Design sensitivity analysis and shape optimization of ...user.engineering.uiowa.edu/~kkchoi/DSA_Opt_A.22.pdf · Design sensitivity analysis and shape optimization of structural components

Design sensitivity analysis and shape optimization of structuralcomponents with hyperelastic material

K.K. Choi *, W. Duan 1

Center for Computer-Aided Design and Department of Mechanical Engineering, College of Engineering, The University of Iowa,

Iowa City IA 52242, USA

Received 12 April 1997; received in revised form 5 August 1998

Abstract

A continuum-based design sensitivity analysis (DSA) method is developed for structural components with hyperelastic (incom-

pressible) material. A mixed variational principle (MVP) and the total Lagrangian formulation are used for nonlinear analysis. E�ects

of large displacements, large strains, and material nonlinearities are included in the analysis model, using appropriate kinematics and

constitutive relations. The material property and shape DSA using both the direct di�erentiation method (DDM) and the adjoint

variable method (AVM) are discussed. For shape DSA, the material derivative concept is used to compute e�ects of the shape

variation. The boundary displacement and isoparametric mapping methods are employed to compute the design velocity ®eld. Both

hydrostatic pressure and structural sti�ness are considered as constraints for design optimization, which is carried out by integrating

shape design parameterization, design velocity computation, DSA, nonlinear analysis, and the optimization method. Examples such as,

an engine mount and a bushing demonstrate the feasibility of the proposed optimization method for designing structural components

using hyperelastic material. Ó 2000 Elsevier Science S.A. All rights reserved.

1. Introduction

Hyperelastic material, such as rubber and rubber-like material, is very versatile and adaptable, and haslong been used successfully in numerous engineering applications. Rubber possesses inherent damping,which is particularly bene®cial when a resonant vibration is encountered, and it can store more elasticenergy than steel. For hyperelastic material, the bulk modulus, which is associated with the volume changeof the structural component, is much larger than the shear modulus, which is associated with the shapedeformation of the structural component.

These properties make the hyperelastic material conducive to a wide range of applications in modernindustry, such as weather-stripping for insulation, and bushings and engine mounts for noise, vibration,and harshness (NVH) control. To obtain a meaningful shape optimal design of a hyperelastic solid, it isnecessary to accurately describe the material properties, thoroughly understand the nonlinear structuralanalysis procedure, and to correctly formulate the structural design optimization problem. Many workshave been published in the ®rst two areas. For design optimization, DSA, which deals with the e�ect ofchange of design variables on the structural response, plays an important role. Various DSA methods forlinear structures with sizing and shape design variables have been developed and are well documented [1±3].For nonlinear structures, Choi and Santos [4], Santos and Choi [5,6], and Choi [7] developed sizing and

www.elsevier.com/locate/cmaComput. Methods Appl. Mech. Engrg. 187 (2000) 219±243

* Corresponding author.

E-mail address: [email protected] (K.K. Choi).1 Presently at Lord Corp., Erie, PA.

0045-7825/00/$ - see front matter Ó 2000 Elsevier Science S.A. All rights reserved.

PII: S 0 0 4 5 - 7 8 2 5 ( 9 9 ) 0 0 1 2 1 - 8

shape DSA methods using the continuum approach. The method was extended to handle geometricnonlinear analysis with linear incompressible material [8]. Using the control volume concept, Tortorelli [9]developed a DSA method for nonlinear structures with incompressible material.

It was shown by Gent that a failure occurs when a structural component with the hyperelastic materialcomes under a certain hydrostatic tensile stress [10]. On the other hand, the sti�ness characteristic is animportant design consideration in shape optimal design of the engine mount or bushing for vibrationisolation [11]. That is, to reduce the NVH of a vehicle system [12,13], the optimum design can be obtainedusing a two-level approach. First, the system level DSA and optimization can be carried out to ®nd op-timum gages and topology of the vehicle body, and optimum sti�ness and damping characteristics of enginemounts and bushings. In this design formulation, the weight can be considered as the cost, whereas theNVH are treated as design constraints. Once optimum sti�ness and damping characteristics of enginemounts and bushings are determined from the system level design, then shapes of engine mounts andbushings can be optimized to yield these sti�ness and damping characteristics.

The objective of this paper is to develop material property and shape DSA and optimization methodsusing the continuum approach for hyperelastic structural components. The Mooney±Rivlin energy densityfunction is employed to describe the material property. The FEA code ABAQUS [14] that utilizes the MVPis used for nonlinear structural analysis. Numerical examples are presented to demonstrate the e�ciencyand accuracy of the proposed method.

For the optimization procedure, the data of shape design parameterization, design velocity computation,nonlinear analysis, and shape DSA are integrated. In this paper, a geometric and ®nite element modelgeneration tool PATRAN [15] and VMA Engineering's design optimization tool DOT [16] are used fordesign optimization.

2. Governing equations for structures with hyperelastic materials

In this paper, conventional notations [17] of nonlinear structural analysis are used. The motion of a bodyin a ®xed Cartesian coordinate system is shown in Fig. 1. The body may experience large displacements,large rotations, and large strains, with the material nonlinear constraint. The coordinate of a point P on thebody is �0x1;

0 x2;0 x3� at time 0, �tx1;

t x2;t x3� at time t, and �t�Dtx1;

t�Dt x2;t�Dt x3� at time t � Dt, where the

left superscript indicates the con®guration time at which the quantity occurs. If the quantity under con-sideration occurs in the same con®guration in which it is also measured, then the left subscript is omitted,e.g., t�Dt

t�Dtzi � t�Dtzi. For the derivative of a quantity with respect to a variable, the left subscript indicates thecon®guration in which the variable is measured. For example, for the ith component tzi of the displacement,

Fig. 1. Motion of the body in ®xed cartesian coordinate system.

220 K.K. Choi, W. Duan / Comput. Methods Appl. Mech. Engrg. 187 (2000) 219±243

t0zi;j denotes its derivative with respect to the coordinate 0xj. The increments of the displacement from time tto t � Dt are denoted as

zi � t�Dtzi ÿ tzi; i � 1; 2: �1�

For nonlinear hyperelastic material, the stresses are not completely determined by the deformation. The®nite element approximation loses accuracy when Poisson's ratio approaches one-half which leads to`locking'. The MVP is therefore introduced in structural analysis to obtain an accurate solution. In theMVP, the general equilibrium equation for a body with volume t�DtX and boundary t�DtC at time t � Dt canbe expressed as

Z Zt�DtW

t�DtW t�DtdX � t�DtR; �2�

where t�DtW and t�DtR represent the energy density function and the virtual work performed on the body byexternally applied forces through kinematically admissible virtual displacement in the con®guration at timet � Dt, respectively. In Eq. (2), t�DtW is a function of independent variational variables of displacement andhydrostatic pressure. The overbar ``±'' indicates the ®rst-order variation of the quantity.

The virtual work done on the body by externally applied forces is

t�DtR �Z Z

t�DtX

t�Dtfi �z t�Dti dX�

Zt�DtC

t�DtTi �z t�Dti dC; �3�

where t�Dtfi and t�DtTi are the components of the externally applied body force and surface traction, re-spectively. The notation �z denotes a kinematically admissible virtual displacement. In the following deri-vations, it is assumed that t�Dtfi and t�DtTi are deformation-independent or conservative static loading.

De®ne the energy form as

a�t�Dtr; �r� �Z Z

t�DtX

t�DtW t�DtdX �4�

and the load linear form as

`��r� � t�DtR; �5�

where t�Dtr � �t�Dtz1;t�Dtz2;

t�Dtz3;t�Dtp� are three components of the displacement and the separately in-

terpolated hydrostatic pressure at time t � Dt, and �r � ��z1; �z2; �z3; �p� are three components of the kine-matically admissible virtual displacement and the hydrostatic pressure. Then, the equilibrium Eq. (2) can berewritten as

a�t�Dtr; �r� � `��r� 8�r 2 Z � P ; �6�

where Z and P are spaces of kinematically admissible virtual displacements and hydrostatic pressure, re-spectively. The equilibrium Eq. (6) cannot be solved directly, since the con®guration at time t � Dt is un-known. An incremental formulation with its linearized form needs to be introduced to obtain anapproximation solution.

