Lecture Notes in Physics Edited by J. Ehlers, Munchen, K. Hepp, Zurich and H. A. WeidenmUller, Heidelberg Managing Editor: W. Beiglbock, Heidelberg
21 Optimization and Stability Problems in Continuum Mechanics
Lectures Presented at the Symposium on Optimization and Stability Problems in Continuum Mechanics Los Angeles, California, August 24, 1971 Edited by P. K. C. Wang University of California, Los Angeles, CNUSA
Springer-Verlag Berlin· Heidelberg · New York 1973
ISBN 3-540-06214-9 Springer-Verlag Berlin . Heidelberg . New York ISBN 0-387-06214-9 Springer-Verlag New York· Heidelberg' Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other the amount of the fee to be determined by agreement with the publisher. © by Springer Verlag Berlin' Heidelberg 1973. Library of Congress Catalog Card Number 73-78080. Printed in Germany. Offsetprinting and bookbinding: Julius Beltz, Hemsbach/Bergstr.
CONTENTS
PART 1. OPTIMIZATION PROBLEMS H.Halkin: The Method of Dubovitskii-Milyutin in Mathematical Programming .••••••••••.••••••••••••••.••••••••••.• 1 R.T.Shield: Optimum Design of Structures Through Variational Principles •.••••.•••••••••.••••••.••••.••••••••.•• 13 T.Y.Wu ,A.T.Chwang and P.K.C.Wang: Optimization Problems in Hydrofoil Propulsion .••••••••••••••••.•••.•.•••.•• 38 PART 2. STABILITY PROBLEMS E.F.Infante: Stability Theory for General Dynamical Systems and Some Applications ••.•••••••••••..•••••••••.••• 63 E.M.Barston: Stability of Dissipative Systems with Applications to Fluids and Magnetofluids ••••••••...••••••••••.• 83
PREFACE
The five papers in this volume represent expanded versions of the l~ctures
presented at the Symposium on Optimization and Stability Prob-
lems in Continuum Mechanics at the University of Southern California, Los Angeles, August 24,1971.
The Symposium was held in conjunction with
the Western Applied Mechanics Conference sponsored by the Applied Mechanics Division of the American Society of Mechanical Engineers with the cooperation of the University of Southern California. The objectives of this Symposium were twofold,namely,to introduce recent results in general optimization and stability theories which have potential applications to continuum mechanical systems and to present new results dealing with specific classes of systems.
It is felt that
there is a wealth of new and interesting optimization and stability problems in continuum mechanics.
Hopefully, these lectures will help to
stimulate further research in this relatively new area. The idea for this Symposium was originally conceived by Professor C.S.Hsu of the University of California,Berkeley, who also presided over the Stability Session of this Symposium.
Professor G.H.Hegemier of the
University of California, San Diego, served as the Co-Chairman of this Symposium.
Los Angeles,California April,1972
P.K.C.Wang
THE METHOD OF DUBOVITSKII-MILYUTIN IN MATHEMATICAL PROGRAMMING * Hubert Halkin** Department of Mathematics University of California at San Diego La Jolla,California
1. INTRODUCTION
I want to give here a brief description of a very attractive formalism in optimization theory: the method of Dubovitskii and Milyutin [1] and the relate some recent extensions of that method, Halkin [2], with the necessary condition of Fritz John[3]. The first step in the method of Dubovitskii-Milyutin is to notice that, in any optimization problem, to say that some solution is optimal is equivalent of saying that a certain family of sets {Si: i E I} have no points in common i. e.
ni E:. I Si=ll.
Consider for example the optimization problem consisting in minimizing a function f 2 over all point of the plane R where a function g is nonpositive. A.
2
To say that an
A
element x 1n R with g(x)
XER2,f(x)
In general we can do very little with two disjoint sets Sl and S2'
But if Sl and S2 happen to be convex and nonempty then we can separate them by an 2 hyperplane, i.e. we can find a nonzero vector p in R such that sup
p' x
~
inf
p • x •
Here the scalar product of two vectors p and x is denoted by p·x.
We shall show
below that this separation, when possible, leads to interesting results. The second step in the method of Dubovitskii-Milyutin is 'to repace each set S. by 1
a set
~, 1
which is convex and such a good approximation of S, ( in a sense to be pre1
cised later) that from the fact that the sets S. have no point in common, i.e. 1
niE. I Si=ll,we can prove that the sets ~i will have no point in common,Le'C\E I~i=ll·
*
This research was supported by the Air Force Office of Scientific Research, under Grant No. AFOSR-68-l529C. **On leave for the academic year 1971-1972 at CORE, University of Louvain,Belgium.
2
In the simple example given above. if f and g are differentiable at x, we could consider the convex sets
...." A" x E R2 .f(x)+(x-x)·grad f(x)
1\
A
A A} xER 2 , (x-x)'grad f(x)
A
~2={x : xER .g(x)+(x-x)·grad g(x)
The third step in the method of Dubovitskii-Milyutin is to go from the fact that the sets
w.1
~. 1
have no point in common. Le.
niEI
~i=<1> to the fact that the sets
: i E: I} can be separated. We shall define later what we mean by separating more
than two convex sets. we have only two sets sets
~l
and
~2
For the time being let us go back to our simple example where ~l
and
~2
and let us see how the separation of the two convex
is equivalent to the known necessary condition for that problem: if '" x
is optimal and grad g(Q)~O, then for some A ~ 0 we have Ag(~) Agrad g(~) = O.
o and -grad f(~) +
The set ~l will always be empty if grad f(~) ~ 0 and the set ~2 will
always be empty if grad g(~) ~ O.
We shall temporarily assume that grad f(k) and
grad g(~) are different from zero.
A
Since for i=l and 2 the point x belongs to
-
~., 1
(the closure of ~.). i.e. since there are points in ~. arbitrarily close to ~. then 1
1
the hyperplane separating ~l and ~2 must pass through vector p such that sUPxE:~lP'x = p.x = infxE~/·x.
2.
i.e. there exists a nonzero
In other words, for all xE~l'
i.e. for all x such that (x-2).grad f(~) < 0 we have p'x ~ p.~. i.e. (x-~).p ~ O. Since p ~ O. this means Moreover. for all
that for
xE ~2'
sorre a > 0 we must have
grad f(~) = ap.
i.e. for all x such that g(Q) + (x-~)'grad g(~) < 0,
we have p'x ~ p,~; i.e. (x-~),p ~ O.
Since p ~ 0, this means that g(~)=O and that
for some b >0, we have p=-b grad g(~) and bg(~)=O.
Combining those two results. we
obtain that, under the assumptions grad f(~)~O and grad g(Q)~O, we have -grad f(2) + Agrad g(~) = 0 and Ag(~)=O for some A=-ab < O.
In the case where grad g(~) ~ 0 and
grad f(~) = O. we can state trivially that -grad f(~) + Agrad g(~) by letting A=O.
=0
and Ag(~) = 0
We have thus obtained the classical result already mentioned above:
if ~ is optimal and grad g(~) ~ 0, then for some A~O we
ve Ag(~)
=0
and -grad f(~)
+ Agrad g(~) = O. Without the assumption grad g(x) "
~
0, this classical result would be incorrect as
one can see in the following simple optimization problem on the real line Rl : mini-
3
2
mize f(t)=t subject to the constraint g(t)=t <0, A
t=O,
The optimal polution is obviously
But since grad f(O)=l and grad g(O)=O, it is impossible to find a real number
A(O such that -grad f(O) + Agrad g(O)=O, i,e. such that -1 + A'O
~
should realize that in this simple pathologival problem the sets
~l
theless disjoint since
~l
is the set {t:t < a} and
~2
O.
The reader ~2
and
are never-
is the empty set.
"-
The assumption grad g(x) i a is the most benign form of a general class of assumptions known as constraint qualifications; I shall come back to that topic in Section 5,
If we do not assume that grad g(~)iO, then the necessary condition for that prob-
lem takes the form: if ~ is optimal, then for some (a,S)iO with a and S~O, we have a grad f(~) + Sgrad g(~)=O and Bg(~)=O.
Indeed if grad g(Q)iO, then we let a=-l and
6=A, where A is the number given earlier; if on the other hand we have grad g(£)=O, A then either g(x)=O or g (Ax)
If g(Q)=O, we let a=O and 6=-1.
A
If g(x)
fact that the sets ~l and ~2 are disjoint implies that grad f(x)~O and in that case we let a=-l and S=O. In this paper, I shall always use the n-dimensional Euclidean space basic reference space.
R
n
as the
Although this is sufficient for most applications to mathe-
matical programming, this is not the case in the theory of optimal control in which we must consider infinite-dimensional spaces of trajectories.
However, the reader
should be aware that everything stated here can be extended to general normed linear spaces, Balkin [2], in which optimal control problems can be treated. 2, SEPARATING
SEVERAL CONVEX SETS
n If P is a vector in R and if a is a real number, then the function f defined over n n R by the relation f(x)=p'x + a is called an affine function on R • speaking, an affine function is a linear-plus-a-constant function.
Colloquially If pia, then the
affine function p.x + a is said to be an affine nonconstant function,
A finite fa-
mily {~. ; iEI} of convex sets will be said to be separated, if there exists a finite l
family of affine functions {w. : i E I} such that l
(i) L
iEI
wi =0,
(ii) w. (x) ~ a for all i E: I and all x E l
(iii) w. is nonconstant for some i E I. l
R, l
4
Let us show that in the case of ~ sets ~l and ~2' this new definition of separation coincides with the classical definition of separation.
~l
If
and
~2
are (classically)
separated, then there exists a nonzero vector p such that sup E ~ p.x x 1 I f we let wI (x)=-p'x
+ sUPxE ~/'x and w2 (x)=p'x -
~
inf E ~ p.x. x 2
sUPxE~lP'x, we thus obtain
(i) wI + w2= 0, (it) w.(x) ~ 0 for all iE{l,2} and all xE~., 1
1
(iii) both WI and w are nonconstant. 2 In other words, if
~l
and
~2
~l
and
~2
are separated according to the new definition.
Conversely,
are separated according to the new definition, i.e. if for some WI (x)=
+ w2=0,
(i) WI
(it) W. (x) :;, 0 for all iE{l,2} 1
and all xE~., 1
(iii) at least one of the vectors PI or P2 is different from zero. Then, by (i), we have P2 = -PI' a = -a and, by (iii), we have P2= -PI ~ O. 2 l
If we
let p = P2 = -PI' we obtain sU P xE ~/.x ~ inf x E~/'x and the two sets ~l and ~2 are separated according to the classical definitions. n We know that two disjoint nonempty convex sets in R can be separated, but it is not correct to say that two separated convex sets in the sets ~l
=
R
n
are disjoint.
For instance
2
{(x ,x ): xl~O} and ~2 ={(x ,x ): xl~O} in the plane R are not disl 2 l 2
joint, since ~l n ~2
= {(xl ,x 2 ):
xl=O} ~~
~O, we have sUPXE~lx,P = infxE~2x.P'
that the fact that
~l
and
~2
but they are separated since for p=(l,O)
If either ~l or ~2 is open however, we know
are separated will imply that
~l
and
~2
are disjoint.
The same result can be extended to several convex sets in the following manner: n Theorem 2.1. If {~. : iEI} is a finite family of nonempty convex sets in R such 1
that
niEI ~i is
empty, then the family {~i ; iEI} can be separated,
ly, i f Wi : i E: I} is a fin:j.. te separated family of convex sets in R most one of them fails to be open, then
ni € I
n
Converse-
and i f at
~i is empty.
The proof of Theorem 2.1 can be found in Balkin [2]. In several applications, we shall assume that 0 E that case,
i f the family
W.: 1
iEI}
of
IT.1
for each i E 1.
In
5
convex sets is separated by the family {w.: i E I} 1
of affine functions, then we shall
w.(O)~
have
for each iEI and Z'EI w.(O)=O which imply w.(O)=O for each iEI and 1 1 1 1
hence the functions w. are not only affine but linear,i.e. of the form 1 We can thus state that a family {rl.:iEI}
w.(x)~p .•
1
1
x.
of convex sets with OE'?f for each iEI
1
1
is separated if and only if there exists a finite set of vectors {p.:iEI} such that 1
(ii) p .• x 1
~
0 whenever iE I and x
E: rl., 1
(iii) p. :f 0 for some i E 1. 1
3. CONVEX APPROXIMATIONS OF SETS We shall consider three different types of convex approximations of sets;(i)the interior convex approximation (ii)the tangent convex approximation. and (iii) the simplicial
k
convex approximation where k is a positive integer.
In mathematical program-
ming, the two concepts of interior convex approximation (asociated with the objective function and the inequality constraints) and of tangent convex approximation (associated with the equality constraints) are the most useful.
The simplicial
k
convex
approximation is used chiefly in optimal control theory and is associated with operator constraints (i.e. when one requires a trajectory to satisfy some differential equations).
however, in the case
k~l,i.e. in the case of the simplicial l convex ap-
proximation, this concept is also used in mathematical programming under the form of the Abadie Sequential Constraint Qualification, Abadie [4]. We should normally give all the definitions under the form: the set rl is an interior (resp. tangent or simplicial if .•..
k
convex approximation around a point
~ to a set S,
For the sake of simplicity of notation, we shall give all those definitions
. "-x=O. with respect to the p01nt
To go back to the general case, we shall use the fol-
k lowing convention: the set rl is an interior (resp. tangent or simplicial ) convex approximation around a point ~ to the set S, if the set rl-~ is an interior (resp. tangent or simplicial
k
convex approximation to the set
S-~). If A is the set in Rn
n and a is a vector in R , then we use the notation A-a to denote the set {x-a:x EA}. Let us specify some further notations. length of x.
n If xER , then [x] will be the Euclidean
I f AERn, then coA will be the convex hull of A.
A set {xl' .•• , xl} in
n R is said to be in general position if the vectors x2-xl,x3-xl, ••• ,Xt-Xl are linearly independent.
6
n Definition 3.1. A subset ~ of R is an interior convex approximation to a subset S n of R if (i) ~ is open, (ii) ~ is convex, (iii) oEIT and (iv) for all iE~ there exists an E>O such that flxES whenever !x-i!<E
and flE(O,E).
n Definition 3.2. A subset ~ of R is a tangent convex approximation to a subset S of n R if there exists a neighborhood V of 0 and a continuous real-valued function ¢ defined on V, differentiable at x=0 and such that (i) grad ¢(O) ; 0, (ii) ¢(O)=O, (iii) ~ ={x:xERn,x'grad ¢(O)=O}and (iv) h:xEV,¢(x)=O}CS.
n Definition 3.3. If k is a positive integer, we shall say that a subset ~ of R is a simplicial
k
n convex approximation to a subset S of R , i f (i)
~ is convex, (ii) OE~
and (iii) for any set {xl'" .,x } with l~k elements in general position in ~ and for l any real number E>O there exists a number flE(O,E) and a continuous function s from n co{xl' ••. ,xt } into R such that!S<x)-X!:::E and fls(x)E.S whenever xEco{xl, ••. ,xRL l Remark: As I mentioned before, the concept of simplicial convex approximation is related to Abadie Sequential Constraint Qualification. have: a subset
Indeed from Definition 3.3,we
~ of Rn is a simplicial l convex approximation to a subset S of Rn if
(i) ~ is convex, (ii) oE i1 and (iii) for each xE~ and each real number E>O, there exists a number flE(O,E) and an element yER condition can be rewritten as: for each
n
xE~,
such that !y-X!~E and flyES.
The last
there exists a sequence of positive
real numbers fl ,fl 2 , ••• and a sequence of elements Yl'Y2"" l
n in R such that limi~
ly.-x!=O, lim.-joQOfl.=O and fl.y.Es for all i=1,2, ..•• 1
1
1
1
1
n Examples of Convex Approximations. If ¢ is a real-valued function defined on R such that (i) ¢(O)~O and (ii) grad ¢(O) exists and is different from zero, then ~={x:xER~ x. grad ¢ (O)
is an interior convex approximation to each of the sets S={x: x ERn,
and S={x:xERn,¢(x)~O}, (the proof of that fact is not too hard).
I f more-
over,¢(O)=O and ¢ is continuous in some neighborhood of 0, then ~ *={x;xERn ,x.grad ¢(O)=O } is a tangent convex approximation to the set S*={x:xER n ,¢(x)=O} , (there is nothing to prove here, just apply the definition).
As I mentioned before,simplicial
k
convex approximations are used in optimal control theory to handle operator constraint of the type xES where S is the set of all trajectories which are solutions of a given family of ordinary differential equations.
It is very hard to express opera-
tor constraint in terms of inequality and/or equality constraint(s) and even when it
7
is possible the function describing those constraints are not "smooth" enough to apply the concepts of interior convex approximation and/or tangent convex approximation. This is the reason why it is convenient to keep operator constraints under their given forms and to define a special type of convex approxiamtion adapted to those operator constraints.
This special type of convex approximation is the simplicial
approximation.
In optimal control theory, the simplicial
k
k
convex
~
convex approximation
to
the set S will be the set of all solutions of a certain linearization of the given For more details, see Halkin [5] and
family of ordinary differential equations.
[6].
Balkin-Neustadt 4.
LuE THiOREM OF DUBOVITSKII
~h L eorem
i
MILYUTIN
n 4 .1• -Le t S_~, ••. , S-1' S 0' S 1 b e subsets of R such that I('\+.1 '" 11--~ S i=w,
sume that we have convex sets tion to 5
M~D
for each
i--~, ..• ,O
Then, the sets
~
-~
~
-~
•.•.
'~l
such
and such that
""'~l
that~. 1
~l
set~.
the
1
l
convex approximation
are disjoint and hence separated.
x which
joint, then there exists an element
E. {-~ ••••• a}
is an interior convex approxima-
is a simplicial
The proof of Theorem 4.1 is particularly simple.
i
If the sets
~
-~
.••.
belongs to each of them.
'~l
are not dis-
Since for each
is an interior convex approximation to the set S .• we know 1
that there exists an €1.>0 such that nxEs. whenever Ix-xl<€. and nE(O.E:.). 1 1 1 €=min{€ Since
-~
As-
If we let
••.•• €O} we see that nxE.S. whenever iE.{-~ ..... O} .[x-x[<€ and n E(O.d.
x~~ 1
1
and since
~l
is a simplicial
1
convex approximation to Sl' we know that
there exists a numbernE:(O.€) and an element yE.R
n
such that Iy-xj<€ andnyESl.
SincenE.(O,€). the elementny belongs to every S. for all iE{-~ ••.•• O}. 1
This con-
tradiction concludes the proof of Theorem 4.1. The results of Theorem 4.1 can be applied directly to the following mathematical n programming problem: given a set 5 CR and given functions ¢-~""'¢-1'¢0 defined l n over R , find an element xfSl \"hich minimizes ¢O(x) subject to constraints ¢i (x)~O A
for i--~ ••..• -l.
We assume that an optimal solution x exists for that problem.that
~l
l
is a simplicial
ferentiable at~.
convex approximation to Sl around
~
and that
¢-~""'¢O
are dif-
We then obtain
Theorem 4.2. Under the preceeding assumptions. there exist numbers A
-~
••••• AO.not
8
all zero. such that (i) A.:S 0 for each iE:{-]1 •.••• 0}. 1
(ii) A.¢.(~) ~ 0 for each i E{-]1 ••••• -l}. 1
1
(iii) '£? A. grad ¢. (~). 1""-]1 1 1
(x-~):;;;, 0 for all x E:Jl •
Proof of Theorem 4.2: Without loss of generality. we assume that ~=O and that ¢O(O) =0.
.let S.~{x:¢.(x)
For each i€.{-]1 ••••• 0}
According to Theorem 4.1. the sets {:J.:i=-]1••••• l} 1
are disjoint and hence separated.
By construction. we have OG.~. for all i E{-]1 .....+l} 1
of vectors {p
-]1
and hence there exists a set
••••• Pl}. not all zero. such that
(i) P"X30 whenever xE:J. and iE{-]1 ..... +l}. 1
1
(ii) ,£:1 p.=O. 1=-]1 1 Since ,£:1
P.=O and since at least one of the vectors p ••.•• Pl is different from 1 -JJ.
