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A stress versus compliance constraint in a minimum weight design
Compu~rrs & Structures Vol. 18, No Pnnted in Great Britain.
I. pp
9-13,
1984
MMS7949/84 Pergamon
A STRESS VERSUS COMPLIANCE CONSTRAINT MINIMUM WEIGHT DESIGN?
s3.00 + .oo Press Ltd.
IN A
ANTONI ZOCHOWSKI: Systems Research Institute, Warsaw, Newelska 6, Poland and KOICHI MIZUKAMI Faculty of Integrated Arts and Sciences, University of Hiroshima, Hiroshima 730, Japan (Received 27 September 1982; received for publication 25 November 1982)
Abstract-The paper contains a study of the stress-constrained minimum-weight design of the beam in the framework of [I]. The existence of optimal shape is considered. The numerical examples are given and stress-constrained designs are compared to compliance-constrained ones.
I. INTRODUCTION
d,v, 0 1
Here
In [l] the implementation of some recent results[2-4] to compliance constrained minimum weight design was proposed. The following paper extends the approach in order to incorporate the stress constraints and to give a sketch of the theoretical justification of the algorithm. In Section 3 we study the existence of optimal shapes to such problems, under the assumption of a sufficient regularity of the solution to the elasticity equations. Section 4contains considerations about the convergence of the approximate solutions. In numerical experiments we have tried to compare complianceand stress-constrained designs and find out the reason for similarities.
A =
0
[
y a, differential
operator,
a,. a,
g = (g,, gJr-load force vector; and n = (n,, n,)Toutward versor normal to the boundary. If we treat a stress also as a vector, it is defined by the equation
2. THE PROBLEM Before proceeding further, we must recall the subject of our study. It has the form of a 2-dimensional elastic structure R, limited by the boundaries S,, S,, .S,, S, (see Fig. 1). The part S, is loaded with the certain force, the part S, is clamped, and on S, and S, there is no force and no displacement conditions. If we define the Hilbert space V(Q) as
u = (c,, g2, a,#
= DAu.
For such a structure, which may be called one-sidedly clamped beam, we have formulated minimization problem:
(3)
the the
V(Q) = {u = (U,, U*)? U,, z$EH’(CI,) and
min J(Q)
(1)
(4)
u,, u2 = 0 on S,} with constraints then the displacement field u = (u,, u,)~ is the solution of the foliowing boundary value problem: find UE V(Q) such that a,(~, 4) =
(@)%A
u dR =
4 ‘g dS
R,ErI
(5)
UC V(Q)
(6)
(2)
IQ
a,(u,4 1=
for all Q E V(Q).
9’4 dS for all Q E V(Q)
(7) (8)
iThis research was done as a part of a Poland-Japan joint research project on “Numerical methods of optimization and game theory” which was supported by Polish Academy of Sciences, Japan Ministry of Education and Japan Society for Promotion of Sciences. $On leave at Hiroshima University.
The part S, of the boundary may change here and serves as a design variable. By II we denote the family of admissible domains, which will be discussed in Section 3 in detail. The sup-norm has been already replaced by L,,-norm, p P 1. 9
IO
A.
ZOCHOWSKI
and K. MEXJKAMI
In the solving process the formula F(u, r) dR =
2 F(u, z) da s R,dt r F(u, t)sn, dS + 4
m
J
-52
was applied to both the state equation (2) and constraint (8), under the assumption that the part S, of the boundary moves in the direction of the y-axis with the velocity s(x, c). It was possible then to write the directional derivative of the functionalj,(u) in the form: j;(u) = s, { s
(Aw)‘DAu-t-Ilu(l;}snzdS
(10)
where w is a solution of the adjoint boundary value problem: find WE?‘@,) such that (A~)%44
dR=
p~~~j~,p-~ur~ dR
(11)
I n,
s n,
for all f#~E V(Q). Now we shall also take into account stresses. Usually it is done by introducing the so called yield function Y(a) and assuming that if the inequality (12)
Y(a) 5 Yin,, is fulfilled than the linear strain-stress
relations hold (the beam does not break or bend irreversibly). Because the sufficient regularity of u has been assumed, we are able to treat the new constraint (12) in a similar way. We shall replace (12) by _Gu) = (1WA
“,(I,” 5 CL
and introduce the second adjoint VEV(Q) such that
sQ
(Av)~D@ dQ=
(13) problem:
dR
2
I
XI
Xn=L
x2 -Sl
Fig. 1. Geometry of the problem and triangulation. y,=f,(x,),
i = 1,2,. . , n (Fig. 1). Therefore the following finite-dimensional optimization problem in terms of y = (_v,,y,, . . ,yn)’ is well defined.
mincry=0.5y,+yz+~~~+y,_,+0.5y,
(17)
Kh(y)uh = bh
(18)
_h(“h)5 &3x
(19)
j,(u*) 5 Y”,,,
(20)
L(y) < 1.