The foundation of mathematical theory for hyperelastic material was established by Mooney and Rivlin[18,19]. The Rivlin's strain energy density function for incompressible material is expressed as a function ofthe ®rst two strain invariants as

t�Dt0W � t�Dt

0I1;t�Dt

0 I2

ÿ � � X1m�n�1

Cmnt�Dt

0I1

ÿ ÿ 3�m t�Dt

0I2

ÿ ÿ 3�n �7�

K.K. Choi, W. Duan / Comput. Methods Appl. Mech. Engrg. 187 (2000) 219±243 221

with the incompressibility condition t�Dt0I3 � 1 � 0, where

t�Dt0I1 � 3� 2t�Dt

0eii

t�Dt0I2 � 3� 4t�Dt

0eii � 2 t�Dt0eii

t�Dt0ejj

ÿ ÿ t�Dt0eij

t�Dt0eji

�t�Dt

0I3 � 1� 2t�Dt0eii � 2 t�Dt

0eiit�Dt

0ejj

ÿ ÿ t�Dt0eij

t�Dt0eji

�� 3

4eijkerst

t�Dt0eir

t�Dt0ejs

t�Dt0ekt

�8�

with the summation convention over repeated indices,

t�Dt0eij � 1

2t�Dt

0zi;j

ÿ � t�Dt0zj;i � t�Dt

0zk;it�Dt

0zk;j

� �9�

are components of the Green±Lagrange strain tensor, and eijk are permutation symbols [20]. Severalsimpli®ed forms are obtained by truncating the Rivlin strain energy density function [21]. The most popularone is the Mooney±Rivlin energy density function which is de®ned as

t�Dt0W � t�Dt

0I1;t�Dt

0 I2

ÿ � � C10t�Dt

0I1

ÿ ÿ 3�� C01

t�Dt0I2

ÿ ÿ 3�: �10�

To avoid the sensitivity of strain invariants t�Dt0I1 and t�Dt

0I2 with respect to the magnitude of the volumeratio, Penn [22], Ogden [23], Sussman and Bathe [24], and Chang et al. [25] proposed various energy densityfunctions by using the reduced strain invariants t�Dt

0Ji�i � 1; 2; 3� instead of the t�Dt0Ii�i � 1; 2; 3� in the Rivlin

strain energy density function. The reduced stress invariants t�Dt0Ji�i � 1; 2� are mutually independent and

noncontributory to the hydrostatic pressure. These invariants are de®ned as

t�Dt0J1 � t�Dt

0 I1t�Dt

0I3

ÿ �ÿ1=3


0 I2t�Dt

0I3

ÿ �ÿ2=3


0I3

ÿ �1=2:

�11�

Another class of energy density functions is developed based on an assumption that the bulk modulus ofthe material is several thousand times larger than the shear modulus. Therefore, it is reasonable to assumethat the material is nearly incompressible [21,24]. The description of nearly incompressible material isobtained by including the work done by the hydrostatic pressure in the energy density function. The energydensity function of nearly incompressible material at con®guration time t � Dt can thus be written as

t�Dt0W � �

X1m�n�1

Cmnt�Dt

0J1

ÿ ÿ 3�m t�Dt

0J2

ÿ ÿ 3�n � 1

2K t�Dt

0J3

ÿ ÿ 1�2; �12�

where Cmn and K are material constants and bulk modulus, respectively. For small strain, 2�C10 � C01�represents the shear modulus; 6�C10 � C01� represents the equivalent Young's modulus E for the 3-D solid;and 8�C10 � C01� represents the equivalent Young's modulus E for the 2-D plane strain problem.

In this paper, the total Lagrangian formulation is employed for nonlinear analysis. In this formulation,all variables are referred to the initial undeformed con®guration of the body at time 0, and the 2nd Piola±Kichho� stress tensor and the Green±Lagrange strain tensor are used. Eq. (6) is then transformed to

a�t�Dt0r; 0�r� �

Z Z0X

t�Dt0 W 0dX �

Z Z0X

t�Dt0 fi 0�zi

0dX�Z

0C

t�Dt0 Ti 0�zi

0dC

� `�0�r�; 8 0�r 2 0Z � 0P : �13�To introduce independent variables of hydrostatic pressure in the MVP, the energy density function

t�Dt0W � is modi®ed by adding an unknown energy term t�Dt

0Q which is a function of displacements andhydrostatic pressure. The modi®ed energy density function [24] needs to satisfy three basic physical re-quirements. These requirements guarantee that the inaccurate pressure computed from the single dis-placement method is removed and the separately interpolated pressure has the correct physical meaning


provided that the Babuska±Brezzi [26] condition is satis®ed. Generally, the modi®ed energy densityfunction can be written as

t�Dt0W � t�Dt

0 W � �t�Dt0 Q: �14�

Using the Lagrange multiplier method, the modi®ed Mooney±Rivlin energy density function for incom-pressible material is implemented in ABAQUS [14] as

t�Dt0W � C10

t�Dt0J1

ÿ ÿ 3�� C01

t�Dt0J2

ÿ ÿ 3�ÿ t�Dt

0 p t�Dt0J3

ÿ ÿ 1�: �15�

Once the energy density function is de®ned, the constitutive equation can be obtained as

t�Dt0Sij �

o t�Dt0W

ÿ �o t�Dt

0eij

ÿ � � t�Dt0 W;eij

� o t�Dt0W

ÿ �o t�Dt

0J1

ÿ � o t�Dt0J1

ÿ �o t�Dt

0eij

ÿ �� o t�Dt0W

ÿ �o t�Dt

0J2

ÿ � o t�Dt0J2

ÿ �o t�Dt

0eij

ÿ �� o t�Dt0W

ÿ �o t�Dt

0J3

ÿ � o t�Dt0J3

ÿ �o�t�Dt

0eij�; �16�

where t�Dt0Sij are Cartesian components of the 2nd Piola±Kichho� stress tensor. Then, the energy form and

load linear form of Eqs. (4) and (5) can be rewritten, respectively, as

a�t�Dt0r; 0�r� �

Z Z0X

t�Dt0 W 0dX

�Z Z

0X

t�Dt0 Sij

t�Dt0�eij

0dX�Z Z

0X

t�Dt0 Q;p

t�Dt0 �p 0dX �17�

and

`�0�r� � t�Dt0 R �

Z Z0X

t�Dt0 fi 0�zi

0dX�Z

0C

t�Dt0 Ti 0�zi

0dC; �18�

where t�Dt0�eij and t�Dt

0 �p are variations of Cartesian components of the Green±Lagrange strain tensor and theseparately interpolated hydrostatic pressure in the con®guration at time t � Dt referred to the initial con-®guration at time zero, respectively. Eq. (2) then yields

a t�Dt0r; 0�r

� �� `�0�r � 8 0�r 2 0Z � 0P : �19�

Eq. (19) cannot be solved and the incremental formulation needs to be introduced. The increments of thesecond Piola±Kichho� stress and the Green±Lagrange strain tensor at time t � Dt are de®ned, respectivelyas

0Sij � t�Dt0 Sij ÿ t

0Sij �20�and

0eij � t�Dt0 eij ÿ t

0eij � 0eij � 0gij; �21�where

0eij � 1

2�0zi;j � 0zj;i � t

0zk;i0zk;j � t0zk;j0zk;i�;

0gij �1

2�0zk;i 0zk;j�;

0zi � t�Dt0 zi ÿt

0 zi;

0p � t�Dt0 p ÿt

0 p:

�22�


In Eqs. (21) and (22), 0eij and 0gij are the linear and the nonlinear parts of 0eij; 0zi is the incremental dis-placement; and 0p is the incremental hydrostatic pressure. If t is the current equilibrium con®guration time,then t

