Automation of The Guiding Center Expansion

We report on the use of the recently-developed Mathematica package \emph{VEST} (Vector Einstein Summation Tools) to automatically derive the guiding center transformation. Our Mathematica code employs a recursive procedure to derive the transformation order-by-order. This procedure has several novel features. (1) It is designed to allow the user to easily explore the guiding center transformation's numerous non-unique forms or representations. (2) The procedure proceeds entirely in cartesian position and velocity coordinates, thereby producing manifestly gyrogauge invariant results; the commonly-used perpendicular unit vector fields $e_1,e_2$ are never even introduced. (3) It is easy to apply in the derivation of higher-order contributions to the guiding center transformation without fear of human error. Our code therefore stands as a useful tool for exploring subtle issues related to the physics of toroidal momentum conservation in tokamaks.Comment: 34 page

The breakage prediction for hydromechanical deep drawing based on local bifurcation theory

A criterion of sheet metal localized necking under plane stress was established based on the bifurcation theory and the characteristics theory of differential equation. In order to be capable to incorporate the directional dependence of the plastic strain rate on stress rate, Ito-Goya’s constitutive equation which gave a one to one relationship between stress rate component and plastic strain rate component was employed. The hydromechanical deep drawing process of a cylindrical cup part was simulated using the commercial software ABAQUS IMPLICIT. The onset of breakage of the part during the forming process was predicted by combining the simulation results with the local necking criterion. The proposed method is applied to the hydro-mechanical deep drawing process for A2219 aluminum alloy sheet metal to predict the breakage of the cylindrical cup part. The proposed method can be applied to the prediction of breakage in the forming of the automotive bodies

Microwave spectroscopy on a double quantum dot with an on-chip Josephson oscillator

We present measurements on microwave spectroscopy on a double quantum dot with an on-chip microwave source. The quantum dots are realized in the two-dimensional electron gas of an AlGaAs/GaAs heterostructure and are weakly coupled in series by a tunnelling barrier forming an 'ionic' molecular state. We employ a Josephson oscillator formed by a long Nb/Al-AlO$_x$/Nb junction as a microwave source. We find photon-assisted tunnelling sidebands induced by the Josephson oscillator, and compare the results with those obtained using an externally operated microwave source.Comment: 6 pages, 4 figure