(21)
By Kh we denote a stiffness matrix and (21) is a set of linear inequalities ensuring that Q: belongs to the set of admissible domains lI” (to be discussed in part 4). We shall only briefly recall that the algorithm presented in [l] was based on local linearization of the constraints (19), (20) by means of formulae (lo), (15) and then solving locally the LP problem in terms of AY= (AY,, A.Y~, . . , AYJ: min c *Ay
(22)
find
n,~IjY(a)li;-’ s x VY(u)DA4
I
(9)
(14)
V&(Y)AYI UP,, -j,(y)
(23)
V~~(Y)AY < YPmar -j,(y)
(24)
L@Y) S I-
L(Y)
\Ayi(
(25) n.
(26)
for all 4 E v(n,>. (vYy(x,, x2, XJ = (8, Y, a,Y,a,Y)j. Then the derivative of the yield constraint can be The constant 6 puts limits on the variation of the free written in the following form: part of the boundary in every step of the algorithm.
iXu) =
s
{ - (Av)‘DA u + 11Y(a)\~;}sn, dS.
S,
3. EXISTENCEOF THE SOLUTION
(15)
In numerical computations the space V(Q) is approximated with the finite elements defined on the triangulation of the domain Q,“, which is close to Q, (see Fig. 1). The important fact worth stressing is that the formula (9) is really applied to the finite element approximant II”, the solution of the discretized problem, not to II, the solution of the original problem. After discretization, the domain n, = {(X,y): 0 5 x I L, 0 S y iJ(x)) becomes
the
polygon
Q,“(_v,, y,,
(16)
. , y,,) where
There is a strict equivalence (16) between the function 1; end the domain R,, so we shall carry on the considerations in terms of functions f, only. The reasons connected with the solvability of the boundary problem (2) and the possibility of the uniform with respect to the family {f,) error assessment I\“*(Q) - u’@,)(/v S Ch/u*(%)/j vt
(27)
have led us to impose on f, the following conditions: (i) Ymin sS,tx)
5 ymax
(ii) IS:(x)/ S C,,
x EIO, Lcl.
WI
(29)
A stress versus compliance constraint in a minimum weight design Here Q,, is a reference domain corresponding to f; = const. However, these two restrictions do not ensure the existence of the solution to the optimization problem, as it will be seen later. Now we shall use results obtained in [6]. Let us denote by F’ the space of functions fdefined on [0, L] such that f(x) fulfils (i) and a( f) fulfils uniform cone condition (see [6]). After translating into our terms, in [6] it was proved that:
F, = v,:f, fulfils (i), (ii), (iii)}
F, = {.&Fe: 11 f’WWJ)~~p From [6] the continuity of the known. Now we must assume that belongs to WI”, p 2 2. Then from theorem (see, e.g. [7]) one obtains
(32) 2 Km&
(33)
functional (30) is the solution u of (2) Sobolev imbedding the inequality
(lull, 5 KllulIv* 2 IP < a
(34)
where the constant K depends only on y,,,, y,,,,,, and C,,. It means that the functional J,(j)
(35)
= Jlu(f)JJ,
is continuous on F, and therefore F, is closed. The same reasoning can be modified to yield the continuity xwI,p, so also F, is closed if Y(.) has a of II~mllwl~P sufficient regularity. Hence Fad= F,fl Fu fl F, is compact in F’. Because also J(f) is continuous on FC, by taking L-f= {Q,(f;) :&F&j
OF DISCRETE
(37)
lYi+, -Y,lI
i=l,...,n-1.
(38)
C&G
for the family &}
I) I W)h
(39)
where the constant a(h) is defined by the particular choice of the family {f;}. Equations (37)-(39) together form a set of linear inequalities (21). In this way we have described Ftd and II* = {Q*(y): Y satisfies (37), (38), (39)).
(40)
From the properties of Fadit follows that for every
fe&d /If-f” )II c,h2
(41)
[J(f) - J(fh)( 5 d2
(42)
and that for every fho Fadthere exists f E Fadsuch that (41) holds. Recalling again Sobolev imbedding theorem it is easy to see (43) and if the yield function is sufficiently regular then the functional j2 is also approximated: lj,(u(f;)) -j,(u*(f:))[ IjMS,))
-j,(~“(f:))I
I c,h I c,h”“- ‘).
