Disc cutters plane layout design of the full-face rock tunnel boring machine (TBM) based on different layout patterns

Computers & Industrial Engineering 61 (2011) 1209–1225

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Disc cutters plane layout design of the full-face rock tunnel boring machine (TBM) based on different layout patterns Junzhou Huo a,⇑, Wei Sun a, Jing Chen b, Xu Zhang a a b

School of Mechanical Engineering, Dalian University of Technology, Dalian 116024, PR China School of Navigation, PLA Dalian Naval Academy 116018, PR China

a r t i c l e

i n f o

Article history: Received 20 November 2010 Received in revised form 14 July 2011 Accepted 15 July 2011 Available online 30 July 2011 Keywords: Full-face tunnel boring machine Disc cutters plane layout Co-evolutionary Spiral Star Stochastic

a b s t r a c t The quality of plane layout design of the disc cutters for the full-face rock tunnel boring machine (TBM) directly affects the balance of force distribution on the cutter head during the excavating. Various layout patterns have been adopted in practice during the layout design of the disc cutters. Considering the engineering technical requirements and the corresponding structure design requirements of the cutter head, this study formulates a nonlinear multi-objective mathematical model with multiple constraints for the disc cutters plane layout design, and analyses the characteristics of a multi-spiral layout pattern, a dynamic star layout pattern and a stochastic layout pattern. And then a genetic algorithm is employed to solve a disc cutters’ multi-spiral layout problem, and a cooperative co-evolutionary genetic algorithm (CCGA) is utilized to solve a disc cutters’ star or stochastic layout problems. The emphasis was put on the study of superiority of three different layout patterns. Finally, an instance of the disc cutters’ plane layout design was solved by the proposed methods using three different kinds of layout patterns. Experimental results showed the effectiveness of the method of combining the mathematical model with the algorithms, and the pros and cons of the three layout patterns. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction The full-face rock tunnel boring machine (TBM) is a large special engineering machine for tunnel boring. TBM has been widely applied to the subway, the railway, the highway and the waterelectricity projects, etc. Optimizing the layout of the disc cutters is the most effective way to improve the global performance of a TBM (Gertsch, 2000), and directly affects TBM’s boring performance, service life, main bearing of the cutter head, the degree of the vibration and noise. The main task of the layout design of the disc cutters is to balance the force distribution on the cutter head during the excavation. Once the disc cutter layout pattern has been developed, the amount of eccentric forces and moments of the cutter head is determined. The optimal design of a cutter head is to make the amount of eccentric forces and eccentric moments be zero. This ideal situation is best for the main bearing and cutter head support, but the reality is that there are some levels of eccentricity in the forces and moments due to the complex engineering technical requirements, the dissimilar rock boundary conditions and the corresponding cutter head structure design requirements. It is very important to study the layout of the disc cutters on the cutter head to reduce these problems and improve the cutter head performance and the cutters’ life. ⇑ Corresponding author. Tel.: +86 411 84707435; fax: +86 411 84708812. E-mail address: [email protected] (J. Huo). 0360-8352/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2011.07.011

Various layout patterns of the disc cutters have been adopted to maintain the balance the force distribution of the cutter head to make the excavation more efficient and improve the performance of the TBM, such as the multi-spiral layout pattern, the star layout pattern and the stochastic layout pattern as shown in Fig. 1. Layout of the disc cutters includes the spacing design and the plane (circumferential) layout design of disc cutters. In studying the spacing of the disc cutters, many researchers adopted the numerical simulation method (Gong, Jiao, & Zhao, 2006; Gong, Zhao, & Hefny, 2006; Gong, Zhao, & Jiao, 2005) and the Linear Cutting Machine (LCM) experimental method (Gertsch, Gertsch, & Rostami, 2007; Moon, 2006; Ozdemir, Miller, & Wang, 1977; Rostami, Ozdemir, & Nilson, 1996; Rostami, 1997; Rostami, & Ozdemir, 1996; Snowdon, Ryley, & Temporal, 1982). Song, Yan and Wang (2005) studied review on the interaction between disk cutter and rock. This part is excluded in this study. After determining the spacing of the disc cutters, the next step is to perform the circumferential layout of the disc cutters on the plane of the cutter head. The layout design of the disc cutters on the cutter head in this stage should meet many engineering technical requirements and the cutter head’s structure design requirements. The relative engineering technical requirements include the minimizing of the eccentric forces, the eccentric moments of the cutter head and the overlapping area among the disc cutters, and the maximizing of the number of the successively cutting between two adjacent disc cutters. The structure design requirements of the

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Fig. 1. Different layout patterns of the disc cutters.

Fig. 2. A disc cutters’ layout scheme (Qiao, Mao, & Liu, 2005).

cutter head include the muck buckets layout requirement, the manholes layout requirement and the multi-segments assemble requirement. All these requirements conflict with the layout design of the disc cutters. The CSM computer model (Cigla, Yagiz, & Ozdemir, 2001) for hard rock TBMs is based on the cutter head profile and rock properties. The model utilizes semi-theoretical formulas developed at EMI over the last 25 years to estimate the cutting forces. The output of this model consists of the cutter head geometry and profile, chromosome cutting forces, thrust, torque, power requirements, eccentric forces, moments, and variation of cutting forces as the cutter head rotates. Rostami (2008) studied the methods of the hard rock TBM cutter head modeling. The model they built is base on the estimation of the cutting forces and has been

proved to be a successful tool for the cutter head design optimization as well as for the performance estimation. Zhang (1996) studied the multi-spiral layout pattern of the disc cutters and presented a simplified equation of the cutting force distribution on the cutter head. The disc cutters plane layout problem belongs to a complex engineering layout problem, which contains multi-conflicting engineering technical requirements. Recently, engineering layout problems of different fields have been studied more widely, which mainly consists of three kinds of research: the layout algorithms or methods(Adel El-Baz, 2004; Maniya & Bhatt, 2001; Mir & Imam, in press), the solving strategies or frameworks of layout problems (Liu & Teng, 2008; Taghavi & Murat, 2011) and the professional

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Fig. 3. Forces exerted on a normal cutter and a gage cutter.

