Shape optimization of elastic bars in torsion - GIREF

SHAPE OPTIMIZATION OF ELASTIC BARS IN TORSION Jean W. Hou, Edward J. Haug, and Robert L. Benedict Center for Computer Aided Design College of Engineering The University of Iowa Iowa City, Iowa 52243 ABSTRACT The problem of shape optimal design for multiply-connected elastic bars in torsion is formulated and solved numerically.

A variational formulation for the

equation is presented in a Sobol~v space setting and the material derivative idea of Continuum Mechanics is used for the shape design s e n s i t i v i t y analysis.

The f i n i t e

element method is used for a numerical solution of the variational state equation and is integrated into an i t e r a t i v e optimization algorithm.

Numerical results are

presented for both simply- and doubly-connected bars, with prescribed bounds on admissible location of both inner and outer boundaries. ACKNOWLEDGEMENT This research was supported by NSF Grant No. CEE 80-05677. INTRODUCTION The use of material derivative in the so called "speed method" was introduced by J. Cea, J.P. Zolesio, see [9], [7] and [8].

and B. Rousselet in a series of papers. For

the details

Here we apply i t d i r e c t l y to the optimization of bars in

torsion, and use the f i n i t e element technique to obtain specific numerical results for multiply-connected cross sections. Consider the torsion problem for an elastic bar shown in Figure 1.

The material

of the bar is homogenous and isotropic and the cross section may have a void, thus resulting in a multiply-connected domain ~.

Torsional stiffness of the bar is defined

by the following boundary-value problem for the stress function z (See reference [ 1 ] ) : Az = -2,

in ~

l

(1)

z : O,

on Fo

I

(2)

z = q,

on Fi

f

~z ~-dS = -2Ai ,

F. 1

(3)

(4)

32

where Fo is the outer boundary of ~ and Fi is the inner boundary, enclosing the domain Qi"

Here, the constant q is to be determined as part of the solution to the

problem and Ai is the area of ~i"

As shown in [ 1 ]

the torsional r i g i d i t y is then

given by

K: 2Hzd

+ 2nAi

Jf x V,

,

IS)

where x is a position vector in ~. Polya and Weinstein [2], have proved the following assertion:

"Of a l l doubly-

connected cross sections with given areas of ~ and ~ i ' the ring bounded by two concentric circles has the maximumtorsional r i g i d i t y . " Banichuk [3] and Kurshin and Onoprienko [ 4 ] have also investigated optimal shape of a bar with doubly-connected cross section.

They hold the inner boundary

Fi fixed and seek a shape for the outer boundary that maximizes torsional r i g i d i t y . The area of ~ is given.

In addition to Equations 1-4, they obtain the following

optimality condition for Fo: ~z = ~n C,

on F0

(6)

Taking account of this excess condition, the boundary Fo is then determined so that the constant C matches the isoparametric constraint on area of ~; i . e . , the problem is treated as an inverse boundary-value problem. Banichuk uses a perturbation technique to obtain approximate solutions of this problem. He is able also to deduce some properties of an optimum contour.

For example, wall thickness of the bar of

optimum shape decreases as one moves along the inner boundary Fi in a direction of increasing curvature. By restricting the cross section to be symmetric with respect to the coordinate axes, Kurshin and Onoprienko apply complex variable theory and solve Equations 1-6, with an isoparametric condition on the area of ~.

A system of nonlinear equations

is obtained to determine the unknown coefficients of a complex function that describes the unknown boundary Fo.

This system is solved by the Newton-Raphson

method. Somenumerical results are presented. Quite recently, Dems [5] used a boundary perturbation analysis for a bar with doubly-connected cross section to maximize torsional r i g i d i t y , with the inner boundary held fixed.

The optimality c r i t e r i a obtained is the same as Equation 6.

The shape optimization problem is formulated by defining shape of the boundary with a set of parameter-dependent, piecewise linear functions.

The reduced problem is

solved by means of the f i n i t e element method and an i t e r a t i v e algorithm based on the optimality condition.

Several numerical examples are included.

The discussion thus far has focused on doubly-connected bars.

I f the cross

section of the bar is simply-connected, Fi is a point (Ai = 0), and the value of q

33 is immaterial.

Thus, the boundary-value problem f o r the stress function z reduces

to Equations I and 2 on the simply-connected domain 9 and the t o r s i o n a l r i g i d i t y is given by Equation 5.

I t is i n t e r e s t i n g that Equation 6 remains v a l i d as an optima-

l i t y c r i t e r i o n f o r the shape of 9 to maximize K with a given area of Q [3]. D i f f i c u l t i e s in solving the torsion problem f o r a bar with a doubly-connected cross section are associated with the boundary conditions of Equations 3 and 4. Usually, q in Equation 3 is determined from Equation 4.

