research article design optimization of a concrete face...

Research ArticleDesign Optimization of a Concrete Face Rock-Fill Dam byUsing Genetic Algorithm

Yanlong Li Jing Wang and Zengguang Xu

State Key Laboratory Base of Eco-Hydraulic Engineering in Arid Area Xirsquoan University of Technology Xirsquoan 710048 China

Correspondence should be addressed to Yanlong Li liyanlongxauteducn and Zengguang Xu xuzengguangxauteducn

Received 20 November 2015 Accepted 21 February 2016

Academic Editor Filippo Ubertini

Copyright copy 2016 Yanlong Li et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper combined with the adaptive principle to improve the genetic algorithms (GA) and applied it to optimal design of theshape of the concrete face rock-fill dam (CFRD) Based on the improved GA a mathematical model was established for the designoptimization of CFRD CFRD utilizes dam cost as objective function and dam slope and geometries of the dam material partitionas design variables Dam stability stress displacement and stress level are used as the main condition constraints The calculationprocedures were prepared and the GA was used to optimize the design of Jishixia CFRD Results show that the GA could solve theglobal optimal solution problem of complex optimization design such as the high degree of nonlinearity and the recessiveness ofconstraint conditions and using the GA to optimize the CFRD design can reduce the quantities of projects and engineering safetycosts

1 Introduction

The objective function and constraint conditions for thedesign optimization of a concrete face rock-fill dam (CFRD)are nonlinear functions of design variables Constraint con-ditions are implicit functions of design variables howeverderiving these constraint conditions is difficult [1] Thecurrent design optimizationmethods for CFRDmainly focusonmathematical programming and criterionmethodswhichinclude the full stress criterion complexmethod and penaltyfunction method These methods have several limitationssuch as slow convergence and low efficiency [2]

Holland [3] first proposed a genetic algorithm (GA) thathas main features operating directly on object structuresThe continuity of the GA derivatives and functions has nolimitations GA has inherent implicit parallelism and goodglobal optimization capability GA also uses the probabilityoptimization method which can automatically access andguide the optimized search space can adjust the searchdirection adaptively and does not require rule determination

Ackley [4] proposed a strategy called ldquostochastic iteratedgenetic hill-climbingrdquo (SIGH) that uses a complex probabilityelection mechanism composed of 119872 ldquovotersrdquo to decide thevalue of the new individual (ldquo119872rdquo refers to the size of groups)

Experimental results show that four of the six test functionsof the SIGHwith single-point crossover namely the uniformcrossover neural GA exhibited better performance than thatof other algorithms Overall the SIGH ismore competitive inthe speed of solving comparedwithmany existing algorithmsIn another study Whitley et al [5] proposed a crossoveroperator on the basis of the cross field This crossoveroperator is specifically for crossing individuals by usingnumbers to represent genes The application of the crossoveroperator to the travelling salesman problem (TSP) has beenverified by experiments [6]

Bersini and Seront [7] combined GA with the simplexmethod and formed a single operation called ldquomulti-parentcrossover operatorrdquo This operator produces new individualsthrough two maternal individuals and an additional indi-vidual The cross results are consistent with those of threeindividuals with elections Three crossover operators havebeen compared with the subpoint crossover and uniformcrossover operators Results show that the three crossoveroperators have better performance than that of the other twooperators

Many experts and scholars in China have improvedthe crossover operator of GA For instance Dai et al [8]presented an algorithm of pattern extraction approach to

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 4971048 11 pageshttpdxdoiorg10115520164971048

2 Mathematical Problems in Engineering

the mutation mechanism for Genetic Algorithm The direc-tion of the mutation of individual is guided by two con-secutive evolution directions of the individual such as themutation probability different mutation operators to searchfor a variable space and migration of operators amongpopulations to exchange genetic information and solve theproblems of the classical GA converging to a local optimumvalue Zhao et al [9] proposed the building-block-codedparallel GA which is encoded by gene block This GA isutilized when the simple GA is used on large-scale combi-natorial optimization problems with low searching efficiencyThis method is based on the coarse-grained parallel GA andaims to identify possible gene block among the chromosomepopulation The identified gene block is used as a new geneblock to recode the chromosome and to generate chromo-someswith shorter lengthsThe recoded chromosome groupsare then used as the initial population that progresses tothe next round of evolution Jiang [10] utilized a flexiblestrategy to maintain the diversity of groups for parallelGA to solve the TSP which enables the algorithm to crossthe local convergence obstacles and reach the evolution ofthe global optimum direction Zhang et al [11] constructedthe analysis and inverse analysis system of temperaturestress simulation The system realizes the automation andintellectualization of complex data analysis and preparationwork in simulation process and complex data adjustment inthe inverse analysis process which can facilitate the real-time tracking simulation and feedback analysis of concretetemperature stress in construction process Gu et al [12]proposed a new method to inverse the actual initial zoningdeformation modulus and to determine the inversion objec-tive function for the actual zoning deformation modulusbased on the dam displacement measured data and finiteelement calculation results Furthermore based on the chaosgenetic optimization algorithm the inversion method forzoning deformation modulus of dam dam foundation andreservoir basin is proposed Combined with the projectcase the feasibility and validity of the proposed method areverified The internal relation between concrete dam crackbehavior abnormality and statistical change point theory isdeeply analyzed by Li et al [13] from the model structureinstability of parametric statistical model and change ofsequence distribution law of nonparametric statistical modelAnd the nonparametric change point diagnosis method ofconcrete dam crack behavior abnormality is used in theactual project demonstrating the effectiveness and scientificreasonableness of the method established Rezaiee-Pajandand Tavakoli [14] introduced an efficient procedure for crackdetection in concrete gravity dams A genetic algorithmand finite element modeling are employed to perform theoptimization tasks Moreover a genetic algorithm approachfor crack identification is proposed which can identify thelocation andmagnitude of cracks in concrete gravity dams Byminimizing the difference between the analytical responsesgiven by Rezaiee-Pajand and Tavakolirsquos element and themeasured ones the genetic algorithm identifies the crack

GA has achieved great success in practical applicationsHowever on a biological basis the GA has mathematicalflaws [15 16] A number of defects exist when GA is used

in calculations which include slow speed and immatureconvergence Moreover GA encounters problems on searchefficiency and time [17ndash19] These deficiencies seriouslyimpede the promotion and application of GA In this paperGA was improved based on adaptive principle [20] andapplied for the design optimization of CFRD

This paper combined with the adaptive principle basedon the genetic algorithm and applied it to optimal design ofthe shape of the concrete face rock-fill dam

