geomechanical parameters identification by particle swarm optimization and support

Applied Mathematical Modelling 33 (2009) 3997–4012

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Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

Geomechanical parameters identification by particle swarmoptimization and support vector machine

Hong-bo Zhao a,*, Shunde Yin b

a School of Civil Engineering, Henan Polytechnic University, Jiaozuo 454003, People’s Republic of Chinab Department of Chemical and Petroleum Engineering, University of Wyoming, Laramie, WY 82071, USA

a r t i c l e i n f o

Article history:Received 7 January 2008Received in revised form 12 January 2009Accepted 22 January 2009Available online 3 February 2009

Keywords:Back analysisGeomechanical parameters identificationSupport vector machineParticle swarm optimization

0307-904X/$ - see front matter � 2009 Elsevier Incdoi:10.1016/j.apm.2009.01.011

* Corresponding author.E-mail address: [email protected] (H.-b. Z

a b s t r a c t

Back analysis is commonly used in identifying geomechanical parameters based on themonitored displacements. Conventional back analysis method is not capable of recognizingnon-linear relationship involving displacements and mechanical parameters effectively.The new intelligent displacement back analysis method proposed in this paper is the com-bination of support vector machine, particle swarm optimization, and numerical analysistechniques. The non-linear relationship is efficiently represented by support vectormachine. Numerical analysis is used to create training and testing samples for recognitionof SVMs. Then, a global optimum search on the obtained SVMs by particle swarm optimi-zation can lead to the geomechanical parameters identification effectively.

� 2009 Elsevier Inc. All rights reserved.

1. Introduction

Displacement back analysis is commonly used in establishing geomechanical parameters in rock mechanics and engi-neering [1–7]. There are mainly three types of displacement back analysis methods: inverse solving method, atlas methodand direct (i.e. optimal) method [6]. Because of the special advantages, the optimal methods are more and more extensivelyemployed in solving engineering problems [8–10]. For example, Levenber–Marquardt method, Gauss–Newton method,Bayesian method, Powell method, Rosenbork method, and genetic algorithm have been proposed to obtain optimal valuesof parameters from measured displacement data. Neural network and genetic algorithm were applied to geotechnical engi-neering back analysis [11–15], and later on neural network was replaced by support vector machine [16–18]. This has beena new way for displacement back analysis. However, there are still two problems associated with the optimal method unre-solved completely. One is that the relation between the displacement and the mechanical parameters is highly non-linearand complex, although support vector machine can provide appropriate techniques to learn and represent this non-linearrelation [19,20]. Another is that the search for the estimated parameter values is in a large space and is highly multi-modal.Some existing techniques such as calculus-based and enumerative techniques are mostly insufficient to handle thisproblem.

In this paper, the particle swarm optimization [21] was chosen for its biological and evolutionary appeal in finding the setof unknown parameters that best matches the modeling prediction with the measured displacement data. Then a new intel-ligent displacement back analysis method incorporating a support vector machine and a particle swarm optimization is pre-sented in detail. By using this method, the estimation of the geomechanical parameters at the permanent shiplock at theThree Gorges Project is illustrated.

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3998 H.-b. Zhao, S. Yin / Applied Mathematical Modelling 33 (2009) 3997–4012

2. Support vector machine and particle swarm optimization

2.1. Support vector machine

The SVM was proposed by Vapnik [19]. It is used to train non-linear relationships based on the structural risk minimiza-tion principle that seeks to minimize an upper bound of the generalization error rather than to minimize the empirical errorimplemented in neural networks. This induction principle is based on the fact that the generalization error is bounded by thesum of the empirical error and a confidence interval term that depends on the Vapnik–Chervonenkis (VC) dimension. Usingthis principle, the SVM will achieve an optimal model structure by striking the right balance between the empirical error andthe VC-confidence interval, eventually giving a better generalization than the neural networks. Another merit of the SVM isthat the training is a uniquely solvable quadratic optimization problem. The SVM uses non-linear mapping based on an inter-nal integral function to transform an input space to a high dimension space and then looks for a non-linear relationship be-tween inputs and outputs in that space. The SVM not only has theoretical support but also can find global optimum solutionsfor problems with small training samples, high dimensions, non-linear and local optima. A wide variety of applications suchas pattern recognition, non-linear regression, etc. have empirically shown the SVM’s ability to generalize.

Suppose that we are given a set of observation data (samples) (X1,y1), (X2,y2), . . . , (Xk,yk), Xi 2 Rn, yi 2 R. For the regressionproblem based on the SVM, our goal is to find a linear function f(X) = w � X + b that has at most e deviations from the actuallyobtained targets yi for all the training data, and is also as flat as possible. We describe the linear function f, in the form

f ðXÞ ¼ w � Xþ b: ð1Þ

One way to ensure this is to minimize the squared Euclidean norm, i.e. kwk2. Formally we can write this problem as aconvex optimization problem by requiring:

minimize12kwk2

;

subject toyi �w � Xi � b 6 e;w � Xi þ b� yi 6 e;

�i ¼ 1; . . . ;n:

ð2Þ

According to statistics learning theory, we want to obtain the optimal generalization. Allowing for some errors, we canintroduce the slack variables ni P 0 and n�i P 0. Hence we arrive at the following:

minimize12kwk2 þ C

Xk

i¼1

ðni þ n�i Þ;

subject toyi �w � Xi � b 6 eþ ni;

w � Xi þ b� yi 6 e;þn�i ;

