Structural Optimization 3, 79-88 (1991)
StructuralOptimization © Springer-Verlag 1991
First- and second-order structures
shape sensitivity
analysis of
K. Dems
L6di Technical University 1-26, ul. Zwirki 36, PL-90924 Lddl, Poland
A b s t r a c t . The first- and second-order sensitivities with respect to varying structural shape are discussed for an arbitrary stress, strain and displacement functional. It is assumed that only the traction-free boundary of a structure can undergo the shape modification described by a set of shape design parameters. The first derivatives of a functional with respect to these parameters are derived using both the direct and adjoint approaches. Next the second derivatives are obtained using the mixed approach in which both the direct and adjoint first-order solutions are used. The general results are particularized for the case of complementary and potential energy of a structure. Some simple examples illustrate the theory presented.
1 Introduction There are numerous problems of mechanics, where we need to assess the change of any local or global structural response characteristic due to variation of the structural shape. For instance, in redesign or optimization problems the variations of state fields and of an objective functional are evaluated at each redesign step. Generally, if G denotes any scalar functional depending on the state fields of a structure and bp constitute a set of shape parameters of a structural boundary, the sensitivity analysis is aimed at expressing the variation of G in terms of bp, for instance in the form of polynomial expansion AG = SpSbp + 1RpqSbpSbq + . . . . 5Gq- 152G+..., where Sp and Rpq are the first-order sensitivity vector and the second-order sensitivity matrix, respectively. This class of problems has been illustrated extensively in the literature. The first-order sensitivity analysis was generally discussed, for instance, by Haug and Arora (1978), Dems and Mr6z (1984), Dems and Haftka (1989) and others. The second-order sensitivity analysis with respect to stiffness properties was presented by Haug (1981), Haftka
(1982), Dems and MrSz (1985) and Haftka and Mrdz (1986). The first- and second-order shape sensitivities were considered by Fujii (1986). The present paper extends the analysis presented by Dems and Mrdz (1984) and Dems and Haftka (1989), and discusses the general equations for the calculation of first- and second-order sensitivities for physically nonlinear elastic structures. As in Dems and Haftka (1989), the concept of total material derivatives with respect to shape parameters bp is used. The analysis is limited to the shape modification of a tractionfree boundary part of a structure, although the results obtained can be easily extended to the case of loaded and/or supported boundary parts modification.
2 F i r s t - o r d e r sensitivities for a v a r y i n g free b o u n d ary Similarly to Dems and Haftka (1989), let us consider a primary structure occupying a domain V with an external boundary S (see Fig. 1). The structure is subjected to body forces f within its domain V, surface traction T ° on a part S T of boundary S and prescribed displacement u ° on a part Su. The remaining portion So of the external boundary is traction-free. The stress, strain and displacement fields a, ¢ and u satisfy equilibrium, compatibility and boundary conditions given in the form
1(
diva+f=0,
¢= ~ Vu+VTu
u=u ° onSu,
a.n=T
° on ST,
)
withinV, a.n=0
on So.(1)
Thus our analysis will be confined to small strain theory but a nonlinear stress-strain relation, given in a general form
= E(~),
(2)
where E(¢) is generated by a potential rule associated with
80 the specific strain energy U(¢) as E = OU/Oz. Besides a deformation process due to the applied load, consider a transformation process x t = x + p (x, bp) (cf. Fig. 1), where p can be considered as a given transformation field depending on a set of design parameters bp, p = 1, 2 , . . . , P. This field modifies the structural shape satisfying the condition p (x, bp) = 0 on S T U Su. Thus, only the traction-free portion So of external boundary S is subject to shape modification in the transformation process. Consider now the infinitesimal transformation 5p from the initial configuration of a structure caused by the infinitesimal variation 5bp of design parameters. The variation 5~ can then be expressed as
O~k
=
= v~hbp ,
(3)
where ~k (k = 1, 2, 3) denotes the k-th component of the transformation vector ~ with respect to the fixed Cartesian coordinate system, and v~ is the k-th component of the transformation velocity field associated with shape parameter bp, treated as a time-like parameter. The transformation velocity field should satisfy the condition v p(x,bp) = 0
forxESTVSu.
