SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION FOR STRUCTURAL...

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 40, 1719—1743 (1997)

SHAPE DESIGN SENSITIVITY ANALYSIS ANDOPTIMIZATION FOR STRUCTURAL DURABILITY

KUANG-HUA CHANG*, XIAOMING YUs AND KYUNG K. CHOIt

Center for Computer-Aided Design and Department of Mechanical Engineering, College of Engineering,¹he ºniversity of Iowa, Iowa City, IA 52245, º.S.A.

SUMMARY

In this paper, a design sensitivity analysis (DSA) method for fatigue life of 3-D solid structural components ofmecanical systems with respect to shape design parameters is presented. The DSA method uses dynamicstress DSA obtained using an analytical approach to predict dynamic stress increment due to designchanges; computes fatigue life of the component, including crack initiation and crack propagation, using thepredicted dynamic stress; and uses the difference of the new life and the original life at the same critical pointto approximate the sensitivity of fatigue life. A tracked vehicle roadarm is presented in this paper todemonstrate accuracy and efficiency of the DSA method. Also, this method is applied to support designoptimization of the tracked vehicle roadarm considering crack initiation lives as design constraints. ( 1997by John Wiley & Sons, Ltd.

KEY WORDS: design sensitivity analysis; shape design optimization; fatigue life prediction; multi-axial crack initiation;structural durability

1. INTRODUCTION

Structural fatigue due to fluctuation of stresses generated in the service life of mechanical systemsis the primary concern in structural design for durability. Currently, in structural design practice,static stress concentration factors, instead of the fatigue life, are used widely as the criteria fordurability designs.1,2 The reason is that stress fluctuation occurring in structural componentsduring the service life of mechanical systems that contribute to fatigue is difficult to obtain. Theworst-case design with stress concentration factors criteria is usually employed to obtain anoptimum design for durability. In this case, stresses used for design constraints are determined byapplying a set of critical static loading to the structural component of interest at a specific time ofthe service life of mechanical systems. The trouble is that wrong criteria may be selected todetermine the optimum design since high stress areas identified at a specific time of the service lifemay not match critical areas where a crack first initiates. Recently, an optimum shape design fora minimum fatigue notch factor was proposed for dynamically loaded machine parts.3 However,the fatigue notch factor is a simple and rough indicator of structural durability. Moreover, it is

*Current address: Assistant Professor, Department of Mechanical Engineering, Northern Illinois University, DeKalb,IL 61105—2854, U.S.A.sCurrent address: Technical Staff, Engineering Department, CSAR Corporation, 28035 Dorothy Drive, Agoura Hills,CA 91301, U.S.A.tProfessor and Director

This article is a U.S. Government work and, as such, is in the public domain in the United States of America.

CCC 0029—5981/97/091719—25$17.50 Received 12 February 1996( 1997 by John Wiley & Sons, Ltd. Revised 27 September 1996

difficult to treat the fatigue notch factor as a design constraint since the upper bound of the fatiguenotch factor is very hard to determine. Since it is widely recognized that about 80 per cent failureof mechanical/structural components and systems are related to fatigue,4 a design optimizationmethodology to increase durability life must be developed for dynamically loaded machine partsand assemblies.

For ground vehicles and heavy equipment, crack initiation life is usually considered as thefailure criteria for durability design.4 For aircraft and offshore oil platforms, crack propagationlife is considered as the design criteria.4 Methods proposed in this paper are intended to supportdurability design of both crack initiation and crack propagation lives of structural components.However, this paper focuses on design of ground vehicles in which the crack initiation life is theprimary concern.

Objectives of this research are to (1) develop an efficient and accurate design sensitivity analysis(DSA) method for the fatigue life of structural components, and (2) apply the DSA method tosupport design optimization considering the structural fatigue life as the design criteria. Thecomputational flow of design optimization for fatigue life is shown in Figure 1.

To generate a representative load history, including inertia forces and external forces ( jointreaction forces and torques), for accurate dynamic stress computation, multibody dynamicanalysis5 is performed for the mechanical system under a typical duty cycle. Quasi-static finiteelement analyses (FEA) are then performed to obtain stress influence coefficients for the struc-tural component. These stress influence coefficients are superposed with the external and inertialoading histories produced in the multibody dynamic analysis to obtain dynamic stress history.The stress history is then employed to predict the fatigue life of the component using a strain-based crack initiation life prediction method and the linear elastic fracture mechanics (LEFM)4for crack propagation life.

The continuum DSA method is extended to compute the dynamic stress design sensitivityanalytically. The proposed method of durability DSA uses dynamic stress DSA to predictdynamic stress history due to design changes, computes life of the component using the predicteddynamic stress, and uses the difference of the new and original lives to approximate the designsensitivity of the structural component life. In this approach, loading history is assumed to beindependent of design changes. Design Optimization Tool (DOT)6 is employed in this paper tosupport design optimization.

Figure 1. Computational flow of design optimization

1720 K.-H. CHANG, X. YU AND K. K. CHOI

INT. J. NUMER. METHODS ENG., VOL. 40: 1719—1743 (1997) ( 1997 by John Wiley & Sons, Ltd.

The rest of the paper is organized as follows. In Section 2, structural fatigue life predictionmethod with emphasis on dynamic stress computation is presented. The DSA method for thestructural fatigue life is described in Section 3. A 3-D tracked vehicle roadarm is presented inSection 4 to demonstrate the proposed DSA method. Design optimization of the tracked vehicleroadarm considering crack initiation life as constraints is discussed in Section 5. Conclusions aregiven in Section 6.

