Struct Multidisc OptimDOI 10.1007/s00158-014-1143-6

INDUSTRIAL APPLICATION

Structural optimization methods and techniques to designlight and efficient automatic transmission of vehicleswith 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 forautomatic transmission of vehicles to achieve light weightand low radiated noise. Light weight design is a fun-damental requirement for protecting the environment andimproving fuel economy. In addition, quietness is anotherrequirement for comfortable drive. However, in the designof automatic transmission, these two requirements are usu-ally in trade-off relationship and engineers spend a longtime to reach a desired design. This paper deals with thedesign approaches using structural optimization method forminimizing the radiation noise and the mass of automatictransmission. The weakly coupled analysis of elastic andacoustic problem are considered for evaluating the radiatednoise problem, where the modal frequency analysis is firstsolved using the finite element method and the acousticproblem for computing a noise radiated from the surface ofthe automatic transmission is then solved using the bound-ary element method. Three different structural optimizationmethods, topometry, topography and freeform optimization,are applied for the design of outer casing of automatic trans-mission. The optimization results show that the optimizationmethods successfully found the light weight and low radi-ated noise design of outer case, and can be used at the earlystage of the design process of automatic transmissions. Thefreeform optimization gives better solution compared with

T. Ide (�) · M. OtomoriAISIN AW Co., LTD. Fujii-cho, Takane 10, Anjo,Aichi, 444-1192, Japane-mail: I24824 IDE@aisin-aw.co.jp

J. P. Leiva · B. C. WatsonVanderplaats Research, Development, Inc.,41700 Gardenbrook, Suite 115, Novi, MI 48375, USA

the result of topography optimization from the standpoint ofthe sound pressure reduction effect while the mass reduc-tion effect is reduced in freeform optimization to satisfy thesound pressure constraint.

Keywords Structural optimization · Structural acoustics ·Industrial application · Automatic transmission

1 Introduction

Energy consumption and comfortable driving are impor-tant factors of vehicle performance (Vanderplaats 2004).Automatic transmissions play a significant role in thesetwo factors. Because automobile engines cannot adjust therotation speed and decoupling, they rely on automatic trans-mission to automatically change gears depending on therunning conditions. The appropriate gear ratio leads to fuelconsumption efficiency. Furthermore, finding lightweightdesigns is another essential factor of fuel consumption effi-ciency. On the other hand, in recent years, more attentionis being placed on NVH (Noise, Vibration, and Harshness)performance and quietness has become one of the essentialfactors of comfortable driving. Therefore the reduction ofradiated noise is one of the key considerations of new auto-matic transmission designs. Due to the fact that automatictransmissions have a very complicated structure resultingfrom the use of precise machinery (e.g. AISIN AW CO.,LTD. product lineup), it is typically very hard to find alightweight and a low noise level design. Traditionally, engi-neers have relied on their intuition based on experimentalresults and have updated the designs to improve the per-formance. As a consequence, conventional design methodsrequire extensive studies that take a long time to reachdesired results.

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One of the possible approaches to overcome this diffi-culty is to apply a structural optimization method. Basedon the degree of design flexibility, structural optimiza-tion method can be broadly categorized into sizing, shapeand topology optimization. The most fundamental methodis a sizing optimization which determines the optimumsizes of the structure, such as thickness, width and length.Schmit (1960) first proposed the structural optimizationmethod 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 theirouter boundaries and the shapes of inner holes. Zienkiewiczand Campbell (1973) first presented the shape optimiza-tion method applying to the design of dam structure, wherethe location of the finite element nodes are optimized dur-ing the procedure. Imam (1982) pointed out that directlychanging the location of the finite element nodes causesthe oscillation of the obtained structure. Several techniquesto avoid this oscillation are proposed, such as, the designelement technique (Imam 1982), super-curves technique(Imam 1982; Bennett and Botkin 1985), the techniqueto superpose the predefined shapes (Vanderplaats 1979;Imam 1982), the method that uses adaptive finite elementmethod (Kikuchi et al. 1986) and uses smoothed sensi-tivities (Azegami et al. 1997). On the other hand, topol-ogy optimization allows topological changes that includeincreasing the number of holes in the design domain,in addition to changes in a structure’s shape. Severalapproaches are proposed such as Homogenization DesignMethod (HDM) (Bendsøe and Kikuchi 1988) and densityapproaches (Bendsøe 1989; Sigmund 1997; Wang et al.2011; Kawamoto et al. 2011), level set-based approaches(Allaire et al. 2002; 2004; Wang et al. 2003; Wei and Wang2009; Yamada et al. 2010).