0zi is known. Thus t0�zi � 0 and t

0�eij � 0, and

t�Dt0�eij � 0�eij � 1

2�0�zi;j � 0�zj;i � t

0zk;i0�zk;j � t0zk;j0�zk;i � 0�zk;i0zk;j � 0zk;i0�zk;j�: �23�

The incremental constitutive law is

0Sij � t0W �

;eij ;ers

�� t

0Q;eij;ers

�0ers � t

0Q;eij;p0p � 0Cijrs 0ers � t0Q;eij ;p0p; �24�

where 0Cijrs is the increment of the material response tensor referred to the initial con®guration. Using thesede®nitions, we can rewrite the nonlinear Eq. (19) in incremental form asZ Z

0X

t0W �

;eij ;ers

�h� t

0Q;eij;ers

�0ers � t

0Q;eij ;p0pi

0�eij0dX�

Z Z0X

t0W �

;eij

�� t

0Q;eij

�0�gij

0dX

�Z Z

0X

t0Q;eij;p 0eij 0 �p 0dX�

Z ZX

t0Q;p;p 0p 0 �p 0dX

� `�0�r� ÿZ Z

X

t0W �

;eij

�� t

0Q;eij

�0�eij

0dXÿZ Z

X

t0Q;p 0 �p 0dX; 8 0�r 2 0Z � 0P : �25�

By linearizing the incremental strain using 0eij � 0eij, the linearized incremental form of the equilibriumequation is obtained asZ Z

0X

t0W �

;eij ;ers

�h� t

0Q;eij;ers

�0ers � t

0Q;eij ;p0pi

0�eij0dX�

Z Z0X

t0W �

;eij

�� t

0Q;eij

�0�gij

0dX

�Z Z

0X

t0Q;eij;p 0eij 0 �p 0dX�

Z Z0X


� `�0�r� ÿZ Z

0X

t0W �

;eij

�� t

0Q;eij

�0�eij

0dXÿZ Z

0X

t0Q;p 0 �p 0dX 8 0�r 2 0Z � 0P �26�

where

0eij� t0z; � � � 1

2��i;j � ��j;i � t

0zk;i��k;j � t0zk;j��k;i�;

0gij� 0�z; �� 1

2� 0�zk;i��k;j � 0�zk;j��k;i�:

Eq. (26) is solved a number of times and residual forces are updated until the out-of-balance virtual worksatis®es a certain convergence tolerance.

For DSA, it is convenient to de®ne an energy bilinear form, from Eq. (26), as

a��t0r; 0r; 0�r� �Z Z

0X

t0W �

;eij ;ers

�h� t

0Q;eij;ers

�0ers

t0z; 0zÿ �� Q;eij ;p 0p�

i0eij

t0z; 0�z� �

0dX

�Z Z

0X

t0W �

;eij

�� t

0Q;eij

�0gij�0�z; 0z� 0dX�

Z Z0X

t0Q;eij;p 0eij

t0z; 0zÿ �

0 �p 0dX

�Z Z

0X


�Z Z

0X0Cijrs0ers

� � t0Q;eij;p0p

�0eij

t0z; 0�z� �

0dX�Z Z

0X

t0Cijrs

t0ers 0gij 0�z; 0z

� �0dX

�Z Z

0X

t0Q;eij ;p 0eij 0 �p 0dX�

Z Z0X

t0Q;p;p 0p 0 �p 0dX: �27�


Since the incremental material response tensor 0Cijrs and the material response tensor t0Cijrs are symmetric

with respect to their indices, the energy bilinear form a��t0r ; �; �� in Eq. (27) is symmetric with respect toits arguments.

3. Material property design sensitivity analysis

Consider a structural system that is in the ®nal equilibrium con®guration at time t with a given design u.When the design is perturbed to u� sdu, the system reaches another equilibrium con®guration at timet � Dt, as shown in Fig. 2. Using the total Lagrangian formulation, the equilibrium equation of the per-turbed design in its ®nal equilibrium con®guration at time t � Dt and referred to initial con®guration attime 0 can be written as

au�sdut�Dt

0r; 0�r� �

� `u�sdu 0�r� �

8 0�r 2 0Z � 0P ; �28�

where the subscript u� sdu is used in Eq. (28) to denote dependency of these terms on the material propertydesign u. As the design perturbation becomes smaller, the difference between the ®nal equilibrium of theoriginal and the perturbed one also becomes smaller i.e., Dt! 0 as s! 0.

The ®rst-order variation of the nonlinear energy form and the load linear form of Eq. (28) with respect toits explicit dependence on the design variation u are de®ned as:

a0dut0r; 0�r� �

� d

dsau�sdu

t0~r; 0�r� ��

s�0�29�

and

`0du 0�r� �

� d

ds`u�sdu�0�r�

��s�0; �30�

where ``�'' is used to indicate explicit dependence on s is suppressed and 0�r is independent of s.The ®rst-order variation of the solution of the equilibrium (28) with respect to the material property

design variable u is de®ned as

t0r0 � d

dst�Dt

0 r�u� sdu�js�0 � lims!0

t�Dt0r�u� sdu� ÿ t

0r�u�s

: �31�

Using this de®nition, the order of the variation with respect to the design and the partial derivative withrespect to the coordinate can be interchanged [4] i.e.,

Fig. 2. Motion of original and perturbed bodies.


t0ri;jÿ �0 � t

0r0iÿ �

;j ; i; j � 1; 2; 3: �32�By the chain rule of di�erentiation, the ®rst-order variation of the energy form with respect to the

material property design variable u is obtained as

d

ds�au�sdu�t�Dt

0r�u� sdu�; 0�r��s�0� a0du�t0r; 0�r� � a�u�t0r; 0r0; 0�r�; �33�

where a�u is the ®rst-order variation of auwith respect to the design variable u implicitly through t0r, which is

obtained by using the linearization process from Eqs. (19)±(27). Taking the ®rst-order variation on bothsides of Eq. (28), and using Eqs. (30) and (33),

a�u�t0r; 0r0; 0�r� � `0du�0�r� ÿ a0du�t0r; 0�r� 8 0�r 2 0Z � 0P : �34�A general structural performance measure of the perturbed design u� sdu in the ®nal equilibrium

con®guration at time t � Dt and referred to the initial con®guration at time 0 can be expressed as

t�Dt0ws �

Z Z0X

g�t�Dt0r; 0rt�Dtr; u� sdu� 0dX; �35�

where

0rt�Dtr � 0rt�Dtz1; 0rt�Dtz2; 0rt�Dtz3; 0rt�Dtp� � �36�

and

0r�� o��o0x1

;o��o0x2

;o��o0x3

� �: �37�

Taking the ®rst-order variation of the performance measure with respect to the material property designvariable,

t0w0 � d

ds

Z Z0X

g t�Dtr�uÿ�� sdu�; 0rt�Dtr�u� sdu�; u� sdu

�0dX

��s�0

�Z Z

0X�gt

0r0r0 � g

0rt r0rr0 � gudu� 0dX; �38�

where

g0rt r �

ogot

0ri;j

� �is the partial derivative of g with respect to t

0ri;j. To obtain the design sensitivity, the expression of t0w0 must

be written in terms of du. That is, the ®rst two terms on the right-hand side of Eq. (38) need to be expressedexplicitly in du.

