Table Of ContentOther Titles in the IF AC Proceedings Series
ATHERTON: Multivariable Technological Systems
BANKS & PRITCHARD: Control of Distributed Parameter
Systems
CICHOCKI & STRASZAK: Systems Analysis Applications to
Complex Programs
CRONHJORT: Real Time Programming 1978
CUENOD: Computer Aided Design of Control Systems
De GIORGO & ROVEDA: Criteria for Selecting Appropriate
Technologies under Different Cultural, Technical and Social Conditions
DUBUISSON: Information and Systems
GHONAIMY: Systems Approach for Development
HARRISON: Distributed Computer Control Systems
HASEGAWA & INOUE: Urban, Regional and National Planning
- Environmental Aspects
ISERMANN: Identification and System Parameter Estimation
LAUBER: Safety of Computer Control Systems
LEONHARD: Control in Power Electronics and Electrical Drives
MUNDAY: Automatic Control in Space
NIEMI: A Link Between Science and Applications of Automatic
Control
NOVAK: Software for Computer Control
OSHIMA: Information Control Problems in Manufacturing
Technology (1977)
REMBOLD: Information Control Problems in Manufacturing
Technology (1979)
RIJNSDORP: Case Studies in Automation related to
Humanization of Work
SAWARAGI & AKASHI: Environmental Systems Planning,
Design and Control
SINGH & TITLI: Control and Management of Integrated
Industrial Complexes
SINGH & TITLI: Large Scale Systems: Theory and Applications
SMEDEMA: Real Time Programming 1977
TOMOV: Optimization Methods - Applied Aspects
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Publisher at Pergamon Press
CONTROL APPLICATIONS
OF NONLINEAR PROGRAMMING
Proceedings of the IF AC Workshop, Denver, Colorado, USA
21 June 1979
Edited by
Η. E. RAUCH
Lockheed Palo Alto Research Laboratory, California, USA
Published for the
INTERNATIONAL FEDERATION OF AUTOMATIC CONTROL
by
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Copyright© IFAC 1980
All Rights Reserved. No part of this publication may be
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First edition 1980
British Library Cataloguing in Publication Data
IFAC Workshop on Control Applications of
Nonlinear Programming, Denver, 1979.
Control applications of nonlinear
programming.
1. Automatic control - Mathematical models
- Congresses.
2. Nonlinear programming - Congresses
I. Title II. Rauch, Η Ε
III. International Federation of
Automatic Control.
629.8'312 TJ213 80-49944
ISBN 0-08-024491-2
These proceedings were reproduced by means of the
photo-offset process using the manuscripts supplied by the
authors of the different papers. The manuscripts have
been typed using different typewriters and typefaces. The
lay-out, figures and tables of some papers did not agree
completely with the standard requirements; consequently
the reproduction does not display complete uniformity.
To ensure rapid publication this discrepancy could not be
changed; nor could the English be checked completely.
Therefore, the readers are asked to excuse any deficiencies
of this publication which may be due to the above men-
tioned reasons.
The Editor
Printed in Great Britain by A. Wheaton & Co. Ltd, Exeter
PREFACE
During the last decade there has been extensive theoretical development of
numerical methods in nonlinear programming for parameter optimization and
control. The increasing use of these methods has led to the organization
of an international Working Group on Control Applications of Nonlinear
Programming, under the auspices of the International Federation of Automatic
Control. The purposes of the Working Group are to exchange information on
the application of optimal and nonlinear programming techniques to real-life
control problems, to investigate new ideas that arise from these exchanges,
and to look for advances in optimal and nonlinear programming which are
useful in solving modern control problems.
This volume contains the Proceedings of the first Workshop which was held in
Denver, Colorado, U.S.A., on June 21, 1979. It represents the latest work
of fifteen invited specialists from the U.S.A., U.S.S.R., Federal Republic
of Germany, France, and Great Britain. The volume covers a variety of spe-
cific applications ranging from microprocessor control of automotive engines
and optimal design of structures to optimal aircraft trajectories, system
identification, and robotics. These significant contributions to numerical
methods in control reflect the great amount of work done by the authors.
