6 th World Congresses of Structural and Multidiscilinary Otimization Rio de Janeiro, 30 May - 03 June 2005, Brazil Toology Otimization of Three Dimensional Structures under Self-weight and Inertial Forces Rafael Acedo Loes 1, Fernando Viegas Stum 2, Emílio Carlos Nelli Silva 2 (1) Voith Siemens Hydro Power Generation, São Paulo, Brazil (1) rafael.loes@vs-hydro.com (2) Deartment of Mechatronics and Mechanical Systems Engineering - University of São Paulo, São Paulo, Brazil fernando.stum@oli.us.br ; ecnsilva@us.br 1. Abstract In the structural design, usually only concentrated loads are considered. However, among mechanical structures, there are some in which the body forces have a major imortance in its design. This is the case of rotational machines such as energy generator rotors, turbine runners and flywheels. Unfortunately, the otimal design of these structures is not intuitive, once both load and stiffness deend on the material distribution. Therefore, toology otimization method can be alied to hel to obtain the concetual design these arts. In this wor, a toology otimization formulation for designing three-dimensional structures under self-weight and inertial forces is develoed. As objective function, the traditional mean comliance design roblem is considered, where the objective is to find the material distribution that minimizes the mean comliance for a certain volume constraint. Material models which arameterize stiffness and density roerties were imlemented based on the well-nown SIMP model. Otimality criteria method is alied as the otimization algorithm. A continuation method is alied to avoid local minima. Concerning industry alication, the otimization algorithm, as well as the toology otimization rocedure, was imlemented in ANSYS TM by using the APDL (ANSYS TM Parametric Design Language). This allows us to tae all advantages of FEM code caabilities resented in the commercial code increasing the design tool flexibility. To demonstrate the algorithm otentiality, some examles of classical roblems were synthesized and the results resented. At conclusion, the design of a hydro generator rotor comonent subjected to self-weight and centrifugal force is shown and discussed. 2. Keywords: hydro generator rotor, self-weight, inertial forces, toology otimization, SIMP, ANSYS TM 3. Introduction In heavy mechanical industries, the structural concet is a consequence of about a century of develoment. Such structural solutions have been exhaustively investigated and few chances of imrovements alying conventional design methods remain. Nowadays, the great majority of new develoments is based on the exerience of a design grou, or modifications uon an existing roject. In this way, the structural otimization methods come as a owerful tool in the design of these comonents. In structural design, usually only concentrated loads are considered. However, there are imortant mechanical structures in which the body forces have a major imortance in the design. It is the case of rotational machines such as energy generator rotors, turbine runners and flywheels. The otimal design of these structures is not intuitive, once both load and stiffness deend on the material distribution. In this way, toology otimization methods can be alied in concetual design. The toology otimization methods search for an ideal material distribution of a structure, such that the objective function is otimized. However, even though there are lenty of articles discussing toology otimization structural design, only a few of them consider the body force influence. The wors of Liu, Pars and Clarson [1] cite Donath as the first to investigate the otimization of rotating structures with centrifugal forces. Donath aroached the structure by a series of diss with constant thicness. Stodola [2] resented the rotor hub of a steam turbine. His wors considered a dis, without central hole, subjected to centrifugal force. Stodola s wor was based in the idea of constant stress level in the whole dis. Stodola suggested a hyerbolic curve to describe the best rofile of the structure. By the advent of comutational analysis methods based on mathematical rogramming, Bhaviatti and Ramarishnan [3] alied the method of sequential linear rogramming to solve the otimization roblem of a dis subject to the centrifugal force. While, Cheu [4] alied with success the method of "feasible direction" for the solution of the same roblem. Kress [5] then alied the method "feasible direction" to otimize the thicness variation of a steering wheel with central hole. In this wor, the Toology Otimization Method is alied to otimize the hub of a hydro ower generator. The formulation of maximum stiffness with volume constraint is alied. The roblem is arametrized with the SIMP material model, and the otimization is solved by the otimality criteria method. This aer is organized as follows: In section 4, the basic Toology Otimization theory is resented, In section 5, the roblem formulation emloyed and the otimization algorithm is described considering density deendent body forces. In section 6. the study case of the otimization of a hydro generator shaft subjected is resented in a two and three dimensional aroach. Section 8 resents the conclusion of this wor 4. Basic toology otimization theory Toology otimization is based on two main concets [6]: the extended design domain and the relaxation of the design domain. The extended design domain is a large fixed domain that must contain the whole structure to be determined by the otimization rocedure. The objective of toology otimization is to determine the holes and connectivity of the structure by adding and removing material in this domain. In turn, toology otimization roblem is defined as a roblem of finding the otimal distribution of material