2. STRUCTURAL DURABILITY ANALYSIS

In structural durability analysis, structural fatigue lives, including crack initiation and crackpropagation, at critical points are calculated. The shortest life among these critical points isconsidered the fatigue life of the structural component. Structural fatigue life computationconsists of two parts, dynamic stress computation and fatigue life prediction. Dynamic stress canbe obtained either from experiment (mounting sensors or transducers on a physical component)or from simulation. Using simulation, a number of quasi-static FEA’s of the component ofinterest are performed first. The stress influence coefficients obtained from these quasi-staticFEA’s are then superposed with the dynamic analysis results, including external forces, acceler-ations, and angular velocities to compute dynamic stress history. Sanders and Tesar7 showed thatthe quasi-static deformation evaluation was a valid form of approximation for most industrialmechanisms that are stiff and operate substantially below their natural frequencies. Note that intheir work, they assumed that deformation caused by applied external and inertia forces aresmall, compared with the geometry of the structural component. It is further assumed that thematerial from which the component is fabricated behaves in a linear elastic fashion. In this paper,same assumptions are employed.

Multibody dynamic analysis methods, which have typically been used for dynamic motion analysis,can be used for dynamic load analysis of mechanical systems,5 e.g., an nb body system connected byjoints shown in Figure 2. In this paper, all bodies of the dynamic model are assumed to be rigid. If theflexibility of bodies is large, such as the hull of a tracked vehicle, a flexible body dynamic model8 mustbe employed. For suspension components of a vehicle, the rigid-body assumption usually yieldsreasonably accurate analysis results to support structural design for durability.

The finite element model of the component of interest corresponds to a body in the multibodydynamic model. Also, it is convenient to create the finite element model on the body reference

Figure 2. A multibody mechanical system

SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION 1721

INT. J. NUMER. METHODS ENG., VOL. 40: 1719—1743 (1997)( 1997 by John Wiley & Sons, Ltd.

frame x@1i!x@

2i!x@

3iso that loading, accelerations, and velocities generated from dynamic

analysis can be applied to the structural finite element model directly.Since dynamic stress histories contain very large amounts of data, it is generally necessary to

reduce or condense the amount of data by, e.g., peak-valley editing, before performing crackinitiation and propagation life computation.9 These values are then used to perform a cyclecounting procedure to transform variable amplitude stress or strain histories into a number ofconstant amplitude stress or strain histories. These histories are used to compute crack initiationlife of the component. In this paper, a multi-axial fatigue model using von Mises equivalent strainfailure criteria is employed.4,9

The edited dynamic stress histories (without cycle counting) at the critical point can also beused for crack propagation life prediction. In this work, NASA/FLAGRO17 is employed tosupport crack propagation life computation. FLAGRO takes edited dynamic stress historiesas inputs to compute stress intensity factors, and then uses the stress intensity factors to calculatecrack propagation life using approximation and empirical equations. The computationprocess for crack initiation and propagation lives is illustrated in Figure 3. In this section, onlydynamic stress computation, including quasi-static loading for both external and inertia forces,is discussed. Computational methods of other components in life prediction can be foundin References 9—17.

2.1. Dynamic stress computation

For a component subjected to external forces ( joint reaction forces and torques) and inertiaforces obtained from multibody dynamic analysis, the quasi-static equation in a matrix form ofthe finite element method can be written as follows:

Kz"F%(t)!F

*(t) (1)

where K is the stiffness matrix, z is a vector of nodal displacements, and F%(t) and F

*(t) are vectors of

external and inertia force histories, respectively, obtained from dynamic analysis. Since the loadingcondition can vary with time in a dynamic system, dynamic stress can be calculated as follows:

r (t)"DBK~1(F%(t)!F

*(t) ) (2)

where D is the elasticity matrix, and B is the strain—displacement matrix.

Figure 3. Computation process for fatigue life



The quasi-static method separates the external forces and inertia forces acting on the compon-ent as two parts: time-dependent (external and inertia force histories) and time-independent(quasi-static loading), and treats the quasi-static loading as static forces. The stress influencecoefficients are obtained by performing FEA for each quasi-static loading separately. Thedynamic stresses can be calculated by using the superposition principle, i.e. external and inertialforce histories are multiplied by the corresponding stress influence coefficients.

2.2. Quasi-static loading for external forces

A set of unit loads is used to calculate the stress influence coefficients corresponding to jointreaction forces and torques. The unit loads are applied at a given point x in all degrees-of-freedomwhere joint reaction forces and torques act. For example, if a set of joint reaction forces andtorques acts on the kth finite element node, the corresponding quasi-static loads qk are three unitforces and three unit torques in the body reference frame of the jth body x@

1j!x@

2j!x@

3japplied to

the kth node as six loading cases. Therefore, the stress influence coefficients rkSIC

can be obtainedusing FEA,

rkSIC

"DBK~1qk (3)

2.3. Quasi-static loading for inertia forces

The inertia body force applied to a point x in the component due to accelerations, angularvelocities and angular accelerations, as shown Figure 4, can be expressed as18,19

fi(x)"f a

i(x)#f r

i(x)#f t

i(x)

"!o (x)ai!o (x)ar

i#o (x)at

i(4)

where o (x) is mass density; fi(x) is the x@

i-component of the inertia body force per unit mass; f a

i(x),

f ri(x) and f t

i(x) are inertia body forces per unit mass in the translational, radial and tangential

directions, respectively; aiis the instantaneous translational acceleration and is independent of

the location of point x; ariis the centripetal acceleration toward the instantaneous axis of the

rotation and is perpendicular to it; and atiis the tangential acceleration.

