research article optimization design and application of ...basic design parameters of bifurcation...

Research ArticleOptimization Design and Application of UndergroundReinforced Concrete Bifurcation Pipe

Chao Su1 Zhenxue Zhu1 Yangyang Zhang1 and Niantang Jiang2

1College of Water Conservancy and Hydropower Engineering Hohai University No 1 Xikang Road Nanjing 210098 China2Shanghai Investigation Design amp Research Institute No 388 Yixian Road Shanghai 200434 China

Correspondence should be addressed to Yangyang Zhang zhangyangyang615163com

Received 18 September 2014 Accepted 12 December 2014

Academic Editor Chenfeng Li

Copyright copy 2015 Chao Su et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Underground reinforced concrete bifurcation pipe is an important part of conveyance structure During construction the workloadof excavation and concrete pouring can be significantly decreased according to optimized pipe structure and the engineering qualitycan be improved This paper presents an optimization mathematical model of underground reinforced concrete bifurcation pipestructure according to real working status of several common pipe structures from real cases Then an optimization design systemwas developed based on Particle SwarmOptimization algorithm Furthermore take the bifurcation pipe of one hydropower stationas an example optimization analysis was conducted and accuracy and stability of the optimization design system were verifiedsuccessfully

1 Introduction

With the development of large hydropower stations especiallypumped storage power stations more and more bifurcationpipes are constructed In conveyance system of hydropowerstations bifurcation pipe usually locates on the end ofpressure pipe and is near the powerhouse so it undertakeshigh internal and external water pressure Bifurcation pipeis a complex space structure and complicated stresses arecaused when it suffers from different water conditions Thereare many constraints when it is designed and requirementsof hydraulic power and structure are needed to be satisfiedhowever there is certain contradiction between these twoaspects they cannot be met mutually In 1990 Zagars et al[1] analyzed reinforced concrete bifurcation pipe of Guxustorage power station using structuralmechanicalmethod In1996 linear and nonlinear finite element methods were usedfor evaluating stress status of surrounding rock lining andreinforcing steel bars of bifurcation pipe under internal andexternal effect by Lu and Liu [2] In 2001 Xiao [3] developedthe optimization analysis method used for assessing the best

structure of large-scale underground reinforced concreteconduit based on failure conditions of reinforced concreteand elastoplastic principle of rock energy dissipation In2006 Haixia and Zhefei [4] established an optimizationdesign model of crescent rib branches for steel pipe anddeveloped a corresponding program In 2009 Ren et al[5] analyzed stress and deformation of surrounding rockand reinforcement of lining material stress and crack inreinforced concrete bifurcation structure of Pushihe pumpedstorage power station

In bifurcation pipe design design safety and reasonablestructure based on real structure layout are themain purposesof most designers Bifurcation pipe optimization can reducenot only structure volume but also cavern excavation volume[6] However most of the current optimization designs aretraditional they only make simple comparisons of someschemes without mathematical programming method andmaterial properties cannot be fully used A reasonable andefficient method is needed and will be significant for opti-mization design of bifurcation pipe

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 948584 7 pageshttpdxdoiorg1011552015948584

2 Mathematical Problems in Engineering

R1R1

R2R2

R3R3

a1a1

L1

L 2

L3

A 12059621205962

12059631205963

B

CbR

R0

0

a2a2

a3

a3

yy

x

2

bb

2

b3

A

B

C

D

E

I

G

Q

H

F

A998400

B998400

x2

Figure 1 Diagrams of Y-type bifurcation pipe and shape-control points

2 Parameterized FEM Model andMesh Program

Shape-control parameters should be defined according to dif-ferent bifurcation pipe structures before establishing modeland then the coordinates of shape-control points can becalculated by shape-control parameters Three-dimensionalmodel containing reinforced concrete blocks of bifurcationpipe is built and an automaticmesh program is developed [7ndash9]

Basic design parameters of bifurcation pipe usually aremain inlet pipe diameter 119877

1 branch pipe outlet diameter

11987721198773 bifurcation angle 119887 joint tangent sphere radius119877main

half cone angle 1198861 half cone angle of branch cone 119886

21198863 waist

turning angle 12059621205963 wall thickness 119905

111990521199053(not shown in

Figure 1) and so forth (see Figure 1) Shape-control pointsand geometric sizes can be calculated according to geometryrelations Shape-control point calculation of bifurcation pipeis done Calculation of some intermediate variables andauxiliary geometric sizes is necessary [10]

