Research Article Back Analysis of Geomechanical Parameters ... · Introduction Numerical analysis...
Transcript of Research Article Back Analysis of Geomechanical Parameters ... · Introduction Numerical analysis...
Research ArticleBack Analysis of Geomechanical Parameters in UndergroundEngineering Using Artificial Bee Colony
Changxing Zhu Hongbo Zhao and Ming Zhao
School of Civil Engineering Henan Polytechnic University Jiaozuo 454003 China
Correspondence should be addressed to Hongbo Zhao bxhbzhaohotmailcom
Received 13 April 2014 Accepted 26 June 2014 Published 17 July 2014
Academic Editor Minghuwi Horng
Copyright copy 2014 Changxing Zhu et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Accurate geomechanical parameters are critical in tunneling excavation design and supporting In this paper a displacementsback analysis based on artificial bee colony (ABC) algorithm is proposed to identify geomechanical parameters from monitoreddisplacements ABC was used as global optimal algorithm to search the unknown geomechanical parameters for the problem withanalytical solution To the problem without analytical solution optimal back analysis is time-consuming and least square supportvector machine (LSSVM) was used to build the relationship between unknown geomechanical parameters and displacement andimprove the efficiency of back analysisThe proposed method was applied to a tunnel with analytical solution and a tunnel withoutanalytical solution The results show the proposed method is feasible
1 Introduction
Numerical analysis plays an important role in constructionand design of geotechnical engineering [1] Geomechani-cal parameters such as Youngrsquos modulus and cohesion arecritical to numerical analysis and are difficult to determinebecause of the complexity and uncertainty of geotechnicalengineering Back analysis is a reliable approach to esti-mate the geomechanical parameters and is used widely ingeotechnical engineering [2] Because the deformation ofrock masses induced by excavation can be measured easilyand reliably displacement-based back analysis techniquesas a practical engineering tool are nowadays frequentlyused in geotechnical engineering problems to determine theunknown geomechanical parameters [3ndash9]
There are mainly three types of displacement back anal-ysis methods inverse solving method atlas method anddirect (ie optimal) method [7] Because of the specialadvantages the optimal methods are more and more exten-sively employed in solving engineering problems [10ndash12]Optimization method is important to optimal back anal-ysis Levenber-Marquardt method Gauss-Newton method
Bayesian method Powell method Rosenbork method softcomputing and particle swarm optimization have been pro-posed and applied to back analysis [12ndash14] To the practicalgeotechnical engineering optimal back analysis needs tocall numerical analysis many times This procedure is time-consuming Neural network and support vector machinewere applied to back analysis to replace the numerical analysis[14ndash17] This has been a new way for displacement backanalysis
In this paper artificial bee colony (ABC) algorithmwas chosen for its biological and evolutionary appeal infinding the set of unknown parameters that best matchesthe modeling prediction with the measured displacementdata Least square support vector machine (LSSVM) wasused to replace numerical analysis to present the relationshipbetween unknown geomechanical parameters and displace-ment of geotechnical structure Firstly the idea and algorithmof ABC were presented in Section 2 In Section 3 ABC wasadopted to search geomechanical parameters in displacementback analysisThe procedure of ABC-based back analysis waspresented to the tunnel with analytical solution and applied toa circular tunnel with hydrostatic stressThen to the complex
Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 693812 13 pageshttpdxdoiorg1011552014693812
2 The Scientific World Journal
geotechnical engineeringwithout analytical solution LSSVMwas used to present the relationship between geomechanicalparameters and displacement LSSVM model replaced thenumerical analysis to improve the efficiency of back analysisBack analysis based on LSSVM and ABC combination wasproposed in Section 4 LSSVM and the procedure of theproposed method were presented in brief Lastly someconclusion was listed in Section 5
2 Artificial Bee Colony Algorithms
The artificial bee colony (ABC) algorithm was originallydeveloped in 2005 by Karaboga [18] In ABC algorithmthe colony of artificial bees contains three groups of beesemployed bees onlookers and scouts Employed bees searchfor specific food sources (solution) and calculate the amountof nectars (fitness value) Onlooker bees choose a food sourcebased on the nectars shared by employed bees and determinethe source to be abandoned and allocate its employed beeas scout bees Scout bees randomly search for a new foodsource The position of a food source represents a possiblesolution for the problem under consideration and the nectaramount of a food source represents the quality of the solutionrepresented by the fitness value [19 20] To the minimumproblem the fitness can be computed by the target function
In the algorithm the first half of the colony consists ofemployed artificial bees and the second half constitutes theonlookersThe number of the employed bees or the onlookerbees is equal to the number of solutions in the population Atthe first step theABCgenerates a randomly distributed initialpopulation of SN solutions and calculates the fitness of eachsolution Consider
119909 (119894 119895) = 119909119895
min + rand (0 1) (119909119895max minus 119909119895
min) (1)
where 119909(119894 119895) is the candidate solution of problem 119894 =
1 2 1198781198732 and 1198781198732 denotes the size of population 119895 =1 2 119863 and 119863 is the dimension number of each solutionrand(01) is a random number between [0 1] 119909119894min and 119909
119894max
are the upper and lower bound of each solutionOnce initialization is completed the artificial bees are
used to conduct the search for the best food resource(solution) Procedures can be described as follows [20]
(i) Employed bees determine a food source within theneighborhood of the food source through their mem-ory
(ii) Employed bees share their information with onlook-ers within the hive and then the onlookers select oneof the food sources
(iii) Onlookers select a food source within the neighbor-hood of the food sources chosen by them to produceand exploit the new food resources
(iv) An employed bee of the sources that have beenabandoned by onlookers becomes a scout and startsto search for a new food source randomly
In the ABC algorithm a candidate food position can beproduced from the memory of bees which is defined as
V (119894 119895) = 119909 (119894 119895) + 120593119894119895 (119909 (119894 119895) minus 119909 (119896 119895)) (2)
where k used to be different from 119894 is randomly chosenindexes from 1 2 1198781198732 j is also randomly chosenindexes from 1 2 119863 and 120593ij is a random numberin [minus1 1] and controls the generation of neighbor foodsources around 119909(119894 119895) and represents the comparison oftwo food positions seen by a bee As can be seen from(2) the perturbation on the position 119909(119894 119895) decreases whenthe difference between the parameters of 119909(119894 119895) and 119909(119896 119895)decreases so that the step length is adaptively reduced
An artificial onlooker bee chooses a food source basedon the probability of food source The probability of beingselected for fitness pi can be expressed as
119901119894 =fitness119894
sum119878119873119899=1 fitness119899
(3)
where fitness119894 is the fitness of the solutionIn ABC algorithm a food source whose position cannot
be improved further through a predetermined number ofcycles is assumed to be abandoned by onlookers 119909(119894 119895) usedto represent the abandoned source is replaced with 1199091015840(119894 119895)that is a new food source the scout bees find which isconducted by (1)
Each candidate source position V(119894 119895) produced by 119909(119894 119895)can be evaluated using the comparison between 119909(119894 119895) and itsold source positionThe old food source will be replaced withthe new food source when it is equal to or better than the oldfood source Otherwise the old food source is retained in thememory
There are three control parameters in the ABC thenumber of food sources which is equal to the number ofemployed or onlooker bees (SN2) the value of limit and themaximum cycle number (MCN) The following is the briefprocedure of artificial bee colony (ABC) algorithm
Step 1 Determine the value of control parametersSN2 MCN and ldquolimitrdquo of ABC algorithmStep 2 Generate the initial population 119909(119894 119895) by (1)and evaluate the fitness of each solutionStep 3 Produce new solution V(119894 119895) for each employedbee by using (2) In the meantime the fitness isevaluatedStep 4 Calculate the probability 119901119894 for the solution119909(119894 119895) by (3)Step 5 Select a solution 119909(119894 119895) for each onlookerbee according to 119901119894 Then a new solution V(119894 119895) isgenerated by (2)Step 6 Calculate the fitnessStep 7 If there is an abandoned solution for the scoutit will be replaced by using a new solution which israndomly produced by (2)Step 8 Trace the best solutionStep 9 Repeat Steps 3 to 8 until the cycle reaches themaximum cycle number (MCN)
The Scientific World Journal 3
p0
p0
pi
rP
r0
PlasticElastic
Figure 1 A circular tunnel subjected to hydrostatic far field stressand uniform support pressure
3 ABC-Based Back Analyses
Optimization algorithm is critical to back analysis In thissection ABC-based back analysis was presented to identifythe geomechanical parameters of a circular tunnel withanalytical solution
31 The Analytical Solution of Circular Tunnel A circulartunnel is excavated in a continuous homogeneous isotropicinitially elastic rock mass and subjected to a hydrostatic farfield stress p0 and uniform support pressure pi as shown inFigure 1
According to the Mohr-Coulomb criterion the normalstress pcr at the plastic-elastic zone interface is given [21] asfollows
119901119888119903 =2119901119900 minus 120590119888
119896 + 1
119896 =
1 + sin1205931 minus sin120593
120590119888 =119888 (119896 minus 1)
tan120593
(4)
where 120593 is the friction angle and c is the cohesion If theuniform support pressure pi is less than the critical pressurepcr the plastic zone exists The plastic zone radius R is given[22] as follows
119877 = 119903119900 lowast [2 (119901119900 + 119904)
(119896 + 1) (119901119894 + 119904)]
1(119896minus1)
(5)
in which
119904 =
120590119888
119896 minus 1
(6)
and 119903119900 is the radius of the tunnel
The deformation of surrounding rock of tunnel is asfollows
Elastic zone
119906119903 =
(119901119900 sin120593 + 119888 sdot cos120593) (1198772119903)
2119866
(7)
Plastic zone
119906119903 =119903
2119866
sdot 120594 (8)
where E is the deformation modulus and 120583 is Poissonrsquos ratio
120594 = (2120583 minus 1) (119901119900 + 119888 sdot ctg120593)
+ (1 minus 120583) [(1198702119901 minus 1) (119870119901 + 119870119901119904)]
times (119901119894 + 119888 sdot ctg120593) (119877
119903119900
)
(119870119901minus1)
(
119877
119903
)
(119870119901119904+1)
+ [
(1 minus 120583) (119870119901119870119901119904 + 1)
(119870119901 + 119870119901119904)
sdot 120583]
times (119901119894 + 119888 sdot ctg120593) (119903
119903119900
)
(119870119901minus1)
119870119901119904 =(1 + sin120595119904)(1 minus sin120595119904)
119866 =
119864
2 (1 + 120583)
(9)
32 Error Function An error function in this work isdefined as the minimum error between the displacementspredicted by the analyticalmodel based identified parametersand the actualmeasured displacements It can be expressed as
fitness = radicsum119899119894=1 (119910119901119894 minus 119910119894)
2
119899
(10)
where n is the number of key points 119910119894 is the monitoreddisplacement of the ith key points and 119910119901119894 is the predicteddisplacement of ith key point
33 The Procedure of ABC-Based Back Analysis ABC-basedback analysis is combined ABC with the analytical solution(see (7) and (8)) ABC produces population of artificial beesincluding employer bees onlooker bees and scout bees Thefitness values can be computed by (10) The displacementof (10) can be computed by (7) and (8) Based on the ABCalgorithm the new population was produced ABC-basedback analysis algorithm can be described as follows (seeFigure 2)
Step 1 Collect the information of engineering such asgeology conditions and engineering size
4 The Scientific World Journal
Start
Determine engineering condition andselect the computing model
Initiate the ABC algorithm
Generate the initial population by (1) andcompute the displacement by (7) and (8)and the fitness of each solution by (10)
Generate the new population by (2) and(3) and compute the displacement by (7)
and (8) the fitness of each solution by (10)
Memorize the best solution
Maximum cycle meets
Get the geomechanical parameters
End
No
Yes
Figure 2 Flowchart of ABC-based back analysis
Step 2 Select the appropriate model according to theabove informationStep 3 Determine the error functionStep 4 Activate the ABC algorithm (see Section 2) toproduce the initial population 119909(119894 119895) by (1) Displace-ments are computed using (7) and (8)Step 5 The fitness of each solution is calculated by(10)Step 6 Generate the new population based on ABCalgorithm (see (2) and (3)) and compute the displace-ment (see (7) and (8))Step 7 Trace the best solution according to the ABCalgorithm
Table 1 Parameters of tunnel model
1199010(MPa) 119864 (MPa) 119888 (MPa) Φ (∘) 119901119894 (Mpa) 120595 (∘)300000 70000000 34500 300000 0 0
Table 2 Identified parameters using ABC-based back analysis
119864 (Mpa) 119888 (Mpa) 120593 (∘)ABC-based back analysis 689304951 35065 2999284Actual value 70000000 34500 300000Relative error () 15279 minus16377 00239
Step 8Repeat Steps 5 to 7 until finding the solution orreaching the maximum cycle
34 Verification The displacement of monitored point oftunnel can be computed by the above formula In this studysix monitored points were used in circular tunnel to monitorthe displacements at the horizontal direction for ABC searchThe distance between central of tunnel and 6 monitoredpoints is 10m 11m 13m 15m 17m and 21m respectively(see Figure ) The radius of tunnel is 1m The parameterof rock is listed in Table 1 ABC-based back analysis is usedto identify geomechanical parameters (eg Youngrsquos modulusE cohesion c and friction angle 120593) from displacements ofsix monitored points The recognized parameters and theirerror are listed in Table 2 The maximum relative error is16 It shows the recognized parameters agree well withthe real parameters The comparison between recognizedand real parameters about the displacement and stress ofsurrounding rock of tunnel is shown in Figures and Theresults show stresses and displacements of surrounding rockidentified by ABC are in well agreement with real stresses anddisplacements of surrounding rock and ABC is an excellentoptimization method The relationship between fitness andcycle is shown in Figure 3The relationship between identifiedparameters and cycle is shown in Figure 4They show that theperformance and convergence of ABC are good and quick foridentification of geomechanical parameters using ABC
341 Effect of Searching Range Theperformances of ABC aredemonstrated with different searching ranges (Table 3) Theresults of different searching ranges are shown in Figure 5To the smaller range the convergence is quicker than thebigger range But to the bigger range the fitness is the sameas the smaller range It shows ABC has strong capabilityof global searching and makes it possible to find the rockmass parameters in a big global space which enables theback analysis to be applied to more complex engineeringproblems
342 Effect of Population Size Population size is key param-eters of ABC To study the effect of the colony size on the
The Scientific World Journal 5
1 2 3 4 5 6
1m 13m
11m
15m 17m 21m
Figure 3 Position of monitored point in circular tunnel
0
02
04
06
08
1
12
1 2 3 4 5 6 7 8 9 10Distance to the center of tunnel (m)
Computed displacement using recognized parametersComputed displacement using actual parameters
Disp
lace
men
t (10
minus2
m)
Figure 4 The comparison of displacement between actual andrecognized parameters
convergence rate of theABC algorithm five different coloniesthat consisted of 20 50 100 200 and 400 bees were usedThefitness versus cycle numbers is shown in Figure 6 It can beseen that the convergence rates increasewith greater numbersof bees and population size of 200 or 400 bees is enough inthis study
4 Back Analysis Based on LSSVM and ABC
In the above section ABC-based back analysis was used tothe circular tunnel with analytical solution To the practicalengineering it is difficult to get the analytical solutionThe procedure with numerical solution is time-consumingRegression analysis is a good approach to build the rela-tion between geomechanical parameters and field moni-tored information In this study least square support vectormachine (LSSVM) was adopted to present the relationship
0
5
10
15
20
25
30
35
40
45
50
1 3 5 7 9
Stre
ss (M
Pa)
Distance to the center of tunnel (m)
Computed radial stress using recognized parametersComputed radial stress using actual parametersComputed tangential stress using recognized parametersComputed tangential stress using actual parameters
Figure 5 The comparison of stress between actual and recognizedparameters
0 200 400 600 800 1000
Fitn
ess
Cycle
100E minus 04
900E minus 05
800E minus 05
700E minus 05
600E minus 05
500E minus 05
400E minus 05
300E minus 05
200E minus 05
100E minus 05
000E + 00
Figure 6 Relationship between fitness value and cycle
between geomechanical parameters and displacement basedon numerical analysis
41 Least Square Support Vector Machine The least squaresupport vector machine (LSSVM) was originally developedby Suykens andVandewalle [21] Consider a given training setofN data points 119909119896 119910119896 (119896 = 1 2 119873)with input data xk isinRN and output yk isin r where RN is the N-dimensional vector
6 The Scientific World Journal
600000
620000
640000
660000
680000
700000
720000
0 200 400 600 800 1000
Fitn
ess
Cycle
E (MPa)
(a) 119864
200
250
300
350
400
450
500
550
0 200 400 600 800 1000
Fitn
ess
Cycle
c (MPa)
(b) 119888
2000
2200
2400
2600
2800
3000
3200
3400
0 200 400 600 800 1000
Fitn
ess
Cycle
120593 (∘)
(c) 120593
Figure 7 The variation of identified parameter with the cycle
space and r is the one-dimensional vector space Accordingto the LSSVM algorithm LSSVMmodel becomes
119910 (119909) =
119873
sum
119896=1
120572119896119870(119909 119909119896) + 119887 (11)
where 119870(119909 119909119896) is kernel functions and 120572 and b meet thefollowing equation
[
0 1119879
1 Ω + 120574minus1119868
] [
119887
120572] = [
0
119910] (12)
where 119910 = [1199101 119910119873] 1 = [1 1] 120572 = [1205721 120572119873]and Mercerrsquos theorem is applied within the Ω matrix
Ω=120593(119909119896)119879120593(119909119897) = 119896(119909119896 119909119897) 119896 119897 = 1 119873 Then the
analytical solution of 120572 and b is given by
[
119887
120572] = Φ
minus1[
0
119910] (13)
42 Representation of Nonlinear Relationship LSSVM is usedin this study to map the nonlinear relationship betweengeomechanical parameters such as Youngrsquos modulus cohe-sion geostress coefficients and monitored displacements
The Scientific World Journal 7
0
000005
00001
000015
00002
000025
0 200 400 600 800 1000
Fitn
ess
Cycle
Range 1Range 2Range 3
Figure 8The performance of ABCusing different searching ranges
0
000005
00001
000015
00002
0 200 400 600 800 1000
Fitn
ess
Cycle
SN2 = 20
SN2 = 50
SN2 = 100
SN2 = 200
SN2 = 400
Figure 9 The convergence of different population size
The mathematical model of least square support vectormachine is defined as
LSSVM (X) 119877119899 997888rarr 119877
Y = LSSVM (X) X = (1199091 1199092 119909119899)
Y = (1199101 1199102 119910119899)
(14)
0 15
minus5 10
minus5 0 5 0
5 10
10MPa 20MPa
30∘
Failure criterion Mohr-Coulomb
Youngrsquos modulus E 20000MPa
Cohesion c 105MPa
Friction angle 120593 35∘
Poissonrsquos ratio 120583 02
Figure 10 The cross section of tunnel and parameters
Table 3 The ranges of identified parameters
Range 1 Range 2 Range 3119864 (Mpa) [2000 12000] [4000 1000] [5000 8000]119888 (Mpa) [05 7] [1 6] [3 7]120593 (∘) [5 60] [10 50] [20 40]
Table 4 Identified in situ stress and angle in different stages
1198751 1198752 AngleActual 200000 100000 300000Stage 1 199583 100614 300104Stage 2 206493 108171 333676Stage 3 200252 100376 30623
where 119909119894(119894 = 1 2 119899) is geomechanical parameters forexample Youngrsquos modulus friction angle geostress coeffi-cients and so forth and 119910119894(119894 = 1 2 119899) is displacementsof the key points
In order to obtain LSSVM(X) a training process basedon the known data set is needed Necessary training samplesare created in this work by using numerical analysis (egFEM model) which is used to obtain displacements of rockmass of key points corresponding to the given set of tentativegeomechanical parameters The geomechanical parametersare defined as input of LSSVM The displacement is definedas output of LSSVM
8 The Scientific World Journal
Table5Training
samples
andmod
elparameterso
fLSSVM
Num
bero
fsamples
1198751(M
pa)
1198752(M
pa)
120593(∘ )
Disp
lacement
120572
MP1
MP2
MP3
MP1119909
MP1119910
MP2119909
MP2119910
MP3119909
MP3119910
119909119910
119909119910
119909119910
110000
0500
0020000
0minus08380
minus13
600
15500
minus00231
minus20200
minus15
100
14473
20149
minus08992
minus03815
15989
22484
210000
075
000
25000
0minus04990
minus23300
13900
minus006
87minus16
700
minus15
800
16424
08880
minus09801
minus03294
16348
19749
310000
010000
030000
0000
00minus31300
1400
0minus14
400
1400
0minus14
400
21479
02439
minus09786
minus16
870
49088
21843
412500
012500
035000
0000
00minus39100
17500
minus18
000
minus17
500
minus18
000
20307
minus03980
minus05684
minus18
560
14959
17655
515000
015000
040000
0minus000
01minus47000
20900
minus21600
minus21000
minus21700
200
40minus10
849
minus02202
minus21514
1240
414
127
615000
0500
0025000
0minus200
00minus14
700
20800
08610
minus31900
minus27200
02187
18194
minus03108
05215
03286
09391
715000
075
000
30000
0minus16
800
minus25600
18300
01890
minus27700
minus28200
05089
06915
minus05137
minus0119
105530
07972
815000
010000
035000
0minus12
300
minus34700
1740
0minus05740
minus24200
minus27500
06722
006
83minus05353
minus05142
07871
07058
915000
012500
040000
0minus064
20minus41900
18300
minus13
800
minus21800
minus25200
10483
minus03389
minus05100
minus10
033
10326
08752
1015000
015000
020000
0minus000
01minus47000
20900
minus21600
minus21000
minus21700
22964
minus12
063
minus04593
minus24334
16207
16580
1120000
0500
0030000
0minus34100
minus19
500
22700
18500
minus42500
minus43300
minus09584
14147
minus01741
13821
minus05279
minus04169
1220000
075
000
35000
0minus30700
minus32100
19200
1100
0minus360
00minus43700
minus04940
02093
minus046
0505409
01538
minus03109
1320000
010000
040000
0minus25800
minus42600
1740
002750
minus31500
minus43100
minus01430
minus06938
minus06499
minus01060
05071
minus03365
1420000
012500
020000
0minus12
600
minus36100
30300
minus07560
minus37300
minus29900
09442
minus00545
04200
minus09120
00125
07845
1520000
015000
025000
0minus09990
minus46500
27900
minus13
700
minus33400
minus31500
12917
minus11019
01791
minus16
037
044
3806994
1625000
0500
0035000
0minus50300
minus28100
22000
29600
minus53200
minus62900
minus23159
07126
minus02344
23232
minus14
578
minus20741
1725000
075
000
40000
0minus45700
minus43400
17200
20100
minus42700
minus62200
minus17
211
minus08042
minus07054
12559
minus03011
minus18
612
1825000
010000
020000
0minus25600
minus25500
40000
06760
minus53900
minus38400
minus006
8108422
12115
02427
minus13
147
01347
1925000
012500
025000
0minus25800
minus38300
35200
02050
minus48900
minus42900
minus01085
minus02739
07151
minus01537
minus08165
minus02612
2025000
015000
030000
0minus23100
minus50100
32000
minus03910
minus44100
minus45200
02387
minus13
780
05252
minus08035
minus04328
minus040
6121
30000
0500
0040000
0minus70
100
minus42700
19200
42500
minus63900
minus85200
minus44142
minus07485
minus05555
37206
minus25680
minus440
6722
30000
075
000
20000
0minus41800
minus15
000
51000
21700
minus74
200
minus48300
minus16
564
19159
23915
16943
minus34243
minus08180
2330000
010000
025000
0minus43200
minus306
0043800
18900
minus65200
minus55400
minus16
996
03741
15582
1364
6minus23453
minus14
311
2430000
012500
030000
0minus41800
minus45200
37600
13100
minus58600
minus59500
minus15
641
minus08404
10366
08632
minus18
027
minus17
517
2530000
015000
035000
0minus38900
minus58500
33200
05890
minus52300
minus61900
minus13
480
minus22716
07182
01455
minus13
269
minus21053
119887mdash
mdashmdash
mdashmdash
mdashmdash
mdashmdash
minus24124
minus34816
25241
03809
minus37541
minus39253
The Scientific World Journal 9
Com
pute
d di
spla
cem
ent u
sing
iden
tified
par
amet
ers
Monitored displacement (mm)
00
50
100
150
00 50 100 150minus150 minus100 minus50
minus150
minus100
minus50
(a) Stage 1
Com
pute
d di
spla
cem
ent u
sing
iden
tified
par
amet
ers
Monitored displacement (mm)
00
50
100
150
00 50 100 150minus150 minus100 minus50
minus150
minus100
minus50
(b) Stage 2
Stage 1Stage 2Stage 3
00
50
100
150
00 50 100 150
Com
pute
d di
spla
cem
ent u
sing
iden
tified
par
amet
ers
Monitored displacement (mm)minus150 minus100 minus50
minus150
minus100
minus50
(c) Stage 3
Figure 11 Comparison between monitored displacement and predicted displacement using identified parameters
43 Procedure of Back Analysis Algorithm Based on LSSVMand ABC After the LSSVM model representing the non-linear relation between the displacement and a parameteris obtained it can be used to predict displacements atmonitored points instead of numerical analysis ABC is usedto search the optimal parameter to be identified based on theerror function (see (10)) The back analysis technique basedon LSSVM-ABC combination can be described as follows
Step 1 Determine ABC parameters and the range ofparameters to be recognized
Step 2 Generate randomly 119899 group of parameters attheir given range Each individual represents an initialsolution
Step 3 Input a set of rock mass parameters to themodel LSSVM(X) obtained above to calculate thedisplacement values at given monitoring points
Step 4 Use (10) to evaluate the fitness of the currentindividuals that is the reasonability of the parameterset
10 The Scientific World Journal
0000e + 000
5000e + 000
1000e + 001
1500e + 001
2000e + 001
2500e + 001
3000e + 001
3500e + 001
4000e + 001
4500e + 001
5000e + 001
5500e + 001
6000e + 001
Use
r dat
a120590
XX
(a) 120590119909 using theory parameters
0000e + 000
5000e + 000
1000e + 001
1500e + 001
2000e + 001
2500e + 001
3000e + 001
3500e + 001
4000e + 001
4500e + 001
5000e + 001
5500e + 001
6000e + 001
Use
r dat
a120590
XX
(b) 120590119909 using identified parameters
Use
r dat
a120590
YY
0000e + 000
4000e + 000
8000e + 000
1200e + 001
1600e + 001
2000e + 001
2400e + 001
2800e + 001
3200e + 001
3600e + 001
4400e + 001
4000e + 001
4800e + 001
(c) 120590119910 using theory parameters
Use
r dat
a120590
YY
0000e + 000
4000e + 000
8000e + 000
1200e + 001
1600e + 001
2000e + 001
2400e + 001
2800e + 001
3200e + 001
3600e + 001
4400e + 001
4000e + 001
4800e + 001
(d) 120590119910 using identified parameters
Figure 12 Calculated stress comparison between using theory value and identified value at stage 3
Step 5 If all individuals are evaluated then go to Step6 Otherwise go to Step 3Step 6 If the maximum cycle is reached or the bestindividuals (the parameter to be back recognized)are obtained then the cycle ends and outputs bestindividuals Otherwise go to Step 7Step 7Update the individuals according to (2) and (3)Step 8 Repeat Step 7 until all 119899 new individuals aregenerated They are used as offspringStep 9 Go to Step 3
