T. Ide et al. One of the possible approaches to overcome this difficulty is to apply a structural optimization method. Based on the degree of design f

Transcription

1 Struct Multidisc Optim DOI /s INDUSTRIAL APPLICATION Structural optimization methods and techniques to design light and efficient automatic transmission of vehicles with low radiated noise Takanori Ide Masaki Otomori Juan Pablo Leiva Brian C. Watson Received: 23 September 2013 / Revised: 28 February 2014 / Accepted: 9 April 2014 Springer-Verlag Berlin Heidelberg 2014 Abstract This paper discusses design methodologies for automatic transmission of vehicles to achieve light weight and low radiated noise. Light weight design is a fundamental requirement for protecting the environment and improving fuel economy. In addition, quietness is another requirement for comfortable drive. However, in the design of automatic transmission, these two requirements are usually in trade-off relationship and engineers spend a long time to reach a desired design. This paper deals with the design approaches using structural optimization method for minimizing the radiation noise and the mass of automatic transmission. The weakly coupled analysis of elastic and acoustic problem are considered for evaluating the radiated noise problem, where the modal frequency analysis is first solved using the finite element method and the acoustic problem for computing a noise radiated from the surface of the automatic transmission is then solved using the boundary element method. Three different structural optimization methods, topometry, topography and freeform optimization, are applied for the design of outer casing of automatic transmission. The optimization results show that the optimization methods successfully found the light weight and low radiated noise design of outer case, and can be used at the early stage of the design process of automatic transmissions. The freeform optimization gives better solution compared with T. Ide ( ) M. Otomori AISIN AW Co., LTD. Fujii-cho, Takane 10, Anjo, Aichi, , Japan I24824 IDE@aisin-aw.co.jp J. P. Leiva B. C. Watson Vanderplaats Research, Development, Inc., Gardenbrook, Suite 115, Novi, MI 48375, USA the result of topography optimization from the standpoint of the sound pressure reduction effect while the mass reduction effect is reduced in freeform optimization to satisfy the sound pressure constraint. Keywords Structural optimization Structural acoustics Industrial application Automatic transmission 1 Introduction Energy consumption and comfortable driving are important factors of vehicle performance (Vanderplaats 2004). Automatic transmissions play a significant role in these two factors. Because automobile engines cannot adjust the rotation speed and decoupling, they rely on automatic transmission to automatically change gears depending on the running conditions. The appropriate gear ratio leads to fuel consumption efficiency. Furthermore, finding lightweight designs is another essential factor of fuel consumption efficiency. On the other hand, in recent years, more attention is being placed on NVH (Noise, Vibration, and Harshness) performance and quietness has become one of the essential factors of comfortable driving. Therefore the reduction of radiated noise is one of the key considerations of new automatic transmission designs. Due to the fact that automatic transmissions have a very complicated structure resulting from the use of precise machinery (e.g. AISIN AW CO., LTD. product lineup), it is typically very hard to find a lightweight and a low noise level design. Traditionally, engineers have relied on their intuition based on experimental results and have updated the designs to improve the performance. As a consequence, conventional design methods require extensive studies that take a long time to reach desired results.