Regarding some optimization methods commerciallyavailable for industrial application, Leiva (2004) presentsa topometry optimization method which designs structuraldimensions or properties of individual elements (e.g. thick-ness of shell). This method can be recognized as element-by-element sizing optimization. On the other hand, Leiva(2003) presents a topography optimization method, whichuses automatically generated perturbation vectors to designthe location of surface grids on shells or composite ele-ments. The topography optimization method is typicallyused to find optimal bead patterns. In addition, Leiva (2010)presents a freeform optimization, in which a given pertur-bation is split into multiple perturbations on a grid-by-gridbasis. The possible distortions of the finite element mesh areavoided by automatically generated distortion constraintsand the use of mesh smoothing. Topography and freeformoptimization methods can be recognized as shape opti-mization method that uses the technique to superpose thepredefined shapes.

Structural optimization has been applied to the design ofautomatic transmission. Bos (2006) proposed a method tofind 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 mean-squared values of normal surface velocity to excitation forceover the frequency. Dai and Ramnath (2007) proposed themethod for reducing the radiated noise of vehicle trans-mission using topography optimization. Their approach isto minimize dynamic velocities for overall shell structure.However, the approaches mentioned above do not guar-antee a reduction of the peak value of sound pressure inthe desirable frequency range, since the surface velocitiesonly represent partial components of radiated noise. Tamariand Miyashita (2012) succeeded to reduce radiated noiseof vehicle transmission. They considered front engine reardrive type automatic transmission. Their method includesacoustic analysis using infinite element method. However,in their approach they only used topometry optimization toreinforce the structure in which the thickness distribution ofthe shell element that is attached on the original model isoptimized; as a consequence, they could not reduce the totalmass of automatic transmission.

In this paper, we apply different structural optimizationmethods such as topometry, topography, and freeform opti-mization, to find lightweight and low radiated noise leveldesigns. In the radiated noise reduction problem, we mustconsider a coupling of elastic problem and a sound pres-sure problem, since the surface vibration of the structurecauses the radiated noise, which is governed by acousticproblem, and the surface vibration is excited by the con-tact force at the internal gear, which is governed by elasticproblem. Our procedure includes modal frequency analysisusing the finite element method and acoustic analysis usingthe boundary element method. In the early stage of design-ing automatic transmission gearboxes, we use large scaleoptimization methods such as topometry, topography, andfreeform optimization.

The rest of this paper is organized as follows: In Section2, a brief explanation of general optimization is given, and inSection 3, commonly used structural optimization types aredescribed. A mathematical formulation for evaluating radi-ated noise in numerical analysis is then presented in Section4. In Section 5, design considerations of vehicle automatictransmission are discussed, and the results of different typesof optimization techniques are shown in Section 6. Wewill summarize our previous work that employs topome-try (Ide et al. 2010) and topography optimization (Ide et al.2012). Furthermore, we will show the numerical result usingfreeform optimization (Leiva 2010). Then we will comparethe mass change and the radiated noise with topometry,topography, and freeform optimization. Finally, in Section7, we present the conclusion of this work.

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2 Optimization problem

The optimization problem can be stated as:

min f (x1, x2, · · · , xn) or max f (x1, x2, · · · , xn)(1)

subject to:

gj (x1, x2, · · · , xn) � 0; j = 1, 2, · · · , m (2)

ai � xi � bi; i = 1, 2, · · · , n, (3)

where m and n is the number of constraint functions anddesign variables, respectively. f is the objective function,gj are the constraints, xi are the design variables and ai andbi are the side constraints associated to the design variables(Vanderplaats 2007; 2011).

