FEM analysis of dielectric loaded waveguides with additive hierarchical singular vector elements

Summary form only given. For problems with smooth surfaces or other regular features, high order hierarchical bases successfully improve accuracy and efficiency. However, for geometries with edges or corners where unbounded fields or other singular types of behavior occur, special bases that incorporate the singular field behavior are better at improving the solution accuracy. Recently, the authors (together with other co-authors) proposed additive basis sets that offer improved generality for high order expansions. These additive bases retain the entire original polynomial set and augment it with additional singular basis functions that define the so-called Meixner subset. Additive bases are more flexible than other type of bases (e.g., those employing substitutive basis functions) and can model appropriate field behavior even if the expected singularity is not excited by the source, or if the cells are electrically large.A drawback to the additive approach is that the resulting system of equations is often poorly conditioned, due to the lack of linear independence between the regular and Meixner subspaces except in the immediate vicinity of the singularity. For 2D problems, a scalar basis set incorporating the appropriate singularities for the longitudinal field was described in [(1) R.D. Graglia, A.F. Peterson, L. Matekovits, “Singular, hierarchical scalar basis functions for triangular cells,” IEEE TAP, vol. 61, no. 7, pp. 3674-3692, 2013]. Vector singular bases have been recently developed in [(2) R.D. Graglia, A.F. Peterson, L. Matekovits, P. Petrini, “Hierarchical additive basis functions for the Finite Element treatment of corner singularities,” Special Issue on Finite Elements for Microwave Engineering, Electromagnetics, vol. 34, no. 3 & 4, 2014; (3) R.D. Graglia, A.F. Peterson, L. Matekovits, P. Petrini, “Singular hierarchical curl-conforming vector bases for triangular cells,” submitted to IEEE TAP]. T- ese singular bases are formed using orthogonal functions to reduce the ill-conditioning of the resulting system matrix. Our new bases have three distinguishing features: (a) the Meixner vector basis functions are subdivided from the outset into two different groups of edge and face-based functions; (b) in each group, each basis function is obtained from mutually orthogonal scalar functions; (c) the hierarchical vector functions are either symmetric or antisymmetric with respect to the local parent variables that describe each cell. In this presentation, the additive hierarchical bases for triangular cells are used to study inhomogeneous dielectric-loaded waveguides using the transverse-longitudinal-field method in terms of the three components of the electric field. Results for the dispersion diagram and field topographies of various waveguides obtained with the new singular bases will be compared with those obtained by using purely polynomial bases.