2 T. Ide et al. One of the possible approaches to overcome this difficulty is to apply a structural optimization method. Based on the degree of design flexibility, structural optimization method can be broadly categorized into sizing, shape and topology optimization. The most fundamental method is a sizing optimization which determines the optimum sizes of the structure, such as thickness, width and length. Schmit (1960) first proposed the structural optimization method in which the size for the design of three trusses. Shape optimization (Zienkiewicz and Campbell 1973; Imam 1982) deals with the shape of structures, such as their outer boundaries and the shapes of inner holes. Zienkiewicz and Campbell (1973) first presented the shape optimization method applying to the design of dam structure, where the location of the finite element nodes are optimized during the procedure. Imam (1982) pointed out that directly changing the location of the finite element nodes causes the oscillation of the obtained structure. Several techniques to avoid this oscillation are proposed, such as, the design element technique (Imam 1982), super-curves technique (Imam 1982; Bennett and Botkin 1985), the technique to superpose the predefined shapes (Vanderplaats 1979; Imam 1982), the method that uses adaptive finite element method (Kikuchi et al. 1986) and uses smoothed sensitivities (Azegami et al. 1997). On the other hand, topology optimization allows topological changes that include increasing the number of holes in the design domain, in addition to changes in a structure s shape. Several approaches are proposed such as Homogenization Design Method (HDM) (Bendsøe and Kikuchi 1988) and density approaches (Bendsøe 1989; Sigmund 1997; Wangetal. 2011; Kawamotoetal.2011), level set-based approaches (Allaire et al. 2002; 2004;Wangetal.2003; Wei and Wang 2009; Yamada et al. 2010). Regarding some optimization methods commercially available for industrial application, Leiva (2004) presents a topometry optimization method which designs structural dimensions or properties of individual elements (e.g. thickness of shell). This method can be recognized as elementby-element sizing optimization. On the other hand, Leiva (2003) presents a topography optimization method, which uses automatically generated perturbation vectors to design the location of surface grids on shells or composite elements. The topography optimization method is typically used to find optimal bead patterns. In addition, Leiva (2010) presents a freeform optimization, in which a given perturbation is split into multiple perturbations on a grid-by-grid basis. The possible distortions of the finite element mesh are avoided by automatically generated distortion constraints and the use of mesh smoothing. Topography and freeform optimization methods can be recognized as shape optimization method that uses the technique to superpose the predefined shapes. Structural optimization has been applied to the design of automatic transmission. Bös (2006) proposed a method to find the optimal thickness distribution of a gearbox model. Their approach is to minimize the structural-borne sound, which is the averaged value of the ratio of the meansquared values of normal surface velocity to excitation force over the frequency. Dai and Ramnath (2007) proposed the method for reducing the radiated noise of vehicle transmission using topography optimization. Their approach is to minimize dynamic velocities for overall shell structure. However, the approaches mentioned above do not guarantee a reduction of the peak value of sound pressure in the desirable frequency range, since the surface velocities only represent partial components of radiated noise. Tamari and Miyashita (2012) succeeded to reduce radiated noise of vehicle transmission. They considered front engine rear drive type automatic transmission. Their method includes acoustic analysis using infinite element method. However, in their approach they only used topometry optimization to reinforce the structure in which the thickness distribution of the shell element that is attached on the original model is optimized; as a consequence, they could not reduce the total mass of automatic transmission. In this paper, we apply different structural optimization methods such as topometry, topography, and freeform optimization, to find lightweight and low radiated noise level designs. In the radiated noise reduction problem, we must consider a coupling of elastic problem and a sound pressure problem, since the surface vibration of the structure causes the radiated noise, which is governed by acoustic problem, and the surface vibration is excited by the contact force at the internal gear, which is governed by elastic problem. Our procedure includes modal frequency analysis using the finite element method and acoustic analysis using the boundary element method. In the early stage of designing automatic transmission gearboxes, we use large scale optimization methods such as topometry, topography, and freeform optimization. The rest of this paper is organized as follows: In Section 2, a brief explanation of general optimization is given, and in Section 3, commonly used structural optimization types are described. A mathematical formulation for evaluating radiated noise in numerical analysis is then presented in Section 4. In Section 5, design considerations of vehicle automatic transmission are discussed, and the results of different types of optimization techniques are shown in Section 6. We will summarize our previous work that employs topometry (Ide et al. 2010) and topography optimization (Ide et al. 2012). Furthermore, we will show the numerical result using freeform optimization (Leiva 2010). Then we will compare the mass change and the radiated noise with topometry, topography, and freeform optimization. Finally, in Section 7, we present the conclusion of this work.