Objective function Any of the responses can be used asthe objective function for minimization or maximization.Often mass, sound pressure or natural frequencies are usedas objective functions.

Constraint functions Any of the responses can be con-strained to satisfy prescribed desirable maximum or mini-mum values. Typical constraints are: mass, stress, displace-ments and dynamic displacements, velocities, and accelera-tions in industrial applications.

Design Variables Design variables are numericalinputs that can be changed during the optimization. Instructural optimization, design variables are typicallyparameters that can change, directly or indirectly, thedimension of elements, grid locations, and/or materialproperties.

3 Structural optimization

Structural optimization is a kind of optimization used toimprove structures. In structural optimization, the responsesare obtained solving the governing equation of the prob-lem, using numerical methods such as the finite elementmethod, and the design variables correspond to parametersthat describe the structure.

3.1 Structural optimization types

The structural optimization can be categorized into sizing,shape, and topology optimizations. A brief description ofeach type is presented below.

3.1.1 Sizing optimization

Sizing optimization is a structural optimization type usedto design specific dimensions or properties of structuralmembers (e.g. thickness of shells). Sizing optimization isthe most fundamental structural optimization method firstlyproposed by Schmit (1960).

Topometry optimization: element-by-element sizing opti-mization Topometry optimization is a structural optimiza-tion type used to design structural dimensions or propertiesof individual elements (e.g. thickness of shell). This methodcan be used to find optimal thickness distributions on shellelements (Leiva 2004).

3.1.2 Shape optimization

Shape optimization is a structural optimization type used todesign the shape of structural boundaries of the structure bymodifying the locations of grids.

Zienkiewicz and Campbell (1973) first presented theshape optimization method applying to the design of damstructure, where the location of the finite element nodesare optimized during the procedure. Imam (1982) pointedout that directly changing the location of the finite elementnodes causes the oscillation of the obtained structure. Sev-eral techniques to avoid this oscillation are proposed, suchas, the design element technique (Imam 1982), super-curvestechnique (Imam 1982; Bennett and Botkin 1985), the tech-nique to superpose the predefined shapes (Vanderplaats1979; Imam 1982), the method that uses adaptive finiteelement method (Kikuchi et al. 1986) and uses smoothedsensitivities (Azegami et al. 1997).

Topography optimization Topography optimization is astructural optimization type used to design surface grids onshells or composite elements. This method is recognizedas shape optimization method that uses the technique tosuperpose the predefined shapes (Vanderplaats 1979; Imam1982). In this type of optimization method, the generalshape Y is defined as follows.

Y = a1Y1 + a2Y

2 + · · · + anYn (4)

where ai is participation coefficients and Y i is basis vec-tor that defines the predefined shapes, which can be alsogiven in the form of Y i = Y 0 + dY i , where Y 0 is originalshape and dY i is called perturbation vector that defined theperturbed shape from original shape.

This method is typically used to find optimal bead pat-terns and it uses automatically generated perturbation vec-tors (Leiva 2003). Topography optimization can also be usedto indirectly design solid elements by placing shell elementson their faces.

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Freeform optimization Freeform optimization is anothertype of shape optimization that uses the technique to super-pose the predefined shapes. In this method, the programsplits given perturbation into multiple perturbations on agrid-by-grid basis. This split increases the variability ofthe design space when compared with traditional shapeoptimization. The possible distortions of the finite elementmesh are avoided by automatically generated distortion con-straints and the use of mesh smoothing. Further details areavailable in Leiva (2010).