in the extended domain. As the extended domain is fixed, the Finite Element model domain is not changed during the otimization rocess. The relaxation of the design roblem is associated to the change of material from solid (one) to void (zero). The discrete roblem, where the amount of material in each element can assume only values equal to either one or zero, is an ill-osed roblem, that is, it does not resent a solution. The roblem must be relaxed by allowing the material to assume intermediate roerty values during the otimization rocedure which can be achieved by defining a material model. Essentially, the material model aroximates the material distribution by defining a function of a continuous arameter (design variable) that determines a mixture of material and void. By allowing the aearance of intermediate (or comosite) materials rather than only void or full material - in the final solution, this rovides enough relaxation for the design roblem. 5. Toology otimization considering self-weight and inertia forces: In this wor, the traditional formulation for stiffness design roblem is considered as objective function, where the objective is to find the material distribution that minimizes the mean comliance (1): Γ C = tudγ + udω (1) mean where t, and u denote the traction, body forces and dislacements, resectively, and Γ reresents the boundary of the domain Ω Then, the general continuous form of the toology otimization roblem for stiffness design can be defined as: Minimize η C mean Ω such that (2) η ( X) dω V Ω 0 < η η η MIN MAX Equilibrium equation where η is the design variable which, for the SIMP material model, can be interreted as the volumetric material fraction. To consider the body forces the material model must be extended to also arametrize the material density. Here the following relation was alied: ρ E f = η ρ 0 (3) = η E 0 (4) In this material model, two enalization factors f and are introduced. In the literature [7][8][9], the arameter is usually set equal to 3 or 4. This enalization avoids the gray-scale regions ( η around 0.5) in the otimized solutions. From a hysical oint of view, the density ρ has a linear relation with the volumetric fraction η, thus the arameter equal to 1. However, as discussed by Bruyneel and Duysinx [10] with the set f = 1 and = 3 f should be set the results, in general, resents surious structures. This haens because when the volume fraction tends to its inferior limit ( η = 10 3 MIN ) the ratio between the density deendent force () and the material stiffness ( E ) increase, locally. To revent this roblem, we roose to aly a enalization f different from 1. Analyzing the relation between enalizations factors, f should avoid the resence of surious structures, however, the final solution tends to resent gray scale because for intermediate values of η because,as can be seen in Figure 1, the ration between density deendent force () and the material stiffness ( E ) is favorable to decrease the mean comliance. Regarding this situation, some numerical test was erformed, and it was concluded that the enalizations factors must be equal f = to obtain results without gray scale or surious structures. f and
0.8 η 1 η 2 η 3 η 4 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 Figure 1: Interolations functions To avoid local minima a continuation method was emloyed, and the schedule of enalization was defined as f = = 1, f = = 2 and f = = 3. Regarding the checerboard instability, the discretization of the density field was made in an element base, while the dislacement field was aroximated by a nine-node element, in the case of two dimensional roblems. This aroach is roven to be stable [11]. In the three dimensional roblem, the same aroach was alied and a twenty-node element emloyed. 5.1 Otimization algorithm To solve the otimization roblem, the otimality criteria [7][12] was alied. Due to the ossibility of ositive values of the objective function sensitivity small changes in the algorithm have been made, following the imlementation of the otimality criteria for the design of flexible mechanism 1, in which the same roblem is faced. The variable udating rule below was alied: η + 1 max{( η ς ), η = η Be min{( η + ς ), η min max if if if max{( η η B e ς ), η max{( η ς ), η min η B min{( η + ς ), η e min{( η + ς ), η max min η B max (5) where B e is given by: 0.3 mean B e (6) C = max 0, η This udating rule allows the method to obtain otimal solutions when the volume constraint finishes inactive. 5.2 Numerical Imlementation The otimization algorithm and the toology otimization rocedure were imlemented in ANSYS by using the APDL (ANSYS Parametric Design Language). This allows us to tae all advantages of FEM code caabilities resented in the commercial code increasing the design tool flexibility. The rocedure imlemented is characterized by some advantages: Multidiscilinary Analysis and Otimization - An advantage of using ANSYS as a FEA solver in otimization is that it is multidiscilinary. Considering the architecture mentioned above, the otimization rogram needs only a slight change when additional ind of analysis is added. Customization and Maintenance - An in-house otimization code integrated with ANSYS assures indeendent maintenance for each rogram. Another asect is the ossibility of customizing roblem arameters (such ass tye of element, oututs)
