Magnetic-field modeling with surface currents. Part I. Physical and computational principles of bfieldtools

Abstract

Surface currents provide a general way to model magnetic fields in source-free volumes. To facilitate the use of surface currents in magneto-quasistatic problems, we have implemented a set of computational tools in a Python package named bfieldtools. In this work, we describe the physical and computational principles of this toolset. To be able to work with surface currents of the arbitrary shape, we discretize the currents on triangle meshes using piecewise-linear stream functions. We apply analytical discretizations of integral equations to obtain the magnetic field and potentials associated with the discrete stream function. In addition, we describe the computation of the spherical multipole expansion and a novel surface-harmonic expansion for surface currents, both of which are useful for representing the magnetic field in source-free volumes with a small number of parameters. Lastly, we share examples related to magnetic shielding and the surface-coil design using the presented tools.

Planar Coil Optimization in a Magnetically Shielded Cylinder

“…Other optimization methods, such as linear programming, could be used to determine optimal designs to allow for more direct constraints on the field fidelity at specific target points [21,22]. Alternatively, the optimal Fourier coefficients for a specific target region could be found using a numerical procedure such as a particle swarm optimization [36].…”

“…Boundary element methods (BEMs) can be used to optimize magnetic fields generated by surface currents on a triangular mesh [19][20][21][22] to generate arbitrary target magnetic fields. BEMs are extremely powerful and flexible since they can be used to define active systems with complex geometries inside passive shields.…”

Magnetic field compensation coil design for magnetoencephalography

“…This can be expressed as which defines a scalar stream function S 11 . This stream function concept can be generalized to arbitrary surfaces, where is replaced by the tangential gradient operator on the surface and is replaced by the unit surface normal vector 12 . Stream lines of the surface current density are, by definition, parallel to and hence perpendicular to .…”

Magnetic Field Compensation Coil Design for Magnetoencephalography

“…which defines a scalar stream function S [11]. This stream function concept can be generalized to arbitrary surfaces, where ∇ is replaced by the tangential gradient operator on the surface and e z is replaced by the unit surface normal vector [12]. Stream lines of the surface current density J are, by definition, parallel to ∇S × e z and hence perpendicular to ∇S.…”

“…Stream lines of the surface current density J are, by definition, parallel to ∇S × e z and hence perpendicular to ∇S. That means, contour lines of S are stream lines of J [12]. Since wire paths are stream lines of a current density, contour lines of S defined our coil wire paths.…”