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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL.
40, 1719—1743 (1997)
SHAPE DESIGN SENSITIVITY ANALYSIS AND
OPTIMIZATION 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 of
mecanical systems with respect to shape design parameters is presented. The DSA method uses dynamic
stress DSA obtained using an analytical approach to predict dynamic stress increment due to design
changes; computes fatigue life of the component, including crack initiation and crack propagation, using the
predicted dynamic stress; and uses the difference of the new life and the original life at the same critical point
to approximate the sensitivity of fatigue life. A tracked vehicle roadarm is presented in this paper to
demonstrate accuracy and efficiency of the DSA method. Also, this method is applied to support design
optimization of the tracked vehicle roadarm considering crack initiation lives as design constraints. ( 1997
by 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 systems
is 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 for
durability designs.1,2 The reason is that stress fluctuation occurring in structural components
during the service life of mechanical systems that contribute to fatigue is difficult to obtain. The
worst-case design with stress concentration factors criteria is usually employed to obtain an
optimum design for durability. In this case, stresses used for design constraints are determined by
applying a set of critical static loading to the structural component of interest at a specific time of
the service life of mechanical systems. The trouble is that wrong criteria may be selected to
determine the optimum design since high stress areas identified at a specific time of the service life
may not match critical areas where a crack first initiates. Recently, an optimum shape design for
a 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.
s Current address: Technical Staff, Engineering Department, CSAR Corporation, 28035 Dorothy Drive, Agoura Hills,
CA 91301, U.S.A.
t Professor 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
( 1997 by John Wiley & Sons, Ltd.
Received 12 February 1996
Revised 27 September 1996
1720
K.-H. CHANG, X. YU AND K. K. CHOI
difficult to treat the fatigue notch factor as a design constraint since the upper bound of the fatigue
notch factor is very hard to determine. Since it is widely recognized that about 80 per cent failure
of mechanical/structural components and systems are related to fatigue,4 a design optimization
methodology to increase durability life must be developed for dynamically loaded machine parts
and assemblies.
For ground vehicles and heavy equipment, crack initiation life is usually considered as the
failure criteria for durability design.4 For aircraft and offshore oil platforms, crack propagation
life is considered as the design criteria.4 Methods proposed in this paper are intended to support
durability 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 the
primary 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 to
support design optimization considering the structural fatigue life as the design criteria. The
computational flow of design optimization for fatigue life is shown in Figure 1.
To generate a representative load history, including inertia forces and external forces ( joint
reaction forces and torques), for accurate dynamic stress computation, multibody dynamic
analysis5 is performed for the mechanical system under a typical duty cycle. Quasi-static finite
element analyses (FEA) are then performed to obtain stress influence coefficients for the structural component. These stress influence coefficients are superposed with the external and inertia
loading 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 strainbased crack initiation life prediction method and the linear elastic fracture mechanics (LEFM)4
for crack propagation life.
The continuum DSA method is extended to compute the dynamic stress design sensitivity
analytically. The proposed method of durability DSA uses dynamic stress DSA to predict
dynamic stress history due to design changes, computes life of the component using the predicted
dynamic stress, and uses the difference of the new and original lives to approximate the design
sensitivity of the structural component life. In this approach, loading history is assumed to be
independent of design changes. Design Optimization Tool (DOT)6 is employed in this paper to
support design optimization.
Figure 1. Computational flow of design optimization
INT. J. NUMER. METHODS ENG., VOL. 40: 1719—1743 (1997)
( 1997 by John Wiley & Sons, Ltd.
SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION
1721
The rest of the paper is organized as follows. In Section 2, structural fatigue life prediction
method with emphasis on dynamic stress computation is presented. The DSA method for the
structural fatigue life is described in Section 3. A 3-D tracked vehicle roadarm is presented in
Section 4 to demonstrate the proposed DSA method. Design optimization of the tracked vehicle
roadarm considering crack initiation life as constraints is discussed in Section 5. Conclusions are
given in Section 6.
2. STRUCTURAL DURABILITY ANALYSIS
In structural durability analysis, structural fatigue lives, including crack initiation and crack
propagation, at critical points are calculated. The shortest life among these critical points is
considered the fatigue life of the structural component. Structural fatigue life computation
consists of two parts, dynamic stress computation and fatigue life prediction. Dynamic stress can
be 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 of
interest are performed first. The stress influence coefficients obtained from these quasi-static
FEA’s are then superposed with the dynamic analysis results, including external forces, accelerations, and angular velocities to compute dynamic stress history. Sanders and Tesar7 showed that
the quasi-static deformation evaluation was a valid form of approximation for most industrial
mechanisms that are stiff and operate substantially below their natural frequencies. Note that in
their work, they assumed that deformation caused by applied external and inertia forces are
small, compared with the geometry of the structural component. It is further assumed that the
material 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 by
joints shown in Figure 2. In this paper, all bodies of the dynamic model are assumed to be rigid. If the
flexibility of bodies is large, such as the hull of a tracked vehicle, a flexible body dynamic model8 must
be employed. For suspension components of a vehicle, the rigid-body assumption usually yields
reasonably accurate analysis results to support structural design for durability.
The finite element model of the component of interest corresponds to a body in the multibody
dynamic model. Also, it is convenient to create the finite element model on the body reference
Figure 2. A multibody mechanical system
( 1997 by John Wiley & Sons, Ltd.
