In the area of microelectronics, it is of interest to compute the lumped inductance matrix of a group of coils or conductors. The geometry of the coils is usually ‘open’ and the modeling of a closed current loop is not of interest. In this case, the ability of modeling non-divergence free currents is desired. As a complement to the The Magnetic Fields Interface which can only be used to model divergence-free currents, The Magnetic Fields, Currents Only Interface is designed to support both divergence-free and non-divergence free currents. In free space it returns the value of the Biot-Savart integral.

Start with Ampère’s law for static cases ∇ × H = J and assume an uniform permeability in free space (B = μ0 H), the magnetostatic problem reads

Using the definitions of magnetic potential, B = ∇ × A, rewrite Equation 5-6 as

Applying the Coulomb gauge ∇ · A = 0 and vector calculus identity ∇ × ∇ × A = −∇2A + ∇(∇ · A), Equation 5-7 reduces to

Equation 5-8 implies that The Magnetic Fields, Currents Only Interface is a div-grad formulation, different from The Magnetic Fields Interface which is a curl-curl formulation. Taking the divergence of both sides of Equation 5-8, it is noticed that the divergence of the current is not necessarily equal to zero. Therefore, Equation 5-8 is able to model open coils or conductors.

Different from the The Magnetic Fields Interface which uses the Curl element as shape functions in 3D, The Magnetic Fields, Currents Only Interface employs the Lagrange element as shape functions. In this way, The Magnetic Fields, Currents Only Interface is able to handle the continuity over non-conformal mesh elements.