However, once an admissible

function space and v a r i a t i o n a l formulation can be defined, i t is seen that Equation 4 becomes a defining equation f o r a natural boundary condition.

Therefore, the

F i n i t e Element technique can be employed to solve the problem numerically.

In

section 2, such a v a r i a t i o n a l formulation and admissible function space are defined and the equivalence between the v a r i a t i o n a l formulation and the Equations 1-4.

We

also prove the existence and uniqueness o f the s o l u t i o n . In section 3, the material d e r i v a t i v e concept is employed to obtain the d i r e c t i o n a l d e r i v a t i v e of t o r s i o n a l r i g i d i t y with respect to the shape of the domain by allowing both Fi and Fo to vary.

Optimality c r i t e r i a f o r the simply- and

doubly- connected domains are obtained. An i t e r a t i v e numerical method f o r optimizing shape of simply- and doublyconnected shaft cross sections is outlined in Section 4.

Numerical c a l c u l a t i o n s

are carried out using the f i n i t e element method f o r analysis of the designs and a nonlinear programming method f o r optimization.

Examples of both simply- and

multiply-connected bars, with constraints on admissible location of the boundaries Fi and £o, are presented in Section 5. 2. VARIATIONAL FORMULATIONOF BOUNDARY-VALUEPROBLEMS Suppose 9 Fi and Fo.

is a doubly-connected open set in R2, bounded by regular boundaries

The outer normals of the boundary curves are represented by n.

The

f o l l o w i n g b i l i n e a r and l i n e a r forms play a key role in the v a r i a t i o n a l formulation of the problem: a(z,

f

v) = ; vz • vv d~

(7)

(x, vv) = f x • vv d£ 9

(8)

where x £ ~ c R 2 is a position vector, v denotes the gradient operator, and x.u = XlU I + x2u2.

The v a r i a t i o n a l equation f o r the torsion problem is given by:

a(z, v) + (x, vv) : O,

for all veV,

where V = { v ~ H I ( ~ ) I v = 0 on r o and v = ~ on Fi , f o r some BeR I} ,

(9)

84

and where HI(~) is the Sobolev space of order one [6]. One may define a formal operator A as, Aw(x) = -Aw(x), where XEQ and A is the Laplace operator.

The domain of this formal operator is defined as

HI(~, A) = {wEHI(Q) I Aw~L2(~)}

(II)

Moreover, one may define the f u n c t i o n space V(A) = { w E H I ( ~ , A) I w = 0 on Fo and w = B on Fi , f o r some B 6 R1} In the d e f i n i t i o n s

(12)

of spaces in Equations 10-12, a f u n c t i o n w evaluated on a boun-

dary F is i n t e r p r e t e d as a trace yw defined in H½(F) (See [ 6 ] ) .

One may proceed to

prove the f o l l o w i n g p r o p o s i t i o n : Proposition.

The f o l l o w i n g problems are e q u i v a l e n t :

Problem (a); Find z e H i ( Q , A) and q ~ R 1 to s a t i s f y Az = 2,

in

13)

z = O,

on F 0

14)

z = q,

on Fi

15)

fFi

fFi x • n dS = 0

16)

dS +

where f x • , dS is equal to twice of area enclosed in Fi" Fi Problem (b); Find zEV such that a(z, v) + (x, Vv) : O, f o r a l l vEV •

17)

Proof: (a + b).

Suppose zEHI(~, A) is a solution of problem (a).

Then, z~V(A).

Since z6V(A) CHI(~, A) and vEVCHI(~), Green's formula for z~V(A) and any v~V, is (as given in [6]) a(z, v) = (Az, v) +



= (2, v) + f

~z v dS 2--6-

Fi = (2,

Fo

v) + B

~

Fi

dS

(18)

35

where 6 z ~ H - I / 2 ( r ) i s

an extension of ~~z .

The second and t h i r d e q u a l i t i e s in

Equation 18 are deduced from the facts that Az = 2 in Q, v = 0 on r o, and v = ~ on r i , f o r some constant B. Further, div x = V'x m 26L2(~).Hence, x E H I ( ~ , d i v ) = { u ~ L 2 ' 2 ( ~ ) I d i v u~L2(~)~ and (x, Vv) = - ( 2 , v) + f x • n v dS FiU r o (19)

x • . dS

= - ( 2 , v) + B j

ri Adding Equations 18 and 19 and using Equation 16, i t follows t h a t

x • , dS + j

a(z, v) + (x, Vv) : B( ?i • for a l l vEV. (b~

~z dS) = 0

(20)

ri

Thus, z is a solution of Problem (b).

a).