The radial and tangential accelerations ari(x) and at

i(x) at point x can be written as

ari(x)"u

iju

jkxk

(5)

Figure 4. Inertia forces applied to a component



and

ati(x)"a

ijxj

(6)

where xkis the kth-co-ordinate of point x, u

ijis the instantaneous angular velocity, and a

ijis the

instantaneous angular acceleration. Hence, the inertia body force at point x is

fi(x)"o (x) (!a

i!a

ijxj#u

iju

jkxk) (7)

Equation (7) can be expressed in a matrix form as

f1(x)

f2(x)

f3(x)

"o (x)

!

a1

a2

a3

!

0 a3

!a2

!a3

0 a1

a2

!a1

0

x1

x2

x3

#

!u22!u2

3u

1u

2u

1u

3u

1u

2!u2

1!u2

3u

2u

3u

1u

3u

2u

3!u2

2!u2

1

x1

x2

x3

"o (x)

!

a1

a2

a3

!

0 !x3

x2

x3

0 !x1

!x2

x1

0

a1

a2

a3

#

x2

x3

0 !x1

0 0

x1

0 x3

0 !x2

0

0 x1

x2

0 0 !x3

u1u

2u

1u

3u

2u

3u2

2#u2

3u2

1#u2

3u2

1#u2

2

(8)

It can be seen from equation (8) that the inertia force, f (x), is linearly dependent on components ofthe acceleration a and the angular acceleration a. However, the inertia force is not linearlydependent on components of the angular velocities x. Instead, it depends linearly on thecombinations of components of the angular velocities x, such as u

1u

2. Therefore, to compute the

quasi-static loading of inertia forces, the loading cases listed in Table I are assumed.A consistent mass matrix for 3-D isoparametric finite elements can be obtained as

mij"PPP)

e

oNiN

jd)

e(9)

where mij

is an entry of element mass matrix, )eis the finite element domain, and N

iis an element

shape function. From the principle of virtual work,20 the load linear form due to inertia forces for



Table I. Quasi-static loading cases for inertial forces

AngularAcceleration acceleration Angular velocity

Loadingcase a

1a2

a3

a1

a2

a3

u1u

2u

2u

3u

1u

3u2

2#u2

3u2

1#u2

3u2

1#u2

2

1 1·0 0 0 0 0 0 0 0 0 0 0 02 0 1·0 0 0 0 0 0 0 0 0 0 03 0 0 1·0 0 0 0 0 0 0 0 0 04 0 0 0 1·0 0 0 0 0 0 0 0 05 0 0 0 0 1·0 0 0 0 0 0 0 06 0 0 0 0 0 1·0 0 0 0 0 0 07 0 0 0 0 0 0 1·0 0 0 0 0 08 0 0 0 0 0 0 0 1·0 0 0 0 09 0 0 0 0 0 0 0 0 1·0 0 0 0

10 0 0 0 0 0 0 0 0 0 1·0 0 011 0 0 0 0 0 0 0 0 0 0 1·0 012 0 0 0 0 0 0 0 0 0 0 0 1·0

an element with consistent mass matrix can be written as

l)(z6 )"PPP)e

fi(x)zN

id)

e

"PPP)e

o (x)(!ai!a

ijxj#u

iku

kjxj)zN

id)

e

"PPP)e

o (x)(!ailNl#(!a

ij#u

iku

kj)Nlxnlj

)(NmzN nmi

) d)e

"o (!ail#(!a

ij#u

iku

kj)xnlj

) PPP)e

NlNmd)

ezN nmi

"(!ail#(!a

ij#u

iku

kj)xn

jl)mlm

z6 nmi

(10)

where xnljis the location of the element’s lth node in the x

j-direction, zN n

miis the virtual

displacement of the element’s mth node in the xi-direction, and o is assumed to be constant. For

a finite element with diagonalized mass matrix, such as ANSYS,21 MSC/NASTRAN,22 andABAQUS,23 equation (10) can be written as

l)(z6 )"(!ail#(!a

ij#u

iku

kj)xn

jl)mDllz6 nmi

(11)

where mDll"mllS/D, and where S"+ijm

ijand D"+

im

ii. Note that equation (11) can be

extended to support finite elements with lumped mass matrix as well.18 In this work, diagonalizedmass matrix is assumed. Thus, the load vector of a finite element is

q*/%ei

"(!ail#(!a

ij#u

iku

kj)xn

jl)mDll (12)

Substituting the 12 cases listed in Table I to equation (11), the load linear form correspondingto element quasi-static loading can be obtained as listed in Table II. Note that the stress influencecoefficients of the first six quasi-static loads can be obtained by applying unit accelerations(instead of evaluating equation (12)) to perform FEA directly, using established FEA codes,such as ANSYS. However, equation (12) must be evaluated to obtain equivalent nodal forces



Table II. Load linear form for quasi-static loads corres-ponding to inertia forces

Loading Quasi-staticcase load l)(z6 )

1 a1"1 !mDllzN nl1

2 a2"1 !mDllzN nl2

3 a3"1 !mDllzN nl3

4 a1"1 xn

3lmDllzN nl2!xn

2lmDllzN nl3

5 a2"1 xn

3lmDllzN nl1!xn

1lmDllzN nl3

6 a3"1 !xn

2lmDllzN nl1#xn

1lmDllzN nl2

7 u1u

2"1 xn

3lmDllzN nl2#xn

2lmDllzN nl3

8 u2u

3"1 !xn

3lmDllzN nl1!xn

1lmDllzN nl3

9 u1u

3"1 !xn

2lmDllzN nl1!xn

1lmDllzN nl2

10 u22#u2

3"1 xn

1lmDllzN nl1

11 u21#u2

3"1 xn

2lmDllzN nl2

12 u21#u2

2"1 xn

3lmDllzN nl3

corresponding to the last six quasi-static loads involving angular velocities, which can be appliedto the finite element model as external nodal forces. The stress influence coefficients r */%

SICdue to

inertia forces can be obtained using FEA,

r*/%SICl

"DBK~1q*/%l , l"1, . . . , 12 (13)

where q*/%l "[q*/%i

]l , and q*/%i

is the summation of equation (12) over all finite elements in thestructure.