To eliminate the dependence on 0r0 for DSA in Eq. (38), an adjoint equation associated with eachperformance measure is introduced. Replacing 0r0 in Eq. (38) by a virtual response, 0

�/ � �0 �k1; 0�k2; 0

�k3; 0�q�and equating terms involving 0r0 in Eq. (38) to the energy bilinear form a�0X�t0r; 0/; 0

�/�, the adjointequation is de®ned as

a�u�t0r; 0/; 0�/� �

Z Z0X

gt0r0

�/h

� g0rt r 0r �/

i0dX 8 0

�/ 2 0Z � 0P ; �39�

where the adjoint solution 0/ is desired. Since 0r0 2 0Z � 0P , evaluate Eq. (39) at 0�/ � 0r0 to obtain

a�ut0r; 0/; 0r0ÿ � � Z Z

0Xgt

0r0r0

�� g

0rt r0rr0�

0dX: �40�


Similarly, since both 0�r and 0/ are in 0Z � 0P , Eq. (34) can be evaluated at 0�r � 0/ to yield

a�ut0r; 0r0; 0/ÿ � � `0du 0/� � ÿ a0du

t0r; 0/ÿ �

: �41�Using the symmetry of a�u�t0r ; �; ��,Z Z

0Xgt

0r0r0

�� g

0rt r0rr0�

0dX � `0du 0/� � ÿ a0dut0r; 0/ÿ �

; �42�

where the right side is explicit and linear in du, and can be evaluated once t0r and 0/ are obtained. The ®rst-

order variation of the general functional with respect to the material property design variable can be writtenexplicitly in terms of du as

t0w0 �Z Z

0Xgudu 0dXs � `0du 0/� � ÿ a0du

t0r; 0/ÿ �

: �43�

This equation can be used for material property DSA of hyperelastic nonlinear structural systems withlarge displacements, large rotations, and large strains.

For the DDM, Eq. (34) is the variational equation for 0r0. Once the solution 0r0 is obtained, the designsensitivity can be evaluated using Eq. (38). As in the adjoint Eq. (39), the tangent sti�ness matrix at the ®nalcon®guration of the original analysis is employed as the tangent sti�ness matrix for Eq. (34). In the DDM,the ®ctitious load on the right side of Eq. (34) needs to be calculated for each design parameter. For eachdesign parameter, one direct di�erentiation analysis needs to be performed to obtain the ®rst-order vari-ation of the structural response with respect to the design parameter.

The general results obtained in this section can be used to derive the material property design sensitivity.In this paper, the material constants C10 and C01 are chosen as design variables. For the totally incom-pressible material model that has been implemented in ABAQUS, the energy form and its ®rst-ordervariation with respect to the material property design variable, respectively, are

aX�t0r; 0�r� �Z Z

0X

t0Sij

t0�eij

�� t

0Q;pt0 �p�

0dX

�Z Z

0X

t0W �

;eij

�h� t

0Q;eij

�t0�eij ÿ

�t0J3 ÿ 1

�t0 �pi

0dX �44�

and

a0du�t0r; 0�r� �Z Z

0X�t0J1�;eij

dC10

n� �t0J2�;eij

dC01

o0dX

�Z Z

0X�t0I3�ÿ1=3�t0I1�;eij

��ÿ 1

3�t0I3�ÿ4=3�t0I1��t0I3�;eij

�dC10

� �t0I3�ÿ2=3�t0I2�;eij

�ÿ 2

3�t0I3�ÿ5=3�t0I2��t0I3�;eij

�dC01

�0dX; �45�

where

t0W � t

0J1;t0J2

ÿ � � C10t0J1

ÿ ÿ 3�� C01

t0J2

ÿ ÿ 3�; �46�

t0Q � ÿt

0p t0J3

ÿ ÿ 1� �47�

and

t0I1

ÿ �;eij� 2dij

t0I2

ÿ �;eij� 4 dij 1

ÿÿ � t0ell

�ÿ t0eji

�t0I3

ÿ �;eij� 2dij 1

ÿ � 2t0ell

�ÿ 4t0eji � 4eipqet

jst0eqt

�48�

and dij is the Dirac delta.


4. Shape design sensitivity analysis

Consider the original domain (initial design) 0X at time s� 0, as shown schematically in Fig. 3. Theshape change from the initial design 0X to the perturbed design 0Xs can be viewed as a dynamic process withthe parameter s playing the role of design time. The mapping T : 0x! 0xs�0x�; 0x 2 0X, from 0X to 0Xs isone-to-one and onto where

0xs � T 0x; sÿ �

0Xs � T 0X; sÿ �

0Cs � T 0C; sÿ � �49�

and 0x � 0x1;0x2;

0x3� �T . After the shape design perturbation, the material point 0x 2 0X moves to 0xs 2 0Xs.Treating of s as time, a design velocity ®eld can be de®ned as

V 0xs; sÿ � � d0xs

ds� dT 0x; s� �

ds� oT 0x; s� �

os: �50�

In a neighborhood of s � 0, under certain regularity hypotheses,

T 0x; sÿ � � T 0x; 0

ÿ �� sdT 0x; 0� �

ds�O�s2� � 0x� sV 0x; 0

ÿ ��O�s2�; �51�

where 0x � T 0x; 0� � and V 0x� � � V 0x; 0� �.The variational equilibrium equation for the perturbed domain 0Xs in its ®nal equilibrium con®guration

at time t � Dt is

a0Xs

t�Dt0rs; 0�rs

� ��Z Z

0Xs

t�Dt0 Sij

t�Dt0�eij

0dXs �Z Z

0Xs

t�Dt0 Q;p

t�Dt0 �p 0dXs

�Z Z

0Xs

t�Dt0 f 0

si 0�zsi0dXs �

Z0Cs

t�Dt0 T 0

si 0�zsi0dCs

� `0Xs�0�rs� 8 0�rs 2 0Zs � 0Ps; �52�

Fig. 3. Shape design perturbation and nonlinear deformation.


where 0Zs and 0Ps are the spaces of kinematically admissible virtual displacement and hydrostatic pressureon 0Xs, respectively. In Eq. (52), the subscript 0Xs is used to denote dependency of these terms on the shapedesign. Assuming that the solution t�Dt

0rs of Eq. (52) in the ®nal equilibrium con®guration of the perturbeddomain is a smooth function of the design, the mapping t�Dt

0rs�0xs� � t�Dt0 rs�0x� sV �0x�� can be de®ned on

0X. Then the solution t�Dt0rs depends on s in two ways. First, t�Dt

0rs�0xs� is the solution of the equilibriumequation of (52) on 0Xs. Second, t�Dt

0rs�0xs� is evaluated at a point 0xs that moves with s. Thus, the pointwisematerial derivative of t

0r at 0x 2 0X, if it exists, is de®ned as

0 _r�0x� � d

dst�Dt

0rs0xÿ� � sV 0x

ÿ ��s�0

� lims!0

t�Dt0rs

0x� sV 0x� �� ÿ t0 r�0x�

s:

�53�

If the solution t�Dt0rs has a regular extension to a neighborhood 0Us of the closed set 0 �Xs, then

0 _r 0xÿ � � 0r0 0x

ÿ �� 0rtr TV 0xÿ �

; �54�where

0r0�0x� � d

dst�Dt

0rs0xÿ �ÿ ��

s�0

� lims!0

t�Dt0rs

0x� � ÿ t0r 0x� �

s�55�

is the partial derivative of t0r with respect to s and 0rtr � �0rtz1; 0rtz2; 0rtz3; 0rtp�. Using the above

de®nitions, it can be seen that the partial derivatives with respect to 0x commute with the derivative withrespect to s [7] i.e.,