44 Verification To verify the model we suppose there isa tunnel (see Figure 7) The size of tunnel geomechanicalparameters and in situ stress are listed in Figure 7 The valuein Figure 7 is theoretical values Displacement values for somekey points indicated by nodes are calculated by elastic finiteelement method The suggested algorithm above is used toidentify initial geostress components P1 and P2 and anglebetween P1 and P2 We used orthogonal experiment design
to create 25 sets of tentative geostresses P1 and P2 and anglebetween P1 and P2 The training samples will be obtainedthrough computing the displacement of each set of tentativegeostresses Then the LSSVMmodel was build based on (13)The training samples and model parameters of LSSVM arelisted in Table 5 In situ stresses P1 and P2 and angle atdifferent stages can be identified according to the procedureof Section 43 Identified in situ stress P1 and P2and angleat different stages are listed in Table 4 The comparisonbetween displacement of the key points using the theoreticalparameters and displacements identified by back analysisbased on ABC and LSSVM is shown in Figure 8 Stresses ofsurrounding rock are shown in Figure 9 after stage 3 Resultsshow the proposed method can effectively identify the in situstress
45 Discussions
451 Performance of LSSVM The performance of LSSVM isvery important to back analysis The predicted displacement
The Scientific World Journal 11
00000
10000
20000
30000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
minus10000
minus20000
minus30000
minus40000
minus50000
(a) Stage 1
00000
20000
40000
60000
80000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
minus20000
minus60000
minus80000
minus100000
minus40000
MP4
-x
MP4
-y
MP5
-x
MP5
-y
(b) Stage 2
Monitored displacementPredicted displacement using LSSVMComputed displacement using FEM
00000
50000
100000
150000
minus50000
minus100000
minus150000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
MP4
-x
MP4
-y
MP5
-x
MP5
-y
MP6
-x
MP6
-y
MP7
-x
MP7
-y
(c) Stage 3
Figure 13 Predicted displacement using LSSVM with calculated displacement using theory and identified parameters
using LSSVM is in well agreement with the calculateddisplacement using theory and identified parameters (shownin Figure 10) It shows the LSSVM model presents wellthe relationship between geomechanical parameters anddisplacement It improves the efficiency of back analysis usingLSSVM
452 Effect of Kernel Parameters In this study the RBFkernel functionwas adoptedThe relationship between fitnessand cycle is listed in Figure 11 with 120590 = 10 and 120590 = 1 Theperformance of LSSVM is listed in Figure 12 using 120590 = 10 and120590 = 1 Its show selecting the appropriate kernel parametersis important to back analysis But there is not any guide toselect kernel function and its parameters according to LSSVMtheory It can be acquired by error-and-trial
5 Conclusions
The paper presents a new methodology called back analysisbased on ABC ABC is used to identify the geomechanicalparameters based on monitored displacements Results ofcircular tunnel with the analytical solution illustrate clearlythat ABC is effectively able to search parameters of geo-material and has proved ABC has powerful global optimalperformance To improve the efficiency of back analysisLSSVMwas used to present the relationship between geome-chanical parameters and displacement instead of numericalanalysis Results of horseshoe tunnel without the analyticalsolution demonstrate that LSSVMpresents well the nonlinearrelationship between geomechanical parameters and moni-tored displacements The proposed approach improves the
12 The Scientific World Journal
0
005
01
015
02
025
0 200 400 600 800 1000
Fitn
ess
Cycle
120590 = 10
120590 = 1
Figure 14 Fitness with different parameters of kernel function
00000
50000
100000
150000
00000 50000 100000 150000
Com
pute
d di
spla
cem
ent u
sing
FEM
bas
ed o
n LS
SVM
(mm
)
Monitored displacement (mm)
120590 = 10
120590 = 1
minus150000
minus100000
minus50000
minus150000 minus100000 minus50000
Figure 15The performance of LSSVMwith different parameters ofkernel function
efficiency and precision of back analysis andmakes it possibleto be applied to more complex engineering problem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was also supported by the National Fund ofScience in China (no 41072224 51104057)
References
[1] L Jing ldquoA review of techniques advances and outstandingissues in numerical modelling for rock mechanics and rockengineeringrdquo International Journal of Rock Mechanics andMining Sciences vol 40 no 3 pp 283ndash353 2003
[2] P Oreste ldquoBack-analysis techniques for the improvement ofthe understanding of rock in underground constructionsrdquoTunnelling and Underground Space Technology vol 20 no 1 pp7ndash21 2005
[3] G Gioda and L Jurina ldquoNumerical identification( back cal-culation) of soil-structure interaction pressuresrdquo InternationalJournal for Numerical amp Analytical Methods in Geomechanicsvol 5 no 1 pp 33ndash56 1981
[4] S Sakurai and K Takeuchi ldquoBack analysis of measured dis-placements of tunnelsrdquo Rock Mechanics and Rock Engineeringvol 16 no 3 pp 173ndash180 1983
[5] S Sakurai N Dees Wasmongkol and M Shinji ldquoBack analysisfor determining material characteristics in cut slopesrdquo inProceedings of the International Symposium on ECRF pp 770ndash776 Beijing China 1986
[6] S Sakurai ldquoInterpretation of the results of displacement mea-surements in cut slopesrdquo in Proceedings of the 2nd InternationalSymposium on Field Measurements in Geomechanics (FMGMrsquo87) pp 2528ndash2540 Kobe Japan 1987
[7] Z L Feng and R W Lewis ldquoOptimal estimation of in-situground stresses from displacement measurementsrdquo Interna-tional Journal for Numerical amp Analytical Methods in Geome-chanics vol 11 no 4 pp 391ndash408 1987
[8] B Pichler R Lackner and H A Mang ldquoBack analysis ofmodel parameters in geotechnical engineering by means ofsoft computingrdquo International Journal for Numerical Methods inEngineering vol 57 no 14 pp 1943ndash1978 2003
[9] F Xia-Ting and J A Hudson Rock Engineering Design CRCPress New York NY USA 2011
[10] T Okabe K Hayashi N Shinohara and S Takasugi ldquoInversionof drilling-induced tensile fracture data obtained from a singleinclined boreholerdquo International Journal of Rock Mechanics andMining Sciences vol 35 no 6 pp 747ndash758 1998
[11] W-G William and Y S Yoon ldquoAquifer parameter identifi-cation with optimum dimension in parameterizationrdquo WaterResources Research vol 17 no 3 pp 664ndash672 1981
[12] A Cividini G Maier and A Nappi ldquoParameter estimation ofa static geotechnical model using a Bayesrsquo approachrdquo Interna-tional Journal of Rock Mechanics and Mining Sciences vol 20no 5 pp 215ndash226 1983
[13] S VardakosM Gutierrez andC Xia ldquoParameter identificationin numerical modeling of tunneling using the DifferentialEvolution Genetic Algorithm (DEGA)rdquo Tunnelling and Under-ground Space Technology vol 28 no 1 pp 109ndash123 2012
[14] H Zhao and S Yin ldquoGeomechanical parameters identificationby particle swarm optimization and support vector machinerdquoApplied Mathematical Modelling vol 33 no 10 pp 3997ndash40122009
[15] X Feng H Zhao and S Li ldquoA new displacement backanalysis to identify mechanical geo-material parameters basedon hybrid intelligent methodologyrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 28 no11 pp 1141ndash1165 2004
[16] Y Yu B Zhang and H Yuan ldquoAn intelligent displacementback-analysis method for earth-rockfill damsrdquo Computers andGeotechnics vol 34 no 6 pp 423ndash434 2007
The Scientific World Journal 13
[17] J H Deng and C F Lee ldquoDiplacement back analysis for a steepslope at the Three Gorges Project siterdquo International Journal ofRockMechanics andMining Sciences vol 38 no 2 pp 259ndash2682001
[18] D Karaboga ldquoAn idea based on honey bee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University 2005
[19] D Karaboga and C Ozturk ldquoA novel clustering approachartificial Bee Colony (ABC) algorithmrdquoApplied Soft ComputingJournal vol 11 no 1 pp 652ndash657 2011
[20] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008
[21] J A K Suykens and J Vandewalle ldquoLeast squares supportvector machine classifiersrdquo Neural Processing Letters vol 9 no3 pp 293ndash300 1999
[22] M E Duncan Fama ldquoNumerical modeling of yield zones inweak rocksrdquo in Comprehensive Rock Engineering J A HudsonEd vol 2 pp 49ndash75 Pergamon Oxford UK 1993
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
2 The Scientific World Journal
geotechnical engineeringwithout analytical solution LSSVMwas used to present the relationship between geomechanicalparameters and displacement LSSVM model replaced thenumerical analysis to improve the efficiency of back analysisBack analysis based on LSSVM and ABC combination wasproposed in Section 4 LSSVM and the procedure of theproposed method were presented in brief Lastly someconclusion was listed in Section 5
2 Artificial Bee Colony Algorithms
The artificial bee colony (ABC) algorithm was originallydeveloped in 2005 by Karaboga [18] In ABC algorithmthe colony of artificial bees contains three groups of beesemployed bees onlookers and scouts Employed bees searchfor specific food sources (solution) and calculate the amountof nectars (fitness value) Onlooker bees choose a food sourcebased on the nectars shared by employed bees and determinethe source to be abandoned and allocate its employed beeas scout bees Scout bees randomly search for a new foodsource The position of a food source represents a possiblesolution for the problem under consideration and the nectaramount of a food source represents the quality of the solutionrepresented by the fitness value [19 20] To the minimumproblem the fitness can be computed by the target function
In the algorithm the first half of the colony consists ofemployed artificial bees and the second half constitutes theonlookersThe number of the employed bees or the onlookerbees is equal to the number of solutions in the population Atthe first step theABCgenerates a randomly distributed initialpopulation of SN solutions and calculates the fitness of eachsolution Consider
119909 (119894 119895) = 119909119895
min + rand (0 1) (119909119895max minus 119909119895
min) (1)
where 119909(119894 119895) is the candidate solution of problem 119894 =
1 2 1198781198732 and 1198781198732 denotes the size of population 119895 =1 2 119863 and 119863 is the dimension number of each solutionrand(01) is a random number between [0 1] 119909119894min and 119909
119894max
are the upper and lower bound of each solutionOnce initialization is completed the artificial bees are
used to conduct the search for the best food resource(solution) Procedures can be described as follows [20]
(i) Employed bees determine a food source within theneighborhood of the food source through their mem-ory
(ii) Employed bees share their information with onlook-ers within the hive and then the onlookers select oneof the food sources
(iii) Onlookers select a food source within the neighbor-hood of the food sources chosen by them to produceand exploit the new food resources
(iv) An employed bee of the sources that have beenabandoned by onlookers becomes a scout and startsto search for a new food source randomly
In the ABC algorithm a candidate food position can beproduced from the memory of bees which is defined as
V (119894 119895) = 119909 (119894 119895) + 120593119894119895 (119909 (119894 119895) minus 119909 (119896 119895)) (2)
where k used to be different from 119894 is randomly chosenindexes from 1 2 1198781198732 j is also randomly chosenindexes from 1 2 119863 and 120593ij is a random numberin [minus1 1] and controls the generation of neighbor foodsources around 119909(119894 119895) and represents the comparison oftwo food positions seen by a bee As can be seen from(2) the perturbation on the position 119909(119894 119895) decreases whenthe difference between the parameters of 119909(119894 119895) and 119909(119896 119895)decreases so that the step length is adaptively reduced
An artificial onlooker bee chooses a food source basedon the probability of food source The probability of beingselected for fitness pi can be expressed as
119901119894 =fitness119894
sum119878119873119899=1 fitness119899
(3)
where fitness119894 is the fitness of the solutionIn ABC algorithm a food source whose position cannot
be improved further through a predetermined number ofcycles is assumed to be abandoned by onlookers 119909(119894 119895) usedto represent the abandoned source is replaced with 1199091015840(119894 119895)that is a new food source the scout bees find which isconducted by (1)
Each candidate source position V(119894 119895) produced by 119909(119894 119895)can be evaluated using the comparison between 119909(119894 119895) and itsold source positionThe old food source will be replaced withthe new food source when it is equal to or better than the oldfood source Otherwise the old food source is retained in thememory
There are three control parameters in the ABC thenumber of food sources which is equal to the number ofemployed or onlooker bees (SN2) the value of limit and themaximum cycle number (MCN) The following is the briefprocedure of artificial bee colony (ABC) algorithm
Step 1 Determine the value of control parametersSN2 MCN and ldquolimitrdquo of ABC algorithmStep 2 Generate the initial population 119909(119894 119895) by (1)and evaluate the fitness of each solutionStep 3 Produce new solution V(119894 119895) for each employedbee by using (2) In the meantime the fitness isevaluatedStep 4 Calculate the probability 119901119894 for the solution119909(119894 119895) by (3)Step 5 Select a solution 119909(119894 119895) for each onlookerbee according to 119901119894 Then a new solution V(119894 119895) isgenerated by (2)Step 6 Calculate the fitnessStep 7 If there is an abandoned solution for the scoutit will be replaced by using a new solution which israndomly produced by (2)Step 8 Trace the best solutionStep 9 Repeat Steps 3 to 8 until the cycle reaches themaximum cycle number (MCN)
The Scientific World Journal 3
p0
p0
pi
rP
r0
PlasticElastic
Figure 1 A circular tunnel subjected to hydrostatic far field stressand uniform support pressure
3 ABC-Based Back Analyses
Optimization algorithm is critical to back analysis In thissection ABC-based back analysis was presented to identifythe geomechanical parameters of a circular tunnel withanalytical solution
31 The Analytical Solution of Circular Tunnel A circulartunnel is excavated in a continuous homogeneous isotropicinitially elastic rock mass and subjected to a hydrostatic farfield stress p0 and uniform support pressure pi as shown inFigure 1
According to the Mohr-Coulomb criterion the normalstress pcr at the plastic-elastic zone interface is given [21] asfollows
119901119888119903 =2119901119900 minus 120590119888
119896 + 1
119896 =
1 + sin1205931 minus sin120593
120590119888 =119888 (119896 minus 1)
tan120593
(4)
where 120593 is the friction angle and c is the cohesion If theuniform support pressure pi is less than the critical pressurepcr the plastic zone exists The plastic zone radius R is given[22] as follows
119877 = 119903119900 lowast [2 (119901119900 + 119904)
(119896 + 1) (119901119894 + 119904)]
1(119896minus1)
(5)
in which
119904 =
120590119888
119896 minus 1
(6)
and 119903119900 is the radius of the tunnel
The deformation of surrounding rock of tunnel is asfollows
Elastic zone
119906119903 =
(119901119900 sin120593 + 119888 sdot cos120593) (1198772119903)
2119866
(7)
Plastic zone
119906119903 =119903
2119866
sdot 120594 (8)
where E is the deformation modulus and 120583 is Poissonrsquos ratio
120594 = (2120583 minus 1) (119901119900 + 119888 sdot ctg120593)
+ (1 minus 120583) [(1198702119901 minus 1) (119870119901 + 119870119901119904)]
times (119901119894 + 119888 sdot ctg120593) (119877
119903119900
)
(119870119901minus1)
(
119877
119903
)
(119870119901119904+1)
+ [
(1 minus 120583) (119870119901119870119901119904 + 1)
(119870119901 + 119870119901119904)
sdot 120583]
times (119901119894 + 119888 sdot ctg120593) (119903
119903119900
)
(119870119901minus1)
119870119901119904 =(1 + sin120595119904)(1 minus sin120595119904)
119866 =
119864
2 (1 + 120583)
(9)
32 Error Function An error function in this work isdefined as the minimum error between the displacementspredicted by the analyticalmodel based identified parametersand the actualmeasured displacements It can be expressed as
fitness = radicsum119899119894=1 (119910119901119894 minus 119910119894)
2
119899
(10)
where n is the number of key points 119910119894 is the monitoreddisplacement of the ith key points and 119910119901119894 is the predicteddisplacement of ith key point
33 The Procedure of ABC-Based Back Analysis ABC-basedback analysis is combined ABC with the analytical solution(see (7) and (8)) ABC produces population of artificial beesincluding employer bees onlooker bees and scout bees Thefitness values can be computed by (10) The displacementof (10) can be computed by (7) and (8) Based on the ABCalgorithm the new population was produced ABC-basedback analysis algorithm can be described as follows (seeFigure 2)
Step 1 Collect the information of engineering such asgeology conditions and engineering size
4 The Scientific World Journal
Start
Determine engineering condition andselect the computing model
Initiate the ABC algorithm
Generate the initial population by (1) andcompute the displacement by (7) and (8)and the fitness of each solution by (10)
Generate the new population by (2) and(3) and compute the displacement by (7)
and (8) the fitness of each solution by (10)
Memorize the best solution
Maximum cycle meets
Get the geomechanical parameters
End
No
Yes
Figure 2 Flowchart of ABC-based back analysis
Step 2 Select the appropriate model according to theabove informationStep 3 Determine the error functionStep 4 Activate the ABC algorithm (see Section 2) toproduce the initial population 119909(119894 119895) by (1) Displace-ments are computed using (7) and (8)Step 5 The fitness of each solution is calculated by(10)Step 6 Generate the new population based on ABCalgorithm (see (2) and (3)) and compute the displace-ment (see (7) and (8))Step 7 Trace the best solution according to the ABCalgorithm
Table 1 Parameters of tunnel model
1199010(MPa) 119864 (MPa) 119888 (MPa) Φ (∘) 119901119894 (Mpa) 120595 (∘)300000 70000000 34500 300000 0 0
Table 2 Identified parameters using ABC-based back analysis
119864 (Mpa) 119888 (Mpa) 120593 (∘)ABC-based back analysis 689304951 35065 2999284Actual value 70000000 34500 300000Relative error () 15279 minus16377 00239
Step 8Repeat Steps 5 to 7 until finding the solution orreaching the maximum cycle
34 Verification The displacement of monitored point oftunnel can be computed by the above formula In this studysix monitored points were used in circular tunnel to monitorthe displacements at the horizontal direction for ABC searchThe distance between central of tunnel and 6 monitoredpoints is 10m 11m 13m 15m 17m and 21m respectively(see Figure ) The radius of tunnel is 1m The parameterof rock is listed in Table 1 ABC-based back analysis is usedto identify geomechanical parameters (eg Youngrsquos modulusE cohesion c and friction angle 120593) from displacements ofsix monitored points The recognized parameters and theirerror are listed in Table 2 The maximum relative error is16 It shows the recognized parameters agree well withthe real parameters The comparison between recognizedand real parameters about the displacement and stress ofsurrounding rock of tunnel is shown in Figures and Theresults show stresses and displacements of surrounding rockidentified by ABC are in well agreement with real stresses anddisplacements of surrounding rock and ABC is an excellentoptimization method The relationship between fitness andcycle is shown in Figure 3The relationship between identifiedparameters and cycle is shown in Figure 4They show that theperformance and convergence of ABC are good and quick foridentification of geomechanical parameters using ABC
341 Effect of Searching Range Theperformances of ABC aredemonstrated with different searching ranges (Table 3) Theresults of different searching ranges are shown in Figure 5To the smaller range the convergence is quicker than thebigger range But to the bigger range the fitness is the sameas the smaller range It shows ABC has strong capabilityof global searching and makes it possible to find the rockmass parameters in a big global space which enables theback analysis to be applied to more complex engineeringproblems
342 Effect of Population Size Population size is key param-eters of ABC To study the effect of the colony size on the
The Scientific World Journal 5
1 2 3 4 5 6
1m 13m
11m
15m 17m 21m
Figure 3 Position of monitored point in circular tunnel
0
02
04
06
08
1
12
1 2 3 4 5 6 7 8 9 10Distance to the center of tunnel (m)
Computed displacement using recognized parametersComputed displacement using actual parameters
Disp
lace
men
t (10
minus2
m)
Figure 4 The comparison of displacement between actual andrecognized parameters
convergence rate of theABC algorithm five different coloniesthat consisted of 20 50 100 200 and 400 bees were usedThefitness versus cycle numbers is shown in Figure 6 It can beseen that the convergence rates increasewith greater numbersof bees and population size of 200 or 400 bees is enough inthis study
4 Back Analysis Based on LSSVM and ABC
In the above section ABC-based back analysis was used tothe circular tunnel with analytical solution To the practicalengineering it is difficult to get the analytical solutionThe procedure with numerical solution is time-consumingRegression analysis is a good approach to build the rela-tion between geomechanical parameters and field moni-tored information In this study least square support vectormachine (LSSVM) was adopted to present the relationship
0
5
10
15
20
25
30
35
40
45
50
1 3 5 7 9
Stre
ss (M
Pa)
Distance to the center of tunnel (m)
Computed radial stress using recognized parametersComputed radial stress using actual parametersComputed tangential stress using recognized parametersComputed tangential stress using actual parameters
Figure 5 The comparison of stress between actual and recognizedparameters
0 200 400 600 800 1000
Fitn
ess
Cycle
100E minus 04
900E minus 05
800E minus 05
700E minus 05
600E minus 05
500E minus 05
400E minus 05
300E minus 05
200E minus 05
100E minus 05
000E + 00
Figure 6 Relationship between fitness value and cycle
between geomechanical parameters and displacement basedon numerical analysis
41 Least Square Support Vector Machine The least squaresupport vector machine (LSSVM) was originally developedby Suykens andVandewalle [21] Consider a given training setofN data points 119909119896 119910119896 (119896 = 1 2 119873)with input data xk isinRN and output yk isin r where RN is the N-dimensional vector
6 The Scientific World Journal
600000
620000
640000
660000
680000
700000
720000
0 200 400 600 800 1000
Fitn
ess
Cycle
E (MPa)
(a) 119864
200
250
300
350
400
450
500
550
0 200 400 600 800 1000
Fitn
ess
Cycle
c (MPa)
(b) 119888
2000
2200
2400
2600
2800
3000
3200
3400
0 200 400 600 800 1000
Fitn
ess
Cycle
120593 (∘)
(c) 120593
Figure 7 The variation of identified parameter with the cycle
space and r is the one-dimensional vector space Accordingto the LSSVM algorithm LSSVMmodel becomes
119910 (119909) =
119873
sum
119896=1
120572119896119870(119909 119909119896) + 119887 (11)
where 119870(119909 119909119896) is kernel functions and 120572 and b meet thefollowing equation
[
0 1119879
1 Ω + 120574minus1119868
] [
119887
120572] = [
0
119910] (12)
where 119910 = [1199101 119910119873] 1 = [1 1] 120572 = [1205721 120572119873]and Mercerrsquos theorem is applied within the Ω matrix
Ω=120593(119909119896)119879120593(119909119897) = 119896(119909119896 119909119897) 119896 119897 = 1 119873 Then the
analytical solution of 120572 and b is given by
[
119887
120572] = Φ
minus1[
0
119910] (13)
42 Representation of Nonlinear Relationship LSSVM is usedin this study to map the nonlinear relationship betweengeomechanical parameters such as Youngrsquos modulus cohe-sion geostress coefficients and monitored displacements
The Scientific World Journal 7
0
000005
00001
000015
00002
000025
0 200 400 600 800 1000
Fitn
ess
Cycle
Range 1Range 2Range 3
Figure 8The performance of ABCusing different searching ranges
0
000005
00001
000015
00002
0 200 400 600 800 1000
Fitn
ess
Cycle
SN2 = 20
SN2 = 50
SN2 = 100
SN2 = 200
SN2 = 400
Figure 9 The convergence of different population size
The mathematical model of least square support vectormachine is defined as
LSSVM (X) 119877119899 997888rarr 119877
Y = LSSVM (X) X = (1199091 1199092 119909119899)
Y = (1199101 1199102 119910119899)
(14)
0 15
minus5 10
minus5 0 5 0
5 10
10MPa 20MPa
30∘
Failure criterion Mohr-Coulomb
Youngrsquos modulus E 20000MPa
Cohesion c 105MPa
Friction angle 120593 35∘
Poissonrsquos ratio 120583 02
Figure 10 The cross section of tunnel and parameters
Table 3 The ranges of identified parameters
Range 1 Range 2 Range 3119864 (Mpa) [2000 12000] [4000 1000] [5000 8000]119888 (Mpa) [05 7] [1 6] [3 7]120593 (∘) [5 60] [10 50] [20 40]
Table 4 Identified in situ stress and angle in different stages
1198751 1198752 AngleActual 200000 100000 300000Stage 1 199583 100614 300104Stage 2 206493 108171 333676Stage 3 200252 100376 30623
where 119909119894(119894 = 1 2 119899) is geomechanical parameters forexample Youngrsquos modulus friction angle geostress coeffi-cients and so forth and 119910119894(119894 = 1 2 119899) is displacementsof the key points
In order to obtain LSSVM(X) a training process basedon the known data set is needed Necessary training samplesare created in this work by using numerical analysis (egFEM model) which is used to obtain displacements of rockmass of key points corresponding to the given set of tentativegeomechanical parameters The geomechanical parametersare defined as input of LSSVM The displacement is definedas output of LSSVM
8 The Scientific World Journal
Table5Training
samples
andmod
elparameterso
fLSSVM
Num
bero
fsamples
1198751(M
pa)
1198752(M
pa)
120593(∘ )
Disp
lacement
120572
MP1
MP2
MP3
MP1119909
MP1119910
MP2119909
MP2119910
MP3119909
MP3119910
119909119910
119909119910
119909119910
110000
0500
0020000
0minus08380
minus13
600
15500
minus00231
minus20200
minus15
100
14473
20149
minus08992
minus03815
15989
22484
210000
075
000
25000
0minus04990
minus23300
13900
minus006
87minus16
700
minus15
800
16424
08880
minus09801
minus03294
16348
19749
310000
010000
030000
0000
00minus31300
1400
0minus14
400
1400
0minus14
400
21479
02439
minus09786
minus16
870
49088
21843
412500
012500
035000
0000
00minus39100
17500
minus18
000
minus17
500
minus18
000
20307
minus03980
minus05684
minus18
560
14959
17655
515000
015000
040000
0minus000
01minus47000
20900
minus21600
minus21000
minus21700
200
40minus10
849
minus02202
minus21514
1240
414
127
615000
0500
0025000
0minus200
00minus14
700
20800
08610
minus31900
minus27200
02187
18194
minus03108
05215
03286
09391
715000
075
000
30000
0minus16
800
minus25600
18300
01890
minus27700
minus28200
05089
06915
minus05137
minus0119
105530
07972
815000
010000
035000
0minus12
300
minus34700
1740
0minus05740
minus24200
minus27500
06722
006
83minus05353
minus05142
07871
07058
915000
012500
040000
0minus064
20minus41900
18300
minus13
800
minus21800
minus25200
10483
minus03389
minus05100
minus10
033
10326
08752
1015000
015000
020000
0minus000
01minus47000
20900
minus21600
minus21000
minus21700
22964
minus12
063
minus04593
minus24334
16207
16580
1120000
0500
0030000
0minus34100
minus19
500
22700
18500
minus42500
minus43300
minus09584
14147
minus01741
13821
minus05279
minus04169
1220000
075
000
35000
0minus30700
minus32100
19200
1100
0minus360
00minus43700
minus04940
02093
minus046
0505409
01538
minus03109
1320000
010000
040000
0minus25800
minus42600
1740
002750
minus31500
minus43100
minus01430
minus06938
minus06499
minus01060
05071
minus03365
1420000
012500
020000
0minus12
600
minus36100
30300
minus07560
minus37300
minus29900
09442
minus00545
04200
minus09120
00125
07845
1520000
015000
025000
0minus09990
minus46500
27900
minus13
700
minus33400
minus31500
12917
minus11019
01791
minus16
037
044
3806994
1625000
0500
0035000
0minus50300
minus28100
22000
29600
minus53200
minus62900
minus23159
07126
minus02344
23232
minus14
578
minus20741
1725000
075
000
40000
0minus45700
minus43400
17200
20100
minus42700