Suppose zEV is a solution of Problem (b).

Since z~V, the boundary

conditions in Equations 14 and 15 of Problem (a) are satisfied.

One may f i r s t con-

sider only those v~V such that v = 0 on r i ; i . e . , vEH~(~)CVCHI(~) • Recall that z6VCHI(~) and xEHI(~, div). For this class of v, Green's formula (see [6~ is a(z,

v) = (Az, v)

(21)

,

where Az ~ H-I(~) and (x, Vv) = (-2, v)

(22)

Adding Equations 21 and 22, i t follows that a(z, v) + (x, Vv) : (Az-2, v) f o r a l l v~H~(Q).

I t is given that the l e f t side of t h i s e q u a l i t y vanishes, so

(Az-2, v) = 0 f o r a l l vGH~(~).

This implies t h a t Az-2 : 0 in H-I(Q).

Az = 2EL 2, so i t follows t h a t zEHI(Q, A). v a l i d f o r a l l v E V C H I ( ~ ) , giving ~z Tn v dS

a(z, v) = (Az, v) +

riUr o ~-dS

= (Az, v) + B

ri

But

Since z ~ H l ( ~ , A), Green's formula is

36 and (x, Vv) : (-2, v) + f

x • ndS

Fi U Fo = (-2,

v) + ~ /

x • n dS

Fi

Adding, i t follows that a(z,

v) + (x, Vv) = (Az-2,

(~z + x • n)dS], for a l l v~V

v) + B []" F.

1

The l e f t

side o f t h i s equation vanishes and i t was shown t h a t Az = 2, so

~ (~-~+ x " n)dS = 0 Fi for a l l vEV, equivalently for any BERI.

Thus,

(az ~- + x . m)dS = 0 Fi and the last condition (Equation 16) of Problem (a) is satisfied.

Q.E.D.

^

One may define ~ = ~U~iU?i with the definition z,

and extend the function zEV(~) to z in

in

z = q E R1, in ~i U Fi An example of such a function is shown in Figure 2.

This extended function belongs

to H~(~). For a l l vEH~(~), Poincare's inequality implies that

f( v. vld > fv where ~ > O.

,

Adding a(v, v) ~ ~ vv • Vv d~ to both sides of the above inequality /

^

and dividing by two, one has

37

v>>

½Fv (23)

> min (~, ½) I Ivll2Hl(~ )

I t is evident that c : min(m/2, 1/2) is greater than zero for m > O. constant in Qi' Vz = 0 in Qi"

Because z is

Therefore,

^

a(z, z) : ~ Vz • Vz d~ J^

= ~ Vz • Vz d£ = a ( z ,

z)

J

Furthermore, zll2V(~) + q2(mes ~i ) zl 2

Iv(m)

Substituting these results into Equation 23, one f i n a l l y has c > 0 and a(z, z) > c l l z l l ~¥ ,

(24)

for a l l zEV

Having proved V - e l l i p t i c i t y

of a(z, z) (Equation 24), the Lax-Milgram Theorem

(as used in [6]) ensures existence and uniqueness of a solution of the Problem (b). The proposition proved above implies that t h i s solution is the unique solution of Problem (a). 3.

SHAPE DESIGN SENSITIVITY ANALYSIS Since the domain ~

is to be varied, i t is convenient to t r e a t i t as a con-

tinuum and u t i l i z e the idea of material d e r i v a t i v e , as introduced

in continuum

mechanics, to find the domain variation of the functionals concerned. of defining a v a r i a t i o n in the domain ~

One method

is to l e t V(X), XE~, be a vector f i e l d

that may be thought of as a "design v e l o c i t y " . domain may then be defined by the mapping

A one parameter family perturbed

38

x = X + tV(X),

X~,

tER ]

(25)

One may denote the deformed domain as ~ ( t ) , with x e ~(t). I f z is the solution of Equation 17, which depends on the shape of ~ ( t ) , then z depends on t z = z(x, t ) .

both through the position x = X + tV(X) and e x p l i c i t l y ; i . e . , Under certain regularity hypothesis on ~ and the vector f i e l d V(X)

[7,8], one can define

~(x) ~

lim [z(X+tV~- z(X)]

t+o

= z'(X)

+ vz(X)

(26)

• A(X)

where ~ is the material derivative and z' is the partial derivative, defined as

lim [zlX, t,,),,T,,z(X, 0)] z'(X) ~ t-~O t

(27)

I f z ~ H l ( ~ ) , with smoothness assumptions on the domain and velocity f i e l d V(X) [7], then z ' E Hl(fl) [9 ],and z#H1(fl) F7, 8].

