Department of Engineering,University of Cambridge,Cambridge, CB2 1PZ, UK
Optimization of Turbine DiskProfiles by MetamorphicDevelopmentA novel topology/shape optimization method, Metamorphic Development, is appliedaxisymmetric thermo-elasticity design problem. Based on solid modeling and finitement analysis, optimal profiles of minimum mass turbine disks are sought by growindegenerating simple initial structures subject to both response and geometric constrRadial stress, axial stress, hoop stress and von Mises stress are analyzed throughoptimization and a constraint is imposed on von Mises stress everywhere in the disoptimal structures are developed metamorphically in specified infinite design domusing both quadrilateral and triangular axisymmetric finite elements. Comparisonsmade of the results obtained for different optimization scenarios: (a) with and witthermal loading; (b) with and without centrifugal body forces; (c) with and without apressure on the inner surface of the hub; and (d) operating at different rotatiospeeds.@DOI: 10.1115/1.1467079#
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1 IntroductionHigh-speed rotating disks are commonly used in rotating m
chinery such as flywheels, gears and rotors in turbines and cpressors. A major constraining factor on the development of thcomponents is the magnitude and distribution of stresses uoperating conditions. Common sense dictates that using a unicross-section for rotating disks is very uneconomic.
Design problems of this sort fall within the category of thstructural shape optimization. However, it is generally difficand expensive to calculate accurate gradients of the functionsensitivity analysis, since rotating disks are generally axisymmric continuum structures and their responses to changestopology/shape are usually non-linear and implicit functions ofvariables. Given the importance of minimizing their mass andpotentially catastrophic effects of disk failures, the optimal desof high-speed rotating disks to achieve low cost and high permance has long been a significant topic in industry, especiallygas turbine industry. In an early approach, Donath@1# approxi-mated the actual disk using a series of rings with uniform thiness. Through a comprehensive analysis of the problem, Sto@2# suggested a hyperbolic curve for the profile of the disk. Ro@3# investigated the effects of peripheral loading and gave chfor the design of such disks. With the advent of the computer aMathematical Programming~MP! methods were used to solve thdesign problem, for example:
• The Gradient Search~GS! method, see Seireg and Surana@4#,in which the disk was defined by a system of finite rings wdifferent thicknesses and the stresses in each ring wereculated from approximate expressions.
• the Sequential Linear Programming~SLP! method, see Bha-vikatti and Ramakrishnan@5#, in which thicknesses at selected radial levels were used as design variables andshape of the disk cross-section was assumed to be anbraic function of theses design variables;
• the Complex method, see Luchi et al.@6#, in which the diskprofile was defined by spline interpolation;
• the Feasible Direction~FD! method, see Cheu@7#, in whichthe coordinates of the selected points on the disk contowere used as design variables.
Contributed by the Design Automation Committee for publication in the JOUR-NAL OF MECHANICAL DESIGN. Manuscript received January 2001. Associate Etor: G. M. Fadel.
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In most of these methods, the Finite Element~FE! method hasbeen a popular analysis tool. As an alternative to FE, the bounelement ~BE! method based shape design sensitivity analy~SDSA! method has also successfully applied to the design prlem ~Lee @8#!.
The last decade has seen a proliferation of structural topoland shape optimization methods based on heuristic interpretaof natural processes. The original Evolutionary Structural Optimzation ~ESO! method ~Xie and Steven@9,10#! mimics the intu-itively simple concept of evolutionary processes in nature,which unnecessary material is gradually removed from a strucand the residual structure evolves towards an optimum. In itstial implementations the criterion for material removal was bason the maximum~Xie and Steven@9#! or mean~Hinton and Sienz@11#! von Mises stress. These ideas have been investigatednumber of researchers, who have all introduced variants onoriginal ESO concept~Rosko@12#; Papadrakakis et al.@13#; VanKeulen and Hinton@14#; Reynolds et al.@15#!. Other naturallyinspired heuristic methods have been inspired by botanical gro~Mattheck and Burkhardt@16#! and bone growth~Tanaka et al.@17#!.
A potential disadvantage of the ESO approach is the comptional effort required in solving problems. To overcome this tBi-directional ESO~BESO! method was developed~Querin et al.@18#!. The BESO entails adding material where the structureover-stressed and removing material where the structure is unstressed. However, since the element removal ratio and the insion ratio of BESO are small, the optimization process is sslow. Metamorphic Development~MD! ~Liu et al. @19#; Liu et al.@20#!, which was developed independently, adopts an approconceptually similar to that of BESO, but uses a dynamic growfactor, which is an adaptive function of the structural responconstraints, to control the growth and degeneration each iterathus accelerating convergence.
