11th International Workshop on Finite Elements for Microwave Engineering - FEM2012, June 4-6, 2012, Estes Park, Colorado, USA
Session 8 - Adaptive FEM, Higher Order Bases, and Advanced FEM Formulations
Rules for Adoption of Expansion and Integration Orders in FEM Analysis
Using Higher Order Hierarchical Bases on Generalized Hexahedral Elements
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Nada J. Šekeljić* 1, Slobodan V. Savić2, Milan M. Ilić1,2, and Branislav M. Notaroš1
Colorado State University, Electrical & Computer Engineering Department, Fort Collins, CO, USA
2
University of Belgrade, School of Electrical Engineering, Belgrade, Serbia
[email protected], [email protected], [email protected], [email protected]
There has lately been a noticeable interest within the computational electromagnetics (CEM) community
for higher order techniques, which rely on basis functions of high orders set on electrically large
(whenever possible) elements (B. M. Notaroš, “Higher Order Frequency-Domain Computational
Electromagnetics,” IEEE Trans. Antennas Propagat., Vol. 56, August 2008). Additionally, higher order
techniques allow for efficient utilization of irregular, elongated, and curved elements, defined using
Lagrange interpolating polynomials, Bézier curves, or NURBS curves, which can be tremendously
beneficial in creating effective meshes and in yielding the final systems of equations with significantly
fewer number of unknowns. Most importantly, higher order bases and elements enable efficient p- and
hp-refinements, as well as adaptive CEM schemes.
However, the great modeling flexibility of higher order elements, basis and testing functions, and
integration procedures, which is the principal advantage of the higher order CEM, is also its greatest
shortcoming. Namely, it poses numerous dilemmas, uncertainties, options, and decisions to be made on
how to actually use these elements and functions. In other words, with the additional degrees of freedom
in modeling, a user has to handle many more parameters in building a CEM model, which requires a
great deal of modeling experience and expertise, and possibly considerably increases the overall
simulation (modeling plus computation) time. Examples of questions that need to be addressed are how
large the elements can be, what polynomial orders of bases should be used in particular cases, and how
accurate the integration has to be depending on the adopted polynomial orders in different directions.
Within efforts to answer these questions, we have recently established and validated general guidelines
and instructions, and as precise as possible quantitative rules, for adoptions of optimal higher order
parameters for electromagnetic modeling using the method of moments (MoM) in single precision
computations. For instance, we have concluded that (in cases with well-behaved fields) the Lagrange-type
(curved or flat) quadrilaterals as large as up to two wavelengths on a side, with polynomial orders of the
equivalent surface current approximation of N=6 in each direction and the number of points in GaussLegendre quadrature integrations of about N+2, are generally optimal.
In our continued study of higher order parameters in CEM, this paper addresses the same problem and
defines similar rules for the finite element method (FEM) using p- and hp-refined higher order
hierarchical generalized hexahedral elements and models in double precision computations. To define
these rules and draw some general conclusions, we carefully and systematically study a diverse and
comprehensive set of higher order FEM simulations of three-dimensional (3-D) cavities, 2-D waveguides,
and 3-D scatterers (with the FEM domain closed using the first order absorbing boundary condition or a
fixed high-order boundary integral), applying exhaustive sweeps in frequency, polynomial orders of the
field approximation, and numbers of points in the Gauss-Legendre integration (integration accuracy).
Preliminary results show that, while it can generally be adopted that curved Lagrange-type hexahedral
finite elements can be as large as two wavelengths on a side (analogously to the MoM analysis),
somewhat larger numbers of integration points can be used in each direction than those reported in the
MoM study. However, higher integration accuracy results in possibly substantially longer matrix filling
times; hence, in this work, we also discuss a compromise between the two requirements.
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