## Integral equation analysis of electromagnetic wave propagation in periodic structure and error analysis of various basis functions in projection of plane waves
Hu, Fu-Gang
(2010)
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## AbstractIn the first part of this dissertation, the integral equation approaches are developed to analyze the wave propagation in periodic structures. Firstly, an integral equation approach is developed to analyze the two-dimensional (2-D) scattering from multilayered periodic array. The proposed approach is capable of handling scattering from the array filled with different media in different layers. Combining the equivalence principle algorithm and connection scheme (EPACS), it can be avoided to find and evaluate the multilayered periodic Green's functions. For 2^N identical layers, the elimination of the unknowns between top and bottom surfaces can be accelerated using the logarithm algorithm. More importantly, based on EPACS, an approach is proposed to effectively handle the semi-infinitely layered case in which a unit consisting of several layers is repeated infinitely in one direction. Secondly, the integral-equation (IE) method formulated in the spatial domain is employed to calculate the scattering from the doubly periodic array of three-dimensional (3-D) perfect electric conductor (PEC) objects. The special testing and basis functions are proposed to handle the problem with non-zero normal components of currents at the boundary of one period. Moreover, a relationship between the scattering from the PEC screen and its complementary structure is established. In order to efficiently compute the matrix elements from the IE approach, an acceleration technique with the exponential convergence rate is applied to evaluate the doubly periodic Green's function. The formulations in this technique are appropriately modified so that the new form facilitates numerical calculation for the general cases. In the second part of this dissertation, the error analysis of various basis functions in projection of the plane wave was conducted, including pulse basis, triangular basis, the basis of their higher-order version, and the divergence-conforming basis on rectangular and triangular elements. The projection error is given analytical, asymptotically, and numerically. The application of the p-th order one-dimensional (1D) basis can result in the projection error which is asymptotically proportional to (p+1)-th power of the density of unknowns. Based on the analytical projection errors in 1D case, it is found when the expansion basis is fixed, the application of different testing functions only affect the constant coefficient of the projection error rather than the order. Generally, the error of divergence-conforming basis in projection of curl-free vectors is less than that of divergence-free vectors.
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