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Fractional Differential Equations in the Exploration of Geological and Mineral Construction

Publicado en línea: 15 Jul 2022
Volumen & Edición: AHEAD OF PRINT
Páginas: -
Recibido: 10 Feb 2022
Aceptado: 14 Apr 2022
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Revista
eISSN
2444-8656
Primera edición
01 Jan 2016
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2 veces al año
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Inglés
Introduction

Subgrade settlement prediction method based on measured data is a hot research topic in traffic and civil engineering. It is also a major problem in theoretical research on settlement control calculation methods for soft foundations. This is because it is difficult to fit the complex form of the actual consolidation law of weak foundations with general mathematical curves [1]. The engineering community has conducted extensive research on settlement prediction methods based on measured data. The representative ones are a hyperbolic method, three-point method, exponential function method, Asaoka method, grey model prediction method, etc. The purpose is to better predict the development law of roadbed settlement through actual measured data. By comparing the prediction results of the above method with the actual settlement data, it is found that the calculation results of the prediction method still have a large error with the actual value. Therefore, the traditional forecasting model and calculation method need to be further improved.

Some scholars have improved the traditional hyperbolic prediction model and proposed a modified hyperbolic model with higher accuracy. Some scholars have studied the time interval value in the Asaoka method based on the illustrative interpolation method and the linear least square method [2]. This method has been successfully applied to the settlement prediction of multi-level load embankment engineering. Some scholars apply the adaptive neuro-fuzzy inference system (ANFIS) to the settlement prediction of weak foundations and propose the ANFIS weak foundation settlement prediction model. This model overcomes the shortcomings of local minima, and its prediction accuracy is higher than that of the traditional growth curve prediction model. Some scholars pointed out that the morbidity of the least-squares method will lead to failure in solving the settlement prediction model parameters. We can improve the prediction accuracy by using regularized unbiased estimation of the processing parameters. Some scholars established the Logistic model of the whole soft foundation settlement prediction process based on the growth curve and determined the model parameters and their influence on the foundation settlement in different stages [3]. Some scholars have improved the calculation of the single prediction model settlement and the error description method of the measured value and fitted a new target error function. In this way, a simple and practical method for predicting settlement after construction is obtained. The settlement law under complicated working conditions is not obvious, so to predict settlement more accurately, many scholars use gray theory to predict foundation settlement. Some scholars have extended the grey forecasting model based on univariate to the grey forecasting model of multivariate. Some scholars used the parameter accumulation method instead of the least square method to modify the traditional GM (1, 1) land subsidence model and built a gray subsidence prediction model with parameter estimation. In addition, some scholars also proposed related calculation models for the gray theory to predict settlement and gave an error analysis [4]. Although the settlement mentioned above calculation model based on gray theory has a certain application value, the differential equations solved are all integer orders. Some scholars set yxa y_x^a to mean that the function y finds the derivative of order a concerning the independent variable x. When a is a fraction (such as a = 0.3), it is a fractional differential. Some scholars have used the fractional-order model to model the thermal change system. Some scholars use fractional representation for the Kelvin-Voigt model to identify its parameters. Some scholars have carried out a fractional simulation of the particle motion equation. Some scholars have used the fractional-order model to analyze the chaotic system [5]. The analysis results all show that the calculation effect of the fractional-order model is better than that of the integer-order model. Although the calculation of the fractional model is more complicated, there are three definitions of fractional calculus. The content includes Caputo differential definition, Grunwald Letnikov differential definition, and Riemann-Lionville calculus definition. Fractional differentials are usually expressed approximately by high-order integer-order differentials [6]. The purpose is to reduce the order of high-order differential equations to achieve model calculations.

This paper takes the gray prediction model as the research object, replaces the first-order differential with the fractional differential, and proposes a gray prediction model based on the fractional order. Compared with the traditional grey prediction model, we increase the identification of fractional-order to increase the difficulty of parameter identification. For this reason, the author established a gray model with fractional order based on the fractional-order system model [7]. Engineering examples verify the rationality of the model, and the error of the fractional gray prediction model is calculated.

