, ν_max]. When the particle achieves its maximum algebra, it travels to a new location and restarts the process. The main parameters of the PSO algorithm include population size and algebra. The topological structure of the group and the number of particles in the group both affect how quickly the algorithm converges. For example, tightly linked groups with bigger domains converge more quickly than weakly connected neighborhoods. Is it feasible to develop a special set of algorithmic parameters that are effective in every situation? The following empirical program is discovered to be effective in practice based on the aforementioned factors [35].

3.2. Support Vector Machine (SVM)

A brand-new class of machine algorithm frequently employed for pattern recognition is the support vector machine. The salient features of SVM are the lack of local minima, the sparsity of the solution, and the feature space using kernel functions. Most classifiers were previously classified using hyperplanes. However, support vector machines extend the concept of classification to situations where linear classification is not possible by mapping predictors to a new dimensional space. In most cases, only the wrong kernel will appear in these different categories. Therefore, it can be considered that the optimal position of the decision plane is found only in the calculation, and the linear boundary is obtained by the nonlinear transformation to obtain the optimization problem of the kernel function creation [35].

3.3. Algorithm Design

PSO-SVM algorithm construction:

(1) Selection of data: According to the vibration table test data, the analysis of the factors affecting liquefaction under ground motion was carried out, and the factors suitable for the prediction model were obtained.

(2) Sample establishment: The parameters of the appropriate influencing factors were examined based on the factor analysis, and the relevant results were then standardized. Then, the historical step number m and the predicted step number n were determined, thereby establishing learning and test samples, respectively.

(3) Initialization of data: Based on the particle swarm method, the initialization of the particle swarm parameters is performed, that is, the population size, the total number of iterations to be performed, the initial random values of the particles ${x^0}_{\mathrm{i}}$ and the velocity ${v^0}_{\mathrm{i}}$ were set. This included a particle vector representing an SVM model that corresponded to different penalty factors C and kernel function parameters g.

(4) Determine the search interval of (C, g).

(5) Found the fitness function of each particle: Calculate the particle fitness value in the group according to the objective function, compared the fitness value with the adaptation value of the more and more positions that were experienced, and obtain the best adaptation value:

$$f(x)=\min \left[\max \left(\frac{\left|x_j-{x^{\ast }}_j\right|}{x_j^{\prime }}\right)\right],j=1,2,\cdots ,n$$

(2)

where $x_j$ the measured value of the j-th sample; ${x^{\ast }}_j$ the predicted value of the j-th sample; and l—the number of samples.

(6) According to the particle swarm algorithm, the fitness value f_i calculated according to Formula (2) is compared with the optimal solution $f\left(p_{bes{t}_i}\right)$ of the self. If so, $f\left(p_{{\mathit{best}}_i}\right)>f_i$ the f_i at this time replaces the optimized value of the previous process and simultaneously updates the particles, namely: $p_{bes{t}_i}=x_i,f\left(p_{bes{t}_i}\right)=f_i$

(7) Compare the fitness value $f\left(p_{{\mathit{best}}_i}\right)$ with the fitness value of the optimal position $f\left(g_{best}\right)$ traversed by the population. If the result is smaller than the fitness value of the optimal position traversed by the population (i.e., $f\left(p_{bes{t}_i}\right)<f\left(g_{\mathit{best}}\right)$), the position is taken as the optimal position and stored.

(8) Determined whether the f_i or the number of iterations meets the preset criteria. At this point, the ideal kernel function parameter g and penalty parameter C will be chosen.

(9) On this basis, the PSO-SVM prediction mode is established, and the output data are obtained. The proposed model is implemented on the MATLAB R2018a platform.

4. Factor Analysis of Influence Parameters of Tailings Sand Liquefaction

The factors impacting liquefaction under the influence of ground motion were examined in accordance with the shaking table test results to get features acceptable for the prediction model. SPSS software was used for factor dimensionality reduction analysis to obtain the most suitable features for factor analysis.

The liquefaction of tailings sand after an earthquake was investigated using the shaking table test under the effect of different time–space characteristics. To find the key factors and to make a corresponding comprehensive prediction, it is necessary to analyze the influence of different predictors on the possibility of liquefaction. According to the results of existing experiments and the above results of factor analysis and correlation among features, this paper selects eight types of impact features.