Thus, (Vz-V) ~ HI(~).

I t is shown in

References 7-9 that the following properties of the material derivative, which are well known in continuum mechanics, are valid in the Sobolev space setting:

(Vz)' : V(z')

(28)

and for an integral functional

fj F(z, x)d.

(29)

the material derivative is

=

aF

z' d~ + !

F VndS

(30)

i U ro where Vn : V.n is the normal component of V on the boundary of ft. More fundamental is the question of existence of the material and partial derivatives ~ and z' of the solution z of the variational equation of Equation 17. Under hypotheses of strong e l l i p t i c i t y of the energy bilinear form a(-, . ) , proved in Equation 24, i t is shown in References 8-10, that z is differentiable with respect to shape. With this knowledge and the material derivative formulas of Equations 26, 28, and 30, one can now study the torsional shape optimal design problem.

39 The f i r s t order domain variation o f torsional r i g i d i t y i s , from Equation 5 and 30, = - ( x , Vz') -

F

(31)

x • Vz Vn dS

iUro Selecting v = z in Equation 17, one has, (32)

a(z, z) + (x, Vz) : 0 Taking the material derivative of both sides of this equation gives 2a(z, z ' ) + (x, Vz') : - f

Vz • Vz V ndS

- [

a(z,

z')

(33)

r i Ur o

riU r o Since z ' ~ H l ( ~ )

x- Vz Vn d S

and Vz = -2 in R, Green's formula y i e l d s

+ (x, v z ' )

= (2, x') + f

a-~ az z , dS

Fi

ro

+ (-2, z') + {

( x • n z')dS

ri~1 r o ~Z Z' dS + f

= f ?iIIC o

~ z')dS

(x

FlU Fo

Substituting this result into Equation 33, i t follows that 2[ ~z z' dS + 2C x • z' dS - (z, Vz') 3 ~n j ?i (/ ro ?i {! ?o + f

VZ ° VZ V ndS

r i I ~ro

+ f I~i I I

x " VZ V ndS

= 0

~Z

On boundaries ?i and Fo' z is a constant, so ~ = = z' + Vz • V = O, because z = 0 on ?o" I t thus follows that

(34)

r° O.

Furthermore, on Fo

However, on r i z = q, z = z' + Vz • V = q.

40

I

~z - ~-V

Z I

on F o

n,

=

[

on Fi

q - ~ V n,

and Bz x = Vz : x • ~ -~-~, on F~Urol-

Therefore, Equation 31 becomes ~z x . n dn~--Vn

-

= -(x, Vz')

!iUro

dS

t

(35)

and Equation 34 may be w r i t t e n as f~zl2 ,~-~, Vn dS - #

- ( x , Vz') - f

Fi U Fo

jr i ~

+ 2q {

az X " a ~ V n dS

FiU ?o dS +

fr i

x • n dS} = 0

(36)

S u b s t i t u t i n g from Equation 36 i n t o Equation 35 and considering Equation 16, one has the desired r e s u l t

=~

(~z) ~n

Vn dS

.

(37)

FiUF o

Note t h a t any monotone outward movement o f the boundary; i . e . , increase in K, which is to be expected. simply-connected domain ~

Vn > O, y i e l d s an

I t is easy to repeat the arguments f o r a

to see t h a t Equation 37 is v a l i d w i t h F i suppressed.

I f the c r o s s - s e c t i o n a l area A of the bar and the area Ai o f the hole are given, isoparametric c o n s t r a i n t s on the shape o f ~ are : [

~i : ;

dQ - A : 0

d~ - A : 0

(38)

(39)

41 Taking the d e r i v a t i v e of both sides of these equations gives $ = f

VndS : 0

(40)

riU'r o $i = f

(41)

Vn dS = 0

Fi

The necessary condition for maximal torsional r i g i d i t y (equivalently the minimum of negative torsional r i g i d i t y ) ,

with shape variations consistent with Equations

38 and 39, is thus -~i .U r o

~-~ VndS + X ~ VndS + Xi f Vn dS : 0 (@z)2 r i UF° ri

(42)

for a r b i t r a r y Vn, where X and ~i are Lagrange m u l t i p l i e r s corresponding to cons t r a i n t s of Equations 38 and 39.

Under the assumption that Vn is smooth and

a r b i t r a r y , provided no intersection of Fi and Fo occurs, one has the following necessary conditions of o p t i m a l i t y : -caz~2"an' + X + ~i = O,

(az~2 ~, + ~ = O,

on Ci

(43)

on Fo

(44)

I t is clear that concentric circles for Fi and F° tions.

satisfy these necessary condi-

This special case is proved in Reference 2.