3.1.3 Topology optimization

Topology optimization is a structural optimization typeused to find the optimal material distribution or materiallayout within a designable space (Bendsøe and Kikuchi1988). Topology optimization allows topological changesthat include increasing the number of holes in the designdomain, in addition to changes in a structure’s shape.The basic ideas of topology optimization are to extendthe design domain to a fixed design domain and toreplace the optimization problem by a material distribu-tion problem, using the characteristic function which hasa discontinuous function that has a value of 1 in mate-rial domain and 0 in void domain. This discontinuitycauses a ill-poseness of optimization problem. To overcomethis problem, Homogenization Design Method (HDM)(Bendsøe and Kikuchi 1988), and density approaches(Bendsøe 1989) are used, in which optimized configu-rations are represented as homogenized material prop-erty or density distributions that assume continuous val-ues. The obtained configurations therefore often includegrayscale areas where the density is an intermediatevalue between 0 and 1, and there also exist the prob-lems of checkerboards and mesh-dependency. To allevi-ate these problem, several filtering schemes (e.g. Sig-mund 1997; Wang et al. 2011; Kawamoto et al. 2011)are widely used. Moreover, to fundamentally solve thegrayscale problem, level set-based topology optimizationare also proposed (Allaire et al. 2002, 2004; Wang et al.2003; Wei and Wang 2009; Yamada et al. 2010). Detailedreviews of topology optimization are in the literature(Sigmund and Maute 2013).

3.2 Simple example of topometry, topography and freeformoptimization

3.2.1 Topometry optimization

Figure 1 shows an example using a plate model. The shellplate is subject to vertical force applied in the center of theplate. Four corners of the plate are constrained. The struc-ture is designed to be as stiff as possible for the applied

Fig. 1 Analysis model and boundary conditions of plate model fortopometry optimization

load. The design problem is to find the optimal thickness ofthe plate that minimizes the strain energy. Figure 2 showsthe optimal thickness distribution by topometry optimiza-tion. The region colored in red represents the area where thethickness is increased by optimization and blue representsthe area where the thickness has not increased. The rib-likethickness distribution is obtained in the center of the domainthat connects to the four corners to support the load appliedin center.

3.2.2 Topography optimization

Figure 3 shows an example using a plate model (GENESISUsers Manual 2011). The shell plate is subject to verticalforce applied in the right corner of the plate. Three cor-ners, on left, front, and rear of the plate, are constrained.The structure is designed to be as stiff as possible for theapplied load. The design problem is to find the optimalshape of the plate that minimizes the strain energy. Figure4 shows the optimal shape obtained by topography opti-mization. Several beads are obtained to stiffen the initialflat plate.

3.2.3 Freeform optimization

Figure 5 shows an example using a solid plate (GENESISUsers Manual 2011). The solid plate is subject to multipletorsional loading conditions. In the first loadcase (Fig. 5a),

2.0

1.6

1.2

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0.6

1.0

1.4

1.8

[mm]

Fig. 2 Optimal thickness distribution obtained by topometryoptimization

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Fig. 3 Analysis model and boundary conditions of plate model fortopography optimization

the front and back corners of the structure are constrainedand a torsional load is applied in the mid section. In thesecond loadcase (Fig. 5b), the front corners of the struc-ture are constrained and a torsional load is applied in theback section. In the third loadcase (Fig. 5c), the back cor-ners of the structure are constrained and a torsional loadis applied in the front section. The structure is designed tobe as stiff as possible for the applied loadcases. The designproblem is to find the shape of the plate that minimizesthe strain energy by moving up to 45% of the designablegrids and ensuring three mirror symmetries along the center.Figure 6 shows the perturbation vector that is acting on thetop face.

Figure 7 shows the final freeform optimized shape. Inthe figure, red represents the highest possible movement andblue the lowest. The result produces ribs that when viewfrom the top have a double x shape. The final shape makessense because it supports the torsional loads applied in themid and side sections.

3.3 Use of structural optimization

“Form follows function” is a Louis Sullivan’s principleassociated with modern architecture, industrial and auto-mobile design which has been present for more than acentury. The principle indicates that the structure or theshape of a building or any object should be primarily basedupon its intended function or purpose. This principle also

0.8

0.5

0.30.20.10.0

0.4

0.60.7

[mm]

Fig. 4 Optimal shape using topography optimization

holds for automatic transmission designs. However, design-ers are not free to just design, but they have to take intoaccount what methods of fabrication exist. In addition, ifthey use optimization software, they need to know whichof them they use. Therefore, it is important to link thedifferent structural optimization types with the differentmanufacturing types. Table 1 shows what types of opti-mization method are appropriate for some manufacturingconstraints. The automatic transmission gearbox is casting.The inside of gearbox is filled with automatic transmissionfluid. We cannot use topology optimization to avoid oil leak-ing. Therefore we use topometry, topography, and freeformoptimization.