1~-]1
zero. we must have at least two of the vectors p zero.and hence at least one of the vectors p Since
p.,x~ 1
•..•• Pl which are different from
•.••• PO must be different from zero.
whenever ¢.(O) + grad ¢i(O).x
P.=A. grad ¢. (0). 1 1 1
1
1
We remark here that we may choose A.=O for all i E{-]1 ••••• -l} such 1
that ¢i(O)
-]1
-]1
P1'~0.
o
We have Pl= -'£. p. and hence the 1""-]1 1
P1"x~0 for all xE:Jl may be written under the form
(,£
?
1=-]1
P.)·x~O for all 1
The last inequality is equivalent to relation (iii) under the assumption ~""O.
xE:J • l
This concludes the proof of Theorem 4.2. Remark 1. The inequality (iii) of Theorem 4.2 may be written under the form of a Maximum Principle: ('£?
1~-]1
A. grad ¢. (~». ~ 1 1
~
('£?
1=-]1
A. grad ¢. (~» x for all x E :J • 1 1 l
.
Remark 2. If the point x'" is an interior point of :J • then the inequality (iii) bel comes (iii)
* n
This will always be the case for the problems where Sl=:Jl=R •
This last form of
Theorem 4.2 is known as Fritz John Theorem [3]. 5. CONSTRAINT QUALIFICATIONS We remark here that Theorem 4.2 contains no information about A besides the fact O
9
that
AO~O.
If we would know that AO
by the positive number -l/A
O
and we would obtain the same type of necessary conditions
with some vector A* for which we would have A* = -1. O
A great variety of conditions
(Constraint Qualifications) can be imposed on the problem which would allow us to guarantee that there exists some vector A with AO
One of the major shortcomings
of the traditional presentation of necessary conditions in the mathematical programming literature is, in my opinion. that the concept and the choice of those Constraint Qualifications influence the entire development of the theory of necessary conditions instead of being introduced at the last minute and used only to prove a variety of. practically important but theoretically easy, corollaries to 7heorem 4.2.
For example.
a very general Constraint Qualification for the problem of Section 4 is to assume
n CI_ 11+l n ... n rl_ l n rl l is a simplicial 1 convex approximation to n ... n S_l n S+l. In the case rll=Sl=Rn , this Constraint Qualification
that rl_
11
S-11
n S-11+1
is known as
the Abadie Sequential Constraint Qualification. (See the remark following Definition
3.3). 6. LIMITATIONS OF THE METHOD OF DUBOVITSKII AND MILYUTIN The method of Dubovitskii-Milyutin is not well adapted to problems with equality constraints.
I shall examplify those difficulties by considering the following opti-
2 mization problem in the plane R : minimize ¢O(x ,x )=x subject to the constraints l l 2 ¢l (x ,x )=x =O and ¢2(x ,x )=x -xi=O. 2 l 2 2 l 2 solution of that problem.
The point
(xl ,Q2)=(O,O)
is the obvious optimal
The sets SO={(x ,x ):¢O(x .x )< ¢O(Ql'x )}={(x .x ):x
Sl={(xl .x 2 ):¢1 (xl.x2)=O}={(xl.x2):x2=O} and S2={(xl.x2):¢2(xl.x2)=O}={(xl.x2):x2=xl} have no point in common. but the sets rlO.rl
l
and rl
is the interior convex approximation to SO' and rl to Si for i=l and 2).
2 i
have points in common (here. rl
O
is the tangent convex approximation
Indeed,rl ={(x .x ):x
(\i=O.l.Z rli={(xl'xZ):xl
Of course such "accidents" could be ruled out
by conditions resembling some Constraint Qualifications.
In the simple example given
above for instance.we could require that the set of gradients of the equality constraints be linearly independent at the optimal point.
If operator constraints are
present in the problem, the situation would still be more complex and one would need further types of Constraint Qualifications.
In the next section.I will present a
10
theory of necessary conditions for optimization problems with equality and operator constraints ( and also inequality constraints, but they never present any difficulties) which will be independent of any sort of constraint qualifications. 7.
T~
CASE OF
AND OPERATOR CONSTRAINTS
I~EQUALITY!EQUALITY
The central part of this section is the following result. Theorem 7.1. I f 1={-]1, ... ,m"l-l} and i f {s . :iEI} 1
and {S""2.:iEI} are families of 1
n subsets of R such that (i)(\iEISi=qJ, (ii) for i=-]1, •.. ,O, the set S""2 convex approximation to the set S., (iii) for i=l, •.• ,m, the set 1
S""2.
1
i
is an interior
is a tangent con-
vex approximation to the set Si' and (iv) S""2m+l is an (m+l)-convex approximation to the set Sm+l' then the family Wi:iE I}
is separated,
We remark immediately that in the case m=O, Theorem 7.1 coincides with Theorem
4,1.
From the counterexample given in Section 6, we know that under the assumptions
of Theorem 7.1, it would be incorrect to say (as in Theorem 4,1) that
niEI
Si=qJ im-
plies that (IiEI S""2 =qJ ,but we can still assert that the family {S""2 :iE:I} is separati i ed and this last statement is all that is needed to obtained appropriate necessary conditions.
The proof of Theorem 7,1, given in Balkin [2], makes a critical use of
Brouwer Fixed Point Theorem. Let us now assume that we are faced with the following optimization problem:given n a subset S of R and functions ¢
-]1
, ••• ,¢
m
n defined over R , find an element ~€Rn
which minimizes ¢O(~) subject to (a) the inequality constraints ¢.(x)~o for i=-]1, ••. ,-l, 1
(S) the equality constraints ¢.(x)=O for i=l •••.• m. 1
(y) the operator constraint ~Es.
. The optimality of such e l ement A x canb e expressed by writ1ng t h at nm+. 1 S .-_'"~
1=-P.
1
where the sets S_]1'" "Sm+l are defined by S.",{x:xERn.¢.(x):;:O} 1
1
for i=-]1 •..•• -l.
n SO={x:xER .¢ (x)<¢ (x)}. o 0 S.",{x:xE:Rn.¢.(x)=O} 1
1
for i"'l ••••• m
and Sm+l=S, Let us assume that the functions ¢
-]1
••.•• ¢
m
are differentiable at ~ and that the
functions ¢l •.••• ¢m are continuous in a neighborhood of
2.
We then define convex
11
sets
~
-11
, .••
,~
m
by the relations
~ "" {x:XERn,cr,(~) + grad cr,(~)'(x-~)
1
1
~O'" {x:x€Rn,grad crO(Q)'(x-Q)
1
for i=l, •.• ,m.
As was mentioned in Section 3, the sets tions around the point
...x
~
-11
, ••.
,~O
are interior convex approxima-
to the sets S_11""'SO and the sets ~l""'~m are tangent
convex approximations around the point ~ to the sets Sl"",Sm' we are given a set
~m+l
which is a simplicial m+l convex approximation around the
~
point x to the set Sm+l=S, be separated.
Let us assume that
From Theorem 7.1, we know that the sets ~-11""'~m+l will
If we translate this last result in terms of the functions cr
-11
, ••• ,cr
m
/\
and their gradients at the point x, then, by an argument similar to the argument followed in the proof of Theorem 4.2. we obtain Theorem 7.2. If ~ is an optimal solution of the given problem. then there exist numbers A , ••• ,A ,not all zero, such that -11 m (i)
A.~O
for i=-11, ••• ,O;
1
(ii) A.cr. (~)=O for i=-11 •••• ,-l; 1
1
(iii) Z, A,grad cr. (~).(x-Q)~O for all x EDm+l' 1=-11, ••• ,m 1 1 We conclude by making two remarks similar to the remarks made at the end of Section 4. Remark 1. The inequality (iii) of Theorem 7.2 may be written under the form a Maximum Principle: Z, A, grad cr.(Q)·~ ~ Z,_ A,grad cr,(i).x 1=-11, •.• ,m 1 1 1--11 ••••• m 1 1
...
Remark 2. If the point x is an interior point of
Z._
1--11, ••• ,m
~.
for all xE~l'
then the inequality (iii) becomes
A. grad cr. (~) = O. 1 1
This will always be the case for the problems where
n Sl~~l=R
•
REFERENCES [1] Dubovitskii,A.Ya. and Milyutin,A.A •• Extremum Problems in the Presnece of Restrictions.U.S.S.R. Computational Mathematics and Mathematical Statistics,i,1965.l-79. [2] Halkin.H •• A Satisfactory Treatment of Equality and Operator Constraints in the Dubovitskii-Milyutin Optimization Formalism. Journal of Optimization Theory and Applications.i. 1970.l38-l49.
12
[3] John,F., Extremum Problems with Inequalities as Subsidiary Conditions, in "Studies and Essays:Courant Anniversary Volume", (K.O.Friedricks,O.E.Neugebauer,and J.J. Stoker, (eds.»,pp.l87-204)Interscience Publishers)New York,l948. [4] Abadie,J., On the Kuhn-Tucker Theorem, in "Nonlinear Programming",J.Abadie(ed.), pp.l9-36,North-Holland,l967. [5] Ralkin)H., Optimal Control as Programming in Infinite Dimensional Spaces. in "C.I.M.E.:Calculus of Variations.Classical and Modern",pp.l79-192,Eidizioni Cremonese)Roma,l966. [6] Halkin.H. and Neustadt.L.W.) Control as Programming in General Normed Linear Spaces) Lecture Notes in Operations Research and Mathematical Economics.Springer Verlag,ll)l969.23-40.
OPTIMUM DESIGN OF STRUCTURES THROUGH VARIATIONAL PRINCIPLES
by RICHARD T. SHIELD Department of Theoretical and Applied Mechanics University of Illinois, Urbana, U. S. A.
1.
INTRODUCTION
The application of the calculus of variations to the design of structures for minimum volume leads to necessary conditions for the structural volume to be stationary, and a local or global minimum is not guaranteed.
However, if an appropriate variational principle ap-
plies for the class of structures under consideration, design criteria can be established which lead to structures of minimum volume. A direct design procedure was first given by Michell [1] for framed structures composed of a material of limited strength.
For per-
fectly-plastic structures, direct design procedures were introduced by Drucker and Shield [2,3,4] and here the upper bound theorem of limit analysis provided the appropriate variational principle.
For elastic structures, variational principles can provide direct design
methods for design for a given stiffness, for a given buckling load or for given fundamental frequency of vibration (see Prager and Taylor [5] and Shield and Prager [6]). The major part of this paper surveys the direct design procedures which have been developed through the use of variational principles. Section 2 describes the procedures for minimum-volume design of structures of perfectly -plastic materials which are required to carry a given set of loads. Section 3 discusses uniform strength designs in which the structural material is required to be stressed within a certain range under a given system of loads.
The stress range may be chosen to
14
ensure that the stresses remain in the elastic range, for example, or to ensure that an appreciable amount of creep will not occur. Section 4 treats elastic design for a given stiffness in order to illustrate the design procedures for elastic structures. Minimum-volume framed structures of material of limited strength in tension and compression are considered in Section 5.
The Michell design method fails when kinematic constraints are present (except when
the tensile and compressive strengths are equal) but an alternative approach [7] does not have this limitation. An example is given to show that minimum-volumE7 frames are not necessarily unique, and some new plane structures of the Michell type are described, including the layout for pure bending. This paper is not intended to provide a comprehensive survey of the literature on optimum design.
The reader will find additional references in [7,8,9,10].
2,
PLASTIC DESIGN OF STRUCTURES
In this section we discuss the optimum design of structures composed of perfectlyplastic materials. A restricted formulation of the problem is indicated in Figure 1. It has the advantage of ensuring that the structure will consist of the usual structural elements, such as frames, plates and shells, and the limitations also increase the chances of determining the optimum solution. We suppose that the structure is to have a prescribed middle surface A.
The loading is prescribed and is distributed over A and its boundary. At sup-
ports either the components of displacement and rotation are prescribed to be zero or the corresponding components of edge traction and moment are given.
For a solid shell, prob-
lem (i), the structure is formed by placing a certain thickness h of a given material at points of the middle surface.
For a sandwich type structure, problem (ii), we suppose that
the shell has a core of prescribed thickness H.
The core carries shear force only, and
bending moments and force resultants are carried by membrane stresses in two thin identical face sheets of thickness h.
In both cases we wish to design the shell, that is choose h,
so that the shell is just at collapse under the given loading and is optimum for a given criterion,
Here we design so that the volume V
= 1hdA
is minimized but the methods extend readily to minimization of the functional
15
Figure 1. Solid and sandwich shells
) hf (X) dA, where f (X) is a non -negative function of position over A.
The extension allows for minimum-
weight design when different materials are to be used in different parts of A, and also admits design for minimum moment of inertia about an axis. The generalized stresses, such as bending moments and stress resultants across a section, will be denoted by Q l' Q2' •• , Q when the stresses
~
N
or simply Qno
Yielding can occur in the shell
lie on the yield surface F
(~;
in N-dimensional stress space.
h)
:=
(1)
0
For stress states Q
n
represented by points on the yield
surface, purely plastic strain rates are possible; and we use
~
(n
:=
1, 2,
'0,
N) to denote
the generalized strain rates, such as rates of curvature change and extension of the middle surface. The vector representing q
n
is normal to the yield surface at regular points, while
at singular points the vector lies between adjacent normals.
For a convex yield surface,
the plastic rate DA of dissipation of energy per unit area of the middle surface is then uniquely determined by the values of
~,
(2)
For the solid shell, DAis quadratic in h in general; while for the sandwich shell, DAis
16
directly proportional to h. Shear forces have little influence on yielding, even for highly localized loading [11], and they are not included in the generalized stresses
~.
The theorems of limit analysis [12,13] can be used to provide information about the volume of a design which can carry the loads. Lower-bound theorem.
The appropriate theorems are the following:
If the applied loads can be carried by an equilibrium distribution of
moments and force resultants Q
n
in the shell which are at or below yield, the loading is at .
or below the collapse loading. Upper-bound theorem.
If the applied loads are such that a deformation of the shell can be
found for which the rate at which the applied loads do work exceeds the rate of internal energy dissipation, the loading is above collapse. The lower-bound theorem can be used to determine upper bounds on the volume of the minimum-volume design for given loads [3].
If a stress distribution Q
librium with the applied loads, we can choose the thickness h is satisfied everywhere on A.
Since the design h
s
s
n
over A is in equi-
so that the yield condition (1)
is then a permissible design, the minimum
volume Vm must be less than or equal to Vs' Vm ::: V s == ) h s d A.
The upper - bound theorem can be used to provide lower bounds for V , as in [3], but m
the theorem also leads to a direct design procedure [4].
For a .shell h
s
which is at or
below collapse under the applied loads and for any kinematically admissible velocity field u. (i 1
= 1,2,3)
we have (' D
j
A
(q ; h ) dA - CT. u. dA "= 0, n s ) 1 1
(3)
for otherwise the use of the upper-bound theorem with the deformation u. would predict that 1
the loading exceeds collapse.
In (3), T
i
are the components of the applied loads, the re-
peated index i implies summation over 1, 2, 3, and the integral of T. u. is to include the 1
rate of work at the edge of the shell.
1
Inequality (3) is a variational principle for the permis-
sible design h ; equality holds in (3) only when u. is a collapse mode for h. 1 s s
The applica-
tion of this principle to the determination of a minimum-volume design is much more direct than the use of the calculus of variations. Suppose that a design h
e
for a solid shell, prob-
lem (i), is just at collapse under the loads in a collapse mode u.; and suppose that a neigh1
boring design h
s
== h
c
+ 6 h is also a permissible design.
If we neglect second order terms,
17
the dissipation rate for the shell h
in the defonnation mode u is l
s
assuming that DAis continuously differentiable in h.
Applying the variational principle (3)
to the design h , we conclude that
s
(
J0 h
8D ahA dA
because equality holds in (3) for the design h. c
~
0
It now follows that if the design h
c
is such
that (4)
over A then
~ ohdA so that the design h
c
2:
(5)
0,
provides a relative minimum for the volume of permissible designs.
Thus, assuming that the neglect of the second -order terms is permissible, the variational principle leads to a direct design procedure.
Mroz [14] has given an example in which the
application of (4) leads only to a stationary value for the volume. The case when DAis directly proportional to h, as for the ideal sandwich shell of problem (ii), is simpler and a stronger result is possible. We again suppose that the design h
c
is at collapse in a deformation mode u., and we use the mode u. in the variational prin1
ciple (3) for another permissible design h. s
~ DA (~; h s ) dA
2:
1
We obtain
~ T i ui dA
= ) DA (~;
h c ) dA.
Since D (q ; h ) A n s
we can conclude that if the design h
c
is such that
= constant over A, then
(6)
18
Not only does the condition (6) lead to an absolute minimum for the design volume but the condition (6) is much easier to use than condition (4) because (6) does not involve the design thickness directly. In order to illustrate the use of these design methods we consider the minimum-volume design of a circular plate with a built-in edge under uniform pressure loading on its upper face (for problems involving other symmetric and non -symmetric pressure distributions see
[15,16]). When the Tresca yield condition is assumed, the yield condition on the radial bending moment M and the circumferential bending moment N is the familiar hexagon in (M, N) space, max (IMI ' INI ' 1M - NI) = M o ' in which M
o
vature rates
=
(J
K,
H h for the sandwich plate and M
;:;:
i
(J
0 0 0
>--
h
2
for the solid plate.
The cur-
in the radial and circumferential directions are derived from the downward
deflection rate w of the middle surface through
1 dw r
dr
where r measures distance from the center. For the sandwich plate, the design criterion (6) does not involve the design thickness h, and it is readily found that (6) can only be satisfied for a finite range of r when either (i)
M=N=±M,
o
or
(ii)
M=±M,
o
N
= O.
For these regimes, condition (6) reduces to
(i)
K
+
>--
=±
0.,
or
(ii)
K
= ±o.,
where 0. is a constant when the core thickness H is constant.
The curvature rates
K,
>--
must have the same sign in regime (i) while in regime (ii) they are of opposite sign and
IKI
2:: I >--1 •
For a plate with a built-in edge, regime (i) with the positive sign will apply in
a central region r :S a and regime (ii) with the negative sign will apply in the remaining portion a :S r ter.
:S
R.
At the built-in edge wand d w /d r are zero and d w /d r is zero at the cen-
In order to have wand d w /d r continuous at the junction r
= a,
it is found that we
19
(0 )
R
( b)
Figure 2.
Minimum -volume designs for built-in circular plate under uniform pressure (a) sandwich plate (b) solid plate
must have a ;:; 2 R/3, and at this radius M must vanish because
K
changes sign. With M
;:; 0 at r ;:; 2 R/3, the moment distribution in the two regions of the plate, and hence the thickness h;:; lMI /(5 H, can now be found from equilibrium. o
The design is indicated in
Figure 2 (a). For the solid plate, we again assume that M and N have the fully plastic value M for o for r s: a and that N is zero for a s: r s: R, with M
:=
0 at r ;;;; a.
From equilibrium the
moment distribution and therefore the thickness h(r),
can be found for a general value of a. In order to satisfy the design criterion (4) we must have
h
h
in the inner and outer regions, respectively.
The continuity of dw /d r at r
(see [15])
a
R
~ rdr ) h (r) ;:; a
~
o
a
dr h(r) ,
a leads to
20
and this equation serves to determine the junction radius a. a
= 0.664 R
For uniform pressure loading
and the design is as indicated in Figure 2 (b).
Minimum -volume design for other one -dimensional situations, such as symmetrically loaded circular cylindrical shells of sandwich type [4], is also relatively straight-forward, but design problems which are two-dimensional can be much more difficult to treat. So far designs for non-circular plates with built-in edges have only been obtained by an inverse method [17]. The design procedure can be modified to include body forces (such as weight) which act only when material is present (see [4]). Also the procedure has been extended to the design of multi -purpose structures which are to support different systems of loads at different times [18], and to the quasi-static design of structures under moving loads [19].
3.
UNIFORM STRENGTH DESIGNS
The methods for plastic design have been extended [7] to materials which are not perfectly plastic, so that design limitations other than plastic collapse are involved.
For ex-
ample, we may use a work-hardening material and in order to minimize the possibility of fracture we may wish to design the shell so that it remains elastic everywhere.