Four main factors affecting the liquefaction of saturated sand were analyzed (dynamic load conditions, soil conditions, burial conditions and static conditions). Considering the above factors, combined with the experimental conditions and research purposes of this article, the following eight influencing indicators were selected: (1) Peak ground motion acceleration (PGA) is used to characterize seismic intensity; (2) Seismic types (Category II, III and IV seismic waves) to characterize the waveform; (3) Depth from the surface (buried depth) to represent the overlying effective pressure; (4) Actual acceleration (refers to the peak acceleration collected by each acceleration sensor); (5) Holding time (refers to the total duration of the earthquake); (6) Earth pressure (refers to the vertical stress component); whether the drainage characteristics and (7) Dynamic shear stress ratio is characterized by the following: (8) pore pressure as the influence characteristics of tailings liquefaction flow threshold under earthquake are suitable as the influence characteristics of liquefaction occurrence.

For these eight factors, the original data results of tailings sand liquefaction are shown in

Table 4. Raw data of tailings liquefaction.

	PGA (g)	Type of Earthquake	Buried Depth (cm)	Peak Acceleration (g)	Holding Time (s)	Peak Pore Pressure (MPa)	Peak Earth Pressure (MPa)	Dynamic Shear Stress Ratio	Liquefaction State
1	0.1	Traft	−75	0.12	30.9	0.0001	——	——	×
2	0.1	Traft	−60	0.224	31.52	0.0008	1.2932	0.6578	○
3	0.1	Traft	−45	0.141	29.38	0.0012	0.8931	0.8068	○
4	0.1	Traft	−30	0.139	35.02	0.0013	1.0849	0.84091	○
5	0.1	Traft	−15	0.149	38.46	0.0011	1.0104	0.8117	×
6	0.1	El Centro	−75	0.199	24.12	0.0002	——	——	○
7	0.1	El Centro	−60	0.143	33.26	0.0007	1.8774	0.9701	○
8	0.1	El Centro	−45	0.261	30.46	0.0005	1.4663	1.3297	×
9	0.1	El Centro	−30	0.147	32.22	0.0008	1.7283	1.5168	×
10	0.1	El Centro	−15	0.201	39.66	0.001	2.139	1.5355	×
11	0.1	Shanghai wave	−75	0.138	16.82	0.0001	——	——	×
12	0.1	Shanghai wave	−60	0.134	18.56	0.0008	1.3646	1.4841	○
13	0.1	Shanghai wave	−45	0.135	16.88	0.001	2.4964	1.7746	○
14	0.1	Shanghai wave	−30	0.12	22.92	0.0014	1.9191	1.7892	○
15	0.1	Shanghai wave	−15	0.128	19.84	0.001	1.018	1.9084	×
16	0.2	Traft	−75	0.226	38.82	0.0002	——	——	○
17	0.2	Traft	−60	0.218	31.82	0.001	1.2664	1.3091	○
18	0.2	Traft	−45	0.222	27.48	0.0013	1.026	1.6283	○
19	0.2	Traft	−30	0.209	36.38	0.0015	1.3019	1.5671	○
20	0.2	Traft	−15	0.278	41.68	0.0018	2.0589	1.7761	○
21	0.2	El Centro	−75	0.255	23.84	0.0002	——	——	○
22	0.2	El Centro	−60	0.215	25.48	0.0006	1.3066	1.0313	○
23	0.2	El Centro	−45	0.269	26.04	0.0008	2.1932	1.53	×
24	0.2	El Centro	−30	0.234	32.3	0.0009	2.0239	1.5746	×
25	0.2	El Centro	−15	0.225	27.34	0.0011	1.6395	1.627	×
26	0.2	Shanghai wave	−75	0.214	19.04	0.0001	——	——	×
27	0.2	Shanghai wave	−60	0.322	22.44	0.0007	1.1371	1.5094	○
28	0.2	Shanghai wave	−45	0.271	21.58	0.0012	2.8246	1.9063	○
29	0.2	Shanghai wave	−30	0.25	32.3	0.0018	2.2371	1.843	○
30	0.2	Shanghai wave	−15	0.322	27.34	0.001	0.8388	1.7258	×
31	0.3	Traft	−75	0.316	29.64	0.0005	——	——	×
32	0.3	Traft	−60	0.303	29.96	0.0009	2.8406	1.7432	○
33	0.3	Traft	−45	0.35	29.64	0.0014	1.7487	2.2737	○
34	0.3	Traft	−30	0.289	30.48	0.0017	2.0538	2.2511	○
35	0.3	Traft	−15	0.33	31.2	0.0018	3.6069	2.2024	○
36	0.3	El Centro	−75	0.466	23.7	0.0002	——	——	○
37	0.3	El Centro	−60	0.493	24.7	0.0006	0.5975	2.8199	○
38	0.3	El Centro	−45	0.349	25.16	0.001	1.1091	3.2321	○
39	0.3	El Centro	−30	0.313	28.62	0.0016	1.0587	3.6079	○
40	0.3	El Centro	−15	0.416	28.46	0.0009	1.323	3.9402	×
41	0.3	Shanghai wave	−75	0.451	17.4	0.0001	——	——	×
42	0.3	Shanghai wave	−60	0.378	20.58	0.0005	1.1371	2.4138	×
43	0.3	Shanghai wave	−45	0.359	21.94	0.001	2.8246	2.779	○
44	0.3	Shanghai wave	−30	0.318	26.22	0.0013	2.2371	2.8106	○
45	0.3	Shanghai wave	−15	0.37	26.86	0.0009	0.8388	2.8	×

