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In this paper we propose a Newton method for shape functions defined on animage set generated by the (Micheletti) metric group. We review basicproperties of the metric group and a quotient associated with the metric groupand a fixed domain. Taking into account the special structure of the secondshape derivative and its symmetric part allows us to distinguish between twoHessians, the domain shape Hessian and the boundary shape Hessian. Using thedomain Hessian we define a Newton method on the metric group by discretisingthe tangent space of the quotient via approximate normal functions usingreproducing kernels. Under suitable assumptions we are able to show superlinearconvergences of the Newton iterations and additionally convergence of theshapes in the metric group. Finally we verify our findings in a number ofnumerical experiments including a thorough numerical study of the impact of thediscretisation on the convergence speed.

Convergence of Newton's method in shape optimisation via approximate normal functions