An equilibrium multi-objective optimum design for non-circular clearance hole of disk with discrete variables

Jiaxin HAN,Haiding GUO

College of Energy and Power Engineering,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China Jiangsu Province Key Laboratory of Aerospace Power System,Nanjing 210016,China

1.Introduction

The turbine disk of aircraft engine rotates at very high speed and under high temperature,which usually results in severe stress situations,especially on the region near the holes.Lots of practical and theoretical researches have shown that the failure of disks caused by stress concentration is one of the most major reasons for the reduction of its service life.1,2

One effective way to lessen stress concentration is to use non-circular hole instead of circular one,which has already been applied to the turbine disk of CFM56-III,see Fig.1.3In Fig.1.In Ref.4,a geometrical model for the non-circular hole was given by Chen et al.and an optimization model was proposed too.4It can also be seen from Fig.1 that the non-circular hole is biaxial symmetrical.The profile consists of 8 arcs,ie.main arcs(R1)and transition arcs(R2).However,further researches show that the maximum stress around the hole will decrease monotonically when the upper bound of the radius of the main arc increases,which means the profile of the hole tends to be a ‘square” and that is not a good option in most cases.5,6To introduce balanced design ideas into the optimization is an effective way to solve this problem.

The turbine shaft,labyrinth disk and turbine disk are connected by 48 long-bolts and nuts.Fig.2(a)shows 1/48 sector model of the disks.The bolt joints of turbine components are illustrated in Fig.2(b).The turbine shaft and turbine disk connected with bolts have similar local connecting structures and rotate at the same speed.According to the widely accepted equal-life design criteria,to design two connected components with similar service life or similar stress level might be a better choice.Such balanced designed structure can effectively avoid over-designs and features better economy.7Take the turbine components of CFM56-III for example,the stress levels around the clearance holes of the turbine shaft and turbine disk are designed to be with almost the same value,which should have adhered to the equal-life design principle.This may offer a reference for the optimization of non-circular clearance holes on turbine disks.

Less profile variation of the clearance hole for the turbine disk will offer larger contact area for the plate nuts;see Fig.2(b).The contact condition between bolt and the hole will not be deteriorated.On the other hand,a relative ‘conservative”option(less profile variation of the hole)will lead to a‘confident” design,and will be benefit to the processing,testing and assembling.Therefore,a compromise design is needed.The stress reduction and the least profile variation can be considered concurrently.

In order to guarantee machining precision,the dimensions of the non-circular hole should be rounded to meet the requirements of industry specification.This means the profile of non-circular hole will be optimized as the one with specified dimensions,rather than that with casual discrete ones.8So far,several optimization methods were already used in dealing with discrete variables,such as Brand-and-Bound,9simulated annealing algorithm,10harmony search,11Genetic Algorithms(GA),12ant colony algorithm13and some other nature-inspired methods.14Yet variables discretization processing are still tedious and inaccurate,and rounding design variables of new design to allowable dimensions usually needs an overcomplicated algorithm.15–17Moreover,there always lacks an effective way to expurgate the unreasonable samples produced in those algorithms.18

In this paper,we introduced an Equilibrium Multiobjective Optimum Model(EMOM),in which balanced design ideas are proposed,for a compromise design between the stress reduction and the least profile variation of the hole on the turbine disk.Also,the dimensions of the non-circular hole are selected as a group of discrete variables to meet the industry specification,and a Surrogate Genetic Coding Algorithm(SGCA)is proposed to solve the non-circular-hole optimization problems.In this study,an indirect coding method and a check model are also applied to check the feasibility and eliminate redundant fitness evaluations.

Fig.1 Non-circular clearance hole on turbine disk.

2.Construction of equilibrium multi-objective optimum model

2.1.Structural analysis of turbine disk based on FEM

The loads acted on the clearance hole of the front flange of the turbine disk are quite complicated.Actually,the centrifugal load,torque,interference fit,pretension of the bolt,axial load and thermal load can be the candidates which affect the stress conditions of the hole.Among them,the centrifugal load is the major load which dominates the stress level of the hole.To build a feasible and efficient optimization model,the complex loads could be reasonably predigested and the factors that have less influences on the stress of the hole could be ignored temporarily.As discussed in Refs.4 and 6,a simplified mechanical model has been proposed and only centrifugal load were considered.Researches have shown that the non-circular hole optimized with such simplified model still has the best performance when complicated load conditions of the turbine disk are considered.5

The Finite Element Model(FEM)for the optimization model of the turbine disk is shown in Fig.3.The material of the turbine disk is Ni-based high temperature alloy GH4169.The rotation speed is ωmax=14731 r/min and the working temperature is 450°C.Fig.4 gives the constitutive relationship of the GH4169 at 450°C.