0ri;j

ÿ �0 � 0r0iÿ �

;j: �56�

The assumption that the energy form and load linear form are di�erentiable with respect to the shapedesign variable are used in the following derivation. Taking material derivatives of both sides of equilib-rium (52), and using the chain rule of di�erentiation and the linearization process as in material propertysensitivity analysis,

a0Xt0r; 0�r� �h i0

� a�0Xt0r; 0 _r; 0�r� �

� a0vt0r; 0�r� �

� `0v 0�r� �

8 0�r 2 0Z � 0P : �57�

The material derivatives of the energy form and load linear form are obtained, respectively, as

a0Xt0r; 0�r� �h i0

� a�0Xt0r; 0r0; 0�r� �

� a0Xt0r; 0�r0� �

� a00X

t0r; 0�r� �

� a�0Xt0r; 0 _r�

ÿ 0rtr TV ; 0�r�� a0X

t0r; 0 _�r�

ÿ 0rt�r TV�� a0

0Xt0r; 0�r� �

�58�

and

`0v�0�r� �Z Z

0X

t0f T

0�z0h

� 0r�t0f 0�z�TV � t0f T

0�z div Vi

0dX

�Z

0C

t0T T

0�z0h

� 0r�t0T 0�z�T 0n� 0Ht0T T

0�zi�V T 0n� 0dC

�Z Z

0X

t0f T

0 _�z�hÿ 0rt�zTV

�� 0r�tf 0�z�TV � t

0f T0�z div V

i0dX

�Z

0C

t0T T

0 _�z�hÿ 0rt�zTV

�� 0r�t0T 0�z�T 0n� 0Ht

0T T0�zi�V T 0n� 0dC; �59�

where 0n is a unit vector normal to the boundary 0C, and 0H is the curvature of the boundary 0C in R2 andtwice the mean curvature in R3. For the material derivative of the load linear form in Eq. (59), 0f 0 � 0T 0 � 0


is used. The linearized energy form a�0X�t0r; 0r0; 0�r� in Eq. (58) can be obtained from Eq. (27). Other terms inEq. (58) are

a00X�t0r; 0�r� �

Z Z0Xr t

0Sijt0�eij

h� t

0Q;p0 �piT

V 0dX�Z Z

0X

t0Sij

t0�eij

�� t

0Q;p0 �p�

div V 0dX �60�

and

a0Xt0r; 0�r� �

�Z Z

0X

t0Sij

t0�eij

0dX�Z Z

0X

t0Q;p

t0 �p 0dX: �61�

Since the virtual variable 0�r is arbitrary, it can be chosen so that 0�r is constant along 0xs � 0x� sV �0x�;i.e., 0�r�0x� sV �0x�� 0�r�0x�. Using Eq. (54), it follows that

0 _�r�0x� � 0�r0 0xÿ �� 0r�rTV 0x

ÿ � � 0: �62�

Thus, from Eqs. (57) and (58),

a0v�

t0r; 0�r

�� ÿa�0X

t0r; 0rtrTV ; 0�r� �

ÿ a0Xt0r; 0rt�rTV� �

� a00X

t0r; 0�r� �

�63�

and, from Eq. (59),

`0v�0�r� �Z Z

0X0�z�0rtf T V �h

� t0f T

0�z div Vi

0dX

�Z

0C

(ÿ t

0T T0rt�zT V� �

� 0r t0T T

0�z� �T

0n

"� 0H t

0T T0�z

#V T0n� �)

0dC: �64�

Using these expressions, Eq. (57) can be rewritten in the form

a�0X�

t0r; 0 _r; 0�r

�� `0v 0�r

� �ÿ a0v

t0r; 0�r� �

8 0�r 2 0Z � 0P : �65�

Now, consider a general performance measure given in the domain integral form as

t�Dt0ws �

Z Z0Xs

g t�Dt0r0; 0rt�Dtrs

� �0dXs; �66�

where

0rt�Dtrs � 0rt�Dtzs1; 0rt�Dtzs2

; 0rt�Dtzs3; 0rt�Dtps

� � �67�

and the function g is assumed to be continuously differentiable with respect to its arguments. Taking thematerial derivative of t�Dt

0ws,

t0w0 �

Z Z0X

gt0r0r0

h� g

0rt r 0rr0 � 0rgTV � g div V�

0dX: �68�

Using Eq. (54), Eq. (68) can be rewritten as

t0w0 �Z Z

0Xgt

0r0 _r

h� g

0rt r 0r_r ÿ gt0r 0rtrTV� �

ÿ g0rt r 0r

�0rtrTV

�� 0rgTV � g div V

i0dX: �69�

To obtain an explicit expression of t0w0 in terms of the design velocity ®eld V, the ®rst two terms on the

right side of Eq. (69) must be expressed in terms of V. In the AVM, to eliminate the dependence on 0 _r inEq. (69), the adjoint Eq. (39) is used. Since 0 _r, 0�r, and 0

�/ are in the space of kinematically admissible virtualdisplacement and hydrostatic pressure 0Z � 0P , evaluating Eq. (39) at 0

�/ � 0 _r yields


a�0Xt0r; 0/; 0 _rÿ � � Z Z

0Xgt

0r0 _r

h� g

0rt r 0r_ri

0dX �70�

and evaluating Eq. (65) at 0�r � 0/,

a�0Xt0r; 0 _r; 0/� �

� `0V 0/� �

ÿ a0V�

t0r; 0/

�: �71�

Using the symmetry of the energy bilinear form a�0X�t0r ; �; ��, and Eqs. (70) and (71),Z Z0X

gt0r0 _r

h� g

0rt r 0r_ri

0dX � `0V 0/� �

ÿ a0V�

t0r; 0/

��72�

and

t0w0 � `0V 0/

ÿ �ÿ a0Vt0r; 0/ÿ �� Z Z

0X

�ÿ gt0r�0rtrTV � ÿ g

0rt r 0r�0rtrTV � � 0rgTV � g div V�

0dX:

�73�The right side of Eq. (73) can be evaluated once the structural response t

0r and the adjoint response 0/ areobtained and the design velocity ®eld V is determined.

As an example, consider the hydrostatic pressure performance measure at an isolated point 0x̂,

t0w � t

0p 0x̂� �

�Z Z

0Xd̂�

0xÿ 0x̂�

t0p 0x� �

0dX; �74�

where

t0p � ÿ 1

3tr11

�� tr22 � tr33

��75�

is the hydrostatic pressure and trii, i� 1, 2, 3, are Cauchy stresses at con®guration time t. There exists arelationship between Cauchy stress and second Piola±Kichhoff stress for the hyperelastic material

trij � t0Fik

t0Skl

t0Fjl; �76�

where t0F is the deformation gradient with components t

0Fij � dij � t0zi;j, i; j; k; l � 1; 2; 3. Thus, the material

derivative of the hydrostatic pressure performance measure can be rewritten as

t0w0 � t

0 _p�0x̂� �Z Z

0Xd̂�

0xÿ 0x̂�

t0 _p 0xÿ �

0dX �Z Z

0X

�ÿ 1

3d̂ 0x�ÿ 0x̂

�t _r11

�� t _r22 � t _r33

��0dX

�Z Z

0X

�ÿ 1

3d̂ 0x�ÿ 0x̂

�t0

_Fpkt0Skl

t0Fpl

h� t

0Fqkt0_Skl

t0Fql � t

0Frkt0Skl

t0

_Frl

i�0dX; �77�

where k; l; p; q; r � 1; 2; 3. The adjoint equation for hydrostatic pressure performance measure is then

a�0X

t0r; 0/; 0

�/� �

� ÿZ Z

X

1

3d̂ 0x�ÿ 0x̂

�0�kp;k

t0Skl

t0Fpl

n� t

0Frkt0Skl0

�kr;l

� t0Fqk

t0W �

;ekl ;ers 0ers�t0z; 0�k�

h� t

0Q;ekl;p0�qi

t0Fql

o0dX 8 0

�/ 2 0Z � 0P : �78�

Using the adjoint response 0/, the shape design sensitivity can be written as

t0w0 � `0v 0/� � ÿ a0v

t0r; 0/ÿ � � Z Z

0X

1

3d̂ 0x�ÿ 0x̂

�n0rtzpV;kÿ �

t0Skl

t0Fpl � t

0Frkt0Skl 0rtzrV;l� �

ÿ t0Fqk

t0W �

;ekl ;ers 0ers�t0z; 0rtzT V �n

� t0Q;ekl;p0rtpTV ÿ t

0Sij;kVk

ot0Fql

o0dX: �79�


For the DDM, if t0r is known as the solution of the equilibrium equation at the ®nal con®guration and 0 _r

is the solution of Eq. (65) for each shape design parameter, then Eq. (69) can be used to obtain the shapedesign sensitivity of the functional t

0w.As in the material property DSA, if FEA is used for analysis, thetangent sti�ness matrix at the ®nal equilibrium con®guration for original analysis is used as the sti�nessmatrix of the linear Eq. (65).