Notes: ○: liquefied; ×: unliquefied.

, and the degree of correlation between the factors is shown in

Table 5. Correlation matrix.

	Influencing Factor	PGA	Type of Earthquake	Buried Depth	Peak Acceleration	Holding Time	Pore Pressure	Earth Pressure	Dynamic Shear Stress Ratio
Correlation	PGA	1.000	——	——	——	——	——	——	——
	Type of earthquake	0.000	1.000	——	——	——	——	——	——
	Buried depth	0.000	0.000	1.000	——	——	——	——	——
	Peak acceleration	0.873	−0.083	−0.055	1.000	——	——	——	——
	Holding time	−0.113	0.719	0.388	−0.156	1.000	——	——	——
	Pore pressure	0.139	0.168	0.760	−0.045	0.394	1.000	——	——
	Earth pressure	0.153	−0.054	0.073	−0.005	0.021	0.337	1.000	——
	Dynamic shear stress ratio	0.773	−0.270	0.222	0.727	−0.301	0.163	0.041	1.000
Significant	PGA	——	0.500	0.500	0.000	0.201	0.135	0.187	0.000
	Type of earthquake	0.500	——	0.500	0.338	0.000	0.091	0.377	0.056
	Buried depth	0.500	0.500	——	0.385	0.011	0.001	0.337	0.097
	Peak acceleration	0.000	0.338	0.385	——	0.217	0.349	0.488	0.000
	Holding time	0.201	0.000	0.011	0.217	——	0.015	0.452	0.037
	Pore pressure	0.135	0.091	0.001	0.349	0.015	——	0.021	0.182
	Earth pressure	0.187	0.377	0.337	0.488	0.452	0.021	——	0.407
	Dynamic shear stress ratio	0.000	0.056	0.097	0.000	0.037	0.182	0.407	——

. Among them, when the pore pressure ratio (the ratio of pore pressure to the initial earth pressure at the location of the pore pressure sensor) is greater than or equal to 1, it is considered to be in a liquefied state, and when the pore pressure ratio is less than 1, it is considered to be not liquefied. At the same time, the liquefaction of tailings sand is determined by 8 types of impact features.

The KMO test and Bartlett’s sphericity test were performed on the above variables. The results are shown in

Table 6. KMO and Bartlett test.

KMO Sampling Suitability	Bartlett Sphericity Test
0.523	Approximate chi square	Degree of freedom	Significant
	155.182	28	0.000

.

The KMO value is 0.537 > 0.5, which is barely suitable for factor analysis. According to Bartlett’s sphericity test, the chi-square value is around 155.182, the corresponding probability is roughly 0.000, and the significance is less than 0.05. Bartlett’s sphericity test’s null hypothesis is therefore rejected, and as it is thought to be appropriate for factor analysis. The following statistical analysis is carried out.

As shown in Figure 4, in general, the change in the feature value (refers to the influencing factors) is significant; that is, when the number of common factors increases to 8 feature values, it still shows a downward trend, which indicates that the eight influencing factors are important for factor analysis. Within the allowable range of error, the rate at which the feature values fall is slowed after six common factors, which indicates that the liquefaction possibility can still be described to some extent when analyzed with at least six influencing factors. This will be detailed and analyzed later.

5. Tailwater Sand Liquefaction Threshold Model Based on PSO-SVM

5.1. Normalization of the Influencing Factors

This paper focuses on the PSO-SVM algorithm to predict whether liquefaction will occur. Considering the difference in the dimensions of each influencing factor, the corresponding data magnitudes are also very different. Therefore, in the prediction of tailings liquefaction under earthquake action, each factor must be normalized to ensure the accuracy of the prediction. This paper normalizes using the following formula, where 0 means not liquefied and 1 means liquefaction:

$$x^{\ast }=\frac{x_i-x_{\min }}{x_{\max }-x_{\min }}$$

(3)

where: $x^{\ast }$ is the value of the normalized factor; $x_i$ is the original value of the factor; $x_{\min }$ is the minimum value of the factor; $x_{\max }$ is the maximum value of the factor.