minus62200
minus17
211
minus08042
minus07054
12559
minus03011
minus18
612
1825000
010000
020000
0minus25600
minus25500
40000
06760
minus53900
minus38400
minus006
8108422
12115
02427
minus13
147
01347
1925000
012500
025000
0minus25800
minus38300
35200
02050
minus48900
minus42900
minus01085
minus02739
07151
minus01537
minus08165
minus02612
2025000
015000
030000
0minus23100
minus50100
32000
minus03910
minus44100
minus45200
02387
minus13
780
05252
minus08035
minus04328
minus040
6121
30000
0500
0040000
0minus70
100
minus42700
19200
42500
minus63900
minus85200
minus44142
minus07485
minus05555
37206
minus25680
minus440
6722
30000
075
000
20000
0minus41800
minus15
000
51000
21700
minus74
200
minus48300
minus16
564
19159
23915
16943
minus34243
minus08180
2330000
010000
025000
0minus43200
minus306
0043800
18900
minus65200
minus55400
minus16
996
03741
15582
1364
6minus23453
minus14
311
2430000
012500
030000
0minus41800
minus45200
37600
13100
minus58600
minus59500
minus15
641
minus08404
10366
08632
minus18
027
minus17
517
2530000
015000
035000
0minus38900
minus58500
33200
05890
minus52300
minus61900
minus13
480
minus22716
07182
01455
minus13
269
minus21053
119887mdash
mdashmdash
mdashmdash
mdashmdash
mdashmdash
minus24124
minus34816
25241
03809
minus37541
minus39253
The Scientific World Journal 9
Com
pute
d di
spla
cem
ent u
sing
iden
tified
par
amet
ers
Monitored displacement (mm)
00
50
100
150
00 50 100 150minus150 minus100 minus50
minus150
minus100
minus50
(a) Stage 1
Com
pute
d di
spla
cem
ent u
sing
iden
tified
par
amet
ers
Monitored displacement (mm)
00
50
100
150
00 50 100 150minus150 minus100 minus50
minus150
minus100
minus50
(b) Stage 2
Stage 1Stage 2Stage 3
00
50
100
150
00 50 100 150
Com
pute
d di
spla
cem
ent u
sing
iden
tified
par
amet
ers
Monitored displacement (mm)minus150 minus100 minus50
minus150
minus100
minus50
(c) Stage 3
Figure 11 Comparison between monitored displacement and predicted displacement using identified parameters
43 Procedure of Back Analysis Algorithm Based on LSSVMand ABC After the LSSVM model representing the non-linear relation between the displacement and a parameteris obtained it can be used to predict displacements atmonitored points instead of numerical analysis ABC is usedto search the optimal parameter to be identified based on theerror function (see (10)) The back analysis technique basedon LSSVM-ABC combination can be described as follows
Step 1 Determine ABC parameters and the range ofparameters to be recognized
Step 2 Generate randomly 119899 group of parameters attheir given range Each individual represents an initialsolution
Step 3 Input a set of rock mass parameters to themodel LSSVM(X) obtained above to calculate thedisplacement values at given monitoring points
Step 4 Use (10) to evaluate the fitness of the currentindividuals that is the reasonability of the parameterset
10 The Scientific World Journal
0000e + 000
5000e + 000
1000e + 001
1500e + 001
2000e + 001
2500e + 001
3000e + 001
3500e + 001
4000e + 001
4500e + 001
5000e + 001
5500e + 001
6000e + 001
Use
r dat
a120590
XX
(a) 120590119909 using theory parameters
0000e + 000
5000e + 000
1000e + 001
1500e + 001
2000e + 001
2500e + 001
3000e + 001
3500e + 001
4000e + 001
4500e + 001
5000e + 001
5500e + 001
6000e + 001
Use
r dat
a120590
XX
(b) 120590119909 using identified parameters
Use
r dat
a120590
YY
0000e + 000
4000e + 000
8000e + 000
1200e + 001
1600e + 001
2000e + 001
2400e + 001
2800e + 001
3200e + 001
3600e + 001
4400e + 001
4000e + 001
4800e + 001
(c) 120590119910 using theory parameters
Use
r dat
a120590
YY
0000e + 000
4000e + 000
8000e + 000
1200e + 001
1600e + 001
2000e + 001
2400e + 001
2800e + 001
3200e + 001
3600e + 001
4400e + 001
4000e + 001
4800e + 001
(d) 120590119910 using identified parameters
Figure 12 Calculated stress comparison between using theory value and identified value at stage 3
Step 5 If all individuals are evaluated then go to Step6 Otherwise go to Step 3Step 6 If the maximum cycle is reached or the bestindividuals (the parameter to be back recognized)are obtained then the cycle ends and outputs bestindividuals Otherwise go to Step 7Step 7Update the individuals according to (2) and (3)Step 8 Repeat Step 7 until all 119899 new individuals aregenerated They are used as offspringStep 9 Go to Step 3
44 Verification To verify the model we suppose there isa tunnel (see Figure 7) The size of tunnel geomechanicalparameters and in situ stress are listed in Figure 7 The valuein Figure 7 is theoretical values Displacement values for somekey points indicated by nodes are calculated by elastic finiteelement method The suggested algorithm above is used toidentify initial geostress components P1 and P2 and anglebetween P1 and P2 We used orthogonal experiment design
to create 25 sets of tentative geostresses P1 and P2 and anglebetween P1 and P2 The training samples will be obtainedthrough computing the displacement of each set of tentativegeostresses Then the LSSVMmodel was build based on (13)The training samples and model parameters of LSSVM arelisted in Table 5 In situ stresses P1 and P2 and angle atdifferent stages can be identified according to the procedureof Section 43 Identified in situ stress P1 and P2and angleat different stages are listed in Table 4 The comparisonbetween displacement of the key points using the theoreticalparameters and displacements identified by back analysisbased on ABC and LSSVM is shown in Figure 8 Stresses ofsurrounding rock are shown in Figure 9 after stage 3 Resultsshow the proposed method can effectively identify the in situstress
45 Discussions
451 Performance of LSSVM The performance of LSSVM isvery important to back analysis The predicted displacement
The Scientific World Journal 11
00000
10000
20000
30000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
minus10000
minus20000
minus30000
minus40000
minus50000
(a) Stage 1
00000
20000
40000
60000
80000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
minus20000
minus60000
minus80000
minus100000
minus40000
MP4
-x
MP4
-y
MP5
-x
MP5
-y
(b) Stage 2
Monitored displacementPredicted displacement using LSSVMComputed displacement using FEM
00000
50000
100000
150000
minus50000
minus100000
minus150000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
MP4
-x
MP4
-y
MP5
-x
MP5
-y
MP6
-x
MP6
-y
MP7
-x
MP7
-y
(c) Stage 3
Figure 13 Predicted displacement using LSSVM with calculated displacement using theory and identified parameters
using LSSVM is in well agreement with the calculateddisplacement using theory and identified parameters (shownin Figure 10) It shows the LSSVM model presents wellthe relationship between geomechanical parameters anddisplacement It improves the efficiency of back analysis usingLSSVM
452 Effect of Kernel Parameters In this study the RBFkernel functionwas adoptedThe relationship between fitnessand cycle is listed in Figure 11 with 120590 = 10 and 120590 = 1 Theperformance of LSSVM is listed in Figure 12 using 120590 = 10 and120590 = 1 Its show selecting the appropriate kernel parametersis important to back analysis But there is not any guide toselect kernel function and its parameters according to LSSVMtheory It can be acquired by error-and-trial
5 Conclusions
The paper presents a new methodology called back analysisbased on ABC ABC is used to identify the geomechanicalparameters based on monitored displacements Results ofcircular tunnel with the analytical solution illustrate clearlythat ABC is effectively able to search parameters of geo-material and has proved ABC has powerful global optimalperformance To improve the efficiency of back analysisLSSVMwas used to present the relationship between geome-chanical parameters and displacement instead of numericalanalysis Results of horseshoe tunnel without the analyticalsolution demonstrate that LSSVMpresents well the nonlinearrelationship between geomechanical parameters and moni-tored displacements The proposed approach improves the
12 The Scientific World Journal
0
005
01
015
02
025
0 200 400 600 800 1000
Fitn
ess
Cycle
120590 = 10
120590 = 1
Figure 14 Fitness with different parameters of kernel function
00000
50000
100000
150000
00000 50000 100000 150000
Com
pute
d di
spla
cem
ent u
sing
FEM
bas
ed o
n LS
SVM
(mm
)
Monitored displacement (mm)
120590 = 10
120590 = 1
minus150000
minus100000
minus50000
minus150000 minus100000 minus50000
Figure 15The performance of LSSVMwith different parameters ofkernel function
efficiency and precision of back analysis andmakes it possibleto be applied to more complex engineering problem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was also supported by the National Fund ofScience in China (no 41072224 51104057)
References
[1] L Jing ldquoA review of techniques advances and outstandingissues in numerical modelling for rock mechanics and rockengineeringrdquo International Journal of Rock Mechanics andMining Sciences vol 40 no 3 pp 283ndash353 2003
[2] P Oreste ldquoBack-analysis techniques for the improvement ofthe understanding of rock in underground constructionsrdquoTunnelling and Underground Space Technology vol 20 no 1 pp7ndash21 2005
[3] G Gioda and L Jurina ldquoNumerical identification( back cal-culation) of soil-structure interaction pressuresrdquo InternationalJournal for Numerical amp Analytical Methods in Geomechanicsvol 5 no 1 pp 33ndash56 1981
[4] S Sakurai and K Takeuchi ldquoBack analysis of measured dis-placements of tunnelsrdquo Rock Mechanics and Rock Engineeringvol 16 no 3 pp 173ndash180 1983
[5] S Sakurai N Dees Wasmongkol and M Shinji ldquoBack analysisfor determining material characteristics in cut slopesrdquo inProceedings of the International Symposium on ECRF pp 770ndash776 Beijing China 1986
[6] S Sakurai ldquoInterpretation of the results of displacement mea-surements in cut slopesrdquo in Proceedings of the 2nd InternationalSymposium on Field Measurements in Geomechanics (FMGMrsquo87) pp 2528ndash2540 Kobe Japan 1987
[7] Z L Feng and R W Lewis ldquoOptimal estimation of in-situground stresses from displacement measurementsrdquo Interna-tional Journal for Numerical amp Analytical Methods in Geome-chanics vol 11 no 4 pp 391ndash408 1987
[8] B Pichler R Lackner and H A Mang ldquoBack analysis ofmodel parameters in geotechnical engineering by means ofsoft computingrdquo International Journal for Numerical Methods inEngineering vol 57 no 14 pp 1943ndash1978 2003
[9] F Xia-Ting and J A Hudson Rock Engineering Design CRCPress New York NY USA 2011
[10] T Okabe K Hayashi N Shinohara and S Takasugi ldquoInversionof drilling-induced tensile fracture data obtained from a singleinclined boreholerdquo International Journal of Rock Mechanics andMining Sciences vol 35 no 6 pp 747ndash758 1998
[11] W-G William and Y S Yoon ldquoAquifer parameter identifi-cation with optimum dimension in parameterizationrdquo WaterResources Research vol 17 no 3 pp 664ndash672 1981
[12] A Cividini G Maier and A Nappi ldquoParameter estimation ofa static geotechnical model using a Bayesrsquo approachrdquo Interna-tional Journal of Rock Mechanics and Mining Sciences vol 20no 5 pp 215ndash226 1983
[13] S VardakosM Gutierrez andC Xia ldquoParameter identificationin numerical modeling of tunneling using the DifferentialEvolution Genetic Algorithm (DEGA)rdquo Tunnelling and Under-ground Space Technology vol 28 no 1 pp 109ndash123 2012
[14] H Zhao and S Yin ldquoGeomechanical parameters identificationby particle swarm optimization and support vector machinerdquoApplied Mathematical Modelling vol 33 no 10 pp 3997ndash40122009
[15] X Feng H Zhao and S Li ldquoA new displacement backanalysis to identify mechanical geo-material parameters basedon hybrid intelligent methodologyrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 28 no11 pp 1141ndash1165 2004
[16] Y Yu B Zhang and H Yuan ldquoAn intelligent displacementback-analysis method for earth-rockfill damsrdquo Computers andGeotechnics vol 34 no 6 pp 423ndash434 2007
The Scientific World Journal 13
[17] J H Deng and C F Lee ldquoDiplacement back analysis for a steepslope at the Three Gorges Project siterdquo International Journal ofRockMechanics andMining Sciences vol 38 no 2 pp 259ndash2682001
[18] D Karaboga ldquoAn idea based on honey bee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University 2005
[19] D Karaboga and C Ozturk ldquoA novel clustering approachartificial Bee Colony (ABC) algorithmrdquoApplied Soft ComputingJournal vol 11 no 1 pp 652ndash657 2011
[20] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008
[21] J A K Suykens and J Vandewalle ldquoLeast squares supportvector machine classifiersrdquo Neural Processing Letters vol 9 no3 pp 293ndash300 1999
[22] M E Duncan Fama ldquoNumerical modeling of yield zones inweak rocksrdquo in Comprehensive Rock Engineering J A HudsonEd vol 2 pp 49ndash75 Pergamon Oxford UK 1993
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
The Scientific World Journal 3
p0
p0
pi
rP
r0
PlasticElastic
Figure 1 A circular tunnel subjected to hydrostatic far field stressand uniform support pressure
3 ABC-Based Back Analyses
Optimization algorithm is critical to back analysis In thissection ABC-based back analysis was presented to identifythe geomechanical parameters of a circular tunnel withanalytical solution
31 The Analytical Solution of Circular Tunnel A circulartunnel is excavated in a continuous homogeneous isotropicinitially elastic rock mass and subjected to a hydrostatic farfield stress p0 and uniform support pressure pi as shown inFigure 1
According to the Mohr-Coulomb criterion the normalstress pcr at the plastic-elastic zone interface is given [21] asfollows
119901119888119903 =2119901119900 minus 120590119888
119896 + 1
119896 =
1 + sin1205931 minus sin120593
120590119888 =119888 (119896 minus 1)
tan120593
(4)
where 120593 is the friction angle and c is the cohesion If theuniform support pressure pi is less than the critical pressurepcr the plastic zone exists The plastic zone radius R is given[22] as follows
119877 = 119903119900 lowast [2 (119901119900 + 119904)
(119896 + 1) (119901119894 + 119904)]
1(119896minus1)
(5)
in which
119904 =
120590119888
119896 minus 1
(6)
and 119903119900 is the radius of the tunnel
The deformation of surrounding rock of tunnel is asfollows
Elastic zone
119906119903 =
(119901119900 sin120593 + 119888 sdot cos120593) (1198772119903)
2119866
(7)
Plastic zone
119906119903 =119903
2119866
sdot 120594 (8)
where E is the deformation modulus and 120583 is Poissonrsquos ratio
120594 = (2120583 minus 1) (119901119900 + 119888 sdot ctg120593)
+ (1 minus 120583) [(1198702119901 minus 1) (119870119901 + 119870119901119904)]
times (119901119894 + 119888 sdot ctg120593) (119877
119903119900
)
(119870119901minus1)
(
119877
119903
)
(119870119901119904+1)
+ [
(1 minus 120583) (119870119901119870119901119904 + 1)
(119870119901 + 119870119901119904)
sdot 120583]
times (119901119894 + 119888 sdot ctg120593) (119903
119903119900
)
(119870119901minus1)
119870119901119904 =(1 + sin120595119904)(1 minus sin120595119904)
119866 =
119864
2 (1 + 120583)
(9)
32 Error Function An error function in this work isdefined as the minimum error between the displacementspredicted by the analyticalmodel based identified parametersand the actualmeasured displacements It can be expressed as
fitness = radicsum119899119894=1 (119910119901119894 minus 119910119894)
2
119899
(10)
where n is the number of key points 119910119894 is the monitoreddisplacement of the ith key points and 119910119901119894 is the predicteddisplacement of ith key point
33 The Procedure of ABC-Based Back Analysis ABC-basedback analysis is combined ABC with the analytical solution(see (7) and (8)) ABC produces population of artificial beesincluding employer bees onlooker bees and scout bees Thefitness values can be computed by (10) The displacementof (10) can be computed by (7) and (8) Based on the ABCalgorithm the new population was produced ABC-basedback analysis algorithm can be described as follows (seeFigure 2)
Step 1 Collect the information of engineering such asgeology conditions and engineering size
4 The Scientific World Journal
Start
Determine engineering condition andselect the computing model
Initiate the ABC algorithm
Generate the initial population by (1) andcompute the displacement by (7) and (8)and the fitness of each solution by (10)
Generate the new population by (2) and(3) and compute the displacement by (7)
and (8) the fitness of each solution by (10)
Memorize the best solution
Maximum cycle meets
Get the geomechanical parameters
End
No
Yes
Figure 2 Flowchart of ABC-based back analysis
Step 2 Select the appropriate model according to theabove informationStep 3 Determine the error functionStep 4 Activate the ABC algorithm (see Section 2) toproduce the initial population 119909(119894 119895) by (1) Displace-ments are computed using (7) and (8)Step 5 The fitness of each solution is calculated by(10)Step 6 Generate the new population based on ABCalgorithm (see (2) and (3)) and compute the displace-ment (see (7) and (8))Step 7 Trace the best solution according to the ABCalgorithm
Table 1 Parameters of tunnel model
1199010(MPa) 119864 (MPa) 119888 (MPa) Φ (∘) 119901119894 (Mpa) 120595 (∘)300000 70000000 34500 300000 0 0
Table 2 Identified parameters using ABC-based back analysis
119864 (Mpa) 119888 (Mpa) 120593 (∘)ABC-based back analysis 689304951 35065 2999284Actual value 70000000 34500 300000Relative error () 15279 minus16377 00239
Step 8Repeat Steps 5 to 7 until finding the solution orreaching the maximum cycle
34 Verification The displacement of monitored point oftunnel can be computed by the above formula In this studysix monitored points were used in circular tunnel to monitorthe displacements at the horizontal direction for ABC searchThe distance between central of tunnel and 6 monitoredpoints is 10m 11m 13m 15m 17m and 21m respectively(see Figure ) The radius of tunnel is 1m The parameterof rock is listed in Table 1 ABC-based back analysis is usedto identify geomechanical parameters (eg Youngrsquos modulusE cohesion c and friction angle 120593) from displacements ofsix monitored points The recognized parameters and theirerror are listed in Table 2 The maximum relative error is16 It shows the recognized parameters agree well withthe real parameters The comparison between recognizedand real parameters about the displacement and stress ofsurrounding rock of tunnel is shown in Figures and Theresults show stresses and displacements of surrounding rockidentified by ABC are in well agreement with real stresses anddisplacements of surrounding rock and ABC is an excellentoptimization method The relationship between fitness andcycle is shown in Figure 3The relationship between identifiedparameters and cycle is shown in Figure 4They show that theperformance and convergence of ABC are good and quick foridentification of geomechanical parameters using ABC
341 Effect of Searching Range Theperformances of ABC aredemonstrated with different searching ranges (Table 3) Theresults of different searching ranges are shown in Figure 5To the smaller range the convergence is quicker than thebigger range But to the bigger range the fitness is the sameas the smaller range It shows ABC has strong capabilityof global searching and makes it possible to find the rockmass parameters in a big global space which enables theback analysis to be applied to more complex engineeringproblems
342 Effect of Population Size Population size is key param-eters of ABC To study the effect of the colony size on the
The Scientific World Journal 5
1 2 3 4 5 6
1m 13m
11m
15m 17m 21m
Figure 3 Position of monitored point in circular tunnel
0
02
04
06
08
1
12
1 2 3 4 5 6 7 8 9 10Distance to the center of tunnel (m)
Computed displacement using recognized parametersComputed displacement using actual parameters
Disp
lace
men
t (10
minus2
m)
Figure 4 The comparison of displacement between actual andrecognized parameters
convergence rate of theABC algorithm five different coloniesthat consisted of 20 50 100 200 and 400 bees were usedThefitness versus cycle numbers is shown in Figure 6 It can beseen that the convergence rates increasewith greater numbersof bees and population size of 200 or 400 bees is enough inthis study
4 Back Analysis Based on LSSVM and ABC
In the above section ABC-based back analysis was used tothe circular tunnel with analytical solution To the practicalengineering it is difficult to get the analytical solutionThe procedure with numerical solution is time-consumingRegression analysis is a good approach to build the rela-tion between geomechanical parameters and field moni-tored information In this study least square support vectormachine (LSSVM) was adopted to present the relationship
0
5
10
15
20
25
30
35
40
45
50
1 3 5 7 9
Stre
ss (M
Pa)
Distance to the center of tunnel (m)
Computed radial stress using recognized parametersComputed radial stress using actual parametersComputed tangential stress using recognized parametersComputed tangential stress using actual parameters
Figure 5 The comparison of stress between actual and recognizedparameters
0 200 400 600 800 1000
Fitn
ess
Cycle
100E minus 04
900E minus 05
800E minus 05
700E minus 05
600E minus 05
500E minus 05
400E minus 05
300E minus 05
200E minus 05
100E minus 05
000E + 00
Figure 6 Relationship between fitness value and cycle
between geomechanical parameters and displacement basedon numerical analysis
41 Least Square Support Vector Machine The least squaresupport vector machine (LSSVM) was originally developedby Suykens andVandewalle [21] Consider a given training setofN data points 119909119896 119910119896 (119896 = 1 2 119873)with input data xk isinRN and output yk isin r where RN is the N-dimensional vector
6 The Scientific World Journal
600000
620000
640000
660000
680000
700000
720000
0 200 400 600 800 1000
Fitn
ess
Cycle
E (MPa)
(a) 119864
200
250
300
350
400
450
500
550
0 200 400 600 800 1000
Fitn
ess
Cycle
c (MPa)
(b) 119888
2000
2200
2400
2600
2800
3000
3200
3400
0 200 400 600 800 1000
Fitn
ess
Cycle
120593 (∘)
(c) 120593
Figure 7 The variation of identified parameter with the cycle
space and r is the one-dimensional vector space Accordingto the LSSVM algorithm LSSVMmodel becomes
119910 (119909) =
119873
sum
119896=1
120572119896119870(119909 119909119896) + 119887 (11)
where 119870(119909 119909119896) is kernel functions and 120572 and b meet thefollowing equation
[
0 1119879
1 Ω + 120574minus1119868
] [
119887
120572] = [
0
119910] (12)
where 119910 = [1199101 119910119873] 1 = [1 1] 120572 = [1205721 120572119873]and Mercerrsquos theorem is applied within the Ω matrix
Ω=120593(119909119896)119879120593(119909119897) = 119896(119909119896 119909119897) 119896 119897 = 1 119873 Then the
analytical solution of 120572 and b is given by
[
119887
120572] = Φ
minus1[
0
119910] (13)
42 Representation of Nonlinear Relationship LSSVM is usedin this study to map the nonlinear relationship betweengeomechanical parameters such as Youngrsquos modulus cohe-sion geostress coefficients and monitored displacements
The Scientific World Journal 7
0
000005
00001
000015
00002
000025
0 200 400 600 800 1000
Fitn
ess
Cycle
Range 1Range 2Range 3
Figure 8The performance of ABCusing different searching ranges
0
000005
00001
000015
00002
0 200 400 600 800 1000
Fitn
ess
Cycle
SN2 = 20
SN2 = 50
SN2 = 100
SN2 = 200
SN2 = 400
Figure 9 The convergence of different population size
The mathematical model of least square support vectormachine is defined as
LSSVM (X) 119877119899 997888rarr 119877
Y = LSSVM (X) X = (1199091 1199092 119909119899)
Y = (1199101 1199102 119910119899)
(14)
0 15
minus5 10
minus5 0 5 0
5 10
10MPa 20MPa
30∘
Failure criterion Mohr-Coulomb
Youngrsquos modulus E 20000MPa
Cohesion c 105MPa
Friction angle 120593 35∘
Poissonrsquos ratio 120583 02
Figure 10 The cross section of tunnel and parameters
Table 3 The ranges of identified parameters
Range 1 Range 2 Range 3119864 (Mpa) [2000 12000] [4000 1000] [5000 8000]119888 (Mpa) [05 7] [1 6] [3 7]120593 (∘) [5 60] [10 50] [20 40]
Table 4 Identified in situ stress and angle in different stages
1198751 1198752 AngleActual 200000 100000 300000Stage 1 199583 100614 300104Stage 2 206493 108171 333676Stage 3 200252 100376 30623
where 119909119894(119894 = 1 2 119899) is geomechanical parameters forexample Youngrsquos modulus friction angle geostress coeffi-cients and so forth and 119910119894(119894 = 1 2 119899) is displacementsof the key points
In order to obtain LSSVM(X) a training process basedon the known data set is needed Necessary training samplesare created in this work by using numerical analysis (egFEM model) which is used to obtain displacements of rockmass of key points corresponding to the given set of tentativegeomechanical parameters The geomechanical parametersare defined as input of LSSVM The displacement is definedas output of LSSVM
8 The Scientific World Journal
Table5Training
samples
andmod
elparameterso
fLSSVM
Num
bero
fsamples
1198751(M
pa)
1198752(M
pa)
120593(∘ )
Disp
lacement
120572
MP1
MP2
MP3
MP1119909
MP1119910
MP2119909
MP2119910
MP3119909
MP3119910
119909119910
119909119910
119909119910
110000
0500
0020000
0minus08380
minus13
600
15500
minus00231
minus20200
minus15
100
14473
20149
minus08992
minus03815
15989
22484
210000
075
000
25000
0minus04990
minus23300
13900
minus006
87minus16
700
minus15
800
16424
08880
minus09801
minus03294
16348
19749
310000
010000
030000
0000
00minus31300
1400
0minus14
400
1400
0minus14
400
21479
02439
minus09786
minus16
870
49088
21843
412500
012500
035000
0000
00minus39100
17500
minus18
000
minus17
500
minus18
000
20307
minus03980
minus05684
minus18
560
14959
17655
515000
015000
040000
0minus000
01minus47000
20900
minus21600
minus21000
minus21700
200
40minus10
849
minus02202
minus21514
1240
414
127
615000
0500
0025000
0minus200
00minus14
700
20800
08610
minus31900
minus27200
02187
18194
minus03108
05215
03286
09391
715000
075
000
30000
0minus16
800
minus25600
18300
01890
minus27700
minus28200
05089
06915
minus05137
minus0119
105530
07972
815000
010000
035000
0minus12
300
minus34700
1740
0minus05740
minus24200
minus27500
06722
006
83minus05353
minus05142
07871
07058
915000
012500
040000
0minus064
20minus41900
18300
minus13
800
minus21800
minus25200
10483
minus03389
minus05100
minus10
033
10326
08752
1015000
015000
020000
0minus000
01minus47000
20900
minus21600
minus21000
minus21700
22964
minus12
063
minus04593
minus24334
16207
16580
1120000
0500
0030000
0minus34100
minus19
500
22700
18500
minus42500
minus43300
minus09584
14147
minus01741
13821
minus05279
minus04169
1220000
075
000
35000
0minus30700
minus32100
19200
1100
0minus360
00minus43700
minus04940
02093
minus046
0505409
01538
minus03109
1320000
010000
040000
0minus25800
minus42600
1740
002750
minus31500
minus43100
minus01430
minus06938
minus06499
minus01060
05071
minus03365
1420000
012500
020000
0minus12
600
minus36100
30300
minus07560
minus37300
minus29900
09442
minus00545
04200
minus09120
00125
07845
1520000
015000
025000
0minus09990
minus46500
27900
minus13
700
minus33400
minus31500
12917
minus11019
01791
minus16
037
044
3806994
1625000
0500
0035000
0minus50300
minus28100
22000
29600
minus53200
minus62900
minus23159
07126
minus02344
23232
minus14
578
minus20741
1725000
075
000
40000
0minus45700
minus43400
17200
20100
minus42700
minus62200
minus17
211
minus08042
minus07054
12559
minus03011
minus18
612
1825000
010000
020000
0minus25600
minus25500
40000
06760
minus53900
minus38400
minus006
8108422
12115
02427
minus13
147
01347
1925000
012500
025000
0minus25800
minus38300
35200
02050
minus48900
minus42900
minus01085
minus02739
07151
minus01537
minus08165
minus02612
2025000
015000
030000
0minus23100
minus50100
32000
minus03910
minus44100
minus45200
02387
minus13
780
05252
minus08035
minus04328
minus040
6121
30000
0500
0040000
0minus70
100
minus42700
19200
42500
minus63900
minus85200
minus44142
minus07485
minus05555
37206
minus25680
minus440
6722
30000
075
000
20000
0minus41800
minus15
000
51000
21700
minus74
200
minus48300
minus16
564
19159
23915
16943
minus34243
minus08180
2330000
010000
025000
0minus43200
minus306
0043800
18900
minus65200
minus55400
minus16
996
03741
15582
1364
6minus23453
minus14
311
2430000
012500
030000
0minus41800
minus45200
37600
13100
minus58600
minus59500
minus15
641
minus08404
10366
08632
minus18
027
minus17
517
2530000
015000
035000
0minus38900
minus58500
33200
05890
minus52300
minus61900
minus13
480
minus22716
07182
01455