Fig. 6. Dynamic star + stochastic layout pattern. Fig. 4. Multi-sprial layout pattern.

Fig. 5. Dynamic star layout pattern.

layout system (Pillai, Hunagund, & Krishnan, in press); Maniya and Bhatt (2001) proposed a hybrid optimization approach for layout

Fig. 7. Stochastic layout pattern.

design of unequal-area facilities, in which simulated annealing is used to optimize a randomly generated initial placement on an

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According to the engineering technical requirements and the corresponding cutter head’s structure design requirements, this study made an overall view of the different layout patterns of the disc cutters, and formulated the layout mathematical models. Based on the characteristics of these models, a genetic algorithm (GA) was employed to solve the disc cutters’ multi-spiral layout problem, and a cooperative co-evolutionary genetic algorithm (CCGA) was utilized to solve the disc cutters’ star or stochastic layout problem. Finally, the disc cutters plane layout design instance of the TBM was solved by the proposed methods using three kinds of layout patterns. 2. Problem statement

Fig. 8. Flowchart of a GA for the cutters plane layout problem based on the multispiral layout pattern.

‘‘extended plane’’ considering the unequal-area facilities enclosed in magnified envelop blocks, an analytical method is then applied to obtain the optimum placement of each envelop block in the direction of steepest descent; Liu and Teng (2008) proposed a Human–Algorithm–Knowledge-based layout Design method (HAKD) to deal with the complicated engineering layout problems, in which human provides artificial layout schemes (artificial solutions), layout diagrams afford prior knowledge solutions, and the evolution algorithm produces novel algorithm solutions. All these numerical solutions are expressed by unified encoding strings, which make up of the evolution algorithm population together, and take part in corresponding evolution operations. It can be seen that the optimal computational methods can solve the engineering layout problems efficiently; there is a need to establish a practical computational model and study the advanced computational methods for the disc cutters plane layout.

As shown in Figs. 2 and 3, the discs are so arranged that they contact the entire cutting face in concentric tracks when the cutter head turns. The rotating cutter head presses the discs with high pressure against the cutting face. The discs therefore make a slicing movement across the face. When the pressure at the cutting edge of the disc cutters exceeds the compressive strength of the rock, the rock will be crushed. The cutting edge of the disc cutter pushes the rolling into the rock, until the advance force and the hardness of the rock are in balance. Through this displacement, described as a penetration, the cutter disc creates a high stress locally, which leads to long flat pieces of rock (chips) breaking off. According to the relative reference (e.g. Rostami, 2008) and the practical engineers’ experiences, the technical requirements of disc cutters’ plane layout design are summarized as follows:  The amount of the eccentric forces are expected to be as small as possible.  The amount of the eccentric moments are expected to be as small as possible.  All adjacent disc cutters should crush the rock successively to keep the high cutting efficiency.  All the disc cutters should be contained within the cutter head, with no overlapping among the disc cutters.  The position error of the centroid of the whole system should not exceed an allowable value and the smaller the better.

Fig. 9. Decomposition of the original layout problem.

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Fig. 10. Flowchart of a CCGA for the cutters plane layout problem based on the dynamic star and the stochastic layout patterns.

 All the disc cutters should not interfere with manholes and buckets. 3. Mathematical model and the different layout patterns 3.1. Mathematical model Suppose that the set of disc cutters to be located on the cutter head is:

C ¼ fC 1 ; C 2 ; . . . ; C n g

ð1Þ

where n = the total number of the disc cutters. As shown in Fig. 2, in this study, all the disc cutters are simplified as circles and are regarded as rigid bodies with uniform mass distribution. During the excavation, there are three forces that are exerted on the tip of the disc cutter, which are the normal force, the rolling force and the side force. Please see Fig. 3, where Fv is the normal force, FR is the rolling force and Fs is the side force. So the ith cutter can be formulated as:

C i ðLi ; r i Þ Li ¼ ðqi ; hi ; ci ÞT 2 R3 ;

i ¼ 1; 2; . . . ; n

ð2Þ

where Li = (qi, hi, ci)T e R3 is the position of a reference point (the centroid of the disc cutter) in the coordinate system oxyz; qi e (0, R) is the polar radius of the ith cutter from the center of the cutter   head; hi e [0, 2p) is the position angle of the ith cutter; ci 2 0; p2 is the tilt angle of the ith cutter; ri is radius of the ith cutter. Generally, the tilt angle of the normal cutter is zero. The masses and dimensions of all the disc cutters are given beforehand. Li is the variable to be manipulated in the design process. Thus, a general disc cutters layout scheme can be formulated as:

X ¼ fL1 ; L2 ; . . . ; Ln g

ð3Þ

After the spacing design of the disc the radius qi e (0, R)  cutters,  of the cutters and the tilt angle ci 2 0; p2 of the cutters have been determined. So a general disc cutters plane layout scheme of a cutter head can be formulated as:

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X h ¼ fh1 ; h2 ; . . . ; hn g

ð4Þ

Based on the technical requirements of the disc cutters’ plane layout design, the mathematical model of the disc cutters’ plane layout problem can be formulated as follows: Find a layout scheme Xh 2 R3n, such that

min y ¼ f ðX h Þ ¼ ðf 1 ðX h Þ; f 2 ðX h ÞÞ

ð5Þ

X2D

s.t. The overlapping constraints:

g 1 ðX h Þ ¼

n1 X n X

DV ij 6 0

ð6Þ

i¼0 j¼iþ1

The two adjacent disc cutters’ successive cutting constraints: n1 X g 2 ðX h Þ ¼ ðhiþ1  hi Þ P Dh

ð7Þ

i¼0

The static balance constraints:

g 3 ðX h Þ ¼ jxm  xe j  dxe 6 0

ð8Þ

g 4 ðX h Þ ¼ jym  ye j  dye 6 0 The manholes and buckets constraints:

g 5 ðX h Þ ¼ f8i 2 f1; . . . ; ng : C i \ OP 2 ;g

ð9Þ

where D denotes the feasible region of variable X; f1(Xh) denotes the side force (Fs) of the cutter head; f2(Xh) denotes the eccentric moments of the cutter head; DVij denotes the overlapping area between the cutter Ci and Cj. Dh denotes the allowable angle difference between two adjacent disc cutters. Om(xm, ym) is the real centroid of the whole system and Oe(xe, ye) is the expected position of Om. C i \ OP 2 ; means that the ith cutter is not overlapped with manholes and buckets. In this study, a semi-empirical cutting forces model proposed by Rostami (2008) was adopted to calculate the normal force of each disc cutter. 3.2. Multi-spiral layout pattern As shown in Fig. 4, the multi-spiral layout pattern usually is used to distribute the cutters successively around the head for the medium sized TBMs. The multi-spiral layout pattern is characterized by succession and centrality, and can be easily used to

Table 1 Parameters setting. Content

Value

Punch shear strength of rock (MPa) Uniaxial compressive strength of rock (MPa) Brazilian tensile strength (MPa) Cutter head radius (m) Rotational speed of cutter head (rad/s) Mass of each cutter (kg) Diameter of each cutter (mm) Cutter tip width (mm) Cutter penetration (mm) Cutter edge angle (rad) Number of the center cutter Number of the gage cutter Number of the normal cutter Expected centroid position of a cutter head (mm) xe ye Allowable centroid error of the whole system (mm) dxe dye Number of the manholes Number of the buckets Radius of the manholes (mm)

7–13 50–93.6 2.14–4 4.015 0.6283 200 483 10 7 1.5708 8 10 33 0 0 5 5 4 8 200

Table 2 Locations of the manholes. No.

1

2

3

4

q (mm) h (rad)

2700.000 1.2217

2700.000 2.793

2700.000 4.363

2700.000 5.934

evenly distribute the cutters. As was mentioned in Section 3.1, for a general plane layout problem of the disc cutters, supposed that there are n disc cutters to be located on the face of the cutter head. Suppose the number of spirals is m, then the angle difference between two adjacent spirals is 2p/m, and the multi-spiral layout formulation of the disc cutters can be constructed as follows:

if ðj > ðm  1ÞÞ 2p ; m j ¼ fm; . . . ; n  1g

qj ¼ ðq0 þ i  DpÞ þ aðhj þ h0 Þ þ i  i ¼ f0; 1; . . . ; m  1g; if ðj 6 ðm  1ÞÞ 2p hj ¼ h0 þ  j; m j ¼ f0; 1; . . . ; m  1g

ð10Þ

where q0 is the initial polar radius of the first normal disc cutter, Dp is the cutting spacing of the normal disc cutter, a is a factor, and h0 is the initial angle of the spirals. It can be seen that the cutter’s position angle hj is related with its polar radius qj which has been determined in the previous step – the spacing design of the disc cutters and considered known here. So based on Eq. (10), if the factor a and the initial angle of the spirals h0 are determined, the position angle hj can be determined by Eq. (11). Thus a disc cutters plane layout problem (DCPLP) that includes n variables can be transformed into a problem with two variables (a and h0).

X h ¼ fh1 ; h2 ; . . . ; hn g ¼ fa; h0 g ( qj ðq0 þiDpÞi2mp  h0 if ðj > ðm  1ÞÞ a hj ¼ h0 þ 2mp  j if ðj 6 ðm  1ÞÞ i ¼ f0; 1; . . . ; m  1g;

ð11Þ

j ¼ f0; 1; . . . ; n  1g

3.3. Dynamic star layout pattern As shown in Fig. 2, there are three kinds of disc cutters (the center cutters, the normal cutters and the gage cutters) located on the cutter head. Practical engineering applications show that gage cutters are easy to be damaged because their linear speed is high and their assemble style is tilt. In order to improve the cutting conditions of the gage cutters, they should be placed as many as possible in the transition zone. So it is better to stochastically layout the gage cutters in the transition zone of the cutter head. Based on the above-mentioned analysis, two dynamic star layout patterns were presented as shown in Figs. 5 and 6, the grey colored space is the transition zone to place the gage cutters. The white colored space is the star layout zone to place the center cutters and the normal cutters. According to the dynamic star layout pattern, this study constructed two kinds of dynamic star layout patterns. The first dynamic star layout pattern shown in Fig. 5 (called dynamic star layout pattern-I) dispersed the searching space of the design variables (the positions of the disc cutters) by setting several different searching branches. In constructing the cutters’ layout design scheme, for the normal cutters, each cutter is asked to select one searching branch from of all the feasible searching branches. There are totally q feasible branches for the ith disc cutter to choose from. For the gage cutters, the number of feasible searching branches is

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J. Huo et al. / Computers & Industrial Engineering 61 (2011) 1209–1225 Table 3 Dimensions and locations of the buckets. No.

