A new continuous optimization method of standardized cubic permanent magnets for stellarators

Stellarator configurations can be realized using permanent magnets in combination with tokamak-like planar coils. From the perspective of engineering feasibility and cost-effectiveness, using standardized cubic permanent magnets with a finite number of magnetization directions is particularly advantageous. However, optimizing the discrete magnetization directions of tens of thousands of cubic magnets poses a significant challenge. This paper introduces a novel method that converts the discrete optimization problem into a continuous nonlinear optimization problem, enabling the direct application of well-established nonlinear optimization algorithms. This approach leverages a specially tailored continuous function to represent the magnetization moment and incorporates a penalty term to ensure the direction vector converges to one of the discrete points. Subsequently, a discrete solution for the magnetization direction distribution is derived by truncating the continuous solution and eliminating adjacent magnets with opposing magnetization directions. The proposed optimization method is validated using the ESTELL stellarator configuration as a case study. The results demonstrate that the designed distribution of cubic permanent magnets can accurately reproduce the intended magnetic field configuration.