(43) (44)
The constants c,, c,, c, do not depend on the choice of a particular f;.sF&. However, in (43) we can see a certain conflict: for greater p the sup-norm is better represented by the &-norm, but the accuracy with which one can calculate the value of this &-norm decreases. So some compromise in the choice of p is necessary. In the discretized optimization problem (17)-(21) gradients of the constraints are calculated according to the formula (9). It can be proved that such a method is valid for the finite element solution of a boundary value problem. Let us now choose a certain set Fo, a sequence {c,,}, e,+O, and the corresponding sequence h(c,) selected in such a way that for the resulting discrete boundary value problem the functionals J(f;), j,(J), j2(f,) are approximated with the accuracy at least 6”. Then we can define the following discrete minimization problem, denoted by D(c,):
(36)
we are able to secure the existence of the minimum weight design. 4. CONVERGENCE
i=l,...,n
(31)
is compact in C’([O, L]). For the natural imbedding of C’([O, 15.1)into L,([O, L]) is continuous, so F, is compact in F’. Such a condition is only slightly more general than the requirement of bounded curvature for S,, so it has satisfactory physical interpretation. Let us consider now two sets F,, F,: Ilu(f,)/I, I ~a,}
Ymln5 Yi 5 Ymax?
12Yl-Yi- I -yi+
(30)
is continuous on F, if only bilinear forms a/(~, v) are uniformly coercive, continuous on F’ and functionals Jo(f) commonly bounded on F’. Obviously, functions satisfying (i) and (ii) do not create a compact set in P. Therefore we are compelled to add the third condition: (iii) {f:} forms a family of equicontinuous functions on [0, L]. By applying Ascoli theorem, it is easy to prove that the set F,
Fu = &Fe:
each f~ Fadwe shall take as its interpolant the piecewise linear function passing through points yi =f(x,), (Fig. 1). Conditions (i) and (ii) have then as their counterparts
The requirement of equicontinuity imposes the condition:
--F’ is compact in L,([O, L)] topology, -the functional Jo(f) = IIu(Q,)(IWI,,
II
SOLUTIONS
At the beginning we shall define the family lTh of domains admissible in a discrete optimization. For
min J(hh)
(45)
AhErIh
(46) bh
(47)
j,(nh) 5 W%ix + 6”
(48)
&Uh =
12
A. ZOCHOWSKI and K. MIZUKAMI
j,(d)
5
Ygax+ c,,
h = h(t,).
(49)
Conditions (48) (49) describe the set of admissible designs F~,,(c,,).Its crucial feature lies in the fact that, because of the definition of h(t,), if JEF,~ then .~;“EF:&J Now we must make the most important assumption which states that our method for solving the problem D (6”)allows us to find the optimum value of J with the accuracy at least t,. In other words we can obtain such a,f:(r,) that IJ(.L”) - W*)(
< f”
(50)
where Jcf,“*) is the exact solution of D(c,). Of course it is impossible to prove the convergence of optimal discrete designs to any particular shape because in principle there can be many equivalent shapes. However, it is possible to prove the convergence of the optimal values of the goal function. To this aim let us create a sequence {Jn) = {J@(Q)]. It can be proved that, granted all stated above pr_operties of discrete problems D(c,), the sequence (J,,) is bounded, has a unique condensation point and therefore is convergent to a certain j*. Furthermore, j* cannot be different from J*, the optimal solution of the original problem, because the other assumption leads to contradiction.
5.NUMERICAL EXPERIMENTS WITH YIELD CONSTRAINT
For computations the simplest possible yield function was assumed, namely Y(a) = aru.
(51)
Nevertheless, it can be considered as a certain generalization of the often met criteria which have the form of the quadratic positively definite function of stresses. In this way it has some relation to reality. Of course the other types of Y may be used without any difficulty. The Lame coefficients had equal values, 1 = p, what corresponds to v = l/3. Similarly to [l], the length of the beam was fixed at L = 5 and the initial shape G = Qyb) was equivalent to f0 = 1. The constants limiting the movement of the boundary were taken as Ymln= 0.5, y,,, = 2. In approximation of j, the parameter p = 10 was applied, while for j, it had value p = 5 (necessity of compromise). Two sets of experiments were performed. In a case of the one-sidedly clamped beam with uniform load the discretization parameters had values n = 16, m = 5. Let us denote by u, and Y, the maximum calculated values of displacement and yield function for the initial shape Q,. The computations with the following parameters were performed.
A.,: u,,~ = 00, Y,,,,, = Y&J;
result:
P = 24.5%
A,: u,,, = co. Y,,, = 1.2Y(u,);
result:
P=30;:,
A,: u,,~ = 00, Y,,, = 0.8 Y(u,);
result:
P = 14.7%.