2 Mathematical Models for the DesignOptimization of CFRD

21 Design Variables In the cross-sectional design of CFRDthe dam height the width of the dam crest and other dataare unchanged parameters that are determined in accordancewith the project planning requirements However theseparameters are generally not considered as design variablesThe angle of the upstream dam slope 119909

1 the angle between

the dammaterial boundaries and the horizontal direction 1199092

and the angle of the downstream dam slope 1199093are considered

as design variables of the CFRD cross section

22 Objective Function The CFRD sections consist of dif-ferent material partitions After considering different com-prehensive factors such as exploitation transportation andconstruction the unit price of volume formed by differentmaterials was determined For the unit length section theunit price of each feed zone was compared A material wasselected and the unit price of the material was assumed as10 The ratio of the unit price of the selected material to thatof the price of other materials is price proportional whichis denoted as 119862

119894 The objective function can be expressed as

follows

119891 = sum(119862119894times 119878119894) (1)

where 119891 is the project cost of the dam and 119878119894is the area of the

material in the dam cross section

23 Constraint Conditions The water pressure on theupstreampanel can be directly determined by considering thestructure and state of workability of CFRD Thus the slidingforce produced by the dead weight of the dam and the waterpressure on the panel significantly outweighs the horizontalthrust However this phenomenon does not produce anintegral sliding problem During design optimization theconstraint conditions take the following forms

(1) Geometric constraint conditions 119909119897le 119909 le 119909

119906

(2) Maximum stress constraint conditions 120590 le [120590](3) Preventive plastic shear failure conditions 119878

119871le [119878119871]

(4) Dam slope stability constraint conditions 119865119904ge [119865119904]

119909119897is the lower limit of variables 119909

119906is the upper limit of

variables 120590 is the stress 119878119871is the stress level and 119865

119904is the

safety coefficient of the dam slope stability

Mathematical Problems in Engineering 3

3 Improved GA with the Adaptive Principle

As a kind of stochastic optimization method the basic GAalso have some shortcomings like the weak local searchingability the nonhigh optimization accuracy premature con-vergence andmany other shortcomings In order to make upfor these deficiencies this paper combined with the adaptiveprinciple to improve the GA and applied it to optimal designof the shape of the concrete face rock-fill dam the specificimprovement process is as follows

31 Parameter Coding Coding is the most important issueconcerning GA application Several types of coding methodshave been proposed which can be divided into three cate-gories binary codingmethod floating-point codingmethodand symbol coding method The optimization of CFRDhas many design variables If the binary coding method isused the coding strings will be long and the coding anddecoding processes will be complex which may affect thecomputational efficiencyTherefore the simple floating-pointcoding with high accuracy and simple process was chosen forthis study

32 Generation of the Initial Group To solve the constrainedinitial group an initial feasible individual 119909

1can be randomly

generated if the individual is a boundary point of a feasibleregionThe regeneration is conducted until the initial feasibleindividual 119909

1is produced which is the interior point of the

feasible region Thus another initial individual 1199092can be

randomly generated as expressed in the following equation

1199092= 119886 + 119903

2times (119887 minus 119886) (2)

where 119886 = (1198861 1198862 119886

119899)119879 1199032= (11990321 11990322 119903

2119899)119879 and 119887 =

(1198871 1198872 119887

119899)119879

We need to determine whether the individual 1199092 which

is generated by (2) can satisfy the constraint conditionsIf the condition is met then a new initial individual 119909

3

will be generated Otherwise the value of 1199092continually

approximates that of 1199091 that is an iteration will be conducted

by using the following equation

1199092= 1199091+ 120572 (119909

2minus 1199091) (3)

where 120572 is a coefficient greater than 0 but less than 1 and isgenerally taken as 05

1199092can become a feasible individual after the constant

iterations The individual 1199093can be generated by using the

samemethod as that of 1199092 1199093can be converted into an initial

feasible individual the iteration continues until119898 individualsare generated

33 Treatment of Fitness Function and Constraint ConditionsTo meet the requirements of GA the fitness function shouldbe nonnegative and the problems should be maximum opti-mization problems In the optimization problem of CFRDthe upper and lower limit constraints of design variablesare considered during coding The other constraint condi-tions are used to deal with the penalty function Therefore

the objective function in the optimization problem of CFRDis expressed as follows

119865 (119909) =

119862max minus 119891 (119909) 119891 (119909) lt 119862max

0 119891 (119909) ge 119862max(4)

where 119862max is a large number greater than 119865(119909) and is takenas 50 000

34 Genetic Operation

341 Selection Operation The basic idea of selection propor-tion is that the probability of each individual being selectedis proportional to the degree of its fitness Individuals witha higher degree of fitness have greater probability of passingtheir characteristics to the next generation

342 Crossover Operation The arithmetic crossover opera-tor was used that is two individuals were randomly selectedand then combined linearly to generate two new individuals

119860 = 12057401198831198951+ (1 minus 120574

0)1198831198952

119861 = 12057401198831198952+ (1 minus 120574

0)1198831198952

(5)

where 1205740is a random number uniformly distributed between

0 and 1 Consider

119875119888=

11989613

100381610038161003816100381610038161198911015840minus 119891max

10038161003816100381610038161003816le Vfix

1198961(119891max minus 119891

1015840)

119891max minus 119891ave1198911015840ge 119891ave

1198963

1198911015840lt 119891ave

(6)

Here 119891max is the maximum fitness value among groups 119891 isthe fitness value of the variation individual119891ave is the averagefitness value of the groups of each generation and 119891

1015840 is thelarger fitness value between two individuals conducting theinterlace operation

To improve the quality of crossover the crossover proba-bility was generated by using (6) and 119875

119888was controlled by the

degree of evolution Here 1198961= 1 1198963= 09 119896

13= 05 and Vfix =

0001

343 Mutation Operation A nonuniform mutation oper-ation was used that is every variable can be mutatedindependently according to the following equation

119883119894=

(119883119894max minus 119883

119894) (1 minus 119903

(1minus119905119879)119861

) rad = 0

(119883119894minus 119883119894min) (1 minus 119903

(1minus119905119879)119861

) rad = 1

(7)

where 119905 is algebra of evolution 119903 is a randomnumber between0 and 1 119879 is the maximum evolution algebra 119861 is the systemparameter which is generally taken as 2

To improve the quality of mutation the mutation proba-bility was generated by using (8) that is 119875