Table 1 shows typical optimization methods used fora typical manufacturing process. However, creativity ofthe users and new functions that emerge can change thisclassification (Leiva 2011).

3.4 Techniques to design & reduce radiated noiseof automatic transmission

Topometry, topography, and freeform optimization are usedin this work as preliminary study for designing automatictransmission. Table 2 shows what types of optimizationmethods are appropriate to satisfy different design require-ments for the design of automatic transmission. In thetopometry optimization, the surface shell is attached onthe original solid model and the thickness distribution ofthe attached shell is optimized. Therefore, during the opti-mization, the grid points of the solid model do not movedand the optimal thickness distribution indicates where toput the material to reinforce the structure. In such way,it is not possible to reduce the mass from the origi-nal model and therefor reducing mass using topometryoptimization is not appropriate. On the other hands, intopography and freeform optimization, the grid points aremoved during the optimization and it is possible to reducethe mass.

4 Mathematical formulation for evaluating radiatednoise

The radiated noise problem is a coupled elastic structuraland fluid acoustical problem. In this paper, we considerweak coupling between structure and fluid that allows sepa-rate computation of dynamic velocities and sound pressures.The velocities are calculated first using the finite elementmethod and then the sound pressures at the field points arecalculated using the boundary element method together withpreviously calculated velocities. A mathematical formula-tion to reduce radiated noise is described in Kosaka et al.(2011).

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(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 3

4.1 Modal frequency response analysis

The governing equation for the dynamic frequency responsecan be written as follows:

[M]{u} + [B]{u} + [K]{u} + i[Ks]{u} = [P ] (5)

where [M] is the mass matrix, [B] is the viscous damp-ing matrix, [K] is the stiffness matrix, [Ks ] is the structuraldamping matrix, [P ] is the load vector, and {u} is thedynamic displacement. Here we assume the load vectorand dynamic displacement to be periodic functions, thegoverning (5) is replaced as follows:(−ω2[M] + iω[B] + [K] + i[Ks]

){u} = [P ], (6)

where ω is angular frequency. In this work, structural damp-ing [Ks] is ignored and modal damping is applied. Thereforesolving the eigenvalue problem in (7) yields the natural fre-quencies and the mode shapes [�] that can reduce (6) tomodal space.

([K] − λ[M]) [�] = [0] (7)

The following equation is the finite element scheme of amodal dynamic frequency problem.(−ω2[m] + iω[b] + [k]

){z} = [c]{z} = [p] (8)

Fig. 6 Perturbation vector

where,

[m] = [�]T [M][�], (9)

[b] = 2πfig(fi), (10)

[k] = [�]T [K][�], (11)

[c] = −ω2[m] + iω[b] + [k], (12)

[p] = [�]T [P ], (13)

{u} = [�]{z}. (14)

where g(fi) is the modal damping at a frequency fi . Thedynamic velocity vector {v} can be obtained by solving (8).

4.2 Acoustic analysis

The governing equation of acoustic problem is expressedas Helmholtz equation and the Helmholtz integral equa-tion is expressed as follows in the direct boundary elementformulation (Citarella et al. 2007):

pF = −∫

S

(iρωGvn + ps

∂G

∂n

)dS, (15)

G(r) = e−ikr

4πr, (16)

where pF is the complex sound pressure in the field point,Gis the free space Green’s function, n is the unit normal vectoron the surface of the radiating body, S, directed away from

1.0

1.2

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0.4

0.2

0.0

Fig. 7 Optimal shape using freeform optimization

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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, vn is the surface normal velocity, andps is the sound pressure on S. Once the (15) is discretizedusing boundary element space, it is replaced as follows:

{pF } = [ATM]{vn} (17)

where [ATM] is acoustic transfer matrix and {vn} is surfacevelocity vector which can be obtained by solving (8). For asingle field point, (17) can be described as follows:

pF,i = {ATM}i{vn} (18)

where {ATM}i is acoustic transfer vector that represents thei-th row of the [ATM].

