## Shape optimization of elastic axisymmetric bodies.(English)Zbl 0691.73037

Let us consider a body of revolution S($$\alpha)$$ having the z-axis as axis of revolution and bounded by two planes perpendicular to the z-axis so that $$0<z<1$$, and by a meridian curve of Cartesian equation $$\alpha =\alpha (z)$$ with $$0<\alpha_{\min}<\alpha (z)<\alpha_{\max}$$. The elastic equilibrium problem for S($$\alpha)$$ requires the solution of a variational problem of the type $$a(u,v)=L(v)$$, where a(u,v) is a bilinear form and L(v) a linear functional. Provided that a(u,v) is continuous and coercive, the problem admits a unique solution by Lax-Milgram’s lemma. Let us now assume we take a sequence of functions $$\alpha_ n(z)$$ uniformly converging to $$\alpha$$ (z) and we define a class of subdomains $$0<z<1$$, $$0<r<\alpha (z)-(1/m)$$. Then for each of them the solution $$u(\alpha_ n)$$ weakly converges to u($$\alpha)$$.
Having defined four different cost functionals, denoted by $$j_ i(\alpha;u)$$ $$(i=1,2,3,4)$$, four optimization problems are studied. For each $$\alpha_ n$$ a value of the cost functional corresponds, that is $$j_ i(\alpha_ n;u_ n)$$. As n tends to infinity, it is possible to prove that $$j_ i(\alpha_ n;u_ n)$$ converges to the value $$j_ i(\alpha;u)$$, attained at the limiting values of $$\alpha_ n$$ and $$u_ n$$. But the result is a little more precise: if the $$\alpha_ n(z)$$ are constrained by the condition of belonging to the class of Lipschitz functions on [0,1] with $$d\alpha_ n/dz\leq$$ constant, then by Ascoli- Arzelá’s theorem there is a subsequence of $$\alpha_ n(z)$$ coverging to a limit which can be assumed as the boundary corresponding to minimum cost.
Reviewer: P.Villaggio

### MSC:

 74P99 Optimization problems in solid mechanics 74S30 Other numerical methods in solid mechanics (MSC2010) 49J20 Existence theories for optimal control problems involving partial differential equations 65N99 Numerical methods for partial differential equations, boundary value problems 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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