Control point coordinates calculation program was writ-ten by Visual Basic Bifurcation pipe control points and sur-rounding rock coordinates were input through visualizationparameter input interface and data files such as CGFKTXTKZDTXT were exported According to the defined meshscheme the whole three-dimensional finite element meshwas obtained using automatic mesh program CGWGEXEwritten by FORTRAN language converting the mesh datainto INP format files of ABAQUS and then the whole meshcan be visualized in ABAQUS [11ndash13]

3 Optimization Design Program

The process of optimization design includes randomizationof initial parameters calculation of structure judgment ofconstraints searching better optimization design schemeand obtaining the best optimization design results Imple-mentation usually goes through the following steps (1) estab-lish a reasonable mathematical model and transform realengineering problems into mathematical problems (2) solvemathematical problems by certain reasonable and effective

optimizationmethod (3) develop a program using structuralanalysis method and implement optimization design calcula-tion

31 Mathematical Model Optimization design in this paperwas based on a defined topology structure so design variablesshould be chosen according to the topology Bifurcation pipeshape-control parameters were chosen as design variablesThe least concrete volume was set as objective function inorder to make full use of reinforced concrete bifurcation pipematerial properties

Constraints contained geometric and strength con-straints Geometric constraints were as follows (1) half coneangle is 120579

1lt 119886119894lt 1205792and (2) bifurcation angle is 120579

3lt 119887 lt 120579

4

(3) the shortest bus bar of pipes was no less than 300mmand (4) wall thickness of concrete bifurcation pip is 119889

1lt

119905119894lt 1198892(1205791 1205792were the range of half cone angle 120579

3 1205794were

the range of bifurcation angle 1198891 1198892were the range of wall

thickness) Stress constraints included the requirements thattensile stress should be lower than concrete ultimate tensilestrength design value and compressive stress should be lowerthan concrete ultimate compressive strength design value

32 Optimization Design Algorithm Mathematical model ofbifurcation pipe optimization is often nonlinear program-ming problems Applicable optimization algorithm includesgeneralized reduced gradient method complex methodParticle Swarm Optimization penalty method sequentiallinear programming and sequential quadratic programmingParticle Swarm Optimization algorithm is adopted in thispaper Particle Swarm Optimization has the advantages ofparallel processing good stability and the most probabilityof finding global optimal solution [14]

A group of random particles was firstly initialized in PSOalgorithm and particlersquos searching following optimal particlewas conducted in the solution space In searching processevery particle updated its position and velocity accordingto the current optimal particle position The global optimalsolution was obtained by continuous iteration and searching[15]

The optimization process of 119873 particles group in 119863-dimension searching space was taken as an example to

Mathematical Problems in Engineering 3

describe the process of optimization algorithms The state ofparticle 119894 in time 119905 was set as follows

velocity 119881119896119894= (V1198961198941 V1198961198942 V119896

119894119895 V119896

119894119863)

position119883119896119894= (119909119896

1198941 119909119896

1198942 119909

119896

119894119895 119909

119896

119894119863)

personal best position 119901119887119890119904119905119883119896

119894= (119901119887119890119904119905119909

119896

1198941

119901119887119890119904119905119909119896

1198942 119901119887119890119904119905119909

119896

119894119895 119901119887119890119904119905119909

119896

119894119863)

global best position 119892119887119890119904119905119883119896

119894= (119892119887119890119904119905119909

119896

1198941

119892119887119890119904119905119909119896

1198942 119892119887119890119904119905119909

119896

119894119895 119892119887119890119904119905119909

119896

119863)

where 119894 = 1 2 119873 119895 = 1 2 119863 119896 = 1 2 119894119905max119894119905max is the maximum number of iterations 119894 represents the119894th particle 119895 represents velocity or the 119895th dimension and 119896represents the count of iterations

In every iteration process how to update the positionof particle is the key factor affecting optimization algorithmeffectiveness Every particle updates its position accordingto two optimal solutions (global best position 119892119887119890119904119905 andpersonal best position 119901119887119890119904119905) Updated velocity and newposition can be defined by following formula

V119896+1119894119895

= V119896119894119895+ 1198881lowast 1199031(119901119887119890119904119905119909

119896

119894119895minus 119909119896

119894119895)

+ 1198882lowast 1199032lowast (119892119887119890119904119905119909

119896

119895minus 119909119896

119894119895)

119909119896+1

119894119895= 119909119896

119894119895+ V119896+1119894119895

(1)