minus13
269
minus21053
119887mdash
mdashmdash
mdashmdash
mdashmdash
mdashmdash
minus24124
minus34816
25241
03809
minus37541
minus39253
The Scientific World Journal 9
Com
pute
d di
spla
cem
ent u
sing
iden
tified
par
amet
ers
Monitored displacement (mm)
00
50
100
150
00 50 100 150minus150 minus100 minus50
minus150
minus100
minus50
(a) Stage 1
Com
pute
d di
spla
cem
ent u
sing
iden
tified
par
amet
ers
Monitored displacement (mm)
00
50
100
150
00 50 100 150minus150 minus100 minus50
minus150
minus100
minus50
(b) Stage 2
Stage 1Stage 2Stage 3
00
50
100
150
00 50 100 150
Com
pute
d di
spla
cem
ent u
sing
iden
tified
par
amet
ers
Monitored displacement (mm)minus150 minus100 minus50
minus150
minus100
minus50
(c) Stage 3
Figure 11 Comparison between monitored displacement and predicted displacement using identified parameters
43 Procedure of Back Analysis Algorithm Based on LSSVMand ABC After the LSSVM model representing the non-linear relation between the displacement and a parameteris obtained it can be used to predict displacements atmonitored points instead of numerical analysis ABC is usedto search the optimal parameter to be identified based on theerror function (see (10)) The back analysis technique basedon LSSVM-ABC combination can be described as follows
Step 1 Determine ABC parameters and the range ofparameters to be recognized
Step 2 Generate randomly 119899 group of parameters attheir given range Each individual represents an initialsolution
Step 3 Input a set of rock mass parameters to themodel LSSVM(X) obtained above to calculate thedisplacement values at given monitoring points
Step 4 Use (10) to evaluate the fitness of the currentindividuals that is the reasonability of the parameterset
10 The Scientific World Journal
0000e + 000
5000e + 000
1000e + 001
1500e + 001
2000e + 001
2500e + 001
3000e + 001
3500e + 001
4000e + 001
4500e + 001
5000e + 001
5500e + 001
6000e + 001
Use
r dat
a120590
XX
(a) 120590119909 using theory parameters
0000e + 000
5000e + 000
1000e + 001
1500e + 001
2000e + 001
2500e + 001
3000e + 001
3500e + 001
4000e + 001
4500e + 001
5000e + 001
5500e + 001
6000e + 001
Use
r dat
a120590
XX
(b) 120590119909 using identified parameters
Use
r dat
a120590
YY
0000e + 000
4000e + 000
8000e + 000
1200e + 001
1600e + 001
2000e + 001
2400e + 001
2800e + 001
3200e + 001
3600e + 001
4400e + 001
4000e + 001
4800e + 001
(c) 120590119910 using theory parameters
Use
r dat
a120590
YY
0000e + 000
4000e + 000
8000e + 000
1200e + 001
1600e + 001
2000e + 001
2400e + 001
2800e + 001
3200e + 001
3600e + 001
4400e + 001
4000e + 001
4800e + 001
(d) 120590119910 using identified parameters
Figure 12 Calculated stress comparison between using theory value and identified value at stage 3
Step 5 If all individuals are evaluated then go to Step6 Otherwise go to Step 3Step 6 If the maximum cycle is reached or the bestindividuals (the parameter to be back recognized)are obtained then the cycle ends and outputs bestindividuals Otherwise go to Step 7Step 7Update the individuals according to (2) and (3)Step 8 Repeat Step 7 until all 119899 new individuals aregenerated They are used as offspringStep 9 Go to Step 3
44 Verification To verify the model we suppose there isa tunnel (see Figure 7) The size of tunnel geomechanicalparameters and in situ stress are listed in Figure 7 The valuein Figure 7 is theoretical values Displacement values for somekey points indicated by nodes are calculated by elastic finiteelement method The suggested algorithm above is used toidentify initial geostress components P1 and P2 and anglebetween P1 and P2 We used orthogonal experiment design
to create 25 sets of tentative geostresses P1 and P2 and anglebetween P1 and P2 The training samples will be obtainedthrough computing the displacement of each set of tentativegeostresses Then the LSSVMmodel was build based on (13)The training samples and model parameters of LSSVM arelisted in Table 5 In situ stresses P1 and P2 and angle atdifferent stages can be identified according to the procedureof Section 43 Identified in situ stress P1 and P2and angleat different stages are listed in Table 4 The comparisonbetween displacement of the key points using the theoreticalparameters and displacements identified by back analysisbased on ABC and LSSVM is shown in Figure 8 Stresses ofsurrounding rock are shown in Figure 9 after stage 3 Resultsshow the proposed method can effectively identify the in situstress
45 Discussions
451 Performance of LSSVM The performance of LSSVM isvery important to back analysis The predicted displacement
The Scientific World Journal 11
00000
10000
20000
30000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
minus10000
minus20000
minus30000
minus40000
minus50000
(a) Stage 1
00000
20000
40000
60000
80000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
minus20000
minus60000
minus80000
minus100000
minus40000
MP4
-x
MP4
-y
MP5
-x
MP5
-y
(b) Stage 2
Monitored displacementPredicted displacement using LSSVMComputed displacement using FEM
00000
50000
100000
150000
minus50000
minus100000
minus150000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
MP4
-x
MP4
-y
MP5
-x
MP5
-y
MP6
-x
MP6
-y
MP7
-x
MP7
-y
(c) Stage 3
Figure 13 Predicted displacement using LSSVM with calculated displacement using theory and identified parameters
using LSSVM is in well agreement with the calculateddisplacement using theory and identified parameters (shownin Figure 10) It shows the LSSVM model presents wellthe relationship between geomechanical parameters anddisplacement It improves the efficiency of back analysis usingLSSVM
452 Effect of Kernel Parameters In this study the RBFkernel functionwas adoptedThe relationship between fitnessand cycle is listed in Figure 11 with 120590 = 10 and 120590 = 1 Theperformance of LSSVM is listed in Figure 12 using 120590 = 10 and120590 = 1 Its show selecting the appropriate kernel parametersis important to back analysis But there is not any guide toselect kernel function and its parameters according to LSSVMtheory It can be acquired by error-and-trial
5 Conclusions
The paper presents a new methodology called back analysisbased on ABC ABC is used to identify the geomechanicalparameters based on monitored displacements Results ofcircular tunnel with the analytical solution illustrate clearlythat ABC is effectively able to search parameters of geo-material and has proved ABC has powerful global optimalperformance To improve the efficiency of back analysisLSSVMwas used to present the relationship between geome-chanical parameters and displacement instead of numericalanalysis Results of horseshoe tunnel without the analyticalsolution demonstrate that LSSVMpresents well the nonlinearrelationship between geomechanical parameters and moni-tored displacements The proposed approach improves the
12 The Scientific World Journal
0
005
01
015
02
025
0 200 400 600 800 1000
Fitn
ess
Cycle
120590 = 10
120590 = 1
Figure 14 Fitness with different parameters of kernel function
00000
50000
100000
150000
00000 50000 100000 150000
Com
pute
d di
spla
cem
ent u
sing
FEM
bas
ed o
n LS
SVM
(mm
)
Monitored displacement (mm)
120590 = 10
120590 = 1
minus150000
minus100000
minus50000
minus150000 minus100000 minus50000
Figure 15The performance of LSSVMwith different parameters ofkernel function
efficiency and precision of back analysis andmakes it possibleto be applied to more complex engineering problem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was also supported by the National Fund ofScience in China (no 41072224 51104057)
References
[1] L Jing ldquoA review of techniques advances and outstandingissues in numerical modelling for rock mechanics and rockengineeringrdquo International Journal of Rock Mechanics andMining Sciences vol 40 no 3 pp 283ndash353 2003
[2] P Oreste ldquoBack-analysis techniques for the improvement ofthe understanding of rock in underground constructionsrdquoTunnelling and Underground Space Technology vol 20 no 1 pp7ndash21 2005
[3] G Gioda and L Jurina ldquoNumerical identification( back cal-culation) of soil-structure interaction pressuresrdquo InternationalJournal for Numerical amp Analytical Methods in Geomechanicsvol 5 no 1 pp 33ndash56 1981
[4] S Sakurai and K Takeuchi ldquoBack analysis of measured dis-placements of tunnelsrdquo Rock Mechanics and Rock Engineeringvol 16 no 3 pp 173ndash180 1983
[5] S Sakurai N Dees Wasmongkol and M Shinji ldquoBack analysisfor determining material characteristics in cut slopesrdquo inProceedings of the International Symposium on ECRF pp 770ndash776 Beijing China 1986
[6] S Sakurai ldquoInterpretation of the results of displacement mea-surements in cut slopesrdquo in Proceedings of the 2nd InternationalSymposium on Field Measurements in Geomechanics (FMGMrsquo87) pp 2528ndash2540 Kobe Japan 1987
[7] Z L Feng and R W Lewis ldquoOptimal estimation of in-situground stresses from displacement measurementsrdquo Interna-tional Journal for Numerical amp Analytical Methods in Geome-chanics vol 11 no 4 pp 391ndash408 1987
[8] B Pichler R Lackner and H A Mang ldquoBack analysis ofmodel parameters in geotechnical engineering by means ofsoft computingrdquo International Journal for Numerical Methods inEngineering vol 57 no 14 pp 1943ndash1978 2003
[9] F Xia-Ting and J A Hudson Rock Engineering Design CRCPress New York NY USA 2011
[10] T Okabe K Hayashi N Shinohara and S Takasugi ldquoInversionof drilling-induced tensile fracture data obtained from a singleinclined boreholerdquo International Journal of Rock Mechanics andMining Sciences vol 35 no 6 pp 747ndash758 1998
[11] W-G William and Y S Yoon ldquoAquifer parameter identifi-cation with optimum dimension in parameterizationrdquo WaterResources Research vol 17 no 3 pp 664ndash672 1981
[12] A Cividini G Maier and A Nappi ldquoParameter estimation ofa static geotechnical model using a Bayesrsquo approachrdquo Interna-tional Journal of Rock Mechanics and Mining Sciences vol 20no 5 pp 215ndash226 1983
[13] S VardakosM Gutierrez andC Xia ldquoParameter identificationin numerical modeling of tunneling using the DifferentialEvolution Genetic Algorithm (DEGA)rdquo Tunnelling and Under-ground Space Technology vol 28 no 1 pp 109ndash123 2012
[14] H Zhao and S Yin ldquoGeomechanical parameters identificationby particle swarm optimization and support vector machinerdquoApplied Mathematical Modelling vol 33 no 10 pp 3997ndash40122009
[15] X Feng H Zhao and S Li ldquoA new displacement backanalysis to identify mechanical geo-material parameters basedon hybrid intelligent methodologyrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 28 no11 pp 1141ndash1165 2004
[16] Y Yu B Zhang and H Yuan ldquoAn intelligent displacementback-analysis method for earth-rockfill damsrdquo Computers andGeotechnics vol 34 no 6 pp 423ndash434 2007
The Scientific World Journal 13
[17] J H Deng and C F Lee ldquoDiplacement back analysis for a steepslope at the Three Gorges Project siterdquo International Journal ofRockMechanics andMining Sciences vol 38 no 2 pp 259ndash2682001
[18] D Karaboga ldquoAn idea based on honey bee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University 2005
[19] D Karaboga and C Ozturk ldquoA novel clustering approachartificial Bee Colony (ABC) algorithmrdquoApplied Soft ComputingJournal vol 11 no 1 pp 652ndash657 2011
[20] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008
[21] J A K Suykens and J Vandewalle ldquoLeast squares supportvector machine classifiersrdquo Neural Processing Letters vol 9 no3 pp 293ndash300 1999
[22] M E Duncan Fama ldquoNumerical modeling of yield zones inweak rocksrdquo in Comprehensive Rock Engineering J A HudsonEd vol 2 pp 49ndash75 Pergamon Oxford UK 1993
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
4 The Scientific World Journal
Start
Determine engineering condition andselect the computing model
Initiate the ABC algorithm
Generate the initial population by (1) andcompute the displacement by (7) and (8)and the fitness of each solution by (10)
Generate the new population by (2) and(3) and compute the displacement by (7)
and (8) the fitness of each solution by (10)
Memorize the best solution
Maximum cycle meets
Get the geomechanical parameters
End
No
Yes
Figure 2 Flowchart of ABC-based back analysis
Step 2 Select the appropriate model according to theabove informationStep 3 Determine the error functionStep 4 Activate the ABC algorithm (see Section 2) toproduce the initial population 119909(119894 119895) by (1) Displace-ments are computed using (7) and (8)Step 5 The fitness of each solution is calculated by(10)Step 6 Generate the new population based on ABCalgorithm (see (2) and (3)) and compute the displace-ment (see (7) and (8))Step 7 Trace the best solution according to the ABCalgorithm
Table 1 Parameters of tunnel model
1199010(MPa) 119864 (MPa) 119888 (MPa) Φ (∘) 119901119894 (Mpa) 120595 (∘)300000 70000000 34500 300000 0 0
Table 2 Identified parameters using ABC-based back analysis
119864 (Mpa) 119888 (Mpa) 120593 (∘)ABC-based back analysis 689304951 35065 2999284Actual value 70000000 34500 300000Relative error () 15279 minus16377 00239
Step 8Repeat Steps 5 to 7 until finding the solution orreaching the maximum cycle
34 Verification The displacement of monitored point oftunnel can be computed by the above formula In this studysix monitored points were used in circular tunnel to monitorthe displacements at the horizontal direction for ABC searchThe distance between central of tunnel and 6 monitoredpoints is 10m 11m 13m 15m 17m and 21m respectively(see Figure ) The radius of tunnel is 1m The parameterof rock is listed in Table 1 ABC-based back analysis is usedto identify geomechanical parameters (eg Youngrsquos modulusE cohesion c and friction angle 120593) from displacements ofsix monitored points The recognized parameters and theirerror are listed in Table 2 The maximum relative error is16 It shows the recognized parameters agree well withthe real parameters The comparison between recognizedand real parameters about the displacement and stress ofsurrounding rock of tunnel is shown in Figures and Theresults show stresses and displacements of surrounding rockidentified by ABC are in well agreement with real stresses anddisplacements of surrounding rock and ABC is an excellentoptimization method The relationship between fitness andcycle is shown in Figure 3The relationship between identifiedparameters and cycle is shown in Figure 4They show that theperformance and convergence of ABC are good and quick foridentification of geomechanical parameters using ABC
341 Effect of Searching Range Theperformances of ABC aredemonstrated with different searching ranges (Table 3) Theresults of different searching ranges are shown in Figure 5To the smaller range the convergence is quicker than thebigger range But to the bigger range the fitness is the sameas the smaller range It shows ABC has strong capabilityof global searching and makes it possible to find the rockmass parameters in a big global space which enables theback analysis to be applied to more complex engineeringproblems
342 Effect of Population Size Population size is key param-eters of ABC To study the effect of the colony size on the
The Scientific World Journal 5
1 2 3 4 5 6
1m 13m
11m
15m 17m 21m
Figure 3 Position of monitored point in circular tunnel
0
02
04
06
08
1
12
1 2 3 4 5 6 7 8 9 10Distance to the center of tunnel (m)
Computed displacement using recognized parametersComputed displacement using actual parameters
Disp
lace
men
t (10
minus2
m)
Figure 4 The comparison of displacement between actual andrecognized parameters
convergence rate of theABC algorithm five different coloniesthat consisted of 20 50 100 200 and 400 bees were usedThefitness versus cycle numbers is shown in Figure 6 It can beseen that the convergence rates increasewith greater numbersof bees and population size of 200 or 400 bees is enough inthis study
4 Back Analysis Based on LSSVM and ABC
In the above section ABC-based back analysis was used tothe circular tunnel with analytical solution To the practicalengineering it is difficult to get the analytical solutionThe procedure with numerical solution is time-consumingRegression analysis is a good approach to build the rela-tion between geomechanical parameters and field moni-tored information In this study least square support vectormachine (LSSVM) was adopted to present the relationship
0
5
10
15
20
25
30
35
40
45
50
1 3 5 7 9
Stre
ss (M
Pa)
Distance to the center of tunnel (m)
Computed radial stress using recognized parametersComputed radial stress using actual parametersComputed tangential stress using recognized parametersComputed tangential stress using actual parameters
Figure 5 The comparison of stress between actual and recognizedparameters
0 200 400 600 800 1000
Fitn
ess
Cycle
100E minus 04
900E minus 05
800E minus 05
700E minus 05
600E minus 05
500E minus 05
400E minus 05
300E minus 05
200E minus 05
100E minus 05
000E + 00
Figure 6 Relationship between fitness value and cycle
between geomechanical parameters and displacement basedon numerical analysis
41 Least Square Support Vector Machine The least squaresupport vector machine (LSSVM) was originally developedby Suykens andVandewalle [21] Consider a given training setofN data points 119909119896 119910119896 (119896 = 1 2 119873)with input data xk isinRN and output yk isin r where RN is the N-dimensional vector
6 The Scientific World Journal
600000
620000
640000
660000
680000
700000
720000
0 200 400 600 800 1000
Fitn
ess
Cycle
E (MPa)
(a) 119864
200
250
300
350
400
450
500
550
0 200 400 600 800 1000
Fitn
ess
Cycle
c (MPa)
(b) 119888
2000
2200
2400
2600
2800
3000
3200
3400
0 200 400 600 800 1000
Fitn
ess
Cycle
120593 (∘)
(c) 120593
Figure 7 The variation of identified parameter with the cycle
space and r is the one-dimensional vector space Accordingto the LSSVM algorithm LSSVMmodel becomes
119910 (119909) =
119873
sum
119896=1
120572119896119870(119909 119909119896) + 119887 (11)
where 119870(119909 119909119896) is kernel functions and 120572 and b meet thefollowing equation
[
0 1119879
1 Ω + 120574minus1119868
] [
119887
120572] = [
0
119910] (12)
where 119910 = [1199101 119910119873] 1 = [1 1] 120572 = [1205721 120572119873]and Mercerrsquos theorem is applied within the Ω matrix
Ω=120593(119909119896)119879120593(119909119897) = 119896(119909119896 119909119897) 119896 119897 = 1 119873 Then the
analytical solution of 120572 and b is given by
[
119887
120572] = Φ
minus1[
0
119910] (13)
42 Representation of Nonlinear Relationship LSSVM is usedin this study to map the nonlinear relationship betweengeomechanical parameters such as Youngrsquos modulus cohe-sion geostress coefficients and monitored displacements
The Scientific World Journal 7
0
000005
00001
000015
00002
000025
0 200 400 600 800 1000
Fitn
ess
Cycle
Range 1Range 2Range 3
Figure 8The performance of ABCusing different searching ranges
0
000005
00001
000015
00002
0 200 400 600 800 1000
Fitn
ess
Cycle
SN2 = 20
SN2 = 50
SN2 = 100
SN2 = 200
SN2 = 400
Figure 9 The convergence of different population size
The mathematical model of least square support vectormachine is defined as
LSSVM (X) 119877119899 997888rarr 119877
Y = LSSVM (X) X = (1199091 1199092 119909119899)
Y = (1199101 1199102 119910119899)
(14)
0 15
minus5 10
minus5 0 5 0
5 10
10MPa 20MPa
30∘
Failure criterion Mohr-Coulomb
Youngrsquos modulus E 20000MPa
Cohesion c 105MPa
Friction angle 120593 35∘
Poissonrsquos ratio 120583 02
Figure 10 The cross section of tunnel and parameters
Table 3 The ranges of identified parameters
Range 1 Range 2 Range 3119864 (Mpa) [2000 12000] [4000 1000] [5000 8000]119888 (Mpa) [05 7] [1 6] [3 7]120593 (∘) [5 60] [10 50] [20 40]
Table 4 Identified in situ stress and angle in different stages
1198751 1198752 AngleActual 200000 100000 300000Stage 1 199583 100614 300104Stage 2 206493 108171 333676Stage 3 200252 100376 30623
where 119909119894(119894 = 1 2 119899) is geomechanical parameters forexample Youngrsquos modulus friction angle geostress coeffi-cients and so forth and 119910119894(119894 = 1 2 119899) is displacementsof the key points
In order to obtain LSSVM(X) a training process basedon the known data set is needed Necessary training samplesare created in this work by using numerical analysis (egFEM model) which is used to obtain displacements of rockmass of key points corresponding to the given set of tentativegeomechanical parameters The geomechanical parametersare defined as input of LSSVM The displacement is definedas output of LSSVM
8 The Scientific World Journal
Table5Training
samples
andmod
elparameterso
fLSSVM
Num
bero
fsamples
1198751(M
pa)
1198752(M
pa)
120593(∘ )
Disp
lacement
120572
MP1
MP2
MP3
MP1119909
MP1119910
MP2119909
MP2119910
MP3119909
MP3119910
119909119910
119909119910
119909119910
110000
0500
0020000
0minus08380
minus13
600
15500
minus00231
minus20200
minus15
100
14473
20149
minus08992
minus03815
15989
22484
210000
075
000
25000
0minus04990
minus23300
13900
minus006
87minus16
700
minus15
800
16424
08880
minus09801
minus03294
16348
19749
310000
010000
030000
0000
00minus31300
1400
0minus14
400
1400
0minus14
400
21479
02439
minus09786
minus16
870
49088
21843
412500
012500
035000
0000
00minus39100
17500
minus18
000
minus17
500
minus18
000
20307
minus03980
minus05684
minus18
560
14959
17655
515000
015000
040000
0minus000
01minus47000
20900
minus21600
minus21000
minus21700
200
40minus10
849
minus02202
minus21514
1240
414
127
615000
0500
0025000
0minus200
00minus14
700
20800
08610
minus31900
minus27200
02187
18194
minus03108
05215
03286
09391
715000
075
000
30000
0minus16
800
minus25600
18300
01890
minus27700
minus28200
05089
06915
minus05137
minus0119
105530
07972
815000
010000
035000
0minus12
300
minus34700
1740
0minus05740
minus24200
minus27500
06722
006
83minus05353
minus05142
07871
07058
915000
012500
040000
0minus064
20minus41900
18300
minus13
800
minus21800
minus25200
10483
minus03389
minus05100
minus10
033
10326
08752
1015000
015000
020000
0minus000
01minus47000
20900
minus21600
minus21000
minus21700
22964
minus12
063
minus04593
minus24334
16207
16580
1120000
0500
0030000
0minus34100
minus19
500
22700
18500
minus42500
minus43300
minus09584
14147
minus01741
13821
minus05279
minus04169
1220000
075
000
35000
0minus30700
minus32100
19200
1100
0minus360
00minus43700
minus04940
02093
minus046
0505409
01538
minus03109
1320000
010000
040000
0minus25800
minus42600
1740
002750
minus31500
minus43100
minus01430
minus06938
minus06499
minus01060
05071
minus03365
1420000
012500
020000
0minus12
600
minus36100
30300
minus07560
minus37300
minus29900
09442
minus00545
04200
minus09120
00125
07845
1520000
015000
025000
0minus09990
minus46500
27900
minus13
700
minus33400
minus31500
12917
minus11019
01791
minus16
037
044
3806994
1625000
0500
0035000
0minus50300
minus28100
22000
29600
minus53200
minus62900
minus23159
07126
minus02344
23232
minus14
578
minus20741
1725000
075
000
40000
0minus45700
minus43400
17200
20100
minus42700
minus62200
minus17
211
minus08042
minus07054
12559
minus03011
minus18
612
1825000
010000
020000
0minus25600
minus25500
40000
06760
minus53900
minus38400
minus006
8108422
12115
02427
minus13
147
01347
1925000
012500
025000
0minus25800
minus38300
35200
02050
minus48900
minus42900
minus01085
minus02739
07151
minus01537
minus08165
minus02612
2025000
015000
030000
0minus23100
minus50100
32000
minus03910
minus44100
minus45200
02387
minus13
780
05252
minus08035
minus04328
minus040
6121
30000
0500
0040000
0minus70
100
minus42700
19200
42500
minus63900
minus85200
minus44142
minus07485
minus05555
37206
minus25680
minus440
6722
30000
075
000
20000
0minus41800
minus15
000
51000
21700
minus74
200
minus48300
minus16
564
19159
23915
16943
minus34243
minus08180
2330000
010000
025000
0minus43200
minus306
0043800
18900
minus65200
minus55400
minus16
996
03741
15582
1364
6minus23453
minus14
311
2430000
012500
030000
0minus41800
minus45200
37600
13100
minus58600
minus59500
minus15
641
minus08404
10366
08632
minus18
027
minus17
517
2530000
015000
035000
0minus38900
minus58500
33200
05890
minus52300
minus61900
minus13
480
minus22716
07182
01455
minus13
269
minus21053
119887mdash
mdashmdash
mdashmdash
mdashmdash
mdashmdash
minus24124
minus34816
25241
03809
minus37541
minus39253
The Scientific World Journal 9
Com
pute
d di
spla
cem
ent u
sing
iden
tified
par
amet
ers
Monitored displacement (mm)
00
50
100
150
00 50 100 150minus150 minus100 minus50
minus150
minus100
minus50
(a) Stage 1
Com
pute
d di
spla
cem
ent u
sing
iden
tified
par
amet
ers
Monitored displacement (mm)
00
50
100
150
00 50 100 150minus150 minus100 minus50
minus150
minus100
minus50
(b) Stage 2
Stage 1Stage 2Stage 3
00
50
100
150
00 50 100 150
Com
pute
d di
spla
cem
ent u
sing
iden
tified
par
amet
ers
Monitored displacement (mm)minus150 minus100 minus50
minus150
minus100
minus50
(c) Stage 3
Figure 11 Comparison between monitored displacement and predicted displacement using identified parameters
43 Procedure of Back Analysis Algorithm Based on LSSVMand ABC After the LSSVM model representing the non-linear relation between the displacement and a parameteris obtained it can be used to predict displacements atmonitored points instead of numerical analysis ABC is usedto search the optimal parameter to be identified based on theerror function (see (10)) The back analysis technique basedon LSSVM-ABC combination can be described as follows
Step 1 Determine ABC parameters and the range ofparameters to be recognized
Step 2 Generate randomly 119899 group of parameters attheir given range Each individual represents an initialsolution
Step 3 Input a set of rock mass parameters to themodel LSSVM(X) obtained above to calculate thedisplacement values at given monitoring points
Step 4 Use (10) to evaluate the fitness of the currentindividuals that is the reasonability of the parameterset
10 The Scientific World Journal
0000e + 000
5000e + 000
1000e + 001
1500e + 001
2000e + 001
2500e + 001
3000e + 001
3500e + 001
4000e + 001
4500e + 001
5000e + 001
5500e + 001
6000e + 001
Use
r dat
a120590
XX
(a) 120590119909 using theory parameters
0000e + 000
5000e + 000
1000e + 001
1500e + 001
2000e + 001
2500e + 001
3000e + 001
3500e + 001
4000e + 001
4500e + 001
5000e + 001
5500e + 001
6000e + 001
Use
r dat
a120590
XX
(b) 120590119909 using identified parameters
Use
r dat
a120590
YY
0000e + 000
4000e + 000
8000e + 000
1200e + 001
1600e + 001
2000e + 001
2400e + 001
2800e + 001
3200e + 001
3600e + 001
4400e + 001
4000e + 001
4800e + 001
(c) 120590119910 using theory parameters
Use
r dat
a120590
YY
0000e + 000
4000e + 000
8000e + 000
1200e + 001
1600e + 001
2000e + 001
2400e + 001
2800e + 001
3200e + 001
3600e + 001
4400e + 001
4000e + 001
4800e + 001
(d) 120590119910 using identified parameters
Figure 12 Calculated stress comparison between using theory value and identified value at stage 3
Step 5 If all individuals are evaluated then go to Step6 Otherwise go to Step 3Step 6 If the maximum cycle is reached or the bestindividuals (the parameter to be back recognized)are obtained then the cycle ends and outputs bestindividuals Otherwise go to Step 7Step 7Update the individuals according to (2) and (3)Step 8 Repeat Step 7 until all 119899 new individuals aregenerated They are used as offspringStep 9 Go to Step 3
44 Verification To verify the model we suppose there isa tunnel (see Figure 7) The size of tunnel geomechanicalparameters and in situ stress are listed in Figure 7 The valuein Figure 7 is theoretical values Displacement values for somekey points indicated by nodes are calculated by elastic finiteelement method The suggested algorithm above is used toidentify initial geostress components P1 and P2 and anglebetween P1 and P2 We used orthogonal experiment design
to create 25 sets of tentative geostresses P1 and P2 and anglebetween P1 and P2 The training samples will be obtainedthrough computing the displacement of each set of tentativegeostresses Then the LSSVMmodel was build based on (13)The training samples and model parameters of LSSVM arelisted in Table 5 In situ stresses P1 and P2 and angle atdifferent stages can be identified according to the procedureof Section 43 Identified in situ stress P1 and P2and angleat different stages are listed in Table 4 The comparisonbetween displacement of the key points using the theoreticalparameters and displacements identified by back analysisbased on ABC and LSSVM is shown in Figure 8 Stresses ofsurrounding rock are shown in Figure 9 after stage 3 Resultsshow the proposed method can effectively identify the in situstress