1

2

3

4

5

6

7

8

q (mm)

3700.000 0.611 700.000 300.000

3500.000 1.396 900.000 300.000

3700.000 2.182 700.000 300.000

3500.000 2.967 900.000 300.000

3700.000 3.753 700.000 300.000

3500.000 4.538 900.000 300.000

3700.000 5.323 700.000 300.000

3500.000 6.109 900.000 300.000

h (rad) Length (mm) Width (mm)

Table 4 Performance indices of the optimal scheme and the original scheme. Layout schemes

Mv (KN m)

Fs (KN)

xm (mm)

ym (mm)

Overlapping area

Unsuccessive disc cutters’ number

Runtime (s)

Original layout scheme

154.840

11.558

2.135

0.221

0.000

4

Unknown

The obtained multi-spiral layout pattern Best 37.09 Average 34.72

35.25 37.036

2.45 2.45

0.09 0.09

0.000 0.000

6 6

222.547 224.378

The obtained star layout pattern-I Best 33.36 Average 35.26

13.86 15.24

0.77 0.95

1.05 1.23

0.000 0.000

0 0

1026 865

The obtained star layout pattern-II Best 29.05 Average 36.623

3.4 4.382

1.95 0.832

0.19 0.262

0.000 0.000

0 0

1189 1202

The obtained stochastic layout pattern Best 0.002 Average 0.1

0.001 5.5

0.9 0.38

0.7 0.43

0.000 0.000

0 0

990 848.755

Fig. 11. The original disc cutters layout scheme.

twice of that of the gage cutters. As shown in Fig. 6, each dotted ray denotes one feasible search branch of a disc cutter, which forms a set of searching branches Ei = {1, . . . , q, . . . , 2q} of the ith disc cutter. The second dynamic star layout pattern shown in Fig. 6 (called dynamic star layout pattern-II) allows the gage cutters to be placed stochastically in the transition zone, which can make more gage cutters locate in the transition zone. As shown in Figs. 5 and 6, when the branch construction of all the disc cutters is completed, a branch path linking all the disc cutters comes into being. Suppose that hi(t) 2 R is the position of the ith disc cutter at the t moment, then the next feasible position of hi(t + 1) at the (t + 1) moment can be formulated as:

Fig. 12. The obtained optimal disc cutters layout scheme based on the multi-spiral layout pattern. o 8n 2jp > > > hjh ¼ hi ðtÞ þ q ; j ¼ 1;...;q if ðqi 2 Star layout zoneÞ < n o jp hi ðt þ 1Þ 2 > hjh ¼ hi ðtÞ þ q ; j ¼ 1;...;2q if ðqi 2 Transition zoneÞ and ðStar layout pattern-IÞ > > : f0;2pg if ðqi 2 Transition zoneÞ and ðStar layout pattern-IIÞ

ð12Þ That is, each disc cutter is limited to move among the discrete searching branches evenly distributed around the cutter head. The searching branches are named branch 1, branch 2, . . . , and branch 2q, respectively. It can be seen that for the star layout pattern-I, there are q and 2q possible searching branches for each disc cutter in the star layout zone and in the transition zone, respectively. For the star layout pattern-II, it is same with the pattern-I in the star layout zone, but in the transition zone, there are infinite possible searching branches for each gage disc cutter.

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Fig. 13. The obtained optimal disc cutters layout scheme based on the dynamic star layout pattern-I.

Fig. 15. The obtained optimal disc cutters layout scheme based on the stochastic layout pattern.

Fig. 16. Solution differences of the obtained 30 schemes based on the multi-spiral layout pattern.

4. A GA for the multi-spiral layout design of the disc cutters Fig. 14. The obtained optimal disc cutters layout scheme based on the dynamic star layout pattern-II.

3.4. Stochastic layout pattern Compared with the multi-spiral layout pattern and the dynamic star layout pattern, the stochastic layout pattern allows all the disc cutters to be located at any place on the face of the cutter head if all the technical requirements are satisfied. As shown in Fig. 7, the feasible possible locating space of all the disc cutters is subjected to layout constraints like the locations of the muck buckets and the manholes, and the multi-segments assemble requirements. These constraints divide the whole layout space into two parts, the feasible layout space and the infeasible layout space.