I f Fi is fixed, the necessary condition is only

_ ( ~ ) 2 + k = O,

on Fo ,

(45)

which agrees with the results of Banichuk [ 3 l a n d Dems [5]. Extensions of the preceding o p t i m a l i t y conditions can be e a s i l y obtained using abstract optimization theory, in conjunction with the design s e n s i t i v i t y analysis results of t h i s section.

For example, i f the inner boundary Fi is fixed

and the outer boundary Fo is constrained to l i e within some specified curve F, then at points on FoPIF the only feasible variation of the domain is Vn ~ O.

Thus,

one can prove existence of a m u l t i p l i e r function ~(X) > O, Xe FobF, such that Equation 42 on Fo becomes i d e n t i f i e d with the vanishing of the following boundary integral

42

F (~z~2 + ~ + ~(X)]V ndS = 0

f ~-,~, Fo

for arbitrary Vn on Fn. _(Bz)2

Thus, i t is necessary that

+ ~ + ~(X) : O,

on Fo

p(X) > O,

on Fo~

~(X) = O,

on

(46)

ro/(ro~ F)

While i t is interesting to derive optimality conditions that must hold on the optimum boundary, such as Equations 43-46, i t is d i f f i c l u t to use these conditions to construct optimum shapes. One may view the necessary conditions as part of an inverse boundary-value problem; i . e . , find the boundary of ~ so

that the solution

of a d i f f e r e n t i a l equation on ~ satisfies given boundary conditions and optimality c r i t e r i a , such as Equations 43-46, on the boundary F.

The l a t t e r , excess boundary

conditions may be interpreted as determining the optimum location of the boundary. Banichuk approached a special case of the problem in this fashion in Reference 3, using a perturbation technique.

Such methods are, however, very complicated and

require a great deal of ad-hoc work for each problem treated. A direct i t e r a t i v e optimization method is presented in the next section, based on the design s e n s i t i v i t y results obtained in this section, parameterization of the unknown boundary, and nonlinear programming methods. 4.

ITERATIVE NUMERICAL SHAPE OPTIMAL DESIGN A t y p i c a l shape optimal design problem is to choose a domain ~ to minimize a

cost f u n c t i o n a l o f the form

@0 = J r GO(z) d~

,

(47)

subject to f u n c t i o n a l c o n s t r a i n t s

~i = J J Gi(z) d~

f

= O, i = 1 . . . . .

k'

I

L < O, i = k' + 1 . . . . .

(48) k

where the s t a t e z i s the s o l u t i o n o f a v a r i a t i o n a l equation o f the form o f Equation 17.

I t is f u r t h e r required t h a t the boundary F l i e between F+ and r - ,

Figure 3,

The l a t t e r pointwise c o n s t r a i n t s are w r i t t e n in the form

as shown in

43

dn(F, F+)

> 0

dn(F - , F)

> 0

(49) where d n ( . , . )

is the distance measured along the normal n to F, from the f i r s t

to

the second curve. Using r e s u l t s o f Equations 37, 40, and 41, each o f the f u n c t i o n a l s o f Equations 47 and 48 can be d i f f e r e n t i a t e d 69 = I AiVn dF, l ? where the s e n s i t i v i t y

( l i n e a r i z e d ) to o b t a i n

i = O, i . . . . .

k

(50)

c o e f f i c i e n t s Ai of Vn in Equations 37, 40, and 41 d e f i n e v a r i -

ations in the cost f u n c t i o n a l and each a c t i v e f u n c t i o n a l c o n s t r a i n t . Even though the l i n e a r i z e d f u n c t i o n a l appearing in Equation 50 has been obtained using a v a r i a t i o n a l f o r m u l a t i o n , a f i n i t e the boundary can be introduced to form.

dimensional parameterization of

reduce t h i s l i n e a r i z e d f u n c t i o n a l to parametric

Presume t h a t points on the boundary F are s p e c i f i e d by a vector r(~;b)

from

the o r i g i n of the coordinate system to the boundary, as shown in Figure 4, where is a parameter vector and b is a vector of design parameters b = [b I . . . .

, bmIT.

When the p a r a m e t e r i z a t i o n of F has been d e f i n e d , the domain o p t i m i z a t i o n problem reduces to s e l e c t i o n o f the f i n i t e

dimensional vector b to minimize the cost

f u n c t i o n o f Equation 47, subject to the c o n s t r a i n t s o f Equations 48 and 49.