In order to coordinate the global and local optimizationcapability of PSO algorithm inertia weight coefficient120596 is setto determine the amount of current velocity particles inheritTransform the velocity equation to formula (2) and keep theposition updating equation

V119896+1119894119895

= 120596119896V119896119894119895+ 1198881lowast 1199031(119901119887119890119904119905

119896

119894119895minus 119909119896

119894119895)

+ 1198882lowast 1199032lowast (119892119887119890119904119905

119896

119895minus 119909119896

119894119895)

(2)

In the equation V119896119894119895and 119909119896

119894119895are the 119895th dimension velocity

and position of particle 119894 in the 119896th iteration respectively 1198881

and 1198882are constant usually 119888

11198882isin [0 4] 119903

1and 1199032are random

number between 0 and 1119901119887119890119904119905119909119896119894119895is the personal best position

of particle 119875119894in the 119895th dimension 119892119887119890119904119905119909119896

119894119895is the global best

position of the groupLinear decreasing inertia weight is usually adopted in

PSO iteration calculation process at present It can acceleratethe convergence of iterative calculation and get global bestsolution quickly Y Shi has suggested that inertia weight120596 decreases linearly from 09 to 04 for better algorithmperformance The linear decreasing formula is as follows

120596119896= 120596max minus

120596max minus 120596min119894119905max

lowast 119896 (3)

The position updating can be described as in Figure 2Reinitialization is carried out on particles which is out of

range or do not satisfy constraints during development of theprogram To ensure the minimum objective function value

y

x

Vk+1iVki

Xki

Xk+1i

pbestXki

gbestXki

0

Figure 2 Particle updating schematic diagram

convergence criterion is that the difference between globaloptimal solutions of last generation and next generationshould be less than calculation tolerance 120575 At the same timeto avoid toomany iterations the maximum count of iterationis 1000 in program

33 Structural Analysis Method Selection FEM was used instructure optimization design and calculation in this paper[16 17] FEM is not only with complete and reliable theoret-ical basis but also with simple and standardized basis It canmeet the accuracy requirements of engineering calculationand its convergence can be guaranteed Calculating resultscan show real structure stress and deformation characteris-tics

4 Engineering Applications

41 Project Overview A hydropower station with totalinstalled capacity of 1200MW unit capacity of 300MW hasfour units The depth of 1 bifurcated pipe is approximately414mndash418m and 2 is approximately 415m The impact ofcomplex geological conditions can be considered in the finiteelement method for structural calculation

C30 concrete is used for lining in bifurcation pipes con-struction Lining thickness is 05m ⩽ 119905 ⩽ 1m Two Φ20mmhoop reinforcement scheme is adopted to prevent concretecracking under high water pressure during operation periodThemainmechanical properties of C30 concrete and steel areshown in Table 1

The optimization design calculation in this paper focusedon normal operation cases The influence of gravity of rockwas considered and it was assumed that all gaps betweenbifurcation pipe and rock were ignored Linear calculationwas implemented for bifurcation pipe under internal waterhead around 520m (including water hammer pressure) andgravity of surrounding rockwithout consideration of externalwater pressure Finally structural stress and strain wereanalyzed


R200

195

195

335

335

60

8080

195

195

8080

1571

121 200

yyyy

y yy

y

y

y

y

y

y

y

y

y y

60

3∘ 12998400 19998400998400

3∘ 12998400 19998400998400

60 ∘0 9984000 998400998400

R200

R279

6 ∘29 9984005 998400998400

Figure 3 Plane design drawing of bifurcation pipe

Table 1 Main mechanical properties of C30 concrete and rebar

Mechanical index C30 concrete RebarDensity 25 gcm3 mdashElastic modulus 119864

028GPa 210GPa

Poissonrsquos ratio 120583 0167 03Tensile strength 15MPa 210MPaCompressive strength 15MPa 210MPa

42 Mathematical Model of Bifurcation Pipe According tothe initial design scheme parameters of bifurcation pipesize model were confirmed Structure type is shown inFigure 3 Parameters which can be used as optimizationdesign parameters are half angle of main cone and branchcone 119886

11198862 bifurcation angle 119887 main cone wall thickness 119905

1

and branch cone wall thickness 11990521199053(see Figure 4)

Bifurcation pipe concrete volume was set as objectivefunction According to bifurcation pipe design codes therange of design variables is shown in Table 2