45 Discussions
451 Performance of LSSVM The performance of LSSVM isvery important to back analysis The predicted displacement
The Scientific World Journal 11
00000
10000
20000
30000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
minus10000
minus20000
minus30000
minus40000
minus50000
(a) Stage 1
00000
20000
40000
60000
80000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
minus20000
minus60000
minus80000
minus100000
minus40000
MP4
-x
MP4
-y
MP5
-x
MP5
-y
(b) Stage 2
Monitored displacementPredicted displacement using LSSVMComputed displacement using FEM
00000
50000
100000
150000
minus50000
minus100000
minus150000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
MP4
-x
MP4
-y
MP5
-x
MP5
-y
MP6
-x
MP6
-y
MP7
-x
MP7
-y
(c) Stage 3
Figure 13 Predicted displacement using LSSVM with calculated displacement using theory and identified parameters
using LSSVM is in well agreement with the calculateddisplacement using theory and identified parameters (shownin Figure 10) It shows the LSSVM model presents wellthe relationship between geomechanical parameters anddisplacement It improves the efficiency of back analysis usingLSSVM
452 Effect of Kernel Parameters In this study the RBFkernel functionwas adoptedThe relationship between fitnessand cycle is listed in Figure 11 with 120590 = 10 and 120590 = 1 Theperformance of LSSVM is listed in Figure 12 using 120590 = 10 and120590 = 1 Its show selecting the appropriate kernel parametersis important to back analysis But there is not any guide toselect kernel function and its parameters according to LSSVMtheory It can be acquired by error-and-trial
5 Conclusions
The paper presents a new methodology called back analysisbased on ABC ABC is used to identify the geomechanicalparameters based on monitored displacements Results ofcircular tunnel with the analytical solution illustrate clearlythat ABC is effectively able to search parameters of geo-material and has proved ABC has powerful global optimalperformance To improve the efficiency of back analysisLSSVMwas used to present the relationship between geome-chanical parameters and displacement instead of numericalanalysis Results of horseshoe tunnel without the analyticalsolution demonstrate that LSSVMpresents well the nonlinearrelationship between geomechanical parameters and moni-tored displacements The proposed approach improves the
12 The Scientific World Journal
0
005
01
015
02
025
0 200 400 600 800 1000
Fitn
ess
Cycle
120590 = 10
120590 = 1
Figure 14 Fitness with different parameters of kernel function
00000
50000
100000
150000
00000 50000 100000 150000
Com
pute
d di
spla
cem
ent u
sing
FEM
bas
ed o
n LS
SVM
(mm
)
Monitored displacement (mm)
120590 = 10
120590 = 1
minus150000
minus100000
minus50000
minus150000 minus100000 minus50000
Figure 15The performance of LSSVMwith different parameters ofkernel function
efficiency and precision of back analysis andmakes it possibleto be applied to more complex engineering problem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was also supported by the National Fund ofScience in China (no 41072224 51104057)
References
[1] L Jing ldquoA review of techniques advances and outstandingissues in numerical modelling for rock mechanics and rockengineeringrdquo International Journal of Rock Mechanics andMining Sciences vol 40 no 3 pp 283ndash353 2003
[2] P Oreste ldquoBack-analysis techniques for the improvement ofthe understanding of rock in underground constructionsrdquoTunnelling and Underground Space Technology vol 20 no 1 pp7ndash21 2005
[3] G Gioda and L Jurina ldquoNumerical identification( back cal-culation) of soil-structure interaction pressuresrdquo InternationalJournal for Numerical amp Analytical Methods in Geomechanicsvol 5 no 1 pp 33ndash56 1981
[4] S Sakurai and K Takeuchi ldquoBack analysis of measured dis-placements of tunnelsrdquo Rock Mechanics and Rock Engineeringvol 16 no 3 pp 173ndash180 1983
[5] S Sakurai N Dees Wasmongkol and M Shinji ldquoBack analysisfor determining material characteristics in cut slopesrdquo inProceedings of the International Symposium on ECRF pp 770ndash776 Beijing China 1986
[6] S Sakurai ldquoInterpretation of the results of displacement mea-surements in cut slopesrdquo in Proceedings of the 2nd InternationalSymposium on Field Measurements in Geomechanics (FMGMrsquo87) pp 2528ndash2540 Kobe Japan 1987
[7] Z L Feng and R W Lewis ldquoOptimal estimation of in-situground stresses from displacement measurementsrdquo Interna-tional Journal for Numerical amp Analytical Methods in Geome-chanics vol 11 no 4 pp 391ndash408 1987
[8] B Pichler R Lackner and H A Mang ldquoBack analysis ofmodel parameters in geotechnical engineering by means ofsoft computingrdquo International Journal for Numerical Methods inEngineering vol 57 no 14 pp 1943ndash1978 2003
[9] F Xia-Ting and J A Hudson Rock Engineering Design CRCPress New York NY USA 2011
[10] T Okabe K Hayashi N Shinohara and S Takasugi ldquoInversionof drilling-induced tensile fracture data obtained from a singleinclined boreholerdquo International Journal of Rock Mechanics andMining Sciences vol 35 no 6 pp 747ndash758 1998
[11] W-G William and Y S Yoon ldquoAquifer parameter identifi-cation with optimum dimension in parameterizationrdquo WaterResources Research vol 17 no 3 pp 664ndash672 1981
[12] A Cividini G Maier and A Nappi ldquoParameter estimation ofa static geotechnical model using a Bayesrsquo approachrdquo Interna-tional Journal of Rock Mechanics and Mining Sciences vol 20no 5 pp 215ndash226 1983
[13] S VardakosM Gutierrez andC Xia ldquoParameter identificationin numerical modeling of tunneling using the DifferentialEvolution Genetic Algorithm (DEGA)rdquo Tunnelling and Under-ground Space Technology vol 28 no 1 pp 109ndash123 2012
[14] H Zhao and S Yin ldquoGeomechanical parameters identificationby particle swarm optimization and support vector machinerdquoApplied Mathematical Modelling vol 33 no 10 pp 3997ndash40122009
[15] X Feng H Zhao and S Li ldquoA new displacement backanalysis to identify mechanical geo-material parameters basedon hybrid intelligent methodologyrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 28 no11 pp 1141ndash1165 2004
[16] Y Yu B Zhang and H Yuan ldquoAn intelligent displacementback-analysis method for earth-rockfill damsrdquo Computers andGeotechnics vol 34 no 6 pp 423ndash434 2007
The Scientific World Journal 13
[17] J H Deng and C F Lee ldquoDiplacement back analysis for a steepslope at the Three Gorges Project siterdquo International Journal ofRockMechanics andMining Sciences vol 38 no 2 pp 259ndash2682001
[18] D Karaboga ldquoAn idea based on honey bee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University 2005
[19] D Karaboga and C Ozturk ldquoA novel clustering approachartificial Bee Colony (ABC) algorithmrdquoApplied Soft ComputingJournal vol 11 no 1 pp 652ndash657 2011
[20] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008
[21] J A K Suykens and J Vandewalle ldquoLeast squares supportvector machine classifiersrdquo Neural Processing Letters vol 9 no3 pp 293ndash300 1999
[22] M E Duncan Fama ldquoNumerical modeling of yield zones inweak rocksrdquo in Comprehensive Rock Engineering J A HudsonEd vol 2 pp 49ndash75 Pergamon Oxford UK 1993
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
The Scientific World Journal 5
1 2 3 4 5 6
1m 13m
11m
15m 17m 21m
Figure 3 Position of monitored point in circular tunnel
0
02
04
06
08
1
12
1 2 3 4 5 6 7 8 9 10Distance to the center of tunnel (m)
Computed displacement using recognized parametersComputed displacement using actual parameters
Disp
lace
men
t (10
minus2
m)
Figure 4 The comparison of displacement between actual andrecognized parameters
convergence rate of theABC algorithm five different coloniesthat consisted of 20 50 100 200 and 400 bees were usedThefitness versus cycle numbers is shown in Figure 6 It can beseen that the convergence rates increasewith greater numbersof bees and population size of 200 or 400 bees is enough inthis study
4 Back Analysis Based on LSSVM and ABC
In the above section ABC-based back analysis was used tothe circular tunnel with analytical solution To the practicalengineering it is difficult to get the analytical solutionThe procedure with numerical solution is time-consumingRegression analysis is a good approach to build the rela-tion between geomechanical parameters and field moni-tored information In this study least square support vectormachine (LSSVM) was adopted to present the relationship
0
5
10
15
20
25
30
35
40
45
50
1 3 5 7 9
Stre
ss (M
Pa)
Distance to the center of tunnel (m)
Computed radial stress using recognized parametersComputed radial stress using actual parametersComputed tangential stress using recognized parametersComputed tangential stress using actual parameters
Figure 5 The comparison of stress between actual and recognizedparameters
0 200 400 600 800 1000
Fitn
ess
Cycle
100E minus 04
900E minus 05
800E minus 05
700E minus 05
600E minus 05
500E minus 05
400E minus 05
300E minus 05
200E minus 05
100E minus 05
000E + 00
Figure 6 Relationship between fitness value and cycle
between geomechanical parameters and displacement basedon numerical analysis
41 Least Square Support Vector Machine The least squaresupport vector machine (LSSVM) was originally developedby Suykens andVandewalle [21] Consider a given training setofN data points 119909119896 119910119896 (119896 = 1 2 119873)with input data xk isinRN and output yk isin r where RN is the N-dimensional vector
6 The Scientific World Journal
600000
620000
640000
660000
680000
700000
720000
0 200 400 600 800 1000
Fitn
ess
Cycle
E (MPa)
(a) 119864
200
250
300
350
400
450
500
550
0 200 400 600 800 1000
Fitn
ess
Cycle
c (MPa)
(b) 119888
2000
2200
2400
2600
2800
3000
3200
3400
0 200 400 600 800 1000
Fitn
ess
Cycle
120593 (∘)
(c) 120593
Figure 7 The variation of identified parameter with the cycle
space and r is the one-dimensional vector space Accordingto the LSSVM algorithm LSSVMmodel becomes
119910 (119909) =
119873
sum
119896=1
120572119896119870(119909 119909119896) + 119887 (11)
where 119870(119909 119909119896) is kernel functions and 120572 and b meet thefollowing equation
[
0 1119879
1 Ω + 120574minus1119868
] [
119887
120572] = [
0
119910] (12)
where 119910 = [1199101 119910119873] 1 = [1 1] 120572 = [1205721 120572119873]and Mercerrsquos theorem is applied within the Ω matrix
Ω=120593(119909119896)119879120593(119909119897) = 119896(119909119896 119909119897) 119896 119897 = 1 119873 Then the
analytical solution of 120572 and b is given by
[
119887
120572] = Φ
minus1[
0
119910] (13)
42 Representation of Nonlinear Relationship LSSVM is usedin this study to map the nonlinear relationship betweengeomechanical parameters such as Youngrsquos modulus cohe-sion geostress coefficients and monitored displacements
The Scientific World Journal 7
0
000005
00001
000015
00002
000025
0 200 400 600 800 1000
Fitn
ess
Cycle
Range 1Range 2Range 3
Figure 8The performance of ABCusing different searching ranges
0
000005
00001
000015
00002
0 200 400 600 800 1000
Fitn
ess
Cycle
SN2 = 20
SN2 = 50
SN2 = 100
SN2 = 200
SN2 = 400
Figure 9 The convergence of different population size
The mathematical model of least square support vectormachine is defined as
LSSVM (X) 119877119899 997888rarr 119877
Y = LSSVM (X) X = (1199091 1199092 119909119899)
Y = (1199101 1199102 119910119899)
(14)
0 15
minus5 10
minus5 0 5 0
5 10
10MPa 20MPa
30∘
Failure criterion Mohr-Coulomb
Youngrsquos modulus E 20000MPa
Cohesion c 105MPa
Friction angle 120593 35∘
Poissonrsquos ratio 120583 02
Figure 10 The cross section of tunnel and parameters
Table 3 The ranges of identified parameters
Range 1 Range 2 Range 3119864 (Mpa) [2000 12000] [4000 1000] [5000 8000]119888 (Mpa) [05 7] [1 6] [3 7]120593 (∘) [5 60] [10 50] [20 40]
Table 4 Identified in situ stress and angle in different stages
1198751 1198752 AngleActual 200000 100000 300000Stage 1 199583 100614 300104Stage 2 206493 108171 333676Stage 3 200252 100376 30623
where 119909119894(119894 = 1 2 119899) is geomechanical parameters forexample Youngrsquos modulus friction angle geostress coeffi-cients and so forth and 119910119894(119894 = 1 2 119899) is displacementsof the key points
In order to obtain LSSVM(X) a training process basedon the known data set is needed Necessary training samplesare created in this work by using numerical analysis (egFEM model) which is used to obtain displacements of rockmass of key points corresponding to the given set of tentativegeomechanical parameters The geomechanical parametersare defined as input of LSSVM The displacement is definedas output of LSSVM
8 The Scientific World Journal
Table5Training
samples
andmod
elparameterso
fLSSVM
Num
bero
fsamples
1198751(M
pa)
1198752(M
pa)
120593(∘ )
Disp
lacement
120572
MP1
MP2
MP3
MP1119909
MP1119910
MP2119909
MP2119910
MP3119909
MP3119910
119909119910
119909119910
119909119910
110000
0500
0020000
0minus08380
minus13
600
15500
minus00231
minus20200
minus15
100
14473
20149
minus08992
minus03815
15989
22484
210000
075
000
25000
0minus04990
minus23300
13900
minus006
87minus16
700
minus15
800
16424
08880
minus09801
minus03294
16348
19749
310000
010000
030000
0000
00minus31300
1400
0minus14
400
1400
0minus14
400
21479
02439
minus09786
minus16
870
49088
21843
412500
012500
035000
0000
00minus39100
17500
minus18
000
minus17
500
minus18
000
20307
minus03980
minus05684
minus18
560
14959
17655
515000
015000
040000
0minus000
01minus47000
20900
minus21600
minus21000
minus21700
200
40minus10
849
minus02202
minus21514
1240
414
127
615000
0500
0025000
0minus200
00minus14
700
20800
08610
minus31900
minus27200
02187
18194
minus03108
05215
03286
09391
715000
075
000
30000
0minus16
800
minus25600
18300
01890
minus27700
minus28200
05089
06915
minus05137
minus0119
105530
07972
815000
010000
035000
0minus12
300
minus34700
1740
0minus05740
minus24200
minus27500
06722
006
83minus05353
minus05142
07871
07058
915000
012500
040000
0minus064
20minus41900
18300
minus13
800
minus21800
minus25200
10483
minus03389
minus05100
minus10
033
10326
08752
1015000
015000
020000
0minus000
01minus47000
20900
minus21600
minus21000
minus21700
22964
minus12
063
minus04593
minus24334
16207
16580
1120000
0500
0030000
0minus34100
minus19
500
22700
18500
minus42500
minus43300
minus09584
14147
minus01741
13821
minus05279
minus04169
1220000
075
000
35000
0minus30700
minus32100
19200
1100
0minus360
00minus43700
minus04940
02093
minus046
0505409
01538
minus03109
1320000
010000
040000
0minus25800
minus42600
1740
002750
minus31500
minus43100
minus01430
minus06938
minus06499
minus01060
05071
minus03365
1420000
012500
020000
0minus12
600
minus36100
30300
minus07560
minus37300
minus29900
09442
minus00545
04200
minus09120
00125
07845
1520000
015000
025000
0minus09990
minus46500
27900
minus13
700
minus33400
minus31500
12917
minus11019
01791
minus16
037
044
3806994
1625000
0500
0035000
0minus50300
minus28100
22000
29600
minus53200
minus62900
minus23159
07126
minus02344
23232
minus14
578
minus20741
1725000
075
000
40000
0minus45700
minus43400
17200
20100
minus42700
minus62200
minus17
211
minus08042
minus07054
12559
minus03011
minus18
612
1825000
010000
020000
0minus25600
minus25500
40000
06760
minus53900
minus38400
minus006
8108422
12115
02427
minus13
147
01347
1925000
012500
025000
0minus25800
minus38300
35200
02050
minus48900
minus42900
minus01085
minus02739
07151
minus01537
minus08165
minus02612
2025000
015000
030000
0minus23100
minus50100
32000
minus03910
minus44100
minus45200
02387
minus13
780
05252
minus08035
minus04328
minus040
6121
30000
0500
0040000
0minus70
100
minus42700
19200
42500
minus63900
minus85200
minus44142
minus07485
minus05555
37206
minus25680
minus440
6722
30000
075
000
20000
0minus41800
minus15
000
51000
21700
minus74
200
minus48300
minus16
564
19159
23915
16943
minus34243
minus08180
2330000
010000
025000
0minus43200
minus306
0043800
18900
minus65200
minus55400
minus16
996
03741
15582
1364
6minus23453
minus14
311
2430000
012500
030000
0minus41800
minus45200
37600
13100
minus58600
minus59500
minus15
641
minus08404
10366
08632
minus18
027
minus17
517
2530000
015000
035000
0minus38900
minus58500
33200
05890
minus52300
minus61900
minus13
480
minus22716
07182
01455
minus13
269
minus21053
119887mdash
mdashmdash
mdashmdash
mdashmdash
mdashmdash
minus24124
minus34816
25241
03809
minus37541
minus39253
The Scientific World Journal 9
Com
pute
d di
spla
cem
ent u
sing
iden
tified
par
amet
ers
Monitored displacement (mm)
00
50
100
150
00 50 100 150minus150 minus100 minus50
minus150
minus100
minus50
(a) Stage 1
Com
pute
d di
spla
cem
ent u
sing
iden
tified
par
amet
ers
Monitored displacement (mm)
00
50
100
150
00 50 100 150minus150 minus100 minus50
minus150
minus100
minus50
(b) Stage 2
Stage 1Stage 2Stage 3
00
50
100
150
00 50 100 150
Com
pute
d di
spla
cem
ent u
sing
iden
tified
par
amet
ers
Monitored displacement (mm)minus150 minus100 minus50
minus150
minus100
minus50
(c) Stage 3
Figure 11 Comparison between monitored displacement and predicted displacement using identified parameters
43 Procedure of Back Analysis Algorithm Based on LSSVMand ABC After the LSSVM model representing the non-linear relation between the displacement and a parameteris obtained it can be used to predict displacements atmonitored points instead of numerical analysis ABC is usedto search the optimal parameter to be identified based on theerror function (see (10)) The back analysis technique basedon LSSVM-ABC combination can be described as follows
Step 1 Determine ABC parameters and the range ofparameters to be recognized
Step 2 Generate randomly 119899 group of parameters attheir given range Each individual represents an initialsolution
Step 3 Input a set of rock mass parameters to themodel LSSVM(X) obtained above to calculate thedisplacement values at given monitoring points
Step 4 Use (10) to evaluate the fitness of the currentindividuals that is the reasonability of the parameterset
10 The Scientific World Journal
0000e + 000
5000e + 000
1000e + 001
1500e + 001
2000e + 001
2500e + 001
3000e + 001
3500e + 001
4000e + 001
4500e + 001
5000e + 001
5500e + 001
6000e + 001
Use
r dat
a120590
XX
(a) 120590119909 using theory parameters
0000e + 000
5000e + 000
1000e + 001
1500e + 001
2000e + 001
2500e + 001
3000e + 001
3500e + 001
4000e + 001
4500e + 001
5000e + 001
5500e + 001
6000e + 001
Use
r dat
a120590
XX
(b) 120590119909 using identified parameters
Use
r dat
a120590
YY
0000e + 000
4000e + 000
8000e + 000
1200e + 001
1600e + 001
2000e + 001
2400e + 001
2800e + 001
3200e + 001
3600e + 001
4400e + 001
4000e + 001
4800e + 001
(c) 120590119910 using theory parameters
Use
r dat
a120590
YY
0000e + 000
4000e + 000
8000e + 000
1200e + 001
1600e + 001
2000e + 001
2400e + 001
2800e + 001
3200e + 001
3600e + 001
4400e + 001
4000e + 001
4800e + 001
(d) 120590119910 using identified parameters
Figure 12 Calculated stress comparison between using theory value and identified value at stage 3
Step 5 If all individuals are evaluated then go to Step6 Otherwise go to Step 3Step 6 If the maximum cycle is reached or the bestindividuals (the parameter to be back recognized)are obtained then the cycle ends and outputs bestindividuals Otherwise go to Step 7Step 7Update the individuals according to (2) and (3)Step 8 Repeat Step 7 until all 119899 new individuals aregenerated They are used as offspringStep 9 Go to Step 3
44 Verification To verify the model we suppose there isa tunnel (see Figure 7) The size of tunnel geomechanicalparameters and in situ stress are listed in Figure 7 The valuein Figure 7 is theoretical values Displacement values for somekey points indicated by nodes are calculated by elastic finiteelement method The suggested algorithm above is used toidentify initial geostress components P1 and P2 and anglebetween P1 and P2 We used orthogonal experiment design
to create 25 sets of tentative geostresses P1 and P2 and anglebetween P1 and P2 The training samples will be obtainedthrough computing the displacement of each set of tentativegeostresses Then the LSSVMmodel was build based on (13)The training samples and model parameters of LSSVM arelisted in Table 5 In situ stresses P1 and P2 and angle atdifferent stages can be identified according to the procedureof Section 43 Identified in situ stress P1 and P2and angleat different stages are listed in Table 4 The comparisonbetween displacement of the key points using the theoreticalparameters and displacements identified by back analysisbased on ABC and LSSVM is shown in Figure 8 Stresses ofsurrounding rock are shown in Figure 9 after stage 3 Resultsshow the proposed method can effectively identify the in situstress
45 Discussions
451 Performance of LSSVM The performance of LSSVM isvery important to back analysis The predicted displacement
The Scientific World Journal 11
00000
10000
20000
30000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
minus10000
minus20000
minus30000
minus40000
minus50000
(a) Stage 1
00000
20000
40000
60000
80000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
minus20000
minus60000
minus80000
minus100000
minus40000
MP4
-x
MP4
-y
MP5
-x
MP5
-y
(b) Stage 2
Monitored displacementPredicted displacement using LSSVMComputed displacement using FEM
00000
50000
100000
150000
minus50000
minus100000
minus150000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
MP4
-x
MP4
-y
MP5
-x
MP5
-y
MP6
-x
MP6
-y
MP7
-x
MP7
-y
(c) Stage 3
Figure 13 Predicted displacement using LSSVM with calculated displacement using theory and identified parameters
using LSSVM is in well agreement with the calculateddisplacement using theory and identified parameters (shownin Figure 10) It shows the LSSVM model presents wellthe relationship between geomechanical parameters anddisplacement It improves the efficiency of back analysis usingLSSVM
452 Effect of Kernel Parameters In this study the RBFkernel functionwas adoptedThe relationship between fitnessand cycle is listed in Figure 11 with 120590 = 10 and 120590 = 1 Theperformance of LSSVM is listed in Figure 12 using 120590 = 10 and120590 = 1 Its show selecting the appropriate kernel parametersis important to back analysis But there is not any guide toselect kernel function and its parameters according to LSSVMtheory It can be acquired by error-and-trial
5 Conclusions
The paper presents a new methodology called back analysisbased on ABC ABC is used to identify the geomechanicalparameters based on monitored displacements Results ofcircular tunnel with the analytical solution illustrate clearlythat ABC is effectively able to search parameters of geo-material and has proved ABC has powerful global optimalperformance To improve the efficiency of back analysisLSSVMwas used to present the relationship between geome-chanical parameters and displacement instead of numericalanalysis Results of horseshoe tunnel without the analyticalsolution demonstrate that LSSVMpresents well the nonlinearrelationship between geomechanical parameters and moni-tored displacements The proposed approach improves the
12 The Scientific World Journal
0
005
01
015
02
025
0 200 400 600 800 1000
Fitn
ess
Cycle
120590 = 10
120590 = 1
Figure 14 Fitness with different parameters of kernel function
00000
50000
100000
150000
00000 50000 100000 150000
Com
pute
d di
spla
cem
ent u
sing
FEM
bas
ed o
n LS
SVM
(mm
)
Monitored displacement (mm)
120590 = 10
120590 = 1
minus150000
minus100000
minus50000
minus150000 minus100000 minus50000
Figure 15The performance of LSSVMwith different parameters ofkernel function
efficiency and precision of back analysis andmakes it possibleto be applied to more complex engineering problem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was also supported by the National Fund ofScience in China (no 41072224 51104057)
References
[1] L Jing ldquoA review of techniques advances and outstandingissues in numerical modelling for rock mechanics and rockengineeringrdquo International Journal of Rock Mechanics andMining Sciences vol 40 no 3 pp 283ndash353 2003
[2] P Oreste ldquoBack-analysis techniques for the improvement ofthe understanding of rock in underground constructionsrdquoTunnelling and Underground Space Technology vol 20 no 1 pp7ndash21 2005
[3] G Gioda and L Jurina ldquoNumerical identification( back cal-culation) of soil-structure interaction pressuresrdquo InternationalJournal for Numerical amp Analytical Methods in Geomechanicsvol 5 no 1 pp 33ndash56 1981
[4] S Sakurai and K Takeuchi ldquoBack analysis of measured dis-placements of tunnelsrdquo Rock Mechanics and Rock Engineeringvol 16 no 3 pp 173ndash180 1983
[5] S Sakurai N Dees Wasmongkol and M Shinji ldquoBack analysisfor determining material characteristics in cut slopesrdquo inProceedings of the International Symposium on ECRF pp 770ndash776 Beijing China 1986
[6] S Sakurai ldquoInterpretation of the results of displacement mea-surements in cut slopesrdquo in Proceedings of the 2nd InternationalSymposium on Field Measurements in Geomechanics (FMGMrsquo87) pp 2528ndash2540 Kobe Japan 1987
[7] Z L Feng and R W Lewis ldquoOptimal estimation of in-situground stresses from displacement measurementsrdquo Interna-tional Journal for Numerical amp Analytical Methods in Geome-chanics vol 11 no 4 pp 391ndash408 1987
[8] B Pichler R Lackner and H A Mang ldquoBack analysis ofmodel parameters in geotechnical engineering by means ofsoft computingrdquo International Journal for Numerical Methods inEngineering vol 57 no 14 pp 1943ndash1978 2003
[9] F Xia-Ting and J A Hudson Rock Engineering Design CRCPress New York NY USA 2011
[10] T Okabe K Hayashi N Shinohara and S Takasugi ldquoInversionof drilling-induced tensile fracture data obtained from a singleinclined boreholerdquo International Journal of Rock Mechanics andMining Sciences vol 35 no 6 pp 747ndash758 1998
[11] W-G William and Y S Yoon ldquoAquifer parameter identifi-cation with optimum dimension in parameterizationrdquo WaterResources Research vol 17 no 3 pp 664ndash672 1981
[12] A Cividini G Maier and A Nappi ldquoParameter estimation ofa static geotechnical model using a Bayesrsquo approachrdquo Interna-tional Journal of Rock Mechanics and Mining Sciences vol 20no 5 pp 215ndash226 1983
[13] S VardakosM Gutierrez andC Xia ldquoParameter identificationin numerical modeling of tunneling using the DifferentialEvolution Genetic Algorithm (DEGA)rdquo Tunnelling and Under-ground Space Technology vol 28 no 1 pp 109ndash123 2012
[14] H Zhao and S Yin ldquoGeomechanical parameters identificationby particle swarm optimization and support vector machinerdquoApplied Mathematical Modelling vol 33 no 10 pp 3997ndash40122009
[15] X Feng H Zhao and S Li ldquoA new displacement backanalysis to identify mechanical geo-material parameters basedon hybrid intelligent methodologyrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 28 no11 pp 1141ndash1165 2004
[16] Y Yu B Zhang and H Yuan ldquoAn intelligent displacementback-analysis method for earth-rockfill damsrdquo Computers andGeotechnics vol 34 no 6 pp 423ndash434 2007
The Scientific World Journal 13
[17] J H Deng and C F Lee ldquoDiplacement back analysis for a steepslope at the Three Gorges Project siterdquo International Journal ofRockMechanics andMining Sciences vol 38 no 2 pp 259ndash2682001
[18] D Karaboga ldquoAn idea based on honey bee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University 2005
[19] D Karaboga and C Ozturk ldquoA novel clustering approachartificial Bee Colony (ABC) algorithmrdquoApplied Soft ComputingJournal vol 11 no 1 pp 652ndash657 2011
[20] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008
[21] J A K Suykens and J Vandewalle ldquoLeast squares supportvector machine classifiersrdquo Neural Processing Letters vol 9 no3 pp 293ndash300 1999
[22] M E Duncan Fama ldquoNumerical modeling of yield zones inweak rocksrdquo in Comprehensive Rock Engineering J A HudsonEd vol 2 pp 49ndash75 Pergamon Oxford UK 1993
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
6 The Scientific World Journal
600000
620000
640000
660000
680000
700000
720000
0 200 400 600 800 1000
Fitn
ess
Cycle
E (MPa)
(a) 119864
200
250
300
350
400
450
500
550
0 200 400 600 800 1000
Fitn
ess
Cycle
c (MPa)
(b) 119888
2000
2200
2400
2600
2800
3000
3200
3400
0 200 400 600 800 1000
Fitn
ess
Cycle
120593 (∘)
(c) 120593
Figure 7 The variation of identified parameter with the cycle
