Electromagnetic Interactions GEneRalized (EIGER): Algorithm abstraction and HPC implementation Page: 4 of 9
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To compute the numerical solution of a problem and
numerically enforce the chosen operator on the
geometry, an appropriate set of expansion and testing
functions is needed. The numerical advantages of using
the Nedelec curl-conforming bases in the Helmholtz
operators and the divergence-conforming bases in the
integral operators are now well established 3'. Nedelec
bases not only easily accommodate discontinuities in
material properties, but also eliminate spurious modes
using a minimum number of degrees of freedom per
element for a given order of accuracy. Recently, high
order interpolatory forms of the Nedelec bases have
been constructed which are convenient as 'universal
bases.' The unnormalized form of the divergence-
conforming form of these bases of order p is
For efficient integral equation solution capabilities,
a number of Green's function capabilities are desired.
Both two- and three-dimensional Green's functions and
their gradients are available in EIGER. A wide variety
of problem types may be handled if multi-layered
media Green's functions for both periodic and non-
periodic media are available. The mixed potential
integral equation (MPIE) formulation 7 for such
problems is particularly convenient in practical
computations. A typical potential in (3), say the
magnetic vector potential, is expressed as an integral
over sources J on a domain D as
Ai,(P,)=4, R;(P,)Al( )
where Ap (g) is the usual (zeroth order) divergence-
conforming basis associated with sub-boundary R of an
element and Ri(p,k) is a modified Silvester-Lagrange
polynomial similar to equation 2 but involving
interpolation points shifted away from the element's
Unnormalized curl-conforming bases have the form
(P, = (P,4)A(2O
for a set of bases associated with edge f6 of a two-
dimensional element and
r. i ;^
for a set associated with edges formed by the
intersections of faces y and P of a three-dimensional
element. A4 and f2, are curl-conforming zeroth order
bases associated with the elements.
When singular quantities such as the fields or
currents near edges of conductors or dielectrics are
modeled, higher order bases do not provide the
expected increase in accuracy. To model such cases
accurately, singular higher order bases are needed5.
Such bases, as well as special basis functions for
modeling junctions between surfaces and wires, are
incorporated into EIGER6.
The Green's potential dyad, GA, may in turn be written
GA(r,r')= IG(r,r')+ J ',G0(r,r)
+ AGA (r,r') (11)
where I is the identity dyad, Go is the background
homogeneous medium Green's function for non-
periodic media, I is a dyadic reflection coefficient
representing a quasi-static image located at r; , and
AG^(r,r) is a relatively smooth integral contribution of
Sommerfeld type. The latter integral is efficiently
evaluated using a combination of complex path
deformation and the method of averages. For periodic
media, Go is the homogeneous media periodic Green's
function, an infinite series that may be efficiently
evaluated using the Ewald method9. In this case AGA is
also a rapidly converging series. For complete
generality, it is possible to separately model the
environment on either side of a surface element using
any Green's function available to the code.
Other important Green's functions for applications
are those that may be constructed using reflection or
rotational symmetries. These symmetry operators may
be constructed by appropriately reflecting or rotating
source elements and endowing them with appropriate
signs or phase factors.
ELEMENT MATRIX CONSTRUCTION
The first step in obtaining a matrix approximation to
an operator equation is to form the element matrix. This
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Sharpe, R.M.; Grant, J.B.; Champagne, N.J.; Wilton, D.R.; Jackson, D.R.; Johnson, W.A. et al. Electromagnetic Interactions GEneRalized (EIGER): Algorithm abstraction and HPC implementation, article, June 1, 1998; Albuquerque, New Mexico. (digital.library.unt.edu/ark:/67531/metadc707363/m1/4/: accessed February 22, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.