Fourier series

About: Fourier series is a research topic. Over the lifetime, 16548 publications have been published within this topic receiving 322486 citations.

Free vibrations of functionally graded porous rectangular plate with uniform elastic boundary conditions

Abstract: In this paper, the free vibrations of functionally graded porous (FGP) rectangular plate with uniform elastic boundary conditions is investigated by means of an improved Fourier series method (IFSM). It is assumed that the distributions of porosity are uniform or non-uniformly along a certain direction and three types of the porosity distribution are considered, among which material property of two non-uniform porous distributions was expressed as the simple cosine. The size of the pore in a rectangular plate is determined by the porosity coefficients. Using the first-order shear deformation theory(FSDT), the energy expression of FGP rectangular plate is created. In order to obtain the admissible function of displacement for functionally graded porous rectangular plate, the IFSM is employed. Then, the Rayleigh-Ritz method is used to solve coefficients in the Fourier series which determine natural frequencies and modal shapes. Convergence and comparative research are performed to prove the convergence, reliability and accuracy of the current method. On this foundation, some new results covering the influence of the geometrical parameters subject to classical and elastic boundary condition are presented, and the parametric studies are also investigated in detail, which can provide a reference for future research by other researchers.

The Fourier transform Coulomb method: Efficient and accurate calculation of the Coulomb operator in a Gaussian basis

Abstract: We describe a method for calculating the matrix elements of the Coulomb operator for Gaussian basis sets using an intermediate discrete Fourier transform of the density. Our goals are the same as those of the Gaussian and augmented-plane-wave method of Parrinello and co-workers [M. Krack and M. Parrinello, Phys. Chem. Chem. Phys. 2, 2105 (2000)], but our techniques are quite different. In particular, we aim at much higher numerical accuracy than typical programs using plane wave expansions. Our method is free of the effects of periodic images and yields full precision. Other low-scaling methods for the Coulomb operator are compared to the Fourier transform method with regard to numerical precision, asymptotic scaling with molecular size, asymptotic scaling with basis set size, onset point (the size of the calculation where the method outperforms traditional Gaussian integral techniques by a factor of 2), and the ability to calculate the Hartree–Fock exchange operator. The Fourier transform method is superior to alternatives by most criteria. In particular, for typical molecular applications it has an earlier onset point than fast multipole methods.