IEEE TRANSACTIONS ON MAGNETICS, VOL. 48, NO. 2, FEBRUARY 2012 551 Calculating the Band Structure of Photonic Crystals Through the Meshless Local Petrov-Galerkin (MLPG) Method and Periodic Shape Functions Williams Nicomedes , Renato Mesquita , and Fernando Moreira Dept. of Electronic Engineering, Federal University of Minas Gerais, Belo Horizonte, MG 31270-901, Brazil Dept. of Electrical Engineering, Federal University of Minas Gerais, Belo Horizonte, MG 31270-901, Brazil This paper illustrates how to determine the bandgap structure of photonic crystals through MLPG. This method is akin to the Finite Element Method (FEM), as it also deals with the discretization of weak forms and produces sparse global matrices. The major difference is the complete absence of any kind of mesh. We concentrate in a particular version of it, the MLPG4, also known as Local Boundary Integral Equation Method (LBIE). Since the boundary conditions governing the electromagnetic field are periodic in a unit cell, we develop a special scheme to embed this feature on the shape functions used in the discretization process. As a result, boundary conditions do not need to be imposed on the unit cell. Index Terms—Electromagnetic wave propagation, finite element methods, integral equations, photonic crystals. I. INTRODUCTION M ESHLESS (or meshfree) methods are a paradigm for finding numerical solutions to differential equations de- fined in geometrical domains without resorting to a mesh (like FEM) or to a grid (like finite-difference methods). In Electro- magnetics there have been such appearances [1], where a mesh- less method called Element-Free Galerkin (EFG) is employed. MLPG, the particular method investigated in this paper, differs somehow from the other methods. It is peculiar in the sense that the numerical integrations are carried out in certain local domains, thus dismissing any sort of background cells [2]. Be- sides that, MLPG employs two kinds of functions, shape and test functions, which belong to two different spaces. The shape functions are constructed numerically, whereas there are many choices available to the test functions. As for the latter, we took the solution to Green’s problem for Laplace’s equation, which leads to MLPG4, or LBIE [2]. Recently, MLPG has been applied to situations in 2D electro- magnetic wave propagation [3] and in 3D electrostatics [4]. We now look for applications of MLPG4 to eigenvalue problems, specifically those arising in the analysis of 2D photonic bandgap crystals. After some discussion on periodic shape functions, we present a pair of examples concerning the band structure for the polarization. The results are compared to those in [5] and [6], which solve the same problems by other methods. II. MESHLESS METHODS:OVERVIEW AND SHAPE FUNCTIONS Let be a two-dimensional domain (global boundary ). In order to find a numerical approximation for a function , we begin by spreading nodes across . To each node is ascribed an index, i.e., a natural number . The next step is to define shape functions associated to each node. They do not present Manuscript received July 07, 2011; revised October 26, 2011; accepted Oc- tober 27, 2011. Date of current version January 25, 2012.. Corresponding au- thor: W. L. Nicomedes (e-mail: wlnicomedes@yahoo.com.br). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2011.2175206 analytical expressions; a shape function usually depends on the relative positions of neighboring nodes. Furthermore, they are compactly supported, i.e., different from zero only at a small region surrounding the node (called the node’s influence domain ). So, given a point , there follows: (1) where the global index runs through all nodes whose influence domains include point and each is the nodal parameter (a coefficient to be determined). There are many ways in which the shape functions can be con- structed. In this work, a procedure called Moving Least Squares (MLS) has been employed [7]. Given a point at which the shape functions are to be calculated, one first finds all neigh- boring nodes which extend their influence domains until (e.g., four nodes with global indices 3, 5, 10 and 18). Then one feeds this information to a numerical procedure [4] which returns the shape functions and their derivatives calculated at : (2) where denotes the derivatives with respect to or , is the shape function associated to node 3 (located at ) calculated at , and so on. III. CALCULATING THE BAND STRUCTURE OF A PHOTONIC CRYSTAL:THE MLPG METHOD In LBIE, to each node at is ascribed a test function , in addition to a shape function [4]. This test function acts in a specific region surrounding the node, the test domain . In LBIE, is required to be a circle centered at each node. Conditions on are (a Dirac delta at ) and at the boundary of the test domain, i.e., . A function satisfying the above requirements is given by . In this work, the radii of all test and influence domains are equal. A two-dimensional photonic bandgap crystal is a periodic array of dielectric structures, one of the most remarkable properties of which is that it is able to select what wavelengths 0018-9464/$31.00 © 2012 IEEE