�i ¼ 1; . . . ;n:

ð3Þ

The constant C > 0 determines the tradeoff between the flatness of f and the amount up to which deviations larger than eare tolerated. Using the optimal method, we can obtain the dual optimization problem

maximize Wða;a�Þ ¼ �12

Xn

i;j¼1

ðai � a�i Þðaj � a�j ÞðXi � XjÞ þXn

i¼1

yiðai � a�i Þ � eXn

i¼1

ðai þ a�i Þ; ð4Þ

subject toPni¼1ðai � a�i Þ ¼ 0;

0 6 ai;a�i 6 C;

8<: i ¼ 1;2; . . . ;n: ð5Þ

Solving the above optimal problems, we can obtain the regression function of the SVM

f ðXÞ ¼Xn

i¼1

ðai � a�i ÞðX � XiÞ þ b: ð6Þ

Based on the Karush–Kuhn–Tucker (KKT) conditions for quadratic programming, only a small number of coefficients (ai � a�i Þwill be assumed to have non-zero values, and their data points could be referred to as support vectors.

The non-linear problems can be solved by mapping the data into a high feature dimension space. In high feature dimen-sion space, the inner product can be replaced by the kernel function, i.e. K(Xi,Xj) = / (Xi)/(Xj); we do not need to know thespecific formulation of the non-linear mapping /. So Eqs. (3)–(5) can be changed into the following formulation:

maximize Wða;a�Þ ¼ �12

Xn

i;j¼1

ðai � a�i Þðaj � a�j ÞKðXi � XjÞ þXn

i¼1

yiðai � a�i Þ � eXn

i¼1

ðai þ a�i Þ; ð7Þ

subject toPni¼1ðai � a�i Þ ¼ 0;

0 6 ai;a�i 6 C;

8<: i ¼ 1;2; . . . ;n; ð8Þ

f ðXÞ ¼Xn

i¼1

ðai � a�i ÞKðX � XiÞ þ b: ð9Þ

H.-b. Zhao, S. Yin / Applied Mathematical Modelling 33 (2009) 3997–4012 3999

The following is three general kernel functions:

(1) Polynomial kernel

KðX;YÞ ¼ ððX � YÞ þ 1Þd; d ¼ 1;2; . . . ;n: ð10Þ

(2) Radial kernel

KðX;YÞ ¼ exp � jX � Y j2

r2

( ): ð11Þ

(3) Sigmoid kernel

KðX;YÞ ¼ tanhð/ðX � YÞ þ hÞ: ð12Þ

We can solve the quadratic optimal problem of Eqs. (4), (5), (7), and (8) by a variety of methods, including the interior pointalgorithm, the sequential minimal optimization [20], and decompression algorithms, etc.

2.2. Particle swarm optimization

The particle swarm optimization (PSO) was originally designed by Kennedy and Eberhart and has been compared to ge-netic algorithms for efficiently finding optimal or near-optimal solutions in large search spaces [21]. The technique involvessimulating social behavior among individuals (particles) ‘‘flying” through a multi-dimensional search space, each particlerepresenting a single intersection of all search dimensions. The particles evaluate their positions relative to a goal (fitness)at every iteration. Particles in a local neighborhood share memories of their ‘‘best” positions, and then use those memories toadjust their own velocities and thus subsequent positions. The original formula developed by Kennedy and Eberhart was im-proved by Shi and Eberhart with the introduction of an inertia parameter that increases the overall performance of PSO.

The original PSO formulae define each particle as a potential solution to a problem in D-dimensional space, with particle irepresented Xi = (xi1,xi2, . . . ,xiD). Each particle also maintains a memory of its previous best position, Pi = (pi1,pi2, . . .,piD), and avelocity along each dimension, represented as Vi = (vi1,vi2, . . . ,viD). At each iteration, the P vector of the particle with the bestfitness in the local neighborhood, designated g, and the P vector of the current particle are combined to adjust the velocityalong each dimension, and that velocity is then used to compute a new position for the particle. The portion of the adjust-ment to the velocity influenced by the individual’s previous best position (P) is considered the cognitioncomponent, and theportion influenced by the best in the neighborhood is the socialcomponent.

Regarding the minimum problem, suppose f(X) is the objection function, Xi = (xi1,xi2, . . . ,xin) is the current position of par-ticle, Vi = (vi1,vi2, . . . ,vin) is the current speed of particle, Pi = (pi1,pi2, . . . ,pin) is the best position which particle flied, then thebest position of particle i can be computed according following formulation:

Piðt þ 1Þ ¼PiðtÞ if f ðxiðt þ 1ÞP f ðPiðtÞÞÞ;Xiðt þ 1Þ if f ðxiðt þ 1Þ < f ðPiðtÞÞÞ:

�ð13Þ

If the population is s, and Pg(t) is the global best position which all particle flied the best position, then

PgðtÞ 2 fP0ðtÞ; P1ðtÞ; . . . ; PsðtÞgjf ðPgðtÞÞ ¼minff ðP0ðtÞÞ; f ðP1ðtÞÞ; . . . ; f ðPsðtÞÞg: ð14Þ