Fig.2 Turbine components.

The first principal stress distribution around the circular clearance hole on turbine disk is calculated firstly,Fig.5 illustrates the stress distribution around the hole of the turbine disk.As the main driving force of crack initiation and propagation is the circumferential stress at the hole-edge,approximate to the first principal stress on the surface of the clearance hole,19we set one of the objectives in the equilibrium optimization model to decrease the maximum stress around non-circular hole σmaxto a required level.

Fig.3 FEM model of turbine disk.

Fig.4 Constitutive relationship of GH4169 in 450°C.

Fig.5 Stress distribution of circular clearance hole on turbine disk.

2.2.An equilibrium multi-objective optimum model for noncircular hole

Using a non-circular hole to substitute the circular,one can effectively decrease the maximum stress and the stress concentration around the hole;However,it does not mean the lower the stress on the hole-edge the better,because minimizing the stress around the hole without consideration of stress on the connected components will result in an over-designed solution and bring no more benefit to the life of the whole structure.Besides,the profile of the non-circular hole with much lower stress tends to be a ‘square”,the contact condition between the bolt and the hole will be deteriorated,and result in increasing of the contact stress,which will bring negative effects on the force transition.In this case,a more balanced design which can concurrently meet the requirements of stress reduction around the hole and the least profile variation would be an advisable choice.Therefore,we propose a new equilibrium multi-objective optimum model to handle this problem and offer a balanced design.

On the one hand,the stress levels of the turbine components are firstly considered.As shown in Fig.2(a)and(b),the back flange of the turbine shaft and the front flange of the turbine disk are of the same connecting structure and rotate at the same speed.In consideration of ‘equilibrium”design,the connecting components are usually designed with similar service life,and instead of seeking for the lowest stress around the non-circular hole of the turbine disk,we set σmaxaround the hole to a required level and the local area of the turbine disk will have the similar service life with that of the turbine shaft.Alternatively,the desired value of σmaxcan also be set as the user’s requirements.

On the other hand,the least profile variations are also our objects as mentioned above.Assuming the non-circular hole is bisymmetrical,the sizes of main arcs(R1)and transition arcs(R2)are designed to be as close as possible to the primal radius of the circular one.

Take these requirements into account,an EMOM optimum model can be expressed as Eqs.(1)and(2).

wherefconsists of three individual objects,representing the objective function for the stress decreasef0,and for the least variations of the main arcsf1and transition arcsf2respectively.R1max,R2maxandR1min,R2minare the upper bound of the main arcs and the lower bound of the transition arcs respectively.

2.3.Determination of the self-regulated weighting factors

The objective functions are normalized by Eqs.(3)–(5),with the similar order of magnitude in the evolution process:

where σrrepresents the first principal stress around the original circular hole,σ*is the target of the first principal stress around the non-circular hole obtained from the equal-life idea and can be set as needed.R*presents the radius of primal circular clearance hole.

By means of compromise programming,we construct a combined objective function as

wherek0,k1andk2are the weighting factors of each objective function.

Practically,different designers conduct designs with different domains ofR1orR2.To make the design result more robust,self-regulated weighting factors are designed to balance the importance of different objectives,as in Eqs.(7)and(8).

wherek1is chosen as unity;and α is set as a hundredth of golden mean(0.01618).R1maxandR2maxis the upper bound of main arc and transition arc respectively.Eqs.(7)and(8)offer an evolution way fork0andk2that can adjust themselves to suitable values asR1maxincreases.In this way,a balanced optimum solution can always be obtained.

3.Surrogate genetic coding algorithm

3.1.Basic ideas of SGCA

In structural design,the dimensions of structure usually follow some industry specifications rules,which is beneficial to the process precision and inspection.The design dimensions of aero-engine should also be rounded except for some particular cases.Therefore,in the optimization model,discrete design variables with specified values confirming with industrial standard will help the design be more meticulous and standardized.20As we can find in the turbine disk and the labyrinth disk of CFM56-III,the main arc and the transition arc for non-circular clearance holes were both normalized as standard values,see Fig.13.