C30 concrete lining is used in this project so stress con-trol standards in this optimization design are C30 concreteultimate tensile strength and compressive strength How-ever because of reinforcement tensile stress control stan-dards were calculated using equivalent stress method [18]According to the amount of bifurcation pipe reinforcementcorresponding design value of ultimate concrete tensile stresscan be increased to 7MPa Considering that reinforcementis mainly subject to tensile stress and its contribution tocompressive resistance could be ignored the correspondingconcrete ultimate compressive stress is unchanged Under

Table 2 Value range of design variables

Optimization variables RangeBifurcation angle 119887 50sim70∘

Half angle of main cone 1198861

2sim5∘

Half angle of branch cone 1198862

2sim5∘

Main cone wall thickness 1199051

05sim1mBranch cone wall thickness 119905

205sim1m

Branch cone wall thickness 1199053

05sim1m

these stress constraints bifurcation pipe size should meet thespecification requirements by traditional calculatingmethodBifurcation pipe optimization design implementation flow isshown in Figure 5

43 Optimization Results Analysis Bifurcation pipe of ahydropower station was analyzed using the optimizationdesign system and the better design parameters wereobtained in Table 3 after 178 iterations Bifurcation angleis significantly reduced walls are much thinner and theamount of concrete is also reduced The amount of concreteis 267741m3 before optimization and it becomes 183330m3after optimization so the amount of concrete reduces to432 after shape optimization Reinforcement checkingcalculation was conducted by using design parameters afteroptimization

431 Surrounding Rock The distribution of surroundingrock stress and strain is similar before and after optimizationwith some differences in extreme values After excavation thewhole rock is almost in compressive status only a small range


t1

t2

a2

a1t8

b

y yy y y

yy

y

y

y

y

y

yy

Figure 4 Schematic diagram of optimization design parameters

of tensile status occurs in the bottom of the cavern excavationarea at bifurcation area The maximum tensile stress beforeand after optimization is 048MPa and 059MPa Com-pressive stress concentration occurs in the bifurcation areaand the maximum compressive stress reaches minus4371MPaminus5756MPa Large displacement is caused in top and bottomof cavern toward to the cave Extreme displacement occursat the top of the bifurcation of the cavern and displace-ment before and after optimization is 231mm and 212mmAfter parameters optimization walls are significantly thinnercavern excavation volume decreases and cavern excavationradius is smaller The top and bottom cavern displacementafter excavation has been reduced Cavern excavation radiushas a certain relationshipwith the surrounding rock displace-ment

432 Bifurcation Pipe The distribution of bifurcation pipestress and strain is almost unchanged before and after opti-mization Two layers of Φ20mm hoop reinforcement werebuilt using T3D2 element in modeling and were inserted intoinner side of bifurcation pipe Analysis of bifurcation pipewith hoop reinforcement is as follows There is little changein the whole stress distributionWhole bifurcation stress afterinserting hoop reinforcement is significantly reduced Thevast majority of the tensile stress values is 20MPa or less butthere is still stress concentration phenomenon at the bifurca-tion area andmaximumstress here can be eliminated throughsuitable reinforcement direction arrangement Extreme dis-placement is 0535mm at intersecting line in top and bottomof inner surface All rebars are subject to tensile stress andmaximum stress reaches 752MPa (see Figures 6 and 7)

For the inevitable stress concentration in bifurcationand waist areas some general engineering measures canbe taken such as sharp corners rounded haunch crotchoutsourcing concrete in bifurcation point configuring hoop

Table 3 Comparison of initial shape and optimization shape

Variable Beforeoptimization

Afteroptimization

Bifurcation angle 119887 (∘) 60 4786Half angle of main cone 119886

1(∘) 32 465

Half angle of branch cone 1198862(∘) 32 457

Main cone wall thickness 1199051(m) 06 057

Branch cone wall thickness 1199052(m) 08 063


and axial steels Reducing the external water pressure is alsoan important measure for stabilizing bifurcation pipe

5 Conclusion

Bifurcation pipe is commonly used in water conveyancesystem of hydropower station and pumped storage powerplant Its structural safety and stability are directly relatedto hydropower security and stability so safety and stabil-ity of bifurcation pipe design are very important Three-dimensional FEMparametric reinforced concrete bifurcationpipe program is established in this paper Particle SwarmOptimization method is selected as the bifurcation pipestructure optimization algorithm and corresponding opti-mization program is written FEM calculation is done by call-ingABAQUSStandardmodule through interface technologybetween optimization program and ABAQUS to achievethe optimization goals In the verified project examplethe amount of concrete of optimized bifurcation pipe wassignificantly reduced At the same time the comparison ofstress and displacement results obtained before and afteroptimization proved that the program is feasible and stable