Fractional system model

There are currently three definitions of fractional order. This article uses the Caputo calculus definition. The a order derivative of a function y(t) can be expressed as: Day(t)=day(t)dta=1Γ(ra)0ty(r)(τ)(tτ)(a+1r)dτr1<α<r \matrix{{{D^a}y\left(t \right) = {{{d^a}y\left(t \right)} \over {d{t^a}}} = {1 \over {\Gamma \left({r - a} \right)}}\int_0^t {{{{y^{\left(r \right)}}\left(\tau \right)} \over {{{\left({t - \tau} \right)}^{\left({a + 1 - r} \right)}}}}d\tau}} \hfill \cr {r - 1 < \alpha < r} \hfill \cr}

In the formula: r represents a positive integer. α represents a score. Γ ( ) represents the Gamma function. Γ ( ) is Γ(β)=0τβ1eτdτ \Gamma \left(\beta \right) = \int_0^\infty {{\tau^{\beta - 1}}{e^{- \tau}}d\tau}

The Laplace transform can be expressed as: L[Day(t)]=saY(s)i=0r1sa1iy(i)(0) L\left[ {{D^a}y\left(t \right)} \right] = {s^a}Y\left(s \right) - \sum\limits_{i = 0}^{r - 1} {{s^{a - 1 - i}}{y^{\left(i \right)}}\left(0 \right)}

The zero initial condition considered in this paper, the Laplace transform is: L[Day(t)]=saY(s) L\left[ {{D^a}y\left(t \right)} \right] = {s^a}Y\left(s \right)

The calculation of sa is usually approximated by integer-order integral. In this paper, a phase pre-filter is used to simulate the output result of sa. Assuming that the fitted frequency band is (ωb, ωh), the transfer function of the following filter can be constructed as: saGasi=1N1+s/ωie1+s/ωi {s^a} \approx {{{G^a}} \over s}\prod\limits_{i = 1}^N {{{1 + s/\omega _i^e} \over {1 + s/{\omega _i}}}}

In the formula: 2N+1 represents the selected filter order. Ga=ωb1α {G^a} = \omega _b^{1 - \alpha} , ωi=δωie {\omega _i} = \delta \omega _i^e , ωi+1e=ηωi \omega _{i + 1}^e = \eta {\omega _i} . a=1logδlogδη a = 1 - {{\log \,\delta} \over {\log \,\delta \eta}} . equation (5) can also be expressed in the form of an equation of state: MIzI(t)=AIzI(t)+BIu(t)ω=Cizi(t) \matrix{{{M_I}{z_I}\left(t \right) = {A_I}{z_I}\left(t \right) + {B_I}u\left(t \right)} \hfill \cr {\omega = {C_i}{z_i}\left(t \right)} \hfill \cr}

In the formula zI(t) represents the N+1 dimensional state vector. MI=[10L0δ1MMMM00Lδ1],BI[Gα0M0]AI=[00L0ω1ω1MMOO00LωNωN],zI(t)=[z1(t)z2(t)Mzn+1(t)]CI=[0L01] \matrix{{{M_I} = \left[ {\matrix{1 & 0 & {\rm{L}} & 0 \cr {- \delta} & 1 & {} & {\rm{M}} \cr {\rm{M}} & {\rm{M}} & {\rm{M}} & 0 \cr 0 & {\rm{L}} & {- \delta} & 1 \cr}} \right],\,{B_I}\left[ {\matrix{{{G^\alpha}} \cr 0 \cr {\rm{M}} \cr 0 \cr}} \right]} \hfill \cr {{A_I} = \left[ {\matrix{0 & 0 & {\rm{L}} & 0 \cr {{\omega _1}} & {- {\omega _1}} & {} & {\rm{M}} \cr {\rm{M}} & {\rm{O}} & {\rm{O}} & 0 \cr 0 & {\rm{L}} & {{\omega _N}} & {- {\omega _N}} \cr}} \right],\,\,{z_I}\left(t \right) = \left[ {\matrix{{{z_1}\left(t \right)} \cr {{z_2}\left(t \right)} \cr {\rm{M}} \cr {{z_{n + 1}}\left(t \right)} \cr}} \right]} \hfill \cr {{C_I} = \left[ {0{\rm{L}}\,01} \right]} \hfill \cr}