(4)
Consider now an arbitrary functional G of the form G=
f
~(a,¢,u)dV+
v(~)
f
h(u,T)dS,
(5)
srvs.
where • and h are continuous and differentiable functions of their arguments. The first variation of G with respect to variations of shape parameters bp can be generally expressed by
5G=S'Sb=
-
~pSbp=
d v + . w ]
+f IvY;as+ hD(dS)] Dbp j}Sbp,
(6)
~ -O
X2
+
~
~ "
X
Fig. 1. Structure with varying free boundary part
So
where S = DG/Db is the first-order sensitivity vector. The derivation of components of this vector will be briefly discussed in this section by applying both the direct and adjoint approaches. The results obtained will be used next in Section 3 in order to determine the second-order sensitivity matrix of functional (5). Since the surface tractions T ° on the fixed boundary portion S T and prescribed displacements u ° on the fixed boundary portion Su are independent of shape parameters bp, we can write
DT ° -0 Dbp
on
ST,
Du ° --0 Dbp
on
Su.
(7)
Recalling now the results of Dems and Haftka (1989) and taking into account conditions (4) and (7), the sensitivities of G following from (6) can be written in the form
DG _ I (q2,a +k~,~.~,p+k~,, Dbp "a,p
.u,p) d V +
+lh,,.u,pdST+/h,T'a,p'ndSu÷fq2n'vPdSo,(8) where q2,a, ~ , ¢ , q2,u and h,, , h,w denote the partial derivatives of integrands of (5) with respect to state variables and n is the normal unit vector to boundary S. Furthermore, the quantities a,p, ¢,p and u,p in (8) are the local derivatives of primary state fields with respect to shape parameters bp (ef. Dams and Haftka 1989). The equation (8) constitutes the basic expression in the first-order sensitivity analysis of any functional, to which the direct or adjoint approach can be next applied. When the direct sensitivity method is used, it requires an additional solution of the boundary-value problem for each particular variation of design parameter bp. Thus, for P design parameters we need P additional sensitivity solutions. On the other hand, tl~e adjoint state method requires only one additional solution of an adjoint problem for a specified functional, independently of design variations. The two approaches have received considerable attention in the literature, e.g. by Choi and Haug (1983), Dams and Mrdz (1984), Choi and Seong (1986), Haber (1986), Dems and Haftka (1989) and others. The choice between these two approaches depends on number of objective functionals and design parameters, and also on the relative difficulty of obtaining adjoint or direct solutions. As will be shown in Section 3, when the second-order sensitivities are considered, both of the above methods should be used to generate the adjoint and sensitivity state fields. Applying the direct approach in sensitivity analysis, we should derive the sensitivity state fields which are the local derivatives of primary state fields appearing in (8), associated with each shape parameter bp. To do this, we should solve the additional boundary-value problems associated with particular variation of each design parameter. Equations describing these problems are obtained by dif-
81 ferentiation with respect to each by, the set (1) and (2) describing the primary boundary-value problem. Introducing the notation
Dems and Mr6z 1984)
'
fly=u, v'
~v=¢,p,
bp=a, v'
~v=a, v.n,
2
(9)
(14)
o° : . and assuming that the body forces f are design insensitive, the equations of the p-th additional boundary-value problem have the form (cf. Dems and Haftka 1989) ~ P = I () ~ f i P + - ~Tfi 2
d i v hv = 0 ,
tiP=0
on Su ,
v
TP = b p . n = O on S T ,
:i:v = ~ v . n = - ~ , k .nv~ + ~ . Vv~nk
~P=D-kP,
within V ,
on
So,
p=l,Z,...,P,
(10)
where D = 0 E / 0 z can be regarded as a tangential stiffness matrix at the equilibrium solution a, z of the primary structure. For a stable elastic material D is symmetric and positive definite. Thus, although the primary problem can be physically nonlinear, all additional problems are always linear. Having P design parameters, P + 1 boundary-value problems (1), (2) and (10) must be solved in order to calculate the first-order sensitivity vector of any functional G, independently of the number of functionals. Using the solutions of problems (10), the sensitivities of G can be finally expressed in the form DG = / Db v
+f
(¢,a "OP + ~ , ¢ "~P + ~,u "tiP) d V +
"fiVdST + /
(11)
where OP, ~P and tip are the local sensitivities of primary state fields with respect to design parameter bp and vVn • denotes the normal component of the transformation velocity field associated with bp on varying boundary portion
So. The alternative method for determining the sensitivity vector of G consists of introducing an adjoint structure with imposed fields of body forces and initial strains and stresses f a = ~,u ,
~ai.= ~ , a ,
a ai = g2,¢ ,
f
dr+
-[/(aa'¢,p-fa'u,p)
=
f ua
f
• O',p
.n dSo
ok.