According to the theory of particle swarm optimization, the following equation presents the process of evolutionary:

v iðt þ 1Þ ¼ wv iðtÞ þ c1r1ðtÞðpijðtÞ � xiðtÞÞ þ c2r2ðtÞðpgðtÞ � xiðtÞÞ; ð15Þxijðt þ 1Þ ¼ xijðtÞ þ v ijðt þ 1Þ; ð16Þ

where vi is the velocity for particle i, which represents the distance to be traveled by this particle from its current position; xij

represents the position of particle i; pij represents the best previous position of particle i; pg represents the best positionamong all particles in the population; r1 and r2 are two independently uniformly distributed random variables with range[0,1]; c1 and c2 are positive constant parameters called acceleration coefficients which control the maximum step size; wis the inertia weight that controls the impact of previous velocity of particle on its current one. In standard PSO, Eq. (15)is used to calculate the new velocity according to its previous velocity and to the distance of its current position from bothits own best historical position and its neighbors best position. Generally, the value of each component in vi can be clampedto the range [�vmax,vmax] to control excessive roaming of particles outside the search space. Then the particle flies toward anew position according to Eq. (16). This process is repeated until a user-defined stopping criterion is reached.

3. An intelligent back analysis algorithm

In this section, an intelligent displacement back analysis algorithm based on the integration of particle swarm optimiza-tion (PSO), support vector machine (SVM), and numerical analysis is proposed. The algorithm is described as follows (seeFig. 1).

Generate randomly N particles (SVM parameters)

Calculate the fitness of each particle(Eq.18)

SVM model is OK? Produce N new

particles using

PSO algorithm

SVM learning

SVM model for back analysis

Calculate the fitness of each particle(Eq.19)

Error between monitored

and predicted by SVM is

Minimum?

Get the parameters to be recognized

Produce N new

particles (parameters to

be recognized)

Produce N new

particles using

PSO algorithm

Yes

Yes

No

No

Recognize the SVM model

Recognize rock mass parameter

Fig. 1. Intelligent displacement back analysis based on PSO and SVM.


3.1. Representation of non-linear relationship

The non-linear relationship between displacement and geomechanical parameters can be described using a support vec-tor machine SVM(X) as

SVMðXÞ : Rn ! R; ð17Þ

Y ¼ SVMðXÞ;X ¼ ðx1; x2; . . . ; xnÞ;Y ¼ ðy1; y2; . . . ; ynÞ;

where xi(i = 1,2, . . .,n) is geomechanical parameters, for example, Young’s modulus, friction angle, geo-stress coefficients, etc.yi(i = 1,2, . . . ,n) is displacements of the key points.

In order to obtain SVM(X), a training process based on the known data set is needed. The training of SVM includes cre-ation of training samples using numerical analysis and determination of training parameters of SVM. The former is per-formed by using numerical analysis for the given set of tentative geomechanical parameters to obtain the correspondingdisplacement of rock mass of key points. Considering influence of training parameters on generalization performance ofSVM, particle swarm optimization is adopted to search the training parameters in global space. The algorithm is describedas follows for this purpose:

Step 1: Estimate the valuing ranges of geomechanical parameters to be recognized. A set of tentative geomechanicalparameters is given in their valuing ranges. Numerical analysis is used to calculate the corresponding displacementof key points for the every set of tentative geomechanical parameters. Each set of geomechanical parameters withthe corresponding displacement of key points is considered as a training sample set. In order to obtain the best


generalization performance of SVMs both for training samples and new samples having similar conditions, anotherset of samples, therefore, should be created to test applicability of SVMs. They are called to be the testing sampleset.

Step 2: Initialize parameters of particle swarm optimization such as number of evolutionary generation, population size,inertia weight, acceleration coefficients, range of kernel function and its parameters including C and r.

Step 3: Select randomly a kernel function from common examples of kernel functions such as polynomials, Gaussian radialbase, and sigmoid. Produce randomly a set of C and r in the given valuing ranges. Each selected kernel function andits parameters such as C and r is regarded as an individual of SVM.

Step 4: Use sequential minimal optimization (SMO) algorithm to solve the quadratic programming problems includingeach individual to obtain values of Lagrange multipliers and their support vectors.

Step 5: Use the selected parameters and the obtained support vectors to represent a SVM model. The testing samples isused to test prediction ability of the SVMs. Applicability of the model is measured by fitness as

fitness ¼MaxðjYij � Y 0ijj=YijÞ; ð18Þ

where Yij and Y 0ij are the estimated displacement of tentative SVM and calculation of numerical analysis for keypoint i of rock mass at the jth testing sample.

Step 6: If fitness is accepted then the training procedure of SVM ends and the best SVMs are found. Otherwise, according toEqs. (15) and (16), produce the new particle.

Step 7: If all new individuals of population size are generated then go to Step 4. Otherwise, go to Step 6.

3.2. Objective function

To formulate the particle swarm optimization to this problem, as in any conventional approach to the displacement backanalysis, objective function must be defined. Here the objective function is defined as

fitness ¼ 1n

Xn

i¼1

SVMiðXÞ � yij jð Þ; ð19Þ

where n is the number of key points, yi is the monitored displacement of the ith key points, SVMi(X) is the predicted displace-ment of ith key point.