Equation (6) can be converted to: zI(t)=AzI(t)+Bu(t)ω=CzI(t) \matrix{{{z_I}\left(t \right) = A{z_I}\left(t \right) + Bu\left(t \right)} \hfill \cr {\omega = C{z_I}\left(t \right)} \hfill \cr}

Where: A=MI1AI A = M_I^{- 1}\,{A_I} , B=MI1BI B = M_I^{- 1}\,{B_I} , C = CI

Grey model with fractional order

Suppose the original time series is: y(0) = [y(0) (1) y(0) (2) … y(0) (n)]. Its one-time cumulative generation sequence is y = [y(1) y(2) … y(n)]. Among them, y(k)=i=1ky(0)(i) y\left(k \right) = \sum\limits_{i = 1}^k {{y^{\left(0 \right)}}\left(i \right)} . The gray model can be expressed as: dy(t)dt+b1y(t)=b2 {{dy\left(t \right)} \over {dt}} + {b_1}y\left(t \right) = {b_2}

The undetermined constant b1, b2 in equation (8) is determined by the following equation: [b1b2]T=(E1TE1)1E1EE2 {\left[ {{b_1}\,\,{b_2}} \right]^T} = {\left({E_1^T\,{E_1}} \right)^{- 1}}\,E_1^E\,{E_2}

In the formula: E1=[0.5(y(1)+y(2))10.5(y(2)+y(3))10.5(y(n1)+y(n))1] {E_1} = \left[ {\matrix{{- 0.5\left({y\left(1 \right) + y\left(2 \right)} \right)} & 1 \cr {- 0.5\left({y\left(2 \right) + y\left(3 \right)} \right)} & 1 \cr \vdots & \vdots \cr {- 0.5\left({y\left({n - 1} \right) + y\left(n \right)} \right)} & 1 \cr}} \right] , E2[y(0)(2)y(0)(3)y(0)(n)] {E_2}\left[ {\matrix{{{y^{\left(0 \right)}}\,\left(2 \right)} \cr {{y^{\left(0 \right)}}\,\left(3 \right)} \cr \vdots \cr {{y^{\left(0 \right)}}\left(n \right)} \cr}} \right] . From formula (8), it can be seen that the gray model is an ordinary differential equation without input. Here are two changes to it. One is to increase the input term u. The other is to change the ordinary differential equation to a fraction Order differential equation: day(t)dta+b1y(t)=b2u(t) {{{d^a}y\left(t \right)} \over {d{t^a}}} + {b_1}y\left(t \right) = {b_2}u\left(t \right)

The input term u here can use the unit step function as follows. The purpose is to maintain the characteristics of the original gray model. u(t)={1t00t0 u\left(t \right) = \left\{{\matrix{1 & {t \ge 0} \cr 0 & {t \le 0} \cr}} \right.