119898was controlled by


the degree of evolution Here 1198962= 05 119896

4= 0065 119896

24= 007

and Vfix = 0001 Consider

119875119898=

11989624

100381610038161003816100381610038161198911015840minus 119891max

10038161003816100381610038161003816le Vfix

1198962(119891max minus 119891

1015840)


1198964

1198911015840lt 119891ave

(8)

4 Optimal Design Program of CFRD Based onthe Improved GA

41 Overall Design of the Program Structure The programincludes three modules namely the module of inputting theoriginal data the module of calculating the optimal designand the module of outputting the results The entire programis written by using FORTRAN The initial data are read inthe form of documents and stored in the form of programvariables The final optimization results can be obtainedthrough cycle calculation and the output in the form offiles

42 Function Design of Each Program Module

421 Module of Inputting the Original Data The originaldata can be divided into three parts The first part is usedto calculate the geometry of CFRD which includes twoaspects (1) the operation parameters of genetic operationwhich include the size of group 119872 and the maximum timeof iterations 119879 and (2) the structural dimensions of CFRDwhich include the thickness of panels and cushions thelevel of thickness of transition layers the dam height andthe water level The second part is used to calculate thestability of the dam slope which includes the physical andmechanical indexes of dam materials the upstream anddownstreamwater levels during the construction period andthe earthquake coefficient The third part is used to calculatethe stress and deformation of the dam which included theload series the number of structure partitions and thecalculation parameters of Duncan-Chang stress-dependentbulk modulus (E-B) model of each dam material

422 Module of the Optimal Design Calculation Themoduleof the optimal design calculation is the core of the entireprogram Given that CFRD has many material partitions thecalculation of the structural reanalysis requires a significantamount of work Therefore when writing the program thatcalculates the optimal design of CFRD the calculations ofslope stability and dam stress deformation must be treated assubprograms whereas the improved GA must be treated asthe main program

(1) Calculation of Slope Stability The calculation program ofthe slope stability utilizes the Bishop method for the analysisof the circular-slip surface This program can be used notonly in analyzing the stability of any designated slip-out pointand arbitrary designated sliding arc but also in determining

the stability of all possible slide-out points as artificiallydesignated density interpolates between any two slide-outpoints of the dam slope

(2) Calculation of Stress Deformation The nonlinear finiteelement method is used to analyze the characteristics ofthe dam stress deformation The dam materials utilize thenonlinear Duncan-Chang E-B constitutive model duringsimulation The simulation construction process calculatesand applies loads layer by layer The filling layer and thefilling soil are calculated however the resulting displacementand strain are changed to zero thus leaving the stresses inthe equation The load increment should have the smallestpossible value to reflect the nonlinear characteristics of thedam material

(3) Calculation of Genetic Manipulation The module ofgenetic manipulation is the main program of the improvedGA optimization program The genetic manipulationincludes selection operation crossover operation and muta-tion operation Figure 1 shows the calculation flow chart

423 Module of Outputting Results The output data of thisoptimization program are as follows

(1) The optimal solution and the value of its objectivefunction in which the solution has to meet the con-straint condition include stability and stress deforma-tion

(2) The safety factor of slope stability which meets theconstraint condition and is under the specified calcu-lation condition corresponds to the optimal solution

(3) The calculation results of stress and deformation onthe dam and panel in which the results are underthe specified calculation condition must correspondto the optimal solution and must meet the constraintcondition The horizontal displacement is positivewhen it moves toward the dam downstream whereasthe vertical displacement is positive when it movestoward the dam upstream Stress is positive when itis compressive and negative when it is tensile

5 Example Analyses

51 Project Profile Jishixia CFRD is a large (II) type ofhydraulic complex project in which the dam is a single-level building with crest elevation of 1861m crest width of10m and maximum dam height of 100m The minimumexcavation height of the toe slab is 1761m the slope ratioof the upstream dam slope is 1 14 the comprehensive sloperatio of the downstream dam slope is 1 171 the normal highwater level is 1856m and the corresponding downstreamwater level is 178326m The grade zones of material in thedam from upstream to downstream consist of the followingconcrete face slab cushion zone transition zone main rock-fill area and downstream rock-fill zone Figure 2 shows thediagram of the dam design optimization variables


No

Yes

Inputting the original

Generating the initial group

The evaluation of each individual

Finding out and saving theindividual with the maximal fitness

Finding out and saving the individual with the maximal fitness

Selection operation

Crossover operation

Mutation operation

Individual evaluation

Outputting the results


gen = gen + 1

gen le maxgen

data and parameters

Checking the stability of the slope and the stress and deformation and usingpenalty function to deal with individualaccording to the degree to which theindividual meets the constraint conditions


Figure 1 Module flow chart of GA

52 Basic Information The comprehensive parities of thedammaterials include the downstream rock-fills main rock-fills transition materials cushion material and panel con-crete with ratio of 1 142 181 213 1868

Geometric Constraints Consider 2486 le 1199091le 261 119909

1le 1199092le

1199093 and 0532 le 119909

3le 0656

Condition Constraints The allowable stability safety factoris [119865119904] = 135 the allowable tensile stress is [120590

119871] = 10MPa

the allowable pressure stress is [120590119888] = 100MPa and the

maximum allowable stress level is [119878119871] = 10

The geometric constraints were selected based on theexperience of completed domestic and international con-structionsThe safety factor of stability was selected based on


Table 1 Material parameters simulated by using the Duncan-Chang E-B model

Material 120588(kgsdotmminus3) 120593(∘) Δ120593(∘) C K 119870ur 119870119887

n 119877119891

mCushion 2200 506 7 0 400 800 520 035 078 022Transition layer 2180 525 10 0 850 1700 550 03 077 020Main rock-fill zone 2135 565 13 0 1070 1300 260 036 082 040Secondary rock-fill zone 2121 565 11 0 310 700 110 039 059 030

Table 2 Comparative results of the initial design and the optimal program

Program Iterative step Design variablesrad Objective function1199091

1199092

1199093

119865(119909)

Initial design mdash 2521 1373 0620 2394691Complex algorithm 535 2500 1560 0656 2281905GA 157 2497 1560 0653 2280383

Transition zoneCushion zone

Panel

Main rock-fill zone

Downstreamrock-fill zone

x1 x2 x3

Figure 2 Diagram of the optimal design variables

the relevant provisions of China in Design Code for ConcreteFace Rock-Fill Dams (SL228-98)

Structure analysis must be performed to establish thecondition constraints Dam slope stability analysis which isconducted by using the Swedish slice method and the finiteelement analysis of CFRDdeformation and stress should alsobe conductedThe condition for water storage calculation forthe upstream water level is 1856m whereas the condition forthe downstream water level is 178326m

