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Fourier series
About: Fourier series is a research topic. Over the lifetime, 16548 publications have been published within this topic receiving 322486 citations.
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TL;DR: In this paper, a probabilistic analysis of buckling loads is performed regarding Fourier coefficients as random variables, and Monte Carlo simulations are executed for a set of six similarly manufactured composite shells.
Abstract: The discrepancy between the analytically determined buckling load of perfect cylindrical shells and experimental test results is traced back to imperfections. The most frequently used guideline for design of cylindrical shells, NASA SP-8007, proposes a deterministic calculation of a knockdown factor with respect to the ratio of radius and wall thickness, which turned out to be very conservative in numerous cases and which is not intended for composite shells. In order to determine a lower bound for the buckling load of an arbitrary type of shell, probabilistic design methods have been developed. Measured imperfection patterns are described using double Fourier series, whereas the Fourier coefficients characterize the scattering of geometry. In this paper, probabilistic analyses of buckling loads are performed regarding Fourier coefficients as random variables. A nonlinear finite element model is used to determine buckling loads, and Monte Carlo simulations are executed. The probabilistic approach is used for a set of six similarly manufactured composite shells. The results indicate that not only geometric but also nontraditional imperfections like loading imperfections have to be considered for obtaining a reliable lower limit of the buckling load. Finally, further Monte Carlo simulations are executed including traditional as well as loading imperfections, and lower bounds of buckling loads are obtained, which are less conservative than NASA SP-8007.
59 citations
Patent•
CGG1
TL;DR: In this article, an improved method for analyzing seismic data to obtain elastic attributes is disclosed in one embodiment, a reflectivity series is determined for at least one seismic trace of seismic data obtained for a subterranean formation, where the reflectivities series includes anisotropy properties of a formation.
Abstract: An improved method for analyzing seismic data to obtain elastic attributes is disclosed In one embodiment, a reflectivity series is determined for at least one seismic trace of seismic data obtained for a subterranean formation, where the reflectivity series includes anisotropy properties of a formation One or more synthetic seismic traces are obtained by convolving the reflectivity series with a source wavelet The one or more synthetic seismic traces are inverted to obtain elastic parameters estimates According to one aspect, the data inputs are angle-azimuth stacks According to another aspect, the data inputs are azimuthal Fourier coefficients, un, vn
59 citations
Book•
11 Jun 1992
TL;DR: In this paper, the authors define compatible transforms for boundary and initial value problems in the context of second-order differential equations (SODE) and derive the corresponding corresponding inverse transforms.
Abstract: Part 1 Compatible transforms: the method of separation of variables and the integral transforms compatible transforms classification of the transforms comments on the inverse transforms - tables of the transforms the compatible transform and the adjoint problem constructing the compatible transforms for self-adjoint problems - second-order differential equations the nth-order differential operator. Part 2 Integral transforms: Laplace transforms Fourier exponential transforms boundary and initial value problems - solutions by Fourier transforms signals and linear systems - representation in the Fourier (spectrum) space Fourier sine and cosine transforms higher-dimensional Fourier transforms the Hankel (bessel) tranforms Laplace transform inversion other important integral transforms. Part 3 Finite transforms - Fourier series and coefficients: Fourier (trigonometric) series and general orthogonal expansion Fourier sine and cosine transforms Fourier (exponential) transforms Hankel (bessel) transforms classical orthogonal polynomial transforms the generalized sampling expansion a remark on the transform methods and nonlinear problems. Part 4 Discrete transforms discrete Fourier transforms discrete orthogonal polynomial transforms bessel-type poisson summation formula (for the Bessel-Fourier series and Hankel transforms). Appendix A: basic second-order differential equations and their (series) solutions - special functions. Appendix B: mathematical modeling of partial differential equations - boundary and initial value problems. Appendix C: tables of transforms.
59 citations
TL;DR: In this paper, the problem of computing wavelet coefficients of compactly supported functions from their Fourier samples was studied and generalized sampling was used to obtain a stable and accurate reconstruction.
59 citations
TL;DR: Cubic Schrödinger equations with small initial data (or small nonlinearity) and their spectral semi-discretizations in space are analyzed and it is shown that along both the solution of the nonlinear Schr Ödinger equation as well as the solutions of the semi-Discretized equation the actions of the linear SchröDinger equation are approximately conserved over long times.
Abstract: Cubic Schrodinger equations with small initial data (or small nonlinearity) and their spectral semi-discretizations in space are analyzed. It is shown that along both the solution of the nonlinear Schrodinger equation as well as the solution of the semi-discretized equation the actions of the linear Schrodinger equation are approximately conserved over long times. This also allows us to show approximate conservation of energy and momentum along the solution of the semi-discretized equation over long times. These results are obtained by analyzing a modulated Fourier expansion in time. They are valid in arbitrary spatial dimension.
59 citations