In the paper,we also employed discrete design variables to design arcs radii for non-circular hole with standard values.The optimization problem of clearance hole is to find the best combination of dimensions from a certain sequence which follows industry specifications.Such treatment will also reduce the feasible set to a finite one,in which the optimal design can be obtained with less iterations and the optimization efficiency will be improved.

Among all those optimization algorithms,GA is able to handle discrete variable problem easily and perform better global search ability.21–23During the evolution in traditional GAs,the evolution process does not operate on design variables,but on the codes.Whereas,when concerning the discrete variables,such ordinary binary coding arithmetic could produce unexpected descendants due to genetic operators such as reproduction,crossover and mutation work randomly,which means newly generated design variables may no longer belong to the predefined discrete value set,say feasible set.So far,there are two most acceptable approaches to solve this problem:one is to repeatedly round-off the continuous variables into the nearest discrete ones which belong to the given discrete variable set;the other is to introduce appropriate penalty functions to fix values of variables.17Nevertheless,both of these methods are tedious and the penalty functions are usually too complex to formulate.In the paper,we propose a surrogate genetic coding algorithm,which can be used to get rid of those fussy processes and can successfully keep the discreteness of descendants as desired during genetic operations.

3.2.A surrogated genetic coding algorithm

Other than classical GA algorithm,we introduce an indirect coding method,in which variables are discretized,and a mapping between the available discrete design variables and general nonzero integers are built.Every design,usually a combination of variables,can be transferred into a string of integer numbers,among which the evolution calculation will conduct.This one-to-one mapping relationship between every discrete design variable and each integer can be formulated by Eq.(9).

wherexiis referred to as each design variable with certain discrete values,Niis integer 1,2,...,n.A similar way is given in Ref.24too.

Up still now,no processing standard for the non-circular hole are presented.According to the standardization rules in the Chinese Machine Design Handbook,if the designs have relatively flexible choices of parameters,the priority number systems will be preferred as acceptable standards.25In this paper,R40 priority series based on GB/T 321-2005 will be employed to regularize the arc radii values,and if we set the arcs radii under 100 mm,the Candidate Values Pool(CVP)will be constituted of 82 dimensions.Then all these integer surrogates will be formulated into an Integer Surrogate Pool(ISP)allocated between CVP and Binary Coding Pool(BCP).The coding process and genetic operations such as election,crossover and mutation will be operated in BCP.Because binary strings of integers have closure properties in the operations,the descendants generated will remain integers,and the corresponding discrete design values can always be obtained through the mappinghi-1(Ni).It will be illustrated in Fig.6,whereAirepresents the binary string ofNi.

The process can be explained further in Fig.7 with a twovariable optimization problem.Four discrete design variables from two designsX1andX2in CVP correspond to four integers in ISP,which will be coded into binary series and be deposited into BCP.The evolution operation will be done between two BCPs.

It can be seen from Fig.7 that by introducing the integer surrogate pool,binary strings of the integers after the genetic operations(crossover&mutation)will remain integers.As there is one-to-one mapping relationship between the integers and discrete variables,all of the decedents will remain standard discrete values too and the tedious and complicated round-off process are avoided.

Fig.6 Transmission relationships among CVP,ISP and BCP.

Fig.7 Coding method and genetic operation process in SGCA.

3.3.A check module of SGCA

In the discrete optimization problem,design members in the feasible set are limited,the same design points may appear repetitively,which will result in premature in the evolution.On the other hand,the cross-border designs during the crossover process cannot be avoided.Therefore,a rationality check module is necessary.The repeated designs are identified by the module,and extra fitness evaluation in the descendants are avoided.The cross-border designs can also be found with the module and eliminated and replaced by re-initialization ones.The re-initialization of the design solutions in this case will not lower the convergence rate.The total exceeding possibility can be calculated in Eqs.(10)–(12).

whereUandLare referred to as the values of upper and lower boundaries of the variables.n1andn2are the binary digits of lower and upper bounds of the feasible set.PUandPLshow the possibilities that a parent is at neighborhood of upper or lower boundary respectively.Perepresents how likely the cross-border designs will occur.Results show that the crossborder designs are less than 10%in most cases during the evolutionary process.