2.4. Dynamic stress

The dynamic stress is calculated using the superposition principle as

r (t)"r*/% (t)#r%95 (t) (14)

where

r*/% (t)"3+l/1

r*/%SICl

al (t)#3+l/1

r*/%SIC(l`3)

al (t)

#r*/%SIC7

u1(t)u

2(t)#r*/%

SIC8u

2(t)u

3(t)#r*/%

SIC9u

1(t)u

3(t)

#r*/%SIC10

(u22(t)#u2

3(t))#r*/%

SIC11(u2

1(t)#u2

3(t) )

#r*/%SIC12

(u21(t)#u2

2(t)) (15)

in which r*/%SICl

is obtained from equation (13), and

r%95(t)"n+k/1

rkSIC

Fk (t) (16)

where rkSIC

can be obtained from equation (3), and n is the number of nodes that external forcesFk (t) are applied.



Since most cracks are initiated at the structural surface, and stress computed using displace-ment-based FEA at the surface are usually less accurate, a stress smoothing technique that usesthe least-squares method24 is employed in this work to improve the accuracy of stress at thesurface.

3. DESIGN-SENSITIVITY ANALYSIS FOR DURABILITY

The proposed DSA method for structural fatigue life uses analytical dynamic stress DSA topredict dynamic stress history due to design changes, computes life of the component at the samecritical point using the predicted dynamic stress, and uses the difference of the newlife and the original life to approximate the sensitivity of structural component life. The computa-tion procedure is illustrated in Figure 5. Even though life is not a continuous function of designparameters due to the peak-valley editing and rain-flow counting algorithms embedded in itscomputation as discussed in References 9 and 13, it behaves like a continuous function. It isbecause a large number of dynamic stresses are counted to compute the fatigue life of structuralcomponents, a small design change only affects a small portion of the dynamic stress set includedin the computation. As a result, the affected stress set is usually too small to make thediscontinuity behaviour of the life function recognized numerically.

In this section, a general analytical DSA method for stress performance measures is presentedfirst. To support DSA of stress influence coefficients, variation of load linear forms of quasi-staticloads is derived in Section 3.2. A finite difference approach for fatigue life DSA computation isdescribed in Section 3.3.

3.1. Continuum shape design sensitivity analysis for stress measures

In continuum shape DSA, parameters that determine geometric shape of the structuraldomain are treated as the design. The relationship between shape variation of a continuousdomain and the resulting variation in structural performance measures can be described using thematerial derivative idea in continuum mechanics.25 A general shape design sensitivity expressionand design velocity field is introduced first. The DSA expression is then applied to 3-D solids withinertia forces. The direct differentiation method of shape DSA is used in this paper. For designsensitivity expressions using the adjoint variable method, see Reference 25.

Figure 5. Computation flow for durability DSA



3.1.1. Design velocity field. Considering the structural domain as a continuous medium, andthe process of changing the shape of domain ) to )q in Figure 6 as a dynamic process thatdeforms the continuum with q playing the role of time, a transformation mapping T thatrepresents this process can be defined as25

T : xPx (x), x3)q (17)

with

xq,T (x, q)

)q,T (), q) (18)

!q,T (!, q)

Suppose that a material point x3) in the initial domain at q"0 moves to a new locationxq3)q in the perturbed domain. Then, a design velocity field V can be defined as

V(xq , q),dxqdq

"

dT(x, q)dq

"

LT(x, q)Lq

(19)

In the neighbourhood of initial time q"0, assuming a regularity hypothesis and ignoringhigher-order terms, T can be approximated by

T (x, q)"T (x, 0)#qLT(x, 0)

Lq#O(q2)

+x#qV(x, 0) (20)

where x,T (x, 0) and V(x),V(x, 0).

3.1.2. Continuum shape design sensitivity analysis. A variational governing equation fora structural component with the domain ) can be written as

a) (z, z6 )"l) (z6 ) , for all z6 3Z (21)

where z and z6 are the displacement and virtual displacement fields of the structure, respectively;Z is the space of kinematically admissible virtual displacements; and a) (z, z6 ) and l) (z6 ) are theenergy bilinear and load linear forms, respectively. The subscript ) in equation (21) is used toindicate the dependency of the governing equation on geometric shape of the structural domain.

A general performance measure that depends on the displacement and stress can be written inan integral form as

t"PP)g (z,+z) d) (22)

Figure 6. Deformation process



Using the direct differentiation method, the first variation of the performance measure t can bewritten as25

t@"PP)[g,

zizRi#g,

zi,jzRi,j!g,

zi(z

i, j»j)!g,

zi,j(z

i,j»j), j

] d)#P!g (»

ini) d! (23)

where z5 is the solution of the sensitivity equation obtained by taking the material derivative ofequation (21), i.e.

a)(z5 , z6 )"l@V(z6 )!a@

V(z, z6 ) , for all z6 3Z (24)

The subscript » on the right-hand side of equation (24) is used to indicate the dependency of theterms on the design velocity field.

For 3-D elastic solids, the variational equation (21) can be written as

a) (z, z6 ),PPP)pij(z)e

ij(z6 ) d)

"PPP)fizNid)#PP!2

¹izNid!#q

izNi,l) (z6 ) , for all z6 3Z (25)

where pij(z) and e

ij(z6 ) are the stress and strain tensors of the displacement z and virtual

displacement z6 , respectively; fi

is the xi-component of the structural inertia force; ¹

iis the

xi-component of the surface traction force; and q

iis point forces applied to the structure.