space and r is the one-dimensional vector space Accordingto the LSSVM algorithm LSSVMmodel becomes
119910 (119909) =
119873
sum
119896=1
120572119896119870(119909 119909119896) + 119887 (11)
where 119870(119909 119909119896) is kernel functions and 120572 and b meet thefollowing equation
[
0 1119879
1 Ω + 120574minus1119868
] [
119887
120572] = [
0
119910] (12)
where 119910 = [1199101 119910119873] 1 = [1 1] 120572 = [1205721 120572119873]and Mercerrsquos theorem is applied within the Ω matrix
Ω=120593(119909119896)119879120593(119909119897) = 119896(119909119896 119909119897) 119896 119897 = 1 119873 Then the
analytical solution of 120572 and b is given by
[
119887
120572] = Φ
minus1[
0
119910] (13)
42 Representation of Nonlinear Relationship LSSVM is usedin this study to map the nonlinear relationship betweengeomechanical parameters such as Youngrsquos modulus cohe-sion geostress coefficients and monitored displacements
The Scientific World Journal 7
0
000005
00001
000015
00002
000025
0 200 400 600 800 1000
Fitn
ess
Cycle
Range 1Range 2Range 3
Figure 8The performance of ABCusing different searching ranges
0
000005
00001
000015
00002
0 200 400 600 800 1000
Fitn
ess
Cycle
SN2 = 20
SN2 = 50
SN2 = 100
SN2 = 200
SN2 = 400
Figure 9 The convergence of different population size
The mathematical model of least square support vectormachine is defined as
LSSVM (X) 119877119899 997888rarr 119877
Y = LSSVM (X) X = (1199091 1199092 119909119899)
Y = (1199101 1199102 119910119899)
(14)
0 15
minus5 10
minus5 0 5 0
5 10
10MPa 20MPa
30∘
Failure criterion Mohr-Coulomb
Youngrsquos modulus E 20000MPa
Cohesion c 105MPa
Friction angle 120593 35∘
Poissonrsquos ratio 120583 02
Figure 10 The cross section of tunnel and parameters
Table 3 The ranges of identified parameters
Range 1 Range 2 Range 3119864 (Mpa) [2000 12000] [4000 1000] [5000 8000]119888 (Mpa) [05 7] [1 6] [3 7]120593 (∘) [5 60] [10 50] [20 40]
Table 4 Identified in situ stress and angle in different stages
1198751 1198752 AngleActual 200000 100000 300000Stage 1 199583 100614 300104Stage 2 206493 108171 333676Stage 3 200252 100376 30623
where 119909119894(119894 = 1 2 119899) is geomechanical parameters forexample Youngrsquos modulus friction angle geostress coeffi-cients and so forth and 119910119894(119894 = 1 2 119899) is displacementsof the key points
In order to obtain LSSVM(X) a training process basedon the known data set is needed Necessary training samplesare created in this work by using numerical analysis (egFEM model) which is used to obtain displacements of rockmass of key points corresponding to the given set of tentativegeomechanical parameters The geomechanical parametersare defined as input of LSSVM The displacement is definedas output of LSSVM
8 The Scientific World Journal
Table5Training
samples
andmod
elparameterso
fLSSVM
Num
bero
fsamples
1198751(M
pa)
1198752(M
pa)
120593(∘ )
Disp
lacement
120572
MP1
MP2
MP3
MP1119909
MP1119910
MP2119909
MP2119910
MP3119909
MP3119910
119909119910
119909119910
119909119910
110000
0500
0020000
0minus08380
minus13
600
15500
minus00231
minus20200
minus15
100
14473
20149
minus08992
minus03815
15989
22484
210000
075
000
25000
0minus04990
minus23300
13900
minus006
87minus16
700
minus15
800
16424
08880
minus09801
minus03294
16348
19749
310000
010000
030000
0000
00minus31300
1400
0minus14
400
1400
0minus14
400
21479
02439
minus09786
minus16
870
49088
21843
412500
012500
035000
0000
00minus39100
17500
minus18
000
minus17
500
minus18
000
20307
minus03980
minus05684
minus18
560
14959
17655
515000
015000
040000
0minus000
01minus47000
20900
minus21600
minus21000
minus21700
200
40minus10
849
minus02202
minus21514
1240
414
127
615000
0500
0025000
0minus200
00minus14
700
20800
08610
minus31900
minus27200
02187
18194
minus03108
05215
03286
09391
715000
075
000
30000
0minus16
800
minus25600
18300
01890
minus27700
minus28200
05089
06915
minus05137
minus0119
105530
07972
815000
010000
035000
0minus12
300
minus34700
1740
0minus05740
minus24200
minus27500
06722
006
83minus05353
minus05142
07871
07058
915000
012500
040000
0minus064
20minus41900
18300
minus13
800
minus21800
minus25200
10483
minus03389
minus05100
minus10
033
10326
08752
1015000
015000
020000
0minus000
01minus47000
20900
minus21600
minus21000
minus21700
22964
minus12
063
minus04593
minus24334
16207
16580
1120000
0500
0030000
0minus34100
minus19
500
22700
18500
minus42500
minus43300
minus09584
14147
minus01741
13821
minus05279
minus04169
1220000
075
000
35000
0minus30700
minus32100
19200
1100
0minus360
00minus43700
minus04940
02093
minus046
0505409
01538
minus03109
1320000
010000
040000
0minus25800
minus42600
1740
002750
minus31500
minus43100
minus01430
minus06938
minus06499
minus01060
05071
minus03365
1420000
012500
020000
0minus12
600
minus36100
30300
minus07560
minus37300
minus29900
09442
minus00545
04200
minus09120
00125
07845
1520000
015000
025000
0minus09990
minus46500
27900
minus13
700
minus33400
minus31500
12917
minus11019
01791
minus16
037
044
3806994
1625000
0500
0035000
0minus50300
minus28100
22000
29600
minus53200
minus62900
minus23159
07126
minus02344
23232
minus14
578
minus20741
1725000
075
000
40000
0minus45700
minus43400
17200
20100
minus42700
minus62200
minus17
211
minus08042
minus07054
12559
minus03011
minus18
612
1825000
010000
020000
0minus25600
minus25500
40000
06760
minus53900
minus38400
minus006
8108422
12115
02427
minus13
147
01347
1925000
012500
025000
0minus25800
minus38300
35200
02050
minus48900
minus42900
minus01085
minus02739
07151
minus01537
minus08165
minus02612
2025000
015000
030000
0minus23100
minus50100
32000
minus03910
minus44100
minus45200
02387
minus13
780
05252
minus08035
minus04328
minus040
6121
30000
0500
0040000
0minus70
100
minus42700
19200
42500
minus63900
minus85200
minus44142
minus07485
minus05555
37206
minus25680
minus440
6722
30000
075
000
20000
0minus41800
minus15
000
51000
21700
minus74
200
minus48300
minus16
564
19159
23915
16943
minus34243
minus08180
2330000
010000
025000
0minus43200
minus306
0043800
18900
minus65200
minus55400
minus16
996
03741
15582
1364
6minus23453
minus14
311
2430000
012500
030000
0minus41800
minus45200
37600
13100
minus58600
minus59500
minus15
641
minus08404
10366
08632
minus18
027
minus17
517
2530000
015000
035000
0minus38900
minus58500
33200
05890
minus52300
minus61900
minus13
480
minus22716
07182
01455
minus13
269
minus21053
119887mdash
mdashmdash
mdashmdash
mdashmdash
mdashmdash
minus24124
minus34816
25241
03809
minus37541
minus39253
The Scientific World Journal 9
Com
pute
d di
spla
cem
ent u
sing
iden
tified
par
amet
ers
Monitored displacement (mm)
00
50
100
150
00 50 100 150minus150 minus100 minus50
minus150
minus100
minus50
(a) Stage 1
Com
pute
d di
spla
cem
ent u
sing
iden
tified
par
amet
ers
Monitored displacement (mm)
00
50
100
150
00 50 100 150minus150 minus100 minus50
minus150
minus100
minus50
(b) Stage 2
Stage 1Stage 2Stage 3
00
50
100
150
00 50 100 150
Com
pute
d di
spla
cem
ent u
sing
iden
tified
par
amet
ers
Monitored displacement (mm)minus150 minus100 minus50
minus150
minus100
minus50
(c) Stage 3
Figure 11 Comparison between monitored displacement and predicted displacement using identified parameters
43 Procedure of Back Analysis Algorithm Based on LSSVMand ABC After the LSSVM model representing the non-linear relation between the displacement and a parameteris obtained it can be used to predict displacements atmonitored points instead of numerical analysis ABC is usedto search the optimal parameter to be identified based on theerror function (see (10)) The back analysis technique basedon LSSVM-ABC combination can be described as follows
Step 1 Determine ABC parameters and the range ofparameters to be recognized
Step 2 Generate randomly 119899 group of parameters attheir given range Each individual represents an initialsolution
Step 3 Input a set of rock mass parameters to themodel LSSVM(X) obtained above to calculate thedisplacement values at given monitoring points
Step 4 Use (10) to evaluate the fitness of the currentindividuals that is the reasonability of the parameterset
10 The Scientific World Journal
0000e + 000
5000e + 000
1000e + 001
1500e + 001
2000e + 001
2500e + 001
3000e + 001
3500e + 001
4000e + 001
4500e + 001
5000e + 001
5500e + 001
6000e + 001
Use
r dat
a120590
XX
(a) 120590119909 using theory parameters
0000e + 000
5000e + 000
1000e + 001
1500e + 001
2000e + 001
2500e + 001
3000e + 001
3500e + 001
4000e + 001
4500e + 001
5000e + 001
5500e + 001
6000e + 001
Use
r dat
a120590
XX
(b) 120590119909 using identified parameters
Use
r dat
a120590
YY
0000e + 000
4000e + 000
8000e + 000
1200e + 001
1600e + 001
2000e + 001
2400e + 001
2800e + 001
3200e + 001
3600e + 001
4400e + 001
4000e + 001
4800e + 001
(c) 120590119910 using theory parameters
Use
r dat
a120590
YY
0000e + 000
4000e + 000
8000e + 000
1200e + 001
1600e + 001
2000e + 001
2400e + 001
2800e + 001
3200e + 001
3600e + 001
4400e + 001
4000e + 001
4800e + 001
(d) 120590119910 using identified parameters
Figure 12 Calculated stress comparison between using theory value and identified value at stage 3
Step 5 If all individuals are evaluated then go to Step6 Otherwise go to Step 3Step 6 If the maximum cycle is reached or the bestindividuals (the parameter to be back recognized)are obtained then the cycle ends and outputs bestindividuals Otherwise go to Step 7Step 7Update the individuals according to (2) and (3)Step 8 Repeat Step 7 until all 119899 new individuals aregenerated They are used as offspringStep 9 Go to Step 3
44 Verification To verify the model we suppose there isa tunnel (see Figure 7) The size of tunnel geomechanicalparameters and in situ stress are listed in Figure 7 The valuein Figure 7 is theoretical values Displacement values for somekey points indicated by nodes are calculated by elastic finiteelement method The suggested algorithm above is used toidentify initial geostress components P1 and P2 and anglebetween P1 and P2 We used orthogonal experiment design
to create 25 sets of tentative geostresses P1 and P2 and anglebetween P1 and P2 The training samples will be obtainedthrough computing the displacement of each set of tentativegeostresses Then the LSSVMmodel was build based on (13)The training samples and model parameters of LSSVM arelisted in Table 5 In situ stresses P1 and P2 and angle atdifferent stages can be identified according to the procedureof Section 43 Identified in situ stress P1 and P2and angleat different stages are listed in Table 4 The comparisonbetween displacement of the key points using the theoreticalparameters and displacements identified by back analysisbased on ABC and LSSVM is shown in Figure 8 Stresses ofsurrounding rock are shown in Figure 9 after stage 3 Resultsshow the proposed method can effectively identify the in situstress
45 Discussions
451 Performance of LSSVM The performance of LSSVM isvery important to back analysis The predicted displacement
The Scientific World Journal 11
00000
10000
20000
30000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
minus10000
minus20000
minus30000
minus40000
minus50000
(a) Stage 1
00000
20000
40000
60000
80000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
minus20000
minus60000
minus80000
minus100000
minus40000
MP4
-x
MP4
-y
MP5
-x
MP5
-y
(b) Stage 2
Monitored displacementPredicted displacement using LSSVMComputed displacement using FEM
00000
50000
100000
150000
minus50000
minus100000
minus150000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
MP4
-x
MP4
-y
MP5
-x
MP5
-y
MP6
-x
MP6
-y
MP7
-x
MP7
-y
(c) Stage 3
Figure 13 Predicted displacement using LSSVM with calculated displacement using theory and identified parameters
using LSSVM is in well agreement with the calculateddisplacement using theory and identified parameters (shownin Figure 10) It shows the LSSVM model presents wellthe relationship between geomechanical parameters anddisplacement It improves the efficiency of back analysis usingLSSVM
452 Effect of Kernel Parameters In this study the RBFkernel functionwas adoptedThe relationship between fitnessand cycle is listed in Figure 11 with 120590 = 10 and 120590 = 1 Theperformance of LSSVM is listed in Figure 12 using 120590 = 10 and120590 = 1 Its show selecting the appropriate kernel parametersis important to back analysis But there is not any guide toselect kernel function and its parameters according to LSSVMtheory It can be acquired by error-and-trial
5 Conclusions
The paper presents a new methodology called back analysisbased on ABC ABC is used to identify the geomechanicalparameters based on monitored displacements Results ofcircular tunnel with the analytical solution illustrate clearlythat ABC is effectively able to search parameters of geo-material and has proved ABC has powerful global optimalperformance To improve the efficiency of back analysisLSSVMwas used to present the relationship between geome-chanical parameters and displacement instead of numericalanalysis Results of horseshoe tunnel without the analyticalsolution demonstrate that LSSVMpresents well the nonlinearrelationship between geomechanical parameters and moni-tored displacements The proposed approach improves the
12 The Scientific World Journal
0
005
01
015
02
025
0 200 400 600 800 1000
Fitn
ess
Cycle
120590 = 10
120590 = 1
Figure 14 Fitness with different parameters of kernel function
00000
50000
100000
150000
00000 50000 100000 150000
Com
pute
d di
spla
cem
ent u
sing
FEM
bas
ed o
n LS
SVM
(mm
)
Monitored displacement (mm)
120590 = 10
120590 = 1
minus150000
minus100000
minus50000
minus150000 minus100000 minus50000
Figure 15The performance of LSSVMwith different parameters ofkernel function
efficiency and precision of back analysis andmakes it possibleto be applied to more complex engineering problem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was also supported by the National Fund ofScience in China (no 41072224 51104057)
References
[1] L Jing ldquoA review of techniques advances and outstandingissues in numerical modelling for rock mechanics and rockengineeringrdquo International Journal of Rock Mechanics andMining Sciences vol 40 no 3 pp 283ndash353 2003
[2] P Oreste ldquoBack-analysis techniques for the improvement ofthe understanding of rock in underground constructionsrdquoTunnelling and Underground Space Technology vol 20 no 1 pp7ndash21 2005
[3] G Gioda and L Jurina ldquoNumerical identification( back cal-culation) of soil-structure interaction pressuresrdquo InternationalJournal for Numerical amp Analytical Methods in Geomechanicsvol 5 no 1 pp 33ndash56 1981
[4] S Sakurai and K Takeuchi ldquoBack analysis of measured dis-placements of tunnelsrdquo Rock Mechanics and Rock Engineeringvol 16 no 3 pp 173ndash180 1983
[5] S Sakurai N Dees Wasmongkol and M Shinji ldquoBack analysisfor determining material characteristics in cut slopesrdquo inProceedings of the International Symposium on ECRF pp 770ndash776 Beijing China 1986
[6] S Sakurai ldquoInterpretation of the results of displacement mea-surements in cut slopesrdquo in Proceedings of the 2nd InternationalSymposium on Field Measurements in Geomechanics (FMGMrsquo87) pp 2528ndash2540 Kobe Japan 1987
[7] Z L Feng and R W Lewis ldquoOptimal estimation of in-situground stresses from displacement measurementsrdquo Interna-tional Journal for Numerical amp Analytical Methods in Geome-chanics vol 11 no 4 pp 391ndash408 1987
[8] B Pichler R Lackner and H A Mang ldquoBack analysis ofmodel parameters in geotechnical engineering by means ofsoft computingrdquo International Journal for Numerical Methods inEngineering vol 57 no 14 pp 1943ndash1978 2003
[9] F Xia-Ting and J A Hudson Rock Engineering Design CRCPress New York NY USA 2011
[10] T Okabe K Hayashi N Shinohara and S Takasugi ldquoInversionof drilling-induced tensile fracture data obtained from a singleinclined boreholerdquo International Journal of Rock Mechanics andMining Sciences vol 35 no 6 pp 747ndash758 1998
[11] W-G William and Y S Yoon ldquoAquifer parameter identifi-cation with optimum dimension in parameterizationrdquo WaterResources Research vol 17 no 3 pp 664ndash672 1981
[12] A Cividini G Maier and A Nappi ldquoParameter estimation ofa static geotechnical model using a Bayesrsquo approachrdquo Interna-tional Journal of Rock Mechanics and Mining Sciences vol 20no 5 pp 215ndash226 1983
[13] S VardakosM Gutierrez andC Xia ldquoParameter identificationin numerical modeling of tunneling using the DifferentialEvolution Genetic Algorithm (DEGA)rdquo Tunnelling and Under-ground Space Technology vol 28 no 1 pp 109ndash123 2012
[14] H Zhao and S Yin ldquoGeomechanical parameters identificationby particle swarm optimization and support vector machinerdquoApplied Mathematical Modelling vol 33 no 10 pp 3997ndash40122009
[15] X Feng H Zhao and S Li ldquoA new displacement backanalysis to identify mechanical geo-material parameters basedon hybrid intelligent methodologyrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 28 no11 pp 1141ndash1165 2004
[16] Y Yu B Zhang and H Yuan ldquoAn intelligent displacementback-analysis method for earth-rockfill damsrdquo Computers andGeotechnics vol 34 no 6 pp 423ndash434 2007
The Scientific World Journal 13
[17] J H Deng and C F Lee ldquoDiplacement back analysis for a steepslope at the Three Gorges Project siterdquo International Journal ofRockMechanics andMining Sciences vol 38 no 2 pp 259ndash2682001
[18] D Karaboga ldquoAn idea based on honey bee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University 2005
[19] D Karaboga and C Ozturk ldquoA novel clustering approachartificial Bee Colony (ABC) algorithmrdquoApplied Soft ComputingJournal vol 11 no 1 pp 652ndash657 2011
[20] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008
[21] J A K Suykens and J Vandewalle ldquoLeast squares supportvector machine classifiersrdquo Neural Processing Letters vol 9 no3 pp 293ndash300 1999
[22] M E Duncan Fama ldquoNumerical modeling of yield zones inweak rocksrdquo in Comprehensive Rock Engineering J A HudsonEd vol 2 pp 49ndash75 Pergamon Oxford UK 1993
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
The Scientific World Journal 7
0
000005
00001
000015
00002
000025
0 200 400 600 800 1000
Fitn
ess
Cycle
Range 1Range 2Range 3
Figure 8The performance of ABCusing different searching ranges
0
000005
00001
000015
00002
0 200 400 600 800 1000
Fitn
ess
Cycle
SN2 = 20
SN2 = 50
SN2 = 100
SN2 = 200
SN2 = 400
Figure 9 The convergence of different population size
The mathematical model of least square support vectormachine is defined as
LSSVM (X) 119877119899 997888rarr 119877
Y = LSSVM (X) X = (1199091 1199092 119909119899)
Y = (1199101 1199102 119910119899)
(14)
0 15
minus5 10
minus5 0 5 0
5 10
10MPa 20MPa
30∘
Failure criterion Mohr-Coulomb
Youngrsquos modulus E 20000MPa
Cohesion c 105MPa
Friction angle 120593 35∘
Poissonrsquos ratio 120583 02
Figure 10 The cross section of tunnel and parameters
Table 3 The ranges of identified parameters
Range 1 Range 2 Range 3119864 (Mpa) [2000 12000] [4000 1000] [5000 8000]119888 (Mpa) [05 7] [1 6] [3 7]120593 (∘) [5 60] [10 50] [20 40]
Table 4 Identified in situ stress and angle in different stages
1198751 1198752 AngleActual 200000 100000 300000Stage 1 199583 100614 300104Stage 2 206493 108171 333676Stage 3 200252 100376 30623
where 119909119894(119894 = 1 2 119899) is geomechanical parameters forexample Youngrsquos modulus friction angle geostress coeffi-cients and so forth and 119910119894(119894 = 1 2 119899) is displacementsof the key points
In order to obtain LSSVM(X) a training process basedon the known data set is needed Necessary training samplesare created in this work by using numerical analysis (egFEM model) which is used to obtain displacements of rockmass of key points corresponding to the given set of tentativegeomechanical parameters The geomechanical parametersare defined as input of LSSVM The displacement is definedas output of LSSVM
8 The Scientific World Journal
Table5Training
samples
andmod
elparameterso
fLSSVM
Num
bero
fsamples
1198751(M
pa)
1198752(M
pa)
120593(∘ )
Disp
lacement
120572
MP1
MP2
MP3
MP1119909
MP1119910
MP2119909
MP2119910
MP3119909
MP3119910
119909119910
119909119910
119909119910
110000
0500
0020000
0minus08380
minus13
600
15500
minus00231
minus20200
minus15
100
14473
20149
minus08992
minus03815
15989
22484
210000
075
000
25000
0minus04990
minus23300
13900
minus006
87minus16
700
minus15
800
16424
08880
minus09801
minus03294
16348
19749
310000
010000
030000
0000
00minus31300
1400
0minus14
400
1400
0minus14
400
21479
02439
minus09786
minus16
870
49088
21843
412500
012500
035000
0000
00minus39100
17500
minus18
000
minus17
500
minus18
000
20307
minus03980
minus05684
minus18
560
14959
17655
515000
015000
040000
0minus000
01minus47000
20900
minus21600
minus21000
minus21700
200
40minus10
849
minus02202
minus21514
1240
414
127
615000
0500
0025000
0minus200
00minus14
700
20800
08610
minus31900
minus27200
02187
18194
minus03108
05215
03286
09391
715000
075
000
30000
0minus16
800
minus25600
18300
01890
minus27700
minus28200
05089
06915
minus05137
minus0119
105530
07972
815000
010000
035000
0minus12
300
minus34700
1740
0minus05740
minus24200
minus27500
06722
006
83minus05353
minus05142
07871
07058
915000
012500
040000
0minus064
20minus41900
18300
minus13
800
minus21800
minus25200
10483
minus03389
minus05100
minus10
033
10326
08752
1015000
015000
020000
0minus000
01minus47000
20900
minus21600
minus21000
minus21700
22964
minus12
063
minus04593
minus24334
16207
16580
1120000
0500
0030000
0minus34100
minus19
500
22700
18500
minus42500
minus43300
minus09584
14147
minus01741
13821
minus05279
minus04169
1220000
075
000
35000
0minus30700
minus32100
19200
1100
0minus360
00minus43700
minus04940
02093
minus046
0505409
01538
minus03109
1320000
010000
040000
0minus25800
minus42600
1740
002750
minus31500
minus43100
minus01430
minus06938
minus06499
minus01060
05071
minus03365
1420000
012500
020000
0minus12
600
minus36100
30300
minus07560
minus37300
minus29900
09442
minus00545
04200
minus09120
00125
07845
1520000
015000
025000
0minus09990
minus46500
27900
minus13
700
minus33400
minus31500
12917
minus11019
01791
minus16
037
044
3806994
1625000
0500
0035000
0minus50300
minus28100
22000
29600
minus53200
minus62900
minus23159
07126
minus02344
23232
minus14
578
minus20741
1725000
075
000
40000
0minus45700
minus43400
17200
20100
minus42700
minus62200
minus17
211
minus08042
minus07054
12559
minus03011
minus18
612
1825000
010000
020000
0minus25600
minus25500
40000
06760
minus53900
minus38400
minus006
8108422
12115
02427
minus13
147
01347
1925000
012500
025000
0minus25800
minus38300
35200
02050
minus48900
minus42900
minus01085
minus02739
07151
minus01537
minus08165
minus02612
2025000
015000
030000
0minus23100
minus50100
32000
minus03910
minus44100
minus45200
02387
minus13
780
05252
minus08035
minus04328
minus040
6121
30000
0500
0040000
0minus70
100
minus42700
19200
42500
minus63900
minus85200
minus44142
minus07485
minus05555
37206
minus25680
minus440
6722
30000
075
000
20000
0minus41800
minus15
000
51000
21700
minus74
200
minus48300
minus16
564
19159
23915
16943
minus34243
minus08180
2330000
010000
025000
0minus43200
minus306
0043800
18900
minus65200
minus55400
minus16
996
03741
15582
1364
6minus23453
minus14
311
2430000
012500
030000
0minus41800
minus45200
37600
13100
minus58600
minus59500
minus15
641
minus08404
10366
08632
minus18
027
minus17
517
2530000
015000
035000
0minus38900
minus58500
33200
05890
minus52300
minus61900
minus13
480
minus22716
07182
01455
minus13
269
minus21053
119887mdash
mdashmdash
mdashmdash
mdashmdash
mdashmdash
minus24124
minus34816
25241
03809
minus37541
minus39253
The Scientific World Journal 9
Com
pute
d di
spla
cem
ent u
sing
iden
tified
par
amet
ers
Monitored displacement (mm)
00
50
100
150
00 50 100 150minus150 minus100 minus50
minus150
minus100
minus50
(a) Stage 1
Com
pute
d di
spla
cem
ent u
sing
iden
tified
par
amet
ers
Monitored displacement (mm)
00
50
100
150
00 50 100 150minus150 minus100 minus50
minus150
minus100
minus50
(b) Stage 2
Stage 1Stage 2Stage 3
00
50
100
150
00 50 100 150
Com
pute
d di
spla
cem
ent u
sing
iden
tified
par
amet
ers
Monitored displacement (mm)minus150 minus100 minus50
minus150
minus100
minus50
(c) Stage 3
Figure 11 Comparison between monitored displacement and predicted displacement using identified parameters
43 Procedure of Back Analysis Algorithm Based on LSSVMand ABC After the LSSVM model representing the non-linear relation between the displacement and a parameteris obtained it can be used to predict displacements atmonitored points instead of numerical analysis ABC is usedto search the optimal parameter to be identified based on theerror function (see (10)) The back analysis technique basedon LSSVM-ABC combination can be described as follows
Step 1 Determine ABC parameters and the range ofparameters to be recognized
Step 2 Generate randomly 119899 group of parameters attheir given range Each individual represents an initialsolution
Step 3 Input a set of rock mass parameters to themodel LSSVM(X) obtained above to calculate thedisplacement values at given monitoring points
Step 4 Use (10) to evaluate the fitness of the currentindividuals that is the reasonability of the parameterset
10 The Scientific World Journal
0000e + 000
5000e + 000
1000e + 001
1500e + 001
2000e + 001
2500e + 001
3000e + 001
3500e + 001
4000e + 001
4500e + 001
5000e + 001
5500e + 001
6000e + 001
Use
r dat
a120590
XX
(a) 120590119909 using theory parameters
0000e + 000
5000e + 000
1000e + 001
1500e + 001
2000e + 001
2500e + 001
3000e + 001
3500e + 001
4000e + 001
4500e + 001
5000e + 001
5500e + 001
6000e + 001
Use
r dat
a120590
XX
(b) 120590119909 using identified parameters
Use
r dat
a120590
YY
0000e + 000
4000e + 000
8000e + 000
1200e + 001
1600e + 001
2000e + 001
2400e + 001
2800e + 001
3200e + 001
3600e + 001
4400e + 001
4000e + 001
4800e + 001
(c) 120590119910 using theory parameters
Use
r dat
a120590
YY
0000e + 000
4000e + 000
8000e + 000
1200e + 001
1600e + 001
2000e + 001
2400e + 001
2800e + 001
3200e + 001
3600e + 001
4400e + 001
4000e + 001
4800e + 001
(d) 120590119910 using identified parameters
Figure 12 Calculated stress comparison between using theory value and identified value at stage 3
Step 5 If all individuals are evaluated then go to Step6 Otherwise go to Step 3Step 6 If the maximum cycle is reached or the bestindividuals (the parameter to be back recognized)are obtained then the cycle ends and outputs bestindividuals Otherwise go to Step 7Step 7Update the individuals according to (2) and (3)Step 8 Repeat Step 7 until all 119899 new individuals aregenerated They are used as offspringStep 9 Go to Step 3
44 Verification To verify the model we suppose there isa tunnel (see Figure 7) The size of tunnel geomechanicalparameters and in situ stress are listed in Figure 7 The valuein Figure 7 is theoretical values Displacement values for somekey points indicated by nodes are calculated by elastic finiteelement method The suggested algorithm above is used toidentify initial geostress components P1 and P2 and anglebetween P1 and P2 We used orthogonal experiment design
to create 25 sets of tentative geostresses P1 and P2 and anglebetween P1 and P2 The training samples will be obtainedthrough computing the displacement of each set of tentativegeostresses Then the LSSVMmodel was build based on (13)The training samples and model parameters of LSSVM arelisted in Table 5 In situ stresses P1 and P2 and angle atdifferent stages can be identified according to the procedureof Section 43 Identified in situ stress P1 and P2and angleat different stages are listed in Table 4 The comparisonbetween displacement of the key points using the theoreticalparameters and displacements identified by back analysisbased on ABC and LSSVM is shown in Figure 8 Stresses ofsurrounding rock are shown in Figure 9 after stage 3 Resultsshow the proposed method can effectively identify the in situstress
45 Discussions
451 Performance of LSSVM The performance of LSSVM isvery important to back analysis The predicted displacement
The Scientific World Journal 11
00000
10000
20000
30000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
minus10000
minus20000
minus30000
minus40000
minus50000
(a) Stage 1
00000
20000
40000
60000
80000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
minus20000
minus60000
minus80000
minus100000
minus40000
MP4
-x
MP4
-y
MP5
-x
MP5
-y
(b) Stage 2
Monitored displacementPredicted displacement using LSSVMComputed displacement using FEM
