Second Order Derivatives , Newton Method , Application to Shape Optimization

{ We describe a Newton method applied to the evaluation of a critical point of a total energy associated to a shape optimization problem. The key point of these methods is the Hessian of the shape functional. We give an expression of the Hessian as well as the relation with the second-order Eulerian semi-derivative. An application to the electromagnetic shaping of liquid metals process is studied. The shape gradient and Hessian of this particular total energy are calculated. The unknown surface is represented by piecewise linear closed Jordan curves. Each step of the algorithm requires solving two exterior elliptic boundary values problems. This is done by using an integral representation of solutions on these surfaces. A comparaison with a Quasi-Newton algorithm is worked out. Our purpose is to analyze in detail the realisation of a Newton method in shape optimization problems and to apply it to a particular problem. We compare cost and eeciency of Newton methods with Quasi-Newton methods applied to the same problem. This problem concerns electromagnetic shaping and levitation of molten metals. A lot of litterature about models of this phenomenon appeared in the last years; we refer for example, to 25], 19], 3] , 1], 14]. Under suitable assumptions, the equilibrium liquid metal conngurations are described by a set of equations containing an equilibrium relation at the boundary between electromagnetic and superrcial tension forces (and gravity in three dimensional models). It involves the curvature of the boundary and an elliptic exterior boundary value problem. This equilibrium shape is shown to be the stationary state of the total energy under the constraint that the surface (the volume in 3-d) is prescribed. Optimization techniques to compute a critical point of the energy need evaluation of the rst shape derivative, in the case of the Quasi-Newton method, and of the second order shape derivative in the Newton method case. Section 1 is devoted to the study of diierent approaches of shape derivatives, see 21], 28], 30], 5], 29] and 12]. We give a precise description of the second shape derivative used in our Newton method, in particular we relate them to the second order Eulerian shape semi-derivatives as described in 5]. Using our approach we compute the rst and second order derivatives of the total energy of the electromagnetic shaping problem mentioned above (we remark