For another
material, we may wish to keep the stresses below the level at which an appreciable amount of creep will occur. In both cases there will be a limiting surface in stress space to restrict the stress states in a section of the shell. We shall say that a design is a uniform strength design for the given loading if the stresses everywhere in the shell are on the limiting surface. We seek the uniform strength design which has least volume. Figure 3 indicates a limiting surface in generalized stress space. We suppose that the surface is given by (7)
where L is a known function, and we assume that the surface is convex. In purely elastic design, for example, the limiting surface is determined by those values
~
for which the
yield limit is reached in the outer fibers of the shell. Consider an infinitesimal virtual deformation of the shell defined by middle surface displacements v. and associated generalized strains e • We shall say that a virtual defor1
n
21
Figure 3,
mation with strains e ing e
n
n
Limiting surface in generalized stress space
is compatible with a limiting stress state Q
n
if the vector represent-
is normal to the limiting surface at the stress point Q , Figure 3. At a singular point n
of the limiting surface the strain vectors representing compatible deformations lie in the fan bounded by adjacent normals.
For a given convex limiting surface, the virtual work WA in
a compatible virtual deformation is then uniquely determined by the virtual strains en' W
A
= Qn
e
n
;; W
A
(e
m
; h),
in which the repeated index n implies summation over 1, 2, •• , N. stress state Q
s n
Moreover, for any other
inside or on the limiting surface we have QS e n n
with equality only if e
n
S;
W
A
(e
. h) m'
( 8)
are also compatible with QS. n
The approach to minimum-volume design for uniform strength is similar to that for plastic design, and we shall treat the case when WA is directly proportional to the design thickness h.
If the shell h
c
has limiting stresses under the loads and if there is an admis-
sible compatible virtual deformation vi' then by virtual work we have
l For any other shell h
s
T i vi dA
=
~WA (en; h c ) dA.
(9)
with stresses ~ at or below the limiting values we can use (8) and
22
virtual work to deduce that
where v. is any admissible virtual displacement. We therefore have the variational principle 1
( W (e ; h ) dA - (T. v. d A j 1 1 } A n s for permissible designs and equality holds only when h
s
2::
0
(10)
is a uniform strength design and v.
1
is compatible with the stresses ~ in the shell. We now use the compatible virtual deformation v. for the design h 1
c
in the variational principle (10) and with (9) we derive
Because WA is proportional to h, we see that if the design h
c
is such that
'" constant over A, then the volume of h
c
will be an absolute minimum for all permissible designs.
For the solid shell the design criterion is constant over the shell. Uniform strength designs have been discussed by Save [20].
4.
ELASTIC DESIGN FOR GIVEN STIFFNESS
Direct design methods can be developed in the same way for other problems of optimum design provided that a suitable variational principle holds for the structure under investigation.
This can be the case in the minimum -volume design of an elastic structure which is to
have a given stiffness under a given set of loads (or, equivalently, elastic design for maxi mum stiffness with a given volume of material). Other examples are minimum -volume design for a given buckling load or for a given fundamental frequency of vibration.
Techniques for
design problems such as these that have been developed in a unified way by Prager and Taylor
[5]. Here we outline the procedure in the case of elastic design for a given stiffness.
23
For an elastic shell there is a strain -energy function EA' per unit area of the middle surface. which is uniquely determined by the generalized strains q
n
derived from middle sur-
face displacements u . The strain energy also depends on the design thickness h so that we i write it as EA (Clu; h).
The potential energy U is defined as U {u*; h} ==
1
EA
(~; h) dA
~ Ti ul' dA.
-
where the integral of T. u. represents all the virtual work of the prescribed loads including 1
1
the edge loading and where u: is a displacement field which satisfies any imposed displace1
ment conditions. When E
A
is a positive definite quadratic function of the strains, the Prin-
ciple of Minimum Potential Energy holds.
The principle states that the potential energy U is
minimized by the actual displacements u. produced by the loads. 1
U{u*;hr ~ Ufu;h}. We now define the compliance of the shell for the given loads to be twice the total strainenergy of the shell and we note that
J
2 E A (Clu; h) dA ::: ) T i ui dA. For two designs hand h
s
with the same compliance. we have (11)
where qS are the strains for the design h.
n
s
The inequality in (11) follows from the Principle
of Minimum Potential Energy applied to the design h ' s
When E
A
is directly proportional to
h. we see from (11) that in designing for a given compliance, the design with EA/h constant will have least volume.
For other types of shells the procedure would be to design so that
8 E J8 h is constant over the shell, and the design would provide a relative minimum for the A volume of permissible designs. As a simple example. suppose we have an elastic beam of length 2 £, which is built-in at both ends and which has a transverse point load P at the center. We wish to design the beam so that the central deflection does not exceed 6 and such that the beam has minimum volume.
For a beam of the sandwich type. minimizing the volume is the same as minimizing
the integral of the bending stiffness over the beam. If two beams with stiffnesses s and
s
have the same central deflection 6 under the load. they have the same compliance PO and in
24
the same way that (11) was derived we can use the Principle of Minimum Potential Energy to get
rJ s 2 dx::: r ) s K
where
K
K
2
dX,
is the curvature of the design s under the load P and x measures distance from one
end. We now see that the design s will have least volume if IKI is constant. In order to satisfy the constraints at the ends, the deflection with constant lKI must have inflection points at the quarter points x == £/2, 3 £/2. Since the moment M == s points where
K
K
must vanish at the quarter
changes sign, the moment distribution is now statically determinate and M (x)
and therefore s (x) can be found. The design procedure obtained from the Principle of Minimum Potential Energy applies for design with given compliance.
However, the design criterion does not always coincide
with the compliance. Thus if we have a distributed load over the built-in beam and we wish to limit the central deflection as before, the compliance will not be known in advance. Similarly, if we have an off-center point load P at the section x
==
x
o
and we wish to limit the maximum is the deflection at x == x
deflection of the beam, the compliance is P u , where u
0 0 0
not necessarily the maximum deflection.
and is
These design problems can be approached by using
a variational principle of a different type called the Principle of Stationary Mutual Potential Energy [6].
Let u and i
ui
be two middle surface displacement fields for a design of thick-
ness h and let qn' ~ and ~, Qn' respectively be the associated generalized strains and stresses. We define the mutual strain energy through N L;
1
For two different sets T
i
and T
i
N
Q n qn
L;
1
Qq. n n
of applied loads, the mutual potential energy U
M
is defined
as
where
ut, ut If u
i
and
are kinematically admissible displacement fields.
ui
are the actual displacements that the loads T
i
and T , respectively, would i
induce in the shell, then (12)
25
With the use of the Principle of Virtual Work, it can now be shown (see [6] for details) that (13) If we apply (13) when
u and i
ut
and
u;
are neighboring displacements to the actual displacements
the right-hand side will be zero to first order. Thus U {u~', M
ui '
at the values u: 1
='
U., 1
u:'1 ;:; u.,1
u";
h} is stationary
and this is the Principle of Stationary Mutual Potential Energy.
Suppose we wish to design a structure so that the transverse deflection at a particular point X of the mid-surface is of amount 0 under the loads T • We take the second system o i of loads T. to be a single unit concentrated load P acting normal to the middle surface at the 1
point X ' o
f
From (12) we then see that the value of UM u,
u; h}
is - Po, so that designs
f
which satisfy the design criterion will have the same value for U u, M
u; h}.
We can there-
fore use the Principle of Stationary Mutual Potential Energy in the same way as the Principle of Minimum Potential Energy was used in design for a given compliance. In this way we find that the design such that ()
Cfh E A (qn' ~; h) ;:;; constant over the shell will provide a stationary value for the volume for designs which have transverse deflection of amount 0 at the point X • o Applications to the minimum-volume design of beams for given deflections (or rotations) are described in [6]. Suppose we wish to design a beam of sandwich type and we require the deflection at the section x ;:;; x
o
to be of amount 0 under a certain system of loads.
Let s
and s* be the bending stiffnesses of two designs that satisfy the constraint on the deflection at x , and let u, u* and U, U*' be the corresponding deflections of these designs under the o given loads and under a unit concentrated load P at x. o
From (12) we have
where we have identified the bending stiffnesses sand s" with the design thicknesses hand h", as we may do for sandwich beams. for the design
s~'
K,
K,
•••
u are kinematically admissible
and if we apply (13) to this design we get
U {u, M where
The deflections u,
u; s'~}
- U
M
{u':',
u*; s*};:;;
j s':' (K'~ - K) (i<* - K) dx,
(14)
are the curvatures associated with the deflections u, u, •••• If we replace
26
U
M
\ u'~,
u"; s*}
by U {u, M
u;
s
1in (14) and use the definition of UM we find that
1s* (K* - K) (K* - i<) dx.
\ (s* - s) KK dx ==
(15)
When s* is a neighboring design to s, the right-hand side of (15) is zero to first-order and we see that KK
=
constant
=
c
2
(16)
is a sufficient condition for the design s to provide a stationary value for the volume If M ;::
S
Ss dx.
K and M '" sKare the bending moments for the optimum design s under the two
systems of loads then
The constant c can be determined from U {u,
u; s} '"
- P a and we finally arrive at
1
s
==
(MM)2
(17)
Po
For a statically determinate beam, the moment distributions M, M can be determined directly so that the optimum design (17) is readily found without calculation of deflections. Moreover, for a statically determinate beam it can be shown [6] that the design satisfying (16) actually furnishes an absolute minimum for the design volume. In this case the moment distributions M and M are independent of the stiffnesses, so that M '"
S K
;::
s*
These equations imply that K* - K ;::
(s* - s) s*
K
,
K* -
K
;::
(s'" - s) K
s ;'
Substituting in (15) we obtain, with (16),
} (s'" - s) dx ::::
~ (s* ;"S)2
dx 2': 0,
which shows that the design s satisfying (16) provides an absolute minimum for the design volume. For a statically indeterminate beam, an extra step is required in order to arrive at a design that provides an absolute minimum for the volume.
Consider, for example, a beam of
27
length 2 £ which is built-in at both ends and is loaded by a uniform pressure p along its length.
We wish to restrict the deflection at the center x
=£
to be of amount 6.
For a beam
built-in at both ends, u" (x) must change sign at least twice for otherwise no deflection is· possible; thus M (x) will have at least two zeros. Assuming a symmetrical design, we suppose that the bending moment M (x) is zero at x ;;;; £ ± b.
If we now consider only designs
for which the stiffness vanishes at x ;;;; £ ± b, we have a statically determinate beam and we can determine the design (17) which has least volume in this class of designs. We can now choose b
SO
that the volume will have the least value for all possible designs, and this value
of b is found to be £12.01. When the loading is not symmetric, the maximum deflection may be off center.
Sup-
pose, for example, that we have a simply supported beam of length 2 £ under a system of loads which produces a bending moment M (x). an amount 6. x
= b.
We wish to limit the maximum deflection to
Let M (x) be the bending moment distribution caused by a unit point load
The design (17) will then be optimum for a deflection of amount 6 at x ;;;; b.
ensure that the section x zero at x ;;;; b.
=
P
at
We can
b will have the greatest deflection if we choose b so that u' (x) is
In order to determine b, we note that if u· (b)
=
0 and u
=
0 at the ends,
then b u (b) ;;;;
\
2£
b
~
u" (x) d x d Y
\
b
Y
o
y \
uti (x) dx d y, b
and this implies that 2£
b
~
xKdx;;;;
o
~
.(2 £ - x) Kdx.
(18)
b
1
Because K
Mis
c (M/M)2, we can write (18) as 2£
= \
(2 £ - x) (M/M)! dx,
o and this equation serves to determine b.
To give an example, when the beam is loaded by a
point load P at x = a, the value of b varies from £ to 1.11£ as a varies from £ to 2 £.
28
The procedures described here for elastic design can be extended to design with two or more constraints on deflection or rotation under a single system of loads [6] and to the design of multi -purpose structures [6,21].
5.
MICHELL STRUCTURES
In formulating the problem of optimum design in Section 2, we assumed that the type of the structure and the layout, that is the middle surface A, were specified. A less restrictive fonnulation merely specifies the region in which the given material can be placed and leaves the type and layout of the structure to be determined.
In 1904 Michell published his paper [1]
on the minimum-volume design of framed structures.
He specified that the structure should
consist of tie -bars in tension and struts in compression, but the layout of the structure was not specified.
The material to be used allows a maximum tensile stress (Jt and a maximum
compressive stress (J , and for a design which carries the prescribed loads, the minimum c volume allowable is
v = 2: 2t
f /(Jt t
+
2: 2
(19)
f /(J • c c c
Here f is the tension in any tie-bar of length 2 and f is the thrust in any strut of length t t c 2. c
Michell showed that a framed structure will be of minimum volume if there is a virtual
small deformation of the space such that each tie - bar suffers an extensional strain of amount e and each strut suffers a compressive strain of amount e and no linear element of space suffers a strain numerically greater than e, where e is a constant.
Note that the actual
deformation of the minimum-volume frame under the loads involves extensional and compressive strains of amounts (Jt/E and (Jc/E, respectively, along the frame elements, where E is Young's modulus. In the proof of his results, Michell used a theorem due to Maxwell.
By imposing a uni-
form dilatation on the whole of space, Maxwell showed that for all structures under the same system of applied loads
(The constant is 2: F •
E,
where F is an applied load at a point with position vector !.) How-
ever, Maxwell's theorem does not apply to structures with kinematic constraints imposed by
29
support conditions because the reactions at the supports can vary with the structure. An exception is a structure with one fixed point but in this case the reaction at the support is determined by overall equilibrium.
Because Maxwell's theorem is essential to Michell's proof
when CT f. CT ' the design procedure of Michell will not be valid in general when kinematic t C constraints are imposed.
This limitation on the use of Michell's method does not appear to
have been mentioned explicitly in the literature. An alternative approach, which does not have the limitation of the Michell method, has been given by Shield [7]. The procedure is to design a frame compatible with a virtual small deformation in which the principal strains are of magnitude e/CT if extensional and of magnit tude e/CT c if compressive, the directions of frame elements coinciding with the principal directions of strain as before. The virtual deformation must satisfy any imposed kinematic constraints.
The proof that the procedure leads to a minimum -volume frame is straight-
forward and it makes direct use of the Principle of Virtual Work as in the method of Section 3 for unifonn strength designs.
The proof has been repeated by Hemp [22] and by Hegemier
and Prager [23] for the case CT
t
= CTc
(when the Michell method and the alternative method
become identical). Michell [1] supplied some examples of minimum -volume framed structures and other examples are given in [22,24,25,26].
Cox [27] has shown that a Michell structure has
greater stiffness under the loads than any other structure which is stressed to the limiting values CTt and CTc'
More recently, Hegemier and Prager [23] have shown that an elastic
frame with a specified stiffness (i. e. compliance) has least volume when it has the layout of a Michell structure, and this holds also for frames designed for a given stiffness in stationary creep or for a given fundamental frequency of vibration. In the following we give an example to show that minimum-volume frames are not necessarily unique, and we describe some new additions to the list of Michell structures. The diagram at the top of Figure 4 indicates the layout given by Michell [1] for a single force applied at the midpoint C of the line A B and balanced by equal parallel forces at A and B. The struts AD, E B and the curved bar DE carry a uniform compressive force and a quadrantal fan of tie-bars from C to DE maintains the equilibrium of the curved bar. The layout is symmetrical about A B with tie -bars replacing struts and vice-versa. The virtual deformation with principal strains ± e associated with the layout can be adjusted so that the
30
B
A
B
A
Figure
4.
Load at
C
supported
at
A, B
31
displacement is zero at points A and B. If we assume ITt = ITc' we can use this virtual deformation for the case when we have the same force at C, but now A and B are fixed points of support.
The optimum structure has the same volume as the structure with speci-
fied parallel forces at A, B, but the optimum design is not unique.
For example, the load at
C can be carried by a frame entirely above A B as indicated in the middle diagram of Figure
4. An infinity of optimum designs results from arbitrarily assigning a fraction of the load at C to be carried by a structure above the line A B and the remainder by a structure below the
line AB. We note that if we had specified that the load at C be carried by a beam with centerline AB and built-in at A and B, the optimum design would have bending moments at A and B.
The Michell structure has no moments at the fixed points A, B. The minimum -volume design indicated at the bottom of Figure 4 uses the same virtual
deformation with principal strains ± e, but now it is specified that distributed loads at A and B balance the load at C. Figure 5 shows the optimum layout for pure bending. A bending moment at the point A is to be transmitted to the point B by a framed structure of minimum volume, composed of a material of limited strength (or the structure has an assigned bending stiffness). In the circular regions around the points A and B, the tie -bars and struts follow logarithmic spirals. The spiral regions are connected by a strut G H in compression at the top and a tie -bar carrying the same force at the bottom of the structure.
The associated virtual deformation with
principal strains ± e is, apart from a rigid displacement, purely circumferential in the circular regions.
The regions between the larger circles and the straight-line boundaries (such
as G C, C H) of the upper and lower quadrants which meet at C move as rigid bodies.
In the
quadrants meeting at C, the principal strain directions are vertical and horizontal, and the quadrants deform like a plastic hinge in a beam in pure bending.
The total volume of material
required is
M[1 + 2In ,;[2 _ a ] [.1-- + l.-] r ITt IT o
.
C
Here M is the moment applied at A and B, a is the length of A C or A Band r
o
is the ra-
dius of small circles at A and B over which the forces equivalent to the moments Mare distributed.
32
<1> ~
::J
a. ~
.....o
::J
o >. o
...J
33
Figure 6 indicates the optimum layout when a downward force P is added at the central point C and upward parallel forces P /2 are added at the points A and B.
The moment M
applied at A and B and the force P are related to the angle 2 Q of the fan regions through 4M/pa As the ratio P 1M increases, the angle
Q
=
cotQ -
tends to
Michell structure for three parallel forces.
1T /4
1. and the structure approaches the
It may be noted that the moments at A, B are of
opposite sign to those that would be developed at the ends of a built-in beam by a downward central load.
The optimum layout for the case of reversed moments at A and B remains to
be determined.
In the particular case when there is no moment across the central section,
that is the case M
=
P a/2, the optimum layout is as shown in Figure 7. In the associated
virtual deformation, the space outside the circular regions does not move while inside the circular regions the displacement is purely circumferential.
Acknowledgment.
The author would like to thank D. E. Carlson for helpful discussions.
The manuscript was typed by Mrs, R. A, Mathine.
34
/'
---
........
I/
\
(
""\ \ / lj
\1 ~
f
"0 C
o c
~
+-
c
Q)
o
Cl C
"0 C Q)
CD
Q) ~
::J Cl
LL
/1
/
\
\
\
""
I ..........
----
/' /
/
)\
35
36
REFERENCES The Limits of Economy in Frame-Structures.
Michell, A. G. M. 597 (1904).
2.
Drucker, D. C., and Shield, R. T. Design for Minimum Weight. Proc. 9th International Congress of Applied Mechanics, Brussels 1956, pp. 212-222.
3.
Drucker, D. C., and Shield, R. T. Math. 12, 269-281 (1957).
4.
Shield, R. T.
5.
Prager, W., and Taylor, 35, 102-106 (1968).
6.
Shield, R. T., and Prager, W. Optimal Structural Design for Given Deflection. J. Appl. Math. Phys. (ZAMP) 21, 513-523 (1970).
7.
Shield, R. T. Optimum Design Methods for Structures. Naval Struct. Mechanics, Providence 1960, pp. 580-591.
8.
Wasiuly:hski, Z., and Brandt, A. The Present State of Knowledge in the Field of Optimum Design of Structures. Appl. Mech. Rev.~, 341-350 (1963).
9.
Sheu, C. Y., and Prager, W. Recent Developments in Optimal Structural Design. Appl. Mech. Rev.~, 985-992 (1968).
Bounds on Minimum Weight Design.
On the Optimum Design of Shells.
J.
E.
Phil. Mag.
~,
1.
J.
589-
Q. Appl.
Appl. Mech. 27, 316-322 (1960).
Problems of Optimal Structural Design.
J.
Appl. Mech.
Plasticity, Proc. 2nd Symp.
