A Stream Function Smoothing Method for the Design of MRI Gradient Coils on Non-Developable Surfaces

Insert gradient coils with similar imaging body shapes typically have smaller dimensions and higher spatial efficiency. This often allows the gradient coils the achievement of stronger and faster gradient fields. Thus, improving existing methods to make them applicable to the design of MRI gradient coils on complex surfaces has also become a challenge. This article proposes an algorithm that smooths the implicitly expressed stream function based on the intrinsic surface Laplace–Beltrami operator. This algorithm can be used to simplify the design procedure of MRI gradient coils on non-developable surfaces. The following steps are performed by the proposed algorithm: an initial design of the stream function configuration, extraction of the surface mesh, discretization of the surface smoothing operator, and a smoothing of the contour lines. To evaluate the quality of the smoothed streamline configuration, several technical parameter metrics—including magnetic field accuracy, coil power consumption, theoretical minimum wire spacing, and the maximum curvature of the contour lines—were evaluated. The proposed method was successfully validated in a design gradient coil on both developable and non-developable surfaces. All examples evolved from an initial value with a locally non-smooth and complex topological configuration to a smooth result while maintaining high magnetic field accuracy.