dV-/Ta°'u,pdSTl
=
=
- u°
,k'nv
dSo.
(15)
Substituting (15) into (8), after some transformations, the first-order sensitivities of any functional G, expressed in terms of primary and adjoint solutions, have the final form (cf. Dems and Haftka 1989)
+ f vV. dSo,
p = 1,2 . . . . . P
where a a, ¢a and u a are the state fields of the adjoint structure. Using now (12), (13) and constitutive laws of additional and adjoint boundary-value problems (10) and (14), as well as the boundary conditions of the p-th additional structure, the first three integrals on the right-hand side of the sensitivity expression (8) can be rewritten in the form
(12)
DG
-
/ (~ - a.
+ f . u °) v;.
dSo,
(16)
and are obtained as the results of solutions of the p r i m a r y problem and one additional adjoint problem for each functional to be considered, independently of the number of design parameters. Thus, when we limit ourselves to first-order sensitivity analysis and there are M functionals and P design parameters, then the direct approach would still require P + 1 solutions in order to generate the sensitivities of M functionals, whereas the adjoint approach would need M + 1 solutions.
and subjected to the following set of boundary conditions S e c o n d - o r d e r sensitivities for a v a r y i n g free b o u n d ary 3
u a° = --h,w
on S u ,
a a • n = T a° = h,u
on S T .
(13)
The adjoint state equations can be written in the form (cf.
Consider now the problem of evaluating the second-order
82 sensitivity matrix of an arbitrary functional G (5) with respect to the change of shape of the free boundary So of a structure. It was shown by Haftka and Mrdz (1986) for the case of material parameters variation that the most efficient method for generating the second-order sensitivities of any functional is to u s e both direct and adjoint solutions of first-order sensitivity analysis. For P design parameters and M functionals this mixed approach requires 1 + P + M solutions of boundary-value problems in order to obtain the first- and second-order sensitivities of M functionals with respect to P design parameters. In this section the mixed approach for second-order sensitivity analysis will be applied to the case of structural shape variation. To start our analysis, we calculate the material derivatives of (8) with respect to shape parameter br. Using the notation (9) and denoting the second derivatives of primary state fields by
ll,p r = flPr,
.C,pr = ~pr , a,pr = opr, a,pr .n = ~.pr,
(17)
it follows from (8)
D2G DbpDbr +~,a.
f .I
• (b v" fir + flY. ~r) +
q- / h,~ "fl pr dST + / h,T .Tpr dSu = =/(¢ai.~PrTaai.~pr+fa.fiPr) +/Ta°.fiPrdST-f'rPr.ua°dSu= = f(opr.ea_aa.~pr
~_fa.fipr) d V +
+f Ta°.flPrdST-f ~rPr.ua°dS~,=f TPr.uadSo.(19) @pr
I n writing (19) t h e c o n d i t i o n s
= 0 on S T and
fl pr z 0
on Su were used. These conditions follow from the fact that the shape of loaded and supported boundary portions is fixed and then the load and support conditions on these boundary portions are design insensitive. Furthermore, we should note that T a = 0 on So for the adjoint structure. The surface tractions @pr on So can be expressed in terms of a stress field of primary structure and solutions of additional boundary-value problems defined by (10). In fact, taking the material derivative with respect to design parameter br of both sides of the load condition on So in (10), we can write
~ipr = 5P;nj = -- (aPj,l v~ + ~rj,kVP ) nj + 5Pjv~,j nl+
+ ~ , e e .~v. ~r + @,e,. (~;. fi~ + tip. ~ ) + + g2,u ~ .tip • fir] dV + f h , ~ .tip • fir dST+
dV +
s
+ %.