Here u( Y,) denotes the value achieved by compliance in the design with only the yield constrained and Y(u,) vice versa, while P stands for the payoff. The experiments A, and A, were done in order to compare the optimal shapes of beams with the compliance or yield value the same as the ones for the straight initial shape. It turned out that the compliance constraint is much more restrictive, as is demonstrated by resulting payoffs and shapes (Fig. 2). In experiment A, it was checked if the shape optimal for the yield constraint Y,,,,, = Y, is also optimal for the pair (Y,, u( YO)).The supposition has not been confirmed, but the gain from relaxing the yield constraint was not substantial. Similarly, experiment A, proved that the design obtained in A, is not optimal for the pair (u,, Y(uJ), but the removing of compliance constraint gave the bigger decrease in weight (5.57; vs 1.3%). It proved again that the limitation of displacement is stronger than the limitation of yield function. Finally, experiments A, and A, show the change of the optimal shape corresponding to the variation of the yield constraint, the shape obtained in A, being taken as a standard (Fig. 3). The two-sidedly clamped beam with the load concentrated on a short central interval of the straight surface was also studied. In this case, because of symmetry, the mesh parameters for the half of the beam were n = 16, m = 7. Four experiments were conducted. B,:
Urnax =
Y,,,
u(),
=
5;
result: Y(uO)= 1.18Y,,, P = 8.3’%;
Fig. 2. Comparison
of designs under constraints.
yield and compliance
A,: u,,, = m, Y,,,,, = Y,,;
result: u( YO)= 2.6u,,
P = 38.7%
A,: u,,, = uo, Y,,, = m;
result: Y(u,) = 0.6Y0, P = 19% A,: n,,, = U(Y(J, Y,,, = rJ3; P =40x result:
Fig. 3. Changes
of the optimal shape with the variation the yield constraint.
of
A stress versus
B2: u,,, = 00, y,,, =
yo;
result: u( Y,) = 1.39~,, B,:
u,,,
= SI,
Y,,,,, =
= CG, Y,,,,, =
P =
19.1%
P =
26.7%
P =
8.894.
I.ZY,;
result: B4: u,,,
compliance constraint in a minimum weight design
0.8 Y,;
result:
In B, and B2 the shapes obtained from the straight beam by fixing the compliance and yield were again compared (Fig. 4). The conclusion is the same: displacement constraint is more severe. Besides, there is a distinct difference in the distribution of thickness, because the yield-constrained design tends to concentrate more material in the area of maximal stresses. In Fig. 5 the variation of shape corresponding to the change of Y,,,,, is displayed. Speaking generally, the actions of both types of optimization, yield-constrained and compliance-
13
constrained, have the common tendency to the more even distribution of the yield function (stresses) along the beam, as can be seen in Fig. 6 (which confirms that despite its arbitrariness such a form of Y has a physical meaning). Obviously, there is no obstable to applying two constraints simultaneously, but such a case is not interesting, because usually one of them prevails. In many experiments both constraints were active at the same time for some iterations and it was confirmed that, similarly as in [l], the replacement of the sup-norm by L,-norm gives good accuracy: the trespassing of the bounds was kept in the 2-37; limit. The only difference was in the time of computation: if the compliance-constrained design required about 60 iterations, in case of the yield-constrained it was 20&300 iterations. The reason lies probably in a smaller accuracy with which the stresses are calculated by this particular method used for the solution of the boundary value problem. Maybe also the nature of the set F, is more complicated. 6. CONCLUSIONS
Fig. 4. Yield and compliance constrained sidedly clamped beam.
designs
of two-
It has turned out that the incorporation of the stress constraints into the framework of [I] was possible. The minimum-weight designs obtained in this way again proved to be in accordance with physical reasoning (see, e.g. [8]), especially the changes of stress distribution during the optimization process. It seems also that the assumptions securing the existence of the solution are natural and have physical interpretation. In practice, while conducting the discrete optimization, the conditions (38), (39) can safely be dropped unless the boundary reveals singular behaviour. In order to make a method of practical value it is necessary now to take into account the dynamics of the structure, which will be the subject of further studies. REFERENCES
Fig.
5. Variation of the shape for the yield-constrained two-sidedly clamped beam.
Yk -
inltlal
----
u-constrained
A,
-.---
Y-constrained
A2
Fig. 6. Yield function along the upper straight initial and optimal shapes.
surface
for
I. A. Zochowski and K. Mizukami, Minimum-weight design with displacement constraints in 2-dimensional elasticity. To appear in Compuf. Structures. 2. B. Palmerio and A. Dervieux, Une formule de Hadamard dans des problemes d’optimal design. Proc. 7th IFIP Conf. on Optimization Techniques (Edited by J. Cea), Lecrure Notes in Computer Sciences 41 (1976). 3. A.Dervieux, A perturbation study of a jet-like annular free boundary problem and an application to an optimal control problem. Comm. in Purfiul D@ Eq. 6 (1981). 4. A. Marrocco and 0. Pironneau, Optimum design with Lagrangian finite elements-design of an electromagnet. Compur. Meth. pppl. Mech. Engng 15 (1978). L. A. Rozin, Variational Formulation of Problems for Elastic Systems. Leningrad (I 978). In Russian. D. Chenais, On the existence of a solution in a domain identification problem. J. Math. Anal. Appt. 52 (1975). R. A. Adams. Soboleo Spaces. Academic Press, New York (1975). G. 1. N. Rozvany, Optimal elastic design for stress constraints. Compur. Struclures 8 (1978).