3 Structural optimization methods to design light and efficient automatic transmission 2 Optimization problem Sizing optimization The optimization problem can be stated as: min f(x 1,x 2,,x n ) or max f(x 1,x 2,,x n ) (1) subject to: g j (x 1,x 2,,x n ) 0; j = 1, 2,,m (2) a i x i b i ; i = 1, 2,,n, (3) where m and n is the number of constraint functions and design variables, respectively. f is the objective function, g j are the constraints, x i are the design variables and a i and b i are the side constraints associated to the design variables (Vanderplaats 2007; 2011). Objective function Any of the responses can be used as the objective function for minimization or maximization. Often mass, sound pressure or natural frequencies are used as objective functions. Constraint functions Any of the responses can be constrained to satisfy prescribed desirable maximum or minimum values. Typical constraints are: mass, stress, displacements and dynamic displacements, velocities, and accelerations in industrial applications. Design Variables Design variables are numerical inputs that can be changed during the optimization. In structural optimization, design variables are typically parameters that can change, directly or indirectly, the dimension of elements, grid locations, and/or material properties. 3 Structural optimization Structural optimization is a kind of optimization used to improve structures. In structural optimization, the responses are obtained solving the governing equation of the problem, using numerical methods such as the finite element method, and the design variables correspond to parameters that describe the structure. 3.1 Structural optimization types The structural optimization can be categorized into sizing, shape, and topology optimizations. A brief description of each type is presented below. Sizing optimization is a structural optimization type used to design specific dimensions or properties of structural members (e.g. thickness of shells). Sizing optimization is the most fundamental structural optimization method firstly proposed by Schmit (1960). Topometry optimization: element-by-element sizing optimization Topometry optimization is a structural optimization type used to design structural dimensions or properties of individual elements (e.g. thickness of shell). This method can be used to find optimal thickness distributions on shell elements (Leiva 2004) Shape optimization Shape optimization is a structural optimization type used to design the shape of structural boundaries of the structure by modifying the locations of grids. Zienkiewicz and Campbell (1973) first presented the shape optimization method applying to the design of dam structure, where the location of the finite element nodes are optimized during the procedure. Imam (1982) pointed out that directly changing the location of the finite element nodes causes the oscillation of the obtained structure. Several techniques to avoid this oscillation are proposed, such as, the design element technique (Imam 1982), super-curves technique (Imam 1982; Bennett and Botkin 1985), the technique to superpose the predefined shapes (Vanderplaats 1979; Imam1982), the method that uses adaptive finite element method (Kikuchi et al. 1986) and uses smoothed sensitivities (Azegami et al. 1997). Topography optimization Topography optimization is a structural optimization type used to design surface grids on shells or composite elements. This method is recognized as shape optimization method that uses the technique to superpose the predefined shapes (Vanderplaats 1979;Imam 1982). In this type of optimization method, the general shape Y is defined as follows. Y = a 1 Y 1 + a 2 Y 2 + +a n Y n (4) where a i is participation coefficients and Y i is basis vector that defines the predefined shapes, which can be also given in the form of Y i = Y 0 + dy i,wherey 0 is original shape and dy i is called perturbation vector that defined the perturbed shape from original shape. This method is typically used to find optimal bead patterns and it uses automatically generated perturbation vectors (Leiva 2003). Topography optimization can also be used to indirectly design solid elements by placing shell elements on their faces.

