Differential Boundary Conditions in Electrodynamics

Terje Vold

Continuum Technology, Inc.

Abstract

Surface integral methods are commonly used to solve electrodynamic problems dealing with 3D regions of matter with piecewise homogeneous linear electromagnetic response (permittivity and permeability). At the boundary between homogeneous regions, the well known boundary conditions for the electric and magnetic field values at a point on such a boundary are typically applied.  In this report the additional use of tangential derivatives of the standard boundary conditions on the fields is described. The resulting conditions depend on field values, the normal directional derivative of field values, and the boundary curvature. These additional boundary conditions allow models of real physical problems that are more symmetric, simpler, and free of problems associated with indeterminacies and null spaces, compared with methods that explicitly use only boundary conditions for field values. The development of these symbolic boundary conditions using geometric algebra and Mathematica is discussed and their numerical implementation is illustrated with simple practical applications.