As was described in Section 3.2, if the multi-spiral layout pattern is adopted, the number of variables will decrease from n to 2, and the number of the variables has nothing to do with the number of disc cutters. Although the variables have been greatly decreased, the multiple objectives and mutual-conflicting constraints of the problem have not been changed, and the multi-spiral DCPLP still belongs to a complex layout problem. A GA was adopted in solving the multi-spiral DCPLP. The flowchart of the GA for the multi-spiral DCPLP is shown in Fig. 8. 4.1. Coding and decoding mechanism The real number coding is adopted. The genes of the chromosomes are used as design variable that denote the factor value a and the initial angle h0 of the spirals as shown in Eq. (11). Then a

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solution vector Xh and its corresponding chromosome v can be formulated as below:

X h ¼ ½h1 ; h2 ; . . . ; hn T

v ¼ ½a; h0 T

ð13Þ

So a population can be formulated as:

0 B V ¼B @

a1 ; h01 .. .

am ; h0m

1 C C A

ð14Þ

where m is the number of chromosomes in the population. Each position angle hi of a solution vector Xh can be obtained by decoding the corresponding chromosome v based on Eq. (10). 4.2. Other operators of the GA Fig. 17. Solution differences of the obtained 30 schemes based on the star layout pattern-I.

The standard selection, crossover, and mutation operators were adopted, which are as follows: Selection: The roulette selection method was adopted. In order to keep the best chromosome of each generation to increase the evolutionary speed in the selection, the elitism strategy is adopted. Crossover: The standard two-point crossover is adopted. Mutation: The standard mutation operator was applied. Producing a random real number Rand among the range of (0, 1); If Rand < Pm, select a chromosome randomly, then re-initialize this chromosome. 5. A cooperative co-evolutionary genetic algorithm (CCGA) for the dynamic star and the stochastic layout design of the disc cutters

Fig. 18. Solution differences of the obtained 30 schemes based on the star layout pattern-II.

Fig. 19. Solution differences of the obtained 30 schemes based on the stochastic layout pattern.

The co-evolutionary method, first proposed by Hillis (1990) in 1990, was a new evolutionary method based on the evolution theory over the last decade. It simulates the relationships of competition, predation and symbiosis of the natural species and their complementary evolution. It is an evolution process from low level to advanced level. According to the biology models, the co-evolutionary method can be divided into the competitive co-evolutionary algorithm, the predatism-based co-evolutionary algorithm and the cooperative co-evolutionary algorithm (CCEA). The first two algorithms adopted the mechanism of competitive among the populations. The third method CCEA adopted the mechanism of cooperation among the populations. In 1994, Potter and Jong (1994) proposed a CCEA that adopted multiple cooperating subpopulations to co-evolve subcomponents of solution. Based on the CCEA, genetic algorithm (GA) was tested in solving the high-dimensional function optimization problems. Experimental results showed that CCEA was a promising framework. In 2004, Van-den Bergh and Engelbrecht (2004) proposed a cooperative particle swarm optimization (CPSO) algorithm based on CCEA and results of the simulation experiments showed that the CPSO was superior to the CCEA. In the same year, Jansen and Wiegand (2004) theoretically proved the convergence property and effectivity of the CCEA. When the dynamic star layout pattern and the stochastic layout pattern are used to distribute the disc cutters, the number of variables is equal to the number of the disc cutters. And the variables consist two parts: the position angles of the normal cutters and the position angles of the gage cutters. These two types of cutters are subjected to different forces distribution and are of different layout patterns. It is better to divide the layout design of the disc cutters into two sub-problems. And each sub-problem includes one type of disc cutters that adopts different forces calculating method and

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Table 5 The maximum stresses and deformations of the cutter head of the obtained optimal schemes and the original scheme. Methods

Von stress (MPa)

Original layout scheme The obtained multi-spiral layout pattern The obtained star layout pattern-I The obtained star layout pattern-II The obtained stochastic layout pattern

Max deformation (mm)

Normal load

Full load

Normal load

Full load

78.466 59.649 63.527 66.313 65.546

241.513 179.756 198.371 205.859 199.754

0.326 0.322 0.31 0.324 0.268

1.039 1.007 1.012 1.036 0.843

Fig. 20. Stress and deformation distributions of the cutter head of the original layout scheme under normal load condition.

Fig. 21. Stress and deformation distributions of the cutter head of the original layout scheme under full load condition.

different layout patterns. Therefore the CCEA was used to solve the dynamic star and the stochastic layout problems of the disc cutters in this study. In order to use the CCEA to solve the disc cutters’ layout problem more efficiently, four key technical issues of the CCEA are to be solved properly: r decomposition of the original problem; s coordination of the subpopulations; t selection of the cooperative chromosomes; u evaluation of the chromosome fitness of each subpopulation.

5.1. Decomposition of the original problem For the decomposition of the original problem, Potter and Jong (1994) used a static decomposition strategy that took every variable as a sub-problem to solve the function optimization problem. In this study, the number of the variables is about 50. If a static decomposition strategy was used, the layout problem would have to be decomposed into about 50 sub-problems. And a huge amount of computing time would need to be invested. Furthermore, the

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Fig. 22. The obtained stress and the deformation distributions of the cutter head based on the multi-spiral layout pattern under normal load condition.

Fig. 23. The obtained stress and the deformation distributions of the cutter head based on the multi-spiral layout pattern under full load condition.

Fig. 24. The obtained stress and the deformation distributions of the cutter head based on the dynamic star layout pattern-I under normal load condition.

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Fig. 25. The obtained stress and the deformation distributions of the cutter head based on the dynamic star layout pattern-I under full load condition.

layout problem includes different types of the disc cutters that should be based on the different layout patterns. In order to saving the computing time and be in accordance with the classification of the disc cutters, the original cutters’ plane layout problem is decomposed into two sub-problems according to the cutters’ types. As shown in Fig. 9, a solution of the sub-problem No. 1 consists of the position angles of all the normal disc cutters; a solution of the sub-problem No. 2 consists of the position angles of all the gage disc cutters. n1 is the number of the normal cutters, and n2 is the number of the gage cutters. In this study, base on the framework of the CCEA, a GA is used to solve each layout sub-problem, named cooperative co-evolutionary genetic algorithm (CCGA).

