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In most practically relevant applications, the objective functional depends on the shape of a (sub-)domain via the solution to a partial differential equation (PDE). Thus, one is facing a problem of PDE-constrained shape optimisation of the form:

In this paper, we present an alternative framework for AD of PDE-constrained problems of type (2). There exist several approaches for the rigorous derivation of the shape derivative of PDE-constrained shape functionals; see Sturm (2015b) for an overview. The main idea, however, is always similar. After transforming the perturbed setting back to the original domain, shape differentiation in the direction of a given vector field reduces to the differentiation with respect to the scalar parameter t which now enters via the corresponding transformation and its gradient. It is shown in Sturm (2015a) that the shape derivative for a nonlinear PDE-constrained shape optimisation problem can be computed as the derivative of the Lagrangian with respect to the perturbation parameter. We will illustrate this systematic procedure for a number of different applications and utilise symbolic differentiation provided by the finite element software package NGSolve (Schöberl 2014) to obtain the shape derivative for different classes of PDE-constrained optimisation problems. NGSolve allows for the fast and efficient numerical solution of a large number of different boundary value problems. The aim of this paper is to extend NGSolve by the possibility of semi-automatic and fully automatic shape differentiation and optimisation.

We stress that the choice of p as the solution of the adjoint equation is important in order for the second equality in (26) to hold. The verification of this equality depends on the specific PDE under consideration and can be accomplished by different methods. We refer the reader to Sturm (2015b) for an overview and remark that (26) holds for a large class of nonlinear PDE-constrained shape optimisation problems; see Sturm (2015a).

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