INT. J. NUMER. METHODS ENG., VOL. 40: 1719—1743 (1997)
1722
K.-H. CHANG, X. YU AND K. K. CHOI
frame x@ !x@ !x@ so that loading, accelerations, and velocities generated from dynamic
1i
2i
3i
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 crack
initiation and propagation life computation.9 These values are then used to perform a cycle
counting procedure to transform variable amplitude stress or strain histories into a number of
constant amplitude stress or strain histories. These histories are used to compute crack initiation
life of the component. In this paper, a multi-axial fatigue model using von Mises equivalent strain
failure criteria is employed.4,9
The edited dynamic stress histories (without cycle counting) at the critical point can also be
used for crack propagation life prediction. In this work, NASA/FLAGRO17 is employed to
support crack propagation life computation. FLAGRO takes edited dynamic stress histories
as inputs to compute stress intensity factors, and then uses the stress intensity factors to calculate
crack propagation life using approximation and empirical equations. The computation
process for crack initiation and propagation lives is illustrated in Figure 3. In this section, only
dynamic stress computation, including quasi-static loading for both external and inertia forces,
is discussed. Computational methods of other components in life prediction can be found
in References 9—17.
2.1. Dynamic stress computation
For a component subjected to external forces ( joint reaction forces and torques) and inertia
forces obtained from multibody dynamic analysis, the quasi-static equation in a matrix form of
the 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 loading
condition can vary with time in a dynamic system, dynamic stress can be calculated as follows:
r(t)"DBK~1(F (t)!F (t))
%
*
where D is the elasticity matrix, and B is the strain—displacement matrix.
(2)
Figure 3. Computation process for fatigue life
INT. J. NUMER. METHODS ENG., VOL. 40: 1719—1743 (1997)
( 1997 by John Wiley & Sons, Ltd.
SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION
1723
The quasi-static method separates the external forces and inertia forces acting on the component 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 influence
coefficients are obtained by performing FEA for each quasi-static loading separately. The
dynamic stresses can be calculated by using the superposition principle, i.e. external and inertial
force 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 joint
reaction forces and torques. The unit loads are applied at a given point x in all degrees-of-freedom
where joint reaction forces and torques act. For example, if a set of joint reaction forces and
torques acts on the kth finite element node, the corresponding quasi-static loads qk are three unit
forces and three unit torques in the body reference frame of the jth body x@ !x@ !x@ applied to
1j
2j
3j
the kth node as six loading cases. Therefore, the stress influence coefficients rk can be obtained
SIC
using FEA,
rk "DBK~1qk
SIC
(3)
2.3. Quasi-static loading for inertia forces
The inertia body force applied to a point x in the component due to accelerations, angular
velocities and angular accelerations, as shown Figure 4, can be expressed as18,19
f (x)"f a (x)#f r (x)#f t (x)
i
i
i
i
(4)
"!o(x)a !o(x)ar#o(x)at
i
i
i
where o(x) is mass density; f (x) is the x@-component of the inertia body force per unit mass; f a (x),
i
i
i
f r (x) and f t (x) are inertia body forces per unit mass in the translational, radial and tangential
i
i
directions, respectively; a is the instantaneous translational acceleration and is independent of
i
the location of point x; ar is the centripetal acceleration toward the instantaneous axis of the
i
rotation and is perpendicular to it; and at is the tangential acceleration.
i
The radial and tangential accelerations ar (x) and at (x) at point x can be written as
i
i
(5)
ar (x)"u u x
i
ij jk k
Figure 4. Inertia forces applied to a component
( 1997 by John Wiley & Sons, Ltd.
INT. J. NUMER. METHODS ENG., VOL. 40: 1719—1743 (1997)
1724
K.-H. CHANG, X. YU AND K. K. CHOI
and
(6)
at (x)"a x
i
ij j
where x is the kth-co-ordinate of point x, u is the instantaneous angular velocity, and a is the
k
ij
ij
instantaneous angular acceleration. Hence, the inertia body force at point x is
f (x)"o(x) (!a !a x #u u x )
i
i
ij j
ij jk k
(7)
Equation (7) can be expressed in a matrix form as

f (x)
1

f (x) "o(x) !
2

f (x)

3
a
0
a
!a
1
3
2
a ! !a
0
a
2
3
1
a
a
!a
0
3
2
1
x
1
x
2
x
3
u u
u u
!u2!u2
3
1 2
1 3
2
u u
!u2!u2
# u u
3
2 3
1
1 2
u u
u u
!u2!u2
1
2
1 3
2 3




"o(x)  !




x
a
0
!x
x
1
3
2
a ! x
0
!x
2
3
1
a
!x
x
0
3
2
1
x
0 !x
0
0
3
1
0 x
0
!x
0
# x
1
3
2
0 x x
0
0
!x
1
2
3
2
x
1
x
2
x
3





a
1
a
2
a
3
u u
1 2
u u
1 3
u u
2 3
u2#u2
3
2
u2#u2
1
3
u2#u2
2
1









(8)
It can be seen from equation (8) that the inertia force, f (x), is linearly dependent on components of
the acceleration a and the angular acceleration a. However, the inertia force is not linearly
dependent on components of the angular velocities x. Instead, it depends linearly on the
combinations of components of the angular velocities x, such as u u . Therefore, to compute the
1 2
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
PPP
m "
ij
)
e
oN N d)
i j e
(9)
where m is an entry of element mass matrix, ) is the finite element domain, and N is an element
ij
e
i
shape function. From the principle of virtual work,20 the load linear form due to inertia forces for
INT. J. NUMER. METHODS ENG., VOL. 40: 1719—1743 (1997)
( 1997 by John Wiley & Sons, Ltd.
