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Keywords
(11)
Adaptive Refinement
Cell Size
Computational Electromagnetics
Helmholtz Equation
Inner Product
Linear Independence
Spectral Properties
System of Equations
Three Dimensional
Higher Order
Magnetic Field Integral Equation
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Highorder vector bases for computational electromagnetics
Highorder vector bases for computational electromagnetics,10.1109/CEM.2011.6047355,Roberto D. Graglia
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Highorder vector bases for computational electromagnetics
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Roberto D. Graglia
New families of hierarchical curl and divergence conforming vector bases for the most commonly used two and threedimensional cells are directly constructed from orthogonal scalar polynomials to enhance their linear independence, which is a simpler process than an orthogonalization applied to the final vector functions. These functions span the mixedorder (or reduced) spaces of N´ ed´ elec and can be used to deal with structures meshed by a mixture of cells of different geometry. I. SUMMARY Vector basis functions find wide application in electromag netics for volumetric discretizations of the vector
Helmholtz equation
in 2D and 3D and surface discretizations of the elec tric and magnetic field integral equations in 3D. These basis functions can be interpolatory, with coefficients that represent specific field components at interpolation points, or they can form hierarchical sets in order to facilitate
adaptive refinement
procedures. In contrast to interpolatory bases, hierarchical bases often exhibit poor
linear independence
as the order of the representation is increased, resulting in an illconditioned system of equations. The behavior of the numerical error as a function of the used cellsize and of the bases order is first discussed in rather general terms by considering interpolation procedures of different order of a scalar quantity, as well as the
spectral properties
of the functions reconstructed by the various interpolation procedures. This study permits to assess the superior quality of higherorder representations for a decreasing (total) number of basis functions on cells of increasing dimensions. The presentation will then review new hierarchical families of vector bases recently developed that are able to alleviate the loss of
linear independence
usually encountered for increasing representation order. We consider N´´ elec vector bases of the curl and of the divergence conforming kind for triangular and quadrilateral cells, and for tetrahedral, brick, and prism cells. The new bases have four distinguishing features: (a) the vector basis functions are subdivided from the outset into three different groups of edge, face, and volumebased functions; (b) each basis function is obtained by using one generating edge, face, or volume based polynomial whose analytical expression involves all the dependent parent variables that describe the element; (c) in each group, all the generating polynomials are mutually orthogonal, independent of the definition domain of the inner product, i.e., either the volume, the face, or the edge of the cell; (d) the hierarchical vector functions are either symmetric or antisymmetric with respect to the parent variables that describe each edge and face of the cell. The four features outlined above yield the following outcomes, respectively: (a) different individual polynomial orders can be used on each edge, face, and volumetric element of a given mesh, thereby facilitating the use of vector bases of different orders together in the same mesh (padaption); (b) the generating polynomials for the edge, the face, and the volumebased vector functions can be implemented in routines that can be used without modification to evaluate the 2D or the 3D vector functions on cells of different shape; this greatly simplifies the implementation of the numerical codes required to deal with 2D or 3D structures; (c) our higherorder bases maintain excellent
linear independence
because they are derived after an analytical orthogonalization of the generating scalar polynomials, and this is done in the element parent domain; (d) the procedure to enforce the conformity of the approximation across element interfaces is drastically simplified. Several numerical results obtained with the new hierarchical bases will be presented and discussed.
Conference:
Computational Electromagnetics Workshop  CEM
, 2011
DOI:
10.1109/CEM.2011.6047355
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References
(6)
Higher order interpolatory vector bases for computational electromagnetics
(
Citations: 263
)
Roberto D. Graglia
,
Donald R. Wilton
,
Andrew F. Peterson
Journal:
IEEE Transactions on Antennas and Propagation  IEEE TRANS ANTENNAS PROPAGAT
, vol. 45, no. 3, pp. 329342, 1997
Higher order interpolatory vector bases on prism elements
(
Citations: 40
)
Roberto D. Graglia
,
Donald R. Wilton
,
Andrew F. Peterson
,
IoanLudovic Gheorma
Journal:
IEEE Transactions on Antennas and Propagation  IEEE TRANS ANTENNAS PROPAGAT
, vol. 46, no. 3, pp. 442450, 1998
CurlConforming Hierarchical Vector Bases for Triangles and Tetrahedra
(
Citations: 5
)
Roberto D. Graglia
,
Andrew F. Peterson
,
Francesco P. Andriulli
Journal:
IEEE Transactions on Antennas and Propagation  IEEE TRANS ANTENNAS PROPAGAT
, vol. 59, no. 3, pp. 950959, 2011
Hierarchical vector polynomials for the triangular prism
(
Citations: 2
)
Roberto D. Graglia
,
Andrew F. Peterson
Conference:
International Conference on Electromagnetics in Advanced Applications  ICEAA
, 2010
Scale Factors and Matrix Conditioning Associated With TriangularCell Hierarchical Vector Basis Functions
(
Citations: 7
)
Andrew F. Peterson
,
Roberto D. Graglia
Journal:
IEEE Antennas and Wireless Propagation Letters  IEEE ANTENN WIREL PROPAG LETT
, vol. 9, pp. 4043, 2010