00000
50000
100000
150000
minus50000
minus100000
minus150000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
MP4
-x
MP4
-y
MP5
-x
MP5
-y
MP6
-x
MP6
-y
MP7
-x
MP7
-y
(c) Stage 3
Figure 13 Predicted displacement using LSSVM with calculated displacement using theory and identified parameters
using LSSVM is in well agreement with the calculateddisplacement using theory and identified parameters (shownin Figure 10) It shows the LSSVM model presents wellthe relationship between geomechanical parameters anddisplacement It improves the efficiency of back analysis usingLSSVM
452 Effect of Kernel Parameters In this study the RBFkernel functionwas adoptedThe relationship between fitnessand cycle is listed in Figure 11 with 120590 = 10 and 120590 = 1 Theperformance of LSSVM is listed in Figure 12 using 120590 = 10 and120590 = 1 Its show selecting the appropriate kernel parametersis important to back analysis But there is not any guide toselect kernel function and its parameters according to LSSVMtheory It can be acquired by error-and-trial
5 Conclusions
The paper presents a new methodology called back analysisbased on ABC ABC is used to identify the geomechanicalparameters based on monitored displacements Results ofcircular tunnel with the analytical solution illustrate clearlythat ABC is effectively able to search parameters of geo-material and has proved ABC has powerful global optimalperformance To improve the efficiency of back analysisLSSVMwas used to present the relationship between geome-chanical parameters and displacement instead of numericalanalysis Results of horseshoe tunnel without the analyticalsolution demonstrate that LSSVMpresents well the nonlinearrelationship between geomechanical parameters and moni-tored displacements The proposed approach improves the
12 The Scientific World Journal
0
005
01
015
02
025
0 200 400 600 800 1000
Fitn
ess
Cycle
120590 = 10
120590 = 1
Figure 14 Fitness with different parameters of kernel function
00000
50000
100000
150000
00000 50000 100000 150000
Com
pute
d di
spla
cem
ent u
sing
FEM
bas
ed o
n LS
SVM
(mm
)
Monitored displacement (mm)
120590 = 10
120590 = 1
minus150000
minus100000
minus50000
minus150000 minus100000 minus50000
Figure 15The performance of LSSVMwith different parameters ofkernel function
efficiency and precision of back analysis andmakes it possibleto be applied to more complex engineering problem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was also supported by the National Fund ofScience in China (no 41072224 51104057)
References
[1] L Jing ldquoA review of techniques advances and outstandingissues in numerical modelling for rock mechanics and rockengineeringrdquo International Journal of Rock Mechanics andMining Sciences vol 40 no 3 pp 283ndash353 2003
[2] P Oreste ldquoBack-analysis techniques for the improvement ofthe understanding of rock in underground constructionsrdquoTunnelling and Underground Space Technology vol 20 no 1 pp7ndash21 2005
[3] G Gioda and L Jurina ldquoNumerical identification( back cal-culation) of soil-structure interaction pressuresrdquo InternationalJournal for Numerical amp Analytical Methods in Geomechanicsvol 5 no 1 pp 33ndash56 1981
[4] S Sakurai and K Takeuchi ldquoBack analysis of measured dis-placements of tunnelsrdquo Rock Mechanics and Rock Engineeringvol 16 no 3 pp 173ndash180 1983
[5] S Sakurai N Dees Wasmongkol and M Shinji ldquoBack analysisfor determining material characteristics in cut slopesrdquo inProceedings of the International Symposium on ECRF pp 770ndash776 Beijing China 1986
[6] S Sakurai ldquoInterpretation of the results of displacement mea-surements in cut slopesrdquo in Proceedings of the 2nd InternationalSymposium on Field Measurements in Geomechanics (FMGMrsquo87) pp 2528ndash2540 Kobe Japan 1987
[7] Z L Feng and R W Lewis ldquoOptimal estimation of in-situground stresses from displacement measurementsrdquo Interna-tional Journal for Numerical amp Analytical Methods in Geome-chanics vol 11 no 4 pp 391ndash408 1987
[8] B Pichler R Lackner and H A Mang ldquoBack analysis ofmodel parameters in geotechnical engineering by means ofsoft computingrdquo International Journal for Numerical Methods inEngineering vol 57 no 14 pp 1943ndash1978 2003
[9] F Xia-Ting and J A Hudson Rock Engineering Design CRCPress New York NY USA 2011
[10] T Okabe K Hayashi N Shinohara and S Takasugi ldquoInversionof drilling-induced tensile fracture data obtained from a singleinclined boreholerdquo International Journal of Rock Mechanics andMining Sciences vol 35 no 6 pp 747ndash758 1998
[11] W-G William and Y S Yoon ldquoAquifer parameter identifi-cation with optimum dimension in parameterizationrdquo WaterResources Research vol 17 no 3 pp 664ndash672 1981
[12] A Cividini G Maier and A Nappi ldquoParameter estimation ofa static geotechnical model using a Bayesrsquo approachrdquo Interna-tional Journal of Rock Mechanics and Mining Sciences vol 20no 5 pp 215ndash226 1983
[13] S VardakosM Gutierrez andC Xia ldquoParameter identificationin numerical modeling of tunneling using the DifferentialEvolution Genetic Algorithm (DEGA)rdquo Tunnelling and Under-ground Space Technology vol 28 no 1 pp 109ndash123 2012
[14] H Zhao and S Yin ldquoGeomechanical parameters identificationby particle swarm optimization and support vector machinerdquoApplied Mathematical Modelling vol 33 no 10 pp 3997ndash40122009
[15] X Feng H Zhao and S Li ldquoA new displacement backanalysis to identify mechanical geo-material parameters basedon hybrid intelligent methodologyrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 28 no11 pp 1141ndash1165 2004
[16] Y Yu B Zhang and H Yuan ldquoAn intelligent displacementback-analysis method for earth-rockfill damsrdquo Computers andGeotechnics vol 34 no 6 pp 423ndash434 2007
The Scientific World Journal 13
[17] J H Deng and C F Lee ldquoDiplacement back analysis for a steepslope at the Three Gorges Project siterdquo International Journal ofRockMechanics andMining Sciences vol 38 no 2 pp 259ndash2682001
[18] D Karaboga ldquoAn idea based on honey bee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University 2005
[19] D Karaboga and C Ozturk ldquoA novel clustering approachartificial Bee Colony (ABC) algorithmrdquoApplied Soft ComputingJournal vol 11 no 1 pp 652ndash657 2011
[20] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008
[21] J A K Suykens and J Vandewalle ldquoLeast squares supportvector machine classifiersrdquo Neural Processing Letters vol 9 no3 pp 293ndash300 1999
[22] M E Duncan Fama ldquoNumerical modeling of yield zones inweak rocksrdquo in Comprehensive Rock Engineering J A HudsonEd vol 2 pp 49ndash75 Pergamon Oxford UK 1993
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
8 The Scientific World Journal
Table5Training
samples
andmod
elparameterso
fLSSVM
Num
bero
fsamples
1198751(M
pa)
1198752(M
pa)
120593(∘ )
Disp
lacement
120572
MP1
MP2
MP3
MP1119909
MP1119910
MP2119909
MP2119910
MP3119909
MP3119910
119909119910
119909119910
119909119910
110000
0500
0020000
0minus08380
minus13
600
15500
minus00231
minus20200
minus15
100
14473
20149
minus08992
minus03815
15989
22484
210000
075
000
25000
0minus04990
minus23300
13900
minus006
87minus16
700
minus15
800
16424
08880
minus09801
minus03294
16348
19749
310000
010000
030000
0000
00minus31300
1400
0minus14
400
1400
0minus14
400
21479
02439
minus09786
minus16
870
49088
21843
412500
012500
035000
0000
00minus39100
17500
minus18
000
minus17
500
minus18
000
20307
minus03980
minus05684
minus18
560
14959
17655
515000
015000
040000
0minus000
01minus47000
20900
minus21600
minus21000
minus21700
200
40minus10
849
minus02202
minus21514
1240
414
127
615000
0500
0025000
0minus200
00minus14
700
20800
08610
minus31900
minus27200
02187
18194
minus03108
05215
03286
09391
715000
075
000
30000
0minus16
800
minus25600
18300
01890
minus27700
minus28200
05089
06915
minus05137
minus0119
105530
07972
815000
010000
035000
0minus12
300
minus34700
1740
0minus05740
minus24200
minus27500
06722
006
83minus05353
minus05142
07871
07058
915000
012500
040000
0minus064
20minus41900
18300
minus13
800
minus21800
minus25200
10483
minus03389
minus05100
minus10
033
10326
08752
1015000
015000
020000
0minus000
01minus47000
20900
minus21600
minus21000
minus21700
22964
minus12
063
minus04593
minus24334
16207
16580
1120000
0500
0030000
0minus34100
minus19
500
22700
18500
minus42500
minus43300
minus09584
14147
minus01741
13821
minus05279
minus04169
1220000
075
000
35000
0minus30700
minus32100
19200
1100
0minus360
00minus43700
minus04940
02093
minus046
0505409
01538
minus03109
1320000
010000
040000
0minus25800
minus42600
1740
002750
minus31500
minus43100
minus01430
minus06938
minus06499
minus01060
05071
minus03365
1420000
012500
020000
0minus12
600
minus36100
30300
minus07560
minus37300
minus29900
09442
minus00545
04200
minus09120
00125
07845
1520000
015000
025000
0minus09990
minus46500
27900
minus13
700
minus33400
minus31500
12917
minus11019
01791
minus16
037
044
3806994
1625000
0500
0035000
0minus50300
minus28100
22000
29600
minus53200
minus62900
minus23159
07126
minus02344
23232
minus14
578
minus20741
1725000
075
000
40000
0minus45700
minus43400
17200
20100
minus42700
minus62200
minus17
211
minus08042
minus07054
12559
minus03011
minus18
612
1825000
010000
020000
0minus25600
minus25500
40000
06760
minus53900
minus38400
minus006
8108422
12115
02427
minus13
147
01347
1925000
012500
025000
0minus25800
minus38300
35200
02050
minus48900
minus42900
minus01085
minus02739
07151
minus01537
minus08165
minus02612
2025000
015000
030000
0minus23100
minus50100
32000
minus03910
minus44100
minus45200
02387
minus13
780
05252
minus08035
minus04328
minus040
6121
30000
0500
0040000
0minus70
100
minus42700
19200
42500
minus63900
minus85200
minus44142
minus07485
minus05555
37206
minus25680
minus440
6722
30000
075
000
20000
0minus41800
minus15
000
51000
21700
minus74
200
minus48300
minus16
564
19159
23915
16943
minus34243
minus08180
2330000
010000
025000
0minus43200
minus306
0043800
18900
minus65200
minus55400
minus16
996
03741
15582
1364
6minus23453
minus14
311
2430000
012500
030000
0minus41800
minus45200
37600
13100
minus58600
minus59500
minus15
641
minus08404
10366
08632
minus18
027
minus17
517
2530000
015000
035000
0minus38900
minus58500
33200
05890
minus52300
minus61900
minus13
480
minus22716
07182
01455
minus13
269
minus21053
119887mdash
mdashmdash
mdashmdash
mdashmdash
mdashmdash
minus24124
minus34816
25241
03809
minus37541
minus39253
The Scientific World Journal 9
Com
pute
d di
spla
cem
ent u
sing
iden
tified
par
amet
ers
Monitored displacement (mm)
00
50
100
150
00 50 100 150minus150 minus100 minus50
minus150
minus100
minus50
(a) Stage 1
Com
pute
d di
spla
cem
ent u
sing
iden
tified
par
amet
ers
Monitored displacement (mm)
00
50
100
150
00 50 100 150minus150 minus100 minus50
minus150
minus100
minus50
(b) Stage 2
Stage 1Stage 2Stage 3
00
50
100
150
00 50 100 150
Com
pute
d di
spla
cem
ent u
sing
iden
tified
par
amet
ers
Monitored displacement (mm)minus150 minus100 minus50
minus150
minus100
minus50
(c) Stage 3
Figure 11 Comparison between monitored displacement and predicted displacement using identified parameters
43 Procedure of Back Analysis Algorithm Based on LSSVMand ABC After the LSSVM model representing the non-linear relation between the displacement and a parameteris obtained it can be used to predict displacements atmonitored points instead of numerical analysis ABC is usedto search the optimal parameter to be identified based on theerror function (see (10)) The back analysis technique basedon LSSVM-ABC combination can be described as follows
Step 1 Determine ABC parameters and the range ofparameters to be recognized
Step 2 Generate randomly 119899 group of parameters attheir given range Each individual represents an initialsolution
Step 3 Input a set of rock mass parameters to themodel LSSVM(X) obtained above to calculate thedisplacement values at given monitoring points
Step 4 Use (10) to evaluate the fitness of the currentindividuals that is the reasonability of the parameterset
10 The Scientific World Journal
0000e + 000
5000e + 000
1000e + 001
1500e + 001
2000e + 001
2500e + 001
3000e + 001
3500e + 001
4000e + 001
4500e + 001
5000e + 001
5500e + 001
6000e + 001
Use
r dat
a120590
XX
(a) 120590119909 using theory parameters
0000e + 000
5000e + 000
1000e + 001
1500e + 001
2000e + 001
2500e + 001
3000e + 001
3500e + 001
4000e + 001
4500e + 001
5000e + 001
5500e + 001
6000e + 001
Use
r dat
a120590
XX
(b) 120590119909 using identified parameters
Use
r dat
a120590
YY
0000e + 000
4000e + 000
8000e + 000
1200e + 001
1600e + 001
2000e + 001
2400e + 001
2800e + 001
3200e + 001
3600e + 001
4400e + 001
4000e + 001
4800e + 001
(c) 120590119910 using theory parameters
Use
r dat
a120590
YY
0000e + 000
4000e + 000
8000e + 000
1200e + 001
1600e + 001
2000e + 001
2400e + 001
2800e + 001
3200e + 001
3600e + 001
4400e + 001
4000e + 001
4800e + 001
(d) 120590119910 using identified parameters
Figure 12 Calculated stress comparison between using theory value and identified value at stage 3
Step 5 If all individuals are evaluated then go to Step6 Otherwise go to Step 3Step 6 If the maximum cycle is reached or the bestindividuals (the parameter to be back recognized)are obtained then the cycle ends and outputs bestindividuals Otherwise go to Step 7Step 7Update the individuals according to (2) and (3)Step 8 Repeat Step 7 until all 119899 new individuals aregenerated They are used as offspringStep 9 Go to Step 3
44 Verification To verify the model we suppose there isa tunnel (see Figure 7) The size of tunnel geomechanicalparameters and in situ stress are listed in Figure 7 The valuein Figure 7 is theoretical values Displacement values for somekey points indicated by nodes are calculated by elastic finiteelement method The suggested algorithm above is used toidentify initial geostress components P1 and P2 and anglebetween P1 and P2 We used orthogonal experiment design
to create 25 sets of tentative geostresses P1 and P2 and anglebetween P1 and P2 The training samples will be obtainedthrough computing the displacement of each set of tentativegeostresses Then the LSSVMmodel was build based on (13)The training samples and model parameters of LSSVM arelisted in Table 5 In situ stresses P1 and P2 and angle atdifferent stages can be identified according to the procedureof Section 43 Identified in situ stress P1 and P2and angleat different stages are listed in Table 4 The comparisonbetween displacement of the key points using the theoreticalparameters and displacements identified by back analysisbased on ABC and LSSVM is shown in Figure 8 Stresses ofsurrounding rock are shown in Figure 9 after stage 3 Resultsshow the proposed method can effectively identify the in situstress
45 Discussions
451 Performance of LSSVM The performance of LSSVM isvery important to back analysis The predicted displacement
The Scientific World Journal 11
00000
10000
20000
30000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
minus10000
minus20000
minus30000
minus40000
minus50000
(a) Stage 1
00000
20000
40000
60000
80000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
minus20000
minus60000
minus80000
minus100000
minus40000
MP4
-x
MP4
-y
MP5
-x
MP5
-y
(b) Stage 2
Monitored displacementPredicted displacement using LSSVMComputed displacement using FEM
00000
50000
100000
150000
minus50000
minus100000
minus150000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
MP4
-x
MP4
-y
MP5
-x
MP5
-y
MP6
-x
MP6
-y
MP7
-x
MP7
-y
(c) Stage 3
Figure 13 Predicted displacement using LSSVM with calculated displacement using theory and identified parameters
using LSSVM is in well agreement with the calculateddisplacement using theory and identified parameters (shownin Figure 10) It shows the LSSVM model presents wellthe relationship between geomechanical parameters anddisplacement It improves the efficiency of back analysis usingLSSVM
452 Effect of Kernel Parameters In this study the RBFkernel functionwas adoptedThe relationship between fitnessand cycle is listed in Figure 11 with 120590 = 10 and 120590 = 1 Theperformance of LSSVM is listed in Figure 12 using 120590 = 10 and120590 = 1 Its show selecting the appropriate kernel parametersis important to back analysis But there is not any guide toselect kernel function and its parameters according to LSSVMtheory It can be acquired by error-and-trial
5 Conclusions
The paper presents a new methodology called back analysisbased on ABC ABC is used to identify the geomechanicalparameters based on monitored displacements Results ofcircular tunnel with the analytical solution illustrate clearlythat ABC is effectively able to search parameters of geo-material and has proved ABC has powerful global optimalperformance To improve the efficiency of back analysisLSSVMwas used to present the relationship between geome-chanical parameters and displacement instead of numericalanalysis Results of horseshoe tunnel without the analyticalsolution demonstrate that LSSVMpresents well the nonlinearrelationship between geomechanical parameters and moni-tored displacements The proposed approach improves the
12 The Scientific World Journal
0
005
01
015
02
025
0 200 400 600 800 1000
Fitn
ess
Cycle
120590 = 10
120590 = 1
Figure 14 Fitness with different parameters of kernel function
00000
50000
100000
150000
00000 50000 100000 150000
Com
pute
d di
spla
cem
ent u
sing
FEM
bas
ed o
n LS
SVM
(mm
)
Monitored displacement (mm)
120590 = 10
120590 = 1
minus150000
minus100000
minus50000
minus150000 minus100000 minus50000
Figure 15The performance of LSSVMwith different parameters ofkernel function
efficiency and precision of back analysis andmakes it possibleto be applied to more complex engineering problem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was also supported by the National Fund ofScience in China (no 41072224 51104057)
References
[1] L Jing ldquoA review of techniques advances and outstandingissues in numerical modelling for rock mechanics and rockengineeringrdquo International Journal of Rock Mechanics andMining Sciences vol 40 no 3 pp 283ndash353 2003
[2] P Oreste ldquoBack-analysis techniques for the improvement ofthe understanding of rock in underground constructionsrdquoTunnelling and Underground Space Technology vol 20 no 1 pp7ndash21 2005
[3] G Gioda and L Jurina ldquoNumerical identification( back cal-culation) of soil-structure interaction pressuresrdquo InternationalJournal for Numerical amp Analytical Methods in Geomechanicsvol 5 no 1 pp 33ndash56 1981
[4] S Sakurai and K Takeuchi ldquoBack analysis of measured dis-placements of tunnelsrdquo Rock Mechanics and Rock Engineeringvol 16 no 3 pp 173ndash180 1983
[5] S Sakurai N Dees Wasmongkol and M Shinji ldquoBack analysisfor determining material characteristics in cut slopesrdquo inProceedings of the International Symposium on ECRF pp 770ndash776 Beijing China 1986
[6] S Sakurai ldquoInterpretation of the results of displacement mea-surements in cut slopesrdquo in Proceedings of the 2nd InternationalSymposium on Field Measurements in Geomechanics (FMGMrsquo87) pp 2528ndash2540 Kobe Japan 1987
[7] Z L Feng and R W Lewis ldquoOptimal estimation of in-situground stresses from displacement measurementsrdquo Interna-tional Journal for Numerical amp Analytical Methods in Geome-chanics vol 11 no 4 pp 391ndash408 1987
[8] B Pichler R Lackner and H A Mang ldquoBack analysis ofmodel parameters in geotechnical engineering by means ofsoft computingrdquo International Journal for Numerical Methods inEngineering vol 57 no 14 pp 1943ndash1978 2003
[9] F Xia-Ting and J A Hudson Rock Engineering Design CRCPress New York NY USA 2011
[10] T Okabe K Hayashi N Shinohara and S Takasugi ldquoInversionof drilling-induced tensile fracture data obtained from a singleinclined boreholerdquo International Journal of Rock Mechanics andMining Sciences vol 35 no 6 pp 747ndash758 1998
[11] W-G William and Y S Yoon ldquoAquifer parameter identifi-cation with optimum dimension in parameterizationrdquo WaterResources Research vol 17 no 3 pp 664ndash672 1981
[12] A Cividini G Maier and A Nappi ldquoParameter estimation ofa static geotechnical model using a Bayesrsquo approachrdquo Interna-tional Journal of Rock Mechanics and Mining Sciences vol 20no 5 pp 215ndash226 1983
[13] S VardakosM Gutierrez andC Xia ldquoParameter identificationin numerical modeling of tunneling using the DifferentialEvolution Genetic Algorithm (DEGA)rdquo Tunnelling and Under-ground Space Technology vol 28 no 1 pp 109ndash123 2012
[14] H Zhao and S Yin ldquoGeomechanical parameters identificationby particle swarm optimization and support vector machinerdquoApplied Mathematical Modelling vol 33 no 10 pp 3997ndash40122009
[15] X Feng H Zhao and S Li ldquoA new displacement backanalysis to identify mechanical geo-material parameters basedon hybrid intelligent methodologyrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 28 no11 pp 1141ndash1165 2004
[16] Y Yu B Zhang and H Yuan ldquoAn intelligent displacementback-analysis method for earth-rockfill damsrdquo Computers andGeotechnics vol 34 no 6 pp 423ndash434 2007
The Scientific World Journal 13
[17] J H Deng and C F Lee ldquoDiplacement back analysis for a steepslope at the Three Gorges Project siterdquo International Journal ofRockMechanics andMining Sciences vol 38 no 2 pp 259ndash2682001
[18] D Karaboga ldquoAn idea based on honey bee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University 2005
[19] D Karaboga and C Ozturk ldquoA novel clustering approachartificial Bee Colony (ABC) algorithmrdquoApplied Soft ComputingJournal vol 11 no 1 pp 652ndash657 2011
[20] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008
[21] J A K Suykens and J Vandewalle ldquoLeast squares supportvector machine classifiersrdquo Neural Processing Letters vol 9 no3 pp 293ndash300 1999
[22] M E Duncan Fama ldquoNumerical modeling of yield zones inweak rocksrdquo in Comprehensive Rock Engineering J A HudsonEd vol 2 pp 49ndash75 Pergamon Oxford UK 1993
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
The Scientific World Journal 9
Com
pute
d di
spla
cem
ent u
sing
iden
tified
par
amet
ers
Monitored displacement (mm)
00
50
100
150
00 50 100 150minus150 minus100 minus50
minus150
minus100
minus50
(a) Stage 1
Com
pute
d di
spla
cem
ent u
sing
iden
tified
par
amet
ers
Monitored displacement (mm)
00
50
100
150
00 50 100 150minus150 minus100 minus50
minus150
minus100
minus50
(b) Stage 2
Stage 1Stage 2Stage 3
00
50
100
150
00 50 100 150
Com
pute
d di
spla
cem
ent u
sing
iden
tified
par
amet
ers
Monitored displacement (mm)minus150 minus100 minus50
minus150
minus100
minus50
(c) Stage 3
Figure 11 Comparison between monitored displacement and predicted displacement using identified parameters
43 Procedure of Back Analysis Algorithm Based on LSSVMand ABC After the LSSVM model representing the non-linear relation between the displacement and a parameteris obtained it can be used to predict displacements atmonitored points instead of numerical analysis ABC is usedto search the optimal parameter to be identified based on theerror function (see (10)) The back analysis technique basedon LSSVM-ABC combination can be described as follows
Step 1 Determine ABC parameters and the range ofparameters to be recognized
Step 2 Generate randomly 119899 group of parameters attheir given range Each individual represents an initialsolution
Step 3 Input a set of rock mass parameters to themodel LSSVM(X) obtained above to calculate thedisplacement values at given monitoring points
Step 4 Use (10) to evaluate the fitness of the currentindividuals that is the reasonability of the parameterset
10 The Scientific World Journal
0000e + 000
5000e + 000
1000e + 001
1500e + 001
2000e + 001
2500e + 001
3000e + 001
3500e + 001
4000e + 001
4500e + 001
5000e + 001
5500e + 001
6000e + 001
Use
r dat
a120590
XX
(a) 120590119909 using theory parameters
0000e + 000
5000e + 000
1000e + 001
1500e + 001
2000e + 001
2500e + 001
3000e + 001
3500e + 001
4000e + 001
4500e + 001
5000e + 001
5500e + 001
6000e + 001
Use
r dat
a120590
XX
(b) 120590119909 using identified parameters
Use
r dat
a120590
YY
0000e + 000
4000e + 000
8000e + 000
1200e + 001
1600e + 001
2000e + 001
2400e + 001
2800e + 001
3200e + 001
3600e + 001
4400e + 001
4000e + 001
4800e + 001
(c) 120590119910 using theory parameters
Use
r dat
a120590
YY
0000e + 000
4000e + 000
8000e + 000
1200e + 001
1600e + 001
2000e + 001
2400e + 001
2800e + 001
3200e + 001
3600e + 001
4400e + 001
4000e + 001
4800e + 001
(d) 120590119910 using identified parameters
Figure 12 Calculated stress comparison between using theory value and identified value at stage 3
Step 5 If all individuals are evaluated then go to Step6 Otherwise go to Step 3Step 6 If the maximum cycle is reached or the bestindividuals (the parameter to be back recognized)are obtained then the cycle ends and outputs bestindividuals Otherwise go to Step 7Step 7Update the individuals according to (2) and (3)Step 8 Repeat Step 7 until all 119899 new individuals aregenerated They are used as offspringStep 9 Go to Step 3
44 Verification To verify the model we suppose there isa tunnel (see Figure 7) The size of tunnel geomechanicalparameters and in situ stress are listed in Figure 7 The valuein Figure 7 is theoretical values Displacement values for somekey points indicated by nodes are calculated by elastic finiteelement method The suggested algorithm above is used toidentify initial geostress components P1 and P2 and anglebetween P1 and P2 We used orthogonal experiment design
to create 25 sets of tentative geostresses P1 and P2 and anglebetween P1 and P2 The training samples will be obtainedthrough computing the displacement of each set of tentativegeostresses Then the LSSVMmodel was build based on (13)The training samples and model parameters of LSSVM arelisted in Table 5 In situ stresses P1 and P2 and angle atdifferent stages can be identified according to the procedureof Section 43 Identified in situ stress P1 and P2and angleat different stages are listed in Table 4 The comparisonbetween displacement of the key points using the theoreticalparameters and displacements identified by back analysisbased on ABC and LSSVM is shown in Figure 8 Stresses ofsurrounding rock are shown in Figure 9 after stage 3 Resultsshow the proposed method can effectively identify the in situstress
45 Discussions
451 Performance of LSSVM The performance of LSSVM isvery important to back analysis The predicted displacement
The Scientific World Journal 11
00000
10000
20000
30000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
minus10000
minus20000
minus30000
minus40000
minus50000
(a) Stage 1
00000
20000
40000
60000
80000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
minus20000
minus60000
minus80000
minus100000
minus40000
MP4
-x
MP4
-y
MP5
-x
MP5
-y
(b) Stage 2
Monitored displacementPredicted displacement using LSSVMComputed displacement using FEM
00000
50000
100000
150000
minus50000
minus100000
minus150000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
MP4
-x
MP4
-y
MP5
-x
MP5
-y
MP6
-x
MP6
-y
MP7
-x
MP7
-y
(c) Stage 3
Figure 13 Predicted displacement using LSSVM with calculated displacement using theory and identified parameters
using LSSVM is in well agreement with the calculateddisplacement using theory and identified parameters (shownin Figure 10) It shows the LSSVM model presents wellthe relationship between geomechanical parameters anddisplacement It improves the efficiency of back analysis usingLSSVM
452 Effect of Kernel Parameters In this study the RBFkernel functionwas adoptedThe relationship between fitnessand cycle is listed in Figure 11 with 120590 = 10 and 120590 = 1 Theperformance of LSSVM is listed in Figure 12 using 120590 = 10 and120590 = 1 Its show selecting the appropriate kernel parametersis important to back analysis But there is not any guide toselect kernel function and its parameters according to LSSVMtheory It can be acquired by error-and-trial
5 Conclusions
The paper presents a new methodology called back analysisbased on ABC ABC is used to identify the geomechanicalparameters based on monitored displacements Results ofcircular tunnel with the analytical solution illustrate clearlythat ABC is effectively able to search parameters of geo-material and has proved ABC has powerful global optimalperformance To improve the efficiency of back analysisLSSVMwas used to present the relationship between geome-chanical parameters and displacement instead of numericalanalysis Results of horseshoe tunnel without the analyticalsolution demonstrate that LSSVMpresents well the nonlinearrelationship between geomechanical parameters and moni-tored displacements The proposed approach improves the