Range of design variables

parameters randomly

Establishing parameterized FEM model

Material properties

Loads information

Calling ABAQUS to do FEM calculating

Extracting stress calculating data

Checking stress constraints

Calculating history optimal position of individuals and groups

Outputting matching variables and objective function values

Minimum objective function values or maximum error accuracy or exceeding maximum iterations

Updating particle velocity and position

Setting group history optimal position parameter as optimal parameter

Optimization terminated

Yes No

Yes

No

New generation

of group treatment

Generating N sets of PSO

Figure 5 Flowchart of optimum design program

+7438e + 06

+6647e + 06

+5856e + 06

+5065e + 06

+4273e + 06

+3482e + 06

+2691e + 06

+1900e + 06

+1109e + 06

+3178e + 05

minus4733e + 05

minus1264e + 06

minus2056e + 06

S max principal(avg 75)

x

y

z

Figure 6 Principal tensile stress of inside surface


x

y

z

+7519e + 06

+6905e + 06

+6292e + 06

+5678e + 06

+5064e + 06

+4451e + 06

+3837e + 06

+3224e + 06

+2610e + 06

+1996e + 06

+1383e + 06

+7693e + 05

+1557e + 05

(avg 75)S max principal

Figure 7 Principal tensile stress of hoop rebar

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The present work is supported by the National NaturalScience Foundation of China (no 51279053) and the Fun-damental Research Funds for the Central Universities (no2014B36514)

References

[1] A Zagars C H Yeh and H Lu ldquoDesign of high pressurereinforced concrete bifurcation piperdquo Water Power no 10 pp13ndash18 1990

[2] X Lu and Q Liu ldquoStress analysis for underground reinforcedconcrete partition manifoldrdquo Journal of Hohai University (Nat-ural Sciences) vol 9 pp 24ndash29 1996

[3] M Xiao ldquoOptimization analysis of large-scale undergroundreinforced concrete branching conduitrdquo Journal of HydraulicEngineering vol 32 no 12 pp 8ndash14 2001

[4] Y Haixia and L Zhefei ldquoOptimum design of crescent ribbranches for steel piperdquo Chinese Journal of Solid Mechanics volZ1 pp 180ndash180 2006

[5] X-H Ren H-B Xu and J-Q Shu ldquoNumerical analysis onsurrounding rock and reinforced concrete bifurcation structureof underground tunnelsrdquo Journal of Sichuan University (Engi-neering Science) vol 41 no 4 pp 8ndash13 2009

[6] Q Liu Hydropower Plant China Water Power Press 2004[7] B Zhang H Chen and L Zou ldquoSimple implementation

of complex hydraulic structure finite element discretizationrdquoWater Resources andHydropower Engineering vol 29 no 12 pp4ndash6 1998

[8] Z Yufeng and Z Yiwen ldquoReview for typical methods of finiteelement mesh automatic generation and research topics infuturerdquo Engineering Journal of Wuhan University vol 38 no 2pp 54ndash59 2005

[9] X Peng and X Qian ldquoZoning-direct method of automatic FEmesh generationrdquo Engineering Mechanics no 4 pp 1396ndash1481993

[10] W Hegao Y Xingyi and W Shougen ldquoResearch and appli-cation on system for computer aided design of rib reinforcedbranch piperdquo in Proceedings of theThe 6th National HydropowerPenstock Papers p 6 2006

[11] Z Bofang Principles and Applications of Finite ElementMethodChina WaterPower Press 1998

[12] E Nie and C Su ldquoApplication of VB and TrueGRID to 3Dfinite grid automatic mesh generation for lock headrdquo Port ampWaterway Engineering no 10 pp 138ndash142 2009

[13] S Chao N Jiang and JWang ldquoOptimization design of chamfertriangular gate lock structurerdquo Port amp Waterway Engineeringno 5 pp 124ndash129 2013

[14] Q Feng and Y Li ldquoResearch on the application of IPSO inengineering optimization problemrdquoChinese Journal of ScientificInstrument vol 26 no 9 pp 984ndash990 2005

[15] G Xu J-P Qu and Z-T Yang ldquoImproved adaptive particleswarm optimization algorithmrdquo Journal of South China Univer-sity of Technology (Natural Science) vol 36 no 9 pp 6ndash10 2008