4 T. Ide et al. Freeform optimization Freeform optimization is another type of shape optimization that uses the technique to superpose the predefined shapes. In this method, the program splits given perturbation into multiple perturbations on a grid-by-grid basis. This split increases the variability of the design space when compared with traditional shape optimization. The possible distortions of the finite element mesh are avoided by automatically generated distortion constraints and the use of mesh smoothing. Further details are available in Leiva (2010) Topology optimization Topology optimization is a structural optimization type used to find the optimal material distribution or material layout within a designable space (Bendsøe and Kikuchi 1988). Topology optimization allows topological changes that include increasing the number of holes in the design domain, in addition to changes in a structure s shape. The basic ideas of topology optimization are to extend the design domain to a fixed design domain and to replace the optimization problem by a material distribution problem, using the characteristic function which has a discontinuous function that has a value of 1 in material domain and 0 in void domain. This discontinuity causes a ill-poseness of optimization problem. To overcome this problem, Homogenization Design Method (HDM) (Bendsøe and Kikuchi 1988), and density approaches (Bendsøe 1989) are used, in which optimized configurations are represented as homogenized material property or density distributions that assume continuous values. The obtained configurations therefore often include grayscale areas where the density is an intermediate value between 0 and 1, and there also exist the problems of checkerboards and mesh-dependency. To alleviate these problem, several filtering schemes (e.g. Sigmund 1997; Wang et al. 2011; Kawamoto et al. 2011) are widely used. Moreover, to fundamentally solve the grayscale problem, level set-based topology optimization are also proposed (Allaire et al. 2002, 2004; Wangetal. 2003; Wei and Wang 2009; Yamada et al. 2010). Detailed reviews of topology optimization are in the literature (Sigmund and Maute 2013). 3.2 Simple example of topometry, topography and freeform optimization Topometry optimization Figure 1 shows an example using a plate model. The shell plate is subject to vertical force applied in the center of the plate. Four corners of the plate are constrained. The structure is designed to be as stiff as possible for the applied Fig. 1 Analysis model and boundary conditions of plate model for topometry optimization load. The design problem is to find the optimal thickness of the plate that minimizes the strain energy. Figure 2 shows the optimal thickness distribution by topometry optimization. The region colored in red represents the area where the thickness is increased by optimization and blue represents the area where the thickness has not increased. The rib-like thickness distribution is obtained in the center of the domain that connects to the four corners to support the load applied in center Topography optimization Figure 3 shows an example using a plate model (GENESIS Users Manual 2011). The shell plate is subject to vertical force applied in the right corner of the plate. Three corners, on left, front, and rear of the plate, are constrained. The structure is designed to be as stiff as possible for the applied load. The design problem is to find the optimal shape of the plate that minimizes the strain energy. Figure 4 shows the optimal shape obtained by topography optimization. Several beads are obtained to stiffen the initial flat plate Freeform optimization Figure 5 shows an example using a solid plate (GENESIS Users Manual 2011). The solid plate is subject to multiple torsional loading conditions. In the first loadcase (Fig. 5a), [mm] Fig. 2 Optimal thickness distribution obtained by topometry optimization

5 Structural optimization methods to design light and efficient automatic transmission Fig. 3 Analysis model and boundary conditions of plate model for topography optimization the front and back corners of the structure are constrained and a torsional load is applied in the mid section. In the second loadcase (Fig. 5b), the front corners of the structure are constrained and a torsional load is applied in the back section. In the third loadcase (Fig. 5c), the back corners of the structure are constrained and a torsional load is applied in the front section. The structure is designed to be as stiff as possible for the applied loadcases. The design problem is to find the shape of the plate that minimizes the strain energy by moving up to 45% of the designable grids and ensuring three mirror symmetries along the center. Figure 6 shows the perturbation vector that is acting on the top face. Figure 7 shows the final freeform optimized shape. In the figure, red represents the highest possible movement and blue the lowest. The result produces ribs that when view from the top have a double x shape. The final shape makes sense because it supports the torsional loads applied in the mid and side sections. 