The

l i n e a r i z e d form o f t h i s problem may be w r i t t e n in terms o f v a r i a t i o n 6b by denoting b = b0 + t6b

(51)

?

where b0 is the design at a given i t e r a t i o n . V =

(r(a;b)) = ~6b

The v e l o c i t y f i e l d at the boundary is

,

(52)

Taking the dot product o f V w i t h the u n i t outward normal to the curve F y i e l d s Vn : m • V : [m - ~ r ( ~ b ) ]

6b .

Here, the c o e f f i c i e n t o f 6b can be c a l c u l a t e d at each p o i n t on F

(53) and the r e s u l t

s u b s t i t u t e d i n t o Equation 50 to obtain ~r ~i T 6~i = [ I Ai (~ ' ~ ) dS]6b ~ 6b .

(54)

F More d i r e c t l y , be l i n e a r i z e d as

the pointwise c o n s t r a i n t s on l o c a t i o n o f F in Equation 49 can

44

-dn(F(~), F+) < n(~)

• ~~r

(~) ~b < dn(r - , F(~))

(55)

on F

This c o n s t r a i n t may be implemented over F in several ways, the simplest being to enforce i t at a grid ~j of points. Having defined a f i n i t e dimensional parameterization of the shape optimal design problem and obtained d e r i v a t i v e s of the cost and c o n s t r a i n t functions with respect to design parameters, one can now apply any well known nonlinear programming algorithm to i t e r a t i v e l y optimize the shape.

In each i t e r a t i o n , a f i n i t e

element approximate s o l u t i o n of the boundary-value problem is constructed and used to evaluate design d e r i v a t i v e s of torsional s t i f f n e s s , using Equations 37 and 54. More d i r e c t l y , Equations 40, 41, and 54 are used to calculate d e r i v a t i v e s of @ and @i in Equations 38 and 39.

F i n a l l y , the d e r i v a t i v e s appearing in Equation 55 are

calculated d i r e c t l y . Numerical results presented in the f o l l o w i n g section have been obtained by a recursive quadratic programming algorithm [11] that has been proved to be g l o b a l l y convergent [12].

With the design d e r i v a t i v e s calculated, however, any gradient

based, nonlinear programming algorithm can be used. 5.

NUMERICALEXAMPLES Example 1 The f i r s t

example presented deals with Polya and Weinstein's proof that con-

c e n t r i c c i r c l e s define the optimum shape, i f no constraints are placed on boundary location.

The amount of material is given as 65 units and the area of the hole is

20 u n i t s .

Both conditions are treated as isoparametric constraints.

As an i n i t i a l

design, two concentric c i r c l e s are selected with r a d i i 4.5 and 2.0 u n i t s , respectively. 5.

A regular polygon is used to approximate the boundary, as shown in Figure

The radial distances bi between the iCh vertex and the o r i g i n are chosen as

design variables. For the coarse grid model (96 elements, 64 nodes, and 16 design variables in Figure 5 ( a ) ) , s i x i t e r a t i o n s , r e q u i r i n g 7.93 CPU seconds on a PRIME 750 minicomputer, were required for convergence to the optimum shape.

I t took 7 i t e r a t i o n s

and 51.57 CPU seconds f o r the f i n e r grid model (384 elements, 224 nodes, and 64 design variables in Figure 5(b)) to achieve convergence.

A comparison between the

t h e o r e t i c a l values and the f i n a l optimum results is given in Table 1. Example 2 As a second example, the inner boundary is fixed as an e l l i p s e with semi-radii a = 2.5 and b = 1.0 units.

The amount of material is given as 45 units.

The

i n i t i a l estimate f o r the outer boundary was taken as a c i r c l e with radius 4.5 units. T h i r t y - f i v e i t e r a t i o n s and 270.1CPU seconds on a PRIME 750 minicomputer were

45

equired to achieve convergence to the design shown in Figure 6. r i g i d i t y is 415.83 at the f i n a l s o l u t i o n , while the i n i t i a l

The t o r s i o n a l

value is 604.74.

These

results support Banichuk's claim that wall thickness of the bar at the optimum shape decreases as one moves along the inner boundary in a d i r e c t i o n of increasing cur-

va~ure. Example 3 As a f i n a l example, both the outer and inner boundaries are treated as design variables.

In addition to the constraint on the amount of material available, the

cross section of the bar is required to be in a 10 x 16 unit Two f i n i t e element meshes are used for analysis.

rectangular housing.

One has 384 elements, 224 nodes,

and 64 design variables as in the preceding example. The second mesh has 960 elements, 528 nodes, and 96 design variables.

The i n i t i a l design is taken as two

concentric circles of radii 5, and 2.5 units. With given amounts of material of 85 and 110 units, numerical results are l i s t e d in Table 2. in Figures 7 and 8.

Optimumshapes, for different f i n i t e element meshes, are shown Note that the corners of the housing are not f i l l e d for all

examples, as one might expect.