The impetus for the Workshop was derived from early efforts of Professor
Henry Kelley (U.S.A.), Chairman of the IFAC Mathematics of Control Committee.
We were fortunate to have had as members of the International Program
Committee Professors Arthur Bryson, Jr. (U.S.A.), Faina M. Kirillova
(U.S.S.R,), and R. W. H. Sargent (Great Britain). Finally, it was a great
pleasure for me to have organized the Workshop and served as Chairman of
the Working Group.
Herbert E. Rauch
Palo Alto, California U.S.A.
vii
APPLICATION OF NON LINEAR
PROGRAMMING TO OPTIMUM DESIGN
PROBLEM
C. Knopf-Lenoir*, G. Touzot* and J. P. Yvon**
*Universite de Technologie de Comptegne, BP 233,
60206 Comptegne, France
**IRIA-LABORIA, BP 105, 78105 Le Chesnay, France
Abstract. As a particular application of the optimal control, theory this
paper presents an optimum design problem coming from mechanical engineering.
This problem can be formulated as to determine the shape of the boundary of
an elastic body in order to minimize the stress concentration near this
boundary. This type of problems are solved by using an optimal control
formulation in which the control is the boundary itself and the state is
given by solving the equation of elasticity. The criterion, for instance
the maximum value of stresses, is to be minimized by non linear programming
methods. The paper presents a formulation of the problems and numerical
solutions obtained on a specific example.
Keywords. Optimum design, elasticity, non linear programming.
INTRODUCTION Let ua be the solution of the partial dif-
ferential equation
The theory of optimal control of systems go-
verned by partial differential equations has Aua = f in Ωα (1)
been widely developed during the last ten Bua = g on 8Ωα (2)
years. An important range of applications of
Then for any value of α it is possible to
this theory consists in problems of optimum :
define a functional 3 ^~R ·
design. By an optimum design problem we mean
a problem where the control is the geometry 3-(oO = Φ^α) · (3)
of the domain in which the boundary value
The optimum design problem consists in
problem is solved, see e.g. LIONS (1972). An
abstract formulation of the problem is the Min J (a)
(4)
following one. at A
The space Λ is a space of parameters which
Let Ω^ be a variable domain in R wi th
determine the shape of Γα and the problem
3Ωα = G υ Γα consists in minimizing J by choosing the
shape of Γα,
where Γα is the variable part of the boun-
dary which depends on a family of parameters The difficulties of this type of problems
oteA^being a space to be defined (see Fig.l), are both theoretical and practical. From
a theoretical point of view it is necessary
to settle a convenient functional framework
in order to give a sense to problem (1) (2)
(3) (4) and to obtain first order necessary
conditions for optimality, concerning this
point we refer to PIRONNEAU (1976) and
MURAT-SIMON (1974).
From a practical point of view this problem
requires a very efficient minimization method
It must be noticed that the computation of
3 (ot) requires one solution of the boundary
value problem (1) (2), and because of the
change of domain at each iteration this
Figure 1. represents a great computational effort.
All these aspects of the problem will be
detailed in the sequel on a specific example.
1
2 C. Knopf-Lenoir, G. Touzot and J. P. Yvon
A MECHANICAL EXAMPLE
Compressing stages of engine, in aircrafts,
have fitted disks with which blades are
fitted. The centrifugal force tends to pull
out the bases of blades and there appear very
large stresses in the disk near the boundary = U
η'Ω
of alveoles, (see Fig. 2).
Blow up of
this zone on
Fig. 3. = 0
Figure 2.
