Search results for "Series expansion"
showing 10 items of 42 documents
Numerical evaluation of iterated integrals related to elliptic Feynman integrals
2021
We report on an implementation within GiNaC to evaluate iterated integrals related to elliptic Feynman integrals numerically to arbitrary precision within the region of convergence of the series expansion of the integrand. The implementation includes iterated integrals of modular forms as well as iterated integrals involving the Kronecker coefficient functions $g^{(k)}(z,\tau)$. For the Kronecker coefficient functions iterated integrals in $d\tau$ and $dz$ are implemented. This includes elliptic multiple polylogarithms.
The kite integral to all orders in terms of elliptic polylogarithms
2016
We show that the Laurent series of the two-loop kite integral in $D=4-2\varepsilon$ space-time dimensions can be expressed in each order of the series expansion in terms of elliptic generalisations of (multiple) polylogarithms. Using differential equations we present an iterative method to compute any desired order. As an example, we give the first three orders explicitly.
Investigation of the crack tip stress field in a stainless steel SENT specimen by means of Thermoelastic Stress Analysis
2019
Abstract In this work a Thermoelastic Stress Analysis (TSA) setup is implemented to investigates the Thermoelastic and Second Harmonic signals on a fatigue loaded Single Edge Notched Tension (SENT) specimen made of stainless steel AISI 304L. Three load ratios are in particular applied, R=-1, 0, 0.1. The thermoelastic signal is used to evaluate the Stress Intensity Factor via two approaches, the Stanley-Chan linear interpolation method and the over-deterministic least-square fitting (LSF) method using the Williams’ series expansion. Regarding least-square fitting, an iterative procedure is proposed to identify the optimal crack tip position in the thermoelastic maps. The SIF and T-Stress are…
Series expansion for the effective conductivity of a periodic dilute composite with thermal resistance at the two-phase interface
2019
We study the asymptotic behavior of the effective thermal conductivity of a periodic two-phase dilute composite obtained by introducing into an infinite homogeneous matrix a periodic set of inclusions of a different material, each of them of size proportional to a positive parameter ?. We assume that the normal component of the heat flux is continuous at the two-phase interface, while we impose that the temperature field displays a jump proportional to the normal heat flux. For ? small, we prove that the effective conductivity can be represented as a convergent power series in ? and we determine the coefficients in terms of the solutions of explicit systems of integral equations.
Cross-correlation and cross-power spectral density representation by complex spectral moments
2017
Abstract A new approach to provide a complete characterization of normal multivariate stochastic vector processes is presented in this paper. Such proposed method is based on the evaluation of the complex spectral moments of the processes. These quantities are strictly related to the Mellin transform and they are the generalization of the integer-order spectral moments introduced by Vanmarcke. The knowledge of the complex spectral moments permits to obtain the power spectral densities and their cross counterpart by a complex series expansions. Moreover, with just the aid of some mathematical properties the complex fractional moments permit to obtain also the correlation and cross-correlatio…
On the Stochastic Response of a Fractionally-damped Duffing Oscillator
2012
A numerical method is presented to compute the response of a viscoelastic Duffing oscillator with fractional derivative damping, subjected to a stochastic input. The key idea involves an appropriate discretization of the fractional derivative, based on a preliminary change of variable, that allows to approximate the original system by an equivalent system with additional degrees of freedom, the number of which depends on the discretization of the fractional derivative. Unlike the original system that, due to the presence of the fractional derivative, is governed by non-ordinary differential equations, the equivalent system is governed by ordinary differential equations that can be readily h…
A correction method for the analysis of continuous linear one-dimensional systems under moving loads
2008
A new correction procedure for dynamic analysis of linear, proportionally damped, continuous systems under traveling concentrated loads is proposed; both cases of non-parametric (moving forces) and parametric (moving mass) loads are considered. Improvement in the evaluation of the dynamic response is obtained by separating the contribution of the low-frequency (LF) modes from that of the high-frequency (HF) modes. The former is calculated, as usual, by classical modal analysis, while the latter is taken into account using a new series expansion of the corresponding particular solution. The advantage of the suggested method is immediately shown in the calculation of the stress distribution s…
A Fast Solar Radiation Transfer Code for Application in Climate Models
1983
A method is presented for the calculation of solar heating rates in turbid and cloudy atmospheres. In contrast to other typical two-stream procedures, the system of differential equations describing the radiative transfer is decoupled through the application of a series expansion of the flux densities resulting in a single analytical expression for each flux. The present method (PM) yields a solution for the entire atmosphere instead of individual atmospheric layers. This procedure avoids as part of the solution scheme the inversion of a rather complex matrix thus resulting in high numerical efficiency. The model includes the absorption by atmospheric gases such as water vapor, CO2, O3 and …
NEW DEVELOPMENTS ON INVERSE POLYGON MAPPING TO CALCULATE GRAVITATIONAL LENSING MAGNIFICATION MAPS: OPTIMIZED COMPUTATIONS
2011
We derive an exact solution (in the form of a series expansion) to compute gravitational lensing magnification maps. It is based on the backward gravitational lens mapping of a partition of the image plane in polygonal cells (inverse polygon mapping, IPM), not including critical points (except perhaps at the cell boundaries). The zeroth-order term of the series expansion leads to the method described by Mediavilla et al. The first-order term is used to study the error induced by the truncation of the series at zeroth order, explaining the high accuracy of the IPM even at this low order of approximation. Interpreting the Inverse Ray Shooting (IRS) method in terms of IPM, we explain the previ…
High-temperature series expansion for the relaxation times of the two dimensional Ising model
1995
We derive the high temperature series expansions for the two relaxation times of the single spin-flip kinetic Ising model on the square lattice. The series for the linear relaxation time τl is obtained with 20 non-trivial terms, and the analysis yields 2.183±0.005 as the value of the critical exponentΔl, which is equal to the dynamical critical exponentz in the two-dimensional case. For the non-linear relaxation time we obtain 15 non-trivial terms, and the analysis leads to the resultsΔnl = 2.08 ± 0.07. The scaling relationΔl −Δnl = β (β being the exponent of the order parameter) seems to be fulfilled, though the error bars ofΔnl are still quite substantial. In addition, we obtain the serie…