The linear elastic constitutive model was used to simulatethe concrete face 120576 (elastic modulus) = 2 times 10

10 pa 120583(Poissonrsquos ratio) = 0167 and 120588 (density) = 2450 kgm3 TheDuncan-Chang E-B constitutive model was used to simulatethe dam materials as shown in Table 1

53 Optimizing Results

531 Analysis of the Optimal Program The paper uses thegenetic algorithm and the complex algorithm to conductthe optimizing design of Jishixia concrete face rock-filldam respectively and the calculation results of these twooptimizing design programs are compared with the initialdesign by listing The design variables and the results ofobjective function of the design with genetic algorithm andcomplex algorithm and the initial design are compared inTable 2 The comparative figure about optimal program withgenetic algorithm and the initial design are shown in Figure 3

(1) Table 2 and Figure 3 show that after optimizationthe upstream and downstream slopes of the dam becamesteeper the angle of the upstream slope of the dam decreasedfrom 2521 rad to 2497 rad and the angle of the downstream

Improved GAInitial design

Figure 3 Comparison of the dam section between the initial designand the GA

slope of the dam increased from 0620 rad to 0653 radConsequently the area of the dam section decreased thusdecreasing the dam costThe angle between the dammaterialboundaries and the horizontal direction increased from1373 rad to 1560 rad This increase in angle increased thecontent of the downstream rock-fill thereby reducing thedam cost Compared with the initial design scheme the damcost was reduced by 48 after optimization

(2) From the comparison optimized results of the com-plex algorithm and the genetic algorithm it can be seen thatthe two optimal results of optimization method are basicallythe same it is also shown that the two methods are suitablefor the optimization design of the face rock-fill dam Thegeneration number of the complex algorithm converging tothe optimal solution is 535 However the generation numberof genetic algorithm converging to the optimal solution is 157the consumed time of genetic algorithm is below one-thirdof complex algorithmThereby the genetic algorithm is moreefficient and speedy

(3) The convergence process for objective functionaccording to generation number is reflected in Figure 4 wecan see that the objective function decreases rapidly and thenthe speed of decrease is reduced and gradually comes tostability in the top 15 generation numbers Figures 5ndash7 showthe changing process for three parameters of 119909

1 1199092 and 119909

3

according to generation number The trend of parameter 1199091

and objective function are basically the same in Figure 5We can see that parameter 119909

2and parameter 119909

3rise rapidly


Table 3 Comparative program results before and after optimization

Program Conditions Behavior indicators120590119871MPa 120590

119888MPa 119878

119871120575m 119865

119904

Before optimization Completion period 043 169 068 011 179Impounding period 050 184 058 022

After optimization Completion period 043 172 083 016 135Impounding period 051 185 072 027

Note 119865119904 is the dam slope safety factor of stability under normal operating conditions

20 40 60 80 100 120 140 1600Generation number

20000

21000

22000

23000

24000

25000

26000

Obj

ectiv

e fun

ctio

n

Figure 4 Evolution of objective function along generation number

x1


245

247

249

251

253

255

257

259

Figure 5 Evolution of 1199091along generation number

and then the speed of rise is reduced and gradually comesto stability with increase in the top 15 generation numbers inFigures 6 and 7

532 Stress Deformation Analysis of the Dam Table 3 showsthe results optimum solution optimal value and maincondition index of the genetic optimization algorithm

(1) Results and Analysis of the Displacement CalculationThe vertical displacements of the dam during completionand impoundment periods are reflected in Figures 8 and 9respectively


x2

12

13

14

15

16



x3

060

061

062

063

064

065

066


Y

X

minus085

minus072

minus064minus056minus048minus040minus032

Figure 8 Contour map of the vertical displacement during thecompletion period (unit m)


Y

X

minus086

minus072

minus064minus056

minus048minus040minus032

Figure 9 Contour map of the vertical displacement during theimpoundment period (unit m)

000

0025

00500750100

0125015

Y

X

minus0025minus0055

Figure 10 Contour map of the horizontal displacement during thecompletion period (unit m)

After optimization the largest settlement of the damduring the completion period was 085m which accounts forabout 085of the damheight and occurs in about two-thirdsof the dam height Given that the downstream rock-fill of thedam is relatively weak the settlement of the dam displace-ment is biased toward the dam downstream and the locationof the maximum settlement of the dam was in the secondaryrock-fill zone Consequently the displacement of the damcrest was slightly biased toward the dam downstream Thedammainly bears dead weight during the completion periodThe area of the dam section decreased after optimizationand the dead weight was reduced Thus the dam settlementdecreased slightly after the completion period comparedwiththat before optimization

Compared with the completion period the impound-ment period bore water load and the dam settlementincreased slightly In addition the maximum settlementduring the impoundment period was 086m which accountsfor about 086 of the dam heightThemaximum settlementincreased about 001 compared with that of the completionperiod The maximum settlement occurred in about two-thirds of the dam height which is biased toward the damdownstream Compared with the design before optimizationthe dam settlement decreased slightly during the impound-ment period

The horizontal displacements of the dam during thecompletion and impoundment periods are shown in Figures10 and 11 respectively

During the completion period the dam is mainly underthe action of dead weight load the horizontal displacementis not symmetrically distributed and the upstream partof the dam tends to be displaced upstream In additionthe maximum value of the displacement is 0055m which

0025

005

010

007501950125

015

Y

X

Figure 11 Contour map of the horizontal displacement during theimpoundment period (unit m)

occurred in one-third of the dam height near the upstreamface The downstream part of the dam tends to be displacedupstream The maximum value of displacement is 015mwhich occurred in one-half of the dam height near thedownstream surface The tendency of displacement duringthe impoundment period is the same as that during thecompletion period However under the combined actions ofwater load and dead weight load the distribution of dam hor-izontal displacement significantly changed the zone of thehorizontal displacement directing to the upstream side of thedam was significantly reduced and the maximum horizontaldisplacement was reduced to 000m Moreover the horizon-tal displacement directing to the downstream increased andthe maximum horizontal displacement increased to 0195mwhich is located in one-half of the dam height near thedownstream surface

Compared with the design before optimization the dis-placement trends of the dam during the completion andstorage periods are the same but the horizontal upstreamand downstream displacement significantly changed Duringthe completion period the horizontal upstreamdisplacementof the dam increased to 168 cm whereas the horizontaldownstream displacement of the dam increased to 78 cmDuring the impounding period the horizontal upstreamdisplacement of the dam increased to 078 cm whereas thehorizontal downstream displacement of the dam increasedto 798 cm This result is mainly caused by the increase inthe proportion of downstream rock-fill in the section afteroptimizationThe relatively weak downstream rock-fill of thedam also caused this increase in horizontal displacement