6. Numerical Results In this wor, we have studied the toology otimization of a hydro generator shaft subjected to centrifugal force. Two and three dimensional examles are resented here to examine the configuration of the otimal solution for the roblem. In all simulations, the roerties of the isotroic material are: Young modulus = 210 MPa Poisson ratio = 0,3 Base material density = 7.850 g/m3 In all cases, the alied loads are: Angular seed = 136 rad/s (1300 rm) Gravity acceleration = 10 m/s 2 6.1 Descrition of the design roblem of a generator rotor The generator rotor is basically comosed by the shaft, oles, fan, rotating arts of the bearings, brae disc and sli ring (as in Figure 2). In the resent design, the generator shaft is has a segmented form. Two shaft arts are available (uer and lower), being directly flanged resectively at the to and at the bottom of the intermediate art. The shafts are made of forged carbon steel. The lower shaft has a flange integrally forged for direct couling to turbine shaft with re-tensioned bolts and taered ins. The shaft transmits the torque from the turbine to the hydro generator. For couling with turbine, a couling flange is forged integrally with the shaft at the drive-end. The oles (in and orange arts) are directly fitted onto the generator shaft (light green). The ole fitting slots are obtained by direct longitudinal machining on the shaft. Figure 2 Generator Rotor The main dimensions of the studied shaft are: Shaft outer diameter (central art) = 1.360 mm Shaft outer diameter (uer and lower art) = 360 mm Shaft inner diameter = 150 mm Couling flange outer diameter = 600 mm Central art length = 1.360 mm Total shaft length = 4.700 mm The shaft function is to transmit the torque from the turbine to the generator. But, in this wor, the generator was considered oerating in runaway seed. At runaway, the generator is electrically disconnected from the ower grid, and there is no torque acting on the shaft. Then, in this oerating condition, the generator rotor comonents are subjected only to inertial forces. The runaway seed is about twice the nominal seed. 6.2 Two-dimensional model Firstly, the hub geometry was synthesized considering a two-dimensional design domain shown in Figure 2 (a) it is considered that the dis is rigid fixed in the inner art of the shaft and concentrates loads are alied in the external art, simulating the ole forces due to centrifugal forces. The finite element analysis emloyed 8-node quadrilateral elements (PLANE82). Once a middle shaft section is simulated the hyothesis of lain strain has been adoted. To save comutational time, radial symmetry conditions were alied and only 1/10 of the dis is otimized. Figure 2 (a) shows the couled degree of freedom.
Pole force Simmetry conditions (a) (b) Figure 2: Two-dimensional aroach: (a) Design domain and boundary condition considering the load axisymmetric (b) Synthesized structure considering external forces due to the ole mass (Solution for volume constraint of 30%). Four structures were synthesized using a two dimensional aroach, one without inertia forces and three considering the inertial force due to angular seed with different volume fraction constraints. The otimal structure, without inertia forces, is resented in Figure 2 (b). In this roblem only the external forces alied to the hub were considered and the volume constraint of 30%. (a) (b) Figure 3: Otimal structures: (a) Solution for volume constrained equal to 30% (b) Solution for volume constraint equal to 70% Comaring the solution considering body forces (Figure 3 (a)) and without consider (Figure 2 (b)) it is observed that both solutions have similar geometries, however, when the inertia force is considered, the ring formed in external art of the of the dis is slightly different. Analyzing all the two dimensional structures it is observed that that the center of the structure remains undefined. However, the center art has an intermediary densities indicating that the stiffness should be fined by the geometry in the axis erendicular to the lane considered. The three dimensional results show how the stiffness must be distributed inside the hub. 6.3 Three-dimensional model For the three dimensional models, the design domain is discretized in 20-node solid elements (SOLID95) subjected by inertial force due angular seed. In site of the method being able to consider the self-weight of the structure, it was decided to neglect this load. In this roblem, the self-weight magnitude is much lower than the inertia load. Then, the symmetry boundary condition was set at middle of the intermediate shaft. The uer and lower boundary of the design domain (only in the couling region) was rigid suort. In this case, it was also alied cyclic symmetry condition. Figure 4 (a) shown the design domain where boundary conditions and loading are as indicated.