vTn
-
s vr,k
+
+aij,k v pkv Ir ,in I + aij,l vPk,j vrn k - (crij,k v~) , l v r n j pr pr --aij,k v k nj + aijv k ,# nk,
(20)
+ / h,TT "rP "~rr dSu+ where v kVr denotes the local derivative OvPk/Obr. Using now (20) in (19), after some transformations, we obtain
4 : / (~b,~r 'b pr + q2,¢ .~pr + ~,~ .£lPr) dV~-
f (~,a'avr+ ¢,e'W + ¢,o'fiVr) dV + f h,u'aPrdST+ f
+ f [(¢,~ .op + ~,~ .~p + ~,~ .av) r + + ( ~ , a "br + q2,e .~r + ~2,~ .fir)vpn + (gSvpk) ,kv~ ] dSo .(18) The fourth, fifth and sixth of (18) involve the second primary structure. These using the solution for the of (12)-(14), it follows
integrals on the right-hand side derivatives of state fields of the integrals can be eliminated by adjoint structure. Making use
f (¢,a "bw + ~,¢ .~vr + ~,~ .fivr) d V +
f
br vPn) . gaq_
q-(er • ,avkjP~ ,k vnr ÷ a • ¢av~rnk] dSo,
(21)
and then the integrals of (18) involving the second derivatives of primary state fields are expressed in terms of prim a r y and adjoint solutions and first-order sensitivity fields. ' Now let the traction-free surface So of a structure be parameterized by an orthogonal curvilinear system. Then (a, fl) coincides with the principal curvature lines on So (see Fig. 2), where a and/3 are the curvature lines parameters, so that dS = dad/~. The unit tangent vectors to So along lines a and/3 are denoted by t a and t~ and the unit normal vector n = t a × t¢~. Assuming that So is a
83 + (~ - ~ ~ ) (vkpr ~k + ~ v k p, ~ v~r - v p~ v r~ , ~ -
,
~o
/~ + d f l ~ -vZvn'flP r
Fig. 2. Parameterization of piecewise regular surface So piecewise regular surface, as shown in Fig. 2, denote by F the intersection curve between two adjacent parts of So and introduce the two unit vectors v - and t, + defined by t,- = t × n - ,
/)+ = ( - t )
×n +,
(22)
where t is the unit vector tangent to F and n - and n + are unit normal vectors to both parts of So along F. The transformation velocity vector vP associated with shape parameter bp can be decomposed on any part of So as follows: vPa = v p ' t o ,
v~ = v p . t ~ ,
vp = v p . n ,
(23)
whereas along the intersection curve F the following components of v p are introduced
~ V : ~ " ~-,
v~ + : v p - ~ + .