3.3. An intelligent displacement back analysis algorithm based on particle swarm optimization

If support vector machine model representing the non-linear relation between the displacement and a parameter is ob-tained, the model can be used to recognize parameters. Particle swarm optimization is used to search for the best parametersystem having the minimum error between the predicted displacements as predicted by the model and the actual measureddisplacement. This back analysis algorithm can be described as follows:

Step 1: Determine the particle swarm optimization parameters and the range of parameters to be recognized.Step 2: Generate randomly n group of parameters at their given range. Each individual represents an initial solution.Step 3: Input a set of rock mass parameters to the model SVM(X) obtained above to calculate the displacement values at

given monitoring points.Step 4: Use Eq. (19) to evaluate the fitness of the current individuals, i.e. the reasonability of the parameter set.Step 5: If all individuals are evaluated, then go to Step 6. Otherwise, go to Step 3.Step 6: If the given evolutionary generation is reached, or the best individuals (the parameter to be back recognized) are

obtained, then the evolutionary process ends. Otherwise, go to Step 7.Step 7: Update the individuals according to Eqs. (15) and (16).Step 8: Repeat Step 7 until all n new individuals are generated. They are used as offspring.Step 9: Go to Step 3.

4. Verification of the algorithm

To verify the model, we suppose there is an infinite large plate with a hole of radius 1 m in centre (Fig. 2). Suppose thereare theoretical values for Poisson’s ratio l = 0.25, equably distribution of initial geo-stress with rx = rz = 0.98 MPa, sxz = 0 andYoung’s modulus E = 98 MPa. Displacement values for some key points, indicated by nodes 1–5 in Fig. 2, are calculated byelastic finite element method. The suggested algorithm above is used to identify initial geo-stress components rx,rz,sxz.

Before the training of SVMs, totally 25 set of training samples and five testing samples are created using finite elementanalysis (Table 1). The established SVM gave its recognized initial geo-stress rx,rz,sxz of about 0.9787, 0.9762, 0 MPa, respec-tively, having maximum relative error 0.39% corresponding to the theoretical values (Table 2). The comparison with evolu-

Fig. 2. A quarter calculation model for FEM. The number is no. of key points to be compared.

Table 1Training and testing samples.

rx0 (Mpa) ry0 (Mpa) sxy (Mpa) Displacement (cm)

1 2 3 4 5

u1x u2x u2y u3x u3y u4x u4y u5y

Training samples1 �0.6 �0.58 �0.2 �0.45136 �0.41508 �0.17422 �0.31696 �0.32117 �0.17419 �0.41845 �0.450772 �0.6 �0.78 �0.4 �0.55592 �0.53144 �0.25745 �0.44528 �0.51236 �0.26666 �0.71794 �0.798003 �0.6 �0.98 0 �0.10498 �0.14153 �0.13103 �0.19138 �0.32242 �0.14968 �0.52279 �0.608424 �0.6 �1.18 0.2 0.16394 0.08339 �0.07435 �0.06289 �0.25972 �0.10227 �0.49379 �0.599535 �0.6 �1.38 0.4 0.43526 0.31123 �0.01762 0.06706 �0.19744 �0.05474 �0.46618 �0.592346 �0.8 �0.58 �0.4 �0.80309 �0.71490 �0.26542 �0.50479 �0.44891 �0.25595 �0.53430 �0.553637 �0.8 �0.78 0 �0.35680 �0.32912 �0.13868 �0.25213 �0.25661 �0.13882 �0.33510 �0.360728 �0.8 �0.98 0.2 �0.09064 �0.10639 �0.08172 �0.12404 �0.19242 �0.09122 �0.30398 �0.350379 �0.8 �1.18 0.4 0.17793 0.11925 �0.02473 0.00553 �0.12864 �0.04349 �0.27425 �0.3417410 �0.8 �1.38 �0.2 �0.29721 �0.34020 �0.24894 �0.38353 �0.57823 �0.27703 �0.90586 �1.0447011 �1 �0.58 0 �0.60731 �0.51657 �0.14672 �0.31386 �0.19180 �0.12839 �0.14738 �0.1118512 �1 �0.78 0.2 �0.34388 �0.29602 �0.08950 �0.18617 �0.12615 �0.08060 �0.11416 �0.1000613 �1 �0.98 0.4 �0.07804 �0.07257 �0.03224 �0.05700 �0.06087 �0.03268 �0.08232 �0.0899914 �1 �1.18 �0.2 �0.54849 �0.52643 �0.25648 �0.44313 �0.51266 �0.26589 �0.71962 �0.7988015 �1 �1.38 �0.4 �0.64953 �0.63941 �0.33821 �0.56879 �0.70056 �0.35697 �1.01470 �1.1419016 �1.2 �0.58 0.2 �0.59583 �0.48551 �0.09769 �0.24929 �0.06089 �0.07041 0.07569 0.1514317 �1.2 �0.78 0.4 �0.33270 �0.26425 �0.04017 �0.12053 0.00586 �0.02231 0.10964 0.1629218 �1.2 �0.98 �0.2 �0.79844 �0.71250 �0.26442 �0.50370 �0.44809 �0.25517 �0.53336 �0.5517019 �1.2 �1.18 �0.4 �0.89848 �0.82379 �0.34603 �0.62810 �0.63634 �0.34600 �0.83021 �0.8969920 �1.2 �1.38 0 �0.45379 �0.44090 �0.22157 �0.37929 �0.45013 �0.23111 �0.63911 �0.7110421 �1.4 �0.58 0.4 �0.58607 �0.45578 �0.04850 �0.18506 0.07155 �0.01238 0.30163 0.4170422 �1.4 �0.78 �0.2 �1.04710 �0.89844 �0.27276 �0.56523 �0.38451 �0.24487 �0.34708 �0.3034223 �1.4 �0.98 �0.4 �1.14610 �1.00800 �0.35425 �0.68837 �0.57310 �0.33544 �0.64572 �0.6509324 �1.4 �1.18 0 �0.70607 �0.62926 �0.22946 �0.44081 �0.38454 �0.22041 �0.45057 �0.4616725 �1.4 �1.38 0.2 �0.44256 �0.40954 �0.17328 �0.31483 �0.32189 �0.17366 �0.42130 �0.45313