1725
SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION
Table I. Quasi-static loading cases for inertial forces
Acceleration
Loading
case
1
2
3
4
5
6
7
8
9
10
11
12
Angular
acceleration
Angular velocity
a
1
a
2
a
3
a
1
a
2
a
3
u u
1 2
u u
2 3
u u
1 3
u2#u2
2
3
1·0
0
0
0
0
0
0
0
0
0
0
0
0
1·0
0
0
0
0
0
0
0
0
0
0
0
0
1·0
0
0
0
0
0
0
0
0
0
0
0
0
1·0
0
0
0
0
0
0
0
0
0
0
0
0
1·0
0
0
0
0
0
0
0
0
0
0
0
0
1·0
0
0
0
0
0
0
0
0
0
0
0
0
1·0
0
0
0
0
0
0
0
0
0
0
0
0
1·0
0
0
0
0
0
0
0
0
0
0
0
0
1·0
0
0
0
0
0
0
0
0
0
0
0
0
1·0
0
0
u2#u2 u2#u2
1
3 1
2
0
0
0
0
0
0
0
0
0
0
1·0
0
0
0
0
0
0
0
0
0
0
0
0
1·0
an element with consistent mass matrix can be written as
PPP
"
PPP
"
PPP
l) (z6 )"
)
)
)
e
e
e
f (x)zN d)
i
i e
o(x)(!a !a x #u u x )zN d)
i
ij j
ik kj j i e
o(x)(!a l Nl#(!a #u u )Nl xnl )(N zN n ) d)
e
j m mi
i
ij
ik kj
"o(!a l#(!a #u u )xnl )
i
ij
ik kj j
PPP
)
e
NlN d) zN n
m e mi
"(!a l#(!a #u u )xnl )ml z6 n
(10)
i
ij
ik kj j
m mi
where xnl is the location of the element’s lth node in the x -direction, zN n is the virtual
mi
j
j
displacement of the element’s mth node in the x -direction, and o is assumed to be constant. For
i
a finite element with diagonalized mass matrix, such as ANSYS,21 MSC/NASTRAN,22 and
ABAQUS,23 equation (10) can be written as
l) (z6 )"(!a l#(!a #u u )xnl )mDll z6 n
(11)
mi
i
ij
ik kj j
where mDll"mll S/D, and where S"+ m and D"+ m . Note that equation (11) can be
i ii
ij ij
extended to support finite elements with lumped mass matrix as well.18 In this work, diagonalized
mass matrix is assumed. Thus, the load vector of a finite element is
q*/%"(!a l#(!a #u u )xnl )mD
(12)
ll
ei
i
ij
ik kj j
Substituting the 12 cases listed in Table I to equation (11), the load linear form corresponding
to element quasi-static loading can be obtained as listed in Table II. Note that the stress influence
coefficients 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
( 1997 by John Wiley & Sons, Ltd.
INT. J. NUMER. METHODS ENG., VOL. 40: 1719—1743 (1997)
1726
K.-H. CHANG, X. YU AND K. K. CHOI
Table II. Load linear form for quasi-static loads corresponding to inertia forces
Loading
case
Quasi-static
load
l) (z6 )
1
2
3
4
5
6
7
8
9
10
11
12
a "1
1
a "1
2
a "1
3
a "1
1
a "1
2
a "1
3
u u "1
1 2
u u "1
2 3
u u "1
1 3
u2#u2"1
2
3
u2#u2"1
1
3
u2#u2"1
1
2
!mD
llzN n
l
1
!mDllzN nl
2
!mDllzN nl
3
xn l mD
zN n !xn l mDll zN nl
3 ll l2
2
3
xn l mD
zN n !xn l mDll zN ln
3 ll l1
1
3
!xn l mDllzN nl #xn l mD
ll zN n
l
2
1
1
2
n #xn l mDll zN ln
xn l mD
llzN l
3
2
2
3
zN n
!xn l mDllzN ln !xn l mD
3
1
1 ll l3
n
!xn l mDllzN ln !xn l mD
ll zN l
2
1
1
2
xn l mDll zN nl
1
1
xn l mDll zN nl
2
2
xn l mDllzN ln
3
3
corresponding to the last six quasi-static loads involving angular velocities, which can be applied
to the finite element model as external nodal forces. The stress influence coefficients r*/% due to
SIC
inertia forces can be obtained using FEA,
r*/% l"DBK~1q*/%
l"1, . . . , 12
(13)
l ,
SIC
where q*/%
l "[q*/% ]l , and q*/% is the summation of equation (12) over all finite elements in the
i
i
structure.
2.4. Dynamic stress
The dynamic stress is calculated using the superposition principle as
r(t)"r*/%(t)#r%95(t)
(14)
where
3
3
r*/%(t)" + r*/% l al (t)# + r*/% l a l (t)
SIC
SIC ( `3)
l
l
/1
/1
#r*/% u (t)u (t)#r*/% u (t)u (t)#r*/% u (t)u (t)
SIC7 1
SIC8 2
SIC9 1
2
3
3
#r*/% (u2 (t)#u2 (t))#r*/% (u2 (t)#u2 (t))
SIC11 1
3
3
SIC10 2
#r*/% (u2 (t)#u2 (t))
(15)
SIC12 1
2
in which r*/% l is obtained from equation (13), and
SIC
n
(16)
r%95(t)" + rk Fk(t)
SIC
k/1
where rk can be obtained from equation (3), and n is the number of nodes that external forces
SIC
Fk(t) are applied.