(24)
Note that the pair (vPu- , vPu+) of the tangential surface
velocities a l o n g / ' is, by simple geometry, uniquely defined ¢
by the pair ( v ~ - , v~+ ) of the normal surface velocities along this line, provided that the intersection angle is not equal to ~ or 0. Substituting now (21) into (18) and performing some transformations, the components of the second-order sensitivity matrix of the functional G are expressed in the form X
dSo + E
(~b - a. ca) [vvvn] dF ,
(25)
where H denotes the mean curvature of So and the sum of the line integrals is taken over all edges of surface So; vPv v r] nJ denotes here the jump of proper components of transformation velocity fields associated with shape parameters bp and br taken as the difference on both sides of the intersection line F. As can be seen from (25), the second-order sensitivity of G is expressed as the sum of the volume integral over the structure domain, surface integrals over fixed surfaces Su and S T and surface and line integrals on varying piecewise regular surface So and it depends on derivatives of integrands of G and direct and adjoint solutions described by (10) and (12)-(14), respectively. When, in particular, the varying surface So is smooth, all line integrals on the right-hand side of (25) vanish. For P design parameters, the first-order sensitivity vector and second-order sensitivity matrix of an arbitrary functional G are obtained as the result of only 2 + P solutions. Occasionally, it may be more convenient to have the second-order sensitivity matrix of G expressed only in the form of surface and line integrals on varying surface So. To obtain such results the first three integrals on the righthand side of (25) should be eliminated. To do this P additional boundary-value problems must be solved in order to calculate the first-order sensitivity fields flap, ~ap, ~rap of adjoint fields with respect to shape parameters, by using the direct approach. The equations describing these problems are obtained in a similar way as for the primary structure, namely by material differentiating the equations (12)-(14) with respect to bp. Thus, introducing for local derivatives of adjoint state fields the notation similar to (9), the equations of the p-th additional boundary-value problem for the adjoint structure, following from (13)(14), have the form div
~ap ÷ ~ap = O,
~ap -~ 21 ( v f i a p + v T f i a P )
withinV,
flap : U , ; O -h,tt "TP on Su,
D2G
DbpDbr - f [q2,aa "8p • ~r + ~,a~" (bP ' U + ~P . ~r) + ~ap =- T,pao = h,u u .tip
on ST,
+ ~ , a u • (#P • fir + tip. ~r) + ~,z¢ "~v " U + +tI,,¢, • (~P • fi~ + tip. U) + ,I~,~ .tiP. fir] d V +
~
= ~,~ .~ = - . , ~ .~v~ + ~. v ~ k
on So,
+ f h,u~.fiv.fi~dST+ f h,~T.~P.~d&+ + f { ( f a . ~, _
-
~o. ~.)v~ + (f°. ~" - ~a. ~.)
In. ¢a,n + an" ¢ , n - f a " u,n + 2 H ( k ~ - er. ca)lvnvnr
where the body forces ~ap and imposed fields of initial strains and stresses, in view of (12), are expressed by
~ap = "~,.a "~P + ~,u¢ "~P + ~,uu "tiP,
84
~aip = k~,aa "Op + q2,ag .~P + q2,au "tip, Oalp = q2,ga .Op + ~,gg "¢P + ~,gu "tiP.
(27)
Using now (26)-(27), the first three integrals on the righthand side of (25) can be retransformed as follows:
f [k~,aa .Op" Or q- ffJ,ag. (O p • ~r q_ ~p. or) 4_ +~,a,
• (0 p • fir + tip. Or) + ~,~g .~p. ~r+
+ 9 , g u • (~P • fir + tip. ~r) + g',u~ "tiP" fir] d r + +
/ h,u~ .tip.fir dS T + f
G coincides with the potential or complementary energy of a structure. In this case the expressions for first- and second-order sensitivities are simplified considerably, since the adjoint solutions can be easily expressed in terms of p r i m a r y solutions. Then only one primary solution and P sensitivity solutions need to be calculated using the direct approach. In considering the sensitivities of potential and complementary energies assume, for simplicity, that vPn is the only nonvanishing component of the transformation velocity field of the smooth varying traction-free surface So, associated with design parameter bp. Consider first the potential energy of a structure with vanishing body forces f, that equals
h,TT'~?P" ~r dSu = IIu
= 21 f (~; . tier+ 9r .f~p . 9at . . tip.
(28) Noting that the surface tractions ~?P and ~av are known as the load conditions of (10) and (26) on the varying surface So and applying t h e m to (28) and next using the result in (25), finally the second-order sensitivity of G expressed in the form of surface and line integrals on varying boundary part So is obtained, namely
D2G
1
DbpDb~-f{[ fa'tip--(a'~p+2
[
+ fa.fir_2
aa
~p)] " J vrnJ-
1 (a.$ar+ aa" }r)]vP- [a" ~a,n~-a --
-fa'u,n+2H(~-a'ga)]v~vn
p r +(~_a.ga)
a
"g,n-
(v kpr nk+
pr v r )}dSo+ +nkvP,n v nr -- VaVn,a --v;~Vn,fl
+
E/
(e
-~
g~)
[~[]
dr.
/ U (g) dV - f T °. u d S T.