The MD optimization procedure starts from a minimal numbof nodes and elements connecting the applied loads and suppoints. The structure is then developed using finite elementscan be of any specified sizes and a design domain may or maybe specified. The optimum is sought through simultaneous groand degeneration, i.e., by adding to and removing from the stture both nodes and elements. Growth is guided to occur in aof high strain energy or high stress. One important feature ofmethod is that it allows the introduction and re-introduction~ifthey have been removed in previous iterations! of nodes and ele-
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Fig. 1 Schematic diagrams of the turbine disk geometries and loadings, boundary conditions and geometricrestrictions: „a… disk geometry, mechanical loading, and displacement boundary conditions; „b… thermalboundary conditions and geometric restriction for optimization
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ments, enabling the topology design space to be robustlyplored. Another original feature of MD is its use of both quadlateral and triangular elements that are ideally suited for modecontinua involving curved boundaries. The MD method canused to solve different kinds of topology/shape optimization prlems, such as
• minimizing mass subject to structural response constraiand
• minimizing mean compliance~or mean stress/strain energ!subject to mass/structural response constraints.
The effectiveness and efficiency of the MD method in solvistructural topology/shape optimization problems has encourathe application of the method to the rotating disk design probleThis paper presents the MD method as applied to axisymmeelastic and thermo-elastic design problems. This capabilitydemonstrated on the shape optimization of a turbine disk rotaat a high speed in a specific temperature environment. The tato find the minimum mass profile of variable thickness of a tbine disk that connects the blades to the rotating shaft and enates the stress peaks in the disk caused by the thermo-mechloads.
The optimization takes into account the centrifugal-type boforces, external forces, fit pressures and the thermal loadingthe loading and boundary conditions are shape-dependent,are updated in structural reanalysis to accommodate the chaSince the displacement field and thermal field are significainfluenced by a shape change of the disk, a coupled steady-thermal-stress analysis is repeatedly performed. Axisymmequadrilateral and triangular finite elements are employed. Thetimization results and the metamorphic development historiespresented for the mass minimization problem of a turbine dwith high temperature on the tip and cooling on the lateral faceaddition to the mechanical loading. Radial stress, axial strhoop stress and von Mises stress are analyzed and the optimtion is subject to a von Mises stress constraint. Optimal shapeobtained and compared for different loading conditions: with awithout thermal loading, with and without centrifugal body forcewith and without fit pressure from the hub, and different rotatiospeeds.
The MD method has been developed to perform both topoland shape optimization. In this paper, only shape optimizatioconsidered. Shape optimization is a specific case addressed bpresented method in that the metamorphic development isstricted to take place only on existing design surfaces and nodesign surfaces are created during the optimization.
2 Optimization ProblemA typical rotating disk with hub and rim is shown in Fig. 1. Th
disk body material is assumed to be isotropic and homogene
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The geometry parameters can best be represented in the symric section on the (r ,z) plane using a cylindrical coordinate system, and the following notation is used to define the disk geoetry:
Rh5 inner radius of hubL5 length between inner radius of hub and outer radius ofLh5thickness of hubLr5thickness of rimZh5half-width of hubZr5half-width of rim
In this study, a typical turbine disk rotating at high speed inthermo-elastic environment is considered for shape optimizausing the MD method. For high-speed rotating turbine and copressor disks, mass minimization is one of the most importdesign goals, and the magnitude of the stresses must be mtained below allowable limits, depending on the disk materHence the task is to find a minimum mass structure with an omal disk section profile under multiple thermo-mechanical loaings and with both temperature and displacement/rotation bouary conditions imposed. In addition to the external~blades!loading, the body forces and thermal loading should be takenaccount in the shape optimization. In some cases, the inner deter of the hub needs to be designed slightly smaller thandiameter of the shaft, so that the disk~after being heated, slottedonto the shaft and then cooled! fits tightly. This would give rise toa fit pressure between the disk and shaft, which should be conered as another type of loading for the disk design. Therefore,loadings to be considered are the blade load due to high-sprotation, the disk centrifugal body force, the temperature distrition, and possibly the fit pressure between hub and shaft.obvious that both loadings and boundary conditions are shdependent. However, the design task can be considered as asymmetric problem on account of the symmetry of the loadinand the geometry. Generally, von Mises stress can be usedstress constraint criterion in shape optimization problemscoupled steady-state thermal-stress analysis should be perfoto reveal the distributions of the stresses and temperature andmutual effects.