[16] C Peijiang B Tengfeng andY Peng ldquo3Dfinite element analysisof underground embedded steel bifurcation piperdquo HydropowerAutomation and DamMonitoring vol 3 pp 64ndash66 74 2006

[17] Q Yan ldquo3D finite element analysis of underground embeddedsteel bifurcation structurerdquoWater Power vol 6 pp 29ndash32 2003

[18] F Zuoxin and Q Xiangdong ldquoApplication of finite elementmethod to the design of arch damsrdquo Journal of Hohai University(Natural Sciences) vol 19 no 2 pp 8ndash15 1991

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


R1R1

R2R2

R3R3

a1a1

L1

L 2

L3

A 12059621205962

12059631205963

B

CbR

R0

0

a2a2

a3

a3

yy

x

2

bb

2

b3

A

B

C

D

E

I

G

Q

H

F

A998400

B998400

x2

Figure 1 Diagrams of Y-type bifurcation pipe and shape-control points

2 Parameterized FEM Model andMesh Program

Shape-control parameters should be defined according to dif-ferent bifurcation pipe structures before establishing modeland then the coordinates of shape-control points can becalculated by shape-control parameters Three-dimensionalmodel containing reinforced concrete blocks of bifurcationpipe is built and an automaticmesh program is developed [7ndash9]

Basic design parameters of bifurcation pipe usually aremain inlet pipe diameter 119877

1 branch pipe outlet diameter

11987721198773 bifurcation angle 119887 joint tangent sphere radius119877main

half cone angle 1198861 half cone angle of branch cone 119886

21198863 waist

turning angle 12059621205963 wall thickness 119905

111990521199053(not shown in

Figure 1) and so forth (see Figure 1) Shape-control pointsand geometric sizes can be calculated according to geometryrelations Shape-control point calculation of bifurcation pipeis done Calculation of some intermediate variables andauxiliary geometric sizes is necessary [10]

Control point coordinates calculation program was writ-ten by Visual Basic Bifurcation pipe control points and sur-rounding rock coordinates were input through visualizationparameter input interface and data files such as CGFKTXTKZDTXT were exported According to the defined meshscheme the whole three-dimensional finite element meshwas obtained using automatic mesh program CGWGEXEwritten by FORTRAN language converting the mesh datainto INP format files of ABAQUS and then the whole meshcan be visualized in ABAQUS [11ndash13]

3 Optimization Design Program

The process of optimization design includes randomizationof initial parameters calculation of structure judgment ofconstraints searching better optimization design schemeand obtaining the best optimization design results Imple-mentation usually goes through the following steps (1) estab-lish a reasonable mathematical model and transform realengineering problems into mathematical problems (2) solvemathematical problems by certain reasonable and effective

optimizationmethod (3) develop a program using structuralanalysis method and implement optimization design calcula-tion

31 Mathematical Model Optimization design in this paperwas based on a defined topology structure so design variablesshould be chosen according to the topology Bifurcation pipeshape-control parameters were chosen as design variablesThe least concrete volume was set as objective function inorder to make full use of reinforced concrete bifurcation pipematerial properties

Constraints contained geometric and strength con-straints Geometric constraints were as follows (1) half coneangle is 120579

1lt 119886119894lt 1205792and (2) bifurcation angle is 120579

3lt 119887 lt 120579

4

(3) the shortest bus bar of pipes was no less than 300mmand (4) wall thickness of concrete bifurcation pip is 119889

1lt

119905119894lt 1198892(1205791 1205792were the range of half cone angle 120579

3 1205794were

the range of bifurcation angle 1198891 1198892were the range of wall

thickness) Stress constraints included the requirements thattensile stress should be lower than concrete ultimate tensilestrength design value and compressive stress should be lowerthan concrete ultimate compressive strength design value

32 Optimization Design Algorithm Mathematical model ofbifurcation pipe optimization is often nonlinear program-ming problems Applicable optimization algorithm includesgeneralized reduced gradient method complex methodParticle Swarm Optimization penalty method sequentiallinear programming and sequential quadratic programmingParticle Swarm Optimization algorithm is adopted in thispaper Particle Swarm Optimization has the advantages ofparallel processing good stability and the most probabilityof finding global optimal solution [14]

A group of random particles was firstly initialized in PSOalgorithm and particlersquos searching following optimal particlewas conducted in the solution space In searching processevery particle updated its position and velocity accordingto the current optimal particle position The global optimalsolution was obtained by continuous iteration and searching[15]