INT. J. NUMER. METHODS ENG., VOL. 40: 1719—1743 (1997)
( 1997 by John Wiley & Sons, Ltd.
SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION
1727
Since most cracks are initiated at the structural surface, and stress computed using displacement-based FEA at the surface are usually less accurate, a stress smoothing technique that uses
the least-squares method24 is employed in this work to improve the accuracy of stress at the
surface.
3. DESIGN-SENSITIVITY ANALYSIS FOR DURABILITY
The proposed DSA method for structural fatigue life uses analytical dynamic stress DSA to
predict dynamic stress history due to design changes, computes life of the component at the same
critical point using the predicted dynamic stress, and uses the difference of the new
life and the original life to approximate the sensitivity of structural component life. The computation procedure is illustrated in Figure 5. Even though life is not a continuous function of design
parameters due to the peak-valley editing and rain-flow counting algorithms embedded in its
computation as discussed in References 9 and 13, it behaves like a continuous function. It is
because a large number of dynamic stresses are counted to compute the fatigue life of structural
components, a small design change only affects a small portion of the dynamic stress set included
in the computation. As a result, the affected stress set is usually too small to make the
discontinuity behaviour of the life function recognized numerically.
In this section, a general analytical DSA method for stress performance measures is presented
first. To support DSA of stress influence coefficients, variation of load linear forms of quasi-static
loads is derived in Section 3.2. A finite difference approach for fatigue life DSA computation is
described 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 structural
domain are treated as the design. The relationship between shape variation of a continuous
domain and the resulting variation in structural performance measures can be described using the
material derivative idea in continuum mechanics.25 A general shape design sensitivity expression
and design velocity field is introduced first. The DSA expression is then applied to 3-D solids with
inertia forces. The direct differentiation method of shape DSA is used in this paper. For design
sensitivity expressions using the adjoint variable method, see Reference 25.
Figure 5. Computation flow for durability DSA
( 1997 by John Wiley & Sons, Ltd.
INT. J. NUMER. METHODS ENG., VOL. 40: 1719—1743 (1997)
1728
K.-H. CHANG, X. YU AND K. K. CHOI
3.1.1. Design velocity field. Considering the structural domain as a continuous medium, and
the process of changing the shape of domain ) to ) in Figure 6 as a dynamic process that
q
deforms the continuum with q playing the role of time, a transformation mapping T that
represents this process can be defined as25
T : xPx(x),
x3)
q
(17)
with
x ,T(x, q)
q
) ,T(), q)
(18)
q
! ,T(!, q)
q
Suppose that a material point x 3 ) in the initial domain at q"0 moves to a new location
x 3 ) in the perturbed domain. Then, a design velocity field V can be defined as
q
q
dx
dT(x, q) LT(x, q)
V(x , q), q"
"
(19)
q
dq
dq
Lq
In the neighbourhood of initial time q"0, assuming a regularity hypothesis and ignoring
higher-order terms, T can be approximated by
T(x, q)"T(x, 0)#q
LT(x, 0)
#O(q2)
Lq
+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 for
a structural component with the domain ) can be written as
a) (z, z6 )"l ) (z6 ), for all z6 3 Z
(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 the
energy bilinear and load linear forms, respectively. The subscript ) in equation (21) is used to
indicate 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 in
an integral form as
PP g(z, +z) d)
t"
(22)
)
Figure 6. Deformation process
INT. J. NUMER. METHODS ENG., VOL. 40: 1719—1743 (1997)
( 1997 by John Wiley & Sons, Ltd.
SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION
1729
Using the direct differentiation method, the first variation of the performance measure t can be
written as25
PP [g,z zR i#g,z zR i,j!g,z (zi,j»j)!g,z (zi,j»j ), j ] d)#P g(»ini) d!
t@"
i
)
i,j
i
i,j
(23)
!
where z5 is the solution of the sensitivity equation obtained by taking the material derivative of
equation (21), i.e.
(24)
a) (z5 , z6 )"l@ (z6 )!a@ (z, z6 ), for all z6 3 Z
V
V
The subscript » on the right-hand side of equation (24) is used to indicate the dependency of the
terms on the design velocity field.
For 3-D elastic solids, the variational equation (21) can be written as
PPP pij (z)eij (z6 ) d)
"
PPP fizN i d)#PP ¹izN i d!#qizN i,l (z6 ),
a) (z, z6 ),
)
)
!2
)
for all z6 3Z
(25)
where p (z) and e (z6 ) are the stress and strain tensors of the displacement z and virtual
ij
ij
displacement z6 , respectively; f is the x -component of the structural inertia force; ¹ is the
i
i
i
x -component of the surface traction force; and q is point forces applied to the structure.
i
i
In equation (24) l@ (z6 ) and a@ (z, z6 ) can be derived for 3-D structural components as
V
V
PPP [ f @izN i#zN i ( fi,j»j)#[ fizN i div V] d)
#
PP M!¹i (zi,j»j)#[(¹izN i), j nj#H(¹izN i)](»ini)N d!#q@izN i
l@ (z6 )"
V
)
(26)
!2
and
PPP M[pij(z)(zN i,k»k,j)#pij (z6 )(zi,k»k,j)]
a@ (z, z6 )"
V
)
![p (z)e (z6 )] div VN d)
(27)
ij
ij
where div V and V, are the divergence and derivative of the design velocity field with respect to
j
x ; n and H are the unit normal vector and curvature of the boundary, respectively; and !2 is the
j
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 for
z5 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 efficiently
since it requires only evaluation of the solution of the same set of finite element matrix equations
with different fictitious loads.