4.3 Sensitivity analysis

The sensitivities of the sound pressure responses withrespect to corresponding design variables are calculatedusing the adjoint method because in topometry, topography,and freeform optimization there is a large number of designvariables (Wu 2000).

5 Approximation problem

With the finite element analysis results and the sensitiv-ity results we construct a model that approximates theresponses of sound pressures. This approximated model isoptimized using the large scale BIGDOT optimizer thatis imbedded in the GENESIS software (GENESIS User’sManual Version 12.1 2011). To avoid over extension of theapproximated responses, we use move limits that limit howmuch each design variable can move. The approximatedproblem is recreated in each design cycle to keep accuracy.

Table 2 Types of structural optimization versus appropriate objectivefunctions for the design of automatic transmission

Sound Pressure Lightweight

Topometry Yes -

Topography Yes Yes

Freeform Yes Yes

Response approximations For most of our approximationswe use the conservative approximation approach first devel-oped by Starnes and Haftka (1979) and later refined byFleury and Braibant (1986):

G(X) = G(X0)+�hi(xi) (19)

where,

hi(xi) =⎧⎨⎩

∂G∂xi

∣∣∣X=X0

(xi − x0i) if xi∂G∂xi

∣∣∣X=X0

> 0

− ∂G∂xi

∣∣∣X=X0

(1xi− 1

x0i

)x2

0i if xi∂G∂xi

∣∣∣X=X0

� 0

(20)

G(X) is the function being approximated. X0 is the vectorof intermediate design variables where the approximation isbased, xi is the i-th intermediate design variable, x0i is thebase value of the i-th intermediate design variable.

6 Numerical examples

As a demonstrative problem, we consider a front engine andfront wheel drive type (herein after, we call FF type) auto-matic transmission. Figure 8 shows the cut model of FF typeautomatic transmission (AISIN AW CO., LTD. 1969).

The inside of the automatic transmission has little designfreedom 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 torepresent high frequency modes. To precisely representthe complicated geometry of the automatic transmission,including variations of wall thickness and ribs, a finite ele-ment model was generated. Our finite element model usesthree-dimensional solid elements. Figure 9 shows the modelof our automatic transmission (Ide et al. 2010, 2012). The

Fig. 8 FF type automatic transmission

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Fig. 9 Finite element model of FF type automatic transmission

model consists of 1,435,381 elements (tetra and hexahe-dron) with 1,100,219 grids and 3,492,156 DOF’s.

6.2 Boundary element model

The boundary element analysis is computationally veryexpensive when applied to a large degree of freedom prob-lem, so relatively coarse boundary element models are usedin general. Figure 10 shows the boundary element modelfor our automatic transmission, which is created based onthe FEM model. The model consists of 4,700 quadrilateralelements with 4,702 grids and 28,212 DOF’s.

6.3 Optimization problem

Acoustic responses depend on loading frequencies and weneed to minimize the peak values over all the applied load-ing frequencies. However, the peak frequency can easilyshift from one frequency to another, when changing thesize or shape of the structure during the optimization. To

Fig. 10 Boundary element model

overcome this difficulty, we introduce an artificial designvariable called beta and add additional constraint equationsusing this beta.

We first consider minimizing radiated noise using topom-etry optimization. The objective of optimization is set tominimize beta, and sound pressures at each frequencies areconstrained to be less than beta. If beta is reduced, the peak(maximum) value of the dynamic response will be reducedin order to satisfy the beta constraints. This method is calledthe beta method (Taylor 1984) and it is widely used tosolve the min-max problem (Vanderplaats 2007, 2011). Wenote that we apply the constraint screening technique whereonly limited numbers of responses are retained to make thesensitivity computation less expensive. In this paper, weminimize radiated noise at a single field point, although themethod can easily be extended to multi field points. Thesound pressure for a single field point is expressed as equa-tion (18). Here pF,1 is the field point at the top of automatictransmission. First, define β as follows:

β =def.

