Hybrid envelope-to-rectangle (HER) three-stage optimization method for ultra-high field MRI superconducting magnets

This paper introduces a novel Hybrid Envelope-to-Rectangle (HER) optimization methodology addressing critical design challenges in ultra-high field magnetic resonance imaging (MRI) superconducting magnets. Traditional two-stage magnet design methods, which integrate linear programming (LP) with nonlinear programming (NLP), frequently produce irregular non-zero current clusters that necessitate manual intervention for coil regularization. This approach is computationally inefficient, particularly in ultra-high field systems where electromagnetic constraints are highly sensitive to geometric deviations. The HER methodology comprises three distinct phases: 1) initialization through linear programming with multi-layer current density constraints, 2) geometric conversion using elliptical envelope parameterization and dual-field matching, and 3) constrained nonlinear optimization incorporating harmonic suppression. The HER method systematically converts irregular current clusters into rectangular coils through dual geometric-field optimization, achieving a peak-to-peak homogeneity of 0.31 parts per million (ppm) within a 120 mm diameter spherical volume (DSV). This approach enhances computational efficiency while maintaining robustness, thereby advancing automation and precision in ultra-high field magnet engineering.