For the totally incompressible material model, the ®rst-order shape variation (63) is

a0Vt0r; 0�r� �

� ÿa�0Xt0r; 0rtrTV ; 0�r� �

ÿ a0Xt0r; 0r�rTV� �

� a00X

t0r; 0�r� �

; �80�

where

a00X

t0r; 0�r� �

�Z Z

0X

r t0Sij0�eij

hÿ t

0J3

ÿ ÿ 1�

0 �piT

V 0dX

�Z Z

0X

t0Sij0�eij

hÿ �t0J3 ÿ 1� 0 �p

idiv V 0dX:

�81�

5. Numerical examples of design sensitivity

In this section, two numerical examples, the engine mount and bushing, are employed to verify theaccuracy and e�ciency of the proposed material property and shape DSA methods. For shape DSA, thevelocity computation method proposed by Choi and Chang [27] is used. Both the boundary displacementand isoparametric methods are employed. The geometric modeling tool, PATRAN [15], is used for gen-erating the geometric and FEA models. The structural analysis is carried out using the isoparametric planestrain element CPE4RH and 3-D solid element C3D8RH in ABAQUS [14], where the bilinear or trilinearshape functions for the displacement and constant hydrostatic pressure on each element are used. Thetangent sti�ness matrix at the ®nal equilibrium con®guration of the original structural analysis is stored forthe AVM or DDM to obtain the adjoint response or the ®rst-order variation of the structural response,respectively.

5.1. Engine mount

The engine mount shown in Fig. 4 is used as the ®rst numerical example. Due to symmetry, one-half of itis modeled. The engine mount is treated as a plane strain problem. The outer boundary of the engine mountis attached to a metal frame of the vehicle body and thus ®xed. In this model, the material in the shadowpart is metal with Young's modulus E � 2:068� 107 N/cm2, and the material in other part is rubber withmaterial constants C10� 19.31 N/cm2 and C01� 8.27 N/cm2. Three concentration forces, F1� 88.97 N,F2� 177.94 N, and F3� 88.97 N, act at nodes 94,107, and 121, respectively, as shown in Fig. 4(b). Themount is meshed using 154 4-node plain strain elements CPE4RH with a total of 207 nodes and 325 de-grees-of-freedom. The displacement, stress, and hydrostatic pressure are chosen as performance measures.

The material constants C10 and C01 are chosen as material property design variables. Table 1 showsmaterial property design sensitivity results that are compared with central ®nite difference results wherew�uÿ du� and w�u� du� are values of the performance measure at perturbed designs uÿ du and u� du,respectively; Dw is the central ®nite difference result; and w0 is the predicted perturbation obtained from theproposed DSA method. In this paper, a design perturbation of 1% (i.e., du� 0.01 u) is used for thecomputation of Dw. As can be seen in this table, design sensitivity results predict central ®nite differenceresults very accurately.

For shape DSA, the geometry of the engine mount is shown in Fig. 4(a). The boundary curves C1±C4 areselected as the shape design boundaries and locations of grid points G1±G19 are selected as shape designparameters. To avoid stress concentration, the design boundaries C1±C4 are modeled using cubic geometriccurves and C1 ± continuity is maintained at joints. In this example, the isoparametric mapping method isused to obtain the design velocity on the boundary and the boundary displacement method [27] is used to


compute the domain design velocity ®eld. In Tables 2±4, grid points G4, G5, G9, and G13 are perturbed in they-direction. Like the material property DSA case, these design sensitivity results predict the central ®nitedifference results very accurately.

5.2. Bushing

The bushing shown in Fig. 5 is used as the second numerical example. Half of the bushing is modeled dueto symmetry. The material constants are C10� 80 N/cm2 and C01� 100 N/cm2. The outer surface of thebushing is formed by revolving boundaries C1±C4. The design boundaries C1±C4 are modeled using C1

continuous lines and parameterized by cubic geometric curves. The inner surface of the bushing is formedby revolving boundaries C5±C8. The radii of the outer surface at grid points G1±G5 are 3.0 cm and the radiusof the inner surface is 1.0 cm. The length of the bushing is 5.0 cm.

Table 1

Displacement sensitivity with respect to material property design variable C10

Node ID Function w(u ) du) (cm) w(u + du) (cm) Dw (cm) w0 (cm) w0/Dw (%)

34 z1(x) 0.10460E+00 0.10452E+00 )0.40023E)04 )0.40021E)04 100.48

34 z2(x) 0.34028E+00 0.34015E+00 )0.67819E)04 )0.67839E)04 100.03

58 z1(x) 0.18232E)01 0.18230E)01 )0.52766E)06 )0.48702E)06 92.30

58 z2(x) 0.21078E+00 0.21009E+00 )0.45448E)04 )0.45418E)04 99.93

62 z1(x) 0.15433E)01 0.15429E)01 )0.19279E)04 )0.19832E)04 102.87

62 z2(x) 0.7485E+00 0.74813E+00 )0.21928E)03 )0.21924E)03 99.98

76 z1(x) )0.33971E)01 )0.33948E)01 0.11479E)04 0.11421E)04 99.49

76 z2(x) 0.97432E+00 0.97374E+00 )0.29648E)03 )0.29642E)03 99.98

135 z1(x) )0.13172E+01 )0.13177E)01 )0.21328E)04 )0.21438E)04 100.52

135 z2(x) 0.11756E+01 0.11749E+01 )0.35042E)03 )0.35032E)03 99.98

Fig. 4. 2-D engine mount.


To simulate ®rst the assembly process where the bushing is being squeezed into a cylindrical metal pipe,a prescribed displacement constraint is imposed at the outer surface of the bushing to deform it to acylindrical shape with the outer radius equal to 2.7 cm. At the second step, two forces with magnitudes of1000 N are applied at node 1 in axial and radial directions as shown in Fig. 5(b). The bushing is modeledusing 1,024 8-node solid elements C3D8RH with a total of 1445 nodes and 4164 degree-of-freedoms.

The boundary curves C1±C4 are chosen as shape design boundaries, and locations of grid points G1±G5

in the radial direction are selected as shape design parameters. To make the design symmetric, three

Table 3

Stress sensitivity with respect to location of grid point G13 in y-direction

Element ID Function w(x ) dx) (N/cm2) w(x + dx) (N/cm2) Dw (N/cm2) w0 (N/cm2) w0/Dw (%)

21 S11(x) 0.43700E+02 0.43936E+02 0.11827E+00 0.11807E+00 99.83

21 S12(x) )0.38917E+01 )0.40364E+01 )0.72382E)01 )0.72372E)01 99.98

21 S22(x) )0.55881E+00 )0.73272E+00 )0.86955E)01 )0.86959E)01 100.00

44 S11(x) )0.77567E+02 )0.77109E+02 0.22996E+00 0.22929E+00 99.71

44 S12(x) )0.66236E+02 )0.66667E+02 )0.21421E+00 )0.21486E+00 100.30

44 S22(x) )0.54213E+02 )0.54057E+02 0.78692E)01 0.78480E)01 99.73

102 S11(x) 0.43879E+02 0.43620E+02 )0.12936E+00 )0.13087E+00 101.66

102 S22(x) 0.29872E+02 0.29534E+02 )0.17107E+00 )0.16996E+00 99.18

117 S12(x) )0.41429E+02 )0.41511E_02 )0.41027E)01 )0.42710E)01 104.10

117 S22(x) )0.24404E+02 )0.24244E+02 0.80613E)01 0.75457E)01 93.60

Table 2

Displacement sensitivity with respect to locations of grid points G4 and G5 in y-direction

Node ID Function w(x ) dx) (cm) w(x + dx) (cm) Dw (cm) w0 (cm) w0/Dw (%)