U imposed
η ,
The problem consists in optimizing the shape Figure 3.
of the alveole in order to minimize a function
of stresses. The domain is pictured in Fig. 10
where Ω2 denotes a part of the disk (the
structure of the disk is periodic) and Ωι is
the basis of the blade. Boundary conditions
are given on Fig, 3. In particular an imposed To the solution u of (5) corresponds the
force is applied on the boundary AB. The stress cr
shape of the boundary Γ (which is the part of
the alveole with which the blade is fitted) at = (νσ , σ , τ )
χ' y' xy
is to be optimized. As we can neglect the
effect of finite thickness of the disk,a two and the strain
dimensional problem may be considered. The ϋ
ε = (ε , ε , γ )
material is assumed linearly elastic, homoge- χ y xy
neous and isotropic. Plane strain is assumed
where
to calculate stresses. - -Ρ-
a χ
ε Y +
x ax y xy 9y 8x
The boundary Γ being given, equation of dis-
y 3
placements u = (u , u ) is given by the
x y
. .
virtual works principle : The elastic law of the medium gives
I ot ε dx = J u* F dx + / u1 Τ ds (5) σ . D ε
Ω{υΩ2 ^ ΩιυΩ2 A
where ε is the strain vectortassociated with where D is the 3x3 matrix
the virtual displacement u (x denotes the
transpose of χ), F is a distributed force di d2 0
(centrifugal) and Τ is the traction due to
the blade. D = di 0
0 0 d3
wi th
E(l-V) EV
1 " (1 + YH1-2V) d2ou = (1+V)(1-2V) 2(1+V)
Application to Optimum Design Problem 3
f
(E is the Young's modulus and V the Poissons Here we present a very simple method which
coefficient). reduces the number of unknowns. If we assume
It is classical that the variational equation that the initial guess of Γα is not too far
(5) has a unique solution belonging to a sub- from t*he optimum it is possible to
space of the Hubert space associate to {Γ^}^ a family of fixed
2 curves ί^}^ nearly orthogonal to
V = [Η^Ωχ όΩ2)]
{Γα} . Theses curves will be called
(cf. for instance ODEN-REDDY (1976)). "meridians" in the sequel.
Then the boundary Γ α will be determi-
ned by a finite set of parameters
{ou}J^j, where ou is the curvilinear
abscissaon Δ. of the intersection
ι
point D^ of Δ, with Γ α . The finite
elements approximation is then based
on a moving mesh which uses the moving
boundary nodes D^ i = 1, 2,..., ρ
(see Fig. 8) .
Nevertheless in order to avoid a
complete modification of the finite
element mesh for any change in the
Figure 4. position of {D^} it is necessary to
divide Ω^ into two parts (see Fig.9):
a fixed part where the mesh is fixed
and a moving part in which the mesh
depends on the position of Γ
On the boundary Γ the normal stress Ο and
shear stressτ vanish (cf. Fig. 4). n The displacement of internal nodes in
nt
the moving part is given by simple
From a mechanical point of view the main
rules. For instance the curvilinear
problem is to avoid appearance of cracks on
abscissa of an internal node on Δ. is
the boundary. The cause of cracks is the ι
stresses concentration at near Γ. A crude way an affine function of ot. .
to solve this problem is to minimize a func-
ι
tion of af.c
FINITE DIMENSIONAL
Then a natural criterion is the following
PROBLEM
max Iσ (α)| (6) Using the classical quadratic seren-
yeT
dipity elements, see ZIENKIEWICZ
(1971), associated to the mesh of Fig,
Nevertheless in order to avoid the dificulties
fl 15, equation (5) is written in the
of "min-max problems we have also considered
following form :
a "smoothed" version of (6) :
7S (D • f k. (a) |2P (7) Κ (a)ua = f(a)
3z
Ρ where K(ot) is an r χ r matrix (stiff-
Γ
ness matrix) and u eR .
which is clearly an approximation of (6) for
α
ρ large enough.
Criterion(7) is written as :
Furthermore additional restrictions on the
2 f
shape of Γ must be considered for several 3Q(<*) = Σ !c^(a)-ua| dY (9)
reasons. For instance the convexity of Γ j J
must be constant in order to avoid difficul- where c..(a)eRr, j = 1,2,..., q.
ties for manufacturing. These points will be
detailed below. The main step is to calculate the
a
gradient of Jp ( ) · This can be done
by using an adjoint state which is
PARAMETERIZATION
usual in optimal control theory. We
have :
The general approach is to combine the finite
elements approximation of problem (5) and the
parametrization of the variable boundary Γ .
