On the Duality of Global Finite Element Discretization Error-Control in Small Strain Newtonian and Eshelbian Mechanics

In this paper global a posteriori error estimators are presented for the error obtained during the finite element discretization of the linear elasticity problem. Thereby, the duality of global error measures is established that are chosen within the framework of traditional Newtonian mechanics as well as within the framework of Eshelbian mechanics. In Newtonian mechanics we are concerned with the (physical) Cauchy stress tensor that reflects the internal resistance of an elastic body to an applied physical force, whereas in Eshelbian mechanics the applied force is a material force acting on a defect of the elastic body with associated (material) (Newton-)Eshelby stress tensor which is the dual stress tensor of the Cauchy stress tensor. The derivation of a posteriori error estimators is based on the well-established strategy of solving an auxiliary dual problem in order to control the global error measures defined in terms of bounded non-linear functionals. In this paper, two principally different strategies are presented to estimate the error measure. The first strategy rests upon an equilibrated residual error estimator based on local Neumann problems, whereas the second one makes use of averaging techniques. The paper is concluded by a numerical example that illustrates our theoretical results

Heating and Trapping of Electrons in ECRIS from Scratch to Afterglow

Plasmas in Electron Cyclotron Resonance Ion Sources (ECRIS) are collisionless and can therefore be simulated by just following the motion of electrons in the confining static magnetic and oscillating microwave (MW) electric field of ECRIS. With a powerful algorithm the three-dimensional trajectories of 104 ECR-heated and confined electrons are calculated in a standard ECRIS with a deep minimum of |B| and a new ECRIS with a very flat minimum of |B|. The spatial electron (plasma) densities and electron energy densities deduced from these trajectories yield new and surprising insight in the performance of ECRIS. With computer animation we plan to present: The energy increase of certain electrons on extremely stable trajectories, the power dependence of the electron energy density up to the X-ray collapse, the time dependent build up of the electron density and energy density distributions, and the time evolution of these electron distributions under afterglow conditions

Eccentric binary black holes: Comparing numerical relativity and small mass-ratio perturbation theory

The modelling of unequal mass binary black hole systems is of high importanceto detect and estimate parameters from these systems. Numerical relativity (NR)is well suited to study systems with comparable component masses, m_1\simm_2, whereas small mass ratio (SMR) perturbation theory applies to binarieswhere $q=m_2/m_1as a function of mass ratio for eccentric non-spinning binary black holes. Weproduce$52$NR simulations with mass ratios between$1:10$and$1:1$andinitial eccentricities up to$0.7$. From these we extract quantities likegravitational wave energy and angular momentum fluxes and periastron advance,and assess their accuracy. To facilitate comparison, we develop tools to mapbetween NR and SMR inspiral evolutions of eccentric binary black holes. Wederive post-Newtonian accurate relations between different definitions ofeccentricity. Based on these analyses, we introduce a new definition ofeccentricity based on the (2,2)-mode of the gravitational radiation, whichreduces to the Newtonian definition of eccentricity in the Newtonian limit.From the comparison between NR simulations and SMR results, we quantify theunknown next-to-leading order SMR contributions to the gravitational energy andangular momentum fluxes, and periastron advance. We show that in the comparablemass regime these contributions are subdominant and higher order SMRcontributions are negligible.<br

Saturable discrete vector solitons in one-dimensional photonic lattices

Localized vectorial modes, with equal frequencies and mutually orthogonal polarizations, are investigated both analytically and experimentally in a one-dimensional photonic lattice with saturable nonlinearity. It is shown that these modes may span over many lattice elements and that energy transfer among the two components is both phase and intensity dependent. The transverse electrically polarized mode exhibits a single-hump structure and spreads in cascades in saturation, while the transverse magnetically polarized mode exhibits splitting into a two-hump structure. Experimentally such discrete vector solitons are observed in lithium niobate lattices for both coherent and mutually incoherent excitations.Comment: 4 pages, 5 figures (reduced for arXiv

Integrated Detector Control and Calibration Processing at the European XFEL

The European X-ray Free Electron Laser is a high-intensity X-ray light source currently being constructed in the area of Hamburg, that will provide spatially coherent X-rays in the energy range between $0.25\,\mathrm{keV}$ and $25\,\mathrm{keV}$. The machine will deliver $10\,\mathrm{trains/s}$, consisting of up to $2700\,\mathrm{pulses}$, with a $4.5\,\mathrm{MHz}$ repetition rate. The LPD, DSSC and AGIPD detectors are being developed to provide high dynamic-range Mpixel imaging capabilities at the mentioned repetition rates. A consequence of these detector characteristics is that they generate raw data volumes of up to $15\,\mathrm{Gbyte/s}$. In addition the detector's on-sensor memory-cell and multi-/non-linear gain architectures pose unique challenges in data correction and calibration, requiring online access to operating conditions and control settings. We present how these challenges are addressed within XFEL's control and analysis framework Karabo, which integrates access to hardware conditions, acquisition settings (also using macros) and distributed computing. Implementation of control and calibration software is mainly in Python, using self-optimizing (py) CUDA code, numpy and iPython parallels to achieve near-real time performance for calibration application.Comment: Proceeding ICALEPS 201

Extending PT symmetry from Heisenberg algebra to E2 algebra

The E2 algebra has three elements, J, u, and v, which satisfy the commutation relations [u,J]=iv, [v,J]=-iu, [u,v]=0. We can construct the Hamiltonian H=J^2+gu, where g is a real parameter, from these elements. This Hamiltonian is Hermitian and consequently it has real eigenvalues. However, we can also construct the PT-symmetric and non-Hermitian Hamiltonian H=J^2+igu, where again g is real. As in the case of PT-symmetric Hamiltonians constructed from the elements x and p of the Heisenberg algebra, there are two regions in parameter space for this PT-symmetric Hamiltonian, a region of unbroken PT symmetry in which all the eigenvalues are real and a region of broken PT symmetry in which some of the eigenvalues are complex. The two regions are separated by a critical value of g.Comment: 8 pages, 7 figure

Periodic orbits for classical particles having complex energy

This paper revisits earlier work on complex classical mechanics in which it was argued that when the energy of a classical particle in an analytic potential is real, the particle trajectories are closed and periodic, but that when the energy is complex, the classical trajectories are open. Here it is shown that there is a discrete set of eigencurves in the complex-energy plane for which the particle trajectories are closed and periodic.Comment: 12 pages, 9 figure

CPT-symmetric discrete square well

A new version of an elementary PT-symmetric square well quantum model is proposed in which a certain Hermiticity-violating end-point interaction leaves the spectrum real in a large domain of couplings $\lambda\in (-1,1)$. Within this interval we employ the usual coupling-independent operator P of parity and construct, in a systematic Runge-Kutta discrete approximation, a coupling-dependent operator of charge C which enables us to classify our P-asymmetric model as CPT-symmetric or, equivalently, hiddenly Hermitian alias cryptohermitian.Comment: 12 pp., presented to conference PHHQP IX (http://www.math.zju.edu.cn/wjd/