In equation (24) l@V(z6 ) and a@

V(z, z6 ) can be derived for 3-D structural components as

l@V(z6 )"PPP)

[ f @izNi#zN

i( f

i, j»j)#[ f

izNidiv V] d)

#PP!2

M!¹i(z

i,j»j)#[(¹

izNi), j

nj#H (¹

izNi)](»

ini)Nd!#q@

izNi

(26)

and

a@V(z, z6 )"PPP)

M[pij(z) (zN

i,k»k, j

)#pij(z6 )(z

i,k»k, j

)]

![pij(z)e

ij(z6 )] divVNd) (27)

where divV and V,jare the divergence and derivative of the design velocity field with respect to

xj; n and H are the unit normal vector and curvature of the boundary, respectively; and !2 is the

boundary where the traction force is applied.To evaluate the design sensitivity expression of equation (23) using the finite element analysis

results, a finite element matrix equation corresponding to equation (24) must be solved forz5 m-times, where m is the number of load cases multiplies with the number of design parameters.If the finite element matrix equation that corresponds to the variational equation of equation (25)is used to find the original response z, the solution z5 of equation (24) can be obtained efficientlysince it requires only evaluation of the solution of the same set of finite element matrix equationswith different fictitious loads.

A pointwise stress performance measure, such as stress at a Gauss point, can be defined as

t"PPP)g(r (z))dK (x!x̂) d) (28)



where g(r (z)) is a stress measure, such as von Mises stress, dK (x!x̂) is a direct delta function, andx̂ is the location where the stress is measured.

The design sensitivity expression of the stress performance measure defined in equation (28) canbe obtained as

t@"PPP)[g,pij

(z)pij(z5 )dK (x!x̂) d)

!PPP)g,pij(z)Cijkl

(zk,m

»m,l

)dK (x!x̂) d) (29)

where pij(z5 ) can be evaluated using z5 obtained from equation (24), C

ijklis the elastic modulus

tensor that satisfies symmetry relations Cijkl

"Cklij

and Cijkl

"Cjilk

, zk,m

is the gradient of zk,

and g,pijis derivative of the stress function g with respect to the stress components p

ij(z). Effort

required to evaluate equation (29) is minimal once z5 is obtained.Note that each integrand in equations (26) and (27) contains either the design velocity field V or

the derivative of the design velocity field V,j. The design velocity field computation depends on

the shape design parameters defined for the structural geometric model. Proper generation ofdesign velocity fields is an important step in obtaining accurate shape design sensitivity informa-tion. The design parameterization method and numerical method of velocity field computationfor 3-D structural components are discussed in References 26 and 27.

3.2. »ariation of load linear form

With no traction force at the design boundary, a variation of the load linear form of equation(26) can be written as25

l@V(z6 )"PPP)

[ f @izNi#zN

i( f

i,j»j)#[ f

izNidiv»] d)#q@

izNi

(30)

where

f @i,lim

q?0

fiq(x)!f

i(x)

q"0 (31)

since fiq(x)"f

i(x) due to the fact that inertia force evaluated at a fixed material point x before and

after design changes is constant. Note that a variation of qizNiis zero since q

i(corresponding to

a joint reaction force) is assumed to be independent of design changes. Therefore, q@i"0 for

quasi-static loads corresponding to joint reaction forces. For the variation of quasi-static loadlinear form corresponding to inertia forces, the second and third integrands of equation (30) mustbe extended, using equation (23) as

fi,j»j"Mo[!a

i#(!a

il#u

iku

kl)xl]N,

j»j

"o (!ail#u

iku

kl)xl,j

»j

"o (!aij#u

iku

kj)»

j(32)

and

fizNidivV"o[!a

i#(!a

ij#u

iku

kj)x

j]zN

idivV (33)



where the assumption o is constant in ) and the fact xl, j"dlj have been used. Therefore,

equation (30) can be rewritten as

l@V(z6 )"o PPP)

[!aidivV#(!a

ij#u

iku

kj)(»

j#x

jdivV)]zN

id) (34)

For an element with a consistent mass matrix, equation (34) can be written as

l@V(z6 )"(!a

ij#u

iku

kj) (» n

jlmlm

#oxnjlPPP)

NlNmdivV d))zN n

mi!a

io PPP)

NlNmdivVd) zN n

mi

(35)

For an element with a diagonalized mass matrix, equation (34) can be written as

l@V(z6 )"[(!a

ij#u

iku

kj)(» n

jlmlm

#xnjlmR Dll)!a

imR Dll]zN nli (36)

where

mR Dll"mR llS

D#mll

SQD!mll

S

D2DQ (37)

and SQ "+ijmR

ij, DQ "+

imR

ii, and mR

ij"o :::

)NlNm

divV d). Equation (35) can be used to supportfinite elements with diagonalized or lumped mass matrix.18

Using equation (36) variations of the load linear form for the 12 quasi-static loading cases thatcorrespond to inertia forces can be summarized as shown in Table III.

3.3. Design sensitivity of fatigue life

A finite difference approach is used in this paper to compute the design sensitivity of thecomponent fatigue life. An analytical approach for fatigue life DSA is not possible since thefatigue life cannot be obtained as a continuous function of design parameters.