Testing samples26 �0.6 �0.78 0.2 0.00039 �0.02212 �0.04618 �0.05931 �0.12642 �0.05562 �0.21782 �0.2578627 �0.8 �1.18 0 �0.19435 �0.22405 �0.16593 �0.25474 �0.38713 �0.18465 �0.60749 �0.6996628 �1 �0.58 �0.4 �0.97049 �0.84716 �0.28662 �0.56582 �0.44890 �0.26781 �0.48342 �0.4753129 �1.2 �0.98 �0.2 �0.79844 �0.71250 �0.26442 �0.50370 �0.44809 �0.25517 �0.53336 �0.5517030 �1.4 �1.38 0.4 �0.25997 �0.24143 �0.10329 �0.18678 �0.19302 �0.10381 �0.25441 �0.27414


tionary support vector machine is shown in Table 2. Comparing displacement at x and y-directions of five key points usingthe theoretical and the identified values of mechanical parameters are shown in Fig. 3. The convergence process of algorithmis shown in Fig. 4. Variation of the recognized parameters with generations is shown in Fig. 5.

Table 2Comparison of back analysis results and theoretic solutions.

Parameters to be recognized (MPa) Theoretical SVM + PSO SVM + GA [18]

Recognized Absolute error Relative error(%) Recognized Absolute error Relative error (%)

rx0 �0.98 �0.97874 �0.00126 0.12827 �0.97442 �0.00558 0.56908ry0 �0.98 �0.97618 �0.00382 0.39000 �0.96808 �0.01192 1.21602sxy 0 0.00000 0.00000 �0.00560 0.00560

-0.5

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0u1x u2x u2y u3x u3y u4x u4y u5y

disp

lace

men

t (m

m)

Key points

calculated displacement using theoerical parameters

Calculated displacement using recognized parameters

Fig. 3. Comparison of displacements from back analysis and finite element calculation.

0

0.05

0.1

0.15

0.2

0.25

Fit

ness

Generation

Fig. 4. Variation of fitness values with generations.


-1.06

-1.04

-1.02

-1

-0.98

-0.96

-0.94

-0.92

σx0(

Mpa

)

Generation

σy0(

Mpa

)

b

a

c

xy(M

pa)

-1.005-1

-0.995-0.99-0.985-0.98-0.975-0.97-0.965-0.96-0.955-0.95

1 9 17 58 79 167

173

192

198

203

216

218

237

279

Generation

-0.05-0.045-0.04-0.035-0.03-0.025-0.02-0.015-0.01-0.005

0

Generatiom

Fig. 5. Variation of the recognized parameters with generations: (a) rx0, (b) ry0, and (c) sxy.


5. Case studies

The permanent shiplock is one of the major components of Three Gorges Project in China. It is one of the largest artificialnavigation structures excavated in a rock mass in the world. The permanent shiplock is located on the right side of the Yan-gtze River. The permanent shiplock is constructed along an azimuth direction of 118� as a double channel with five stages(Fig. 6) and a total length of 1617 m. The single shiplock room is excavated in granite and is 280 m in length, 34 m in width,and 5 m in depth for storing water. Both sides of the shiplock room are high and steep granite slopes. The sidewall of theshiplock room is vertically cut with a height of 40–50 m. The deepest excavation is about 170 m deep. The section 17–17is located at the head of the third shiplock room. Its slope is the highest one of the permanent shiplock area. Also, the geo-logical structures there are the most complex and there is a fault (F215) in this area. The design and stability analysis of thisslope is crucial in the construction of the permanent shiplock. For both analysis and design, proper recognition of the rockmass mechanical parameters is key problem.

The section 17–17 is located at the top of the third shiplock room. The corresponding engineering geological zones areshown in Fig. 6. They consist mainly of a hard and intact rock mass slightly weathered in some places. It is suitable for deepexcavation of high and steep slopes. The mechanical properties of the rock mass do not vary at the slightly weathered andfresh zones. The upper portion of the strata is completely weathered thin strata and next to it is a moderately weathered

Fig. 6. Engineering geology of section 17–17 (scale: 1:500).


zone. Therefore, mechanical parameters for these natural rock mass zones, such as the slightly and non-weathered zone, thecompletely weathered zone, and the moderately weathered zone are considered to be recognized. The in situ stress is dis-turbed due to excavation of the slope, which generates unloading and relaxation zones. Because of excavation and blasting, adamage zone forms at the boundary of the excavation. Therefore, there are two other zones related to engineering activities,unloading and damaged zones, to be considered in the back analysis of displacement.