So the populations of each sub-problem can be formulated as:

0

h11 ; h12 B .. B P1 ¼ @ . hm1 ; hm2 0 h11 ; h12 B .. P2 ¼ B . @ hm1 ; hm2

1 h1n1 .. C C . A    hmn1 1 . . . h1n2 .. .. C C . . A    hmn2 ... .. .

ð16Þ

where m is the number of chromosomes in the subpopulation.

5.2. Coding mechanism

5.3. Coordination of the subpopulations

In each subpopulation, the real number coding is adopted. The genes of the chromosomes are used as design variables that denote the position angle of the disc cutter. There are two subpopulations in this study. The solution vectors and their corresponding chromosomes of each subpopulation can be formulated as:

The coordination mechanisms of the subpopulations for the CCGA are various. The main purpose of the various coordination mechanisms is to keep all the subpopulations converging on a global consistency objective. There are two classical coordination mechanisms, the cooperative mechanism and the competitive mechanism. The traditional evolutionary algorithms simulate more on the competition among the natural species, and less of the cooperation among them (Liu, 2004). For the practical layout problems, the competition and cooperation usually exist simultaneously. It is important for evolutionary algorithms to bal-

X 1h ¼ v 1 ¼ ½h1 ; h2 ; . . . ; hn1 T X 2h ¼ v 2 ¼ ½h1 ; h2 ; . . . ; hn2 T n ¼ n1 þ n2

ð15Þ

Fig. 26. The obtained stress and the deformation distributions of the cutter head based on the dynamic star layout pattern-II under normal load condition.

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Fig. 27. The obtained stress and the deformation distributions of the cutter head based on the dynamic star layout pattern-II under full load condition.

ance the cooperation and competition among the populations, and to reach to a global balance status. Because GA is used to solve the sub-problem, in order to keep a good balance between the cooperation and competition, this study adopted the cooperative mechanism among the subpopulations.

chromosomes at the interval of every t generations during the optimization process. The above-mentioned hybrid method can better balance the global search and the local search and keep the sustained searching capability of the algorithm. 5.5. The chromosome’s fitness evaluation of each subpopulation

5.4. Selection of the cooperative chromosomes The chromosome’s fitness evaluation of each subpopulation needs some cooperative chromosomes provided by other subpopulations. There are several methods to select cooperative chromosomes, e.g. the random selection method, the best selection method and the cooperative population selection method. Although the random selection method can maintain the diversity of each subpopulation, it will result in slow convergence. The best selection method can accelerate the convergence speed and is suit for the final local convergence of the latter searching process, but it easily falls into local searching area once the best selection method is adopted early. For the cooperative population selection method, it is hard to be operated and controlled the cooperative population selection method. In this study, a hybrid random-best selection method is adopted, which means that the random selection method and the best selection method are transferred to be used to evaluate the

The CCGA exchanges the information among the subpopulations by the mechanism of adopting the cooperative chromosome method to evaluate the chromosome’s fitness. A chromosome of each subpopulation is a part of the whole solution of the problem, which is shown in Fig. 9. Suppose that Xij is the jth chromosome of the ith subpopulations. When calculating the fitness value of Xij, first a cooperative chromosome set X select ðk ¼ 1; 2; . . . ; i  1; i þ 1; . . . ; qÞ is selected k from other subpopulations, where q = the number of subpopulations. In this study, q = 2. The chromosome Xij and the cooperative chromosome set X select form a whole solution X of the problem, k which can be formulated as:

n o X ¼ X select ; X select ; . . . ; X select ; . . . ; X ij ; . . . ; X select 1 2 k q

ð17Þ

The flowchart of a cooperative co-evolutionary genetic algorithm is shown in Fig. 10.

Fig. 28. The obtained stress and the deformation distributions of the cutter head based on the stochastic star layout pattern under normal load condition.

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Fig. 29. The obtained stress and the deformation distributions of the cutter head based on the stochastic star layout pattern under full load condition.

6. Strategies for handling constraints and objectives CCGA needs scalar fitness information to work. At present, based on the idea of the aggregating functions, the approach of combining multiple objectives into a single objective to construct the fitness function has been studied widely. The aggregating function methods include the weighted sum approach and the goal programming, etc. The advantages of the above-mentioned methods lie in their efficiency and their ability to be easily implemented in engineering practice. The weighted sum approach is used here to transfer the multiple-constrained mathematical model into unconstrained mathematical model. The chromosome Xi of each subpopulation cannot be evaluated independently, because Xi is not a complete solution of the whole problem. As shown in Eq. (17), the chromosome Xi and the cooperative chromosomes X select are combined into a complete solution X k of the whole layout problem. Then Eq. (18) is used to evaluate the complete solution X, and the fitness value of the whole solution X is regarded as the fitness value of the chromosome Xi of the subpopulation.

