Optimal discretization based adaptive finite element analysis for electromagnetics with vector tetrahedra

Abstract

Abstract-Efficient functional derivative formulas suitable for optimal discretization based refinement criteria are developed for 3-D adaptive finite element analysis (FEA) with vector tetrahedra. Results for generalized vector Helmholtz systems are derived directly from first principles, and confirmed numerically through fundamental benchmark evaluations. Practical adaption applications are illustrated for selected FEA refinement models.Index Terms-Adaptive systems, electromagnetic analysis, error analysis, finite element methods.

Optimal discretizations in adaptive finite element electromagnetics

“…More recently, a number of extended variations of the grading function approach, designed for 2-D, 3-D, and even approximation problems, were reported in References [36; 38-40]. Overall, the most promising of these contributions were subsequently found to be potentially unreliable, or at best inconclusive, for practical ÿnite element applications [44]. Finally, two other EP methods were proposed in References [29; 41].…”

“…The counterparts of the optimization equations derived for 1-D systems may be developed in a common form for 2-D and 3-D simplex elements. However, while the principal steps of the development are the same, the speciÿcs of their implementation can be substantially di erent, and in some instances considerably more complex for the 3-D case [44]. Consequently, the more general 3-D case is considered ÿrst, followed by a brief explanation for extracting the corresponding 2-D equations.…”

“…Ultimately, the matrix equations corresponding to this formulation will be solved, using standard optimization methods, in order to compute truly optimal ÿnite element solutions for a set of benchmark electromagnetic systems. These solutions will be of the highest accuracy possible for the level of problem reÿnement speciÿed, for the variational approach used, because the variational principle underlying the problem formulation is itself used as the basis for deriving the optimization equations [44].…”

“…after some simpliÿcation, as explained in Reference [44]. Finally, the true solution to Equation (6) over the problem domain , subject to the boundary conditions in Equation (7), is the admissible function u for which the above functional F is stationary.…”

“…Overall, and still consistent with the ÿndings of Reference [33], the current body of published knowledge on optimal discretizations is insu cient to provide a reliable operational characterization of optimal ÿnite element discretizations for electromagnetics analysis. A fully expanded review and detailed discussion of the abridged survey presented above is provided in Reference [44] …”