In conclusion,the rationality check module is of two functions.Firstly,the calculations of repeated design points are skipped when dealing with nonlinear finite element calculations,so the computer time can be effectively saved.Secondly,cross-border designs are eliminated.This not only avoids degradation by keeping that every generation has same number of solutions but also increases the diversity of the population and suppresses the premature phenomena in some degree.

The whole optimization algorithm can be illustrated by Fig.8,which illustrates the optimization procedure for the non-circular clearance hole of the turbine disk.

4.Results and discussion

4.1.Optimization results of the non-circular hole

The proposed EMOM+SGCA model is used to optimize the biaxial symmetric non-circular clearance hole of the turbine disk as shown in Fig.2(a)and(b).The target value of stress decrease around the non-circular hole is set based on the equal-life requirement(it can also be set as needed).By assuming the connecting components in Fig.2 working under the similar situations,we choose to decrease the maximum stress of the clearance hole on turbine disk to a certain level,which is equal to the stress of clearance hole on turbine shaft.In this case,19.0%decrease of the stress level around the hole of the turbine disk is employed.The boundaries ofR1andR2are set as 6 mm-R1maxand 2–5 mm respectively,whereR1maxcan be adjusted too based on designer’s experience and preference.Actually,R1maxcan be any values among 30–100 mm,which will have no critical influences on the optimization result.The weighting factor in EMOM will be self-regulated asR1maxchanges.The evolutionary histories with four referencedR1maxare listed below in Fig.9(a)and(b)to show the stability of the proposed optimization method Fig.9(b)presents the scattering of design points during the evolution.

Fig.8 Flowchart of EMOM+SGCA for the optimization of non-circular hole of turbine disk.

Fig.9 Optimization results of non-circular clearance hole.

Fig.10 Stress distribution of the optimized non-circular hole.

T

able 1 Comparisons of two different non-circular hole optimization methods.

Fig.9(a)illustrates the whole process of evolution.At the beginning,designs with bigger stress decrease rates is obtained.Along with the evolutionary process,these over-designed solutions are gradually replaced by a more balanced one,which has less profile variation and with desired stress decrease rate concurrently.Besides,the optimization will stably converge on the design with ideal stress decrease rate after 8 generations,no matter how theR1maxchanges.Fig.9(b)presents the scattering of design points during the evolution,in whichR1max=100 mm.According to Fig.9(b),all design points based on the proposed method are only generated within the feasible set with the standardized discrete values,which means that the introduced SGCA can effectively manage the improper solutions and expurgate invalid fitness evaluations.It can also be found that solutions are clustered around the final optimal design(noted by the red star).Fig.10 presents the profile and stress distribution of the optimum non-circular clearance hole.

In Fig.10,the main arc and transition arc of the optimized non-circular hole isR1=19.00 mm andR2=3.35 mm respectively.Compared with the stress around the primal circular hole(see Fig.4),the optimized non-circular one has more uniform stress distribution and lower stress level of σmax.

4.2.Comparison with the other optimization method of noncircular holes

The optimization procedure of EMOM+SGCA is compared to the one with singular objective and continuous variables in literature 4.The optimization strategies and results of these two procedures are listed in Table 1.

According to Table 1,the model and optimization procedure proposed in Ref.4 will offer several different noncircular rounded holes with lower stresses.But they are clearly over-designed and with excessively large profile variations.Moreover,the results are unstable when different domains ofR1maxare applied,which usually result in designers’confusion in practice.The method based on the EMOM+SGCA,however,can successfully lead to a compromise solution,which satisfies both requirements of stress reduction and the least profile variations.The method proposed is with a better performance in the robustness no matter how the upper bound of the design variableR1maxchanges,and tedious trials for variables’domain are avoided.

We can also find in Table 1 that the SGCA is with much higher efficiency in optimization procedures,which could be attributed to two reasons:one is the applications of discrete variables,in which the evolutions are limited in a finite feasible set.The other is the introduction of check module,in which redundant fitness evaluations are eliminated and therefore computer time is saved.

5.Conclusions

In the study,an equilibrium multi-objective optimization model and a surrogate genetic coding algorithm are proposed to find an equilibrium design of non-circular bolt hole on the turbine disk with discrete variables.The conclusions are:

(1)A stable balanced design can be obtained with proposed EMOM module which can meet the requirements of stress reduction and the least profile variation of the non-circular hole,and the design robustness can be guaranteed.