A pointwise stress performance measure, such as stress at a Gauss point, can be defined as
PPP g(r(z)) dK (x!x̂) d)
t"
( 1997 by John Wiley & Sons, Ltd.
(28)
)
INT. J. NUMER. METHODS ENG., VOL. 40: 1719—1743 (1997)
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K.-H. CHANG, X. YU AND K. K. CHOI
where g(r(z)) is a stress measure, such as von Mises stress, dK (x!x̂) is a direct delta function, and
x̂ is the location where the stress is measured.
The design sensitivity expression of the stress performance measure defined in equation (28) can
be obtained as
PPP [g,p (z)pij (z5 ) dK (x!x̂) d)
!
PPP g,p (z)Cijk (zk,m»m, )dK (x!x̂) d)
t@"
ij
)
)
ij
l
l
(29)
where p (z5 ) can be evaluated using z5 obtained from equation (24), C l is the elastic modulus
ij
ijk
tensor that satisfies symmetry relations C l"C l and C l"C l , z is the gradient of z ,
ijk
k ij
ji k k,m
ijk
k
and g, is derivative of the stress function g with respect to the stress components p (z). Effort
ij
pij
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, . The design velocity field computation depends on
j
the shape design parameters defined for the structural geometric model. Proper generation of
design velocity fields is an important step in obtaining accurate shape design sensitivity information. The design parameterization method and numerical method of velocity field computation
for 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
PPP [ f @i zN i#zN i ( fi,j»j)#[ fi zN i div » ] d)#q@izN i
l@ (z6 )"
V
(30)
)
where
f (x)!f (x)
i "0
f @,lim iq
i q?0
q
(31)
since f (x)"f (x) due to the fact that inertia force evaluated at a fixed material point x before and
iq
i
after design changes is constant. Note that a variation of q zN is zero since q (corresponding to
i i
i
a joint reaction force) is assumed to be independent of design changes. Therefore, q@"0 for
i
quasi-static loads corresponding to joint reaction forces. For the variation of quasi-static load
linear form corresponding to inertia forces, the second and third integrands of equation (30) must
be extended, using equation (23) as
f » "Mo[!a #(!a l#u u l )xl ]N, »
i,j j
i
i
ik k
j j
"o(!a l#u u l )xl »
ik k
i
,j j
"o(!a #u u )»
ij
ik kj j
(32)
and
f zN div V"o [!a #(!a #u u )x ]zN div V
i i
i
ij
ik kj j i
INT. J. NUMER. METHODS ENG., VOL. 40: 1719—1743 (1997)
(33)
( 1997 by John Wiley & Sons, Ltd.
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SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION
where the assumption o is constant in ) and the fact x l "dl have been used. Therefore,
,j
j
equation (30) can be rewritten as
l@ (z6 )"o
V
PPP
)
[!a div V#(!a #u u )(» #x div V)] zN d)
i
ij
ik kj j
j
i
(34)
For an element with a consistent mass matrix, equation (34) can be written as
l@ (z6 )"(!a #u u )(» nl m l #oxnl
j
j
V
ij
ik kj
m
PPP N Nm div V d))zN nmi!ai o PPP N Nm div V d) zN nmi
)
l
)
l
(35)
For an element with a diagonalized mass matrix, equation (34) can be written as
l@ (z6 )"[(!a #u u )(» nl m l #xnl mR Dll )!a mR Dll ]zN nl
j m
j
V
ij
ik kj
i
i
(36)
where
D "mR ll
mR ll
S
SQ
S
#mll !mll
DQ
D
D
D2
(37)
and SQ "+ mR , DQ "+ mR , and mR "o :::) NlN div V d). Equation (35) can be used to support
m
ij
ij ij
i ii
finite elements with diagonalized or lumped mass matrix.18
Using equation (36) variations of the load linear form for the 12 quasi-static loading cases that
correspond 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 the
component fatigue life. An analytical approach for fatigue life DSA is not possible since the
fatigue life cannot be obtained as a continuous function of design parameters.
Table III. Variation of load linear form corresponding to inertia forces
Loading
case
Quasi-static
load
1
2
3
4
5
6
7
8
9
10
11
12
a "1
1
a "1
2
a "1
3
a "1
1
a "1
2
a "1
3
u u "1
1 2
u u "1
2 3
u u "1
1 3
u2#u2"1
2
3
u2#u2"1
1
3
u2#u2"1
1
2
( 1997 by John Wiley & Sons, Ltd.