3 Optimization MethodologyIn this study, the optimal shapes of the structures under con
eration are determined using the Metamorphic Development omization procedure. The design problem can be stated as:
Minimize compliance f 1~Ti ! and/or massf 2~Ti ! (1a)
subject to geometric and response constraints (b)
The optimization method is powered with mechanisms for bgrowing and degenerating a structure in order to modify its p
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formance. Growth is guided to occur only in certain ‘‘growtcones,’’ a growth cone being a local section of structural ‘‘suface’’ where high strain energy, high compliance and/or histress occur. The growth ‘‘grammars’’ implemented througrowth cones, as shown in Fig. 2, are based on the followreasoning:
• Adding new elements can create new load paths;• Adding structural material to these areas can disperse h
strain energy and reduce high compliance;• High stress can be decentralized by adding more elemen
Conversely, elements that carry only a small load are conered to be used inefficiently, and thus can be removed. The lsection of structural ‘‘surfaces’’ occupied by these less-streselements may be called ‘‘degeneration cones,’’ and these aresentially inverse growth cones. The grammars of the growthdegeneration cones are implemented in such a way that geomirregularities can not be introduced by their use.
A dynamic growth factor~DGF! is used to control growth anddegeneration. The general form of the DGF is shown in Fig.The values of max(i)1, min(i)1, min(i)2 and max(i)2 varythroughout the optimization process. These specify the maximand minimum number of elements that may be added or remoeach iteration. They depend on factors which can vary in eiteration such as the scale of the structure~numbers of nodes andelements!, the total size of structural surface, and the availabilof structural symmetry in one or more directions. The waywhich these factors are combined is specific to each design plem and may need prior experience of using the MD method.
The DGF is used to regulate dynamically the rates of growand degeneration~i.e., to control the sizes of the growth cones athe parts to be removed!. The DGF is related to the current structural performance, the closeness to the imposed responsestraints and the calculated stress~and/or strain energy!. In Fig. 3,G(Ti) represents a hybrid constraint function and is determinby comparing structural responses~such as stress or deflection!with specified limits:
G~Ti !5(j 51
m
wjgj~Ti !5(j 51
m
wj~ uRj~Ti !u2uRj* u! (2)
whereRj (Ti) is j -th structural response,m is the number of theresponse constraints,Rj* is a user-specified target forj-th re-sponse, andwj is a user-selected weighting function dependentthe importance of the particular constraint. The values ofG(Ti) atpointsA andB are specified as:
A5(j 51
m
wj uRj* u (3a)
Fig. 2 Growth cones
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Thus the DGF, the value of which depends through simpleeraging on the values of the limits max(i)1, min(i)1 etc., is apiecewise linear function ofG(Ti). The values and factors thadetermine the growth factor may vary in each iteration. Therefothe DGF is an adaptive function changing from one structureanother. The adaptive nature of the DGF improves the algorithspeed of convergence, allowing more elements to be adderemoved each iteration when the structure is far from optimthus making the MD method very computationally efficientcomparison to other heuristic methods~Liu et al. @20#!.
4 The Optimization ProcessStructural shape optimization is inherently an iterative proc
dure. The MD optimization process consists of the followinsteps:Step 1: Form an optimization modelDefine all the kinematic boundary constraints, loads and mateproperties that are expected under service conditions. Specifyoptimization criteria to be used and the design constraints todirectly and explicitly imposed to optimize the structure.Step 2: Define a finite or infinite design domainIt is not necessary to specify a finite design domain. This is escially helpful if the best domain for the design is not knowa priori. Nevertheless it is possible to impose restrictions on strtural growth both in terms of location and direction, if appropriaStep 3: Choose appropriate finite elementsFinite elements appropriate to the design problem under consiation must be chosen. They can be linear or non-linear elemwith various shape functions. In the application detailed in thpaper, both triangular and quadrilateral axisymmetric elementsused. The use of triangular elements produces relatively smoboundaries, thus enabling a relatively coarse mesh to be uwhile still capturing detailed shape information~Liu et al. @20#!.Step 4: Generate an initial structureThe initial FE structure can be the simplest possible geomeconnecting the loads to the supports, providing that it has a nsingular FE solution. In fact, the optimization process can sfrom any degree of structural development from the simplest psible structure to a heavy ground mesh. Like other heuristic meods, MD cannot guarantee a global optimum solution, but it hbeen shown to be able to successfully find the optima on stand
Fig. 3 Structural dynamic growth factor versus hybridconstraint function
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test problems~Liu et al. @20#!, and, based on our experience,most cases the choice of initial structure will not make mudifference to the final design obtained. Generally, only a rou~far from optimal! initial design is needed. It does not matter tomuch if an extremely poor initial solution is used, although toptimization process will take longer. However, if the user hasexisting design to use as a starting point or can estimate an inshape that is close to the optimum, this will certainly reducenumber of iterations needed to reach the optimum.Step 5: Perform a structural finite element analysisFEA is performed to find the stress/displacement distribution othe whole structure for a strength/stiffness related optimizaproblem. Based on the analysis, growth cones, the parts ofstructural surface that are heavily over-stressed, are locaMeanwhile, heavily under-stressed regions are also located.information obtained in this step is used in MD to alter the exing configuration as it evolves towards an optimal design.Step 6: GrowÕdegenerate the structureIn this step, the structure is allowed to grow and degenerate vmetrically. Growth and degeneration take place simultaneouslevery iteration. Material is added in overloaded locations~growthcones! and removed from underloaded locations. The growth ris controlled by the DGF, which can be both positive and negaduring the course of an optimization.