(2)In SGCA,the introduction of Integer Surrogate Pool(ISP)gets rid of the tedious round-off process and brings an efficient coding method for specific discrete variables optimization problems.

(3)A rationality check module can manage the improper solutions and avoid redundant fitness evaluations which saves lots of computer time.

1.Yuan HX.Connecting techniques of the modern long life aircraft.J Eng Technol2002(3);46–7[Chinese].

2.Ye DR.Strength and fatigue life calculation of an engine turbine disk.J Aeroeng2006;32(4):16–8[Chinese].

3.Chen G.Aviation engine structure design and analysis.2nd ed.Beijing:Beihang University Press;2006.p.176[Chinese].

4.Chen QR,Guo HD,Zhang C,Liu XG.Structural optimization of uniaxial symmetry non-circular bolt clearance hole on turbine disk.Chin J Aeronaut2014;27(5):1142–8.

5.Zhang C.Stress field analysis and structural optimization of noncircular hole on flange of turbine disks[dissertation].Nanjing:Nanjing University of Aeronautics and Astronautics;2014[Chinese].

6.Chen QR,Guo HD,Liu XG.Modeling and optimization for the structure of biaxial symmetry non-circular hole of turbine disk.J Aerospace Power2013;28(6):1250–6[Chinese].

7.Tian GF,Qiao YY.Equal life design and life prediction of the shaft parts.Adv Mater Res2014;875–877:962–7.

8.Jalili S,Hosseinzadeh Y,Taghizadieh N.A biogeography-based optimization for optimum discrete design of skeletal structures.Eng Optimiz2015;48(9):1491–514.

9.D’Ariano A,Pacciarelli D,Pranzo M.Discrete optimization a branch and bound algorithm for scheduling trains in a railway network.Eur J Operat Res2007;183(2):643–57.

10.Alberdi R,Khandelwal K.Comparison of robustness of metaheuristic algorithms for steel frame optimization.Eng Struct2015;102:40–60.

11.Zong WG,Kim JH,Loganathan GV.A new heuristic optimization algorithm:harmony search.Simul Trans Soc Model Simul Int2001;76(2):60–8.

12.Borna K,Khezri R.A combination of genetic algorithm and particle swarm optimization method for solving traveling salesman problem.Cogent Math2015;2(1):1048581.

13.Sabour MH,Eskandar H,Salehi P.Imperialist competitive ant colony algorithm for truss structures.World Appl Sci J2011;12(1):105.

14.Dede T.Application of teaching-learning-based-optimization algorithm for the discrete optimization of truss structures.KSCE J Civil Eng2014;18(6):1759–67.

15.Bhuvana J.A hybrid evolutionary algorithm for discrete optimization.Res J Appl Sci Eng Technol2015;9(9):770–7.

16.Balabanov VO,Venter G.Response surface optimization with discrete variables.Reston:AIAA;2004.Report No.:AIAA-2004-1872.

17.Stolpe M.To bee or not to bee–comments on discrete optimum design of truss structures using artificial bee colony algorithm.Struct Multidiscipl Optimiz2011;44:707–11.

18.Stolpe M.Truss optimization with discrete design variables:a critical review.Struct Multidisciplin Optimiz2015;53(2):349–74.

19.Gong M,Zhao JH,Dong BH,Wang XF,Li CZ.Initiation and propagation of fatigue crack in edge region of hole in a sheet with central hole.Acta Aeronaut et Astronaut Sinica2002;23(3):202–5[Chinese].

20.Huang MW,Arora JS.Engineering optimization with discrete variables.Reston:AIAA;1995.Report No.:AIAA-1995-1333.

21.Pisinger D,Ropke S.A general heuristic for vehicle routing problems.Comput Operat Res2010;34(8):2403–35.

22.Surekha P.Solution to multi-depot vehicle routing problem using genetic algorithms.World Appl Prog2011;1(3):118–31.

23.Melnik IM.Genetic algorithm for solving the problem of an optimum regression model construction as a discrete optimization problem.J Automat Informat Sci2008;40(6):60–71.

24.Lin CY,Hajela P.Genetic algorithms optimization problems with discrete and integer design variables.Eng Optimiz1992;19(4):309–27.

25.Xu H.Mechanical design handbook.2nd ed.Beijing:Chinese Machine Press;1992.p.8–9[Chinese].