26 z1(x) )0.25217E)01 )0.18194E)01 0.35111E)02 0.34754E)02 98.98

26 z2(x) 0.85632E)01 0.96006E)01 0.51850E)02 0.51029E)02 98.42

38 z1(x) 0.11592E+00 0.10214E+00 )0.68903E)02 )0.70839E)02 102.81

38 z2(x) 0.12597E+01 0.12077E+01 )0.25970E)01 )0.25188E)01 96.99

56 z1(x) )0.56575E)01 )0.36721E)01 0.99271E)02 0.98651E)02 99.38

56 z2(x) 0.55896E+00 0.50555E+00 )0.26707E)01 )0.28257E)01 105.80

124 z1(x) 0.13566E)05 0.17610E)05 0.20216E)06 0.21723E)06 107.45

124 z2(x) 0.13783E+01 0.13110E+01 )0.33655E)01 )0.34142E)01 101.45

184 z1(x) 0.97032E)02 0.11079E)01 0.68800E)03 0.66645E)03 96.87

184 z2(x) 0.16834E+00 0.13368E+00 )0.17326E)01 )0.16698E)01 96.38

Table 4

Hydrostatic pressure sensitivity with respect to location of grid point G9 in y-direction


24 P(x) 0.22460E+02 0.22076E+02 )0.19169E+00 )0.19241E+00 100.38

39 P(x) 0.16327E+02 0.16545E+02 0.10870E+00 0.11088E+00 102.00

44 P(x) 0.30002E+02 0.30119E+02 0.58995E)01 0.57544E)01 97.54

52 P(x) 0.21579E+02 0.22375E+02 0.39774E+00 0.39818E+00 100.11

56 P(x) 0.16378E+02 0.17579E+02 0.60058E+00 0.60009E+00 99.92

99 P(x) )0.25940E+02 )0.26230E+02 )0.14447E+00 )0.14498E+00 100.35

106 P(x) 0.40072E+00 0.35791E+00 )0.21409E)01 )0.21423E)01 100.06

110 P(x) 0.60769E+01 0.60863E+01 0.47011E)02 0.46289E)02 98.46

118 P(x) 0.39136E+01 0.39120E+01 )0.82409E)03 )0.80016E)03 97.10

139 P(x) )0.20402E+02 )0.20556E+02 )0.77555E)01 )0.77534E)01 99.98


independent design parameters G1±G3 are selected. The grid point G4 is linked to G2, and grid point G5 islinked to G1. Thus, the curve C3 is linked to C2, and C4 is linked to C1. Curves C1 and C2 are chosen asindependent shape design boundaries. For this example, the isoparametric mapping method is used toobtain the design velocity ®eld.

The displacement and hydrostatic pressure are selected as performance measures. Perturbations of threeshape design parameters are considered for the veri®cation of the design sensitivity results, which are shownin Tables 5±7. The results in Tables 5 and 6are for the loading case 1 (prescribed displacement at the outersurface) and the results in Table 7 are for the loading case 2 (1000 N at node 1 in axial and radial direc-tions). In Table 5, the location of grid point G1 is perturbed in the radial direction and other design pa-rameters are ®xed. In Table 6, the grid point G3 is perturbed in the radial direction and other designparameters are ®xed. In Table 7, the grid point G2 is perturbed in the radial direction and the other designparameters are ®xed. In all tables, accurate design sensitivity predictions are obtained using the proposedDSA method.

Fig. 5. 3-D bushing.

Table 5

Displacement sensitivity with respect to location of grid point g1 in radial direction (loading case 1)

Node ID Function w(x ) dx) (cm) w(x + dx) (cm) Dw (cm) w0 (cm) w0/Dw (%)

18 z1(x) 0.14687E+00 0.14491E+00 )0.97871E)03 )0.98127E)03 100.26

97 z2(x) 0.41321E)02 0.43055E)02 0.86724E)04 0.81724E)04 94.23

159 z1(x) 0.20836E+00 0.20321E+00 )0.25747E)02 )0.26247E)02 101.94

330 z2(x) )0.24633E+00 )0.24106E+00 0.26368E)02 0.25868E)02 98.10

472 z3(x) 0.37932E+00 0.37386E+00 )0.27281E)02 )0.26775E)02 98.14

518 z1(x) )0.76902E)01 )0.75748E)01 0.57714E)03 0.60214E)03 104.33

653 z2(x) )0.20124E)01 )0.25025E)01 )0.24506E)02 )0.23505E)02 95.92

857 z1(x) 0.11932E)01 0.12405E)01 0.23649E)03 0.23149E)03 97.89

964 z2(x) )0.25085E+00 )0.24709E+00 0.18807E)02 0.19309E)02 102.66

1021 z3(x) 0.29931E+00 0.29680E+00 )0.12549E)02 )0.13049E)02 103.98


6. Design optimization

In this paper, the volume or area of structure is selected as the objective function. Two types of con-straints are considered, the tensile hydrostatic stress and sti�ness. It is known that when the tension hy-drostatic stress is larger than 5E=6, the structure fails [10]. The mathematical formulation for shape designoptimization of structural components with the hyperelastic material is

Minimize F �b� �Z Z

0X

0dX; �82�

Subject to :

gi � ÿ ri11 � ri

22 � ri33

3

� �� 5E

6P 0:0; i � 1ÿ Ni

�83�

hj � F j

zjÿ aj � 0; aj > 0; j � 1ÿ Nj �84�

with side constraints,

blk 6 bk 6 buk; k � 1ÿ Nb; �85�where rii is the Cauchy stress at the ®nal equilibrium con®guration; Fj the jth external force actingon certain nodes which has displacement zj in the direction of the force; and blk and buk are lower- and

Table 7

Hydrostatic pressure sensitivity with respect to location of grid point G2 in radial direction (loading case 2)


47 p(x) 0.62921E+03 0.62237E+03 )0.34227E+01 )0.34726E+01 101.45

126 p(x) 0.14731E+03 0.14602E+03 )0.64883E+00 )0.66884E+00 103.08

257 p(x) 0.25136E+03 0.24887E+03 )0.12422E+01 )0.12372E+01 99.60

381 p(x) 0.59043E+02 0.58534E+02 )0.25448E+00 )0.25149E+00 98.82

439 p(x) 0.25182E+03 0.24963E+03 )0.10952E+01 )0.11453E+01 104.57

562 p(x) 0.30912E+03 0.31199E+03 0.14334E+01 0.14635E+01 102.10

636 p(x) 0.32628E+03 0.32410E+03 )0.10918E+01 )0.10721E+01 98.20

784 p(x) 0.75927E+03 0.74586E+03 )0.67098E+01 )0.66999E+01 99.85

938 p(x) 0.64487E+03 0.63535E+03 )0.47605E+01 )0.47655E+01 100.11

1012 p(x) 0.58867E+03 0.58122E+03 )0.37265E+01 )0.37216E+01 99.84

Table 6

Hydrostatic pressure sensitivity with respect to location of grid point G3 in radial direction (loading case 1)


58 p(x) 0.23076E+03 0.23021E+03 )0.27514E+00 )0.25117E+00 91.29

134 p(x) 0.16589E+03 0.16577E+03 )0.59841E)01 )0.60075E)01 100.39

275 p(x) 0.59266E+03 0.59285E+03 0.94326E)01 0.96002E)01 101.78

462 p(x) )0.18890E+03 )0.18877E+03 0.65184E)01 0.63129E)01 96.85

521 p(x) 0.70882E+03 0.71367E+03 0.24251E+01 0.24199E+01 99.79

687 p(x) 0.59102E+03 0.59291E+03 0.94506E+00 0.98253E+00 103.96

743 p(x) )0.11460E+03 )0.11575E+03 )0.57524E+00 )0.56658E+00 98.49

816 p(x) 0.56446E+03 0.56389E+03 )0.28501E+00 )0.28269E+00 99.19

983 p(x) 0.60772E+03 0.60765E+03 )0.36248E)01 )0.36956E)01 101.95

1021 p(x) 0.61936E+03 0.62090E+03 0.77152E+00 0.77160E+00 100.01


upper-bounds of the kth design parameter, respectively. In Eq. (84), ai is the stiffness requirement that maybe obtained from system level design optimization. Also, Ni, Nj, and Nb are the numbers of tensile hy-drostatic stress constraints, stiffness constraints, and design parameters, respectively. During the optimi-zation process, the equality constraint (84) is converted to two inequality constraints with a tolerance e,

�gj1 �F j

zjajÿ 1:06 e; �86�

�gj2 � ÿF j

zjaj� 1:06 e: �87�

For design optimization, the modi®ed feasible direction (MFD) in DOT [16] is used. The ¯owchart of thedesign optimization process is shown in Fig. 6.