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Step 7: Update the structureThe FE model is now modified to trace and accommodatechanges to the structure made in step 6 and to avoid posssingularities caused by them. This entails updating the structelements, nodes etc.
Fig. 4 Flow chart of the MD method
Fig. 5 The metamorphic development convergence history for the turbine disk design: „a… iteration 0 „initial …; „b…iteration 5; „c… iteration 10; „d… iteration 15; „e… iteration 20; „f… iteration 25 „optimized …
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Fig. 6 Contours of various stresses, temperature and heat flux distributions in the optimized turbine disk; „a… Mises stress;„b… radial stress; „c… axial stress; „d… hoop stress; „e… temperature; „f… heat flux
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Steps 5 to 7 are repeated until a suitably optimized structureevolved. This process will result in a homogeneous distributionelastic energies~or stresses! in the structure, indicating that thpeak stresses~and/or strain energies! are adequately reducedHowever, this distribution does not identify the optimal topolog
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shape for the specified loads and design constraints. The opzation process continues until a minimum mass structure sating all the structural response constraints has been achieveuntil design limitations prevent a further increase~or decrease! inthe size of the structure. Convergence is deemed to have occu
Fig. 7 Stress distributions on the non-design surface of the optimized disk structure
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Fig. 8 Stress distributions on the design surface of the optimized disk structure
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if the performance of the structure cannot be improved for a spfied number of consecutive iterations. In this manner, the inidesign evolves into an optimized design, which is a minimmass structure satisfying all the constraints.
The optimization algorithm employs a hierarchical structuFirst, objectivef 1(Ti) ~the compliance! is minimized and objec-tive f 2(Ti) ~the mass! is disregarded until the structural responconstraints are satisfied. Thenf 2(Ti) is minimized subject to con-straints on the structural response. In the first stage, a posgrowth factor is used and more elements are added than remeach iteration. Conversely, in the next stage, a negative grofactor is used and more elements are removed than addediteration. Depending on the current structural performance, thtwo optimization schemes may be adopted alternately. If a wdeveloped structure is used as a starting design that satisfieconstraints, the first optimization stage may not be necessarflow chart for the optimization procedure is given in Fig. 4. Tprocedure used does not require gradient-based sensitivity asis, as the surface strain-energy/stress distribution is the solerameter considered. Although there is no mathematical proofthis hierarchical approach and other similar approaches~e.g.,Querin et al.@18#! will result in convergence to the global optmum, practical experience indicates that they do in fact ususucceed in doing so~Liu et al. @20#!.
Since this method avoids difficult formulation and programing of complicated relationships between various elementsshape variables, commercial FE packages can be used as aers. The optimization algorithm is used as a design tool in cjunction with commercial FE codes and ABAQUS software@21#is currently used to perform the required FEA.
5 Examples
5.1 Specifications. The design problem has been outlinedsection 2. Because of symmetry, only one half of the diskz>0) and the symmetric portion (r>0) about the symmetry axis(r 50) of the cross-section of the disk needs to be modeled. Asymmetric boundary conditions are imposed. The geometricrameters for the disk, as shown in Fig. 1(a), areRh520 mm, L5100 mm, Lh515 mm, Lr510 mm, Zh540 mm and Zr540 mm.