I. pp
9-13,
1984
MMS7949/84 Pergamon
A STRESS VERSUS COMPLIANCE CONSTRAINT MINIMUM WEIGHT DESIGN?
s3.00 + .oo Press Ltd.
IN A
ANTONI ZOCHOWSKI: Systems Research Institute, Warsaw, Newelska 6, Poland and KOICHI MIZUKAMI Faculty of Integrated Arts and Sciences, University of Hiroshima, Hiroshima 730, Japan (Received 27 September 1982; received for publication 25 November 1982)
Abstract-The paper contains a study of the stress-constrained minimum-weight design of the beam in the framework of [I]. The existence of optimal shape is considered. The numerical examples are given and stress-constrained designs are compared to compliance-constrained ones.
I. INTRODUCTION
d,v, 0 1
Here
In [l] the implementation of some recent results[2-4] to compliance constrained minimum weight design was proposed. The following paper extends the approach in order to incorporate the stress constraints and to give a sketch of the theoretical justification of the algorithm. In Section 3 we study the existence of optimal shapes to such problems, under the assumption of a sufficient regularity of the solution to the elasticity equations. Section 4contains considerations about the convergence of the approximate solutions. In numerical experiments we have tried to compare complianceand stress-constrained designs and find out the reason for similarities.
A =
0
[
y a, differential
operator,
a,. a,
g = (g,, gJr-load force vector; and n = (n,, n,)Toutward versor normal to the boundary. If we treat a stress also as a vector, it is defined by the equation
2. THE PROBLEM Before proceeding further, we must recall the subject of our study. It has the form of a 2-dimensional elastic structure R, limited by the boundaries S,, S,, .S,, S, (see Fig. 1). The part S, is loaded with the certain force, the part S, is clamped, and on S, and S, there is no force and no displacement conditions. If we define the Hilbert space V(Q) as
u = (c,, g2, a,#
= DAu.
For such a structure, which may be called one-sidedly clamped beam, we have formulated minimization problem:
(3)
the the
V(Q) = {u = (U,, U*)? U,, z$EH’(CI,) and
min J(Q)
(1)
(4)
u,, u2 = 0 on S,} with constraints then the displacement field u = (u,, u,)~ is the solution of the foliowing boundary value problem: find UE V(Q) such that a,(~, 4) =
(@)%A
u dR =
4 ‘g dS
R,ErI
(5)
UC V(Q)
(6)
(2)
IQ
a,(u,4 1=
for all Q E V(Q).
9’4 dS for all Q E V(Q)
(7) (8)
iThis research was done as a part of a Poland-Japan joint research project on “Numerical methods of optimization and game theory” which was supported by Polish Academy of Sciences, Japan Ministry of Education and Japan Society for Promotion of Sciences. $On leave at Hiroshima University.
The part S, of the boundary may change here and serves as a design variable. By II we denote the family of admissible domains, which will be discussed in Section 3 in detail. The sup-norm has been already replaced by L,,-norm, p P 1. 9
IO
A.
ZOCHOWSKI
and K. MEXJKAMI
In the solving process the formula F(u, r) dR =
2 F(u, z) da s R,dt r F(u, t)sn, dS + 4
m
J
-52
was applied to both the state equation (2) and constraint (8), under the assumption that the part S, of the boundary moves in the direction of the y-axis with the velocity s(x, c). It was possible then to write the directional derivative of the functionalj,(u) in the form: j;(u) = s, { s
(Aw)‘DAu-t-Ilu(l;}snzdS
(10)
where w is a solution of the adjoint boundary value problem: find WE?‘@,) such that (A~)%44
dR=
p~~~j~,p-~ur~ dR
(11)
I n,
s n,
for all f#~E V(Q). Now we shall also take into account stresses. Usually it is done by introducing the so called yield function Y(a) and assuming that if the inequality (12)
Y(a) 5 Yin,, is fulfilled than the linear strain-stress
relations hold (the beam does not break or bend irreversibly). Because the sufficient regularity of u has been assumed, we are able to treat the new constraint (12) in a similar way. We shall replace (12) by _Gu) = (1WA
“,(I,” 5 CL
and introduce the second adjoint VEV(Q) such that
sQ
(Av)~D@ dQ=
(13) problem:
dR
2
I
XI
Xn=L
x2 -Sl
Fig. 1. Geometry of the problem and triangulation. y,=f,(x,),
i = 1,2,. . , n (Fig. 1). Therefore the following finite-dimensional optimization problem in terms of y = (_v,,y,, . . ,yn)’ is well defined.
mincry=0.5y,+yz+~~~+y,_,+0.5y,
(17)
Kh(y)uh = bh
(18)
_h(“h)5 &3x
(19)
j,(u*) 5 Y”,,,
(20)
L(y) < 1.