J. Optimization Theory and Applic.
~,
10.
Prager, W. Optimization of Structural Design. 1-21 (1970).
11.
Anderson, C. A., and Shield, R. T. On the Validity of the Plastic Theory of Structures for Collapse under Highly Localized Loading. J. Appl. Mech. 33, 629-636 (1966).
12.
Drucker, D. C., Prager, W., and Greenberg, H. J. Extended Limit Design Theorems for Continuous Media. Q. Appl. Math. 2, 381-389 (1952).
13.
Prager, W. General Theory of Limit Design. Applied Mechanics, Istanbul 1952.
14.
Mr6z, Z. The Load Carrying Capacity and Minimum Weight Design of Annular Plates. Rozpr. InZyn. (Engin. Trans., Warsaw) 114, 605-625 (1958).
15.
Onat, E. T., Schumann, W., and Shield, R. T. Design of Circular Plates for Minimum Weight. J. Appl. Math. Phys. (ZAMP) ~, 485-499 (1957).
16.
Prager, W., and Shield, R. T. Minimum Weight Design of Circular Plates under Arbitrary Loading. J. Appl. Math. Phys. (ZAMP) 10, 421-426 (1959).
17.
Shield, R. T.
18.
Shield, R. T. Optimum Design Methods for Multiple Loading. (ZAMP) 14, 38-45 (1963).
19.
Save, M. A., and Shield, R. T. Minimum -Weight Design of Sandwich Shells Subjected to Fixed and Moving Loads. Proc. 11th International Congress of Applied Mechanics, Munich 1964, pp. 341-349.
20.
Save, M. A. Some Aspects of Minimum-Weight Design. Cambridge Univ. Press 1968, pp. 611-626.
Proc. 8th International Congress of
Plate Design for Minimum Weight.
Q. Appl. Math• .!..§., 131-144 (1960).
J.
Appl. Math. Phys.
Engineering Plasticity,
37
Prager, W., and Shield, R. T. Optimal Design of Multi-Purpose Structures. Solids Structures !, 469-475 (1968).
22.
Hemp, W. S. Studies in the Theory of Michell Structures. Proc. 11th International Congress of Applied Mechanics, Munich 1964, pp. 621-628.
23.
Hegemeir, G. A., and Prager, W. 215 (1969).
24.
Prager, W. On a Problem of Optimal Design. Non-homogeneity in Elasticity and Plasticity, Pergamon Press 1959, pp. 125-132.
25.
Hu, T. C., and Shield, R. T. :Minimum-Volume Design of Discs. Phys. (ZAMP) g, 414-433 (1961).
26.
Cox, H. L.
27.
Cox, H. L. The Theory of Design. Great Britain (1958).
On Michell Trusses.
Int.
J.
Int.
J.
21.
Mech. Sci. Q, 209-
J.
Appl. Math.
The Design of Structures of Least Weight, Pergamon Press 1965. Aeronautical Research Council Report 19791,
OPTIMIZATION PROBLEMS IN HYDROFOIL PROPULSION':'
Th. Yao-tsu Wu,
Allen T. Chwang
California Institute of Technology, Pasadena, California and Paul K. C. Wang University of California at Los Angeles
This paper attempts to apply the principle of control theory to investigate the possibility of extracting flow energy from a fluid medium by a flexible hydrofoil moving through a gravity wave in water, or by an airfoil in gust.
The present
optimization consideration has led to the finding that although the flexible hydrofoil may have an infinite number of degrees of freedom, the optimum shape problem is nevertheless a finite-dimensional one.
The optimum shape sought here is the One
which minimizes the required power subject to the constraint of fixed thrust.
A
primary step towards the solution is to reduce the problem to one of minimizing a finite quadratic form;
after this reduction the solution is determined by the method
of variational calculation of parameter s.
It is found that energy extraction is
impossible if the incident flow is uniform, and may be possible when the primary flow contains a wave component having a longitudinal distribution of the velocity component normal to both the mean direction of flight and the wing span.
When
such waves of sufficiently large amplitude are present, not only flow energy but also a net mechanical power can be extracted from the surrounding flow.
':' This paper includes further extension to that which was originally presented at the Symposium.
39
1. Introduction SorTIe previous observations on fish swiIllIlling and bird flight seeIll to suggest that SOIlle species Illay have learned, through experience, to acquire the key to high perforIllance by executing the optiIlluIll IlloveIllent that Illay be of great interest to control theory related to fluid Illechanics.
An especially intriguing aspect of
the optiIllization probleIll concerns with the possibility of extracting energy froIll surrounding flow by an oscillating lifting surface ( such as the fish body and fins, bird wings, and artificial wings like airfoil and hydrofoil) and its associated effect on the control of Illotion. This general probleIll has been explored to various degrees of generality. Based on the approxiIllation of potential flow with sIllall aIllplitude, it has been found by Lighthill (1960) for slender bodies, and by Wu (1961) for two-diIllensional plates, that if the basic flow is uniforIll, energy is always iIllparted by an oscillating wing to the surrounding fluid, and an extraneous Illechanical work IllUSt therefore be continuously supplied to Illaintain the Illotion.
Even though it is iIllpos sible
in this case to extract energy froIll the flow field, the highest possible hydroIllechanical efficiency that can be attained by a wing, subject to delivering a given forward thrust, can be very high, as found by Wu (1971 b,c) for the two-diIllensional plate and a slender lifting surface. As was subsequently pointed out by Wu (1972), the situation becoIlles drastically different when the basic flow is no longer uniforIll, but contains a wave COIllPOnent,such as gravity waves in water, or wavy gust in air.
The contention that the
wave energy stored in a fluid IllediuIll can be utilized to assist propulsion has been suggested by intuitive observations.
Sea gulls and pelicans have been observed
to skiIll ocean waves over a long distance without Illaking noticeable flapping Illotions (save SOIlle gentle twisting) of their wings.
In an extensive study of the
Illigrating salIllon, Osborne (1960) found that the increased flow rate in a swollen river did not slow the salIllon down (for known biocheIllical energy expended during the travel) by that Illuch a Illargin as would be predicted by the law of resistance in proportion to the square of their velocity relative to the flowing water.
Several
possible explanations were conjectured by Osborne, including the prospect that the flow energy as sociated with the eddies in river could be converted to generate thrust.
To explore this possibility Wu (1972) introduced an energy consideration
to an ear lie r
study of WeinbluIll (1954) on the probleIll of heaving and pitching
of a rigid hydrofoil in regular water waves.
It was found that the greatest
possible rate of energy extraction is provided by the OptiIllUIll Illode of heaving and pitching.
When waves of sufficiently large aIllplitude are present, not only
flow energy but also a net Illechanical power can be extracted froIll the wave field.
40
In the present study this problem is further generalized by allowing the hydrofoil to be flexible so as to admit an infinitely many degrees of freedom ( of small amplitudes).
This general problem merits study for several reasons,
First, it is of a theoretical interest to find out how much improvement in the hydromechanical efficiency and energy extraction can be gained by admitting the additional degrees of freedom.
Second, the results of the present study of energy
transfer between an oscillating body and surrounding stream can be useful to the development of control theory for hydrofoil ships and to the analysis of flutter phenomena.
In the case of flutter in a uniform stream, it is usually assumed
that the engine maintains the constant forward speed regardless of the fluttercreated inertial drag,
In a wavy stream, however, the flutter may create a
propulsive thrust, which may amplify further instability and a self -excited flutter may develop.
Some of these aspects have already been observed by K~ssner (1935)
and Garrick (1936, 1957);
this paper is aimed at the general case of propulsive
energy balance. Further, from the standpoint of development of control theory, the present problem also merits study in its own right since it presents some new features and difficulties that-apparently do not confirm with the known classical cases, brief description can be given as follows.
A
Section 2 presents the general (lineariz-
ed) theory for a two-dimensional hydrofoil oscillating in waves, which is applied in Section 3 to the general case of a flexible plate wing.
In Section 4 the problem
of optimum motion is formulated as to find a hydrofoil profile that minimizes the energy loss C
subject to a constrained thrust coefficient C , It is shown that E T although the flexible hydrofoil may have infinitely many degrees of freedom, C E and C can be reduced to quadratic forms of finite dimensions, After this crucial T step the optimization problem reduces to one defined on a three-dimensional vector space (
S l' s2' s 3)'
With this drastic reduction it is possible to show that an optimal
solution does not exist unless appropriate bounds are imposed on the variables
independent
Under this condition the optimal solution is determined and n compared with the previous special cases, It is felt that the present method of SIS,
solution is still heuristic, to some extent, for much of the intuitive physical picture was relied on for guidance.
It is with the hope to stimulate further
development of the general theory for this class of control problem that the present study is presented before this Symposium.
41
2. Two-dimensional Hydrofoil Oscillating in Waves With specific applications in view we consider the basic flow to be a sinusoidal gravity wave of small amplitude in water of finite depth, H, in which a two-dimensional hydrofoil of chord 2J. moves horizontally with velocity U while submerged at a mean depth hI underneath the free surface.
In terms of the body coordinate
system (x, y), the wave profile of the basic flow (see Fig. 1) may be written as
y
= hI + Re
[ a e i (wot - kx ) ]
(1 )
the wave amplitude, a, being assumed small such that ka« velocity (U
+uo,
v ) of the wave field, by clas sical theory, is 0
u o - iv 0 = A."',' cos [ k (x + iy + ih where h
2
The corresponding
1.
2
) - w0 t]
(2 )
= H - hI is the distance from the bottom and w
o
is the encounter frequen-
cy (3) 1
2 w,:' = gk tanh kH
(4 )
A,:J a = (2gk/ sinh 2kH)2
where g is the gravitational acceleration and in (3), the and - sign for following the waves.
+ sign
is for heading sea,
In particular, the y-component wave velocity
at the x-axis (which coincides with the mean position of the hydrofoil), denoted by v (x, 0, t) o
=V 0 (x, t),
is
V ox, ( t)-'A i(wot-kx) _1 0 e
,
(5)
A o = A", ',. sinh kh 2
Here and henceforth, the real part of a complex expression will be understood for physical interpretation. Since the problem of central interest at hand is to determine the effect of a waving stream on the propulsive performance of a hydrofoil in unsteady motion, we shall further assume, for simplicity, that the hydrofoil is located sufficiently far from both the free water surface and solid bottom so as to curtail the complicated (but only secondary) corrections due to these boundary effects.
This condi-
tion would be nearly satisfied if the hydrofoil is at a distance more than two chords away from each of these boundaries, that is for the chord 2J. <
i
max{h , h }, this 1 2 estimate being inferred by the known results of the steady flow case (see Wu, 1954) which is assumed to remain valid in the unsteady case.
As an additional simplify-
= A /U of the magnitude of the orbital wave velocity o to the mean free stream velocity is taken to be small so that the x-component
ing assumption, the ratio
E
42
y
------t----=--i.~=t=~+I.T-------.......... x h2 = H - hi J/~/----r/--:/r-rl'I,..--rl-Ir-?'I+1 1 1 7 7 7 7 7 7 7 777771777177/7777
FIGURE 1
43
orbital velocity, u , may be neglected in comparison with U in formulating the o present linear theory. Although the simple water wave is chosen as a concrete example, it makes little difference to the subsequent discussion if other kinds of wavy streams are considered as long as the transverse velocity of the basic flow can be represented by equation (5).
For arbitrary V{x, t) the result can be obtained
by the Fourier synthesis of this fundamental case. The hydrofoil (or airfoil) is assumed to be thin, though sufficiently rounded at the leading edge to keep the flow from being separated there.
The foil-thickness
effect is then only secondary and will be further disregarded in this study.
For
brevity, the semi-chord, 1., of the hydrofoil will be normalized to unity as the reference length.
The unsteady motion of the hydrofoil assumes the fundamental
form A
Y = h{x, t) = h{x) e
iwt
'"
where the circular frequency w is arbitrary, and h of x (with respect to i denoted by (U
=N in the time factor).
+Uo +u I '
v0
+ vI)'
(6 )
(-I < x< I) ,
may be a complex function
With the resultant flow velocity
the linearized boundary condition that the flow
be always tangential to the moving body surface requires (7a)
(/xl
V{x, t)
D h{x, t)
D=a/at+ua/ax
,
where Vo{x,t) is given by equation (5), and vI±{x,t) signifies vI{x,
(7b)
o±
,t).
Like
in the uniform stream case, it is convenient to use the acceleration potential, defined by (8)
as a new dependent variable, particularly since it is continuous throughout the fluid.
It is related to the velocity by
Du
o
= acP 0 lax,
on linearized theory.
Dv
0
= acP 0 lay -
g
, which gives the pressure distribution o in the primary wave field, can be readily obtained by integration of the first two
equations in (9);
The component cP
it gives no hydrodynamic force (except a bouyancy) or moment
on the hydrofoil since it is continuous across the plate.
The effect of waving
stream on the hydrodynamic performance comes with calculations on cP , explicitly I through the term Vo{x, t) in condition (7). The problem of cP I is specified by the
44
boundary conditions
y =
(Ixl > I,
y =
° °
±),
(lOa)
±),
(lOb)
°
± = at x = I, and that cP I vanishes at I Condition (lOa) follows fronl substituting (7a) into the last equation of (9);
together with the Kutta condition that cP infinity.
(Ixl < I,
and (lOb) is a consequence of the pressure being continuous in the flow and thefact that cP
is odd in y. I The solution to this nlixed-type boundary problenl of cP
197Ia);
is known (see Wu, I in particular, the value of cPI(x, O± ,t) at the plate is given by + U (II +x _x ) cPI-(x,t) = ± -Zao
i ±:;;I ~r I
" .)_1
( 1- x Z
~
)1. 2.
i.(J1( S ,t) dS
S-
(II)
x
X V
i.(JI(X,t)=-DS
-I
1
(IZ)
(s,t)dS I
a
o
I
1
= [b - (b + bI)@(a)] - [ b - (b + bI)@(a )] I o o o I
Z
(13 )
fTr
bn=rrJ
(x=cosB,
V(cosB,t)cosnB dB
(14 )
n=O,I,Z, ... )
o b
r
Z Tr =-; V (cosB,t)cosnBdB n Tr vo 0 I
@(a) = ~(a)
+i
"(a)
a o = (oK)i + K,
,
0
(15 )
a.wi/u,
a
o
=(gf. IU Z )tanh kH,
!!!w
0
f./U
(16)
,
(17 )
K 5. k1
Here the integral in equation (II) aSSUnles its Cauchy principal value;
J (k) is the
Bessel function of the first kind; @(a) is the Theodorsen function, ~and
n
!!J being
its real and inlaginary parts(for a tabulation of @, see Luke and Dengler, 1951);
a is the reduced frequency of the body nlotion, a
o
the wave reduced frequency, both
The function a (K) in equation (17) is the nono dinlensional fornl of equation (3). We shall write K as k since f. = I.
being based on the half-chord f..
The differential lift distribution along the chord is clearly
..,
-
+
ol-(x, t) = p (x, t) - p (x, t) = ZpcP
I
+(x, t)
(Ixl <
I)
(18 )
The integral representation of the lift L and the nlOnlent M (about the nlid-chord, positive in the nose-up sense) are
L =
5~L(x, t) dx
,
(19 )
45
5
1 "r.<X,t) x dx -I
M = -
(20)
.
Also, the formulas for calculating the thrust, T, the power required for maintaining the motion, P, and the kinetic energy imparted to the fluid, E, remain the same as in the uniform stream case (Wu, 1971 a), 1
T =
5
L(x, t) h
-I
x dx + S
5 5
,
12 S = 2: rrp(Re a) , 0
(21)
1
P
=-
-I
E = -
J:.(X, t) h t
dx
(22)
,
I.:!) x , t)(h + Uhx ) dx t
su
(23)
-I
where the subscripts designate partial differentiation, S represents the leadingedge suction, which now includes the contribution from the wave component.
From
the above expressions we note that the energy balance can be expressed as
P
TU
+E
(24 )
,
which is formally the same as in the uniform stream case.
However, unlike the
case of uniform free stream, the time average of E here is no longer always positive, and we shall see that energy can be extracted from the waves when E becomes negative.
3. Flexible Plate Wing We shall consider the general case when the motion is periodic in time, as
....
prescribed by (6), with arbitrary amplitude function h(.), such as anyone that can be performed by a completely flexible plate.
Substituting the differential lift
.!.(x, t) given by (II) - (18) in (19) and (20), we obtain the total lift L, and the moment M, as L=rrpU{a
o
- (b
I
-b I) _lia(b -b )+lia (b I_b I)} 2 2 1 02 00 2
(25 )
(26 )
The corresponding results for the thrust T, power P, and energy loss E can be obtained by substituting (II) - (18) in (21) - (23) and by following the same procedure as that used for the uniform stream case by Wu (1971 a, in deriving hi s equations 44 - 46);
the intermediate manipulation again can be considerably
simplified by making use of the relationship
46
51 fl(X)dx -I
5I(~)i g(s)ds -I1-s
x~)i f_(S)dS
= 51
S-x
gl(x)dx &1 (1-II-S
-I
(27)
s-x
where f(x), g(x) are two arbitrary functions, provided they and their derivatives -I~x~
f'(x), gl(X) are continuous in
-
I.
-
The mean values of thrust T, power P,
and ener gy 10 ssE c an be deduc ed by aver aging T, P, and E over a long time period.
Two different cases arise according as w = wo or w T.4. w0 • (i) When w = w , that is when the wing oscillates at the wave encounter o frequency, the two motions are correlated. In this case we obtain T as
T
•
...1..
...1..
•
1•
..,(
1 •
,-I(
...I"
•
•
.. I..
= ~ p Re{ (a + b - (3 )(a'" - b'I"+ (3 'I' )- b R'I' - b (3 "+ (3 (3 '1° + 2I} 4 0 0 0 0 d0 I 0 2
I = ~ lf
2 51!: 1 (1 - x ~ 2 -1 -1 1 - s
)i
V(x,t)~(s,t)
S-
dxdS
,
,
(28a) (28b)
x
•
where the superscript ':' denotes the complex conjugate, (3 n= df3 n (t)/ dt, and
(3 n (t) =
hence ~
n
= iwf3
n
2 (If
:rr J
(x=cose,
h(x, t) cos ne de
n=0,I,2, ... ),
(28c)
o
when h is given by (6).
The mean power P can be shown, after
some manipulation, to have the following expression (29a)
Sl
CI(I
-1 .)-1 I
2)i av~:'(x,
~~2
t)/at
(29b)
s -x
By substituting in (29b) the relationship
av (x, t)/at o
=-
(w/k) aV lax , 0
(see (5), with w = w for the present case), and applying the formula (27), it o immediately follows that
II = (o)k) I where I is given by (28b). E =
,
Whence, by (24), E = P - UT, or
~ pU Re{(a + b )(b'I:'- a':') + 2(cr/k - I )I} •
4
(29c)
0
0
0
(30)
We note that, upon substituting (13) in (30), the first term on the right-hand side
47
of (30) involves only the first two Fourier coefficients of V, in the particular combination of (b + b ). However, the second term in (30) with I, which results o I from the interaction between the wave action and body motion, involves all the Fourier coefficients of V since the integral I has the following Fourier-Bessel expansion 00
I = A e -iwt \ ' (i)n+I J (k) ( b b ) o ~J n n+I- n-I • n=I
T
We further note that the expressions for also the first two Fourier coefficients, 13
0
and
P
(31 )
,
involve, in addition to the b
n
s,
and 13 I' of h(x, t).
To facilitate the subsequent consideration of the optimum shape problem, it is useful to recast the above expressions for T, P, and E in terms of certain inner products.
Let
J4.- denote
a subset of the complex Hilbert space L
2
[ -I, I ]
(32a)
and let the inner product between f(·) and g(.) on
=.;
SI
-I
f(x)g'~(x)(1- x 2 ) -i
where the weighting function (I - x
2
)-i
J4. be
dx =< g,f>';':,
Jl- if < f,
(f,ge);l.-)
(32b)
is introduced in order to convert the
Fourier coefficients into the inner product form. be said to be orthogonal on
defined by
Any two functions f, g inJ=l.-will
g > = O.