(a) (b) Figure 4: Three-dimensional aroach: (a) Design domain and boundary condition considering the load axisymmetric (b) Synthesized structure considering external forces due to the ole mass. 2 arts of 1/10 of the structure 2 arts of 1/20 of the structure Figure 5: Longitudinal slices of the synthesized structure in the three dimensional aroach. The otimal structure is resented in Figure (4) b. The Figure 5 resents the slices of the same structure, allowing too observe its inner art. Here its ossible to observe how the structure must be defined in the center of the shaft. Forging is the traditional rocess for shaft manufacturing. However, by observing the results obtained in Figure 5, we can suggest alternative shaft design configurations. One feasible solution would be the relacement of the forged shaft by a grou of cast diss. The diss would be fastened by bolts and have the same external diameter. The internal diameter would then vary in a way to reroduce a similar configuration obtained in the otimized solution. 7. Conclusions In this wor, a Toology Otimization formulation for designing three-dimensional structures under self-weight and inertial forces was develoed. Some articularities of toology otimization including inertial forces and a comarison between two and three dimensional results were resented. The toology otimization rocedure was imlemented in ANSYS Language, which allows the above mentioned benefits. The mean comliance design roblem was the objective function and density roerties were imlemented based on SIMP model. An adated otimality criteria method was alied as the otimization algorithm, as the traditional otimality criteria method is not aroriate for density-deendent body forces. The modification was imlemented based on the criteria. The numerical alications have shown that the density-deendent body forces have a strong influence in the toology otimization result deending on their magnitude in relation to the alied concentrated loads. Hence, deending on the oerating
condition of the mechanical art, body forces cannot be neglected to obtain the otimized design. 8. Acnowledgements Voith Siemens Hydro Power Generation is gratefully acnowledged. We also thans University of São Paulo (Brazil), FAPESP - Fundação de Amaro à Pesquisa do Estado de São Paulo; CNPq - Conselho Nacional de Desenvolvimento Científico e Tecnológico - Brazil, and CAPES - Coordenação de Aerfeiçoamento de Pessoal de Nível Suerior for the financial suort. 9. References 1 Liu, J.; Pars, G. & Clarson, P. (2005), 'Toology/shae otimisation of axisymmetric continuum structures - a metamorhic develoment aroach', Structural and Multidiscilinary Otimization 29, 73--83. 2 Stodola, A., Damf- und Gasturbinen. Berlin, Heidelberg, Sringer, 1924 3 Bhaviatti, S.S.; Ramarishnan, C.V.,. Otimum shae design of rotating diss. Comuter and Structures, 1980, (11): 397-401. 4 Cheu, T.C., Procedures for shae otimization of gas turbine diss. Comuter and Structures, 1990 (34), 1-4. 5 Kress, G. R., Shae otimization of a flywheel. Structural and Multidiscilinary Otimization, 2000. (19), 74-81 6 Bendsøe, M., P., & Kiuchi, N., Generating otimal toologies in structural design using a homogenization method. Comuter Methods in Alied Mechanics and Engineering, 1998,.(71)2, 197-224. 7 Bendsøe, M., P., & Sigmund, O., Toology Otimization: Theory, Methods and Alications. Sringer-Verlag, Berlin, 2003 8 Rozvany, G.I.N.; Zhou, M. & Birer, T., 'Generalized Shae Otimization without Homogenization', Structural Otimization, 1992, (4)3-4, 250--252. 9 Bendsøe, M., 'Otimal shae design as a material distribution roblem', Structural Otimization, 1989, (1), 193 202, 10 Bruyneel, M. and Duysinx, P., Note on toology otimization of continuum structures including self-weight, Structural and Multidiscilinary Otimization, 2005, (29)4, 245 256 11 Jog, C.S. and Haber, R.B., 'Stability of finite element models for distributed-arameter otimization and toology design', Comuter Methods in Alied Mechanics and Engineering, 1996. 130(3-4), 203 226, 12 Sigmund, O., A 99 line toology otimization code written in Matlab. Structural and Multidiscilinary Otimization, 2001,(21),2, 120-127.