3.3 Use of structural optimization Form follows function is a Louis Sullivan s principle associated with modern architecture, industrial and automobile design which has been present for more than a century. The principle indicates that the structure or the shape of a building or any object should be primarily based upon its intended function or purpose. This principle also holds for automatic transmission designs. However, designers are not free to just design, but they have to take into account what methods of fabrication exist. In addition, if they use optimization software, they need to know which of them they use. Therefore, it is important to link the different structural optimization types with the different manufacturing types. Table 1 shows what types of optimization method are appropriate for some manufacturing constraints. The automatic transmission gearbox is casting. The inside of gearbox is filled with automatic transmission fluid. We cannot use topology optimization to avoid oil leaking. Therefore we use topometry, topography, and freeform optimization. Table 1 shows typical optimization methods used for a typical manufacturing process. However, creativity of the users and new functions that emerge can change this classification (Leiva 2011). 3.4 Techniques to design & reduce radiated noise of automatic transmission Topometry, topography, and freeform optimization are used in this work as preliminary study for designing automatic transmission. Table 2 shows what types of optimization methods are appropriate to satisfy different design requirements for the design of automatic transmission. In the topometry optimization, the surface shell is attached on the original solid model and the thickness distribution of the attached shell is optimized. Therefore, during the optimization, the grid points of the solid model do not moved and the optimal thickness distribution indicates where to put the material to reinforce the structure. In such way, it is not possible to reduce the mass from the original model and therefor reducing mass using topometry optimization is not appropriate. On the other hands, in topography and freeform optimization, the grid points are moved during the optimization and it is possible to reduce the mass. 4 Mathematical formulation for evaluating radiated noise Fig. 4 Optimal shape using topography optimization [mm] The radiated noise problem is a coupled elastic structural and fluid acoustical problem. In this paper, we consider weak coupling between structure and fluid that allows separate computation of dynamic velocities and sound pressures. The velocities are calculated first using the finite element method and then the sound pressures at the field points are calculated using the boundary element method together with previously calculated velocities. A mathematical formulation to reduce radiated noise is described in Kosaka et al. (2011).

6 T. Ide et al. (a) (b) (c) Fig. 5 Analysis model and boundary conditions of a solid plate for freeform optimization; a loadcase 1, b loadcase 2, c loadcase Modal frequency response analysis where, The governing equation for the dynamic frequency response can be written as follows: [M]{ü}+[B]{ u}+[k]{u}+i[k s ]{u} =[P ] (5) where [M] is the mass matrix, [B] is the viscous damping matrix, [K] is the stiffness matrix, [K s ] is the structural damping matrix, [P ] is the load vector, and {u} is the dynamic displacement. Here we assume the load vector and dynamic displacement to be periodic functions, the governing (5) is replaced as follows: ( ) ω 2 [M]+iω[B]+[K]+i[K s ] {u} =[P ], (6) ERA where ω is angular frequency. In this work, structural damping [K s ] is ignored and modal damping is applied. Therefore solving the eigenvalue problem in (7) yields the natural frequencies and the mode shapes [ ] that can reduce (6) to modal space. ([K] λ[m]) [ ] =[0] (7) The following equation is the finite element scheme of a modal dynamic frequency problem. ( ) ω 2 [m]+iω[b]+[k] {z} =[c]{z} =[p] (8) [m] =[ ] T [M][ ], (9) [b] = 2πf i g(f i ), (10) [k] =[ ] T [K][ ], (11) [c] = ω 2 [m]+iω[b]+[k], (12) [p] =[ ] T [P ], (13) {u} =[ ]{z}. (14) where g(f i ) is the modal damping at a frequency f i.the dynamic velocity vector {v} can be obtained by solving (8). 4.2 Acoustic analysis The governing equation of acoustic problem is expressed as Helmholtz equation and the Helmholtz integral equation is expressed as follows in the direct boundary element formulation (Citarella et al. 2007): ( ) G p F = iρωgv n + p s ds, (15) n S G(r) = e ikr 4πr, (16) where p F is the complex sound pressure in the field point, G is the free space Green s function, n is the unit normal vector on the surface of the radiating body, S, directed away from Glob Fig. 6 Perturbation vector Fig. 7 Optimal shape using freeform optimization

7 Structural optimization methods to design light and efficient automatic transmission Table 1 Structural optimization method vs. manufacturing types Stamping Casting Extrusion Tailored Blank Sizing Yes - - Yes Shape Yes Yes - - Topology Yes Yes Yes - Topometry - Yes - Yes Topography Yes Yes - - Freeform Yes Yes - - the acoustic domain, v n is the surface normal velocity, and p s is the sound pressure on S. Once the (15) is discretized using boundary element space, it is replaced as follows: {p F }=[AT M]{v n } (17) where [AT M] is acoustic transfer matrix and {v n } is surface velocity vector which can be obtained by solving (8). For a single field point, (17) can be described as follows: p F,i ={AT M} i {v n } (18) where {AT M} i is acoustic transfer vector that represents the i-th row of the [AT M]. 