12 The Scientific World Journal
0
005
01
015
02
025
0 200 400 600 800 1000
Fitn
ess
Cycle
120590 = 10
120590 = 1
Figure 14 Fitness with different parameters of kernel function
00000
50000
100000
150000
00000 50000 100000 150000
Com
pute
d di
spla
cem
ent u
sing
FEM
bas
ed o
n LS
SVM
(mm
)
Monitored displacement (mm)
120590 = 10
120590 = 1
minus150000
minus100000
minus50000
minus150000 minus100000 minus50000
Figure 15The performance of LSSVMwith different parameters ofkernel function
efficiency and precision of back analysis andmakes it possibleto be applied to more complex engineering problem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was also supported by the National Fund ofScience in China (no 41072224 51104057)
References
[1] L Jing ldquoA review of techniques advances and outstandingissues in numerical modelling for rock mechanics and rockengineeringrdquo International Journal of Rock Mechanics andMining Sciences vol 40 no 3 pp 283ndash353 2003
[2] P Oreste ldquoBack-analysis techniques for the improvement ofthe understanding of rock in underground constructionsrdquoTunnelling and Underground Space Technology vol 20 no 1 pp7ndash21 2005
[3] G Gioda and L Jurina ldquoNumerical identification( back cal-culation) of soil-structure interaction pressuresrdquo InternationalJournal for Numerical amp Analytical Methods in Geomechanicsvol 5 no 1 pp 33ndash56 1981
[4] S Sakurai and K Takeuchi ldquoBack analysis of measured dis-placements of tunnelsrdquo Rock Mechanics and Rock Engineeringvol 16 no 3 pp 173ndash180 1983
[5] S Sakurai N Dees Wasmongkol and M Shinji ldquoBack analysisfor determining material characteristics in cut slopesrdquo inProceedings of the International Symposium on ECRF pp 770ndash776 Beijing China 1986
[6] S Sakurai ldquoInterpretation of the results of displacement mea-surements in cut slopesrdquo in Proceedings of the 2nd InternationalSymposium on Field Measurements in Geomechanics (FMGMrsquo87) pp 2528ndash2540 Kobe Japan 1987
[7] Z L Feng and R W Lewis ldquoOptimal estimation of in-situground stresses from displacement measurementsrdquo Interna-tional Journal for Numerical amp Analytical Methods in Geome-chanics vol 11 no 4 pp 391ndash408 1987
[8] B Pichler R Lackner and H A Mang ldquoBack analysis ofmodel parameters in geotechnical engineering by means ofsoft computingrdquo International Journal for Numerical Methods inEngineering vol 57 no 14 pp 1943ndash1978 2003
[9] F Xia-Ting and J A Hudson Rock Engineering Design CRCPress New York NY USA 2011
[10] T Okabe K Hayashi N Shinohara and S Takasugi ldquoInversionof drilling-induced tensile fracture data obtained from a singleinclined boreholerdquo International Journal of Rock Mechanics andMining Sciences vol 35 no 6 pp 747ndash758 1998
[11] W-G William and Y S Yoon ldquoAquifer parameter identifi-cation with optimum dimension in parameterizationrdquo WaterResources Research vol 17 no 3 pp 664ndash672 1981
[12] A Cividini G Maier and A Nappi ldquoParameter estimation ofa static geotechnical model using a Bayesrsquo approachrdquo Interna-tional Journal of Rock Mechanics and Mining Sciences vol 20no 5 pp 215ndash226 1983
[13] S VardakosM Gutierrez andC Xia ldquoParameter identificationin numerical modeling of tunneling using the DifferentialEvolution Genetic Algorithm (DEGA)rdquo Tunnelling and Under-ground Space Technology vol 28 no 1 pp 109ndash123 2012
[14] H Zhao and S Yin ldquoGeomechanical parameters identificationby particle swarm optimization and support vector machinerdquoApplied Mathematical Modelling vol 33 no 10 pp 3997ndash40122009
[15] X Feng H Zhao and S Li ldquoA new displacement backanalysis to identify mechanical geo-material parameters basedon hybrid intelligent methodologyrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 28 no11 pp 1141ndash1165 2004
[16] Y Yu B Zhang and H Yuan ldquoAn intelligent displacementback-analysis method for earth-rockfill damsrdquo Computers andGeotechnics vol 34 no 6 pp 423ndash434 2007
The Scientific World Journal 13
[17] J H Deng and C F Lee ldquoDiplacement back analysis for a steepslope at the Three Gorges Project siterdquo International Journal ofRockMechanics andMining Sciences vol 38 no 2 pp 259ndash2682001
[18] D Karaboga ldquoAn idea based on honey bee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University 2005
[19] D Karaboga and C Ozturk ldquoA novel clustering approachartificial Bee Colony (ABC) algorithmrdquoApplied Soft ComputingJournal vol 11 no 1 pp 652ndash657 2011
[20] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008
[21] J A K Suykens and J Vandewalle ldquoLeast squares supportvector machine classifiersrdquo Neural Processing Letters vol 9 no3 pp 293ndash300 1999
[22] M E Duncan Fama ldquoNumerical modeling of yield zones inweak rocksrdquo in Comprehensive Rock Engineering J A HudsonEd vol 2 pp 49ndash75 Pergamon Oxford UK 1993
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
10 The Scientific World Journal
0000e + 000
5000e + 000
1000e + 001
1500e + 001
2000e + 001
2500e + 001
3000e + 001
3500e + 001
4000e + 001
4500e + 001
5000e + 001
5500e + 001
6000e + 001
Use
r dat
a120590
XX
(a) 120590119909 using theory parameters
0000e + 000
5000e + 000
1000e + 001
1500e + 001
2000e + 001
2500e + 001
3000e + 001
3500e + 001
4000e + 001
4500e + 001
5000e + 001
5500e + 001
6000e + 001
Use
r dat
a120590
XX
(b) 120590119909 using identified parameters
Use
r dat
a120590
YY
0000e + 000
4000e + 000
8000e + 000
1200e + 001
1600e + 001
2000e + 001
2400e + 001
2800e + 001
3200e + 001
3600e + 001
4400e + 001
4000e + 001
4800e + 001
(c) 120590119910 using theory parameters
Use
r dat
a120590
YY
0000e + 000
4000e + 000
8000e + 000
1200e + 001
1600e + 001
2000e + 001
2400e + 001
2800e + 001
3200e + 001
3600e + 001
4400e + 001
4000e + 001
4800e + 001
(d) 120590119910 using identified parameters
Figure 12 Calculated stress comparison between using theory value and identified value at stage 3
Step 5 If all individuals are evaluated then go to Step6 Otherwise go to Step 3Step 6 If the maximum cycle is reached or the bestindividuals (the parameter to be back recognized)are obtained then the cycle ends and outputs bestindividuals Otherwise go to Step 7Step 7Update the individuals according to (2) and (3)Step 8 Repeat Step 7 until all 119899 new individuals aregenerated They are used as offspringStep 9 Go to Step 3
44 Verification To verify the model we suppose there isa tunnel (see Figure 7) The size of tunnel geomechanicalparameters and in situ stress are listed in Figure 7 The valuein Figure 7 is theoretical values Displacement values for somekey points indicated by nodes are calculated by elastic finiteelement method The suggested algorithm above is used toidentify initial geostress components P1 and P2 and anglebetween P1 and P2 We used orthogonal experiment design
to create 25 sets of tentative geostresses P1 and P2 and anglebetween P1 and P2 The training samples will be obtainedthrough computing the displacement of each set of tentativegeostresses Then the LSSVMmodel was build based on (13)The training samples and model parameters of LSSVM arelisted in Table 5 In situ stresses P1 and P2 and angle atdifferent stages can be identified according to the procedureof Section 43 Identified in situ stress P1 and P2and angleat different stages are listed in Table 4 The comparisonbetween displacement of the key points using the theoreticalparameters and displacements identified by back analysisbased on ABC and LSSVM is shown in Figure 8 Stresses ofsurrounding rock are shown in Figure 9 after stage 3 Resultsshow the proposed method can effectively identify the in situstress
45 Discussions
451 Performance of LSSVM The performance of LSSVM isvery important to back analysis The predicted displacement
The Scientific World Journal 11
00000
10000
20000
30000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
minus10000
minus20000
minus30000
minus40000
minus50000
(a) Stage 1
00000
20000
40000
60000
80000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
minus20000
minus60000
minus80000
minus100000
minus40000
MP4
-x
MP4
-y
MP5
-x
MP5
-y
(b) Stage 2
Monitored displacementPredicted displacement using LSSVMComputed displacement using FEM
00000
50000
100000
150000
minus50000
minus100000
minus150000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
MP4
-x
MP4
-y
MP5
-x
MP5
-y
MP6
-x
MP6
-y
MP7
-x
MP7
-y
(c) Stage 3
Figure 13 Predicted displacement using LSSVM with calculated displacement using theory and identified parameters
using LSSVM is in well agreement with the calculateddisplacement using theory and identified parameters (shownin Figure 10) It shows the LSSVM model presents wellthe relationship between geomechanical parameters anddisplacement It improves the efficiency of back analysis usingLSSVM
452 Effect of Kernel Parameters In this study the RBFkernel functionwas adoptedThe relationship between fitnessand cycle is listed in Figure 11 with 120590 = 10 and 120590 = 1 Theperformance of LSSVM is listed in Figure 12 using 120590 = 10 and120590 = 1 Its show selecting the appropriate kernel parametersis important to back analysis But there is not any guide toselect kernel function and its parameters according to LSSVMtheory It can be acquired by error-and-trial
5 Conclusions
The paper presents a new methodology called back analysisbased on ABC ABC is used to identify the geomechanicalparameters based on monitored displacements Results ofcircular tunnel with the analytical solution illustrate clearlythat ABC is effectively able to search parameters of geo-material and has proved ABC has powerful global optimalperformance To improve the efficiency of back analysisLSSVMwas used to present the relationship between geome-chanical parameters and displacement instead of numericalanalysis Results of horseshoe tunnel without the analyticalsolution demonstrate that LSSVMpresents well the nonlinearrelationship between geomechanical parameters and moni-tored displacements The proposed approach improves the
12 The Scientific World Journal
0
005
01
015
02
025
0 200 400 600 800 1000
Fitn
ess
Cycle
120590 = 10
120590 = 1
Figure 14 Fitness with different parameters of kernel function
00000
50000
100000
150000
00000 50000 100000 150000
Com
pute
d di
spla
cem
ent u
sing
FEM
bas
ed o
n LS
SVM
(mm
)
Monitored displacement (mm)
120590 = 10
120590 = 1
minus150000
minus100000
minus50000
minus150000 minus100000 minus50000
Figure 15The performance of LSSVMwith different parameters ofkernel function
efficiency and precision of back analysis andmakes it possibleto be applied to more complex engineering problem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was also supported by the National Fund ofScience in China (no 41072224 51104057)
References
[1] L Jing ldquoA review of techniques advances and outstandingissues in numerical modelling for rock mechanics and rockengineeringrdquo International Journal of Rock Mechanics andMining Sciences vol 40 no 3 pp 283ndash353 2003
[2] P Oreste ldquoBack-analysis techniques for the improvement ofthe understanding of rock in underground constructionsrdquoTunnelling and Underground Space Technology vol 20 no 1 pp7ndash21 2005
[3] G Gioda and L Jurina ldquoNumerical identification( back cal-culation) of soil-structure interaction pressuresrdquo InternationalJournal for Numerical amp Analytical Methods in Geomechanicsvol 5 no 1 pp 33ndash56 1981
[4] S Sakurai and K Takeuchi ldquoBack analysis of measured dis-placements of tunnelsrdquo Rock Mechanics and Rock Engineeringvol 16 no 3 pp 173ndash180 1983
[5] S Sakurai N Dees Wasmongkol and M Shinji ldquoBack analysisfor determining material characteristics in cut slopesrdquo inProceedings of the International Symposium on ECRF pp 770ndash776 Beijing China 1986
[6] S Sakurai ldquoInterpretation of the results of displacement mea-surements in cut slopesrdquo in Proceedings of the 2nd InternationalSymposium on Field Measurements in Geomechanics (FMGMrsquo87) pp 2528ndash2540 Kobe Japan 1987
[7] Z L Feng and R W Lewis ldquoOptimal estimation of in-situground stresses from displacement measurementsrdquo Interna-tional Journal for Numerical amp Analytical Methods in Geome-chanics vol 11 no 4 pp 391ndash408 1987
[8] B Pichler R Lackner and H A Mang ldquoBack analysis ofmodel parameters in geotechnical engineering by means ofsoft computingrdquo International Journal for Numerical Methods inEngineering vol 57 no 14 pp 1943ndash1978 2003
[9] F Xia-Ting and J A Hudson Rock Engineering Design CRCPress New York NY USA 2011
[10] T Okabe K Hayashi N Shinohara and S Takasugi ldquoInversionof drilling-induced tensile fracture data obtained from a singleinclined boreholerdquo International Journal of Rock Mechanics andMining Sciences vol 35 no 6 pp 747ndash758 1998
[11] W-G William and Y S Yoon ldquoAquifer parameter identifi-cation with optimum dimension in parameterizationrdquo WaterResources Research vol 17 no 3 pp 664ndash672 1981
[12] A Cividini G Maier and A Nappi ldquoParameter estimation ofa static geotechnical model using a Bayesrsquo approachrdquo Interna-tional Journal of Rock Mechanics and Mining Sciences vol 20no 5 pp 215ndash226 1983
[13] S VardakosM Gutierrez andC Xia ldquoParameter identificationin numerical modeling of tunneling using the DifferentialEvolution Genetic Algorithm (DEGA)rdquo Tunnelling and Under-ground Space Technology vol 28 no 1 pp 109ndash123 2012
[14] H Zhao and S Yin ldquoGeomechanical parameters identificationby particle swarm optimization and support vector machinerdquoApplied Mathematical Modelling vol 33 no 10 pp 3997ndash40122009
[15] X Feng H Zhao and S Li ldquoA new displacement backanalysis to identify mechanical geo-material parameters basedon hybrid intelligent methodologyrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 28 no11 pp 1141ndash1165 2004
[16] Y Yu B Zhang and H Yuan ldquoAn intelligent displacementback-analysis method for earth-rockfill damsrdquo Computers andGeotechnics vol 34 no 6 pp 423ndash434 2007
The Scientific World Journal 13
[17] J H Deng and C F Lee ldquoDiplacement back analysis for a steepslope at the Three Gorges Project siterdquo International Journal ofRockMechanics andMining Sciences vol 38 no 2 pp 259ndash2682001
[18] D Karaboga ldquoAn idea based on honey bee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University 2005
[19] D Karaboga and C Ozturk ldquoA novel clustering approachartificial Bee Colony (ABC) algorithmrdquoApplied Soft ComputingJournal vol 11 no 1 pp 652ndash657 2011
[20] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008
[21] J A K Suykens and J Vandewalle ldquoLeast squares supportvector machine classifiersrdquo Neural Processing Letters vol 9 no3 pp 293ndash300 1999
[22] M E Duncan Fama ldquoNumerical modeling of yield zones inweak rocksrdquo in Comprehensive Rock Engineering J A HudsonEd vol 2 pp 49ndash75 Pergamon Oxford UK 1993
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
The Scientific World Journal 11
00000
10000
20000
30000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
minus10000
minus20000
minus30000
minus40000
minus50000
(a) Stage 1
00000
20000
40000
60000
80000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
minus20000
minus60000
minus80000
minus100000
minus40000
MP4
-x
MP4
-y
MP5
-x
MP5
-y
(b) Stage 2
Monitored displacementPredicted displacement using LSSVMComputed displacement using FEM
00000
50000
100000
150000
minus50000
minus100000
minus150000
MP1
-x
MP1
-y
MP2
-x
MP2
-y
MP3
-x
MP3
-y
MP4
-x
MP4
-y
MP5
-x
MP5
-y
MP6
-x
MP6
-y
MP7
-x
MP7
-y
(c) Stage 3
Figure 13 Predicted displacement using LSSVM with calculated displacement using theory and identified parameters
using LSSVM is in well agreement with the calculateddisplacement using theory and identified parameters (shownin Figure 10) It shows the LSSVM model presents wellthe relationship between geomechanical parameters anddisplacement It improves the efficiency of back analysis usingLSSVM
452 Effect of Kernel Parameters In this study the RBFkernel functionwas adoptedThe relationship between fitnessand cycle is listed in Figure 11 with 120590 = 10 and 120590 = 1 Theperformance of LSSVM is listed in Figure 12 using 120590 = 10 and120590 = 1 Its show selecting the appropriate kernel parametersis important to back analysis But there is not any guide toselect kernel function and its parameters according to LSSVMtheory It can be acquired by error-and-trial
5 Conclusions
The paper presents a new methodology called back analysisbased on ABC ABC is used to identify the geomechanicalparameters based on monitored displacements Results ofcircular tunnel with the analytical solution illustrate clearlythat ABC is effectively able to search parameters of geo-material and has proved ABC has powerful global optimalperformance To improve the efficiency of back analysisLSSVMwas used to present the relationship between geome-chanical parameters and displacement instead of numericalanalysis Results of horseshoe tunnel without the analyticalsolution demonstrate that LSSVMpresents well the nonlinearrelationship between geomechanical parameters and moni-tored displacements The proposed approach improves the
12 The Scientific World Journal
0
005
01
015
02
025
0 200 400 600 800 1000
Fitn
ess
Cycle
120590 = 10
120590 = 1
Figure 14 Fitness with different parameters of kernel function
00000
50000
100000
150000
00000 50000 100000 150000
Com
pute
d di
spla
cem
ent u
sing
FEM
bas
ed o
n LS
SVM
(mm
)
Monitored displacement (mm)
120590 = 10
120590 = 1
minus150000
minus100000
minus50000
minus150000 minus100000 minus50000
Figure 15The performance of LSSVMwith different parameters ofkernel function
efficiency and precision of back analysis andmakes it possibleto be applied to more complex engineering problem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was also supported by the National Fund ofScience in China (no 41072224 51104057)
References
[1] L Jing ldquoA review of techniques advances and outstandingissues in numerical modelling for rock mechanics and rockengineeringrdquo International Journal of Rock Mechanics andMining Sciences vol 40 no 3 pp 283ndash353 2003
[2] P Oreste ldquoBack-analysis techniques for the improvement ofthe understanding of rock in underground constructionsrdquoTunnelling and Underground Space Technology vol 20 no 1 pp7ndash21 2005
[3] G Gioda and L Jurina ldquoNumerical identification( back cal-culation) of soil-structure interaction pressuresrdquo InternationalJournal for Numerical amp Analytical Methods in Geomechanicsvol 5 no 1 pp 33ndash56 1981
[4] S Sakurai and K Takeuchi ldquoBack analysis of measured dis-placements of tunnelsrdquo Rock Mechanics and Rock Engineeringvol 16 no 3 pp 173ndash180 1983
[5] S Sakurai N Dees Wasmongkol and M Shinji ldquoBack analysisfor determining material characteristics in cut slopesrdquo inProceedings of the International Symposium on ECRF pp 770ndash776 Beijing China 1986
[6] S Sakurai ldquoInterpretation of the results of displacement mea-surements in cut slopesrdquo in Proceedings of the 2nd InternationalSymposium on Field Measurements in Geomechanics (FMGMrsquo87) pp 2528ndash2540 Kobe Japan 1987
[7] Z L Feng and R W Lewis ldquoOptimal estimation of in-situground stresses from displacement measurementsrdquo Interna-tional Journal for Numerical amp Analytical Methods in Geome-chanics vol 11 no 4 pp 391ndash408 1987
[8] B Pichler R Lackner and H A Mang ldquoBack analysis ofmodel parameters in geotechnical engineering by means ofsoft computingrdquo International Journal for Numerical Methods inEngineering vol 57 no 14 pp 1943ndash1978 2003
[9] F Xia-Ting and J A Hudson Rock Engineering Design CRCPress New York NY USA 2011
[10] T Okabe K Hayashi N Shinohara and S Takasugi ldquoInversionof drilling-induced tensile fracture data obtained from a singleinclined boreholerdquo International Journal of Rock Mechanics andMining Sciences vol 35 no 6 pp 747ndash758 1998
[11] W-G William and Y S Yoon ldquoAquifer parameter identifi-cation with optimum dimension in parameterizationrdquo WaterResources Research vol 17 no 3 pp 664ndash672 1981
[12] A Cividini G Maier and A Nappi ldquoParameter estimation ofa static geotechnical model using a Bayesrsquo approachrdquo Interna-tional Journal of Rock Mechanics and Mining Sciences vol 20no 5 pp 215ndash226 1983
[13] S VardakosM Gutierrez andC Xia ldquoParameter identificationin numerical modeling of tunneling using the DifferentialEvolution Genetic Algorithm (DEGA)rdquo Tunnelling and Under-ground Space Technology vol 28 no 1 pp 109ndash123 2012
[14] H Zhao and S Yin ldquoGeomechanical parameters identificationby particle swarm optimization and support vector machinerdquoApplied Mathematical Modelling vol 33 no 10 pp 3997ndash40122009
[15] X Feng H Zhao and S Li ldquoA new displacement backanalysis to identify mechanical geo-material parameters basedon hybrid intelligent methodologyrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 28 no11 pp 1141ndash1165 2004
[16] Y Yu B Zhang and H Yuan ldquoAn intelligent displacementback-analysis method for earth-rockfill damsrdquo Computers andGeotechnics vol 34 no 6 pp 423ndash434 2007
The Scientific World Journal 13
[17] J H Deng and C F Lee ldquoDiplacement back analysis for a steepslope at the Three Gorges Project siterdquo International Journal ofRockMechanics andMining Sciences vol 38 no 2 pp 259ndash2682001
[18] D Karaboga ldquoAn idea based on honey bee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University 2005
[19] D Karaboga and C Ozturk ldquoA novel clustering approachartificial Bee Colony (ABC) algorithmrdquoApplied Soft ComputingJournal vol 11 no 1 pp 652ndash657 2011
[20] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008
[21] J A K Suykens and J Vandewalle ldquoLeast squares supportvector machine classifiersrdquo Neural Processing Letters vol 9 no3 pp 293ndash300 1999
[22] M E Duncan Fama ldquoNumerical modeling of yield zones inweak rocksrdquo in Comprehensive Rock Engineering J A HudsonEd vol 2 pp 49ndash75 Pergamon Oxford UK 1993
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
12 The Scientific World Journal
0
005
01
015
02
025
0 200 400 600 800 1000
Fitn
ess
Cycle
120590 = 10
120590 = 1
Figure 14 Fitness with different parameters of kernel function
00000
50000
100000
150000
00000 50000 100000 150000
Com
pute
d di
spla
cem
ent u
sing
FEM
bas
ed o
n LS
SVM
(mm
)
Monitored displacement (mm)
120590 = 10
120590 = 1
minus150000
minus100000
minus50000
minus150000 minus100000 minus50000
Figure 15The performance of LSSVMwith different parameters ofkernel function
efficiency and precision of back analysis andmakes it possibleto be applied to more complex engineering problem
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This research was also supported by the National Fund ofScience in China (no 41072224 51104057)
References
[1] L Jing ldquoA review of techniques advances and outstandingissues in numerical modelling for rock mechanics and rockengineeringrdquo International Journal of Rock Mechanics andMining Sciences vol 40 no 3 pp 283ndash353 2003
[2] P Oreste ldquoBack-analysis techniques for the improvement ofthe understanding of rock in underground constructionsrdquoTunnelling and Underground Space Technology vol 20 no 1 pp7ndash21 2005
[3] G Gioda and L Jurina ldquoNumerical identification( back cal-culation) of soil-structure interaction pressuresrdquo InternationalJournal for Numerical amp Analytical Methods in Geomechanicsvol 5 no 1 pp 33ndash56 1981
[4] S Sakurai and K Takeuchi ldquoBack analysis of measured dis-placements of tunnelsrdquo Rock Mechanics and Rock Engineeringvol 16 no 3 pp 173ndash180 1983
[5] S Sakurai N Dees Wasmongkol and M Shinji ldquoBack analysisfor determining material characteristics in cut slopesrdquo inProceedings of the International Symposium on ECRF pp 770ndash776 Beijing China 1986
[6] S Sakurai ldquoInterpretation of the results of displacement mea-surements in cut slopesrdquo in Proceedings of the 2nd InternationalSymposium on Field Measurements in Geomechanics (FMGMrsquo87) pp 2528ndash2540 Kobe Japan 1987
[7] Z L Feng and R W Lewis ldquoOptimal estimation of in-situground stresses from displacement measurementsrdquo Interna-tional Journal for Numerical amp Analytical Methods in Geome-chanics vol 11 no 4 pp 391ndash408 1987
[8] B Pichler R Lackner and H A Mang ldquoBack analysis ofmodel parameters in geotechnical engineering by means ofsoft computingrdquo International Journal for Numerical Methods inEngineering vol 57 no 14 pp 1943ndash1978 2003
[9] F Xia-Ting and J A Hudson Rock Engineering Design CRCPress New York NY USA 2011
[10] T Okabe K Hayashi N Shinohara and S Takasugi ldquoInversionof drilling-induced tensile fracture data obtained from a singleinclined boreholerdquo International Journal of Rock Mechanics andMining Sciences vol 35 no 6 pp 747ndash758 1998
[11] W-G William and Y S Yoon ldquoAquifer parameter identifi-cation with optimum dimension in parameterizationrdquo WaterResources Research vol 17 no 3 pp 664ndash672 1981
[12] A Cividini G Maier and A Nappi ldquoParameter estimation ofa static geotechnical model using a Bayesrsquo approachrdquo Interna-tional Journal of Rock Mechanics and Mining Sciences vol 20no 5 pp 215ndash226 1983
[13] S VardakosM Gutierrez andC Xia ldquoParameter identificationin numerical modeling of tunneling using the DifferentialEvolution Genetic Algorithm (DEGA)rdquo Tunnelling and Under-ground Space Technology vol 28 no 1 pp 109ndash123 2012
[14] H Zhao and S Yin ldquoGeomechanical parameters identificationby particle swarm optimization and support vector machinerdquoApplied Mathematical Modelling vol 33 no 10 pp 3997ndash40122009
[15] X Feng H Zhao and S Li ldquoA new displacement backanalysis to identify mechanical geo-material parameters basedon hybrid intelligent methodologyrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 28 no11 pp 1141ndash1165 2004
[16] Y Yu B Zhang and H Yuan ldquoAn intelligent displacementback-analysis method for earth-rockfill damsrdquo Computers andGeotechnics vol 34 no 6 pp 423ndash434 2007
The Scientific World Journal 13
[17] J H Deng and C F Lee ldquoDiplacement back analysis for a steepslope at the Three Gorges Project siterdquo International Journal ofRockMechanics andMining Sciences vol 38 no 2 pp 259ndash2682001
[18] D Karaboga ldquoAn idea based on honey bee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University 2005
[19] D Karaboga and C Ozturk ldquoA novel clustering approachartificial Bee Colony (ABC) algorithmrdquoApplied Soft ComputingJournal vol 11 no 1 pp 652ndash657 2011
[20] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008
[21] J A K Suykens and J Vandewalle ldquoLeast squares supportvector machine classifiersrdquo Neural Processing Letters vol 9 no3 pp 293ndash300 1999
[22] M E Duncan Fama ldquoNumerical modeling of yield zones inweak rocksrdquo in Comprehensive Rock Engineering J A HudsonEd vol 2 pp 49ndash75 Pergamon Oxford UK 1993
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
The Scientific World Journal 13
[17] J H Deng and C F Lee ldquoDiplacement back analysis for a steepslope at the Three Gorges Project siterdquo International Journal ofRockMechanics andMining Sciences vol 38 no 2 pp 259ndash2682001
[18] D Karaboga ldquoAn idea based on honey bee swarm for numer-ical optimizationrdquo Tech Rep TR06 Computer EngineeringDepartment Engineering Faculty Erciyes University 2005
[19] D Karaboga and C Ozturk ldquoA novel clustering approachartificial Bee Colony (ABC) algorithmrdquoApplied Soft ComputingJournal vol 11 no 1 pp 652ndash657 2011
[20] D Karaboga and B Basturk ldquoOn the performance of artificialbee colony (ABC) algorithmrdquo Applied Soft Computing Journalvol 8 no 1 pp 687ndash697 2008
[21] J A K Suykens and J Vandewalle ldquoLeast squares supportvector machine classifiersrdquo Neural Processing Letters vol 9 no3 pp 293ndash300 1999
[22] M E Duncan Fama ldquoNumerical modeling of yield zones inweak rocksrdquo in Comprehensive Rock Engineering J A HudsonEd vol 2 pp 49ndash75 Pergamon Oxford UK 1993
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of