Boundary Integral Equations

The boundary integral equations (IE) is an exact transparent condition. The Sommerfeld radiation condition at infinity as required by physics is enforced exactly through the employment of appropriate Green's functions via an integral equation method. This hybridization is commonly known as the finite element-boundary integral (FE-BI) method. Unlike ABC and PML, the IE hybrid region can be of arbitrary shape, both concave and convex thus in some cases allowing the size of the finite element solution domain to be significantly reduced. It can be placed very close to the structures and produce accurate results and from a performance point of view a minimum distance of 0.125 wavelengths is recommended. Since analytic Green's functions are known only for a few specific situations, the IE hybrid region is not allowed to touch any other boundary except an infinite ground plane. In addition the IE hybrid region does not work with symmetry planes. The IE condition is a non-local condition, producing a partial sparse and partial dense system matrix. For this reason when simulating electrically large structures, the IE condition typically requires more RAM and CPU than either ABC or PML assuming all are placed on the same geometry. However by taking advantage of conformal radiation volumes to reduce the overall finite element solution domain utilization of the IE hybrid region will result in more efficient simulation for electrically large open boundary problems.