Substituting (13) - (16) and (28c) in (29) - (30), we obtain the mean coefficients _ - _ I 3 of thrust, power, and energy loss, defined by (C p ' C , C ) = (P, E, TU)/(~ 'TT'pU i), E T in terms of the inner products as
(33)
C E= Re { B(cy)l< v, £1 >
2 1
+ 2E (1- 2@)(W +i W 2)< v, f I> + 2E (CY /k -I )< v, g2 > - 4E 2 W 2 }, I (34 )
where v(x) = e
-iwt
f (x) = I I ® (CY)
V(x, t)/U ,
+x
,
= 3'" (CY) + i ~
A.
_
h(x) = e
iwt
h(x, t) ,
(35a)
gI(x)=(I-@)x+@ ,
(CY) ,
B(CY) =
~ _ ( cg;- 2 +
(35b)
&2
) ,
(35c)
(35d)
48
(35e)
In the above, as well as in the sequel, the argument k of the Bessel functions J (k) n
will be understood unless otherwise designated.
The mean thrust coefficient is
simply (the coefficient form of (24))
(36 ) Another flow quantity of interest is the mean leading-edge-suction coefficient, - I 2 C = S/.;- TrpU 1. From (21), (13), (14) we obtain S cs=l@-+2E(W1-iW2)12 where
f (x)
o
=x
(-l~x~1)
(37a)
,
•
As suggested by Lighthill (1969, 1970), the ratio cSI C
provides a measure of the T relative strength of the leading-edge suction; moderate and large values of cSI C T (as compared to unity) suggest a tendency that the flow would separate, or stall, near the leading edge (such a category of separated flow would be quite different from the completely wetted flow as as sumed here). (ii) w
f: W o - - - In this case the mean product of exp(iwt) and exp(±iwot) vanishes
as the body motion and wave action become uncorrelated.
Consequently the terms
which are linear in E in (33) and (34) drop out of the expressions for C p and C ; E 2 further, W in (34) then assumes its value at ( jo. The corresponding C likewise . s becomes (38 )
The result of this case therefore reduces virtually to the case of uniform stream except for the additional term ( - 4E 2 W 2) in the expression for C
E
and (4E 2 W 2) in
C
' These added terms indicate that energy is invariably being supplied by the S primary wave, through the mechanism of generating a greater leading-edge suction, at no expense of C p ' It thus follows that for C p fixed, C becomes T greater and C smaller (hence higher efficiency) with increasing wave action E
(greater E W).
The energy gain in this case, however, is always accompanied by
an appreciable increase in the leading-edge suction, suggesting an easier leadingedge stall.
When the suction is required to remain reasonably small, the optimum
motion and the corresponding improvement of efficiency are not significantly different from the uniform stream case which has been discussed earlier by Wu
49
(1971b).
For this reason this second case will not be further pursued here,
4. The Optimum Motion (w = wo) The present problem of optimum motion is formulated especially to analyze the interaction between the body motion and wave action;
it can be stated as
follows: Given a reduced frequency CT> 0 (hence also the wave number k, see (3)) and a A
thrust coefficient C set
.J:t--
0> 0, find a velocity profile v, or a hydrofoil profile h in the T, (defined by (32a)) such that C is minimized subject to the constraint E (39)
assuming that the wing oscillates at the wave encounter frequency. It is desirable to choose C
(rather than C p or C ) to be a constrained quanT E tity since a constant thrust is required to overcome the (nearly constant) viscous drag if the uniform forward motion is to be maintained.
No additional constraints
are imposed here on the total lift L and moment M for balancing the rectilinear and angular recoils of the flexible plate (see Wu, 1971a, Eqs. (56a, b)); is made for two reasons.
this choice
First, when a body structure consists of components
other than the flexible plate, the recoil consideration must take the motion of the entire body into account.
Second, even when the wing alone comprises a self-
propelling body in its entirety, there will still be other degrees of freedom left to be used to satisfy the recoil conditions, if desired, as we shall see later. In choosing the independent functionals for the optimization calculation, we note that only two of C p ' C ' C are independent since they are related by (36). E T There are great advantages in the choice of C p and C as the independent functionE A ~ als of v and h because C ' in particular, does not involve h, and C p is also E simpler in expression than CT' In the expression (34) for C ' the first term on E the right-hand side is the same as in the uniform-stream case (see Wu, 1971 b, Eq. (13));
it is always non-negative since B(CT) > 0 for CT> O.
The second and
third terms, which are bilinear in e and v, represent the body-wave interaction, The last term, which is proportional to e 2, is solely due to the wave action.
This
result actually proves the statement that extraction of energy from the surrounding flow by an oscillating flexible wing is impossible if the incident flow is uniform. In the presence of a primary wave, with appropriate v and increasing wave parameter e , the last three terms in (34) may become negative and numerically so large as to reduce C seen later.
at first, and C p eventually, to negative values, as will be E The case of C p < 0 signifies the operation in which a mechanical power
is received by the body, instead of being consumed by it, through a favorable extraction of the wave energy,
In spite of these pos sibilities, we shall still con-
tinue to use the Froude efficiency
50
(40a) as a measure of the hydromechanical performance.
Aside from its usual signifi-
cance for 0 < ,,< I, now we may have new generalized interpretations as follows:
(i) "> I (ii)
"<
0
for
C
for
C
E E
< 0,
C p > 0;
(40b)
< C p < O.
(40c)
Another step of primary importance is to choose the independent function for the optimization calculation.
Although either v or h may serve as an independent
function {since they are related by a differential equation (7b)), the advantage of taking v is clear, as was noted by Wu (197Ib, section 6) in discussing the optimum shape of a flexible plate oscillating in a uniform stream.
As another reason, we
note that in the present formulation, an inner product of h with a given f{.) can be converted into an equivalent one involving v, whereas the converse is generally impossible. Accepting v as the independent function, we proceed to recast the inner product A
A.
< gl,h> in (33) in terms of v. (d/dx
+ ia
By (35a) and (7b), h and v are related by A.
) h{x):: v{x)
(Ix I < I)
(41 a)
,
which has the general integral as
(41 b) "'-
where h_ A
l
is an arbitrary integration constant.
Substituting (4Ib) and (35b) in
< gl' h >, and integrating by parts, we obtain (42a) where
(42b)
(42c) Consequently (33) becomes C p :: R e { - i a [ < v, f I > - 2e ( J I + iJ 0)] [ < g 3 ' v >
+ 2e
+ CI -
iC2 ] I
(a / k) < v, g 2 >}
.
(33 )
51
I
Now the expression for C p in (33) and C in (34) are both expressed in tenns of E v and contain only three inner products: < v,f >, < v,g2>' and < g3' v>. 1 Since f , g2' g3 are not mutually orthogonal on Jt, we next construct a set 1 of three orthogonal functions, f , f , f say{there being no need here to normalize 1 3 2 them), by the Schmidt scheme: (43a) (43b) (43c) such that
(ilk)
The coefficients a
a
n
(44)
•
are determined by the orthogonality condition (44) as
=< g2,f >/ = t< g2,f > = t [ 2J {k) - iJ {k)] 1 1 1 1 1 2
1
a 2 =/ = 3: 2 {@[ l+ilT-e
ilT
(45a)
J o {lT)] +i{1-@)[;-e
a 3 ==-a;' =-3a;a2
ilT
J 1 {lT)]},{45b)
.
(45c)
By separate calculations,
< f 2' f 2 > = < g 2' g 2> - a 1 < f 1 ' g 2> -
< g ,g > 2
2
=
2
~
= 1-
S
1
2 3/2 (I - x ) dx
-1
J~{k)
a~ < g 2 ' f 1 >
S
+ a 1 a;' < f 1 ' f 1 > =< g 2' g 2 > - 3 a 1 a~'
1 e - ikS ds I -1 (1_g 2 )2 (oS-x)
5
1
-1
eikr] dYj 1
(l_Yj2)2{Yj_X)
+ 2Ji{k) - 2J o {k)J {k) , 2
which can be shown by successive interchange of the order of integration and by making use of the Poincar~-Bertrandformula, and hence (45d)
Finally,
52
where I
N (a) = - [ I - @(o')] J (a) - i@(O')J (a) n n n
(n =0,1.2, ..••• ).
(45f)
The above result can be shown by using the series expansion of g2 and g 3as 00
g2(x) = 2isinB)' (-it In(k) sin nB
(x=cosB) ,
.J
n=l 00 g3(x) = 2ie- iO'cosB sinB '\' (i)n[ N (a) _ (i)n N (a)] sin nB /J non n=1
(x=cosB) •
This cOlnpletes the determination of a , hence also the orthogonalization. 3 It is now evident that v can be expres sed as 3
v(x) = "/~, B n f n (x) + vJ..(x) n=1 where B
n
IS
,
(-I~x~l)
,
(46 )
are complex coefficients and v..l.. is any function belonging to the ortho-
gonal complement of the subspace spanned by {f , f , f }. that is, < f , v..l.> = 0 l 2 3 n for n = 1,2,3. For convenience of the subsequent computations. we introduce the real parameters £
n
I S
by
= B n =£2 n- l+ i £2 n
(47a) (47b)
where C + iC is given by (42c). I 2 =
a~«v.fl>
From (43), (46) and (47). we have
+ =
a~«£I+iS2)
+ (£3+ iS 4)
,
= a; (SI+iS2) + a; (S3+iS4) + (S5+iS6) - (C I +iC 2 ).
(48a) (48b)
I
Substitution of (47). (48) in (33)
and (34) yields
(49)
( 50) where a
2
= Al (0')+ iA (a) , 2
a
3
= A (a, k) + iA (a. k) • 3 4
2 PI =-A I J o (k)-A 2 J I (k)+3k JI(k).
P2=AIJI(k)-A2Jo(k)-J2(k)/3k.
(51 a) (SIb)
53
(Sl c) Q1
",(l-2:g;')W1+2~W2+i( ~ -I )J 1 '
Q2=2~W1-(1-2~)W2-}(~-1)J2'
(Sl d) Q3=(a/k)-1,
(Sl e)
The other coefficients appeared here have been given in (3S), (4S). Equations (49), (SO) show that C p depends on only six real parameters {~1' S2"" ~6}, and C E depends on only three parameters {~1' ~2' ~3}' while both C p and C hence also C ' are independent of v.J,.(x). (Note that the orthogonal E' T cOITlpleITlent of the subspace spanned by {f , f 2, f } is infinite diITlensional.) Thus, it is l 3 clear that the optimization problem posed earlier nOw reduces to one defined on a finite-dimensional vector space. Before we proceed with our discussion from this approach, further simplification of the expressions for C p and C can be gained if we first eliminate the terms E linear in ~ I and ~ 2 in (49), (SO) and then reduce the number of quadratic terms in (49) by the following transformation
Sl+ iS 2:::
i
S 3 + is 4:::
(~ 3 + iG 4) + (A 3 + iA4 )(s S + is 6)/ A 2
(A 4 -iA 3 )[
~1+iS2+ k(Q1+ iQ 2)]
,
(S2a) (S2b)
,
(S2c)
SS+iS6=(SS+iS6)+(CS+iC6) , where
(S2d)
Then (49) and (SO) reduce to
2 2 Cp/a =A 2 (sl +S2 )+A(Sl s 3+ s 2 S 4)+E (A S 3 +A 6 S 4 )+E A o
s
2
2
(S3 ) (S4 )
CE=B(Sl+s2)+2EQ3s3-EQo' where AS:::2P3-(A3Q2+A4Q1)/B,
A6:::2P4-(A4Q2-A3Q1)/B,
(SSa)
A o =-2[ Jo+(P3A3+P4A4)/A2] sS+2[ J1+(P3A4-P4A3)/A2] S6 2 2 2 +4E [ JoP2+J1P1-A2(JoQ2+J1Q1)/B ] -EA 2 (Q1+ Q 2)/B, Q o =2Q 3 (A 3
sS - A 4 s6 )/A 2 +4E
2 2 2 W +E (Q1 +Q )/B • 2
(SSb) (SSc)
Thus in the above reduced form, C p depends quadratically on {s I' S 2' S 3' S4 }, C E
54
depends quadratically on {I; 1,1;2} but is independent of 1;4' while both C p and C E depend linearly on { I; 5' I; 6}, When the primary wave is absent (i. e. e =0), equations (53) and (54), or equivalently (49) and (50), reduce to the case of a flexible plate in uniform flow treated earlier by Wu (1971 b, see his equations (79) and (80), which involve also six independent parameters).
The present result of C p and C E is also very similar to that for a flat plate oscillating in waves discussed by Wu (1972, see his equations (50), (51) for the four independent parameters proper to that problem).
Like those simpler cases investigated previously, we note that in
the three-dimensional Euclidean space (1;1,1;2,1;3) (i.e. with 1;4,1;5,1;6 held fixed), the C
= const. surfaces are paraboloids of revolution with its generating axis lying E along the I; 3 -axis, while C p = const. surfaces are oblique hyperboloids, who se
cross -sections with I; 3 = const. planes, if real, are circles.
-=
The optimization problem posed earlier can now be reformulated as follows: Let R
denote the six-dimensional Euclidean space of ordered six-tuples I; 6 1;2,1;3,1;4,1;5,1;6) of real numbers; and let n be a subset of R defined by 6
(So I'
(56)
-
The optimization problem is to find a vector fO E for all I; E
n
such that C E( fO ) ~ C E( f)
n.
FroITl the known geoITletric properties of constant C p and C
surfaces, and hence E also of CT=C , d'0 surface, it follows thatnis an unbounded set in R . Consequently, 6 T it is possible that the optimization problem may not have a solution. It suffices to demonstrate two such cases.
As the first, consider a sequence of points {fk} in
the set Sl c n defined by (57) k
such that Q 1; 3 - - 00 as k 3 on (1;2,1;4,1;~1;6) while C
-00.
It is readily shown that in the set Sl' C
depends T Sl is therefore nonempty.
depends on (1;2,1;3,1;5,1;6); E But, since 1;. 's are all constant for j=2,4, 5, 6 and for all k, we immediately see J -k from (54) that the sequence of values CE(I; ) - - 00, as k - 0 0 , implying the nonexistence of an optimal solution. As the second example, consider another sequence of points {fP. } in the set S2 Co
Q
defined by (58)
such that Q o (I;~, I;~ ) - 00 as P. - 00. It is also easily seen that in the s~P. S2 ' C T = C T ( I; I ' I; 2' I; 3' I; 4 ), C E = C E ( I; I ' I; 2' I; 3' I; 5 ) and cons equently C E ( I; ) - while C remains unchanged as P. - 0 0 , implying again the nonexistence of an T optimal solution.
00
To ensure the existence of an optimal solution that is physically meaningful,
55
we shall minimize C
over a subset
E
'"
n of n which is closed and bounded, i. e. C
'"
where R
6
denote a bounded subset of R
6
6
T ,0
> 0 } ,
(59a)
such that
II;~~M
(59b)
n=l -0 The new optimization problem is to find a vector I; E
-
'"
n
-0-
( I; )~ C ( I; ) E E for all I; En. Evidently, this optirnization problern has a solution since C E is ...., continuous On the closed bounded set such that C
n.
In what follows, we shall consider the particular case where s4' Ss and ~ are treated as free pararneters sO that the optirnization problern reduces to a threedirnensionalone.
Moreover, the cOnstant M in (S9b) is adjusted sO that the opti-
rnurn solution can be deterrnined frorn the points in ~ at which (grad Cp) is proportional to (grad C
E
).
Thus, we set
j:::l, 2, 3,
(60)
where )..., is a Lagrange multiplier, giving (61 a) 1;2=)...AI;4
(61 b)
'
I; I = (e / AB )( 2 A 2Q 3 - A5 B ) + (e Q 3/ AB ) )...
- I
(61 c)
where)... is related to)...' by)... - I = 2 (B)"" - A ). From the three equations (61a-c) 2 we can determine the variables (1;1,1;2,1;3)' which are subject to variation, in terms of (1;4,1;5,1;6' )...).
Finally, the Lagrange multiplier)... can be determined in
terms of 1;4' 1;5' 1;6' CT,o and e by invoking condition (39), indicates that the extremal solution will involve (1;4' I; 5' I; 6' C parameters.
This line of approach T
,
0 '
e ) as free
It is more desirable, however, to adopt
I; 0 = (I;
~ + I; ~ )i
(62 )
rather than 1;4 as a free parameter since this replacement will facilitate computation as well as comparison with the earlier results for the uniform stream case (Wu, 1971b) and those for the rigid plate in waves (Wu, 1972).
Thus, we first
eliminate I; I' I; 2 in (53), (54), and (61 a-c), next we apply condition (39), giving 2 A [ T 2 )...2+
IT)...]
+E:'[ (lTA5-2Q3)z3+lTA6z4]
= CT,o
-e
~o
(63)
56
(64 ) (65) where T 2 ::: CTA
2
z·:=s·/s J J 0
- B
(j
CT,o C T,o I s02
~
o
(66a)
1,2, ... 6 )
= (CTA 0 + Q 0 ) 1 s0
(66b)
Equation (65) follows from the definition of z3' z4 and So as given by (62) and (66a), there being two branches of z4 for given z3' with
I z3 I ~ I.
The three equations (63 )-(65) involve three unknowns, z3' z4' A., and three the "proportional loading factor", e - the T ,0 "proportional wave factor", and ~ - the "complementary mode factor" which o includes the contribution from the mode s5+is6 and that from the waves. parameters, namely C
As for the actual calculation of the Lagrange multiplier A., we note that if equations (64), (65) are substituted for z3' z4 in equation (63), then, upon perfect squaring, there results an eighth degree algebraic equation for A., of which the real solutions (appearing always in even number s) are of interest.
This equation seems
to be too difficult for analytical solutions; resort was then made to numerical methods.
The method which proved to be successful is as follows.
Since the
physically meaningful solutions also require z3' z4 to be both real, equation (65) suggests that z3 can be used effectively for parametric computation of A., with the obvious advantage of having a bounded range - I
~
z3
~
I.
In this parametric form,
both equations (63) and (64) are quadratic equations in A., each giving two solutions of A. in closed form, from which the real solutions of A. satisfying both equations were determined by Newton's method, using z3 as a parameter.
The number of
' E and ~ ; on occasions as many as T ,0 0 eight real solutions were obtained, and there are cases in which two real solutions solutions depend on the values of
are very clo se to each other.
CT,
C
In all the cases tried the two real solutions provid-
ing the highest and lowest efficiencies were taken as the desired optimal solutions. The numerical results of
, as shown in Figs 2 - 5 for a few representamax tive cases, exhibit the following salient features of the optimum solution. For T)
CT,o and E both as small as 10- 3 and with z5 ::: z6 ::: 0,_ ~he maximum efficiency = 1.0 for the reduced frequency CT> 10 The corre sponding max results of the maximum efficiency for a rigid hydrofoil in a uniform streaITl (E ::: 0)
is already
T)
and in regular waves (e > 0) are reproduced (see Wu I97Ib, 1972) in Fig. 3 over
-
-
,although the C and E in those two cases are defined T ,0 T ,0 with reference to the heaving amplitude at mid-chord, whereas the definition of a similar range of C
and E in the present case (by referring to S , see (66), whose physical T ,0 0 significance is not quite so simple) is slightly different. With this qualification,
C
a comparison between Fig. 2 and 3 shows that T)
max
is further improved by the
57
flexible over the rigid foil, in the frequency range of interest. at 10-
3
and
When C
2 T ,0
is kept
E alone is increased (by having a stronger wave) to 10- , 1)max
becomes greater than I, corresponding to the operation in which energy is extracted from the surrounding waves, but a power (somewhat smaller than before) is still required for maintaining the hydrofoil motion.
When E is as large as O. I,
we see that 1)
becomes negative, indicating that both energy and power are max supplied by the exterior wave field. At this high level of E, 1) becomes more max negative as C is increased to 10- 2 This trend implies that mOre energy and T ,0 power can be extracted from stronger waves at higher loadings. This is a quite
remarkable feature since this trend is reversed from that at smaller -2 waves, see the curves with E = 10 ). To summarize the case of Zs
= z6 = 0,
E
(or weaker
we note that the overall features of
1)
of the flexible and rigid hydrofoils are very similar, the difference, after max _ the two definitions of C and E are properly reconciled, being rather small. T ,0 Since the salient features of the optimum motion, including the variation of the leading -edge suction C
s '
feathering of the hydrofoil to the trajectory, etc., have
been thoroughly explored for the rigid plate case (see Wu, 1972), these features will not be further pursued here for the flexible plate.