(30)
q:~p tir) d & . Comparing (30) with (5), it is easy to observe that k9 = U (z) within V, h = - T ° . u on S T and h = 0 on Su. Thus, in view of (12)-(14), the solutions for the adjoint structure are u a = 0, ga = 0 and a a = - a . The firstorder sensitivity following from (16) is now expressed in a very simple form (el. Dems and Mrdz 1984), namely
DIIu
- / Uv p a s o .
The same result is obtained by using the direct approach since the sum of the first three integrals on the right-hand side of (11) vanishes in view of boundary conditions (1) and (10) of primary and additional structures. Similarly, the expression (25) for the second derivatives of Hu is simplified as follows:
Z) pDbr- V' g' P' rdV+ P r ] a S o .(32) vPvrn n _u U (TzkvP: r @ n kvl~,n'Un)
-~- (if" g,n - 2 H e )
(29)
We should note, however, that expression (29) is obtained as the result of solutions for primary and adjoint structures, as well as the sensitivity solutions for both these problems; contrary to expression (25) which follows from the solutions for primary and adjoint structures and sensitivity solutions for the primary structure. Thus the difference between the number of desired solutions for both approaches is equal to the number of design parameters.
4 First- and second-order sensitivities of potential and complementary energies
(31)
The volume integral on the right-hand side of (32) can be easily transformed to the surface integral over So. In view of (2) and (10), we can write
/ -
1/
U, ge .~P . ~r dV = ~
1/ 2
(hr . ~p + hp . ~r) dV -
a . (~Vv r + ~rv~) dSo.
Using now (33) in (32), the second-order sensitivities of Hu are expressed finally in the simple form
D2IIu DbpDbr
/[1
~a" (~Pv r +
nUN (?%kV~r -c nkvPk,n Vrn)] dSo.
Consider now two particular cases in which the functional
(33)
J
+
g,n (34)
85 Assume now t h a t the functional G coincides with the c o m p l e m e n t a r y energy of the same s t r u c t u r e and is given in the form
(as) C o m p a r i n g (35) with (5) we observe now t h a t T = W (a) within V, h = 0 on ST, h = - T . u ° on Su and then the adjoint solutions are u a = u, ga = g and a a = 0 (cf. Dems and Mr6z 1984). Thus, the first-order sensitivity of Ha, following from (16) is expressed by f
DII~ Dbp
! (W - a . g) v~ dSo.
(36)
J
The same result can be o b t a i n e d from (11). The secondorder sensitivities of Ha, in view of (25) and adjoint solutions, take the form
D2 ]-[a
DbpDbr
_fw,~.a~.ardy_f([¢.~,~+2H(W_
Fig. 3. Circular disk with varying outer radius re with a fixed inner radius r i and a variable outer radius r~ (see Fig. 3). The disk is loaded by the uniform internal pressure p and its outer edge is traction-free. Our aim is to d e t e r m i n e the first- and second-order sensitivities of the disk c o m p l e m e n t a r y energy with respect to the varying outer radius r e . This energy is given in the form re
-~
~)] v~,~~ - (w - .
~) (~kv~ r + ~k,~,~ v r) } dSo,
G (re) = Ilcr = ~ 1 /
D2IIa
(39)
rdr,
ri
(37) or, by the same arguments as in (34), they can be expressed in form of a surface integral only, n a m e l y
(cr22~,arat÷a2)
where ~rr a n d crt are the radial and circumferential stress components, while E and u denote the m a t e r i a l constants. Since the only design p a r a m e t e r is the outer radius re, the t r a n s f o r m a t i o n velocity field v associated with this p a r a m e t e r can be assumed in the form
DbpDbr •
--
.
VnV
v -
~ -
- (w - ~. ~) (,~k~; ~ + ~k~;,~ ~ ) } dSo.
(as)
Thus, the first-order sensitivities of Hu and Ha are obt a i n e d as the result of one solution, whereas the secondorder sensitivities require 1 + P solutions, where P is the n u m b e r of shape p a r a m e t e r s modifying the varying traction-free surface So. Moreover, it is easy to verify t h a t DHcr/Dbp = - D I I u / D b p and D 2 I I a / D b p D b r = - D 2 I I u / DbpDbr, since U ÷ W = a - z.
