Accurate surface normal representation to facilitate gradient coil optimization on curved surface

Abstract

The design methods for gradient coils are mostly based on discrete extrinsic methods (e.g., the Biot–Savart integration calculation), for which the surface normal vector strongly influences any numerical calculation of the discretized surface. Previous studies are mostly based on regular or analytical surfaces, which allow normal vectors to be expressed analytically. For certain applications, design methods for extending current-carrying surfaces from developable or analytic geometries to arbitrary surfaces generated from a scanned point cloud are required. The key task is to correctly express the discretized normal vectors to ensure geometrical accuracy of the designed coils. Mathematically, it has been proven that applying a Delaunay triangulation to approximate a smooth surface can result in the discrete elemental normal vectors converging to those of the original surface. Accordingly, this article uses Delaunay triangulation to expand upon previous design methods so that they encompass arbitrary piecewise continuous surfaces. Two design methods, the stream function and the so-called solid isotropic material with penalization (SIMP) method, are used to design circumvolute and noncircumvolute gradient coils on general surfaces.