l@ (z6 )
V
!mR lDl z6 ln
1
!mR Dll z6 ln
2
!mR Dll z6 ln
3
(» n l mDll#xn l mR Dll )z6 nl !(» n l mDll#xn l mR Dll )z6 ln
3
3
2
2
2
3
(» n l mDll#xn l mR Dll )z6 nl !(» n l mDll#xn l mR Dll )z6 ln
3
3
1
1
1
3
!(» n lmD
#xn l mR Dll)z6 nl #(» n lmD
#xn lmR Dll)z6 nl
2 ll
2
1
1 ll
1
2
(» n l mDll#xn l mR Dll )z6 nl #(» n l mDll#xn l mR Dll )z6 ln
3
3
2
2
2
3
!(» n l mDll#xn l mR Dll )z6 nl !(» n l mDll#xn l mR Dll )z6 nl
3
3
1
1
1
3
!(» n l mDll#xn l mR Dll )z6 ln !(» n l mDll#xn l mR lDl )z6 ln
2
2
1
1
1
2
(» n l mDll#xn l mR lDl )z6 ln
1
1
1
(» n l mDll#xn l mR lDl )z6 ln
2
2
2
(» n l mDll#xn l mR lDl )z6 ln
3
3
3
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K.-H. CHANG, X. YU AND K. K. CHOI
Once the sensitivity of stress influence coefficients are obtained using the DSA method
described in Section 3.2, increment of stress influence coefficients can be obtained by
dr
Lr
" SIC db
SIC
j
Lb
j
(38)
where db is the perturbation of the jth design parameter. Note that the design perturbation db
j
j
must be small for linear approximation of the life. On the other hand, in numerical calculation,
db cannot be too small since it introduces numerical noise.
j
The stress influence coefficients of the perturbed design can be approximated by
r
SIC
(b#db )+r (b)#dr
j
SIC
SIC
(39)
A stress time history of the perturbed design can be obtained by superposing r (b#db ) with
SIC
j
the loading history obtained from multibody dynamic analysis. Note that the design perturbation
is assumed to be local so that dynamic behaviour of the mechanical system is not altered. The new
dynamic stress history is then used to calculate the fatigue life of the component with a perturbed
design, ¸(b#db ), using the same life prediction method. The design sensitivity coefficient of
j
component fatigue life with respect to the jth design parameter can be obtained from
L¸ ¸(b#db )!¸(b)
j
+
Lb
db
j
j
(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 are
available.
4. NUMERICAL EXAMPLE—A TRACKED VEHICLE ROADARM
A roadarm of the military tracked vehicle shown in Figure 7 is employed to demonstrate the DSA
method for crack initiation life proposed in this paper. In this section, the multibody dynamic
model of the tracked vehicle and its simulation environment are described first. The structural
finite element model of the roadarm is presented in Section 4.2. Contours of crack initiation life
and von Mises stress at the peak load of the simulation period are presented in Section 4.3. Design
parameterization of the roadarm is discussed in Section 4.4. Design sensitivity analysis and result
verification 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 the
Aberdeen Proving Ground 4 (APG4), as shown in Figure 9, at a constant speed of 20 miles per
hour forward (positive X -direction). The Tracked Vehicle Design Workspace (TVWS)28 is used
2
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 of
results 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 the
analysis. A time history of joint reaction forces at the wheel end is shown in Figure 10.
INT. J. NUMER. METHODS ENG., VOL. 40: 1719—1743 (1997)
( 1997 by John Wiley & Sons, Ltd.
SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION
1733
Figure 7. A military tracked vehicle and roadarm: (a) a military tracked vehicle; (b) roadarm geometric model
Figure 8. The tracked vehicle dynamic model
( 1997 by John Wiley & Sons, Ltd.
INT. J. NUMER. METHODS ENG., VOL. 40: 1719—1743 (1997)
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K.-H. CHANG, X. YU AND K. K. CHOI
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, of
ANSYS are used for the roadarm finite element model. A number of rigid beams are created to
connect nodes at the inner surface of the two holes to end nodes of beam elements to simulate the
roadwheel shaft and torsion bar, respectively. Displacement constraints are defined at the end
nodes of the beam elements that simulate the torsion bar, and joint reaction forces and torques
are applied at the end node of the other beam element that simulates the shaft of the roadwheel, as
shown in Figure 11. The roadarm is made of S4340 steel, with material properties Young’s
modulus E"3·0]107 psi and Poisson’s ratio l"0·3. Note that the co-ordinate systems of the
finite element model is identical to the body reference frame of the roadarm in the tracked vehicle
dynamic model. Therefore, loading history generated from dynamic analysis can be used without
transformation.
INT. J. NUMER. METHODS ENG., VOL. 40: 1719—1743 (1997)
( 1997 by John Wiley & Sons, Ltd.
SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION
1735
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 roadarm
using ANSYS by applying 18 quasi-static loads. Among the loads, the first six that correspond to
external joint forces are three unit forces and three unit torques applied at the centre of the
roadwheel, in x@ , x@ and x@ directions, and the remaining 12 that correspond to inertia forces are
3
1 2
unit accelerations, unit angular accelerations, and unit combinations of angular velocities, as
listed in Table I. The stress influence coefficients obtained from analyses are six component
stresses at finite element nodes in the x@ !x@ !x@ co-ordinates.
3
2
1
Dynamic stresses at finite element nodes are then calculated by superposing stress influence
coefficients with their corresponding external forces and accelerations and velocities in time
domain. To compute multiaxial crack initiation life of the roadarm, the equivalent von Mises
strain approach10 is employed. The fatigue life contour is given in Figure 12. Note that the
spectrum in Figure 12 is the number of blocks to initiate crack in logarithm.
( 1997 by John Wiley & Sons, Ltd.