Some researchers have suggested regressive equations [12] for calculating stress fields from in situ measurement in theregion. In this study, Feng’s assumptions for back analysis were adopted [13]. Therefore, we have the following equations forslightly weathered and fresh rock mass:

Table 3Arrange

Monito

TP/BM1TP/BM1TP/BM2TP/BM2TP/BM2TP/BM2

rx ¼ ax þ 0:01168H ðMpaÞ;ry ¼ ay þ 0:03039H ðMpaÞ;rz ¼ 4:7152þ 0:01027H ðMpaÞ;sxy ¼ 0; syz ¼ 0; szx ¼ 0:

ð20Þ

And the following equations for the strongly weathered rock mass:

rx ¼ lcH;

ry ¼ cH;

rz ¼ cH=ð1� lÞ;sxy ¼ 0; syz ¼ 0; szx ¼ 0;

ð21Þ

where c is the unit weight of the rock mass (�106 N/m3), H is depth below ground surface (m), and ax,ay are coefficients to beback analyzed.

ment of displacement monitoring points for section 17–17 of the permanent shiplock.

ring point no. Height above sea level (m) Location Date at initial monitoring

0GP01 230 North slope June 19951GP01 200 North slope January 19956GP02 170 South slope November 19967GP02 200 South slope November 19958GP02 230 South slope March 19959GP02 245 South slope December 1994

Table 4The learning and testing samples.

Samplesno.

Parameters to be recognized Computational displacement for monitoring points (mm)

Young’s modulus E (Gpa) Coefficientofgeostressfield

TP/BM10GP01

TP/BM11GP01

TP/BM26GP02

TP/BM27GP02

TP/BM28GP02

TP/BM29GP02

Wwz Dz Udz Swf ax ay

Learning samples1 6.0 8.0 15.0 25.0 3.0 0.8 13.11 13.84 17.87 13.47 11.6 11.372 6.0 10.0 18.0 28.0 4.0 1.2 16.44 16.98 21.96 15.98 14.29 14.373 6.0 12.0 20.0 30.0 5.0 0.6 19.79 20.21 26.51 18.97 17.12 17.374 6.0 15.0 23.0 32.0 6.0 0.8 23 23.38 30.51 22.18 19.79 20.035 6.0 18.0 25.0 35.0 7.0 2.0 24.94 25.26 33.14 24.52 21.46 21.566 8.0 8.0 18.0 30.0 6.0 2.0 23.58 23.92 31.19 22.56 20.14 20.67 8.0 10.0 20.0 32.0 7.0 0.8 26.28 26.62 35.16 25.94 22.43 22.738 8.0 12.0 23.0 35.0 3.0 1.2 9.56 10.02 13.18 9.63 8.23 8.39 8.0 15.0 25.0 25.0 4.0 1.6 18.7 19.17 23.96 17.79 16.07 16.2610 8.0 18.0 15.0 28.0 5.0 1.8 20.92 21.4 28.45 19.93 17.89 18.2311 10.0 8.0 20.0 35.0 4.0 1.8 12.81 13.17 17.74 12.33 10.84 11.2312 10.0 10.0 23.0 25.0 5.0 2.0 23.15 23.45 30.17 21.79 19.89 20.2213 10.0 12.0 25.0 28.0 6.0 0.8 25.47 25.78 33.86 24.73 21.94 22.1314 10.0 15.0 15.0 30.0 7.0 1.2 27.62 28.04 37.67 26.98 23.32 23.7515 10.0 18.0 18.0 32.0 3.0 1.6 10.38 10.89 14.24 10.15 8.73 8.9416 12.0 8.0 23.0 28.0 7.0 1.6 29.31 29.51 38.13 28.24 24.94 25.4417 12.0 10.0 25.0 30.0 3.0 1.8 10.87 11.37 14.66 10.59 9.17 9.418 12.0 12.0 25.0 32.0 4.0 2.0 13.8 14.26 19.34 13.39 11.66 12.0219 12.0 15.0 28.0 35.0 5.0 0.8 16.31 16.73 23.62 16.51 14.11 14.3320 12.0 18.0 20.0 25.0 6.0 1.2 28.17 28.47 37.48 26.95 24.17 24.3921 15.0 8.0 25.0 32.0 5.0 1.2 17.49 17.88 24.09 17.17 15.11 15.4822 15.0 10.0 15.0 35.0 6.0 1.6 19.39 19.82 27.55 19.17 16.42 16.9523 15.0 12.0 18.0 25.0 7.0 1.8 32.12 32.48 42.73 30.7 27.36 27.7724 15.0 15.0 20.0 28.0 3.0 2.0 11.45 12.05 15.58 11.22 9.66 9.8525 15.0 18.0 23.0 30.0 4.0 0.8 14.85 15.4 20.58 14.98 13 1326 6.0 8.0 15.0 32.0 7.0 1.8 26.28 26.63 34.85 25.67 22.25 22.6927 6.0 10.0 18.0 35.0 3.0 2.0 9.56 9.98 13.05 9.19 7.97 8.228 6.0 12.0 20.0 25.0 4.0 0.8 18.65 19.29 24.48 18.36 16.39 16.2229 6.0 15.0 23.0 28.0 5.0 1.2 21.51 21.99 28.41 20.76 18.75 18.830 6.0 18.0 25.0 30.0 6.0 1.6 24.75 25.16 32.44 23.81 21.39 21.531 8.0 8.0 18.0 25.0 5.0 1.6 23 23.43 30.34 21.76 19.86 20.232 8.0 10.0 20.0 28.0 6.0 1.8 25.5 25.79 33.48 24.26 21.8 22.1933 8.0 12.0 23.0 30.0 7.0 2.0 28.35 28.54 36.92 27.24 24.13 24.5134 8.0 15.0 25.0 32.0 3.0 0.8 10.58 11.14 14.62 11.21 9.46 9.2135 8.0 18.0 15.0 35.0 4.0 1.2 13.13 13.62 18.59 13.01 11.23 11.4136 10.0 8.0 20.0 30.0 3.0 1.2 10.77 11.32 14.79 10.65 9.2 9.3737 10.0 10.0 23.0 32.0 4.0 1.6 14.17 14.55 19.12 13.62 12.06 12.4138 10.0 12.0 25.0 35.0 5.0 1.8 16.69 16.98 22.79 16.05 14.29 14.7239 10.0 15.0 15.0 25.0 6.0 2.0 28.1 28.49 37.46 26.54 23.92 24.340 10.0 18.0 18.0 28.0 7.0 0.8 29.9 30.24 40.13 29.02 25.44 25.741 12.0 8.0 23.0 35.0 6.0 0.8 19.95 20.27 27.42 19.93 17.05 17.4242 12.0 10.0 25.0 25.0 7.0 1.2 32.97 33.19 42.63 31.6 28.19 28.5543 12.0 12.0 15.0 28.0 3.0 1.6 11.41 12.01 15.71 11.13 9.6 9.8244 12.0 15.0 18.0 30.0 4.0 1.8 14.92 15.39 20.46 14.44 12.68 1345 12.0 18.0 20.0 32.0 5.0 2.0 18.1 18.42 24.97 17.36 15.41 15.8246 15.0 8.0 25.0 28.0 4.0 2.0 15.59 16.07 21 15.07 13.34 13.6647 15.0 10.0 15.0 30.0 5.0 0.8 18.26 18.87 26.34 18.28 15.79 16.0848 15.0 12.0 18.0 32.0 6.0 1.2 21.38 21.77 30 21.02 18.23 18.6549 15.0 15.0 20.0 35.0 7.0 1.6 23.37 23.66 32.48 23.08 19.76 20.2650 15.0 18.0 23.0 25.0 3.0 1.8 12.86 13.59 17.29 12.74 11 11.16