(2) Results and Analysis of the Stress Calculation The majorprincipal stress isoline minor principal stress isoline andstress level isoline during the completion and impoundmentperiods are shown in Figures 12ndash17

During the completion period themaximummajor prin-cipal stress is 170MPa and the maximum minor principalstress is 043MPa in which both stresses are located inthe dam base and are near the dam axis Both stressesare compressive The directions of the major and minorprincipal stresses are close to that of gravityThe vertical stresscomponent is close to the dead weight stress of the overlyingrock-fillThe stress level of the entire dam is less than 10 andthe maximum value is 065

During the impoundment period the dam stress isredistributed the dam is under the combined action of


170162144126108090

072054036

Figure 12 Major principal stress isoline during the completionperiod (unit MPa)

043040035

030

025020

015 Y

X

Figure 13 Minor principal stress isoline during the completionperiod (unit MPa)

the dead weight load and water load the dam stress in theupstream side of the dam axis is more than that in thedownstream side and the dam stress level is less than thatof the completion period in which the maximum value was063 This result shows that impounding is favorable for thestability of the concrete face rock-fill dam The water loadnear the cushion material is perpendicular to the panel andclose to the direction of the minor principal stress in thedead weight stress fieldThe increment of theminor principalstress is greater than that of the major principal stress Thisphenomenon results in directional changes of the principalstress During the impoundment period the maximummajor principal stress is 182MPa and the maximum minorprincipal stress is 051MPa Both stresses are located in thedambase and are near the damaxisThedam stress during theimpoundment period did not significantly change The damstress is mainly caused by the dead weight of the rock-fillsand the dam stress generated by the reservoir water pressureis relatively smallThe stress level of the entire dam is less than10

Compared with the initial design the major and minorprincipal stresses of the dam after the completion perioddid not significantly change The major principal stressincreased to 002MPa whereas the minor principal stressincreased to 006MPa After the impoundment period themajor principal stress decreased to 003MPa whereas theminor principal stress increased to 009MPa The stress levelincreased to 002 in the completion period and increased to005 in the impoundment period The stress level in mostparts of the dam was low Thus the dam optimal design withthe GA is in good working condition

(3) Calculation Results of the Panel Deflection and AnalysisThe deflection curves of the panel during the completionand impoundment periods are shown in Figures 18 and 19respectively

182160

140120100080060040020

Y

X

Figure 14 Major principal stress isoline during the impoundmentperiod (unit MPa)

051045

040

035030025020

Y

X

Figure 15 Minor principal stress isoline during the impoundmentperiod (unit MPa)

060

050040030020

Figure 16 Stress level isoline during the completion period

060

050040030020

Figure 17 Stress level isoline during the impoundment period

0 40 60 80 100 120 140 160 18020The inclined length of panel (m)

000002004006008010012014016

Defl

ectio

n va

lue o

f pan

el (m

)

Figure 18 Deflection curves of a panel during the completionperiod



000

005

010

015

020

025

030

Defl

ectio

n va

lue o

f pan

el (m

)

Figure 19 Deflection curves of a panel during the impoundmentperiod

The following can be seen fromTable 3 and Figures 18 and19

The maximum deflection of the panel during the com-pletion period is 1559 cm which is located in the upper partof the panel In the impoundment period the maximumdeflection of the panel increased to 2671 cm under the effectof the upstream water pressure The entire panel deformeddownstream and the maximum deflection was located inthe middle of the panel which gradually decreases in theupper and lower parts of the panel Analysis results show thatthe panel deflection of the Jishixia project is relatively smallwhich could not trigger the peripheral joints to produce largedisplacement and can meet the requirements of the design

Compared with the initial design the deflection of thepanel increased after optimizationThedeflection of the panelincreased to 434 cm in the completion period and increasedto 448 cm in the impoundment period

6 Conclusion

We conclude the following

(1) We improvedGAwith the adaptive principle by usingthe improved GA optimization program to conductthe optimal design of Jishixia CFRD we have deter-mined that the GA could not only solve the globaloptimal solution problem of complex optimizationdesign such as the high degree of nonlinearity and therecessiveness of constraint conditions but also makeup for deficiencies which exist in the basic geneticalgorithm such as the weak local searching abilitythe nonhigh optimization accuracy and prematureconvergence The GA consumes less time and hashigh efficiency

(2) The comparative analysis results between the initialdesign and optimization show that after optimiza-tion the dam slope became steeper the downstreamrock-fill area increased and the quantities of projectsand project costs were reduced

(3) This paper conducts a finite element calculation ofstress deformation and dam slope stability on JishixiaCFRD Results show that after optimization boththe dam stability and stress deformation conformwith the general rule of CFRD and the maximum

stresses meet the requirementsThese results confirmthat the optimal result is safe reliable economic andreasonable

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research is financially supported by the CRSRI OpenResearch Program (Program SN CKWV2014208KY)the National Natural Science Foundation of China (nos51309190 51409206 and 51579207) Program 2013KCT-15for Shaanxi Provincial Key Innovative Research TeamChina Postdoctoral Science Foundation (no 2014M552470)Open Research Fund Program of State Key Laboratory ofWater Resources and Hydropower Engineering Science(2014SGG03) and the Innovative Research Team of Instituteof Water Resources and Hydroelectric Engineering XirsquoanUniversity of Technology

References

[1] M Zhou and S Sun Genetic Algorithms Theory and Applica-tions National Defense Industry Press Beijing China 1999

[2] C Xin Structure Analysis and Optimization Design of Concrete-Faced Rock-Fill Dam ChinaWater Power Press Beijing China2005

[3] J H Holland Adaptation in Natural and Artificial SystemsUniversity of Michigan Press Ann Arbor Mich USA 1975

[4] D H Ackley A Connectionist Machine for Genetic HillclimbingKluwer Academic Publishers Boston Mass USA 1987

[5] D Whitley K Mathias and P Fitzhorn ldquoDelta coding aniterative search strategy for genetic algorithmsrdquo in Proceedingsof the 4th International Conference on Genetic Algorithms pp77ndash84 San Diego Calif USA 1991

[6] J Dzubera and D Whitley ldquoAdvanced correlation analysis ofoperators for the traveling salesman problemrdquo in Parallel Prob-lem Solving from NaturemdashPPSN III International Conferenceon Evolutionary Computation The Third Conference on ParallelProblem Solving from Nature Jerusalem Israel October 9ndash141994 Proceedings vol 866 of Lecture Notes in Computer Sciencepp 68ndash77 Springer Berlin Germany 1994

