Electromagnetic Interactions GEneRalized (EIGER): Algorithm abstraction and HPC implementation Page: 3 of 9
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r = YrRi(p,4)
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)
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)
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 =
V x (gj -Vx H)-k02Pr'H =
- jtyom - 0 X (sjr' - J)
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.
American Institute of Aeronautics and Astronautics
Y . 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.