6.1. Engine mount

The geometric shape and physical dimensions at the initial design are shown in Fig. 4(a). For this ex-ample, the isoparametric mapping method [27] is used to obtain the design velocity ®eld on the boundaryand domain. Boundary curves C1±C4 de®ned in Table 8 are selected as shape design boundaries. 18 shapedesign parameters are de®ned and listed in Table 9.

Two external loading cases, which correspond to structural sti�ness requirements are considered. The®rst loading case is F 1

1 � 8:9 N, F 12 � 17:79 N, and F 1

3 � 8:9 N. The second loading case is F 21 � 88:97 N,

F 22 � 177:94 N, and F 2

3 � 88:97 N. Other data are Ni � 240, Ni � 2, Nb � 18, a1 � 4:05 N/cm, anda2 � 58:06 N/cm. The maximum allowable tensile hydrostatic stress is 137.9 N/cm2. The side constraints fordesign parameters are listed in Table 10.

Fig. 6. Flowchart of shape design optimization.


Table 9

Design parameters of the engine mount

Design parameter De®nition of design parameter

b1 Position of grid G4 in x-direction

b2 Position of grid G4 in y-direction

















Table 10

Design parameters of the engine mount at initial and optimum design

Design parameter (in.) Initial design (cm) Lower bound (cm) Upper bound (cm) Optimum design (cm)

b1 0.22337E+01 0.20320E+01 0.25400E+01 0.22685E+01

b2 0.64005E+01 0.50800E+01 0.71120E+01 0.69720E+01

b3 0.15011E+01 0.12700E+01 0.20320E+01 0.14102E+01

b4 0.59380E+00 0.50800E+01 0.73660E+01 0.60345E+01

b5 0.15781E+01 0.127000E+01 0.25400E+01 0.15829E+01

b6 0.49047E+01 0.22860E+01 0.76200E+01 0.51597E+01

b7 0.25768E+01 0.12700E+01 0.27940E+01 0.27780E+01

b8 0.52382E+01 0.38100E+01 0.55880E+01 0.53330E+01

b9 0.27424E+01 0.12700E+01 0.27940E+01 0.27935E+01

b10 0.27546E+01 0.12700E+01 0.27940E+01 0.27935E+01

b11 0.18514E+01 0.12700E+01 0.25400E+01 0.19010E+01

b12 0.23696E+01 0.12700E+01 0.25400E+01 0.20826E+01

b13 0.18514E+01 0.20320E+00 0.25400E+01 0.17767E+01

b14 0.11946E+01 0.50800E+00 0.16510E+01 0.15245E+01

b15 0.30455E+01 0.12700E+01 0.33020E+01 0.30475E+01

b16 0.12875E+01 0.25400E+00 0.15240E+01 0.13309E+01

b17 0.25009E+01 0.22860E+01 0.38100E+01 0.29195E+01

b18 0.22098E+01 0.20320E+01 0.38100E+01 0.27104E+01

Table 8

Design boundaries of the engine mount

Design boundary Curves

C1 Curve sections between grid point G2±G6





After 21 iterations, the optimum design is obtained. The cost function history is shown in Fig. 7. Thereduction of the total area is 11.57 %. The history of constraints corresponding to Eqs. (83) and (84) areshown in Fig. 8. The design parameter values at the optimum design are listed in Table 10. The optimumdesign is shown Fig. 9(b). It can be seen from the optimum design shape that the optimum design boundarycurves have less curvature than the initial design, which help reduce stress concentration. The lower arm hasbeen removed signi®cantly, because the tensile hydrostatic stress at this part is lower. Also it can be seenthat the upper arm has been reinforced to satisfy structural sti�ness constraints. The reduction of themaximum tensile hydrostatic stress at the optimum design is 19.34% as shown in Fig. 9.

6.2. Bushing

Three independent shape design parameters shown on the geometric model in Fig. 5(a) are listed inTable 11. The isoparametric mapping method [27] is used to obtain the design velocity ®eld on the

Fig. 7. Cost function history of the engine mount.

Fig. 8. Sti�ness constraint function history of the engine mount.


boundary and domain. At the initial design, structural sti�nesses are a1 � 1:4292� 103 N/cm anda2 � 1:6722� 104 N/cm. The desired structural sti�nesses corresponding to loading cases 1 and 2 are1:80� 103 N/cm and 1:65� 104 N/cm, respectively. The maximum allowable tensile hydrostatic stress is

Table 11

Design parameters of the bushing

Design parameter De®nition of design parameter

b1 Position of grid G1 in radial-direction



Fig. 9. Initial and optimum design of the engine mount with hydrostatic stress.

Fig. 10. Cost function history of the bushing.


600 N/cm2. A total of 2048 tensile hydrostatic stress constraints de®ned on ®nite elements and twostructural sti�ness constraints de®ned at node 1 are imposed in this example. The side constraints are2:76 b16 3:2, 2:76 b26 3:5, and 2:76 b36 3:5.

After 7 iterations, an optimum design is obtained. The volume of 63.99 cm3 at the initial design is re-duced to 59.48 cm3 at the optimum design. The cost function history is shown in Fig. 10. The optimumdesign parameter values are listed in Table 12. The optimum geometric and ®nite element models of thebushing are shown in Fig. 11. The tensile hydrostatic stress contours at initial and optimum designs areshown in Figs. 12 and 13 for loading case 2.

Fig. 11. Geometric and ®nite element models of the bushing at optimum design.

Fig. 12. Tensile hydrostatic stress of the bushing at initial design for loading case 2.

Table 12

Design parameter of the bushing at initial and optimum design

Design parameter Initial design (cm) Lower bound (cm) Upper bound (cm) Optimum design (cm)

b1 0.30000E+01 0.27000E+0 0.32000E+01 0.31836E+01

b2 0.30000E+01 0.27000E+01 0.35000E+01 0.27741E+01

b3 0.30000E+01 0.27000E+01 0.35000E+00 0.28733E+01


7. Conclusion

A general formulation of DSA for 3-D solids with the hyperelastic material is presented. The MVP isemployed to derive material and shape design sensitivity expressions in analytical form. For materialproperty DSA, the coe�cients of the Mooney±Rivlin polynomial energy density function are used as designvariables. For shape DSA, the boundary displacement and isoparametric mapping methods are used toobtain the design velocity ®eld. Both the AVM and DDM are used to obtain design sensitivity expressions.The tangent sti�ness matrix at the ®nal equilibrium con®guration of the original design is used in adjointvariable and direct di�erentiation analysis. The adjoint equation and the direct di�erentiation equation arelinear even though the governing structural equation is nonlinear.

In this paper, the Mooney±Rivlin incompressible model of ABAQUS is used for numerical implemen-tation. The design sensitivity coe�cient is calculated outside ABAQUS by postprocessing the output data.The DSA methods presented in this research yield accurate design sensitivity information in a computa-tionally e�cient manner.

The mathematical model of shape design optimization is de®ned. Due to the material property, thetensile hydrostatic pressure is considered as the failure criterion, and the structural sti�ness is chosen as thedesign requirement. Reasonable optimum designs are obtained for both examples.

Acknowledgements

Research is supported by the Automotive Research Center sponsored by the US Army TARDEC.

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