However, it must be
stressed that the additional degrees of freedom provided by Zs and z6 for flexible plates can alter the 1)
of a flexible plate. As indicated by Figs. 4 and S for max two typical examples, 1) max for the basic case of z S = z6 = 0 can be increased by
suitable choice of Zs and z6 (within the set R of (59)). The trend of the influence 6 by Zs and z6 on the value of 1)max can be seen clearly through the linear dependence of
~
o on Zs and z6 (see (63) - (66)). _ unchanged when the set of parameters (C , T E, 0, 0), where
0
Obviously, 1) max will remain _ ' E , zs' z6) is replaced by (C
1
T
,
0
'
_I
(67)
C T,o = This explains the opposite trend of 1)max from its basic value at Zs = z6 a set of nonvanishing Zs and z6 is reversed in sign.
=0
when
By making use of this proper-
ty' or equivalently the simple formula (67), the utility and interpretation of Fig. 2 is thereby greatly extended.
Acknowledgments This work was partially sponsored by the National Science Foundation, under Grant GK 10216, and by the Office of Naval Research, under Contract NOOOI4 -67A0094 -0012.
58
1.5
,.ok::::::::----r-------.:==:::=========------=~ CT" = 10- 3 e"10- 3
0.5
-0.5
om
FIGURE 2
w "W".
Zs = 0
8 "' 0.2.
Z6 = 0
59
1.2 EO
1.0 -
-3
CT. = 10 0.8
E
=10- 3
7 0.6
0.4
0.2
-0.2
-0.4 . 0 01
FIGURE 3
W
= w.
8
=0.1
60
2.0..-------,-----,---,-,--r--.-,--,.-,------,---,---.,.----,r---r--r--r-...,..,
ero" 1.8
W
=
3
10- ,
;?" 10-
Wo,
23" 0.2
3
FIGURE 4
61
-
ero ~
-3
10
•
~ ~ 10-
W~Wo
•
23 ~0.2
1
FIGURE, 5
62
References Garrick, I. E. 1936 Propulsion of a flapping and oscillating airfoil.
NACA TR
567. Garrick, I. E. 1957 Nonsteady Wing Characteristics. Sect. F., High Speed Aerodynamics and Jet Propulsion, Vol. 7 (ed. A. F. Donovan and H. R. Lawrence), Princeton University Press, Princeton, New Jersey. Kussner, H. G, 1935 Augenblicklicher Entwicklungsstand der Frage des Fll1 g elflatterns.
Luftfahrtforschung, Vol. 6, 193-209.
K{1ssner, H. G. 1940 Das Zweidimensionale Problem der beliebig bewegten Tragflache unter Berllcksichtigung von partialbewegungen der Flussigkeit. Luftfahrtforschung, Vol. 17, 355-361. Lighthill, M. J. 1960 Note on the swimming of slender fish.
J. Fluid Mech.
Vol. 9, 305-317. Lighthill, M. J. 1969 Hydromechanics of aquatic animal propulsion.
Annu. Rev.
Fluid Mech. Vol. 1,413-445. Lighthill, M. J. 1970 Aquatic animal propulsion of high hydromechanical efficiency. J. Fluid Mech. Vol. 44, 265-301. Luke, Y. L. and Dengler, M. A. 1951 Tables of the Theodorsen circulation function for generalized motion.
J. Aero. Sci. Vol. 18, 478-483.
Osborne, M. F. M. 1960 The hydrodynamical performance of migratory salmon. J. Exp. BioI. Vol. 38, 365-390. Sears, W. R. 1940 Some aspects of nonstationary airfoil theory and its practical application.
J. Aero. Sci. Vol. 8, 104-108.
von K~rma:n, Th. and Sears, W. R, 1938 Airfoil theory for non-uniform motion. J. Aero, Sci. Vol. 5, 379-390. Weinblum, G. p. 1954 Approximate theory of heaving and pitching of hydrofoils in . regular shallow waves.
DTMB Report C -479.
Wu, T, Y. 1954 A theory for hydrofoils of finite span.
J. Math. Phys. Vol. 33,
207-248. Wu, T. Y.1961Swimmingofawavingplate.
J. Fluid Mech. Vol. 10, 321-344.
Wu, T. Y. 1971a Hydromechanics of swimming propulsion.
Part 1. Swimming
of a two-dimensional flexible plate at variable forward speeds in an inviscid fluid.
J. Fluid Mech. Vol. 46, 337 -355.
Wu, T. Y. 1971 b Hydromechanics of swimming propulsion. shape problems.
Part 2. Some optimum
J. Fluid Mech. Vol. 46, 521-544.
Wu, T. Y. 1972 Extraction of flow energy by a wing oscillating in waves. Res. Vol. 14, No.1, 66-78.
J. Ship
STABILITY THEORY FOR GENERAL DYNAMICAL SYSTEMS AND SOME APPLICATIONS
E. F. Infante Center for Dynamical Systems Division of Applied Mathematics Brown University Providence, Rhode Island
In this lecture, it will be attempted to describe some recent results in the stability of general dynamical systems from the viewpoint of Liapunov theory, and to illustrate and motivate them through examples. The subject of stability theory is an extremely broad One.
During the last
ten years a large number of developments have taken place in this area; it could be said that, beside Liapunov theory and the classical theory fOr finite dimensional systems, the new branch of the functional analytic approach has sprung during this period. This lecture is purposefully limited to a description of the Liapunov approach, and a rather limited One at that.
To attempt mOre wOuld surely lead to
failure; furthermore, the authOr has wOrked much mOre extensively in this area than in the other ones.
Fortunately, some recent expositions of a very readable nature
are available [4, 13J to those interested in the functional analytic approach; furthermore, at the foundations of the functional analytic approach is the concept of a dynamical system, the central mathematical concept of this lecture. Before embarking On the subject, the authOr feels obligated to apOlogize to the numerous workers in this area which he will neither reference nor acknowledge contributions from. the lecture.
This, it is hoped, is permissible, given the tutorial nature of
Interested readers will find the appropriate references in the few
papers quoted. The development of stability theory, from the time of Poincare and Liapunov, was considered a branch of ordinary differential equations and of mechanics; interest in the stability theory of partial differential equations and functional This research was supported by the Office of Naval Research, NONR N0014-67-A-0191-0009
64
differential equations is rather mOre recent, especially in the engineering literature.
The fundamental idea behind the concept of a dYnamical system is to try to
generalize to broader classes of evolutions
the results known for ordinary differ-
ential equations.
1.
Dynamical Systems and Some Examples Let
the interval
n R
denote an n-dimensional vector space with nOrm and ~ a Banach space with
[0,00)
I -I,
R+
denote
the nOrm of an element in ~.
Ilcpll
Then [9, 11]
Definition 1.1.
A dynamical system in a Banach space
C) is a function u: R+
xf8~~
such that (i) (ii) (iii)
u
is continuous
u(O,cp) '" cP u(t+~,cp)
'" u(t,u(~,cp))
for every
t,~ ~
0, every cp
in~.
Hence, a dynamical system has some continuity properties, the second cOndition states that at
t '"
°
the dynamical system is the identity map and, finally,
that it has the semigroup property.
It will be noted that the definition implies
that a dynamical system is autOnomous and that the map of the dynamical system is defined only forward in time.
Except for this last restriction, it represents, at
a slightly more abstract level, precisely the properties associated with the solutions of ordinary differential equations of autonomous type.
Associated with the
dynamical system we have
Definition 1.2. defined by
The positive orbit
0+ (cp) '"
U
O+(cp)
through
cp
is the set of elements in ~
u(t,Cp).
t>O Let us give some examples of dynamical systems.
Example 1.
Ordinary differential equations.
Consider the equation
65
where
n f: R ~Rn, is continuous for every
U(0,5)
= 5 exists for all t
~
Example 2.
t,~ > 0.
the solution
in
0, is unique and depends continuously upon
Uniqueness of the solution implies fore for all
(1.1)
f(x),
x
Clearly u
U(t+~,5)
=
u(t,U(~,5))
for all
is a dynamical system on
t,5'
t,~
and there_
Let
C
n R .
Functional differential equations of the retarded type.
n R
n c([_r,OJ,R ), r ~ 0, be the space of continuous functions from
[-r,OJ
the uniform convergence topology.
For any continuous function
x
let
c defined by
[-r,s), s >
°
and any
°< t
xt(e) = x(t+e), -r < e < 0.
<s
be the function in
to
defined on
Then the functional differential equation
(1.2)
x(t)
with
f: C ~Rn
continuous and locally lipschitz will have a solution
defined and continuous on ~
in
C.
With
u(t,~)
=
with
[-r,s), s > Xt(~),
°
and
X =
o
~,
x
=
x(~)(t)
the initial value for every
since local existence, uniqueness and continuous
dependence is easily proved [12J then
u
is a dynamical system
on C.
It should be noted that this functional differential equation (simplest example
x(t)
=
ax(t-l)) only defines solutions forward in time, hence the dynamical
system definition.
Example 3. n c([_r,O),R )
Functional differential equations of the neutral type.
Let
C
=
with the same norm as above and, with the same notation consider the
equation
(1.3 )
where
D is a difference operator defined by
66
N
~
where
~
are
n Xn
~(O)
~~(-~k) k=l
(1.4 )
- I
°<
constant matrices and
~k ~
r
~./~ J n
with
rational.
This is a special case of a functional differential equation of the neutral type (for example, x(t) + dX(t-l) + bx(t) + cx(t-l) = continuous and locally lipschitzian in initial value and
~
~
in
and is unique [10].
is such an equation).
=
x(~)
exists, is continuous in
If solutions exist for all
fines a dYnamical system on
C.
If
f
is
C then it is possible to show that with any u(t,~)
C the solution
°
t
~
0, then
u(t,~)
t de-
Note again that the solution can be defined only
forward in time.
Example 4.
Parabolic partial differential equations.
heat equation, with boundary and initial
u
Consider the space [o,rr]
with
t
= u , O<xO xx u(rr,t) = 0, t >
u(x,o)
X of functions
it is well-known that
co~ditions
u(o,t)
In this case, consider the
°
~:
[O,rr]
and with norm
~R
11
u(t,
1
°
(1.5)
continuously differentiable on = sup {1
exists for all
°
is unique and
depends continuously on
t,
in the Banach space
Note that, once again, the solutions are not defined back-
X.
X.
t >
Then
Hence, we have a dynamical system
ward in time - as is well-known the backward solution will not be in
Example 5.
X.
Consider the equation
Vtt = Vxx + f(v,vt,v x )
° ~ x ~ 1,
v(O,x) = ~(x),Vt(O,x)
Hx)
v(t,o) = 0,
v(t,l) = 0,
t >
t >
° (1.6)
°
67
where
f
is analytic in its variables in the whole space.
~ the space of
Let
functions with all generalized derivatives of order less than an equal to integrable in
IIcpl12k
with norm
[0,1]
=
W
and any
~
X
in
~
_lc-l
W2
2 + ... + (cp (k)) ]dx, where
Then [19], i t is known that (1.6)
is thej th generalized derivative of cpo
has a unique generalized solution
2
square
0
2
cp(j)
/[cp2 + (cpl)
k
v(t,x,cp,~) on -~ ~ t ~ ~
for every cp
and that the pair
in
~
belongs to
~-l and is continuous in t,cp,~. Hence, if it is assumed that such a solu-
tion exists for all dynamical system on
t ~ 0, then u(t,~) = [(v(t,x; cp,~), vt(t,x; cp,~)], is a _lc k-l W2 X W2 for any k > 1.
The purpose of these examples has been to illustrate the generality of the concept of dynamical systems.
We shall return to some specific applications of a
physical nature later.
2.
Same Stability Theorems Let us now state, for our general dyQamical system, the fundamental
theorems which we wish to exploit for the determination of stability results.
For
this purpose, let
Definition 2.1. If
u(t,~)
=
~
Let a dynamical system for all
t
~
0, then
u(t,cp)
~
~.
be defined in the Banach space
is an equilibrium solution of the dynamical
system.
The equilibrium solution cp = 0
Definition 2.2. every all
E
> 0 there exists a
t > O.
aCe)
The equilibrium cp
there exists a
y
such that
such that
=0
of
II~II ~
u(t,cp)
is stable, if for
a implies
Ilu(t,~) II ~
E
for
is asymptotically stable if it is stable and
II~II < y
implies
lim
u(t,~) ..... 0
(in the norm in ~).
t ..... ""
Definition 2.3. system
u
A set
if for each
M in
~
¢
M, O+(¢) eM.
in
is a positively invariant set of the dynamical It is invariant if for each
¢
in M
68
there exists a function
U(s,¢), U(O,¢)
u(t,U(s,¢)) = U(t+s,¢)
such that
=
¢
for all
defined and in
M for
-~
< s <
~
and
t > O.
Definitions 2.1 and 2.2 are the natural generalization of the familiar ones. The first part of Definition 2.3 is well-known; the second part of the definition simply uses the device of extending the dynamical system backward, if possible, since the dynamical system is not defined backward. exist only for those
¢
in
~,
function On
If
u
U must
M.
Let us nOw define, in the manner of [9,
Definition 2.4.
Note that the function
l~
~
is a dynamical system on
and
V is a continuous scalar
define
IIm
1 fv(u(t,¢)) _ v(¢)].
t ~0 t
V is said to be a Liapunov functional on a set
G
and if
and let
v(¢) ~ 0
for every
¢
in
G.
M be the largest invariant set in
G in
~
is
S = (¢
Furthermore, let S
V is continuous on in Glv(¢) = O}
for the dynamical system
u.
Then it is possible to prove [9 ]
Theorem 2.1.
Suppose
tionalon
and the orbit
G
Furthermore, if
o+(cp)
u
~.
is a dynamical system on
O+(cp)
belongs to
G
then
belongs to a compact set of
~
If
V is a Liapunov func-
u(t,cp)~S as then
t~~.
u(t,cp) ~M, and M
is nonempty, compact and invariant. This is one of the mOst general stability theorems available. first of all, we always require the orbit to remain in
G;
Note that
secondly, that compact-
ness of the orbit allows much more to be said about the set of points approached if S
contains mOre than one element. In the next examples, we attempt to illustrate the application of this
general theorem. (i)
Note that the elements needed are:
a dynamical system
69
(ii)
a set
GC
~
(iii)
a Liapunov functional On
(iv)
compactness of the orbits
G and, finally, perhaps
3. A Problem of Nonexistence of Oscillations Consider the network shown in Figure 1. tween
o
and
1
is a lossless transmission line with specific capacitance
o
and specific inductance current
i
The
and the voltage
this line are functions of t
In this circuit the section be-
and satisfy the equations
v ~
I
of and
-=- E Figure 1
o<
~
< 1,
t
> O.
(3. 1 )
-C s
The circuits at the ends of the line give rise to the boundary conditions
(3.2)
where
vO(t)
function
f
=
v(O,t), vl(t) = v(l,t), iO(t) = i(O,t)
and
il(t) = i(l,t).
The
which renders the problem nonlinear is pictured in Figure 2 and re-
presents the general characteristic on an Esaki diode. There has been considerable recent interest in circuits of this type, generally called flip-flops, particularly regarding the existence and nonexistence of oscillations.
Moser [16], Brayton [ 2] and Brayton and Miranker [3 ] have COn-
sidered increasingly sophisticated mathematical mOdels for the study of such
70
circuits, from lumped models to the present one.
The equilibrium states of (3.1),
(3.2) are given by
(3.3 )
and, as illustrated in Figure 2, we shall consider only the case of a unique equilibrium point, say
(v*,i*).
Translating
the equilibrium state to the origin and denoting the new variables by the same
v
notation yields Figure 2
L
?N
2li
s
"dt= -d[ ?N
2li
-c s dt= ~ with
g(v l )
=
° = Vo + ROiO' dV
(3.4 ) l
c -+ g(v l ) dt
il ,
f(vl+v*) - f(v*), which is assumed continuously differentiable and
globally lipschitzian. The behavior of the solutions of (3.4) is far from obvious.
What is de-
sired is to determine conditions On the parameters that guarantee the global asymptotic stability of the solution; because of the nature of the circuit, the lossless transmission line, it is suspected that periodic oscillations are possible. To study this problem with some mathematical care it is necessary to have an existence theorem which suggests the appropriate space in which the problem should be viewed; for this purpose it is fairly simple to prove [17]:
Theorem 3.1.
For the system (3.7), let the initial conditions
v(~,O) = ~(~) belong to Cl[O,l]
i(~,O) = i(~)
and satisfy the consistency conditions
and
71
(i)
0
(ii)
0
-~(O) - ROf(O) (0) + ROCS~I (0),
= Lsi>
g..- i, (1)
(iii)
= -5:(1) + f(;(l)),
s
then there exists a unique solution
v(~,t), i(~,t)
in
1
1
C [O,lJ X C [0,00).
Further-
more, this solution has the representation
with
cr
=~
i(~,t)
=
L
1
=
[¢(~-crt) + ~(~+crt)J,
~ [¢(~-crt) 2z
-
(3.5)
~(~+crt)J,
1/2
(2.) C
z
(L C )1/2 ,
1
v(~,t)
s
s s
This theorem yields a representation for the solutions which is very suggestiv~
through the use of this representation it is possible to reduce this problem
to a more tractable One.
Indeed, introducing
(3.5)
into
(3.4),
the wave equation is
automatically satisfied and the boundary conditions become
(3.6)
Eliminating
i
l
and
~l
then yields the neutral functional differential equation
(3.7)
2
r = cr
where A
data
and
R - z _ R + z O
'T;"0~
It is also simple to see that the given initial
A
i(~), v(~)
for (1.7).
k = in
Cl[O,lJ
completely determines the initial data
Furthermore, it is not difficult to see that since
Ikl
< 1
v
l
€
if
1
C [-r,OJ
72
lim
vl(t)
= 0, then
t~co
lim
i(s, t)
=
a
and
t~co
lim v(s, t)
a
unifOrmly in
and
t~co
that therefore oscillations will not exist. The problem has then been reduced to the determination of conditions for the global asymptotic stability (3.7), which is rewritten for convenience of later computations as
(3.8)
where
~
=
~(O)
+ ~(-r), xt(e) = x(t+e)
with
-r
< e < O.
Cruz and Hale [10]
have developed existence, uniqueness and continuous dependence results for this type of neutral functional differential equation. Indeed, it should be noted that this is a functional differential equation of the neutral type of the type described in Example 3.
Within this context and con-
sidering the application of the first part of Theorem 2.1 leads to
Theorem 3.2. on
G = Gp
=
If the
{~
E
D operator is a stable one and V is a Liapunov functional
C: V(~) < p}.
Then, i f
v(~):::: -m(I~I) ::::
a
with
m(s) >
a
S > 0, with m continuous, then every solution of (1.3) approaches zero as t
for ~ co.
The result is precisely the one expected as a generalization of the usual theorems for ordinary differential equations.
NOW, through the use of this theorem
it is not too difficult to obtain same stability results for our problem. it is possible to prove [17].
Theorem 3.3.
If
g
satisfies the sector criterion
sup (1 and
(g~(1)) < (~) ~
+ inf 0
(g~(1)),
Indeed,
73
then the equilibrium solution
°
vi
of Equation (3.8) is globally asymptotically
uniformly stable. The proof of this theorem is straightforward, although the detailed compuIn essence, the Liapunov functional
tations are involved.
a
J°qJ 2 (e)de
=
1
~ -~[D:p]
2
,
~
> 0, are determined.
2
2" [D:p]
is used and conditions for the existence of a nonnegative
-r
. V(t,qJ)
V(qJ)
a
+
such that
These conditions yield the sector criterion
quoted in the theorem. From what has been said above, these sector criteria naturally also imply the nonexistence of oscillations in the original problem. that these criteria are sharp in the following sense.