INT. J. NUMER. METHODS ENG., VOL. 40: 1719—1743 (1997)
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K.-H. CHANG, X. YU AND K. K. CHOI
Figure 12. Contour of crack initiation life
A static stress contour shown in Figure 13 is given to demonstrate that the worst-case scenario
employed for durability design using stresses as performance measures is problematic. Stress
contour shown in Figure 13 is obtained by applying the peak load found at 17·35 s of the 20 s
simulation, including six joint reaction forces at the roadwheel end, accelerations, angular
accelerations, and angular velocities of the roadarm. Note that from Figures 12 and 13, the stress
concentration area identified as the worst case does not conform with the critical areas where the
crack is first initiated.
4.4. Design parameterization
For shape design parameterization, eight design parameters are defined to characterize shapes
of the four cross-sections shown in Figure 7(b). Contour of the cross-sectional shape is composed
of four straight lines and four cubic curves. Side expansions (x@ -direction) of cross-sectional
2
shapes are defined using design parameters b1, b3, b5, and b7 for intersections 1—4, respectively.
Vertical expansions (x@ -direction) of the cross-sectional shapes are defined using the remaining
3
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 design
parameters are computed using the direct differentiation method of continuum DSA. Velocity
fields 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 perturbation of design parameter b . In Table IV, t(b#db) and t(b) are crack initiation lives at the
2
INT. J. NUMER. METHODS ENG., VOL. 40: 1719—1743 (1997)
( 1997 by John Wiley & Sons, Ltd.
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SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION
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
t(b#db)
Life
node
node
node
node
node
node
node
node
1544
1519
1433
1340
1227
918
922
1391
8·9084296E#7
1·4521717E#8
2·7728931E#8
4·8991923E#8
6·6329165E#8
1·0039969E#9
1·0622614E#9
1·5882051E#10
( 1997 by John Wiley & Sons, Ltd.
t(b)
dt
t@
8·9261936E#7 !1·7764E#5 !1·7764E#5
1·4468427E#8
5·3290E#6
5·3290E#6
2·7622349E#8
1·0658E#5
1·0658E#6
4·9062976E#8 !7·1053E#5 !7·1054E#5
6·6187059E#8
1·4211E#6
1·4211E#6
9·5710106E#8
4·6895E#7
4·6896E#7
9·9831251E#8
6·3949E#7
6·3949E#7
1·5609203E#10
2·7284E#8
2·7284E#8
t@/dt % dt/t(b) %
100·00
100·00
100·00
100·00
100·00
100·00
100·00
100·00
!0·20
3·68
0·04
!0·14
0·21
4·90
6·41
1·75
INT. J. NUMER. METHODS ENG., VOL. 40: 1719—1743 (1997)
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K.-H. CHANG, X. YU AND K. K. CHOI
Table V. CPU seconds spent for crack initiation life computation
Tasks
Quasi-static FEA
Dynamic stress
Life computation
Total
CPU s
7080
2
2
7084
Table VI. CPU seconds spent for sensitivity
analysis of crack initiation life for eight design
parameters
Tasks
Dynamic stress DSA
Life DSA
Total
CPU s
4280
24
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 responses
using sensitivity coefficients, i.e. Lt/Lb]db; and t@/dt is the accuracy measurement of sensitivity
coefficients; 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 sensitivity
prediction is more accurate. A small value in the last column suggests that numerical noise may
contribute to an inaccurate finite difference verification, and a large value indicates that nonlinear effect from finite difference results might destroy the sensitivity accuracy. Table IV shows
that the sensitivity coefficients of crack initiation lives at nodes randomly selected are very
accurate.
CPU time spent for the crack initiation life and sensitivity analysis of crack initiation life on
an HP 9000/755 machine are summarized in Tables V and VI, respectively. It is shown in
Table V that the bulk computation of the crack initiation life is in quasi-static FEA. If the overall
finite difference approach is used for sensitivity computation, i.e. perturbing each design parameter, creating finite element model of the perturbed design, and performing the entire computation, 56 672 (7084]8 design parameters) CPU seconds are needed for sensitivity computation, in
addition to human effort spent for finite element modelling. The proposed DSA method needs
only 4304 CPU s (Table VI) to perform the sensitivity computation which is much more efficient
than the overall finite difference approach.
5. DESIGN OPTIMIZATION
Shape of the roadarm is optimized and presented in this section. Section 5.1 describes the
definition of the roadarm design model. Section 5.2 explains how the design optimization is
performed using ANSYS, DRAW,9 Design Optimization Tool (DOT),6 and sensitivity analysis
and design model update programs in the Design Sensitivity Analysis and Optimization (DSO)
tool.29 In Section 5.3, design optimization results are presented.
INT. J. NUMER. METHODS ENG., VOL. 40: 1719—1743 (1997)
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SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION
5.1. Roadarm design model definition
The objective of the roadarm design is to minimize its volume and keep crack initiation lives at
the same level as the initial design. The objective is achieved by changing shapes of the four
cross-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 critical
nodes (with ids shown in Figure 15) are defined as constraints with a lower bound of
9·63]106 blocks. Note that the lower bound defined is equivalent to 20 year service life, assuming
the tracked vehicle is operating 8 hours per day, five days per week. Definition of the cost function
and 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 Modified
Feasible Direction method in DOT, together with the life DSA and design model update methods
provided in DSO. As shown in Figure 16, DOT is treated as a subroutine in the integrated
optimization module. When DOT provides a new design or carries out a line search, the main
program of the optimization module OPTMAIN calls the UPMODEL subroutine to update the
design model corresponding to the new design, and sends the design model to DRAW (ANSYS
for FEA) for fatigue life prediction. When the gradient information is requested by DOT, the life
DSA module is executed to compute sensitivity coefficients of the cost function and e-active
constraints. The process is repeated until an optimum design is obtained or the maximum
iteration number is reached.
