Moderate degree cubature formulas for 3‐D tetrahedral finite‐element approximations

Abstract

SUMMARYSome new moderate degree cubature formulas for tetrahedra are derived. The suitability of various types of rules for the finite-element technique is discussed exhaustively. Consequently, the newly developed formulas supplemented by lower degree known ones are tabulated for the purpose of direct application in finite-element programming. All the formulas are presented in a virtually machine-independent form.

Nonconforming Discretization of the Electric-Field Integral Equation for Closed Perfectly Conducting Objects

“…We show the computed near-field and RCS results for several sharp-edged objects, with either small or moderate electrical dimensions. In all the testing cases, the exciting plane wave is x-polarized, with z-propagation and m. In order to obtain a fair assessment of the observed relative performance, we minimize the sources of error in the MoM-implementations through the following actions: (i) Direct solution of the resulting matrix system; (ii) Computation of the inner surface and volumetric integrals in the impedance elements through the singularity subtraction of the -Kernel contributions [21] and the numerical evaluation of the remaining low-order Kernel contributions with Gaussian numerical rules (9 quadrature points for the surface inner integrals [22] and 11 cubature points for the volumetric inner integrals [23]); (iii) Computation of the outer line and surface integrals with a 9-point quadrature rule; (iv) Computation of the outer volumetric integrals with a cubature rule of 11 points; (v) Swapping of the inner line-integrals with the outer surface or volumetric integrals; (vi) Accurate testing of the incident fields, with a three-point quadrature rule for the surface testing and with a five-point cubature rule for the volumetric testing; (vii) Accurate computation of the source-integrals in the computed RCS with a three-point quadrature rule.…”

“…Therefore, for a given degree of discretization, the computational time required for the matrix generation is higher than for the conventional RWG discretization. Moreover, the computation of the three additional submatrices in (23), (27) and (28) becomes especially time consuming because of the numerical evaluation of volumetric and line integrals in addition to the surface integrals. An equivalent matrix definition to the matrix system in (19) can be obtained by using the original monopolar-RWG basis functions instead of the even-odd monopolar-RWG rearrangement, and by keeping the RWG basis functions in (15) and the volumetric functions in (26) [14] or the electric-magnetic field integral equation (EMFIE) [9], [15].…”

“…1). In view of the normal-continuity of the testing RWG functions, this expression may be simplified as (23) where it is better suited for enhanced numerical accuracy to set the source line-integrals as outer integrals and to extract analytically the quasi-singular Kernel contributions from the inner surface integrals so that (24) In view of Fig. 1, the unit vectors perpendicular to the th edge, and , lying, respectively, on the triangles and are in general independent, except for co-planar triangles, where…”

Improved error and work estimates for high‐order elements

“…It is for this reason that the usual integration work estimates were used here.Polynomials for simplex elements:The ratio of DOFs for simplex (tria/tet) elements compared with tensor‐product Lagrange (quad/hex) elements may be approximated by the following: where the following values provide good fits (Figure ): 2D: a = 0.51, b = 0.50, c = 1.10;3D: a = 0.18, b = 0.82, c = 1.30;implying for the DOF per element the number of matrix entries per element and the number of elements where c v = 2.0 for 2D and c v = 5.5 for 3D.As before, the work required for numerical integration has to be taken into account. In reviewing the literature on cubature formulas , one can approximate the number of Gauss‐points required for a funct...…”

“…As before, the work required for numerical integration has to be taken into account. In reviewing the literature on cubature formulas , one can approximate the number of Gauss‐points required for a function of polynomial order p by where, surprisingly, k = 2.5 for 3D (Figure ).…”