Table III. Variation of load linear form corresponding to inertia forces

Loading Quasi-staticcase load l@

V(z6 )

1 a1"1 !mR Dllz6 nl1



4 a1"1 (» n

3lmDll#xn

3lmR Dll)z6 nl2!(» n

2lmDll#xn

2lmR Dll)z6 nl3

5 a2"1 (» n

3lmDll#xn


1lmDll#xn

1lmR Dll)z6 nl3

6 a3"1 !(» n

2lmDll#xn

2lmR Dll)z6 nl1#(» n

1lmDll#xn

1lmR Dll)z6 nl2

7 u1u

2"1 (» n

3lmDll#xn

3lmR Dll)z6 nl2#(» n

2lmDll#xn

2lmR Dll)z6 nl3

8 u2u

3"1 !(» n

3lmDll#xn


1lmDll#xn

1lmR Dll)z6 nl3

9 u1u

3"1 !(» n

2lmDll#xn


1lmDll#xn

1lmR Dll)z6 nl2

10 u22#u2

3"1 (» n

1lmDll#xn

1lmR Dll)z6 nl1

11 u21#u2

3"1 (» n

2lmDll#xn

2lmR Dll)z6 nl2

12 u21#u2

2"1 (» n

3lmDll#xn

3lmR Dll)z6 nl3



Once the sensitivity of stress influence coefficients are obtained using the DSA methoddescribed in Section 3.2, increment of stress influence coefficients can be obtained by

drSIC

"

LrSIC

Lbj

dbj

(38)

where dbjis the perturbation of the jth design parameter. Note that the design perturbation db

jmust be small for linear approximation of the life. On the other hand, in numerical calculation,db

jcannot be too small since it introduces numerical noise.

The stress influence coefficients of the perturbed design can be approximated by

rSIC

(b#dbj)+r

SIC(b)#dr

SIC(39)

A stress time history of the perturbed design can be obtained by superposing rSIC

(b#dbj) with

the loading history obtained from multibody dynamic analysis. Note that the design perturbationis assumed to be local so that dynamic behaviour of the mechanical system is not altered. The newdynamic stress history is then used to calculate the fatigue life of the component with a perturbeddesign, ¸ (b#db

j), using the same life prediction method. The design sensitivity coefficient of

component fatigue life with respect to the jth design parameter can be obtained from

L¸Lb

j

+

¸ (b#dbj)!¸(b)

dbj

(40)

Note that equations (38)—(40) must be evaluated repeatedly for all the design parameters.This computation is very efficient since design sensitivities of stress influence coefficients areavailable.

4. NUMERICAL EXAMPLE—A TRACKED VEHICLE ROADARM

A roadarm of the military tracked vehicle shown in Figure 7 is employed to demonstrate the DSAmethod for crack initiation life proposed in this paper. In this section, the multibody dynamicmodel of the tracked vehicle and its simulation environment are described first. The structuralfinite element model of the roadarm is presented in Section 4.2. Contours of crack initiation lifeand von Mises stress at the peak load of the simulation period are presented in Section 4.3. Designparameterization of the roadarm is discussed in Section 4.4. Design sensitivity analysis and resultverification are described in Section 4.5.

4.1. Dynamics model and simulation

A 17-body dynamics model shown in Figure 8 is generated to drive the tracked vehicle on theAberdeen Proving Ground 4 (APG4), as shown in Figure 9, at a constant speed of 20 miles perhour forward (positive X

2-direction). The Tracked Vehicle Design Workspace (TVWS)28 is used

to generate the dynamics model and to perform dynamic analysis.A 20 second dynamic simulation is performed at a maximum integration time step of 0·05

seconds. An output interval of 0·05 seconds is predefined for this analysis and a total of 400 sets ofresults are generated. The joint reaction forces applied at the wheel end of the roadarm,accelerations, angular velocities, and angular accelerations of the roadarm are obtained from theanalysis. A time history of joint reaction forces at the wheel end is shown in Figure 10.



Figure 7. A military tracked vehicle and roadarm: (a) a military tracked vehicle; (b) roadarm geometric model

Figure 8. The tracked vehicle dynamic model



Figure 9. Aberdeen Proving Ground 4: (a) Aberdeen Proving Ground; (b) APG4 Computer Model

4.2. Roadarm finite element model

Four beam elements, STIF4, and 310 20-node isoparametric finite elements, STIF95, ofANSYS are used for the roadarm finite element model. A number of rigid beams are created toconnect nodes at the inner surface of the two holes to end nodes of beam elements to simulate theroadwheel shaft and torsion bar, respectively. Displacement constraints are defined at the endnodes of the beam elements that simulate the torsion bar, and joint reaction forces and torquesare applied at the end node of the other beam element that simulates the shaft of the roadwheel, asshown in Figure 11. The roadarm is made of S4340 steel, with material properties Young’smodulus E"3·0]107 psi and Poisson’s ratio l"0·3. Note that the co-ordinate systems of thefinite element model is identical to the body reference frame of the roadarm in the tracked vehicledynamic model. Therefore, loading history generated from dynamic analysis can be used withouttransformation.



Figure 10. Joint reaction forces applied to the wheel end of the roadarm

Figure 11. Roadarm finite element model

4.3. Crack initiation life contour and stress contour at peak loads

Finite element analysis is first performed to obtain stress influence coefficients of the roadarmusing ANSYS by applying 18 quasi-static loads. Among the loads, the first six that correspond toexternal joint forces are three unit forces and three unit torques applied at the centre of theroadwheel, in x@

1, x@

2and x@

3directions, and the remaining 12 that correspond to inertia forces are

unit accelerations, unit angular accelerations, and unit combinations of angular velocities, aslisted in Table I. The stress influence coefficients obtained from analyses are six componentstresses at finite element nodes in the x@

1!x@

2!x@

3co-ordinates.

Dynamic stresses at finite element nodes are then calculated by superposing stress influencecoefficients with their corresponding external forces and accelerations and velocities in timedomain. To compute multiaxial crack initiation life of the roadarm, the equivalent von Misesstrain approach10 is employed. The fatigue life contour is given in Figure 12. Note that thespectrum in Figure 12 is the number of blocks to initiate crack in logarithm.



