Calculating the Band Structure of Photonic Crystals through the Meshless Local Petrov-Galerkin (MLPG) Method and Periodic Shape Functions

摘要

This paper illustrates how to determine the bandgap structure of photonic crystals through meshless methods. These methods are generally akin to the Finite Element Method (FEM), as they also deal with the discretization of weak forms and produce sparse stiffness matrices. The major difference is the complete absence of whatever kind of mesh. We concentrate in a particular method, the Meshless Local Petrov Galerkin 4 (MLPG4), also known as Local Boundary Integral Equation Method (LBIE). Since the boundary conditions governing the electromagnetic fields are periodic in a unit cell, we develop a special scheme to embed this feature on the shape functions used to approximate the field. As a result, the boundary conditions do not need to be imposed on the unit cell.