Testing samples51 6.0 9.0 17.0 29.0 5.4 1.88 22.00 22.41 29.24 20.90 18.91 19.2952 6.9 11.0 20.0 34.0 3.8 1.76 12.82 13.20 17.35 12.28 10.85 11.1653 7.8 13.0 23.0 28.0 6.6 1.64 28.64 28.87 37.25 27.37 24.50 24.7854 8.7 15.0 15.0 33.0 5.0 1.52 17.55 17.99 24.72 17.00 15.02 15.3955 9.6 17.0 18.0 27.0 3.4 1.40 14.12 14.70 18.83 13.64 12.01 12.1956 10.5 8.0 21.0 32.0 6.2 1.28 22.70 23.02 30.45 22.18 19.36 19.7957 11.4 10.0 24.0 26.0 4.6 1.16 21.10 20.55 26.70 19.38 17.39 17.6258 12.3 12.0 16.0 31.0 3.0 1.04 10.30 10.89 14.54 10.39 8.79 8.9359 13.2 14.0 19.0 25.0 5.8 0.92 26.64 27.04 36.09 25.79 22.97 23.1660 14.1 16.0 22.0 30.0 4.2 0.80 15.68 16.20 21.77 15.71 13.68 13.7261 15.0 18.0 25.0 35.0 7.0 2.00 23.70 23.89 32.39 23.23 20.13 20.59



The parameters to be back analyzed are the coefficients ax and ay (Eq. (20)) for the geostress equation and the deformationmodulus for four kinds of rock mass zones (moderately weathered zone, slightly weathered or fresh zone, unloading defor-mation zone and damaged zone). Data for deformation modulus, Poisson’s ratio, and weight for the strongly weathered zoneand the fault F215 were provided by the Yangtze River Water Conservancy Committee. The ranges for these six parametersto be back analyzed are determined based on the field monitoring data as follows [13]:

(1) Deformation modulus for moderately weathered zone Wwz: 6–15 GPa.(2) Deformation modulus for damaged zone Dz: 8–18 GPa.(3) Deformation modulus for unloading deformation zone Udz: 15–25 GPa.(4) Deformation modulus for slightly or non-weathered zone Swf: 25–35 GPa.(5) The geostress coefficients ax: 3–7 and ay: 0.8–2.0.

The rock masses in all zones are considered to be plastic. Their cohesion c and friction angle f are determined directly fromengineering tests and previous monitoring data [13].

Table 5Identified the parameters and its comparisons.

Method Young’s modulus E (Gpa) Coefficient of geostress field

Wwz Dz Udz Swf ax ay

SVM + PSO 6.000 9.498 17.313 29.253 4.355 1.370NN + GA [13] 7.515 9.683 18.95 32.1 4.793 1.599

0

5

10

15

20

25

TP/BM10GP01 TP/BM11 GP01 TP/BM26GP02 TP/BM27 GP02 TP/BM28GP02 TP/BM29 GP02

Dis

plac

emen

t (m

m)

Monitored points

Monitored dis placement

Predicted displacement using SVM

Fig. 7. Comparison of displacements from back analysis and monitored displacement.

Table 6Comparison of measured displacement with predictions from support vector machine and neural network model.

