Giacomo Oliveri and Mirco Raffetto
The paper's purpose is to deduce additional information on the accuracy of finite element simulators for electromagnetic problems involving effective models of metamaterials.
Abstract
Purpose
The paper's purpose is to deduce additional information on the accuracy of finite element simulators for electromagnetic problems involving effective models of metamaterials.
Design/methodology/approach
The objective is achieved by solving, with a well known commercial simulator, many different configurations of two types of electromagnetic problems: a free‐space scattering by a sphere and a waveguide discontinuity problem. Such problems are known to be able to point out the difficulties of numerical simulators. On the other hand, they are representative of two important classes of problems and can provide indications on what can happen in other cases.
Findings
This analysis confirms that the numerical errors can be important just in close proximity of the interface between metamaterials and standard media. Small values of loss tangents can be sufficient to obtain very accurate results. Adaptive mesh generators should not be used in the presence of negligible values of the loss tangents. For more uniform meshes the results are satisfactory, with sufficiently fine meshes. When the magnitude of the real parts of the effective dielectric permittivity of a metamaterial and of the adjacent standard media are significantly different, the accuracy is satisfactory in any case.
Research limitations/implications
The results are obtained by considering problems of two types. There is no guarantee that all the deductions apply to other models.
Practical implications
To design practical devices involving metamaterials reliable electromagnetic simulators are necessary. The reported results seem to indicate that it is possible to adopt some countermeasures against the possible lack of accuracy of finite element simulators in the presence of effective models of metamaterials.
Originality/value
For the first time, to the best of authors' knowledge, an extensive analysis on the accuracy of finite element simulators for critical problems involving metamaterials has been carried out. Some simple suggestions to improve their reliability in these cases are provided.
Details
Keywords
Paolo Fernandes and Mirco Raffetto
To provide sufficient conditions for existence, uniqueness and finite element approximability of the solution of time‐harmonic electromagnetic boundary value problems involving…
Abstract
Purpose
To provide sufficient conditions for existence, uniqueness and finite element approximability of the solution of time‐harmonic electromagnetic boundary value problems involving metamaterials.
Design/methodology/approach
The objectives are achieved by analysing the most simple conditions under which radiation, scattering and cavity problems are well posed and can be reliably solved by the finite element method. The above “most simple conditions” refer to the hypotheses allowing the exploitation of the simplest mathematical tools dealing with the well posedness of variationally formulated problems, i.e. Lax‐Milgram and first Strang lemmas.
Findings
The results of interest are found to hold true whenever the effective dielectric permittivity is uniformly positive definite on the regions where no losses are modelled in it and, moreover, the effective magnetic permeability is uniformly negative definite on the regions where no losses are modelled in it. The same good features hold true if “positive” is replaced by “negative” and vice versa in the previous sentence.
Research limitations/implications
It is a priori known that more sophisticated mathematical tools, like Fredholm alternative and compactness results, can provide more general results. However this would require a more complicated analysis and could be considered in a future research.
Practical implications
The design of practical devices involving metamaterials requires the use of reliable electromagnetic simulators. The finite element method is shown to be reliable even when metamaterials are involved, provided some simple conditions are satisfied.
Originality/value
For the first time to the best of authors' knowledge a numerical method is shown to be reliable in problems involving metamaterials.
Details
Keywords
Paolo Fernandes and Mirco Raffetto
From a theoretical point of view the question of spurious modes has been regarded as a closed problem. However, in this paper we show that even a precise definition of…
Abstract
From a theoretical point of view the question of spurious modes has been regarded as a closed problem. However, in this paper we show that even a precise definition of spurious‐free approximation was lacking. Hence, a sound definition of spurious‐free finite element method is given and a set of necessary and sufficient conditions ensuring that a finite element method is spurious‐free in the defined sense is stated. A critical comparison between the proposed theory and the currently accepted one is then carried out and existing counterexamples to the latter are pointed out. Comparison with an older theory leads to another set of necessary and sufficient conditions providing a better grasp of the key feature a finite element space must have to rule out spurious modes. The impact of the proposed theory is stressed, showing that Nedelec's tetrahedral edge elements of all orders provide spurious‐free approximations in all conditions of practical interest. Finally, it is shown, for the first time to the best of authors’ knowledge, that also many high‐order edge elements, recently proposed in the engineering literature for the analysis of electromagnetic problems, provide the same kind of reliable approximation.