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Extra resources for 13th Int'l Conference on Numerical Methods in Fluid Dynamics
If the discrete set of wave vectors has been chosen such that they form a coherent path around the irreducible Brillouin zone the band structure can be plotted as a band diagram, also known as a dispersion relation (Figure 2-14). The normalised frequencies of the modes are plotted against the wave vector. If a photonic band gap exists for the photonic crystal being modelled this will be seen as a frequency range that is not intersected by any of the bands. 34 Photonic Band Gap Figure 2-14 Example of a band diagram for a photonic crystal.
It is beneficial to number nodes such that their difference in an element is minimised. This reduces the bandwidth of the resulting diagonal matrices thereby increasing the efficiency with which the resulting eigenvalue problem can be solved (Fagan 1992). 10 Reciprocal Lattice Space and the Brillouin Zone The periodicity of the crystal lattice allows us to model a crystal of infinite extent by considering only the unit cell, an area bounded by the lattice vectors, as illustrated in Figure 2-10.
3. g. the two vertex coordinates. 4. g. the three vertex coordinates. Any scalar function of ξ s multiplied by the vector ∇ξ will possess the same tangential continuity characteristics hence it remains to specify a combination of ξ and ∇ξ that are suitable for use as a vector interpolation function. The necessity is that the interpolation functions be linearly independent of one another. First order vector interpolation functions 6 1 to with the required linear independence have been proposed by (Nedelec 1986) and (Bossavit and Verite 1982) and have therefore been named Nedelec-Bossavit edge elements.