##
**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.

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 |