This has been done by many authors (see for d a a =
dc!
instance MORICE (1975), PIRONNEAU (1976)). 2p Σ C4<a) uJ1 2"P! 7H-^u+nJ(1)0
j a j ot ^ ot
j
4 C. Knopf-Lenoir, G. Touzot and J. P. Yvon
where ua is the solution of the linear-
ized problem :
da a
The adjoint state ρ(α) is defined by : Γ
Ω α
ϋ 2n-l
Κ (α)Ρ = Σ |c,(a)u (ID α
J
Then
ptK(a)u = Σ |c*(a)u I 2p-' t
ι J1 a c . u
Figure 5.
t dfr ~ dK(a)
= Ρ Ja [ a da -a au]
Using this last expression in (10) we
get :
d3p()a - , trdf , dK ~ ,
a = 2p ρ [-a- - aua]
,D ,dc (a)
t 2 pI
+ 2p |C j(a)uJ - T^ aua
1
In particular the i*"* component of the
gradient is :
Β3ρ(α) t d±_ 0 t t 3K(a)
= 2p 2p u ρ
8a.ι * 8a.ι a 8a
/ (12)
I t, , ,2P-1 t
+ 2 Ρ
Figure 6.
The gradient is given by one solution
of the adjoint state equation (11).
The crucial point is to calculate
carefully the coefficients :
8K
lm
8a. Let us represent on Fig. 7 a part of
the boundary where is a moving
which give the derivative of the stiff- node associated with a^ and i s the
ness matrix coefficient K- with
oriented angle.
lm
respect to the design parameter a.. Θ. = (oT. DfS >
i+J
THE NON LINEAR PROGRAMMING
PROBLEM χ
As we have mentioned earlier we have
to introduce additional constraints on
y
the variable {a. } ? , in order to keep v
i i =l
a constant convexity of the boundary.
For instance the boundary Γ pictured
on Fig. 5 is unfeasible, we want to
obtain a shape as on Fig. 6.
Figure 7.
Application to Optimum Design Problem 5
The condition of constant convexity imposes can be transformed in the new problem;
θ, ^ θu· min 3
i+l (17)
α» 3
or, if - I < Q± <2 ' Π c.(o) u 1, 2, q (18)
J α
tg Θ. >, tg θ.+ 1 . with the same other constraints (13) and (14),
In this case it is not easy to use the same
This condition can be written obviously as a method as previously because of constraints
function of the coordinates (x., y.) of D. (18) which are not simple at all.
We have used for this last problem the
x.ι- x.l -l, x>l.+.l. - xι. method proposed by several authors like
BIGGS (1975), HAN (1976) and POWELL (1978),
y y y y
i" i-l ' i+l i which is based on augmented Lagrangian.
(in the case of Fig.7).
This last inequality can be rewritten by NUMERICAL RESULTS
using the design parameters a^'s and this
The initial domain before optimization is
gives the following inequality
given on Fig. 10. The result after optimi-
zation (reduced gradient) of criterion 3p>(9)
C.(a) = a.+b.a. ,+c.a.+d.a,· . ,+e.a. ,a.
ι ι I I -I li ii+Ι li-1 ι
with ρ = 8, is shown on fig. II. The fig. 12
(13) represents the same results after optimiza-
+ f ^ a ^ g ^ a ^, < 0 tion of the criterion (16) %(α) by the
Powell's method.
where a^,b^, g^ are constants, i = 1,2, Table 1 gives a comparizon of convergences
of the two algorithmes.
