# Electromagnetic Interactions GEneRalized (EIGER): Algorithm abstraction and HPC implementation Page: 3 of 9

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r = YrRi(p,4)(1)

where i = (i,,iy...,,,) is a multi-index designating

both the order and locations of interpolation points on

curvilinear elements, and = ( i, 2,..., is a

multi-vector of normalized coordinates defined on an

element, one for each sub-boundary (endpoint, edge, or

face of a line segment, surface, or volume element,

respectively) comprising an element t. R; (p, f) is a

Lagrange interpolation polynomial of order p and has

the separable form

R,(p, f)=R(P, )---R,(,,,) (2)

where Ri(p, ) is the Silvester-Lagrange interpolating

polynomial'2. All additional geometrical quantities

(e.g., element jacobian, edge vectors lj, and coordinate

gradient vectors V ) may be obtained from the so-

called unitary basis vectors li = 8r/8 associated with

the independent coordinates ?i. The detailed geometry

of a 3D-prism element, depicting these quantities, is

shown in Figure 1.

Careful examination of the prism element, along

with the other elements of interest, clearly identifies

information that all elements must have knowledge of.

In EIGER, this information is cast into an element class

(a fundamental class for geometry) which contains

some of the following:

" An element type

" An element order

" A set of physical points that define the element

" The number of basis functions on the element

" Pointers to specific basis sets

" Additional attributes (possibly thickness or radius)

OPERATORS

The current development activity grew from

research in integral equation methods. For dynamic

problems, it is assumed that the unknowns associated

with any element may be either equivalent electric or

magnetic currents---or a combination of the two.

Similarly, boundary conditions may involve either the

electric field, the magnetic field, or both. Therefore

electric and magnetic field operators of the following

type are needed:

E(J,M) =- jcoA(J) -V<D(J)

--V x F(M) (3)

andH(J,M)=- j)F(M)-VT(M)

+-VxA(J)

6(4)

These operators are expressed in terms of the

electric and magnetic scalar potentials 4 and 'P and the

magnetic and electric vector potentials A and F due to

equivalent sources J and M, respectively. The potential

formulation minimizes the order of singularities that

appear in the kernels of the associated integro-

differential operators. In order to completely determine

the potentials, appropriate Green's functions, as

discussed below, must also be specified.

In the current development both integral and partial

differential equation formulations as well as hybrid

formulations employing both types of operators are

under way. Finite element method s directly attempt to

solve partial differential equation formulations such as

the vector Helmholtz equations

SX (Ju-' - V x E )- ko26-E =-jcviuJ-VX(ur_1M)

or

V x (gj -Vx H)-k02Pr'H =

- jtyom - 0 X (sjr' - J)(5)

(6)

The forcing functions are the source currents J or M,

which may be actual or equivalent sources.

Alternatively, the excitation may be due to sources

outside a region's boundary. Both the differential and

integral equation operators are enforced in a weak sense

in order to minimize differentiability requirements on

basis and testing functions.

Initially, EIGER was focused on frequency domain

problems. However, the object-oriented structure of

EIGER has facilitated extensions of the code to employ

static operators. The unknown electric and magnetic

currents (J and M) from the dynamic case are replaced

by potentials and gradients of potentials respectively ((D

and WD/3n) otherwise the code structure remains

identical. The code presently has the capability of

modeling perfect electric conductors, perfect magnetic

conductors, and dielectric materials both in 2 and 3

dimensions for static operators. Also, a hybrid FEM

integral equation is available in 2d and 3d. The EIGER

pre-processor is being modified to output the

associations needed for static analysis so once a

structure is meshed it may be analyzed with either static

or dynamic excitations.2

American Institute of Aeronautics and AstronauticsY . IV LL/ VVI...L

I I VL

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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/m1/3/: accessed February 17, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.