[7] H Bersini and G Seront ldquoIn search of a good evolution-optimization crossoverrdquo in Proceedings of the 2nd Conferenceon Parallel Problem Solving from Nature pp 479ndash488 NorthHolland Amsterdam Netherlands 1992

[8] X-M Dai Z-G Chen R Feng X-F Mao and H-H ShaoldquoImproved algorithm of pattern extraction based mutationapproach to genetic algorithmrdquo Journal of Shanghai JiaotongUniversity vol 36 no 8 pp 1158ndash1160 2002

[9] H-L Zhao X-H Pang and Z-MWu ldquoA building block codedparallel genetic algorithm and its application in TSPrdquo Journal ofShanghai Jiaotong University vol 38 no 2 pp 213ndash217 2004

[10] L Jiang ldquoStudy of elastic TSP based on parallel genetic algo-rithmrdquoMicroelectronics vol 22 no 8 pp 130ndash133 2005

[11] L Zhang Y Liu B Li G Zhang and S Zhang ldquoStudyon real-time simulation analysis and inverse analysis systemfor temperature and stress of concrete damrdquo Mathematical


Problems in Engineering vol 2015 Article ID 306165 8 pages2015

[12] H Gu Z Wu X Huang and J Song ldquoZoning modulusinversion method for concrete dams based on chaos geneticoptimization algorithmrdquo Mathematical Problems in Engineer-ing vol 2015 Article ID 817241 9 pages 2015

[13] Z Li C Gu andZWu ldquoNonparametric change point diagnosismethod of concrete dam crack behavior abnormalityrdquo Mathe-matical Problems in Engineering vol 2013 Article ID 969021 13pages 2013

[14] M Rezaiee-Pajand and F H Tavakoli ldquoCrack detection inconcrete gravity dams using a genetic algorithmrdquo Proceedings ofthe Institution of Civil Engineers Structures and Buildings vol168 no 3 pp 192ndash209 2015

[15] X Zongben and L Guo ldquoThe simulated-biology-like algo-rithms for global optimization part I simulated evolutionaryaslgorithmsrdquoChinese Journal of Operations Research vol 14 no2 pp 1ndash13 1995

[16] D B Fogel ldquoIntroduction to simulated evolutionary optimiza-tionrdquo IEEE Transactions on Neural Networks vol 5 no 1 pp3ndash14 1994

[17] D E Goldberg and P Segrest ldquoFinite Markov chain analysisof genetic algorithmsrdquo in Proceedings of the 2nd InternationalConference on Genetic Algorithms pp 1ndash8 Cambridge MassUSA July 1987

[18] H Asoh and H Muhlenbein ldquoOn the mean convergence timeof evolutionary algorithms without selection and mutationrdquoin Parallel Problem Solving from Nature-PPSN III vol 866 ofLecture Notes in Computer Science pp 88ndash97 Springer BerlinGermany 1994

[19] C A Ankenbrandt ldquoAn extension to the theory of convergenceand a proof the time complexity of genetic algorithmsrdquo Foun-dations of Genetic Algorithms pp 53ndash58 1990

[20] G Xuejing andW Pu ldquoAn adaptive genetic algorithm based onreal coded and its applicationrdquo Journal of Beijing University ofTechnology vol 33 no 2 pp 144ndash148 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


the mutation mechanism for Genetic Algorithm The direc-tion of the mutation of individual is guided by two con-secutive evolution directions of the individual such as themutation probability different mutation operators to searchfor a variable space and migration of operators amongpopulations to exchange genetic information and solve theproblems of the classical GA converging to a local optimumvalue Zhao et al [9] proposed the building-block-codedparallel GA which is encoded by gene block This GA isutilized when the simple GA is used on large-scale combi-natorial optimization problems with low searching efficiencyThis method is based on the coarse-grained parallel GA andaims to identify possible gene block among the chromosomepopulation The identified gene block is used as a new geneblock to recode the chromosome and to generate chromo-someswith shorter lengthsThe recoded chromosome groupsare then used as the initial population that progresses tothe next round of evolution Jiang [10] utilized a flexiblestrategy to maintain the diversity of groups for parallelGA to solve the TSP which enables the algorithm to crossthe local convergence obstacles and reach the evolution ofthe global optimum direction Zhang et al [11] constructedthe analysis and inverse analysis system of temperaturestress simulation The system realizes the automation andintellectualization of complex data analysis and preparationwork in simulation process and complex data adjustment inthe inverse analysis process which can facilitate the real-time tracking simulation and feedback analysis of concretetemperature stress in construction process Gu et al [12]proposed a new method to inverse the actual initial zoningdeformation modulus and to determine the inversion objec-tive function for the actual zoning deformation modulusbased on the dam displacement measured data and finiteelement calculation results Furthermore based on the chaosgenetic optimization algorithm the inversion method forzoning deformation modulus of dam dam foundation andreservoir basin is proposed Combined with the projectcase the feasibility and validity of the proposed method areverified The internal relation between concrete dam crackbehavior abnormality and statistical change point theory isdeeply analyzed by Li et al [13] from the model structureinstability of parametric statistical model and change ofsequence distribution law of nonparametric statistical modelAnd the nonparametric change point diagnosis method ofconcrete dam crack behavior abnormality is used in theactual project demonstrating the effectiveness and scientificreasonableness of the method established Rezaiee-Pajandand Tavakoli [14] introduced an efficient procedure for crackdetection in concrete gravity dams A genetic algorithmand finite element modeling are employed to perform theoptimization tasks Moreover a genetic algorithm approachfor crack identification is proposed which can identify thelocation andmagnitude of cracks in concrete gravity dams Byminimizing the difference between the analytical responsesgiven by Rezaiee-Pajand and Tavakolirsquos element and themeasured ones the genetic algorithm identifies the crack

GA has achieved great success in practical applicationsHowever on a biological basis the GA has mathematicalflaws [15 16] A number of defects exist when GA is used

in calculations which include slow speed and immatureconvergence Moreover GA encounters problems on searchefficiency and time [17ndash19] These deficiencies seriouslyimpede the promotion and application of GA In this paperGA was improved based on adaptive principle [20] andapplied for the design optimization of CFRD

This paper combined with the adaptive principle basedon the genetic algorithm and applied it to optimal design ofthe shape of the concrete face rock-fill dam