Furthermore, as we know roughly the position CRITERION NUMBER OF MAXIMUM
of the optimal solution it is possible to ITERATIONS STRESS VALUE
impose restrictions on the values of ou's
J 15 43.71
Ρ
α. ^ α. ^ α. ι = 1, 2,..., ρ (14) J 3 37.58
1 11 00
, m Μ
where a. et a. are given. (initial value of maximum stress : 50.06)
11
Table 1
OPTIMIZATION METHODS
Figures 13 and 14 show the different results
a n (
The nonlinear programming problem consists obtained by minimizing ^ i J*
in minimizing (9) under constraints (13) and
(14). This problem is non convex but it is
possible to use a (generalized) reduced CONCLUDING REMARKS
gradient method. The basis of the method is
that if the constraint Ck(°0 is active, i.e. The efficiency of the Powell's method is
patent. It seems to be very well adapted to
Ck (a) = 0, (15) highly non linear problems.
then it is possible to express in function This fact is very important from pratical
point of view. In optimum design problems the
of a_. for j = k-1, k+1 . This allows to reduce calculation of one value of the criterion
the number of variables by eliminating a^. and/or the gradient is very expensive in
computer time and the feasability of the
This element is quite simple to perform. Of method is closely related to the efficiency
course this method requires some "pivoting" of the optimization method. In the near fu-
steps when one constraint (14) is active. ture a lot of optimum design problems will
As we have mentioned earlier the criterion be solved numerically, because this field
(9) is a "smoothed" form of criterion (6) seems to be very promising in mechanical
which takes the form, in the finite dimen- engineering, electrical engineering, etc...
sional case : (see for instance, PIRONNEAU (1973)(1974),
M0RICE (1974), KELLY and al. (1977),
J (a) = max c^ (a) u^ (16) MIDDLET0N-0WEN (1977)).
i For the case of distributed systems this
As it is classical the minimization of (16) type of problem has given new interesting
extensions of the classical control theory.
6 C. Knopf-Lenoir, G. Touzot and J. P. Yvon
REFERENCES
Biggs, . C (1975). Constrained optimization
using recursive quadratic programming.
L.C.W. DIXON and G.P. SZEGO Eds. Toward
Global Optimization. North-Holland,
Amsterdam.
Han, S.P. (1976). Superlinearly convergent
variable metric algorithms. Math. Program-
ming, 11, pp. 263-282.
Kelly, D.W., Morris, A.J., Bartholomew, P.,
Stafford, R.D. (1977). A review of tech-
niques for automatic structural design.
Comp. Methods in Applied Mechanics and
Engineering, 12, pp. 219-242. fixed part moving part
Lions, J.L., (1972). Some aspects of the
optimal control of distributed parameter
systems. Regional Conference Series in Figure 9.
Applied Math., SIAM n°6.
Middleton, J., Owen, D.R.J. (1977). Automated
design optimization to minimize stearing
stress in axisymetric pressure vessels.
Nuclear Engineering and Design, 44, pp.
357-366.
f
Morice, P. (1975). Une methode doptimisation
de forme. Proc. Conference IFIP-IRIA 1974.
Springer-Verlag, pp. 454-467.
Murat, F., Simon, J. (1977). Etude de quelques
1
problemes doptimisation. Publication
Univ. Paris VI.
Oden, J.T., Reddy, J.N. (1976). Variational
methods in theoretical mechanics.
Universitext, Sringer-Verlag, Berlin.
Pironneau, 0., (1973). On optimum profiles
in stoker flow. J. Fluid Mech., Vol. 59,
pp. 117-128.
Pironneau, 0. (1974). On optimum design in
fluid mechanics. J. Fluid Mech., Vol. 64,
pp. 97-114.
Pironneau, 0., (1976). Ph. D. Thesis, Paris VI.
Powell, M.J.D., (1978). Algorithms for non
linear constraints that use Lagrangian
functions. Math. Programming, 14, pp. 4- 4-
224-248.
Zienkiewicz, O.C. (1971). The finite element
+ 1
method. Mc. Graw-Hill, London.
4- ι
4- 4- 4-
4-4-4-
4—h 4-4-
4—h 4-4- +
4-4-4-4- 4-
Figure 10.
Figure 8.