(21)
By Kh we denote a stiffness matrix and (21) is a set of linear inequalities ensuring that Q: belongs to the set of admissible domains lI” (to be discussed in part 4). We shall only briefly recall that the algorithm presented in [l] was based on local linearization of the constraints (19), (20) by means of formulae (lo), (15) and then solving locally the LP problem in terms of AY= (AY,, A.Y~, . . , AYJ: min c *Ay
(22)
find
n,~IjY(a)li;-’ s x VY(u)DA4
I
(9)
(14)
V&(Y)AYI UP,, -j,(y)
(23)
V~~(Y)AY < YPmar -j,(y)
(24)
L@Y) S I-
L(Y)
\Ayi(
(25) n.
(26)
for all 4 E v(n,>. (vYy(x,, x2, XJ = (8, Y, a,Y,a,Y)j. Then the derivative of the yield constraint can be The constant 6 puts limits on the variation of the free written in the following form: part of the boundary in every step of the algorithm.
iXu) =
s
{ - (Av)‘DA u + 11Y(a)\~;}sn, dS.
S,
3. EXISTENCEOF THE SOLUTION
(15)
In numerical computations the space V(Q) is approximated with the finite elements defined on the triangulation of the domain Q,“, which is close to Q, (see Fig. 1). The important fact worth stressing is that the formula (9) is really applied to the finite element approximant II”, the solution of the discretized problem, not to II, the solution of the original problem. After discretization, the domain n, = {(X,y): 0 5 x I L, 0 S y iJ(x)) becomes
the
polygon
Q,“(_v,, y,,
(16)
. , y,,) where
There is a strict equivalence (16) between the function 1; end the domain R,, so we shall carry on the considerations in terms of functions f, only. The reasons connected with the solvability of the boundary problem (2) and the possibility of the uniform with respect to the family {f,) error assessment I\“*(Q) - u’@,)(/v S Ch/u*(%)/j vt
(27)
have led us to impose on f, the following conditions: (i) Ymin sS,tx)
5 ymax
(ii) IS:(x)/ S C,,
x EIO, Lcl.
WI
(29)
A stress versus compliance constraint in a minimum weight design Here Q,, is a reference domain corresponding to f; = const. However, these two restrictions do not ensure the existence of the solution to the optimization problem, as it will be seen later. Now we shall use results obtained in [6]. Let us denote by F’ the space of functions fdefined on [0, L] such that f(x) fulfils (i) and a( f) fulfils uniform cone condition (see [6]). After translating into our terms, in [6] it was proved that:
F, = v,:f, fulfils (i), (ii), (iii)}
F, = {.&Fe: 11 f’WWJ)~~p From [6] the continuity of the known. Now we must assume that belongs to WI”, p 2 2. Then from theorem (see, e.g. [7]) one obtains
(32) 2 Km&
(33)
functional (30) is the solution u of (2) Sobolev imbedding the inequality
(lull, 5 KllulIv* 2 IP < a
(34)
where the constant K depends only on y,,,, y,,,,,, and C,,. It means that the functional J,(j)
(35)
= Jlu(f)JJ,
is continuous on F, and therefore F, is closed. The same reasoning can be modified to yield the continuity xwI,p, so also F, is closed if Y(.) has a of II~mllwl~P sufficient regularity. Hence Fad= F,fl Fu fl F, is compact in F’. Because also J(f) is continuous on FC, by taking L-f= {Q,(f;) :&F&j
OF DISCRETE
(37)
lYi+, -Y,lI
i=l,...,n-1.
(38)
C&G
for the family &}
I) I W)h
(39)
where the constant a(h) is defined by the particular choice of the family {f;}. Equations (37)-(39) together form a set of linear inequalities (21). In this way we have described Ftd and II* = {Q*(y): Y satisfies (37), (38), (39)).
(40)
From the properties of Fadit follows that for every
fe&d /If-f” )II c,h2
(41)
[J(f) - J(fh)( 5 d2
(42)
and that for every fho Fadthere exists f E Fadsuch that (41) holds. Recalling again Sobolev imbedding theorem it is easy to see (43) and if the yield function is sufficiently regular then the functional j2 is also approximated: lj,(u(f;)) -j,(u*(f:))[ IjMS,))
-j,(~“(f:))I
I c,h I c,h”“- ‘).
