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) 1730 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. 1731 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 INT. J. NUMER. METHODS ENG., VOL. 40: 1719—1743 (1997) 1732 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) 1734 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) 1736 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. 1737 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) 1738 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) ( 1997 by John Wiley & Sons, Ltd. 1739 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) 1740 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. 1741 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. INT. J. NUMER. METHODS ENG., VOL. 40: 1719—1743 (1997) 1742 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. REFERENCES 1. A. Francavilla, C. V. Ramakrishan and O. C. Zienkiewicz, ‘Optimization on shape to minimize stress concentration’, J. Stress Anal., 10, 63—70 (1975). 2. J. C. Thompson and O. K. Bedair, ‘Techniques to minimize the cost of shape optimization of stress concentration’, in S. Hermandez and C. A. Brebbia (eds.), Optimization of Structural Systems and Industrial Applications, OPTI, 1991, pp. 129—140. 3. M. Fanni, E. Schnack and J. 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TVWS, ¹racked »ehicle ¼orkspace, Center for Computer-Aided Design, College of Engineering, The University of Iowa, Iowa City, IA, 1994. K. H. Chang, K. K. Choi, C. S. Tsai, C. J. Chen, B. S. Choi and X. Yu, ‘Design sensitivity analysis and optimization tool (DSO) for shape design applications’, Computing Systems Eng., 6, 151—175 (1995). . ( 1997 by John Wiley & Sons, Ltd. INT. J. NUMER. METHODS ENG., VOL. 40: 1719—1743 (1997)

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