High-order vector bases for computational electromagnetics

High-order vector bases for computational electromagnetics,10.1109/CEM.2011.6047355,Roberto D. Graglia

High-order vector bases for computational electromagnetics  
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New families of hierarchical curl and divergence- conforming vector bases for the most commonly used two- and three-dimensional 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 mixed-order (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 2-D and 3-D and surface discretizations of the elec- tric and magnetic field integral equations in 3-D. 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 ill-conditioned system of equations. The behavior of the numerical error as a function of the used cell-size 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 higher-order 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 volume-based 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 (p-adaption); (b) the generating polynomials for the edge, the face, and the volume-based vector functions can be implemented in routines that can be used without modification to evaluate the 2-D or the 3-D vector functions on cells of different shape; this greatly simplifies the implementation of the numerical codes required to deal with 2-D or 3-D structures; (c) our higher-order 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.
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