Although the values of optimum torsional r i g i d i t i e s

are very close for the two f i n i t e element meshes, optimum shapes of the inner boundaries show significant differences.

I t is apparent that improved stress evaluation,

which gives a better approximation of design s e n s i t i v i t y coefficients, has caused this deviation.

I t is also interesting to see that the bar with a hole has dis-

tributed the material more e f f i c i e n t l y (has higher torsional r i g i d i t y ) than the bar with a solid cross section.

Calculated with a f i n i t e element model of 384 elements,

209 nodes, and 32 design variables, numerical results for the optimum design of a solid bar are listed in Table 3. 6.

The optimum shapes are shown in Figure 9.

CONCLUSIONSAND REMARKS The numerical examples offered here i l l u s t r a t e the wide a p p l i c a b i l i t y of the

i t e r a t i v e numerical schemes for shape optimal design.

Note that the pattern of

f i n i t e element mesh does not change during an iteration.

In each i t e r a t i o n , new

positions of boundary nodes are determined by the algorithm and the positions of i n t e r i o r nodes change accordingly. The s e n s i t i v i t y functional, derived using the variational formulation of the state equation and material derivative, is a boundary integral that contains only the nermal boundary movement (that is Vn) and the stress terms (~z~. "@n' Success in a numerical technique for shape optimal design depends on an accurate evaluation of these stress terms and on the representation of the boundary and i t s normal movement. A more sophisticated choice of elements or of a finer mesh can be introduced in the f i n i t e element method to improve the numerical approximation of stress

46

values.

Instead of l i n e a r piecewise functions, some smoother or more r e s t r i c t e d

classes of functions may be used to describe the boundary shapes.

From engineer's

point of view t h i s w i l l undoubtedly broaden the u t i l i z a t i o n of the shape o p t i m i zation techniques. A d e t a i l e d discussion of some mathematical approaches to the choices of f i n i t e elements and to the grid optimization f o r the f i n i t e element formulation of structural problems is offered in the paper of A.R. Diaz, N. Kikuchi and J.E. Taylor [14] in this volume.

Also see [12] and [8].

For a basic

introduction to this t o p i c , see reference [ 1 3 ] . F i n a l l y , we comment that the basic problem of pure torsion of an e l a s t i c m u l t i p l y connected bar is an important problem in the theory of e l a s t i c i t y and does have a long history.

Large body of l i t e r a t u r e concerning i t goes back to the

o r i g i n a l papers of Saint Venant, Lord Kelvin (Sir William Thompson~and Prandtl. While the numerical aspects and the t h e o r e t i c a l results of this paper d e a l t with a version of this classical problem or rather with the related problem of shape o p t i mization f o r e l a s t i c m u l t i p l y connected bars subjected to pure t o r s i o n , the uses of material d e r i v a t i v e and the other concepts u t i l i z e d here are quite general and are c e r t a i n l y not r e s t r i c t e d to the specific problem stated in the t i t l e of t h i s paper. An elementary introduction to the theory of pure torsion f o r l i n e a r e l a s t i c i sotropic bars may be found in the reference

[i].

47 REFERENCES 1.

Sokolnikoff, loS., Mathematical Theory of Elasticity, McGraw-Hill, New York, (1956).

2.

Polya, G., and Weinstein, A., "On the Torsional Rigidity of Multiply Connected Cross-Sections", Ann. of Math, 52, 154-163, (1950).

3.

Banichuk, N.V., "Optimization of Elastic Bars in Torsion", Int. J. Solids Struct., 12, 275-286, (1976).

4.

Kurshin, L. M., and Onoprienko, P.N., "Determination of the Shapes of DoublyConnected Bar Sections of Maximum Torsional Stiffness", Prik. Math. Mech. (English translation Appl. Mathematics and Mechanics, PMM), 40, 1078-1084, (1976).

5,

Dems, K., "Multiparameter Shape Optimization of Elastic Bars in Torsion", Int. J. Num. Meth. Engng., 15, 1517-1539, (1980).

6.

Aubin, J.P., Applied Functional Analysis, Wiley-lnterscience, New York, (1979).

7.

Zolesio, J.P,, "The Material Derivative (or speed) Method for Shape Optimization", Optimization of Distributed Parameter Structures, Vol. I I , (ed, E.J. Haug and J. Cea) Sitjthoff-Noordhoff, Rockville, Md., 1089-1151, (1981).

8.

Haug, E.J., Choi, K.K., and Komkov, V., Structural Design Sensitivity Analysis, Academic Press, New York, (1984).

9.