5.2. Construction of the Tailings Sand Liquefaction Threshold Model

The threshold model established in this paper mainly uses the data of nine working conditions preset in the experiment. That is, the input conditions of various working conditions and the data collected from each position sensor are set. This paper selects 22 data collection points, five different depths and nine working conditions, corresponding to a total of 45 kinds of results as samples. Take the normalized factor value as the network input and take the occurrence of liquefaction as the network output (0 means no liquefaction, 1 means liquefaction).

The initial parameters of the PSO model are shown in

Table 7. Parameters used in the PSO model.

Initial Parameter	Sizepop	Maxgen	Learning Factor	Learning Factor	Inertia Weight	Penalty Factor C Optimization Range	Kernel Parameter g Optimization Range
Value	20	200	1.5	1.7	0.5	[0, 200]	[0, 1]

.

The PSO-SVM model constructed in this paper is first constructed by the RBF kernel. The two core parameters in the PSO are the penalty factor C and the kernel parameter g.

The entire liquefaction threshold model is established as follows:

After the test data are imported into the program according to the implementation of the preset, the classification and all data normalization are automatically performed first. The data classification (liquefaction classification and influencing factor attribute classification) is shown in Figure 5. Then, the PSO will perform a global search, and the optimal parameters with the highest classification accuracy will be obtained. However, this process requires constant iterations and automatic searches until a satisfactory result position is reached. Table 7 lists the parameters that were utilized in the PSO optimization process. The group size (Sizepop), the number of iterations (Maxgen), and the inertia weight were all set to 20. When the optimal parameters are found, a fitness curve for finding the best parameters is generated. If the best fitness is below the average fitness, the optimization fails; otherwise, the optimization succeeds. The maximum location of the optimal fitness determines how accurate an optimization is. The best parameters (penalty factor and kernel parameter), which are determined once the particle swarm optimization process is finished, are then automatically given to the support vector machine software for the appropriate training. After the completion of the training, the liquefaction condition is automatically predicted, thereby obtaining the actual situation. The test set and the corresponding prediction set are predicted according to the comparison between the prediction set and the test set.

In order to fully verify the proposed optimized PSO-SVM model, in this paper, the data collected and sorted by all monitoring points under the nine working conditions in the test are first tested, with a total of 45 monitoring values. Figure 5 shows the liquefaction of 45 monitored values and the attribute values classified by eight influencing factors. The global penalty of the factor C and the kernel parameter g of the SVM model are derived by the population particles using the MATLAB software of the PSO-SVM. Under the experimental conditions, the SVM model’s optimal penalty factor for seismic liquefaction is 12.7695. The kernel parameter is 0.2916, and the cross-validation rate is 92%, as shown in Figure 6. In order to produce the final classifier in this outcome, the SVM model is trained using the best parameters. Only 3 of the 45 test datasets were incorrectly identified, yielding a classification accuracy percentage of 92.7% overall, as shown in Figure 7. The PSO-SVM model that was created may foresee tailings liquefaction.

5.3. Analysis of the PSO-SVM Tailings Liquefaction Threshold Model

The current genetic algorithm support vector machine (GA-SVM) and grid search technique (GS) optimized support vector machine (GS-SVM) algorithms are compared, and calculations are made to demonstrate the superiority of the PSO-SVM methodology.

The genetic algorithm (GA), as an algorithm for survival of the fittest, natural selection and random genetic inheritance in the rules of source nature, has great advantages over traditional algorithms that cannot solve complex and other common nonlinear problems. At present, the GA has been widely used in various fields such as combinatorial optimization and has achieved some excellent results. Unlike traditional algorithms, this algorithm can search using randomly selected solutions without specifying an initial solution. It is similar to biological gene sequences and iteratively generates new solutions through certain selection, crossover and mutation operations. The use of chromosomes means that, like natural selection, there is a specific solution to each individual problem, and its quality is evaluated by fitness. The best ones are continuously selected to obtain excellent individuals. After that, the same process is used to breed the next generation, until after many generations of continuous updates, the results finally converge to obtain the optimal solution, and sometimes a suboptimal one.