Displacement at monitoring point (mm) Average absolute error

TP/BM10GP01 TP/BM11GP01 TP/BM26GP02 TP/BM27GP02 TP/BM28GP02 TP/BM29GP02

Measurement 16.32 19.11 20.76 16.71 19.1 16.71Prediction of SVM model 17.26 17.75 23 16.71 15.1 14.93 1.72Prediction of NN model [13] 17.36 17.79 23.64 16.85 14.91 15.29 1.83


Feng et al. used the Drucker–Prager model to analyze in the FLAC2D and obtained totally 50 sets of training samples and11 testing samples (Table 4) [13]. The computational model scope was 1000 m in the x-direction, 500 m extended from theisolated rock mass, and 510 m in the y-direction. There are more than 15,000 nodes and 15,000 quadrilateral elements. Infor-mation of displacement monitoring points for section 17–17 is listed in Table 3 and the location is shown in Fig. 6. Based onabove algorithm for recognizing rock mass parameters, the value of geostress coefficients and the deformation module for

0

1

2

3

4

5

6

7

8

9

10

0 20 40 60

Fitn

ess

population

(a) Initial generation

1.718

1.72

1.722

1.724

1.726

1.728

1.73

1.732

1.734

1.736

0 20 40 60

fitn

ess

population

(c) 200th generation

1.718

1.72

1.722

1.724

1.726

1.728

1.73

1.732

1.734

1.736

1.715

1.72

1.725

1.73

1.735

1.74

1.745

1.75

fitn

ess

population

(b) 100th generation

0 20 40 60

0 20 40 6 0

fitn

ess

population

(d) 500th generation

Fig. 8. Variation of fitness values with generations.

5

5.5

6

6.5

7

7.5

8

8.5

Ww

z (G

Pa)

Generation

Fig. 9. Variation of the recognized parameters Wwz with generations.

8

8.2

8.4

8.6

8.8

9

9.2

9.4

9.6

9.8

10

10.2

Dz

(GPa

)

Generation

Fig. 10. Variation of the recognized parameters Dz with generations.

16.95

17

17.05

17.1

17.15

17.2

17.25

17.3

17.35

Udz

(G

Pa)

Generation

Fig. 11. Variation of the recognized parameters Udz with generations.

29

29.1

29.2

29.3

29.4

29.5

29.6

29.7

Swf

(GPa

)

Generation

Fig. 12. Variation of the recognized parameters Swf with generations.


four kinds of rock mass zones are obtained as 4.3546, 1.3697, 6.0002, 9.4975, 17.3129 and 29.2527 GPa, respectively. Itscomparison with the Feng’s results is listed in Table 5. Comparison of displacements at monitoring points between moni-tored and the identified values of mechanical parameters are shown in Fig. 7 and Table 6. We can see the proposed methodis better than NN-based method from Table 6. The convergence process of algorithm is shown in Fig. 8. Variation of the rec-ognized parameters with generations is shown in Figs. 9–14.

4.2

4.25

4.3

4.35

4.4

4.45

4.5

4.55

4.6

4.65

axGeneration

Fig. 13. Variation of the recognized parameters ax with generations.

1.18

1.2

1.22

1.24

1.26

1.28

1.3

1.32

1.34

1.36

1.38

ay

Generation

Fig. 14. Variation of the recognized parameters ay with generations.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

9 363 374 398 417 488 650 840

Min

imum

fitn

ess

C

Fig. 15. Relationship between the parameters C of SVM and minimum fitness of the tentative SVM.


6. Discussions

Relation between rock mass parameters and displacements is a key element in back analysis. It affects directly the resultsof rock mass to be recognized (see Figs. 15 and 16). Support vector machine represents well the relationship between rockmass parameters and displacements (see Fig. 17). Support vector machine model standing of numerical analysis in backanalysis procedure improved the efficiency of back analysis (see Fig. 18). Particle swarm optimization has strong capabilityof global searching, and it improves the generalization performance in searching the support vector machine model. This

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

18 24 25 26 27 27 32 34

Min

imum

fitn

ess

σ

Fig. 16. Relationship between the parameters r of kernel function and minimum fitness of the tentative SVM.

0

5

10

15

20

25

30

35

40

0 5 10 15 20 25 30 35 40

Pred

icte

d di

spla

cem

ent b

y SV

M (

mm

)

Monitored displacement (mm)

TP/BM10GP01 TP/BM11GP01



Fig. 17. Comparison between monitored displacement and predicted displacement by SVM of testing samples.

1.6

1.65

1.7

1.75

1.8

1.85

1.9

fitn

ess

Generation

Fig. 18. Change tendency of the minimum fitness of SVMs versus number of evolutionary generation.


Table 7The results based on different searching bound.

Range pf searching rx0 (Mpa) ry0 (Mpa) sxy (Mpa) Fitness

1 [�1,0] �0.978743 �0.976178 0 0.010312 [�2,0] �0.978743 �0.976178 0 0.010313 [�5,0] �0.978743 �0.976178 0 0.010314 [�10,0] �0.978743 �0.976178 0 0.010315 [�20,0] �0.978743 �0.976178 0 0.01031


makes it possible to find the rock mass parameters in a big global space (see Table 7), which enables the back analysis to beapplied to more complex engineering problems.

7. Conclusions

The paper presents a new intelligent back analysis method for recognizing the geomechanical parameters through comb-ing particle swarm optimization, support vector machine and numerical analysis. Support vector machine is used to buildthe non-linear relationship between geomechanical parameters and displacements, and has proved excellent performanceof non-linear representation based on little samples. Particle swarm optimization is used to improve the generalization per-formance in searching the support vector machine model, and has proved powerful global optimal performance. Overall, theproposed approach improves the efficiency and precision of back analysis, and makes it possible to be applied to more com-plex engineering problems.

Acknowledgement

The financial support from Program for New Century Excellent Talents in University (NCET) and Doctoral Fund of HenanPolytechnic University (No. 648197) are greatly acknowledged.

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geomechanical parameters identification by particle swarm optimization and support

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