Introduction to Shape Sensitivity Three-Dimensional and Surface Systems
In these lectures we present some basic material for the shape optimization of structures. We emphasise the so — called continuous approach with few results on numerical approximation with finite elements or boundary integrals; this approach is traditiona
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INTRODUCTION TO SHAPE SENSITIVITY THREE-DIMENSIONAL AND SURFACE SYSTEMS
B. Rousselet University of Nice, Nice, France
1 Introduction In these lectures we present some basic material for the shape optimization of structures. We emphasise the so - called continuous approach with few results on numerical approximation with finite elements or boundary integrals; this approach is traditional in mathematics and theoretical mechanics, whereas in mechanical engineering the tendency is to fust approximate the behaviour of the structure with fmite elements and afterwards to tackle optimization. The choice of one of these approaches depends on the habits of thought; in many cases, discretisation in the first or second step yields the same results; this has been proved when one uses conformal finite elements (S. Moriano 1988 , M. Masmoudi 1987 ). If one is interested in deriving necessary optimality conditions and fmding explicit solutions, then the continuous approach is necessary; this is the route followed by the Pragerian school. However in connection with finite elements, the continuous approach is quite versatile: it enables the addition of design sensitivity to a commercial fmite element code ( Melao Barros & Mota soares 1987, Chenais & Knopf-Lenoir 1988 ); but it also enables the inclusion of design sensitivity in an open fmite element library such as Modulef (1985) and makes good use of existing software ( Mehrez-Palma-Rousselet 1991 to appear). Moreover, formulae obtained with the continuous approach can be implemented with boundary
element~
(Masmoudi (1987), Mota soares, Rodrigues Choi (1984) ).
G. I. N. Rozvany (ed.), Shape and Layout Optimization of Structural Systems and Optimality Criteria Methods © Springer-Verlag Wien 1992
B. Rousselet
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It should also be pointed out that these techniques may be used and are used in other fields of application; for example in acoustic ( Masmoudi 1987) and in fluid mechanics (Pironneau 1984). However what is shape optimization ? It is an optimal design problem where the design variable is the shape of the domain n occupied by the physical system; the best shape of a fillet in a tension bar will provide a classsical engineering example (Haug,ChoiKomkov 1986): we want to find the best shape of
IQ
to minimize volume with constraints on Von-Mises
yield stress.
One of the first publications seems to originate with Hadamard (1908) but the pioneers of research oriented toward the use of computers seems to be Cea,Gioan, Michel (1974). Since that date many papers have been devoted to this topic; for example Chenais (1977), Murat, Simon (1976), Rousselet (1976,1977,1982), Dems,Mroz (1984), Pironeau (1984). INRIA schools devoted to shape optimization have been organized by Pironneau (1982) and Cea, Rousselet (1983).
2 shape optimization and continuum mechanics
As in conventional optimal design, the clue of the approach is to obtain first- order .;stimates of the variation of a functional of the state of the system; but for shape optimization one soon realizes that the set
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