Gradient-based optimization of permanent-magnet assemblies for any objective

We introduce a semianalytical optimization method that can be applied to find the optimal magnetization for any permanent-magnet system regarding the desired field or objective. Our approach is based on the Landau-Lifshitz equation, which is normally employed to simulate micromagnetic systems. Instead, we recognize that the same mathematical formalism is also applicable to the optimization of macroscopic magnetic assemblies. This point of view has the advantage of being easy to intuitively visualize and thus is also easy to implement. Our approach can also be seen as a special case of the adjoint method, which is a gradient-based constrained optimization method. However, most studies applying the adjoint method to magnet design adopt the point of topology optimization, where the magnetic permeability tensor field depends on the design variables, making the problem highly nonlinear. Instead, we restrict to the less general problem of optimizing the direction of the magnetization in any point of a given design region. Our strategy leads to a more robust and computationally efficient algorithm since the underlying problem is intrinsically simpler. Additionally, since the integration of the underlying magnetostatic equations is performed numerically, the method is still very flexible and versatile. Previously, an equivalent method has been considered only for solving the magnetic inverse problem. In the present work, we fully demonstrate its effectiveness by presenting several prototypical optimization problems and we discuss the strengths and limitations of the approach in relation to the existing optimization methods.