Figure 12. Contour of crack initiation life

A static stress contour shown in Figure 13 is given to demonstrate that the worst-case scenarioemployed for durability design using stresses as performance measures is problematic. Stresscontour shown in Figure 13 is obtained by applying the peak load found at 17·35 s of the 20 ssimulation, including six joint reaction forces at the roadwheel end, accelerations, angularaccelerations, and angular velocities of the roadarm. Note that from Figures 12 and 13, the stressconcentration area identified as the worst case does not conform with the critical areas where thecrack is first initiated.

4.4. Design parameterization

For shape design parameterization, eight design parameters are defined to characterize shapesof the four cross-sections shown in Figure 7(b). Contour of the cross-sectional shape is composedof four straight lines and four cubic curves. Side expansions (x@

2-direction) of cross-sectional

shapes are defined using design parameters b1, b3, b5, and b7 for intersections 1—4, respectively.Vertical expansions (x@

3-direction) of the cross-sectional shapes are defined using the remaining

four design parameters, as shown in Figure 14.

4.5. Design sensitivity analysis and result verification

Design sensitivity coefficients of stress influence coefficients with respect to eight designparameters are computed using the direct differentiation method of continuum DSA. Velocityfields of the roadarm are generated using the isoparametric mapping method.26,27

Selected sensitivity coefficients are verified using finite difference results with 0·01 in perturba-tion of design parameter b

2. In Table IV, t(b#db) and t (b) are crack initiation lives at the



Figure 13. Contour of static von Mises stress at 17·35 s

Figure 14. Cross-sectional shape and design parameter definitions

Table IV. Verification of design sensitivity coefficients of fatigue life

Life t(b#db) t(b) dt t@ t@/dt% dt/t(b)%

node 1544 8·9084296E#7 8·9261936E#7 !1·7764E#5 !1·7764E#5 100·00 !0·20node 1519 1·4521717E#8 1·4468427E#8 5·3290E#6 5·3290E#6 100·00 3·68node 1433 2·7728931E#8 2·7622349E#8 1·0658E#5 1·0658E#6 100·00 0·04node 1340 4·8991923E#8 4·9062976E#8 !7·1053E#5 !7·1054E#5 100·00 !0·14node 1227 6·6329165E#8 6·6187059E#8 1·4211E#6 1·4211E#6 100·00 0·21node 918 1·0039969E#9 9·5710106E#8 4·6895E#7 4·6896E#7 100·00 4·90node 922 1·0622614E#9 9·9831251E#8 6·3949E#7 6·3949E#7 100·00 6·41node 1391 1·5882051E#10 1·5609203E#10 2·7284E#8 2·7284E#8 100·00 1·75



Table V. CPU seconds spent for crack initia-tion life computation

Tasks CPUs

Quasi-static FEA 7080Dynamic stress 2Life computation 2Total 7084

Table VI. CPU seconds spent for sensitivityanalysis of crack initiation life for eight design

parameters

Tasks CPUs

Dynamic stress DSA 4280Life DSA 24Total 4304

perturbed and initial designs obtained using the life prediction method discussed in Section 2;dt are the finite difference results, i.e. t (b#db)!t (b); t@ are the predicted structural responsesusing sensitivity coefficients, i.e. Lt/Lb]db; and t@/dt is the accuracy measurement of sensitivitycoefficients; and dt/t (b) is an index that indicates validity of the finite difference results.Under the t@/dt column, a value that is closer to 100 per cent indicates that the sensitivityprediction is more accurate. A small value in the last column suggests that numerical noise maycontribute to an inaccurate finite difference verification, and a large value indicates that non-linear effect from finite difference results might destroy the sensitivity accuracy. Table IV showsthat the sensitivity coefficients of crack initiation lives at nodes randomly selected are veryaccurate.

CPU time spent for the crack initiation life and sensitivity analysis of crack initiation life onan HP 9000/755 machine are summarized in Tables V and VI, respectively. It is shown inTable V that the bulk computation of the crack initiation life is in quasi-static FEA. If the overallfinite difference approach is used for sensitivity computation, i.e. perturbing each design para-meter, creating finite element model of the perturbed design, and performing the entire computa-tion, 56 672 (7084]8 design parameters) CPU seconds are needed for sensitivity computation, inaddition to human effort spent for finite element modelling. The proposed DSA method needsonly 4304 CPUs (Table VI) to perform the sensitivity computation which is much more efficientthan the overall finite difference approach.

5. DESIGN OPTIMIZATION

Shape of the roadarm is optimized and presented in this section. Section 5.1 describes thedefinition of the roadarm design model. Section 5.2 explains how the design optimization isperformed using ANSYS, DRAW,9 Design Optimization Tool (DOT),6 and sensitivity analysisand design model update programs in the Design Sensitivity Analysis and Optimization (DSO)tool.29 In Section 5.3, design optimization results are presented.



5.1. Roadarm design model definition

The objective of the roadarm design is to minimize its volume and keep crack initiation lives atthe same level as the initial design. The objective is achieved by changing shapes of the fourcross-sections, which is characterized by the eight design parameters defined in Section 4.

At the initial design, the structural volume is 486·7 in3. The crack initiation lives at 24 criticalnodes (with ids shown in Figure 15) are defined as constraints with a lower bound of9·63]106 blocks. Note that the lower bound defined is equivalent to 20 year service life, assumingthe tracked vehicle is operating 8 hours per day, five days per week. Definition of the cost functionand five selected constraint functions are listed in Table VII.

