Local stress in periodic composites via the Riesz summability method

Different summation methods of multidimensional trigonometric Fourier series are adopted for approximating the local strain and stress in elastic periodic composites. In fact, approximating the exact solution of a problem with the original partial sums of Fourier series can produce unwanted effects, such as the Gibbs phenomenon at the jump discontinuities that may occur at the interface between the constituents of the composites. Nevertheless, the Fourier coefficients of the original partial sums contain enough information to provide better estimates via the summability methods. In the means provided by the summability methods, the Fourier coefficients are suitably weighted by reducing windows, involving a regularization of the numerical solution. First, a complete solution method is used in order to determine some Fourier coefficients of the local strain in periodic composites. Next, these Fourier coefficients are used to construct suitable summations and means exhibiting better convergence properties than those of the original partial sums of Fourier series. The proposed method, valid for a great variety of the geometry of the constituents embedded in the matrix, is here applied to unidirectional composites with cylindrical fibers. The behavior of the iterated Fejer partial sums, Fejer and Riesz means is investigated in order to improve the convergence, observing that the Riesz means have better convergence properties than those of the original partial sums of Fourier series.