The optimization process of 119873 particles group in 119863-dimension searching space was taken as an example to



velocity 119881119896119894= (V1198961198941 V1198961198942 V119896

119894119895 V119896

119894119863)

position119883119896119894= (119909119896

1198941 119909119896

1198942 119909

119896

119894119895 119909

119896

119894119863)


119894= (119901119887119890119904119905119909

119896

1198941

119901119887119890119904119905119909119896

1198942 119901119887119890119904119905119909

119896

119894119895 119901119887119890119904119905119909

119896

119894119863)


119894= (119892119887119890119904119905119909

119896

1198941

119892119887119890119904119905119909119896

1198942 119892119887119890119904119905119909

119896

119894119895 119892119887119890119904119905119909

119896

119863)



V119896+1119894119895

= V119896119894119895+ 1198881lowast 1199031(119901119887119890119904119905119909

119896

119894119895minus 119909119896

119894119895)

+ 1198882lowast 1199032lowast (119892119887119890119904119905119909

119896

119895minus 119909119896

119894119895)

119909119896+1

119894119895= 119909119896

119894119895+ V119896+1119894119895

(1)


V119896+1119894119895

= 120596119896V119896119894119895+ 1198881lowast 1199031(119901119887119890119904119905

119896

119894119895minus 119909119896

119894119895)

+ 1198882lowast 1199032lowast (119892119887119890119904119905

119896

119895minus 119909119896

119894119895)

(2)





11198882isin [0 4] 119903







120596119896= 120596max minus

120596max minus 120596min119894119905max

lowast 119896 (3)



y

x

Vk+1iVki

Xki

Xk+1i

pbestXki

gbestXki

0









R200

195

195

335

335

60

8080

195

195

8080

1571

121 200

yyyy

y yy

y

y

y

y

y

y

y

y

y y

60

3∘ 12998400 19998400998400

3∘ 12998400 19998400998400

60 ∘0 9984000 998400998400

R200

R279

6 ∘29 9984005 998400998400




028GPa 210GPa




1







2sim5∘


2sim5∘



205sim1m


05sim1m





t1

t2

a2

a1t8

b

y yy y y

yy

y

y

y

y

y

yy







Afteroptimization


1(∘) 32 465






5 Conclusion




parameters randomly


Material properties

Loads information










Yes No

Yes

No

New generation

of group treatment



+7438e + 06

+6647e + 06

+5856e + 06

+5065e + 06

+4273e + 06

+3482e + 06

+2691e + 06

+1900e + 06

+1109e + 06

+3178e + 05

minus4733e + 05

minus1264e + 06

minus2056e + 06


x

y

z



x

y

z

+7519e + 06

+6905e + 06

+6292e + 06

+5678e + 06

+5064e + 06

+4451e + 06

+3837e + 06

+3224e + 06

+2610e + 06

+1996e + 06

+1383e + 06

+7693e + 05

+1557e + 05





Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











velocity 119881119896119894= (V1198961198941 V1198961198942 V119896

119894119895 V119896

119894119863)

position119883119896119894= (119909119896

1198941 119909119896

1198942 119909

119896

119894119895 119909

119896

119894119863)


119894= (119901119887119890119904119905119909

119896

1198941

119901119887119890119904119905119909119896

1198942 119901119887119890119904119905119909

119896

119894119895 119901119887119890119904119905119909

119896

119894119863)


119894= (119892119887119890119904119905119909

119896

1198941

119892119887119890119904119905119909119896

1198942 119892119887119890119904119905119909

119896

119894119895 119892119887119890119904119905119909

119896

119863)



V119896+1119894119895

= V119896119894119895+ 1198881lowast 1199031(119901119887119890119904119905119909

119896

119894119895minus 119909119896

119894119895)

+ 1198882lowast 1199032lowast (119892119887119890119904119905119909

119896

119895minus 119909119896

119894119895)

119909119896+1

119894119895= 119909119896

119894119895+ V119896+1119894119895

(1)


V119896+1119894119895

= 120596119896V119896119894119895+ 1198881lowast 1199031(119901119887119890119904119905

119896

119894119895minus 119909119896

119894119895)

+ 1198882lowast 1199032lowast (119892119887119890119904119905

119896

119895minus 119909119896

119894119895)

(2)