As shown in Figure 8 and Figure 9, the best (C, g) obtained by the GA-SVM algorithm is (9.778, 0.3343), and the cross-validation rate is 87.8%. In order to get the final classification findings in this result, the SVM model was trained using the optimum parameters. Five samples were incorrectly identified out of the 45 test datasets employed, and the total classification accuracy percentage was 88.9%.

The model selection problem is solved by the GS method by doing an exhaustive search in the provided data space. To discover the ideal area of the SVM parameter, the parameter is initially searched in the coarse range because the SVM parameter’s boundary is known in advance. The fine mesh search is carried out after the coarse mesh search, but in the opposite order. The fact that the time is so lengthy is a drawback.

Table 8. Parameter grid range in the SVM model.

SVM Parameters	Coarse Grid Range	Fine Mesh Range
C	(2−5, 25)	(2−2, 24)
g	(2−10, 210)	(2−4, 24)

lists the parameter ranges for the coarse mesh and fine mesh searches. The SVM will choose the parameter with the best classification accuracy following the grid search.

As shown in Figure 10, the best (C, g) obtained by the coarse grid search using the GS-PSO algorithm is (8, 0.5), and the cross-validation rate is 85.4%. Figure 11 shows that the best (C, g) obtained by the fine mesh search using the GS-PSO algorithm is (21.1121, 0.1436), and the cross-validation rate is 85.4%. In order to attain the ultimate liquefaction classification goal, the SVM model was trained using the ideal parameters in this result. In the 45 test datasets used, 4 samples were classified incorrectly, achieving an overall classification accuracy of 91.1%, as shown in Figure 12.

To further study the accuracy of the PSO-SVM and GA-SVM and GS-SVM algorithms for predicting the seismic liquefaction threshold model, in addition to the test data used above, the working conditions of the seismic station model test are added, which enriches the model database.

The newly used database combines the data used in the previous section with the addition of nine verification tests, a total of 18 trials with a total of 90 feedback data. SC-1 represents all the datasets in the second test (verification test), a total of 45; SP-1 represents part of the dataset in the second test, that is, a total of 36 after removing some missing data points; SP-2 indicates the test data points obtained by removing the factors that are relatively influential between each other (the peak acceleration of one test point is removed in this test) according to the factor reduction analysis and correlation statistics between the influencing factors.

Similarly, AC-1 represents a total of 90 points obtained after the combination of two experimental data (the first test data are used as the test set and the second test data are used as the prediction set); AC-2 indicates that 90 points have been dimension-reduced, that is, to remove the seven influencing factors and 90 points obtained by the relatively large correlation factors; AP-1 indicates the test points with 74 points after removing the missing data; AP-2 indicates that the AP-2 is based on AP-2. Data points after the dimension (seven influencing factors): AP-3 means not distinguishing the two experimental sequences with 90 points obtained by combination arrangement; AP-4 means seven influencing factors obtained after dimension reduction on the basis of the AP-3 data points.

As shown in Figure 13, the outcomes are contrasted using the outcomes of the PSO-SVM, GA-SVM, and GS-SVM. It can be seen that the success rate predicted by the PSO-SVM algorithm is basically higher than that of the other two algorithms. In the least ideal case, the PSO-SVM method is almost similar to the prediction rate of the other two methods. As a result, choosing features and setting parameters in the PSO-SVM are quite important. To compute the classification accuracy rate and assess the built model, this research only uses the results database of the vibrating table indoor test, owing to space constraints. If possible, they should be applied in many ways to fully verify their innovative meaning. Therefore, in future research work, the model developed still has room for improvement.

6. Conclusions

The primary elements impacting liquefaction were taken into consideration while using the prototype tailings sand as the material for this article. Shaking table tests of tailings liquefaction were carried out with a self-designed rigid test-box. The following conclusions were drawn:

(1)

By performing a large number of factor analyses on the selected factors, the influencing factors suitable for the prediction model were obtained, namely, PGA, type of earthquake, buried depth, peak acceleration, holding time, peak pore pressure, peak earth pressure, and dynamic shear stress ratio.

(2)

The support vector machine (SVM) was utilized to forecast the liquefaction scenario after the parameters were optimized using the particle swarm optimization method (PSO). The liquefaction threshold model’s forecast accuracy has increased, and prediction speed has increased.

(3)

In comparison to the current genetic algorithm (GA) and grid node search technique (GS), the revised PSO-SVM algorithm was shown to be not only quickly calculated but also to have a high degree of accuracy, reaching 92.7%, and to have strong application.