5 Illustrative examples In this section we will discuss two simple examples in order to illustrate the analysis presented for evaluating the firstand second-order sensitivities of an a r b i t r a r y functional.
5.1 E x a m p l e 1 Consider a circular disk m a d e of a linearly elastic m a t e r i a l
r -- r i re -- r i
for r i <_ r < re.
(40)
Thus, v vanishes on the fixed inner edge for r = r i and takes the value 1 on the varying outer edge for r = re. The stress d i s t r i b u t i o n within the disk is given in the well-known form
dr-
r2~r2ei \ l - ~ j
,
at = r2e_ r2i ~ 1 + 7 ~ j . (41)
To derive the desired sensitivities of Ha only the direct solutions must be found, since the adjoint solutions are expressed in terms of p r i m a r y solutions and they have, as was shown in Section 4, the form ai
o
~i
ai
a=o
O,
cr~i
o~:o.
0,
fa
O,
(42)
To find the direct sensitivity solutions, we must consider an additional disk of the same form as the p r i m a r y one, loaded on outer edge of radius re, in view of load conditions of (10), by external pressure p given by
86
-
2rJp re (r2 - rJ ) '
(43)
with a traction-free inner edge of radius ri. The stress distribution within this additional disk is given by
2r~rep ar = 0 - r ' r e - -
(
\(r~--~-'~2e z,
2r2rep( rJ~ (,7--,2----)2 l@r2 j "
_
-~t=0-1:'re
DII~ Dre
1 2 re (grCrr-~ gt0-t) r=re •
(45)
2rtrep 2 E(r 2 _ r/2) 2"
(46)
The expression for the second-order sensitivity of the disk complementary energy follows now from (38) and is given by
DJiic~
DreDre
aM
° dA =
dA,
+ 0-0
where 0-M denotes the yon Mises stress and 0-o is a prescribed stress level. Thus, for n tending to infinity, the functional G can be treated as the global measure of stress intensity within the bar cross-section domain. Derive now the first- and second-order sensitivities of G with respect to varying semi-axes bl and b2 of the bar cross-section for n = 2 and 0-o = 1. The transformation velocity fields associated with shape parameters bl and b2 have the components Xl vl : ~-1'
v1 : 0 ,
v2 : 0 ,
x2 v22 : b2-"
(51)
The sensitivity solutions of boundary-value problems (10) following from the direct approach can be written in the form 013 -
cr13 bl '
~rl3 __
30-23 ,
bl
3o13 b2 '
(47)
Relating the primary strains to stresses by Hooke's law and using (41) and (44) in (47), the second derivative of Ha finally has the form
+ 3
[
J
DJH~
DreDre
kg (a) dA =
[ [
(44)
Applying now the solutions (41) and (44) into (45), the first-order sensitivity of II~ is expressed finally by
DII~ _ Dre
/
G=
When the direct approach for first-order sensitivity analysis is applied, the derivative DIIa/Dre follows from (11). On the other hand, using the adjoint approach, it is expressed by (36). In both cases this derivative takes the form
(49)
In writing (49) Saint-Venant's theory of torsion and elementary beam theory were applied. Consider now the following functional:
r2"~ 1 - - r2 j
2Mix2 2Mix1 4MbX2 ~rblb32 , 0-23--~rbalb 2 , a33--~rblb32 •
0-13--
0-23 52 '
a313
a33 bl '
(52)
3a33
b2
The adjoint bar, in view of (12) and (13), is subjected to the imposed field of initial strains
ai E13
=
kI/ o.13= 6o_13,
ai = ffj,a23= 60_23, ~23
2r 4 @2 + 3r 2) p2
-
E(r 2 _ rJ) 3
(48)
The results (46) and (48) can be very easily verified by simply integrating the expression (39) and next calculating the proper derivatives with respect to re.