4.3 Sensitivity analysis The sensitivities of the sound pressure responses with respect to corresponding design variables are calculated using the adjoint method because in topometry, topography, and freeform optimization there is a large number of design variables (Wu 2000). 5 Approximation problem With the finite element analysis results and the sensitivity results we construct a model that approximates the responses of sound pressures. This approximated model is optimized using the large scale BIGDOT optimizer that is imbedded in the GENESIS software (GENESIS User s Manual Version ). To avoid over extension of the approximated responses, we use move limits that limit how much each design variable can move. The approximated problem is recreated in each design cycle to keep accuracy. Response approximations For most of our approximations we use the conservative approximation approach first developed by Starnes and Haftka (1979) and later refined by Fleury and Braibant (1986): G(X) = G(X 0 ) + h i (x i ) (19) where, G X=X0 x i (x i x 0i ) h i (x i ) = ( x G X=X0 1xi i 1 x 0i ) G X=X0 if x i x i > 0 x0i 2 G X=X0 if x i xi 0 (20) G(X) is the function being approximated. X 0 is the vector of intermediate design variables where the approximation is based, x i is the i-th intermediate design variable, x 0i is the base value of the i-th intermediate design variable. 6 Numerical examples As a demonstrative problem, we consider a front engine and front wheel drive type (herein after, we call FF type) automatic transmission. Figure 8 shows the cutmodelof FF type automatic transmission (AISIN AW CO., LTD. 1969). The inside of the automatic transmission has little design freedom because of precise machinery. As a design space, we consider the wall thickness of the gearbox. 6.1 Finite element model The finite element model needs to be detailed enough to represent high frequency modes. To precisely represent the complicated geometry of the automatic transmission, including variations of wall thickness and ribs, a finite element model was generated. Our finite element model uses three-dimensional solid elements. Figure 9 shows the model of our automatic transmission (Ide et al. 2010, 2012). The Table 2 Types of structural optimization versus appropriate objective functions for the design of automatic transmission Sound Pressure Lightweight Topometry Yes - Topography Yes Yes Freeform Yes Yes Fig. 8 FF type automatic transmission

8 T. Ide et al. Fig. 9 Finite element model of FF type automatic transmission model consists of 1,435,381 elements (tetra and hexahedron) with 1,100,219 grids and 3,492,156 DOF s. 6.2 Boundary element model The boundary element analysis is computationally very expensive when applied to a large degree of freedom problem, so relatively coarse boundary element models are used in general. Figure 10 shows the boundary element model for our automatic transmission, which is created based on the FEM model. The model consists of 4,700 quadrilateral elements with 4,702 grids and 28,212 DOF s. 6.3 Optimization problem Acoustic responses depend on loading frequencies and we need to minimize the peak values over all the applied loading frequencies. However, the peak frequency can easily shift from one frequency to another, when changing the size or shape of the structure during the optimization. To overcome this difficulty, we introduce an artificial design variable called beta and add additional constraint equations using this beta. We first consider minimizing radiated noise using topometry optimization. The objective of optimization is set to minimize beta, and sound pressures at each frequencies are constrained to be less than beta. If beta is reduced, the peak (maximum) value of the dynamic response will be reduced in order to satisfy the beta constraints. This method is called the beta method (Taylor 1984) and it is widely used to solve the min-max problem (Vanderplaats 2007, 2011). We note that we apply the constraint screening technique where only limited numbers of responses are retained to make the sensitivity computation less expensive. In this paper, we minimize radiated noise at a single field point, although the method can easily be extended to multi field points. The sound pressure for a single field point is expressed as equation (18). Here p F,1 is the field point at the top of automatic transmission. First, define β as follows: β = Max(p F,1 ). (21) def. Then our optimization problem for topometry optimization is defined as follows: Objective function minβ subject to p F,1 β. Design variable x j 5.0mm (j = 1, 2,,n) Figure 11a and b show the area where the design variables will act, the areas correspond to surface shells generated from the finite element model. Next, we consider topography optimization to reduce both mass and radiated noise. Here, the objective function is set to minimize mass. The radiated noise is dealt using constraint functions where the sound pressure at the measuring field point is constrained to not exceed the sound pressure value obtained in topometry optimization. Then our optimization problem for topography optimization is defined as follows: Objective function minmass subject to p F,1 β. Design variable 5.0 x j 5.0mm (j = 1, 2,,n). Fig. 10 Boundary element model where β is the optimized value of beta obtained in topometry optimization. The design area is the same as in topometry optimization. But, in this case we design the location of the grids.