2 Mathematical Models for the DesignOptimization of CFRD

21 Design Variables In the cross-sectional design of CFRDthe dam height the width of the dam crest and other dataare unchanged parameters that are determined in accordancewith the project planning requirements However theseparameters are generally not considered as design variablesThe angle of the upstream dam slope 119909

1 the angle between

the dammaterial boundaries and the horizontal direction 1199092

and the angle of the downstream dam slope 1199093are considered

as design variables of the CFRD cross section

22 Objective Function The CFRD sections consist of dif-ferent material partitions After considering different com-prehensive factors such as exploitation transportation andconstruction the unit price of volume formed by differentmaterials was determined For the unit length section theunit price of each feed zone was compared A material wasselected and the unit price of the material was assumed as10 The ratio of the unit price of the selected material to thatof the price of other materials is price proportional whichis denoted as 119862

119894 The objective function can be expressed as

follows

119891 = sum(119862119894times 119878119894) (1)

where 119891 is the project cost of the dam and 119878119894is the area of the

material in the dam cross section

23 Constraint Conditions The water pressure on theupstreampanel can be directly determined by considering thestructure and state of workability of CFRD Thus the slidingforce produced by the dead weight of the dam and the waterpressure on the panel significantly outweighs the horizontalthrust However this phenomenon does not produce anintegral sliding problem During design optimization theconstraint conditions take the following forms

(1) Geometric constraint conditions 119909119897le 119909 le 119909

119906

(2) Maximum stress constraint conditions 120590 le [120590](3) Preventive plastic shear failure conditions 119878

119871le [119878119871]

(4) Dam slope stability constraint conditions 119865119904ge [119865119904]

119909119897is the lower limit of variables 119909

119906is the upper limit of

variables 120590 is the stress 119878119871is the stress level and 119865

119904is the

safety coefficient of the dam slope stability






1can be randomly





1199092= 119886 + 119903

2times (119887 minus 119886) (2)

where 119886 = (1198861 1198862 119886

119899)119879 1199032= (11990321 11990322 119903

2119899)119879 and 119887 =

(1198871 1198872 119887

119899)119879



3




1199092= 1199091+ 120572 (119909

2minus 1199091) (3)








119865 (119909) =

119862max minus 119891 (119909) 119891 (119909) lt 119862max

0 119891 (119909) ge 119862max(4)





119860 = 12057401198831198951+ (1 minus 120574

0)1198831198952

119861 = 12057401198831198952+ (1 minus 120574

0)1198831198952

(5)


0 and 1 Consider

119875119888=

11989613

100381610038161003816100381610038161198911015840minus 119891max

10038161003816100381610038161003816le Vfix

1198961(119891max minus 119891

1015840)


1198963

1198911015840lt 119891ave

(6)






13= 05 and Vfix =

0001


119883119894=

(119883119894max minus 119883

119894) (1 minus 119903

(1minus119905119879)119861

) rad = 0

(119883119894minus 119883119894min) (1 minus 119903

(1minus119905119879)119861

) rad = 1

(7)






4= 0065 119896

24= 007


119875119898=

11989624

100381610038161003816100381610038161198911015840minus 119891max

10038161003816100381610038161003816le Vfix

1198962(119891max minus 119891

1015840)


1198964

1198911015840lt 119891ave

(8)














5 Example Analyses



No

Yes






Selection operation

Crossover operation

Mutation operation




gen = gen + 1

gen le maxgen

data and parameters






1le 1199092le

1199093 and 0532 le 119909

3le 0656


119871] = 10MPa







n 119877119891




1199092

1199093

119865(119909)



Panel

Main rock-fill zone


x1 x2 x3














1 1199092 and 119909

3




3rise rapidly




119888MPa 119878

119871120575m 119865

119904





20000

21000

22000

23000

24000

25000

26000

Obj

ectiv

e fun

ctio

n


x1


245

247

249

251

253

255

257

259






x2

12

13

14

15

16



x3

060

061

062

063

064

065

066


Y

X

minus085

minus072




Y

X

minus086

minus072

minus064minus056



000

0025

00500750100

0125015

Y

X

minus0025minus0055






0025

005

010

007501950125

015

Y

X








170162144126108090

072054036


043040035

030

025020

015 Y

X





182160

140120100080060040020

Y

X


051045

040

035030025020

Y

X


060

050040030020


060

050040030020



000002004006008010012014016

Defl

ectio

n va

lue o

f pan

el (m

)




000

005

010

015

020

025

030

Defl

ectio

n va

lue o

f pan

el (m

)





6 Conclusion






Competing Interests


Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














1can be randomly





1199092= 119886 + 119903

2times (119887 minus 119886) (2)

where 119886 = (1198861 1198862 119886

119899)119879 1199032= (11990321 11990322 119903

2119899)119879 and 119887 =

(1198871 1198872 119887

119899)119879



3




1199092= 1199091+ 120572 (119909

2minus 1199091) (3)








119865 (119909) =

119862max minus 119891 (119909) 119891 (119909) lt 119862max

0 119891 (119909) ge 119862max(4)





119860 = 12057401198831198951+ (1 minus 120574

0)1198831198952

119861 = 12057401198831198952+ (1 minus 120574

0)1198831198952

(5)


0 and 1 Consider

119875119888=

11989613

100381610038161003816100381610038161198911015840minus 119891max

10038161003816100381610038161003816le Vfix

1198961(119891max minus 119891

1015840)


1198963

1198911015840lt 119891ave

(6)






13= 05 and Vfix =

0001


119883119894=

(119883119894max minus 119883

119894) (1 minus 119903

(1minus119905119879)119861

) rad = 0

(119883119894minus 119883119894min) (1 minus 119903

(1minus119905119879)119861

) rad = 1

(7)






4= 0065 119896

24= 007


119875119898=

11989624

100381610038161003816100381610038161198911015840minus 119891max

10038161003816100381610038161003816le Vfix

1198962(119891max minus 119891

1015840)


1198964

1198911015840lt 119891ave

(8)














5 Example Analyses



No

Yes






Selection operation

Crossover operation

Mutation operation




gen = gen + 1

gen le maxgen

data and parameters






1le 1199092le

1199093 and 0532 le 119909

3le 0656


119871] = 10MPa







n 119877119891




1199092

1199093

119865(119909)



