A B-spline representation of the stellarator boundary for stellarator optimisation

The geometry of the magnetic field of a stellarator is intrinsically linked to the quality of its confinement. Constructing magnetic fields with desirable properties is difficult or impossible analytically; therefore, numerical optimisation lies central to the art of stellarator design. Given a pressure profile and a rotational transform or toroidal current profile, the magnetic field inside of a stellarator is determined by the shape of its boundary; therefore, stellarator design may be posed as a shape optimisation problem where the degrees of freedom control the boundary shape. We introduce a novel B-spline representation of the stellarator boundary for use in stellarator optimisation. The degrees of freedom in this representation can be easily constrained to an optimisation space of diversified but feasible stellarator shapes; the same cannot be done in the commonly used Fourier series representation. This makes the spline representation amenable to non-convex/global optimisation techniques that involve randomly sampling over a bounded optimisation space, such as Bayesian optimisation. Moreover, it addresses pathologies of the Fourier representation such as non-uniqueness and non-local control, which make it more suitable to the task of mapping the design space of stellarators. We implement the representation in the commonly used open-source stellarator design framework simsopt and present stellarator optimisation results obtained using our parametrization.