9 Structural optimization methods to design light and efficient automatic transmission Fig. 11 Design variables; a housing, b case Finally, we consider freeform optimization to reduce both mass and radiated noise. Again, the objective function is set to minimize mass, and the radiated noise is dealt using constraint functions where the sound pressure at the measuring field point is constrained to not exceed the sound pressure value obtained in topometry optimization. In summary, our optimization problem for freeform optimization is defined as follows: Objective function minmass subject to p F,1 β. Design variable 5.0 x j 5.0mm (j = 1, 2,,n) The design area is the same as topometry optimization and topography optimization cases. In this case, like in topography optimization, we design the location of grids. 6.4 Numerical results In this section, we present the numerical results obtained using topometry, topography, and freeform optimizations, applied to the gearbox design Topometry optimization Topometry optimization can be used to find optimal thickness distributions on an element-by-element basis. In this case, we use this optimization type to find the thickness distribution of additional shell elements added to the surface of the solid mesh to reduce radiated noise. The added shell element represents areas to reinforce the structure. The initial thickness is set in this case to mm. This initial small value allows us to start with an initial design practically equal to the nominal design. Figure 12 shows the optimal thickness distribution of the added shell elements. The red color indicates the upper bound of the design variable thickness (5mm) while the green color indicates the lower bound of the design variable thickness (0.0005mm). We note that although the lower bound of design variable is here set to mm, the lower bound of color bar in the figure is set to - 5mm to facilitate the comparison of thickness distribution with the other results that we will discuss later. Designable areas of outer surfaces are designated and 64,482 shell elements are generated covering the solid elements in the surface. Since each shell element is designed with a unique design variable, 64,482 design variables are used in total on the design. Details can be found in (Ide et al. 2010) Topography optimization Topography optimization can be used to find the optimal location of bead or rib patterns. In this example, topography optimization was used to find the best places that can simultaneously reduce mass and radiated noise. Figure 13 shows the optimal shape obtained using topography optimization. The red color region indicates the upper bound of shape changes (the upper bounds of design variables were set to 5mm) and blue color region indicates the lower bound of shape changes (the lower bounds of design variables were set to -5mm). Designable areas of outer surfaces are designated using a thin skin of shell elements that contain 33,153 grids on the solid element surfaces. Therefore 33,153 design variables are created and are used in this design case. The details are described in (Ide et al. 2012) Freeform optimization Freeform optimization can be used to find the optimal locations of rib patterns. In this case, freeform optimization was

10 T. Ide et al. Top 世 安 [mm] Left Rear Right Fig. 12 Topometry Optimization Results 太 亚 RA PE Bottom cate more mass reduction. The numerical results for the mass, as shown later in Fig. 15, confirm this observation. On the other hand, comparing the thickness distribution obtained using topometry and the other two optimizations, the distribution is quite different. This comes from the fact that in this case topometry optimization has less design freedom as it is only used for adding material which significantly affects the thickness distribution results Comparison of thickness distribution Figure 15 shows the comparison of mass change between baseline model, topometry, topography, and freeform optimization. The three optimization cases took different number of design cycles to finish: Topography took 15 design cycles, freeform took 18 and topometry took 40. It should be mentioned that the optimization terminates ba lo G used to find the best places that can simultaneously reduce mass and radiated noise. Figure 14 shows the resultant optimal shape. The red color region indicates the