Panel

Main rock-fill zone


x1 x2 x3














1 1199092 and 119909

3




3rise rapidly




119888MPa 119878

119871120575m 119865

119904





20000

21000

22000

23000

24000

25000

26000

Obj

ectiv

e fun

ctio

n


x1


245

247

249

251

253

255

257

259






x2

12

13

14

15

16



x3

060

061

062

063

064

065

066


Y

X

minus085

minus072




Y

X

minus086

minus072

minus064minus056



000

0025

00500750100

0125015

Y

X

minus0025minus0055






0025

005

010

007501950125

015

Y

X








170162144126108090

072054036


043040035

030

025020

015 Y

X





182160

140120100080060040020

Y

X


051045

040

035030025020

Y

X


060

050040030020


060

050040030020



000002004006008010012014016

Defl

ectio

n va

lue o

f pan

el (m

)




000

005

010

015

020

025

030

Defl

ectio

n va

lue o

f pan

el (m

)





6 Conclusion






Competing Interests


Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











4= 0065 119896

24= 007


119875119898=

11989624

100381610038161003816100381610038161198911015840minus 119891max

10038161003816100381610038161003816le Vfix

1198962(119891max minus 119891

1015840)


1198964

1198911015840lt 119891ave

(8)














5 Example Analyses



No

Yes






Selection operation

Crossover operation

Mutation operation




gen = gen + 1

gen le maxgen

data and parameters






1le 1199092le

1199093 and 0532 le 119909

3le 0656


119871] = 10MPa







n 119877119891




1199092

1199093

119865(119909)



Panel

Main rock-fill zone


x1 x2 x3














1 1199092 and 119909

3




3rise rapidly




119888MPa 119878

119871120575m 119865

119904





20000

21000

22000

23000

24000

25000

26000

Obj

ectiv

e fun

ctio

n


x1


245

247

249

251

253

255

257

259






x2

12

13

14

15

16



x3

060

061

062

063

064

065

066


Y

X

minus085

minus072




Y

X

minus086

minus072

minus064minus056



000

0025

00500750100

0125015

Y

X

minus0025minus0055






0025

005

010

007501950125

015

Y

X








170162144126108090

072054036


043040035

030

025020

015 Y

X





182160

140120100080060040020

Y

X


051045

040

035030025020

Y

X


060

050040030020


060

050040030020



000002004006008010012014016

Defl

ectio

n va

lue o

f pan

el (m

)




000

005

010

015

020

025

030

Defl

ectio

n va

lue o

f pan

el (m

)





6 Conclusion






Competing Interests


Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










No

Yes






Selection operation

Crossover operation

Mutation operation




gen = gen + 1

gen le maxgen

data and parameters






1le 1199092le

1199093 and 0532 le 119909

3le 0656


119871] = 10MPa







n 119877119891




1199092

1199093

119865(119909)



Panel

Main rock-fill zone


x1 x2 x3














1 1199092 and 119909

3




3rise rapidly




119888MPa 119878

119871120575m 119865

119904





20000

21000

22000

23000

24000

25000

26000

Obj

ectiv

e fun

ctio

n


x1


245

247

249

251

253

255

257

259






x2

12

13

14

15

16



x3

060

061

062

063

064

065

066


Y

X

minus085

minus072




Y

X

minus086

minus072

minus064minus056



000

0025

00500750100

0125015

Y

X

minus0025minus0055






0025

005

010

007501950125

015

Y

X








170162144126108090

072054036


043040035

030

025020

015 Y

X





182160

140120100080060040020

Y

X


051045

040

035030025020

Y

X


060

050040030020


060

050040030020



000002004006008010012014016

Defl

ectio

n va

lue o

f pan

el (m

)




000

005

010

015

020

025

030

Defl

ectio

n va

lue o

f pan

el (m

)





6 Conclusion






Competing Interests


Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












n 119877119891




1199092

1199093

119865(119909)



Panel

Main rock-fill zone


x1 x2 x3














1 1199092 and 119909

3




3rise rapidly




119888MPa 119878

119871120575m 119865

119904





20000

21000

22000

23000

24000

25000

26000

Obj

ectiv

e fun

ctio

n


x1


245

247

249

251

253

255

257

259






x2

12

13

14

15

16



x3

060

061

062

063

064

065

066


Y

X

minus085

minus072




Y

X

minus086

minus072

minus064minus056



000

0025

00500750100

0125015

Y

X

minus0025minus0055






0025

005

010

007501950125

015

Y

X








170162144126108090

072054036


043040035

030

025020

015 Y

X





182160

140120100080060040020

Y

X


051045

040

035030025020

Y

X


060

050040030020


060

050040030020



000002004006008010012014016

Defl

ectio

n va

lue o

f pan

el (m

)




000

005

010

015

020

025

030

Defl

ectio

n va

lue o

f pan

el (m

)





6 Conclusion






Competing Interests


Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119888MPa 119878

119871120575m 119865

119904





20000

21000

22000

23000

24000

25000

26000

Obj

ectiv

e fun

ctio

n


x1


245

247

249

251

253

255

257

259






x2

12

13

14

15

16



x3

060

061

062

063

064

065

066


Y

X

minus085

minus072




Y

X

minus086

minus072

minus064minus056



000

0025

00500750100

0125015

Y

X

minus0025minus0055






0025

005

010

007501950125

015

Y

X








170162144126108090

072054036


043040035

030

025020

015 Y

X





182160

140120100080060040020

Y

X


051045

040

035030025020

Y

X


060

050040030020


060

050040030020



000002004006008010012014016

Defl

ectio

n va

lue o

f pan

el (m

)




000

005

010

015

020

025

030

Defl

ectio

n va

lue o

f pan

el (m

)





6 Conclusion






Competing Interests


Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Y

X

minus086

minus072

minus064minus056



000

0025

00500750100

0125015

Y

X

minus0025minus0055






0025

005

010

007501950125

015

Y

X








170162144126108090

072054036


043040035

030

025020

015 Y

X





182160

140120100080060040020

Y

X


051045

040

035030025020

Y

X


060

050040030020


060

050040030020



000002004006008010012014016

Defl

ectio

n va

lue o

f pan

el (m

)




000

005

010

015

020

025

030

Defl

ectio

n va

lue o

f pan

el (m

)





6 Conclusion






Competing Interests


Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










170162144126108090

072054036


043040035

030

025020

015 Y

X





182160

140120100080060040020

Y

X


051045

040

035030025020

Y

X


060

050040030020


060

050040030020



000002004006008010012014016

Defl

ectio

n va

lue o

f pan

el (m

)




000

005

010

015

020

025

030

Defl

ectio

n va

lue o

f pan

el (m

)





6 Conclusion






Competing Interests


Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











000

005

010

015

020

025

030

Defl

ectio

n va

lue o

f pan

el (m

)





6 Conclusion






Competing Interests


Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article design optimization of a concrete face...

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