  In this paper we propose a Newton method for shape functions defined on an
image set generated by the (Micheletti) metric group. We review basic
properties of the metric group and a quotient associated with the metric group
and a fixed domain. Taking into account the special structure of the second
shape derivative and its symmetric part allows us to distinguish between two
Hessians, the domain shape Hessian and the boundary shape Hessian. Using the
domain Hessian we define a Newton method on the metric group by discretising
the tangent space of the quotient via approximate normal functions using
reproducing kernels. Under suitable assumptions we are able to show superlinear
convergences of the Newton iterations and additionally convergence of the
shapes in the metric group. Finally we verify our findings in a number of
numerical experiments including a thorough numerical study of the impact of the
discretisation on the convergence speed.