-ru,
that is, g(a) = -y
2: -
__ k j ) z1 (1+ l'+"fKr 1
oscillations.
It is of interest to note
If the problem is linear,
then it is a simple exercise to determine that the condition
is a necessary and sufficient condition for the non-existence of
But in the linear case, this is precisely the condition given by
Theorem 3.3, which implies that a type of Aizerman conjecture is valid for this problem.
4.
A Bifurcation Problem A number of applications, especially those arising from chemical reactor
stability problems [ 1] give rise to a problem of the following nature.
Consider
the partial differential equation
U
xx
+ A.f ( u) ,
A.
2:
0,
° ~ x ~ 7r,
t
>
°
(4.1)
which satisfies the boundary and initial conditions
u(o,t)
u(7r,t)
0,
t
2:
0,
(4.2)
u(x,o)
where
f
¢(x),
o<x<7r
is a given function defined on the real line, f(O)
0, uf(u) >
°
for
74
u
I
0
and
f(u)u-
odd and sgn f" (u)
l
~ 0 as
=
-sgn u.
solution of this problem.
Assume for simplicity that With the given hypotheses For
A= 0
u == 0
smooth,
is an equilibrium
it is well known that this solution of the
heat equation is stable in any usual meaning of the word, and the qualitative behavior of the solutions of (4.1), (4.2) is clear. determine how this picture changes as u == 0
the equilibrium solution
What is of interest here is to
A is allowed to increase from zero value; if
loses its property of stability, do there appear
any new equilibrium solutions which inherit this property?
This problem has been
investigated by Matkowsky [15J using formal asymptotic methods under hypothesis differing somewhat from these given here.
The viewpoint here is to interpret (4.1),
(4.2) as a dynamical system in an appropriate Banach space and to apply Liapunov methods of the type developed in [9, 11,14J. the sake of the brevity of exposition.
Again, the details are omitted for
This specific application is more fully
described in [5 J . The first task here is to show that (4.1) - (4.2) defines a dynamical system. ~:
As a first step in this direction, consider the Banach space
[O,~J ~R
norm
continuously differentiable on
1I~lll = sup (I~' (x)l: 0::: x::: ~}.
sup (I ~(x) I: 0 ::: x :::~}
!1~11 1
and
II 11
0
norm.
For any
u(x,t;
Furthermore, if U(¢,A)(t)
B (r)
o
=
~(~)
= 0 and with
Define also the norms !I~IIO = 2 1/2 (f ¢\ (x) dx) , and note that !1~110:::
=
0
be open balls centered at zero with radius
r
in the
Then it is possible to prove [ 5 J:
Theorem 4.1. solutions
Let
~(O)
with
~
W 2
.[~ II ~II 1::: ~II ~111' W2
[O,~J
X of functions
~,A)
~
€
X and
A
denoted by
U(¢,A)(t)
defined for all
E
~
BO(r) E
furthermore, the positive orbit
E
[0,00), Equations (4.1), (4.2) have unique
U(¢,A)(t)
E
for some
r
X defined for then
S(~,A)
X is a dynamical system in
O+(~,A)
of
U(¢,A)(t)
0 < t < S(¢,A) ::: 00.
= 00, the map
X with
II III
(t,~) ~
and
is relatively compact in
this space. Note that, except for the hypothesis that the orbits are bounded in the
75
110
II
norm, the theorem states that we are dealing with a dynamical system; further-
more, that the dynamical system is self-compactifying.
This last property is pre-
cisely the expected result, given the smoothing properties of the heat equation which, this theorem states, are not affected by the nonlinearity. Let us now define for every ¢(x)
7T
J {~
°
¢ (x)2 - A.
relative to
J
°
II III
ficult to see that
A.
f(~)d~}dx for ¢ II II 1 W2
and
VA.(¢) ~oo
€
€
[0,00) X.
the Liapunov functional
Note that
V,
,.
VA.(¢)
is continuous on
X
and that, given the assumptions i t is not too difas
11¢ll
o ~oo.
Furthermore, it is of interest to see
that
These ob-
servations lead to
Theorem 4.2.
For any
system in
normed by
X
¢
€
X and
II Ill.
A.
€
[0,00) the map t,¢
~u(¢,A.)(t)
Furthermore, the positive orbit
is a dynamical
O+(¢,A.)
is
relatively compact in this space. Note that the use of the Liapunov functional was essential in proving global existence.
But now, since the Liapunov function has already been constructed
it is possible to conclude much more. Indeed, all of the conditions for the entire Theorem 2.1 are satisfied. Note that the largest invariant set solutions.
Hence
Theorem 4.3. the norm
M within our context is the set of equilibrium
Every solution of (4.1) - (4.2) approaches an equilibrium solution in
II Ill' Actually, much more can be said about the qualitative picture by analyzing
the equilibrium solutions, which are the solutions of the two point boundary value problem
u" (x) + H(u(x))
0,
u(O)
0,
° < A. < 00.
76
Using methods inspired by the work of Urabe [20J it is possible to prove
Theorem 4.4.
Let
)..
n = 1,2, . . . .
n
Then, for any
)..
E [)..
n
,00)
Equation
+
u~()..)
(4.3) has two solutions
BO(r ) O
E
with the properties that
+
o
u-().. ) n n
(i)
+
(ii)
u-()..) n
have exactly
(iii)
+ u-()..) n +
varies continuously in
)..
E
zeros in
[O,rrJ
relative to the norm
II
[0,00) , (4.1) have no equilibrium points in
X
Ilu~()..)lll ~oo
Furthermore, for any
n + 1
as
)..
III
with
)..~OO.
other than
+
the origin
u
o
0
and those elements
u-()..) , n n
It is then quite clear that i f
sponding solution norm
II Ill'
rium point
u(¢,)..)(t)
~O
as
).. ::; )..1 ~
t
The question arises, given u(¢,)..)(t)
will converge.
1, for which
)..
then for every
¢
~
n < )... E
X
the corre-
00, the convergence naturally being in the
¢
X and
E
)..
E
()..l'oo)
to which equilib-
Again, it is possible to answer, at least
partially, this query by an appropriate analysis of the Liapunov functional.
Indeed,
we have +
Theorem 4.5.
For each integer
n
~
u~()..),
1, let
)..n < ).. < 00 be as in Theorem 4.4. +
Then for any
the origin
is asymptotically stable and for assertions are valid in
).. E
X normed by
u
o
= 0
[).. ,00), n
II
is unstable. n
+ 2, u- ()..) n
>
For any
).. E
is unstable.
[)..l'oo), u~()..)
(All these
Ill).
These five theorems give a rather clear picture of the qualitative behavior + of the solutions. All solutions will, in general, approach either u or u~()..)
o
5.
The General Problem of Thermoelasticity In the previous problem it was possible to find a Banach space in which the
dynamical system was self-compactifying.
It was this property that was heavily
77
exploited and which is essential in the application of invariance principles.
It is
to be suspected that such self-compactifying properties can be expected in dynamical systems which arise from functional differential equations of the retarded type and partial differential equations of parabolic nature.
For hyperbolic partial differ-
ential equations clearly this property would be very surprising.
The example
presented now is of hyperbolic nature, yet it is possible, through a little more work, to still apply the principle. Elastic stability is usually discussed from strictly mechanical considerations; here the concern is with thermodynamic properties of elastic materials.
More
specifically, one may ask what effects the second law of thermodynamics has on the asymptotic stability of equilibrium of otherwise non_dissipative materials [7J. x = (x ,x2 'x ) l 3
A material point is identified by
= yO)'
rium (no stresses, constant temperature t
following an initial disturbance at time
temperature deviation by rium state. regular [ 8
The displacement field at same time
=0
is given by
denotes the density at
u(x,t) x
and the
in the equilib-
n be an open, bounded, connected set in E 3 which is properly
Let
J;
T(x,t); p(x)
t
in its state of equilib-
on
let
denote the boundary of
n.
The constitutive equations of
thermoelasticity are taken then in the form
(c. 'k n )
(5.1)
(m .. T) .,
• -
lJ .., ,J
lJ, J
lJ ,J ,l
where body forces and heat sources have been excluded. Cijk £
Cj ik£
=
Ck£ij' mij
=
.. mj i' K lJ
assumed to be smooth functions of Let now value problem in
to > O.
(5. 2 )
(K .. T .) .;
PGOT + m.lJ'You l, . .J
K..
Jl
In these equations
and
and
K .. lJ
are
x.
By a classical solution of the mixed initial-boundary
n X (O,tO) we mean a pair
(u,T)
satisfying equation (5.1) and
(5.2) together with the boundary conditions
u
o
on
d1 X (0, to)
(clamped boundary),
(5.3 )
78
T= 0
en
on
(O,t ) O
X
(constant temperature);
(5.4)
(u O(x), 110 (x), TO(X))'
(5.5 )
and with initial conditions
(u(x,O),u(x,O),T(X,O))
where
uO(x), uO(x)
and
TO(X)
are given functions On
n.
The generalized solutions of the mixed initial boundary value problem described above can be viewed on an appropriate Banach space as a dynamical system. Once this is done, the application of Theorem 2.1 permits us to draw immediate conClusions On the asymptotic behavior of the solutions of our problem. Consider the Sobolev spaces
(k) W (n) 2
and
(k) ~ W (n), k - 1,2, ... 20
Assume
that
ess inf p(x) > 0, ess inf CD(x) > 0,
(5.6)
K.. ~.~. >
(5.7)
lJ 1 J -
Cl~'~"
1 1
Cl > 0
(the second law of thermodynamics requires
K..
lJ
constant,
positive semidefinite at
we make the stronger assumption of positive definiteness).
J
,j
2 I (v.,w.,R)!O = 1
1
w~) (n).
C. ·kp,v . . vk p,dx::: C2 J v . . v . . dx, lJ 1,J , n 1,J 1,J
J
n
C2 > 0
Also for all
constant
[pw.w. + 1
1
Define the map
tion of the system
nJ
C.. knu k lJ)j
ne
.
. dx =
,)j 1, J
-Jn
[pw.e. - m. . Te . . ]dx 1 1 lJ 1, J
x
E
n;
v.
E
w(l)(n) 20
1
(5. 8 )
79
for every
D,B
i
E
w~~)(n).
The mapping
P
is linear, well-defined on
Ho(n)
m
and One to One. the map
0/
E
P. m
Hm(n)
Hence, defining
Lemma 5.1,
l pm
It is clear that Io/I m =
and define
H
m
Pm
= po pO ... P let H!n (n) denote the range of 0
exists and maps
I p~lo/Io'
H (n) m
onto
HO(n).
Let
Then [6 ],
is a Banach space with norm
algebraically and topologically.
I:
and the imbedding
m
Hm(n) ~Hf',(n)
m > f',
for
Furthermore, H (n) is compact.
Let us now define appropriately a generalized solution of our problem:
Definition 5.1,
will be called a generalized solution of (5.1) - (5.5)
(u. U. T) l,
l,
if for all smooth test functions nand
v. l
vanishing on
(v. R) l,
with compact support on
n X0
to
J nJ
o
((t-toHpu.v. - C. 'knuk /'" , + m.. TV .. + l l lJ '" ,'" l, J lJ l, J
PC D . CD TR + m.. u . . R] + pul,v l' + P - TR + YO lJ l, J YO
(5.9)
+ -
1 + m.. u . . R - -
lJ l, J
YO
t
J 0
(K .. R .) .T dt}dxdt lJ, l ,J
With this definition i t follows that [6]:
Theorem 5.1.
Under assumptions (5.1) - (5.3) the triple
dynamical system on
Hm(n), m
= 0,1,2, ... , where
(u.~. T)
(u,~. T) l,
l,
l,
l,
describes a
is the generalized
solution to the equations of linear thermoelasticity satisfying equation (5.9). Furthermore, for
t
in
(O,t ) O
80
(5.10)
where
T(m) (x,t)
denotes the generalized mth derivative in time of
T(x,t).
The problem of termoelastic stability has now been put in a setting appropriate for the application of Theorem 2.1 which allows us to obtain stability results in a simple and direct manner. For the trajectory
. in
(u.,u.,T) l l
It follows from the definition of the map On
Hm+l(D)
with initial data
P
that
pO(uo.,UO.,TO) l
define
Hm (D)
pO (u.,u.,T) == (U.,u.,'T). l
(u.,u.,'T) l
l
l
is a dynamical system
l
Hm+l(D)
in
l
satisfying (5.9) and
l
Theorem 5.1.
Therefore, Theorem 2.1 and (5.10) imply that for any initial data
(uO.,uO.,T ) O
in
l
Hm(D)
Hm(D)
the trajectory
(ui,ui,'T)(t)
for all
compact set met with
~
t > O.
G of =
Hence by Lemma 5.1 the trajectory ~
H£(D), £
H£ (D) .
m.
For simplicity let
£ = 1
is then
S = ((U., u., T) l
largest invariant set in
Theorem 5.2.
l
E
I (ui , ~i''T) I ~.
From (5.7) 2 -1 J K.. T(l) T(l) dx _< -c31 (0,0,T)1 . 1 "(.0" lJ . .
V =
and
"
S
(ui,ui,T) will lie in a
But then all the hypotheses of Theorem 2.1 are
and (5.10) it immediately follows that The set
will lie in abounded set of
l
V =
',l,J
Hl(D) IT = O}.
The determination of
M, the
S, which is not trivial, then leads to [18J:
For any initial data
(uo"~O.,TO) l
in
l
H (D), m> 1, and under m
assumptions (5.6) - (5.7), (u.,~.,T)(t) approaches the set l l
M = ((w.,w.,Y) l l
in
t
H (D) 1m .. w. . O lJ l , J
0, Y = 0,
-to ~ PWoiVilt=Odx vanishing on
DX 0
J 0DJ (( t - t o )[ pw.l v.l -
0
for all
vi
C.. k nw nV. . J + pW. v.} dxdt lJ h k , h l , J l l
test functions with compact support On
in the norm of the space
Ho(D)
as
t
~
=
D and
00.
It is of interest to note that in this case there is an infinity of solutions in the set
M and that the use of the Liapunov functional allows a very nice
characterization of them; they are the isothermal oscillations of the body, representing pure shear stresses.
It should be noted that to obtain the needed
campactification it is necessary for the problem to represent a dynamical system in
81
two Banach spaces, here, for example, H l H completely continuous. O
and
The boundedness of the trajectories in
that the trajectory is in a compact set in theorem.
H with the imbedding of O
into
then imply
H and allows the application of the O
In this problem, which is linear, the generation of the
quite natural, they are velocity spaces.
H l
H l
H n
spaces is
For nonlinear problems, unfortunately,
this is far from easy.
6.
Summary In this brief lecture an attempt has been made to indicate the power and
difficulties of application of Liapunov stability theory, with emphasis on the invariance principle.
Looking back OVer the three examples, it is quite clear that
the construction of the Liapunov functional is, in general, necessary to obtain the boundedness results required by a dynamical system.
Once this functional is known,
then if its derivative is negative definite in an appropriate domain, then only one equilibrium point will be stable.
If the derivative is negative semidefinite, but
the trajectory lies in a compact orbit, then the invariant subset of the set will be the set approached by the solutions.
V
~
a
In the second example, the equations
of motion were self-compactifying - in the last one they were not and one had to give initial conditions in a subspace which had the property that bounded set in it are compact in the larger space.
82
REFERENCES
[1]
Admvuson, N. R. and L. R. Raymond; AICHE J., 11, 339-362, (1965).
[2]
Brayton, R. K.; Quarterly Appl. Math., 24, (1966).
[3]
Brayton, R. K. and W. L. Miranker; Arch. Rat. Mech. and Anal., (1964).
[4]
Brockett, R. W.; IEEE Tr. Aut. Cont., 11, 596-606, (1966).
[5]
Chafee, N. and E. F. Infante; Applicable Math., to appear.
[6]
Dafermos, C. M.; Arch. Rat. Mech. and Anal., 29, 241-271, (1968).
[7]
Eriksen, J. L.; Int. J. Solids and Structures,
[8]
Fichera, G.; Lectures on Elliptic Boundary Differential Systems and Eigenvalue
~
~
61-73,
573-580, (1966).
Problems, Springer-Verlag, 1965, p. 21. [9]
Hale, J. K.; J. Math. Anal. and Appl., 26, 39-59, (1969).
L
[10]
Hale, J. K. and M. Cruz; J. Diff. Eqns.,
334-355, (1970).
[11]
Hale, J. K. and E. F. Infante; Proc. Nat. Acad. Sci.,
[12]
Hale, J. K. and C. Imaz; Bul. Soc. Mat. Mex., 29-37, (1967).
[13]
Holtzman, J. M.; Nonlinear System Theory, Prentice-Hall, (1970).
[14]
LaSalle, J. P.; Int. Symp. Diff. Eqns. and Dym. Systems, Academic Press,
2..S
405-409, (1967).
1967, p. 277.
J.;
Matkowsky, B.
[16]
Moser,
[17]
Slemrod, M.; J. Math. Anal. and Appl., to appear.
[18]
Slemrod, M. and E. F. Infante; Proc. IUTAM Symp. on Inst. Cont. Systems,
J.;
Bull. A. M.
s.,
[15]
76, 620-625, (1970).
Quarterly Appl. Math., 25, 1-9, (1967).
Springer-verlag, to appear. [19]
Sobolev, S. L.; Appl. of Funct. Anal. in Mat. Physics, Trans. Mat. Monographs, A. M.
[20]
s.,
(1969).
urabe, M.; Army Math. Res. Center T. S. R. #437, (1963).
STABILITY OF DISSIPATIVE SYSTEMS WITH APPLICATIONS TO FLUIDS AND MAGNETO FLUIDS E.M. Barston Courant Institute of Mathematical Sciences New York University, New York, New York Abstract An energy principle is presented which gives necessary and sufficient conditions for exponential stability for a useful class of continuous linear dissipative systems.
The maximal growth rate
~
of an
unstable system is shown to be the least upper bound of a certain tional, providing a variational expression
for~.
fun~
Applications to the
problems of the stability of a stratified viscous incompressible fluid in a gravitational field and the
resistiv~viscous,
incompressible
magnetohydrodynamic sheet pinch are dicussed. Introduction
I.
In attempting to determine the stability characteristics of a given (usually nonlinear) physical system, one is often led stability of a derived (approximate)
linear system.
to consider the Perhaps it is
known that the stability or instability of the original problem can in fact be inferred from the results obtained for the linearized problem; even if this information is not available, the lack of a general systematic method for the construction of Lyapunov functions often leaves one with no alternative, and so one proceeds with a study of the stability of the linear system, at least as a preliminary step in the solution of the problem. Unfortunately, the solution of the derived linear problem itself is often formidable, even for autonomous systems, when the dimension is sufficiently large.
This is particularly true for continuous systems
where the linearized equations contain partial differential operators with spatially varying coefficients. f~r
Perhaps the best one can hope
in such cases is the existence of an energy principle which gives
necessary and sufficient conditions for (exponential) stability.
The
84
existence of such an energy principle for determining the linear stability of the equilibrium states of a conservative dynamical system is well-known, and has been the cornerstone of almost every investigation of the stability of non-trivial equilibria in perfectly conducting, invicid, magneto-hydrodynamics
[5), [6).
In 1903, Kelvin and Tate[8)
proposed an extension of the energy principle to a class of real, finite-dimensional, dissipative linear systems
(Kelvin and Tate did not
prove their assertion; a proof using Lyapunov methods can be found in Ref.
[7).
In recent years, the energy principle has been extended to
a general class of continuous linear dissipative systems, and in the process, a maximum principle for the maximal growth rate of an unstable system has been obtained [1), [3).
We shall briefly discuss these
developments and some applications in this paper. discussion and further applications references
For a more complete
[1)-[4) should be con-
sul ted. We shall begin with a discussion of the problem of the gravitational stability of a stratified viscous incompressible fluid, which will serve to motivate as well as illustrate the theory.
After developing
the energy and maximum principles, we briefly dicuss the application of these results to the problem of the stability of the resistive, viscous, incompressible magneto-hydrodynamic sheet pinch. II.
