# Electromagnetic Interactions GEneRalized (EIGER): Algorithm abstraction and HPC implementation Page: 4 of 9

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GREEN'S FUNCTIONS

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 isFor 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 asA=fG (r,r')-J(r')dD

(10)

Ai,(P,)=4, R;(P,)Al( )

Al9(7)

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

boundaries.

Unnormalized curl-conforming bases have the form(P, = (P,4)A(2O

'p(8)

for a set of bases associated with edge f6 of a two-

dimensional element and(9)

r. i ;^

--rfor 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

as

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. This3

American Institute of Aeronautics and AstronauticsBASES

l'v'j

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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.