In this paper, a Petrov–Galerkin finite element interface method (PGFEIM) is proposed to compute the band structures of 2D photonic crystals (PtCs) with complex scatterer geometry, which is formulated as a generalized eigenvalue problem (GEP) for given wave vectors. The key idea of this method is to choose a piecewise linear function satisfying the jump conditions across the interface to be the solution basis, and choose a special finite element basis independent of the interface to be the test function basis to remove the boundary term upon the assertion of Bloch boundary conditions. Non-body-fitting projected grid is employed to implement this approach. Both isotropic and anisotropic materials are considered and discussed for two- and three-component PtCs with square or triangular lattice. Taking advantage of the PGFEIM in dealing with sharp-edged interfaces, PtCs with various peculiar scatterer geometries are studied. Particularly, some distinctive three-component structures with triple-junction points between different materials are fabricated, and the middle frequencies of its absolute photonic band gaps are higher than conventional three-component structures. Numerical examples demonstrate that the bands move to higher or lower frequency regions, which is determined by the component materials, with the increase of number of sharp corners on the surface of entire scatterer.

本文提出了一种彼得罗夫-加勒金有限元界面方法（PGFEIM）来计算具有复杂散射几何形状的二维光子晶体（PtCs）的能带结构，并将其表示为广义特征值给定波矢的问题（GEP）。该方法的关键思想是选择满足界面上跳跃条件的分段线性函数作为求解基础，并选择独立于接口的特殊有限元基础作为测试函数基础，以消除边界条件。布洛赫边界条件的断言。采用非拟合投影网格来实现此方法。对于具有正方形或三角形晶格的两组分和三组分PtC，均考虑和讨论了各向同性和各向异性材料。利用PGFEIM处理尖锐的界面，研究了具有各种特殊散射体几何形状的PtC。特别是，制造了一些在不同材料之间具有三结点的独特三组分结构，其绝对光子带隙的中频高于传统的三分量结构。数值示例表明，随着整个散射体表面上尖角数量的增加，波段移动到由成分材料决定的更高或更低的频率区域。