Max(pF,1). (21)

Then our optimization problem for topometry optimizationis defined as follows:

Objective functionminβ

subject topF,1 � β.

Design variable0.0005 � xj � 5.0mm (j = 1, 2, · · · , n)

Figure 11a and b show the area where the design vari-ables will act, the areas correspond to surface shells gener-ated from the finite element model.

Next, we consider topography optimization to reduceboth mass and radiated noise. Here, the objective function isset to minimize mass. The radiated noise is dealt using con-straint functions where the sound pressure at the measuringfield point is constrained to not exceed the sound pressurevalue obtained in topometry optimization. Then our opti-mization problem for topography optimization is defined asfollows:

Objective functionminMass

subject topF,1 � β∗.Design variable−5.0 � xj � 5.0mm (j = 1, 2, · · · , n).where β∗ is the optimized value of beta obtained in topom-etry optimization.

The design area is the same as in topometry optimization.But, in this case we design the location of the grids.

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Fig. 11 Design variables; ahousing, b case

Finally, we consider freeform optimization to reduceboth mass and radiated noise. Again, the objective functionis set to minimize mass, and the radiated noise is dealt usingconstraint functions where the sound pressure at the mea-suring field point is constrained to not exceed the soundpressure value obtained in topometry optimization. In sum-mary, our optimization problem for freeform optimization isdefined as follows:

Objective functionminMass

subject topF,1 � β∗.Design variable−5.0 � xj � 5.0mm (j = 1, 2, · · · , n)

The design area is the same as topometry optimizationand topography optimization cases. In this case, like intopography optimization, we design the location of grids.

6.4 Numerical results

In this section, we present the numerical results obtainedusing topometry, topography, and freeform optimizations,applied to the gearbox design.

6.4.1 Topometry optimization

Topometry optimization can be used to find optimalthickness distributions on an element-by-element basis.In this case, we use this optimization type to find thethickness distribution of additional shell elements addedto the surface of the solid mesh to reduce radiatednoise. The added shell element represents areas to rein-force the structure. The initial thickness is set in thiscase to 0.0005mm. This initial small value allows usto start with an initial design practically equal to thenominal design.

Figure 12 shows the optimal thickness distribution ofthe added shell elements. The red color indicates the upperbound of the design variable thickness (5mm) while thegreen color indicates the lower bound of the design vari-able thickness (0.0005mm). We note that although thelower bound of design variable is here set to 0.0005mm,the lower bound of color bar in the figure is set to -5mm to facilitate the comparison of thickness distribu-tion with the other results that we will discuss later.Designable areas of outer surfaces are designated and64,482 shell elements are generated covering the solid ele-ments in the surface. Since each shell element is designedwith a unique design variable, 64,482 design variablesare used in total on the design. Details can be found in(Ide et al. 2010).

6.4.2 Topography optimization

Topography optimization can be used to find the optimallocation of bead or rib patterns. In this example, topogra-phy optimization was used to find the best places that cansimultaneously reduce mass and radiated noise.

Figure 13 shows the optimal shape obtained using topog-raphy optimization. The red color region indicates the upperbound of shape changes (the upper bounds of design vari-ables were set to 5mm) and blue color region indicatesthe lower bound of shape changes (the lower bounds ofdesign variables were set to -5mm). Designable areas ofouter surfaces are designated using a thin skin of shellelements that contain 33,153 grids on the solid element sur-faces. Therefore 33,153 design variables are created andare used in this design case. The details are described in(Ide et al. 2012).

6.4.3 Freeform optimization

Freeform optimization can be used to find the optimal loca-tions of rib patterns. In this case, freeform optimization was

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RearLeft Right

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Fig. 12 Topometry Optimization Results

used to find the best places that can simultaneously reducemass and radiated noise.

