Semi-analytical Expansion Research Articles

Magnetic Field Mitigation Shielding of Underground Power Cables

This paper is a contribution to the analysis of magnetic field mitigation shielding of underground power cables. A hybrid finite-element method (FEM)/semi-analytical expansion is adopted for the numerical treatment of the open boundary magnetic field problem. Different constitutive materials for the magnetic shielding are studied, and comparison is established between conductive (aluminum) and ferromagnetic (steel) cases. A horizontal flat configuration for the underground cable is considered. It is concluded that the aluminum shielding is more efficient as far as magnetic field mitigation is concerned.

QED corrections of orderα(Zα)2EFto the hyperfine splitting ofP1/2andP3/2states in hydrogenlike ions

The hyperfine structure (HFS) of a bound electron is modified by the self-interaction of the electron with its own radiation field. This effect is known as the self-energy correction. In this work, we discuss the evaluation of higher-order self-energy corrections to the HFS of bound P states. These are expressed in a semi-analytic expansion involving powers of Zalpha and ln(Zalpha), where Z is the nuclear charge number and alpha is the fine-structure constant. We find that the correction of relative order alpha (Zalpha)^2 involves only a single logarithm ln(Zalpha) for P_1/2 states [but no term of order alpha (Zalpha)^2 ln^2(Zalpha), whereas for P_3/2 states, even the single logarithm vanishes. By a Foldy-Wouthuysen transformation, we identify a nuclear-spin dependent correction to the electron's transition current, which contributes to the HFS of P states. A comparison of the obtained analytic results to a numerical approach is made.

Extrapolation of the Zα-expansion and two-loop bound-state energy shifts

Quantum electrodynamic (QED) effects that shift the binding energies of hydrogenic energy levels have been expressed in terms of a semi-analytic expansion in powers of Zα and ln[(Zα)−2], where Z is the nuclear charge number and α is the fine-structure constant. For many QED effects, numerical data are available in the domain of high Z where the Zα expansion fails. In this Letter, we demonstrate that it is possible, within certain limits of accuracy, to extrapolate the Zα-expansion from the low-Z to the high-Z domain. We also review two-loop self-energy effects and provide an estimate for the problematic nonlogarithmic coefficient B60.

Numerical Fourier Expansions of the Planetary Disturbing Function

Various Fourier expansions of the planetary disturbing function can be computed numerically with the use of numerical Fourier analysis. The task to compute the most general five-dimensional Fourier expansion of disturbing function has become feasible with typical server-class computers quite recently. In such an expansion two anomalies, two arguments of perihelions and two longitudes of the node are independent angular variables, while two semi-major axes, two eccentricities and two inclinations are fixed numerically. The semianalytical expansion of the disturbing function resulting from numerical Fourier analysis theoretically converges for any values of the parameters except for those sets of parameters which allow the bodies to collide. Various aspects of the numerical computation of the Fourier expansion are discussed. Theoretical and practical convergence of the Fourier series is discussed and illustrated.

Nodal kinetics model upgrade in the Penn State coupled TRAC/NEM codes

The Pennsylvania State University currently maintains and does development and verification work for its own versions of the coupled three-dimensional kinetics/thermal-hydraulics codes TRAC-PF1/NEM and TRAC-BF1/NEM. The subject of this paper is nodal model enhancements in the above mentioned codes. Because of the numerous validation studies that have been performed on almost every aspect of these codes, this upgrade is done without a major code rewrite. The upgrade consists of four steps. The first two steps are designed to improve the accuracy of the kinetics model, based on the nodal expansion method. The polynomial expansion solution of 1D transverse integrated diffusion equation is replaced with a solution, which uses a semi-analytic expansion. Further the standard parabolic polynomial representation of the transverse leakage in the above 1D equations is replaced with an improved approximation. The last two steps of the upgrade address the code efficiency by improving the solution of the time-dependent NEM equations and implementing a multi-grid solver. These four improvements are implemented into the standalone NEM kinetics code. Verification of this code was accomplished based on the original verification studies. The results show that the new methods improve the accuracy and efficiency of the code. The verification of the upgraded NEM model in the TRAC-PF1/NEM and TRAC-BF1/NEM coupled codes is underway.