FðXÞ ¼ k1 C 1 F 1 ðXÞ þ k2 C 2 F 2 ðXÞ þ    þ kk C k F k ðXÞ

ð18Þ

where k1, k2, . . . , kk are the weights of each sub-objective function that reflect the importance of the sub-objectives. F1, . . . , Fk are the sub-objective set. C1, C2, . . . , Ck are the normalizing coefficients of each sub-objective function. In this study, according to the mathematical model of the disc cutters layout problem, Eq. (18) can be transformed into:

FðXÞ ¼ k1 C 1 f1 ðXÞ þ k2 C 2 f2 ðXÞ þ k3 C 3 g 1 ðXÞ þ k4 C 4 g 2 ðXÞ þ k5 C 5 g 3 ðXÞ þ k6 C 6 g 4 ðXÞ þ k7 C 7 g 5 ðXÞ

ð19Þ

7. Application instance Taking the disc cutters layout design of a full-face rock TBM for a water tunnel project as a background, 41 disc cutters are to be located on the cutter head surface shown in Fig. 2. The relative parameters are listed in Table 1. The rock is mainly in granitebased geology. According to engineering requirements, there are four manholes and eight symmetrically distributed buckets on the cutter head, the locations of the manholes and the buckets are listed in Tables 2 and 3, respectively. The problem is to optimize the objective function formulated by Eq. (5), whilst satisfying all the technological constraints given by Eqs. (6)–(9).

The numerical experiments were run on an AT-compatible PC of IntelÒ Core™ 2 CPU 2140 @ 1.6 GHz, and 2.00 GB memory. Parameters set in the FEM calculation are: the calculated FEM platform is ANSYS (R) Release 10.0, unit Shell63 was adopted, the number of units is 63,369, the number of nodes is 62,193, the density is 7850 kg/m3, the elastic modulus E = 2.06  105 MPa, and the Poisson’s ratio is 0.3. The GA was used to solve the DCPLP using the multi-spiral layout pattern. The CCGA utilized to solve the disc cutters’ plane layout problem using the dynamic star and the stochastic layout patterns. Both CCGA and GA run thirty times. In each running time, the maximum number of generations is 2000. The number of the spirals is 8. The number of the branches of the dynamic star layout pattern is 16. The performance indices of the optimal layout scheme based on the four layout patterns and the original layout scheme were listed in Table 4. The original disc cutters’ layout scheme (DCLS) used in the project was illustrated as Fig. 11. The obtained optimal DCLS using the multi-spiral layout pattern was illustrated as Fig. 12. The obtained optimal DCLS using the dynamic star layout pattern-I was illustrated as Fig. 13. The obtained optimal DCLS using the dynamic star layout pattern-II was illustrated as Fig. 14. The obtained optimal DCLS using the stochastic layout pattern was illustrated as Fig. 15. The solution difference of the obtained 30 schemes using the multi-spiral layout pattern was illustrated Fig. 16. The solution differences of the obtained 30 schemes using the dynamic star layout pattern-I was illustrated Fig. 17. The solution difference of the obtained 30 schemes using the dynamic star layout pattern-II was illustrated Fig. 18. The solution difference of the obtained 30 schemes using the stochastic layout pattern was illustrated Fig. 19. The obtained maximum stresses and deformations of the cutter head under two different loading conditions (the normal loading condition and the full loading condition) were listed in Table 5. For the normal loading condition, forces exerted on the tip of the cutters were calculated by a semi-empirical cutting force model proposed by Rostami (2008). For the full loading condition, forces exerted on the tip of the cutters were determined by the maximum normal force of the cutter, which is about 250 KN for the 19 in. disc cutters adopted in this study. The obtained stress and the deformation distributions of the original layout scheme under the normal load condition were shown in Fig. 20. The obtained stress and the deformation distributions of the original layout scheme under the full load condition were shown in Fig. 21. The obtained stress and the deformation distributions of the optimal layout scheme using the multi-spiral layout pattern under the normal load condition were shown in

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Fig. 22. The obtained stress and the deformation distributions of the optimal layout scheme using the multi-spiral layout pattern under the full load condition were shown in Fig. 23. The obtained stress and deformation distributions of the optimal layout scheme using the dynamic star layout pattern-I under the normal load condition were shown in Fig. 24. The obtained stress and deformation distributions of the optimal layout scheme using the dynamic star layout pattern-I under the full load condition were shown in Fig. 25. The obtained stress and deformation distributions of the optimal layout scheme using the dynamic star layout pattern-II under the normal load condition were shown in Fig. 26. The obtained stress and deformation distributions of the optimal layout scheme using the dynamic star layout pattern-II under the full load condition were shown in Fig. 27. The obtained stress and deformation distributions of the optimal layout scheme using the stochastic layout pattern under the normal load condition were shown in Fig. 28. The obtained stress and deformation distributions of the optimal layout scheme using the stochastic layout pattern under the full

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load condition were shown in Fig. 29. The geometry parameters of cutter head were shown in Fig. 30. The data in Table 4 showed that: (1) The obtained scheme using the multi-spiral layout pattern is equivalent to the original layout scheme on the performance indices. The obtained schemes using the dynamic star layout pattern-I, pattern-II and the stochastic layout pattern are superior to the original layout scheme on all the performance indices. It can be seen that the layout problem of the disc cutters belongs to a multi-objectives optimization problem, and when the optimization methods are used to solve this problem, better optimal layout schemes can be obtained. (2) Compared with the obtained layout scheme using the multispiral layout pattern, the obtained layout schemes using the dynamic star layout pattern-I, pattern-II and the stochastic layout pattern are superior on the performance indices of the eccentric forces, the eccentric moments and the succes-

Fig. 30. Geometry parameters for the cutter head.

