A sequential approach combining topology and shape optimization for nonlinear 3D magnetostatics

Abstract Shape and topology optimization is currently enjoying renewed interest, largely thanks to the rise of 3D printing techniques that make it possible to manufacture increasingly complex designs. In nonlinear 3D magnetostatics, the implementation of topological optimization methods still faces substantial mathematical and numerical challenges. Beyond these challenges, in many practical applications, a key difficulty lies in obtaining designs with well-defined boundaries, which is essential to guarantee manufacturability. To address this issue, we propose a sequential optimization strategy that combines topology and shape optimization to overcome the limitations of each method when they are used independently. First, a SIMP density-based topology optimization is performed to explore a wide design space. Next, an isosurface of the resulting material distribution is extracted and used as the initial geometry for a shape optimization phase, based on the Hadamard boundary variation method and a level-set representation. This hybrid approach enables the generation of designs with well-defined boundaries, avoiding both the intermediate densities typical of SIMP and the sensitivity to initialization inherent in pure shape optimization. The effectiveness of the methodology is demonstrated through two three-dimensional examples: the design of a C-core actuator and a preliminary design of a magnetic switch exploiting nonlinear material effects. These results highlight the potential of our approach to produce innovative, high-performance magnetic devices that are physically consistent and provide a solid basis for manufacturing.