Equations for a Viscous Incompressible Fluid in a Gravitational Field
Perhaps the most familiar example of a continuous dissipative system of the type we shall analyze is the problem of the gravitational ity of a stratified, viscous, incompressible fluid.
stabili-
Let us then con-
sider such a fluid occupying a bounded region U (a simply connected open set) with surface dU, satisfying the following set of equations in U;
-+
V-v
~ + dt
=
V- ~p
(2.1)
0 :=
0
(2.2)
85
f d~
~
~1
PLat + (v'V)v
= -
J
2~
-+
;Z;
vp - pg e
+ vV v
z
(2.3)
-+ -+
-+
The quantity p(x,t) denotes the mass density, v(x,t) the fluid veloc-+
ity, p(x,t) the scalar pressure, v the viscosity (a positive constant), -+
g the gravitational acceleration, and e recti on (assumed vertical).
z
the unit vector in the z-di-
The equilibrium values of the fluid vari-+
ables, denoted by the subscript a, are given as follows: v
a
=Po(z) >
on [zl,z2]' Po
E
C
l
[Zl,Z2]' where zl Z
o
~o
(z)
is given by po(z)
=
J
g
=i nf z, x€.U
- OJ P
o
0
z2 = ~up x€.U
=
and
Zi
P (u) du +c onst He linearize Eqs. (2J.)-C2.3) o
zl
-+
about the equilibrium state (in the sequel, the variables v, p, and P without the subscript 0 will denote linearized quantities) and obtain, ++ after introducing the (linear) displacement vector ~(x,t) -+ -+
+ Ux,O) where
-+ -+
V'~
(x,O)
= a
-+
= -
and P (x,O) -+
-+
t J-+-+ v(x,T)dT
o
-+-+
Vp ·Ux,O), o
= a
V'~
(2 .4)
a •
(2.5)
We take dU to be a rigid surface, so that the appropriate boundary con••
dit10n 1S that
-+ ~
vanish on dUo
We assume, of course, that all quanti-
ties are sufficiently smooth so that the indicated operations are
.
defined; in particular, we consider the class of solutions Eq.
(2.5) such that
each t
~
-+ ~
and
-+ ~
•
are both 1n the class D and
-+ -+
~(x,t)
-+
~
-
E.
C(S""2)
-+
-+
0, where D is defined as the set of all functions f(x) -+
the properties that V·f
= a
-+
= a
in U, f
wel~
of for
with
-+
on dU, and f is twice contin-
It is easy to see that the operators P,K, dPo -+ 2-+ -+ -+ -+ vV ~, and H~ - - gd'Z~z e z are Po ~, K~ -
uously differentiable on U. and H defined by
-+
P~
-
-
-+
-+
formally self adjoint on D with respect to the inner product (f,g) ==
=~-+fT. ·gd 3 x -+
-+
-+
(f* denotes the complex conjugate of £) and that P and K -+
are positive. Eq.
(2.5).
We note that (Vp,
-+
~)
-+
=
a
The Energy and Maximum principles
-+
for our solutions ~ of
This follows from the divergence theorem, since
and! vanishes on dUo III.
•
-+
(Vp,O =
-+
V'~ ==
a
86
The preceeding problem is a special case of the more general system
..
p~
.
+ K; + H;(t) + F;
= a ,
t
>
a
( 3.1)
where ;,t,~ and F; are elements of an inner product space E for each fixed t
~
0; P,K, and H are time-independent linear formally self-ad-
joint operators from E into E with domains of definition Dp,D , K
respectively;
P
>
a
on Dp and K
Golution ;(t) of Eq. t > O.
a
~
and~,
on D ; and F;, defined for each K
•
(3.1), has the property that (F;rt;;) =
=
(F;,;)
so~
In the sequel, we restrict our attention to the class S of
utions ; (t) of Eq.
(3.1)
0,
satisfying the following ten conditions: ~ (t)
D -
€
t (t) o.
I,t
P; +
€.
+
n DK nDH D p nD ' K
t
>
a
(3 .2)
t
>
a
(3.3)
t
>
a
(3.4)
a ,
t
>
a
(3.1)
Dp
H; + F;
=
d~
(Lpt)
(i,pt)
+
(t,p~)
t
>
a
(3.5)
d~
(Lp;) =
(Lp~)
+ (t,pt)
t
>
a
(3.6)
t
>
a
(3.7)
t >
a
(3.8)
t
>
a
(3.9)
t
>
a
(3.10)
ddt (; , p ; )
(
d
Lp; )
+ (; , p
•
t) •
dt (;,K;) "" (;,K;) + (;,K;)
d~
(;,H;)
(F;,;)
=
(LH;)
=
(F;,t)
+
(H~,t)
a ,
=
,
The class S may be thought of as the class of suitably "smooth" SCJlutions
of Eq.
(3.1).
Equations
(3.5)-(3.9) are merely the usual rules
for differentiating inner products; Eqs. tions on the solutions of Eq.
(3.2)-(3.4) offer no restric-
(3.1) provided Dp::::lDK:::JD H , but become
additional "smoothness" requirements should the above relation not hold. The precise definition of the t-derivative
t
is not important in the
sequel, provided that the usual rules for differentiating sums and ducts
(of vectors and scalars) are valid.
pr~
Thus one can think of ~ as
87
being defined in the
norm-topology of E, or if E is an n-fold Cartes-
ian product of L 2 -spaces
(as is usually the case in applications),
t
can be taken to be the n-vector obtained by computing the partial der~(t).
ivative with respect to t of each of the n components of
~(t)
In addition to restricting the analysis to solutions
S, we
£
assume that H is bounded below on D and that inf (n, [wP+K)n) D (n, n)
>
0 for
In the circumstance that inf (,Hn~ < 0, we define D n,n {nine D, (n,Hn) < O}, require P > 0 on D, set
all w > O. D ==
for n E
15,
rl == sup Q ,
'"
n
D
y
==
{cp
I
that P1jJ
for each w
=
~(O)
0,
=
Sand 1jJ
£
ynO
"~,, = (~,~)1/2.
~(t)
of Eq.
The function
Dp
£
cp, t(O) = wcp+1jJ}, and assume that sup
The stability of the solutions in terms of
~ (t)
(o,m, there exists
E
Q
n
n DK
such
= sup Q = rl.
D
n
(3.1) will be discussed
~(t),
defined for t .:.. 0, is
said to be exponentially stable if for every E > 0, there exists a constant ME such that
II ~ (t)!I..::.
ME e
Et
for t
If
> O.
~ (t) is not ex-
ponentially stable, we say it is exponentially unstable. lution
~(t)
£
S is exponentially stable, the system
If every so-
(3.1) is called
exponentially stable. wi th the preceeding definitions and hypothesis, we have the following theorem: Theorem 1: (A)
> 0 Let inf (n,Hn) (n,n) •
Then for each
~(t)
S, there exists a con-
€
D
stant B such that "~(t)'1 < B for all t (B)
(C)
(n ,Hn) ;::: Let inf (n, n) o. D (n , Hn) Let inf (n,n) < o.
> O.
Then system (3.1)
is exponentially stable.
Then the system is exponentially unstable
D
with maximal growth rate rl, i.e., given any w
~ (t)
£
S and a positive constant M such that
£
(O,rl), there exists
II ~ (t) I 2.
M e
wt
for all
88
t
~
~
0, and given any
such that Proof:
II ~ {t)1I
Let
E
~(t)
~
d
S. •
dt {{~
Sand
(Q+E) t
e
< M
~
(t)
> 0, there exists a constant ME
€
, t >
o.
Then •
,pU
(L Kt ) - 2 (L Kt )
(F ~ , t )
-2 =
a ,
<
t
> 0,
so that
to -
Where ~o - ~(o),
~(o).
) Let t;, - inf (n(n, Hn , n) .
If t;, > 0, Eq.
(3.11)
D
gives t
Let w > 0, ~(t)
which proves (A). Then ~(t) ~ s{t)e
wt
, and a straightforward
.
w
(3.12)
t >
~
a ,
t
>
a
(3 .13)
-wt _ 2wP+K, H - w2 P+ wK+H, and f = F~ e so that (fs's) w s a for t > O. By analogy with the derivation of Eq. (3.11)
we have
Let t;,
=
inf (n,Hn) (n,n) -D
a•
_ inf (n,Hwn)
II ~ (t)
2
"
which holds for any w > pose that t;, < O. that Q >
o.
~
o.
> 0,
we conclude
(3.14) implies
• p.) + ( H r )] 1/2 ( so' So so'''o wt t;, e ,t > [
w
Thus statement (B) is verified.
a
Now sup-
-
6
is nonempty, and for each n ~ D, Q > 0, so n (a, m. Since sup Q = st, there exists ¢ ~ Y such
Then
Let w
i~f (n,[(~:~~n) .v
> 0, so that Eq.
(n, n)
D
Then since
ynts
n
~(t)
c S
that w < Q¢
2
~o ::: w¢+~,
::: t o -w~ 0
0,
and a where
P~ =
:::~, and Eq.
o.
o.
calculation yields
P, + Kws + Hws + f .,r where K
= ~{t)e-wt,
S, and set s{t)
€
a
>
such that Set
s{t)
(3.14) yields
~
o
::: ¢ ,
_ ~{t)e-wt .
89
The quadratic function g(a) tion of a for 0 < a <
therefore conclude from Eq.
Thus the growth rate ~(t)
inf D Let
~
(¢,H ¢) is a strictly increasing funca
and vanishes for a -- Q¢'• thus (¢,H w¢) < O.
00
I ~II
~
(3.15)
> 0
t
=
We
(3.15) that
II ~II e
(¢,H ¢)11/2 wt .::.- [ 11 w e
wt
J
> 0
t
•
can be approached arbitraily closely for some
S. Finally, suppose that ~ is finite and let E > O. Since [P+K]) . (11,H~+E11) 11,W n > 0 for w > 0, it follows that 11 = lnf >0. (11,11) ~+E D (11,11) €
(
~(t)
Eo
Sand
( 3 • 14 ) give s t
> 0
,
which completes the proof. The derivation of the energy principle given herein has the advantage of being free from any assumptions of completeness imposed on the eigenfunctions of the linear system; in fact, the results are valid for systems with no proper eigenfunctions.
This is important in appli·
cations to systems with a continuous spectrum.
We have basically made
the much weaker assumption that the system (3.1) admits smooth solutions for smooth initial data, and do not require the existence of any solutions of the form ~(t) = 11e~t, where 11 is independent of t. should be clear that in general,
~
It
will not lie in the discrete spec-
trum, i.e., the theorem only guarantees that the growth rate ~ can be approached arbitrarily closely, but does not imply that it can actually be achieved. IV.
Applications
The energy and maximum principles of Theorem 1 are applicable to any system satisfying an equation of the form (3.1) and the associated hypothesis imposed in Sec. III.
(It should be observed from the proof
of Theorem 1 that relatively little of that hypothesis is required to prove exponential stability once H
~
0 on D is known; the entire H
90
hypothesis was used, however, in the proof of the instability results and the maximum principle).
There are two approaches to the rigorous
application of the energy and maximum principles to a given problem. The first, and usually most difficult, requires an existence theorem guaranteeing the existence of the required smooth solutions for smooth initial data.
The second approach, applicable to unstable systems
where the maximal growth rate
lies in the discrete spectrum, is to
~
demonstrate the existence of an eigenvector n (independent of t) such ~(t) =
that
ne
~t
is a solution of (3.1).
the resistive sheet pinch [2].
This approach is valid for
It is to be expected, however, that in
most applications the investigator will simply assume that his system is well-behaved and that the energy and maximum principles apply.
If
the system is based on sound physical principles and the equilibrium data are sufficiently smooth, one would generally expect smooth solutions for smooth initial data. choice of the domain D :
Then the only problem remaining is the
DpnDKnDH'
A guiding principle here is to
take D to be the "maximal I' linear manifold satisfying the conditions that P,K, and H are all well-defined and formally self-adjoint on D (P is formally self-adjoint on D if and only if (n,ps) all n,s
€
n
Of course we require P
€
D.
=
(pn,s)
for
D) and that Pn, Kn, and Hn are reasonably smooth for all ~
0 and K > 0 on D.
Returning to the
problem discussed in Sec. II, we identify P with p , K with - v9 2, and o H with -g
apo
-+
-+
crz; (ez·)e z
ary condition
-+ ~
~
0 on -+ -+
vector functions f(x)
Due to the side condition (2.4) and the bound-
au,
we take D to be the linear manifold of all -+
such that V·f
~
-+
0 in U, f
~
0 on
au,
-+
f is twice
continuously differentiable in U, and the functions defined by the -+
first and second partials of f can be extended to continuous on the closure of U.
=-
g
f
dp
2 3
dp
nential stability.
=
-+
o < 0 on U, dz If, on thl1 other ham,
d:1fzl d x; thus if
U
For n
in U, then we can choose an n
~
f
€
au
so that they are
D, we have (n,Hn)
o
on D and we have expo-
> 0 on some open sphere
D such that (n,Hn) < 0, and we then
"conclude" that the system is exponentially unstable with the maximal
91
growth rate ~
=
s»p Qn'
The maxiMal growth rate ~ will of course de-
D
pend on the viscosity v, the mass density Po' and the domain U. The remainder of this section will be devoted to a brief discussion of the application of the energy principle to the resistive incompressible, magnetohydrodynamic sheet pinch.
viscous,
A detailed discus-
(For an application to the electrohydro-
sion can be found in [2].
dynamic Rayleigh-Taylor bulk instability [9], see
[~]).
We consider an infinite horizontal layer of an incompressible, viscous fluid satisfying the usual incompressible magnetohydrodynamic equations with a viscosity term added to the equation of motion, except for a simple "Ohm's Law" of the form If +
V
addition of a conservation equation %~ + V' (nV) n'
-+
=
-+
x
B
nJ and the
=
0 for the resistiviW
The equilibrium quantities are assumed to be functions of the
vertical coordinate z only, with the equilibriuD fluid velocity identically zero, and the equilibrium magnetic field Bo{z) horizontal. The boundaries of the fluid (located at z
=
0 and z = a) are assumed
The system equations require that
to be rigid, perfect insulators. -+
the equilibrium electric field Eo be constant and horizontal, while -+
Bo{Z) and no{z) are related by x -+ ez -+
z
Io
-1 no (u)du,
where Eo (a) is a constant horizontal magnetic field and meability of free space (mks units).
~o
is the per-
The system equations are linear-
ized about the equilibrium and the linearized variables are Fourier analyzed in the horizontal plane.
After a great leal of algebra, the
following 2 x 2 matrix equation is obtained, which determines the stability of the system: pE; + where E; nent
(E;l{Z't~ E;2{Z,t»)
Kt
+ lIE;
=
0 ,
(4. 1)
with E;l the Fourier coefficient of the z compo-
of the perturbed displacenent vector and E;2 the Fourier coeffi-
cient of the z component of the perturbed magnetic field; the 2 x 2 matrix operators P,K, and
if
have the forn
92
K
P
where L
L
and L
l
3
L ( 02
'=
+ Bl
00)
'
are second-order linear differential operators in z,
is a fourth-order differential operator, and B
2
and B
l
are 2
2
x 2
Hermitian matrix operators whose elements are continuous functions of z on [O,a]
(we assume that all equilibrium quanti ties are twice
continuously differentiable functions of z on [O,a]).
Consideration
of the boundary conditions and smoothness requirements leads us to
~l (z,t)
require that for each t -2 0, =f'(O) = f(a)
:=
f' (a) +kf (a)
0
:=
f'(a)
:=
O} and
= f '( 0) -kf (0)
~2(z,t)
}, where k Thus
horizontal wave number vector.
D l
EO
_ {f(z) If D
E.
2
_
c 4 [0 ,a],
E.
{f(z)~
c2
E
f(O)
[O,a],
denotes the magnitude of the we take D
D l
:=
x D , and find 2
that P,K, and H are all formally self-adjoint on D with P > 0 and K > O.
The energy principle is applicable
(here
F~ -+
sult is that unless no is a constant, the pinch (Eo
~
0), and the re-
~
0)
is always ex-
ponentially unstable (for sufficiently small k) . The theory of Sec. I I I leads us to expect that if the sheet pinch is unstable at the wave number k, then the maximal growth rate of perturbations with this wave number will be given by ~(k) =
S]P
On'
(The
maximal growth rate for arbitrary disturbances, i.e., disturbances of ~(k),
arbitrary wave number, would then be given by sup remum is over all k for which ~(k)
growth rate k)
We now show that the maximal
is actually achieved for the unstable
(at wave number
sheet pinch, i.e., we demonstrate the existence of a nonzero eigen-
~
vector
EO
number k).
i~f (7~~~~
B l ": 0 on E =J-.2[0,a] W
<
00,
< 0
satisfies Eq.
Let k > 0, and
(4.1).
(i.e., the system is unstable for wave
The operators L ;
<
~e~t
D such that
suppose that
o
> 0.)
~(k)
where the sup-
are strictly positive on D , and l ( ~, H w!;) xj.2[0,a], so that F(w) - igf (~,O l
and L
2
is strictly increasing on [0,00).
Now
~
= S;tp
On
> 0, and
D
we have F(W)
< 0 on
[O,~),
F(~)
> O.
compact Hermitian inverse K defined on
The operator L
t
2
3
has a positive
[0 ,a] such that
93
K{J [0,a])Cc[0,a], L K ~ I on C[O,a], and KL ~ I on D . 2 2 3 3
For each
2
+ wL 2 has a positive compact Hermitian inverre 2 K defined onI [0,a] such that K {/ [0,a])Cc[0,a], (W L + wL 2 )K = I 1 2 w w 2 w 2 or C[O,a], Kw(w L +wL 2 ) = I on D , and K is continuous in w on (O,oo). l l w
w
>
0, the operator w L
:::: :::P:C:
l
:~~:t:an(~:~~:~~2T" ::1:K: Wi:) d::::nEDi:::i:Ss::: :::~ =
I on D, S T = I on C[O,a]xC[O,aJ, and w w l 2 T (E)C C[O,a] x C [O,a]. For w > 0, let r - T / , B ~ -wB -B , and 2 l w w w w (¢,[I-rBr]¢) G{w) ~ i£f (¢,¢) w w Note that for l;; E.. D, (cp, [I-rwBwrwJ¢) = T is continuous in w, T S w w w
(l;;,H s), where ¢ w
==
[O,~),
r~S~{D)
and since
r S l;;. Therefore F (w) < w w =
E,
F{~)
~
a
a
on [0
implies
implies G (w) < a on
,~)
G{~)
~
O. .
It therefore (~lr~B~r~¢)
follows from the continuity of G{w)on (O,oo) thatGf
1.
The operator
=
seE, Iisil
1, such that s
x C[O,a], so that 1j!
~ T~B~1j!
EO
r~B~r~is
D.
B~1j!
EO
(¢,¢)
compact and Hermitian, so that there exists
=
r~B~r~l;;.
Hence 1j! == rrl s ~ T~Bn1j!
EO
C[O,a]
C[O,a] x C[O,a], which implies that
Therefore Srl1j!
~
SrlTrlB~1j!
= B~1j!,
i.e., Hrl 1j! = 0, and the
proof is complete. Acknowledgment The work presented here was supported by the Magneto-Fluid Dynamics Division, Courant Institute of Mathematical Sciences., New York University, under U.S. Air Force Grant AFOSR-71-2053. Bibliography 1.
Barston, E .M. , Cornrn. Pure and Appl. Math. ~' 627
2.
Barston, E. M. , Phys. Fluids
2162
(1969) •
3.
Barston, E .M. , J. Fluid Mech. 42, 97
(1970) •
4.
Barston, E .M. , Phys. Fluids 13, 2876
(1970) •
5.
Bernstein, I.B., Frieman, E.A., Kruskal, M.D., and Kulsrud, R.M.,
g,
(1969) •
Proc. Roy. Soc. A 244, 17 (1958). 6.
Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability, Chap. 14 (Oxford University Press, 1961).
94
7.
Chetaev, N.G., The Stability of Motion, Chap. 5 (Pergamon Press, London 1961).
8.
Thompson, W.,
(Lord Kelvin), and Tait, P.G., Treatise on Natural
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Turnbull, R.J., and Melcher, J.R., Phys. Fluids
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