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sive cutting. Because the multi-spiral layout pattern makes all disc cutters link to one another and limits the layout space of the disc cutters. On the contrary, the dynamic star layout patterns and the stochastic layout pattern can provide more layout space for the disc cutters. (3) It also can be seen that the obtained layout scheme using the dynamic star layout pattern-II is a little superior to the obtained scheme using the dynamic star layout patter I. The main reason is that the dynamic star layout pattern-II not only provides infinite layout space for the gage cutters but also allows the gage cutters to distribute stochastically in the transition zone. (4) Among the five obtained layout schemes, the scheme using the stochastic layout pattern can keep the best performance indices on the eccentric forces, the eccentric moments and the successive cutting. Because the stochastic layout pattern allows the normal cutters and the gage cutters to distribute stochastically in the face of the cutter head. And the algorithm can deeply search the solution space to distribute the disc cutters according to this stochastic feature. (5) Because the multi-spiral layout pattern can decrease the number of the variables from n to 2, it cost the shortest runtime. And other layout patterns need relatively longer runtime. As was mentioned above, based on each layout pattern, the algorithm can obtained 30 optimal layout schemes after running 30 times. As shown in Figs. 16–19, the obtained 30 optimal layout schemes using the multi-spiral layout pattern are different with each other, but the differences are not significant. And the obtained 30 optimal schemes using other layout patterns are significantly different with each other. In order to study how the obtained layout schemes influence the strength and the stiffness of the cutter head, the FEM calculations were operated based on the obtained layout schemes under the two loading conditions. The data in Table 5, Figs. 20–29 show that: (1) Under the two loading conditions, according to the obtained optimal layout schemes using the four layout patterns, the cutter head can keep a better uniform deformation and stress distribution than the original layout scheme. (2) Among the obtained four layout schemes, the obtained optimal layout scheme using the multi-spiral layout pattern can make the cutter head have the most uniform stress distribution, and the obtained layout schemes using other three layout patterns are almost the same with each other on the stress distribution. It can be seen that the multi-spiral layout pattern can make the disc cutters evenly distribute on the face of the cutter head (as shown in Fig. 12). The even distribution of the disc cutters has a good effect on the stress distribution of the cutter head. As shown in Figs. 13–15, although other three layout patterns can provide more layout space and can obtain better layout schemes that are superior to the scheme using the multi-spiral layout pattern on the eccentric forces, the eccentric moments and the successive cutting, the freer stochastically layout characteristics of the three layout patterns may result in uneven distribution of the disc cutters. (3) From data in Table 5, it also can be seen that the obtained layout schemes using the multi-spiral and the dynamic star layout patterns are of the same deformation distribution with the original layout scheme. However, the obtained layout scheme using the stochastic layout pattern has the lowest values of the maximum deformation. From data in Table 4, it also can be seen that the obtained optimal layout scheme using the stochastic layout pattern has the lowest values of

the eccentric forces and the eccentric moments, which means that the values of the eccentric forces and the eccentric moments have a big influence on the deformation distribution of the cutter head and a small influence on the stress distribution of the cutter head. The smaller these values are, the better deformation distribution of the cutter head will be. 8. Conclusions Based on the engineering technical requirements and the corresponding cutter head structure design requirements, this paper studied the pres and cons of three existing layout patterns: the multi-spiral layout pattern, the dynamic star layout pattern and the stochastic layout pattern. The models of the three layout patterns were constructed. Based on the characters of each layout pattern, this study adopted a GA to solve the multi-spiral layout problem of the disc cutters and a CCGA to solve the dynamic star and the stochastic layout problems of the disc cutters. The computational results showed that: (1) The layout design of the disc cutters belongs to a complex optimization problem, so there is a need to make use of the advanced optimization methods to solve this problem. (2) The multi-spiral layout pattern can decrease the size of the problem from n to 2, and can save the computing time. But simultaneously, it also limits the layout space of the disc cutters and is hard to obtain the best solution with the best performance indices. (3) The multi-spiral layout pattern can make the disc cutters distribute more evenly on the cutter head and can improve the stress distribution of the cutter head. (4) The dynamic star layout pattern can transfer a continuous disc cutters’ problem into a combinational optimization problem by adjust the number of branches of the star. Based on this transformation, the algorithm can produce the better layout schemes that can improve the stress distribution of the cutter head during the following structure design of the cutter head. (5) The stochastic layout pattern can provide infinite possible solution space for the layout of the disc cutters and can produce the best solutions with the best performance indices. (6) The values of the eccentric forces and the eccentric moments have a big influence on the deformation distribution of the cutter head and a small influence on the stress distribution of the cutter head. According to the above-mentioned analysis, the layout design of the disc cutters not only influences the balance of the cutter head, but also influences the strength and the stiffness of the cutter head. Further research will be conducted on the following issues: (1) The layout design of the disc cutters is coupled with the structure design of the cutter head, so the collaborative optimization design between the layout design of the disc cutters and the structure design of the cutter head should be studied. (2) Only one rock boundary condition is considered in the application instance. The multiple rock boundary conditions should be considered synchronously during the layout design of the disc cutters.

Acknowledgements This work is supported by the National Natural Science Foundation for Young Scholars of China (Grant No. 51005033), the China

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Postdoctoral Special Science Foundation (Grant No. 201003618), the Major State Basic Research Development Program of China (973 Program) (Grant No. 2007CB714000), the Fundamental Research Funds for the Central Universities and Liaoning Key Science and Technology Project (Grant No. 2008220017).

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