We perform Laplace transformation on both sides of equation (10) to obtain the transfer function of the system as: G(s)=Y(s)U(s)=b2sa+b1 G\left(s \right) = {{Y\left(s \right)} \over {U\left(s \right)}} = {{{b_2}} \over {{s^a} + {b_1}}}

If the frequency domain of z(t) is defined as: Z(s)=1sa+b1U(s) Z\left(s \right) = {1 \over {{s^a} + {b_1}}}U\left(s \right)

Then the output variable y(t) can be expressed as: Daz(t)=b1z(t)+u(t)y(t)=b1z(t) \matrix{{{D^a}z\left(t \right) = - {b_1}z\left(t \right) + u\left(t \right)} \hfill \cr {y\left(t \right) = {b_1}z\left(t \right)} \hfill \cr}

The above formula can be equivalently written as: zI(t)=Ga[b1zN+1(t)+u(t)]y(t)=b2zN+1(t) \matrix{{{z_I}\left(t \right) = {G^a}\left[ {- {b_1}{z_{N + 1}}\left(t \right) + u\left(t \right)} \right]} \hfill \cr {y\left(t \right) = {b_2}{z_{N + 1}}\left(t \right)} \hfill \cr} which is: zI(t)=AzI(t)+Bu(t)y=CzI(t) \matrix{{{z_I}\left(t \right) = \,A{z_I}\left(t \right) + Bu\left(t \right)} \hfill \cr {y = C{z_I}\left(t \right)} \hfill \cr}

Where A=Ab1BCB=BC=b2C \eqalign{& A = A - {b_1}BC \cr & \matrix{{B = B} \hfill \cr {C = {b_2}C} \hfill \cr} \cr}

One is that the model recognition with fractional order requires input from the system. Enter according to formula (11) to get the correct result. Second, compared with the conventional gray model identification method, the improved gray model only adds a parameter α (or intermediate parameters δ and η). Still, the appearance of the parameter α greatly increases the difficulty of identification. Only after obtaining formula (16) can the recognition become feasible.

Application of prediction model in settlement prediction

Let's take the K1049+678 section of a certain expressway as an example. This section of the foundation is rich in soft soil and has high compressibility. The water in the soil is drained by driving a plastic drainage board on the foundation, and three points A, B, C on the section are selected for settlement monitoring (Figure 1). The project started in March 2019, and the frequency of settlement monitoring is 30d/time [8]. The project ends on June 22, 2020. We use the traditional hyperbolic settlement prediction model and the fractional gray theory prediction model proposed in this paper to calculate the settlement data of this section. See Table 1 for the original data of settlement observation points.

Figure 1

Schematic diagram of measuring point layout

Raw data of settlement observation.

Monitoring time Cumulative settlement at point A/mm Cumulative settlement at point B/mm Cumulative settlement at point C/mm
2019 3 131 113 138
4 251 218 251
5 347 311 358
6 450 400 464
7 530 484 564
8 606 563 655
9 699 629 752
10 784 698 840
11 849 759 935
12 909 822 1016
2020 1 969 876 1091
2 1024 924 1157
3 1077 965 1214
4 1122 1007 1267
5 1162 1044 1316
6 1202 1079 1357

We use the data from March to October 2019 as the original data for modeling and forecasting. The article predicts the settlement from November 2019 to June 2020 and compares it with the measured value [9]. We use the model proposed in this paper to select the fitting frequency range when using the fractional differential filter approximation is (0.0011000) rad/s, N=4. That is, the order of the filter is 9. The parameter estimates are α = 1.14; b1 = 0.135; b2 = 1.200. s1.14 respectively. The expression of is s1.14=V(s)W(s) {s^{1.14}} = {{V\left(s \right)} \over {W\left(s \right)}} . V(s)=2.63s10+1398s9+1.316×105s8+2.573×106s7+1.075×107s6+9.652×106s5+1.864×106s4+7.692104s3+658.7s2+sW(s)=s9+658.7s8+7.692×104s7+1.864×106s6+9.652×106s5+1.075×107s4+2.573×106s3+7316×105s2+1398s+2.63 \matrix{{V\left(s \right) = 2.63{s^{10}} + 1398{s^9} + 1.316 \times {{10}^5}{s^8} + 2.573 \times {{10}^6}{s^7} + 1.075 \times {{10}^7}{s^6} + 9.652 \times {{10}^6}{s^5}} \hfill \cr {+ 1.864 \times {{10}^6}{s^4} + 7.692{\rm{1}}{{\rm{0}}^4}{s^3} + 658.7{s^2} + s} \hfill \cr {W\left(s \right) = {s^9} + 658.7{s^8} + 7.692 \times {{10}^4}{s^7} + 1.864 \times {{10}^6}{s^6} + 9.652 \times {{10}^6}{s^5} + 1.075 \times {{10}^7}{s^4}} \hfill \cr {+ 2.573 \times {{10}^6}{s^3} + 7316 \times {{10}^5}{s^2} + 1398s + 2.63} \hfill \cr}