Cea, J., "Problems of Shape Optimal Design", Optimization of Distributed Paramemter Structures, Vol. II. (ed. E.J. Haug and J. Cea) Sitjthoff-Noordhoff, Rockvil!e, Md., 1005-1048,~(1981).

i0.

Rousselet, B., and Haug, E.J., "Design Sensitivity Analysis of Shape Variation", Optimization of Distributed Parameter Structures, Vol. I I , (ed. E.J. Haug and J. Cea) Sijthoff-Noordhoff, Rockville, Md., 1397-1442, (1981).

11.

Choi, K.K., Haug, E.J., Hou, J.W., and Sohoni, V.N., "Pshenichny's Linearization Method for Mechanical System Optimization", Trans, ASME, J. Mech. Design, to appear (1984).

12.

Pshenichny, B.N., and Danilin, Y.M., Numerical Methods in Extremal Problems, Mir, Moscow, (1978).

13.

Zienkiewicz, O.C., The Finite Element Method in Engineering Science, (the second expanded edition), McGraw H i l l , London, 1971.

14.

Diaz, A.R., Kikuchi N. and Taylor J.E., Optimal Design Formulation for Finite Element Grid Adaptation, in this volume.

48

LIST OF FIGURES AND TABLES Figure 1

Torsion of a Doubly-Connected Bar

2

Stress Function f o r a Doubly-Connected Bar

3

Pointwise Constraint on Boundary r

4

Parametric Definition of ?

5

F i n i t e Element Methods of Elastic Bar's Cross Section (a) Coarse Grid Model (b) Fine Grid Model Final Optimum Shape Final Optimum Shapes for a Torsion Bar with Coarse Mesh (a) Given Amount of Material is 85 Units (b) Given Amount of Material is i i 0 Units Final Optimum Shapes f o r a Torsion Bar with Fine Mesh (a) Given Amount of Material is 85 Units (b) Given Amount of Material is 110 Units Final Optimum Shapes f o r a Torsion Bar with Solid Cross Section (a) Given Amount of Material is 85 Units (b) Given Amount of Material is i i 0 Units

Table 1

Numerical Results f o r Optimum Shapes

2

Numerical Results f o r a Torsion Bar with Hollow Cross-Section

3

Numerical Results f o r a Torsion Bar with Solid Cross-Section

Table I.

Numerical Results for Optimum Shapes

Optimum Values

Theoretical Values

Coarse Grid

Torsional Rigidity

1086.44

1067.4

Fine Grid

1081.51

Radius of Outer Boundary

5.2016

5.2625 4 b i ~ 5.2716

5.2180 < b i < 5.2186

Radius of Inner Boundary

2.523

2.523 < b I ~ 2.5570

2.53L2 ~ b I ~ 2.5313

49 Table 2.

Numerical Results for a Torsion Bar with Hollow Cross-Section

T

Given Material

Optimum No. of Iterations CPU Seconds for Convergence on PRIME 750 Torsional Rigidit)

~ Grid

I,,Grid

Finer I Coarse Grid .......

Finer Grid

1419.~ 1602. 2457.3

2433.4 I 83.79

82.9

1523.~ 1602. 2826.9

2820.8 1107.8

106.7

Coarse Grid

Finer Grid

coars~ Finer Grid

85 units

704

410

II0 units

755

410

Table 3.

Constraint on Area

Coarse Grid

Numerical Results for a Torsion Bar with Solid Cross-Section

Given Material

No. of Iterations for Convergence

CPU Seconds on PRIME 750

Optimum Torsional Rigidity

85 units

47

247.4

1139.3

110 units

177

923.4

1785,2

Figure I.

Constraint on Area

Torsion of a Doubly-Connected Bar

84.99 108.1

50

Z

CI X2

>

Figure 2.

X1

Stress Function for a Doubly-Connected Bar

51

Figure 3.

Pointwise Constraint on Boundary F

n

/

>X 1

x2 Figure 4.

Parametric Definition of F

52

(a)

(b) Figure 5.

Coarse Grid Model

Fine Grid Model

Finite Element Models of Elastic Bar's Cross Section

53

Figure 6.

(a)

Final Optimum Shape

Given amount of material is 85 units

i/L./z.~-/si

(b) Given amount of material Figure

7.

Final Optimum Shapes with Coarse Mesh

is II0 units for a Torsion

Bar

(a) Given amount of material is 85 units

(b) Given amount of material is ii0 units Figure 8.

Final Optimum Shapes for a Torsion Bar with Fine Mesh

55

HOUSING

J

(a) Given amount of material is 85 units

HOUSING J

(b) Given amount of material is ii0 units Figure 9,,

Final Optimum Shapes for a Torsion Bar with Solid Cross-Section