Figure 14 shows the resultant optimal shape. Thered color region indicates the upper bound of shapechanges (the upper bounds of the design variableswere set to 5mm) and blue color region indicates thelower bounds of shape changes (the lower bounds ofdesign variables were set to -5mm). Designable areas ofthe outer surfaces are designated; they contain 32,864grids on the solid element surfaces so 32,864 corre-sponding design variables are created and used in thedesign.

6.4.4 Comparison of thickness distribution

Comparing the colored thickness distribution obtainedusing topography and freeform optimizations, we observea similar distribution. However, the topography opti-mization results show, larger blue areas that indi-

cate more mass reduction. The numerical results forthe mass, as shown later in Fig. 15, confirm thisobservation.

On the other hand, comparing the thickness distribu-tion obtained using topometry and the other two opti-mizations, the distribution is quite different. This comesfrom the fact that in this case topometry optimization hasless design freedom as it is only used for adding mate-rial which significantly affects the thickness distributionresults.

6.4.5 Comparison of mass change and sound pressure

Figure 15 shows the comparison of mass change betweenbaseline model, topometry, topography, and freeform opti-mization. The three optimization cases took differentnumber of design cycles to finish: Topography took 15design cycles, freeform took 18 and topometry took 40.It should be mentioned that the optimization terminates

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Fig. 13 Topography optimization results

if the changes in design variables are sufficiently smallor the objective function is not significantly improved.In addition, the maximum design cycle was set here to40. Topography and freeform optimization successfullyreduced the total mass of our automatic transmission gear-box model. While topography optimization reduced themass by -2.99kg, freeform reduced it by -1.65kg. On theother hand, topometry optimization could not reduce themass because it was applied to an additional layer of ele-ments, resulting in that the mass increased by approximately+3.10kg.

Figure 16 shows a comparison of the sound pressure fora field point between baseline model, topometry, topogra-phy, and freeform optimization. The red curve shows thesound pressure of the baseline model while the other curvesshow the sound pressure of the optimized design. In allthree optimization cases, the sound pressure was reducedfor almost the entire range of the loading frequencies. Thereduction ratios of the maximum value of sound pressure

(in linear amplitude scale) for topometry, topography andfreeform optimizations are, respectively, 59.81%, 58.95%and 59.33%. This indicates that, although the maximumsound pressure values for topography and freeform opti-mization slightly exceed the maximum value of the appliedconstraint (59.81%), the constraint on sound pressure suc-cessfully functioned to reduce sound pressure in topographyand freeform optimization. While the mass reduction effectis slightly reduced in freeform optimization compared withthe result of topography optimization, to satisfy the soundpressure constraint, the freeform optimization gives bettersolution from the standpoint of the sound pressure reductioneffect.

7 Conclusions

Topometry, topography, and freeform optimization meth-ods have been presented. The use of these methods allows

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Fig. 14 Freeform optimization results

designers to find efficient and innovative designs whichcan not be achieved by traditional manual methods. Themethods presented allow the designers to explore a larger

Design cycle

Mas

s cha

nge

[kg]

TopometryTopographyFreeform

4

3

2

1

0

-1

-2

-3

-4

10 20 30 40

Fig. 15 Comparison of mass change

design space and reduce sound pressure of vehicle auto-matic transmission. First, topometry optimization is appliedfor the design of thickness distribution of automatic trans-

Loading frequency [Hz]

Soun

d pr

essu

re [d

B]

BaselineTopometryTopographyFreeform

200Hz5dB

Fig. 16 Comparison of sound pressure

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mission that minimizes the sound pressure at a single fieldpoint. Next, both topography and freeform optimization areapplied to minimize the total mass, where the optimizationproblem is formulated as to minimize the total mass with theconstraint on the sound pressure so that the sound pressuredoes not exceed the value obtained by topometry optimiza-tion. The optimization results demonstrated that the massreduction effect is slightly reduced in freeform optimiza-tion compared with the result of topography optimization,to satisfy the sound pressure constraint, and the freeformoptimization gives better solution from the standpoint of thesound pressure reduction effect.

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