document: The boundary element method (BEM) has clear advantages when applied to shape optimisation of high-voltage devices, see [4–8] for an introduction to BEM. First of all, BEM relies only on a surface discretisation so that there is no need to maintain an analysis-suitable volume discretisation during the shape optimisation process. Moreover, BEM is ideal for solving problems in unbounded domains that occur in electrostatic field analysis. In gradient-based shape optimisation the shape derivative of the cost functional with respect to geometry perturbations is needed [9–11]. To this purpose, we use the adjoint approach and solve the primary and the adjoint boundary value problems with BEM. The associated linear systems of equations are dense and an acceleration technique, such as the fast multipole method [12,13], is necessary for their efficient solution. For some recent applications of fast BEM in shape optimisation and Bernoulli-type free-boundary problems we refer to [14–16].
extractive_keyphrases: adjoint approach bem boundary element method electrostatic field analysis fast multipole method primary and the adjoint boundary value problems shape derivative shape optimisation solving problems in unbounded domains surface discretisation
abstractive_keyphrases: bernoulli-type free-boundary problems fast bem in shape optimisation gradient-based shape optimisation shape optimisation of high-voltage devices