a~" _ ~,o3~ g33
20-33,
(53)
with vanishing twisting and bending moments Mta and Me. Thus, the solutions for the adjoint bar are given in
5.2 Example 2 Consider a prismatic bar with the elliptical cross-section of semi-axes b1 and bj, subjected to torsion and bending (see Fig. 4). Denoting the twisting and bending moments by Mt and Mb, respectively, the nonvanishing stress components within the bar cross-section are expressed in the form
b
2
~
:Fig. 4. Prismatic bar with elliptical cross-section
/
87 the form
ai
~ 3 = ~13 ----6°"13,
ai
Z~3 : Z23 : 60"23 ,
0.~3 = O, o-~3 ~ O, 6 Concluding remarks
ai
~ 3 ---=~33 = 20"33,
(54)
0"~3 = O.
The first-order sensitivities of the functional G follow from (11) or (16), according to the applied direct or adjoint approach. Using (16), we can write
z)bl
-
(¢ -
DG _ / Db2
(¢ -
dSo .
(55)
Substituting now (49), (51) and (54) into (55) and integrating along the outer edge of the bar cross-section, the first derivatives of functional (50) with respect to bl and b2 can be written in the form
Db 1
This paper supplements the results of previous works and provides a systematic variational approach to sensitivity analysis of first and second orders for structures with varying traction-free boundary. The methods discussed provide an effective tool in generating first and second variations of an arbitrary volume a n d / o r surface integrals which can be useful in solving optimization, redesign or identification problems. Although the analysis presented was limited to the shape modification of the free boundary part, the results obtained can be easily extended to cases of modification of loaded and supported boundary parts.
7 Acknowledgement This research work was carried out with the support of the Polish Academy of Sciences Grant No. CPBP 02. 01.
~ 2 ~rblb
References Similarly, the second derivatives of (50) can be derived by using the expression (25) which is now simplified to the form
D2G / DbpDbr ~ , a a "uP • ~r d A ÷
Choi, K.K.; Haug, E.J. 1983: Shape design sensitivity analysis of elastic structures. J. Struct. Mech. 11,231-269 Choi, K.K.; Seong, H.G. 1986: A domain method for shape design sensitivity analysis of built-up structures. Comp. Meth. Appl. Mech. Eng. 57, 1-15 Dems, K.; Mr6z, Z. 1984: Variational approach by means of adjoint system to structural optimization and sensitivity analysis. Int. J. Solids Struct. 20, 527-552
~_ (~ _ 0.. sa) (rtkv~vr ~_ ~kv~r _ vPsvrn,s ) } d~o, p , r = 1,2,
(57)
where K denotes the curvature of the cross-sectional outer edge. Substituting the primary, direct and adjoint solutions (49), (52) and (54) as well as (51) into (57), after integration, the second derivatives of G equal
Dems, K.; Mr6z, Z. 1985: Variational approach to first- and second-order sensitivity analysis of elastic structures. Int. J. Num. Meth. Eng. 21, 637-661 Dems, K; Haftka, R.T. 1989: Two approaches to sensitivity analysis for shape variation of structures. Mech. Struct. Maeh. 16, 379-400 Fujii, N. 1986: Domain optimization problems with a boundary value problem as a constraint. In: Rauch, H.E. (ed.) Proc. 4th IFAC Symp. Control of Distributed Parameter Systems, pp. 5-9. Los Angeles: Pergamon Haber, R.B. 1986: A new variational approach to structural shape design sensititvity analysis. In: Mota Soares, C.A. (ed.)
DblDb2
7rb464
b_z__j
Computer aided optimal design: structural and mechaniehal systems, pp. 573-587. Berlin, Heidelberg, New York: Springer
88 Haftka, R.T. 1982: Second-order sensitivity derivatives in structural analysis. AIAA J. 20, 1765-1766 Haftka, R.T.; Mrdz, Z. 1986: First- and second-order sensitivity analysis of linear and nonlinear structures. AIAA J. 24, 11871192
Received Apr. 17, 1990
Haug, E.J.; Arora, J.S. 1978: Design sensitivity analysis of elastic mechanical systems. Comp. Meth. Appl. Mech. Eng. 15, 35-62 Haug, E.J. 1981: Second-order design sensitivity analysis of structural systems. AIAA J. 19, 1087-1088