(43) (44)
The constants c,, c,, c, do not depend on the choice of a particular f;.sF&. However, in (43) we can see a certain conflict: for greater p the sup-norm is better represented by the &-norm, but the accuracy with which one can calculate the value of this &-norm decreases. So some compromise in the choice of p is necessary. In the discretized optimization problem (17)-(21) gradients of the constraints are calculated according to the formula (9). It can be proved that such a method is valid for the finite element solution of a boundary value problem. Let us now choose a certain set Fo, a sequence {c,,}, e,+O, and the corresponding sequence h(c,) selected in such a way that for the resulting discrete boundary value problem the functionals J(f;), j,(J), j2(f,) are approximated with the accuracy at least 6”. Then we can define the following discrete minimization problem, denoted by D(c,):
(36)
we are able to secure the existence of the minimum weight design. 4. CONVERGENCE
i=l,...,n
(31)
is compact in C’([O, L]). For the natural imbedding of C’([O, 15.1)into L,([O, L]) is continuous, so F, is compact in F’. Such a condition is only slightly more general than the requirement of bounded curvature for S,, so it has satisfactory physical interpretation. Let us consider now two sets F,, F,: Ilu(f,)/I, I ~a,}
Ymln5 Yi 5 Ymax?
12Yl-Yi- I -yi+
(30)
is continuous on F, if only bilinear forms a/(~, v) are uniformly coercive, continuous on F’ and functionals Jo(f) commonly bounded on F’. Obviously, functions satisfying (i) and (ii) do not create a compact set in P. Therefore we are compelled to add the third condition: (iii) {f:} forms a family of equicontinuous functions on [0, L]. By applying Ascoli theorem, it is easy to prove that the set F,
Fu = &Fe:
each f~ Fadwe shall take as its interpolant the piecewise linear function passing through points yi =f(x,), (Fig. 1). Conditions (i) and (ii) have then as their counterparts
The requirement of equicontinuity imposes the condition:
--F’ is compact in L,([O, L)] topology, -the functional Jo(f) = IIu(Q,)(IWI,,
II
SOLUTIONS
At the beginning we shall define the family lTh of domains admissible in a discrete optimization. For
min J(hh)
(45)
AhErIh
(46) bh
(47)
j,(nh) 5 W%ix + 6”
(48)
&Uh =
12
A. ZOCHOWSKI and K. MIZUKAMI
j,(d)
5
Ygax+ c,,
h = h(t,).
(49)
Conditions (48) (49) describe the set of admissible designs F~,,(c,,).Its crucial feature lies in the fact that, because of the definition of h(t,), if JEF,~ then .~;“EF:&J Now we must make the most important assumption which states that our method for solving the problem D (6”)allows us to find the optimum value of J with the accuracy at least t,. In other words we can obtain such a,f:(r,) that IJ(.L”) - W*)(
< f”
(50)
where Jcf,“*) is the exact solution of D(c,). Of course it is impossible to prove the convergence of optimal discrete designs to any particular shape because in principle there can be many equivalent shapes. However, it is possible to prove the convergence of the optimal values of the goal function. To this aim let us create a sequence {Jn) = {J@(Q)]. It can be proved that, granted all stated above pr_operties of discrete problems D(c,), the sequence (J,,) is bounded, has a unique condensation point and therefore is convergent to a certain j*. Furthermore, j* cannot be different from J*, the optimal solution of the original problem, because the other assumption leads to contradiction.
5.NUMERICAL EXPERIMENTS WITH YIELD CONSTRAINT
For computations the simplest possible yield function was assumed, namely Y(a) = aru.
(51)
Nevertheless, it can be considered as a certain generalization of the often met criteria which have the form of the quadratic positively definite function of stresses. In this way it has some relation to reality. Of course the other types of Y may be used without any difficulty. The Lame coefficients had equal values, 1 = p, what corresponds to v = l/3. Similarly to [l], the length of the beam was fixed at L = 5 and the initial shape G = Qyb) was equivalent to f0 = 1. The constants limiting the movement of the boundary were taken as Ymln= 0.5, y,,, = 2. In approximation of j, the parameter p = 10 was applied, while for j, it had value p = 5 (necessity of compromise). Two sets of experiments were performed. In a case of the one-sidedly clamped beam with uniform load the discretization parameters had values n = 16, m = 5. Let us denote by u, and Y, the maximum calculated values of displacement and yield function for the initial shape Q,. The computations with the following parameters were performed.
A.,: u,,~ = 00, Y,,,,, = Y&J;
result:
P = 24.5%
A,: u,,, = co. Y,,, = 1.2Y(u,);
result:
P=30;:,
A,: u,,~ = 00, Y,,, = 0.8 Y(u,);
result:
P = 14.7%.