5.2. Integration of design optimization codes

Design optimization of the roadarm is performed using ANSYS, DRAW, the ModifiedFeasible Direction method in DOT, together with the life DSA and design model update methodsprovided in DSO. As shown in Figure 16, DOT is treated as a subroutine in the integratedoptimization module. When DOT provides a new design or carries out a line search, the mainprogram of the optimization module OPTMAIN calls the UPMODEL subroutine to update thedesign model corresponding to the new design, and sends the design model to DRAW (ANSYSfor FEA) for fatigue life prediction. When the gradient information is requested by DOT, the lifeDSA module is executed to compute sensitivity coefficients of the cost function and e-activeconstraints. The process is repeated until an optimum design is obtained or the maximumiteration number is reached.

Figure 15. Node ids of fatigue life constraints

Table VII. Cost and selected constraint function definitions

Function Description Lower bound Current design Status

Cost Volume 487·678 in3Constraint 1 Life at node 1216 9·63E#6 (20 y) 9·631E#6 blocks ActiveConstraint 2 Life at node 926 9·63E#6 (20 y) 8·309E#7 blocks InactiveConstraint 3 Life at node 1544 9·63E#6 (20 y) 8·926E#7 blocks InactiveConstraint 4 Life at node 1519 9·63E#6 (20 y) 1·447E#8 blocks InactiveConstraint 5 Life at node 1433 9·63E#6 (20 y) 2·726E#8 blocks Inactive



5.3. Roadarm design optimization results

An optimum design of the roadarm is obtained in six iterations, with 22 crack initiation lifecomputations and six fatigue life DSA computations. The isoparametric mapping method isemployed to compute the design velocity field in the first iteration, which is used during allsubsequent design iterations for life DSA computation. Moreover, the design velocity field is usedto update the finite element mesh for the new design, i.e.

xb`dbi

"xbi#dx

i"xb

i#

n+k/1

» kidbk (41)

where xb`dbi

and xbi

are the locations of the nodes of the perturbed and the current designs,respectively; dx

iis the nodal point movement due to design changes; » k

iand dbk are the velocity field

and perturbation of the kth design parameter, respectively; and n is the number of design parameters.The optimization histories for the cost and constraints are shown in Figure 17. Figure 17(a)

shows that the cost function starts from 487·7 in3 and reduces to 436·7 in3. The convergence

Figure 16. Integrated design optimization codes

Figure 17. Cost and constraint function history (selected): (a) cost; (b) constraints



criterion is a 0·1 per cent relative change in cost function in two consecutive iterations. The costfunction value reduces quickly in the first four iterations, and an optimum design is reached at thesixth iteration. At the optimum design, all lives are above the lower bound and the history of fiveselected normalized constraints is given in Figure 17(b). Also, the cost and the five selectedconstraint function values at the initial and optimum designs are listed in Table VIII. AsFigure 17(b) and Table VIII shows, life at node 1216 (constraint 1) reaches initially the lowerbound and increases to 7·705]107 at the optimum design. The shortest life at the optimum is9·63]106, found at node 926 (constraint 2). Through optimization, the structural volume ofroadarm reduces 10·45 per cent while its fatigue life is kept at the same level as the initial design.

Table VIII. Cost and selected constraint function values at initial and optimum designs

Function Description Initial design Optimum design % changes

Cost Volume 487·678 in3 436·722 in3 !10·5Constraint 1 Life at node 1216 9·631E#6 blocks 7·704E#7 blocks 699·9Constraint 2 Life at node 926 8·309E#7 blocks 9·631E#6 blocks !88·4Constraint 3 Life at node 1544 8·926E#7 blocks 9·678E#6 blocks !89·2Constraint 4 Life at node 1519 1·447E#8 blocks 4·698E#7 blocks !67·5Constraint 5 Life at node 1433 2·726E#8 blocks 4·815E#8 blocks 74·3

Figure 18. Shapes of roadarm at initial and optimum designs: (a) initial design; (b) optimum design

Table IX. Design parameter values at initial and optimum designs

Initial OptimumDes. param. Description design (in) design (in) % changes

b1 Width of Cross Section 1 3·250 2·889 !11·09b2 Height of Cross Section 1 1·968 1·583 !19·56b3 Width of Cross Section 2 3·170 2·911 !8·17b4 Height of Cross Section 2 1·968 1·637 !16·78b5 Width of Cross Section 3 3·170 2·870 !9·45b6 Height of Cross Section 3 2·635 2·420 !8·17b7 Width of Cross Section 4 3·170 2·801 !11·65b8 Height of Cross Section 4 5·057 4·700 !7·06



Figure 18 shows the shape changes of roadarm. Note that the roadarm becomes thinner andnarrower at intersections 1, 2 and 4 at the optimum design. Design parameter values at initial andoptimum designs are listed in Table IX.

6. CONCLUSIONS

In this paper, a design sensitivity analysis (DSA) method for fatigue life of 3-D elastic solidstructural components of mechanical systems with respect to shape design parameters has beenpresented. The proposed DSA method has been demonstrated to be accurate and efficient usinga tracked vehicle roadarm example. Also, this method has been applied to support designoptimization of the tracked vehicle roadarm considering crack initiation life design constraints.The DSA method proposed is applicable to structural components with less flexibility, such assuspension or engine parts. Also, design changes for the structural components is assumed to besmall so that dynamic behavior of mechanical systems before and after design changes are notaltered.

This work is being extended to support durability design of vehicle body or hull with largeflexibility, in which, flexible body behavior of the mechanical systems must be considered indynamic analysis. Design sensitivity analysis of stress induced by additional inertia forces due toconsideration of flexibility of dynamic systems are much more challenging.

ACKNOWLEDGEMENTS

The authors would like to express their appreciation to Drs. Jia-Yi Wang and Jun Tang for theirvaluable input and feedback of this paper. Also, thanks to Mr. Jack Standefer for creatinga dynamic model schematic drawing of the tracked vehicle.

Research is supported by the Automotive Research Center sponsored by the U.S. Army Tankand Automotive Command Center.

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