To make the design model as realistic as possible, four diffetypes of loading are considered:
• Centrifugal loading: The disk is assumed to be rotating atconstant speedv520000 rpm, v518000 rpm, or v516000 rpm. The distributed~throughout the disk body!centrifugal load varies in proportion to the distance from t
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• Blade loading: There is a uniformly distributed load,p5200 MPa, acting radially outwards on the circumferentsurface of the tip. This load simulates the forces inducedthe blades.
• Thermal loading: The thermal boundary conditions arshown in Fig. 1(b). The operating temperature of the diskvaries from 20°C at the hub to 600°C at the tip. The consthigh temperatureTr5600°C atG r ~the outer radius of therim! is caused by the hot gas deriving from the bladestemperature ofTh520°C is maintained atGh ~the inner ra-dius of the hub!. The condition]T/]z50 is imposed on thesymmetric boundaryG0 (z50). The disk is cooled by airflowing past the rotor. A convective boundary conditio]T/]n5h(T2T`), is imposed on the disk surfaceGs wherecooling is assumed to occur, with the convective heat trancoefficienth5100 W/m2K and the ambient temperatureT`520°C.
• Fit pressure loading: A uniformly distributed load resultingfrom the disk and shaft fit pressure,q540 MPa, acts radially~outwards from the axis of rotation! inwards on the circum-ferential surface of the hub.
The Poisson’s ratio, Young’s modulus, density, thermal condtivity and thermal expansion coefficient of the disk material usare taken to be 0.3, 180.36 GPa, 7706 kg/m3, 47 W/mK and1.1731025/K respectively. The von Mises stresses are used asstress design criterion. The upper limit set on the maximumlowed von Mises stress is 1.10 GPa over the whole regioncoupled steady-state thermal-stress analysis is performed in wtwo different fields are active. The elements used have both tperature and displacement degrees of freedom.
5.2 Optimization Results. First, a general case is considered for optimization, in which centrifugal (v520000 rpm),blade and thermal loadings are included. This general case elishes a basic example for the purpose of comparison with oloading cases. The disk cross-section used as the starting dfor the optimization is shown in Fig. 5(a). This initial design issimply a flat plate disk connecting the hub and rim, and the vMises stress in it is as high as 3.18 GPa, which violates the stconstraint by over 270%. The disk can have varying thicknbetween hub and rim, and only the disk surfaceGs needs to bereshaped during the optimization. Engineering design consiations specify geometric parametersZh and Zr , which represent
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the half-widths of the hub and rim respectively. However,Lh andLr can vary during the optimization, but they are not allowed toless than 5 mm~i.e., Lh>Lh min55 mm andLr>Lr min55 mm!.Another geometric constraint imposed is that material cannoremoved if it is located within a cone surfacez511523r andz5592r ~see Fig. 1(b)!.
The optimization takes just 25 iterations to converge. This taonly 15 minutes to run on a Sun Ultra1 workstation running Slaris 2.6. The history of the optimization is shown in Fig. 5. Tmass of the final structure is 4.036 kg. The maximum Misstress, which is 1.10 GPa, occurs at the center of the inner suof the hub~the bottom left corner of the symmetric model! whichis not part of the design surface. From Fig. 6 it can be seenthe stress constraint is satisfied throughout the structure. Thfore, an optimal turbine disk with a minimum mass and satisfythe imposed constraints has been achieved for a practical oping environment~with a variety of loadings!.
Four types of stress, namely radial, axial, hoop and von Mistress, are analyzed throughout the optimization. To illustratedetailed stress distributions on all the surfaces of the optimidisk, some specific points are identified in Fig. 10(a) and thevariations of these stresses along these surfaces are shown in7 and 8.
Figure 7 shows the changes in the four stresses along mo
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the non-design surfaces~from pointsa to d, via b and c!. Frompointsa to b, the inner surface of the hub, all the stresses rise fra value near zero at pointa until their turning values at pointb,with the exception of the radial stress that shows almostchange. The stresses’ turning at pointb is caused by the geometrchange there. The Mises and hoop stresses reach their maxivalues at this point.
Along the symmetric boundary of the disk~points b to c!, allthe stresses show a gentle, smooth fluctuation. Approaching pc, however, the hoop and axial stresses exhibit a rapid decrand a sudden change at pointc. A ‘‘kink’’ in the Mises stress curveis also observed at pointc. The sudden, steep changes in tstresses around pointc are mainly due to the corner geometry anthe action of the blades loading.
From pointc to point d, which is the outer surface of the rimwhere the blades loading acts, all the stresses are basically hea lower level with little variation.