Here u( Y,) denotes the value achieved by compliance in the design with only the yield constrained and Y(u,) vice versa, while P stands for the payoff. The experiments A, and A, were done in order to compare the optimal shapes of beams with the compliance or yield value the same as the ones for the straight initial shape. It turned out that the compliance constraint is much more restrictive, as is demonstrated by resulting payoffs and shapes (Fig. 2). In experiment A, it was checked if the shape optimal for the yield constraint Y,,,,, = Y, is also optimal for the pair (Y,, u( YO)).The supposition has not been confirmed, but the gain from relaxing the yield constraint was not substantial. Similarly, experiment A, proved that the design obtained in A, is not optimal for the pair (u,, Y(uJ), but the removing of compliance constraint gave the bigger decrease in weight (5.57; vs 1.3%). It proved again that the limitation of displacement is stronger than the limitation of yield function. Finally, experiments A, and A, show the change of the optimal shape corresponding to the variation of the yield constraint, the shape obtained in A, being taken as a standard (Fig. 3). The two-sidedly clamped beam with the load concentrated on a short central interval of the straight surface was also studied. In this case, because of symmetry, the mesh parameters for the half of the beam were n = 16, m = 7. Four experiments were conducted. B,:
Urnax =
Y,,,
u(),
=
5;
result: Y(uO)= 1.18Y,,, P = 8.3’%;
Fig. 2. Comparison
of designs under constraints.
yield and compliance
A,: u,,, = m, Y,,,,, = Y,,;
result: u( YO)= 2.6u,,
P = 38.7%
A,: u,,, = uo, Y,,, = m;
result: Y(u,) = 0.6Y0, P = 19% A,: n,,, = U(Y(J, Y,,, = rJ3; P =40x result:
Fig. 3. Changes
of the optimal shape with the variation the yield constraint.
of
A stress versus
B2: u,,, = 00, y,,, =
yo;
result: u( Y,) = 1.39~,, B,:
u,,,
= SI,
Y,,,,, =
= CG, Y,,,,, =
P =
19.1%
P =
26.7%
P =
8.894.
I.ZY,;
result: B4: u,,,
compliance constraint in a minimum weight design
0.8 Y,;
result:
In B, and B2 the shapes obtained from the straight beam by fixing the compliance and yield were again compared (Fig. 4). The conclusion is the same: displacement constraint is more severe. Besides, there is a distinct difference in the distribution of thickness, because the yield-constrained design tends to concentrate more material in the area of maximal stresses. In Fig. 5 the variation of shape corresponding to the change of Y,,,,, is displayed. Speaking generally, the actions of both types of optimization, yield-constrained and compliance-
13
constrained, have the common tendency to the more even distribution of the yield function (stresses) along the beam, as can be seen in Fig. 6 (which confirms that despite its arbitrariness such a form of Y has a physical meaning). Obviously, there is no obstable to applying two constraints simultaneously, but such a case is not interesting, because usually one of them prevails. In many experiments both constraints were active at the same time for some iterations and it was confirmed that, similarly as in [l], the replacement of the sup-norm by L,-norm gives good accuracy: the trespassing of the bounds was kept in the 2-37; limit. The only difference was in the time of computation: if the compliance-constrained design required about 60 iterations, in case of the yield-constrained it was 20&300 iterations. The reason lies probably in a smaller accuracy with which the stresses are calculated by this particular method used for the solution of the boundary value problem. Maybe also the nature of the set F, is more complicated. 6. CONCLUSIONS
Fig. 4. Yield and compliance constrained sidedly clamped beam.
designs
of two-
It has turned out that the incorporation of the stress constraints into the framework of [I] was possible. The minimum-weight designs obtained in this way again proved to be in accordance with physical reasoning (see, e.g. [8]), especially the changes of stress distribution during the optimization process. It seems also that the assumptions securing the existence of the solution are natural and have physical interpretation. In practice, while conducting the discrete optimization, the conditions (38), (39) can safely be dropped unless the boundary reveals singular behaviour. In order to make a method of practical value it is necessary now to take into account the dynamics of the structure, which will be the subject of further studies. REFERENCES
Fig.
5. Variation of the shape for the yield-constrained two-sidedly clamped beam.
Yk -
inltlal
----
u-constrained
A,
-.---
Y-constrained
A2
Fig. 6. Yield function along the upper straight initial and optimal shapes.
surface
for
I. A. Zochowski and K. Mizukami, Minimum-weight design with displacement constraints in 2-dimensional elasticity. To appear in Compuf. Structures. 2. B. Palmerio and A. Dervieux, Une formule de Hadamard dans des problemes d’optimal design. Proc. 7th IFIP Conf. on Optimization Techniques (Edited by J. Cea), Lecrure Notes in Computer Sciences 41 (1976). 3. A.Dervieux, A perturbation study of a jet-like annular free boundary problem and an application to an optimal control problem. Comm. in Purfiul D@ Eq. 6 (1981). 4. A. Marrocco and 0. Pironneau, Optimum design with Lagrangian finite elements-design of an electromagnet. Compur. Meth. pppl. Mech. Engng 15 (1978). L. A. Rozin, Variational Formulation of Problems for Elastic Systems. Leningrad (I 978). In Russian. D. Chenais, On the existence of a solution in a domain identification problem. J. Math. Anal. Appt. 52 (1975). R. A. Adams. Soboleo Spaces. Academic Press, New York (1975). G. 1. N. Rozvany, Optimal elastic design for stress constraints. Compur. Struclures 8 (1978).