11198882isin [0 4] 119903







120596119896= 120596max minus

120596max minus 120596min119894119905max

lowast 119896 (3)



y

x

Vk+1iVki

Xki

Xk+1i

pbestXki

gbestXki

0









R200

195

195

335

335

60

8080

195

195

8080

1571

121 200

yyyy

y yy

y

y

y

y

y

y

y

y

y y

60

3∘ 12998400 19998400998400

3∘ 12998400 19998400998400

60 ∘0 9984000 998400998400

R200

R279

6 ∘29 9984005 998400998400




028GPa 210GPa




1







2sim5∘


2sim5∘



205sim1m


05sim1m





t1

t2

a2

a1t8

b

y yy y y

yy

y

y

y

y

y

yy







Afteroptimization


1(∘) 32 465






5 Conclusion




parameters randomly


Material properties

Loads information










Yes No

Yes

No

New generation

of group treatment



+7438e + 06

+6647e + 06

+5856e + 06

+5065e + 06

+4273e + 06

+3482e + 06

+2691e + 06

+1900e + 06

+1109e + 06

+3178e + 05

minus4733e + 05

minus1264e + 06

minus2056e + 06


x

y

z



x

y

z

+7519e + 06

+6905e + 06

+6292e + 06

+5678e + 06

+5064e + 06

+4451e + 06

+3837e + 06

+3224e + 06

+2610e + 06

+1996e + 06

+1383e + 06

+7693e + 05

+1557e + 05





Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










R200

195

195

335

335

60

8080

195

195

8080

1571

121 200

yyyy

y yy

y

y

y

y

y

y

y

y

y y

60

3∘ 12998400 19998400998400

3∘ 12998400 19998400998400

60 ∘0 9984000 998400998400

R200

R279

6 ∘29 9984005 998400998400




028GPa 210GPa




1







2sim5∘


2sim5∘



205sim1m


05sim1m





t1

t2

a2

a1t8

b

y yy y y

yy

y

y

y

y

y

yy







Afteroptimization


1(∘) 32 465






5 Conclusion




parameters randomly


Material properties

Loads information










Yes No

Yes

No

New generation

of group treatment



+7438e + 06

+6647e + 06

+5856e + 06

+5065e + 06

+4273e + 06

+3482e + 06

+2691e + 06

+1900e + 06

+1109e + 06

+3178e + 05

minus4733e + 05

minus1264e + 06

minus2056e + 06


x

y

z



x

y

z

+7519e + 06

+6905e + 06

+6292e + 06

+5678e + 06

+5064e + 06

+4451e + 06

+3837e + 06

+3224e + 06

+2610e + 06

+1996e + 06

+1383e + 06

+7693e + 05

+1557e + 05





Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










t1

t2

a2

a1t8

b

y yy y y

yy

y

y

y

y

y

yy







Afteroptimization


1(∘) 32 465






5 Conclusion




parameters randomly


Material properties

Loads information










Yes No

Yes

No

New generation

of group treatment



+7438e + 06

+6647e + 06

+5856e + 06

+5065e + 06

+4273e + 06

+3482e + 06

+2691e + 06

+1900e + 06

+1109e + 06

+3178e + 05

minus4733e + 05

minus1264e + 06

minus2056e + 06


x

y

z



x

y

z

+7519e + 06

+6905e + 06

+6292e + 06

+5678e + 06

+5064e + 06

+4451e + 06

+3837e + 06

+3224e + 06

+2610e + 06

+1996e + 06

+1383e + 06

+7693e + 05

+1557e + 05





Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











parameters randomly


Material properties

Loads information










Yes No

Yes

No

New generation

of group treatment



+7438e + 06

+6647e + 06

+5856e + 06

+5065e + 06

+4273e + 06

+3482e + 06

+2691e + 06

+1900e + 06

+1109e + 06

+3178e + 05

minus4733e + 05

minus1264e + 06

minus2056e + 06


x

y

z



x

y

z

+7519e + 06

+6905e + 06

+6292e + 06

+5678e + 06

+5064e + 06

+4451e + 06

+3837e + 06

+3224e + 06

+2610e + 06

+1996e + 06

+1383e + 06

+7693e + 05

+1557e + 05





Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










x

y

z

+7519e + 06

+6905e + 06

+6292e + 06

+5678e + 06

+5064e + 06

+4451e + 06

+3837e + 06

+3224e + 06

+2610e + 06

+1996e + 06

+1383e + 06

+7693e + 05

+1557e + 05





Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article optimization design and application of ...basic design parameters of bifurcation...

Documents