The stress variations along the disk surface~from point d topoint a, via e, f andg! are shown in Fig. 8. From pointsd to e, allthe stresses rise and fall within a limited range of amplitudFrom pointse to f, however, the Mises and radial stresses exhgreater fluctuations and a highest radial stress of 1.1 GPa isserved. These cyclic stress variations are mainly due to the sgeometric discontinuities in the outer surface at transitions
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tween triangular and quadrilateral elements. Although quadreral and triangular elements can approximate a smooth surthey cannot model it perfectly. In this region, hoop and axstresses also oscillate but at a relatively lower level. From poinfto g, all four stresses decrease and vanish at pointg. Although thestresses are low in this region, further removal of material is pvented due to the geometric design constraints~a cone surface, seFig. 1(b)!. Along the line from pointsg to a, the disk surface isbasically stress-free.
If we refer to the loading case above as case A, Fig. 9 showsoptimized structures obtained under five alternative loading caB, C, D, E and F respectively:
• Case B–blades loading plus disk centrifugal loadingv520000 rpm);
• Case C–blades loading only;
Fig. 10 Comparisons of the optimal shapes for different load-ing cases: „a… comparison of optimal shapes for different load-ings; „b… comparison of optimal shapes for different rotationalspeeds; „c… comparison of optimal shapes with and without a fitpressure
Table 1 A summary of the results for all the loading casesconsidered
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• Case D–blades loading plus disk centrifugal loadingv518000 rpm) plus thermal loading;
• Case E–blades loading plus disk centrifugal loadingv516000 rpm) plus thermal loading;
• Case F–blades loading plus disk centrifugal loadingv518000 rpm) plus thermal loading plus fit pressure loadin
To make the results comparable, the same mesh is used ithese models. The optimal shapes for loading cases B and Cand E, and F and D are compared respectively with that for geral case A in Fig. 10. Starting from the same initial structure,shown in Fig. 5(a), all these optimizations are subject to the samresponse and geometric constraints, but, of course, with diffeloading conditions. By plotting these optimal surface profilesgether, the effects of the body force, rotation speed, fit pressand thermal loading on the final results can be clearly identifiThese comparisons suggest that it could be essential to includthese loadings when optimizing turbine disks operating inthermo-mechanical environment because of the significant effeach has. It is shown, for instance in Fig. 10(a), that the optimaldisk profile under thermo-mechanical loading requires a thicdisk than under the mechanical loading only. Table 1 summarall the loading cases in terms of the number of iterations neefor convergence and the final masses of the optimized structu
6 ConclusionsIn this paper the problem of optimizing the design of turbi
disks is tackled by using the MD method, which is a systemaexpansion and contraction approach for adding and removingterials in a structure. A FE-based fully automatic axisymmetshape optimization capability is presented.
The design problem is to minimize the volume of the disunder thermo-mechanical loading with a constraint imposedthe von Mises stresses over the whole structure. The optimizaseeks to eliminate the stress peaks caused by various loadThe disks are subject to thermal loading caused by temperadistribution, blade loading and centrifugal forces caused by hispeed rotation, and a fit pressure caused by the assembly proThe loading cases and boundary conditions are shape depeand therefore are updated in structural reanalysis. The final stture is developed metamorphically in a specified infinite desdomain. In general, the growth and degeneration are based ocurrent objective and constraint functions of the optimizatiproblem under consideration. The results reveal that the minimmass profiles for the thermo-mechanical loading cases exhthicker flanges than those for the mechanical loading case aland similarly for cases with and without body forces. These incate the significant influence of these factors and the importaof including them within the loading case when optimizing tubine disks ‘‘in anger.’’
The results also show that, by using the MD method, shoptimization of axisymmetric continuum structures with compcated contour shapes and loading conditions can be achievedrelatively little computational effort. A marked reduction in anstress concentrations present is sought and minimum mass stures have been obtained. Both quadrilateral and triangularments are used to build the structures. The employment of nrectangular elements makes the surfaces of the resulting strucrelatively smooth. Since in the optimization considered heregrowth and degeneration occur only on the design surfaces, tare no interior geometric discontinuities in any form. More geerally, when MD is used for topology optimization the formultion used for the addition and removal of material throughgrowth and degeneration cones automatically avoids the introdtion of interior geometry discontinuities, such as checkerboardThe linking with commercially available analysis packages eables the method to utilize their versatile capabilities effectivwhen dealing with complicated structures. The results demstrate the success of the method in finding realistic solutions topractical design problem under consideration.
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