Figure 15. Node ids of fatigue life constraints
Table VII. Cost and selected constraint function definitions
Function
Cost
Constraint
Constraint
Constraint
Constraint
Constraint
Description
1
2
3
4
5
Life
Life
Life
Life
Life
( 1997 by John Wiley & Sons, Ltd.
Volume
at node 1216
at node 926
at node 1544
at node 1519
at node 1433
Lower bound
9·63E#6
9·63E#6
9·63E#6
9·63E#6
9·63E#6
(20 y)
(20 y)
(20 y)
(20 y)
(20 y)
Current design
Status
487·678 in3
9·631E#6 blocks
8·309E#7 blocks
8·926E#7 blocks
1·447E#8 blocks
2·726E#8 blocks
Active
Inactive
Inactive
Inactive
Inactive
INT. J. NUMER. METHODS ENG., VOL. 40: 1719—1743 (1997)
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K.-H. CHANG, X. YU AND K. K. CHOI
5.3. Roadarm design optimization results
An optimum design of the roadarm is obtained in six iterations, with 22 crack initiation life
computations and six fatigue life DSA computations. The isoparametric mapping method is
employed to compute the design velocity field in the first iteration, which is used during all
subsequent design iterations for life DSA computation. Moreover, the design velocity field is used
to update the finite element mesh for the new design, i.e.
n
(41)
xb`db"xb#dx "xb# + » k dbk
i
i
i
i
i
k/1
where xb`db and xb are the locations of the nodes of the perturbed and the current designs,
i
i
respectively; dx is the nodal point movement due to design changes; » k and dbk are the velocity field
i
i
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
INT. J. NUMER. METHODS ENG., VOL. 40: 1719—1743 (1997)
( 1997 by John Wiley & Sons, Ltd.
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SHAPE DESIGN SENSITIVITY ANALYSIS AND OPTIMIZATION
criterion is a 0·1 per cent relative change in cost function in two consecutive iterations. The cost
function value reduces quickly in the first four iterations, and an optimum design is reached at the
sixth iteration. At the optimum design, all lives are above the lower bound and the history of five
selected normalized constraints is given in Figure 17(b). Also, the cost and the five selected
constraint function values at the initial and optimum designs are listed in Table VIII. As
Figure 17(b) and Table VIII shows, life at node 1216 (constraint 1) reaches initially the lower
bound and increases to 7·705]107 at the optimum design. The shortest life at the optimum is
9·63]106, found at node 926 (constraint 2). Through optimization, the structural volume of
roadarm 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
Cost
Constraint
Constraint
Constraint
Constraint
Constraint
Description
1
2
3
4
5
Life
Life
Life
Life
Life
Volume
at node 1216
at node 926
at node 1544
at node 1519
at node 1433
Initial design
Optimum design
487·678 in3
9·631E#6 blocks
8·309E#7 blocks
8·926E#7 blocks
1·447E#8 blocks
2·726E#8 blocks
436·722 in3
7·704E#7 blocks
9·631E#6 blocks
9·678E#6 blocks
4·698E#7 blocks
4·815E#8 blocks
% changes
!10·5
699·9
!88·4
!89·2
!67·5
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
Description
Initial
design (in)
Optimum
design (in)
% changes
Width of Cross Section 1
Height of Cross Section 1
Width of Cross Section 2
Height of Cross Section 2
Width of Cross Section 3
Height of Cross Section 3
Width of Cross Section 4
Height of Cross Section 4
3·250
1·968
3·170
1·968
3·170
2·635
3·170
5·057
2·889
1·583
2·911
1·637
2·870
2·420
2·801
4·700
!11·09
!19·56
!8·17
!16·78
!9·45
!8·17
!11·65
!7·06
Des. param.
b1
b2
b3
b4
b5
b6
b7
b8
( 1997 by John Wiley & Sons, Ltd.
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K.-H. CHANG, X. YU AND K. K. CHOI
Figure 18 shows the shape changes of roadarm. Note that the roadarm becomes thinner and
narrower at intersections 1, 2 and 4 at the optimum design. Design parameter values at initial and
optimum designs are listed in Table IX.
6. CONCLUSIONS
In this paper, a design sensitivity analysis (DSA) method for fatigue life of 3-D elastic solid
structural components of mechanical systems with respect to shape design parameters has been
presented. The proposed DSA method has been demonstrated to be accurate and efficient using
a tracked vehicle roadarm example. Also, this method has been applied to support design
optimization of the tracked vehicle roadarm considering crack initiation life design constraints.
The DSA method proposed is applicable to structural components with less flexibility, such as
suspension or engine parts. Also, design changes for the structural components is assumed to be
small so that dynamic behavior of mechanical systems before and after design changes are not
altered.
This work is being extended to support durability design of vehicle body or hull with large
flexibility, in which, flexible body behavior of the mechanical systems must be considered in
dynamic analysis. Design sensitivity analysis of stress induced by additional inertia forces due to
consideration 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 their
valuable input and feedback of this paper. Also, thanks to Mr. Jack Standefer for creating
a dynamic model schematic drawing of the tracked vehicle.
Research is supported by the Automotive Research Center sponsored by the U.S. Army Tank
and Automotive Command Center.
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