  We outline the construction of compatible B-splines on 3D surfaces that
satisfy the continuity requirements for electromagnetic scattering analysis
with the boundary element method (method of moments). Our approach makes use of
Non-Uniform Rational B-splines to represent model geometry and compatible
B-splines to approximate the surface current, and adopts the isogeometric
concept in which the basis for analysis is taken directly from CAD (geometry)
data. The approach allows for high-order approximations and crucially provides
a direct link with CAD data structures that allows for efficient design
workflows. After outlining the construction of div- and curl-conforming
B-splines defined over 3D surfaces we describe their use with the electric and
magnetic field integral equations using a Galerkin formulation. We use B\'ezier
extraction to accelerate the computation of NURBS and B-spline terms and employ
H-matrices to provide accelerated computations and memory reduction for the
dense matrices that result from the boundary integral discretization. The
method is verified using the well known Mie scattering problem posed over a
perfectly electrically conducting sphere and the classic NASA almond problem.
Finally, we demonstrate the ability of the approach to handle models with
complex geometry directly from CAD without mesh generation.