upper bound of shape changes (the upper bounds of the design variables were set to 5mm) and blue color region indicates the lower bounds of shape changes (the lower bounds of design variables were set to -5mm). Designable areas of the outer surfaces are designated; they contain 32,864 grids on the solid element surfaces so 32,864 corresponding design variables are created and used in the design. l Comparing the colored thickness distribution obtained using topography and freeform optimizations, we observe a similar distribution. However, the topography optimization results show, larger blue areas that indi Comparison of mass change and sound pressure

11 Structural optimization methods to design light and efficient automatic transmission Left Top Rear Right [mm] Fig. 13 Topography optimization results Bottom if the changes in design variables are sufficiently small or the objective function is not significantly improved. In addition, the maximum design cycle was set here to 40. Topography and freeform optimization successfully reduced the total mass of our automatic transmission gearbox model. While topography optimization reduced the mass by -2.99kg, freeform reduced it by -1.65kg. On the other hand, topometry optimization could not reduce the mass because it was applied to an additional layer of elements, resulting in that the mass increased by approximately +3.10kg. Figure 16 shows a comparison of the sound pressure for a field point between baseline model, topometry, topography, and freeform optimization. The red curve shows the sound pressure of the baseline model while the other curves show the sound pressure of the optimized design. In all three optimization cases, the sound pressure was reduced for almost the entire range of the loading frequencies. The reduction ratios of the maximum value of sound pressure (in linear amplitude scale) for topometry, topography and freeform optimizations are, respectively, 59.81%, 58.95% and 59.33%. This indicates that, although the maximum sound pressure values for topography and freeform optimization slightly exceed the maximum value of the applied constraint (59.81%), the constraint on sound pressure successfully functioned to reduce sound pressure in topography and freeform optimization. While the mass reduction effect is slightly reduced in freeform optimization compared with the result of topography optimization, to satisfy the sound pressure constraint, the freeform optimization gives better solution from the standpoint of the sound pressure reduction effect. 7 Conclusions Topometry, topography, and freeform optimization methods have been presented. The use of these methods allows

12 T. Ide et al. Top 世 安 [mm] Left Rear 太 RA 亚 PE Bottom Fig. 14 Freeform optimization results designers to find efficient and innovative designs which can not be achieved by traditional manual methods. The methods presented allow the designers to explore a larger ba lo 2 30 Topometry -2 Topography 40 Sound pressure [db] 3 Mass change [kg] design space and reduce sound pressure of vehicle automatic transmission. First, topometry optimization is applied for the design of thickness distribution of automatic trans- G 4 0 Right Freeform -3 5dB 200Hz -4 Design cycle Fig. 16 Comparison of sound pressure l Fig. 15 Comparison of mass change Loading frequency [Hz] Baseline Topometry Topography Freeform

13 Structural optimization methods to design light and efficient automatic transmission mission that minimizes the sound pressure at a single field point. Next, both topography and freeform optimization are applied to minimize the total mass, where the optimization problem is formulated as to minimize the total mass with the constraint on the sound pressure so that the sound pressure does not exceed the value obtained by topometry optimization. The optimization results demonstrated that the mass reduction effect is slightly reduced in freeform optimization compared with the result of topography optimization, to satisfy the sound pressure constraint, and the freeform optimization gives better solution from the standpoint of the sound pressure reduction effect. References AISIN AW CO. LTD. (1969) product lineup jp/en/products/drivetrain/lineup/index.html Allaire G, Jouve F, Toader AM (2002) A level-set method for shape optimization. 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