The definition formula for error analysis through the second-order norm is as follows: yy˜2 {\left\| {y - \tilde y} \right\|_2} y represents the actual measurement accumulated data. y˜ \tilde y represents the estimated value of y. The calculation expression of the traditional hyperbolic settlement prediction model is as follows: St=S0+tt0α+β(tt0) {S_t} = {S_0} + {{t - {t_0}} \over {\alpha + \beta \left({t - {t_0}} \right)}}

In the formula: S0 represents the cumulative settlement at time t0. t0 represents time zero. St represents the accumulated settlement at time t. α, β stands for undetermined parameters, which are the intercept and slope in the tt0StS0~t {{t - {t_0}} \over {{S_t} - {S_0}}}\sim t relationship graph. According to the graphical method:

Observation point A: α = 0.7779, β = 0.0553

Observation point B: α = 0.9071, β = 0.0426

Observation point C: α = 0.8057, β = 0.0273

Substituting, α, β into equation (18) can get the traditional hyperbolic settlement prediction model. When we adopt the grey prediction model, we can estimate its parameters by formula (9). They are:

Observation point A: b1 = 0.0729; b2 = 127.0539

Observation point B: b1 = 0.0759; b2 = 116.4983

Observation point C: b1 = 0.0591; b2 = 131.0533

Substituting the calculated parameters into equation (8) can calculate the integer-order prediction model.

The A, B, C errors calculated by the hyperbolic prediction model according to formula (17) are 282, 139, and 225, respectively [10]. The error of the integer-order model is 186, 202, 93. In this paper, the errors of the method with the fractional gray model are 116, 63, 53, respectively. It can be seen that the calculation results of the fractional-order grey prediction model proposed in this paper are better than the traditional hyperbolic and integer-order prediction models.

We summarize the actual measurement results and the calculation results of the fractional gray theory prediction model included in this paper, the traditional hyperbolic prediction model, and the integer-order in Figure 2. By comparison, it can be seen that the settlement calculation method with the fractional gray theory proposed in this paper can get better prediction results.

Figure 2

Comparison of measured results and predicted results

Conclusion

(1) The traditional gray model is an ordinary differential equation without input. This article makes two improvements to it. One is to increase the input item u. The second is that the ordinary differential equation is a fractional differential equation. (2) Calculate the errors of the fractional gray model, integer gray model, and hyperbolic model based on engineering examples. The error between the prediction results of the fractional-order gray theory and the actual engineering results established in this paper is higher than that of the integer-order gray theory settlement prediction model and hyperbolic prediction model.

Figure 1

Schematic diagram of measuring point layout
Schematic diagram of measuring point layout

Figure 2

Comparison of measured results and predicted results
Comparison of measured results and predicted results

Raw data of settlement observation.

Monitoring time Cumulative settlement at point A/mm Cumulative settlement at point B/mm Cumulative settlement at point C/mm
2019 3 131 113 138
4 251 218 251
5 347 311 358
6 450 400 464
7 530 484 564
8 606 563 655
9 699 629 752
10 784 698 840
11 849 759 935
12 909 822 1016
2020 1 969 876 1091
2 1024 924 1157
3 1077 965 1214
4 1122 1007 1267
5 1162 1044 1316
6 1202 1079 1357

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