Mass Correction Terms Research Articles

Investigation of a perturbation-based model updating approach for structural health and event monitoring

With the intention to detect structural events, for example, occurrence of damage and assessing their severity under service, structural health and event monitoring proved itself as a reliable evaluation tool. While many algorithmic approaches exist, model-based monitoring offers the prominent advantage to enable extensive analysis of damage states based on a structural digital twin. In order to accurately replicate the dynamical behaviour of a real-world structure, inherent physical properties, for example, mass, stiffness and structural damping, of the digital twin might be corrected from nominal values. This can be achieved by numerical model updating procedures. In this study, the application of a model updating approach is presented to eliminate modal discrepancies. Based on the perturbated modal dynamic residual, an updating equation is formulated. Stiffness and mass correction terms with preserved orthogonality and symmetry conditions are determined by the method of least squares. A case study using computationally generated modal data is conducted, evaluating the numerical updating performance in terms of correction accuracy and reproduction of modal parameters. Finally, the transient structural response to an impulse excitation is compared between updated and reference model. The gathered results prove the method’s suitability as accurate and robust updating procedure, fostering its application in model-based monitoring frameworks.

Exact Foldy–Wouthuysen transformation for a Dirac equation describing the interaction of spin-1/2 relativistic particles with an external electromagnetic field

The Thomas–Bargmann–Michel–Telegdi (TBMT) equation is derived using the Exact Foldy–Wouthuysen transformation. Extra new terms were found, and we discuss their possible physical applications. The main point of this work is to detail the procedure to get the general result. We explicitly present the choice of parametrization we used on the initial Hamiltonian and the motivations to take it. We emphasize that the final equations can depend on this choice, and it is possible to prevent the manipulations of the quadratic Hamiltonian that become extremely cumbersome. More importantly, it is done in such a way that the transformed equations allow the direct separation into mass, kinetic, and interaction correction terms to the original TBMT equation.

Dynamic Characteristic Analysis of a Toothed Electromagnetic Spring Based on the Improved Bouc—Wen Model

Electromagnetic spring active isolators have attracted extensive attention in recent years. The standard Bouc–Wen model is widely used to describe hysteretic behavior but cannot accurately describe asymmetric behavior. The standard Bouc–Wen model is improved to better describe the dynamic characteristic of a toothed electromagnetic spring. The hysteresis model of toothed electromagnetic spring is established by adding mass, damping, and asymmetric correction terms with direction. Subsequently, the particle swarm optimization algorithm is used to identify the parameters of the established model, and the results are compared with those obtained from the experiment. The results show that the current has a significant impact on the dynamic curve. When the current increases from 0.5 A to 2.0 A, the electromagnetic force sharply increases from 49 N to 534 N. Under different excitations and currents, the residual points predicted by the model proposed in this work fall basically in the horizontal band region of −20–20 N (for an applied current of 1.0 A) and −40–80 N (for an application of 4.5 mm/s). Furthermore, the maximum relative error of the model is 12.75%. The R2 of the model is higher than 0.98 and the highest value is 0.9993, proving the accuracy of the established model.

Effect of plasma environment on spectral and structural properties of H-like C, N and O ions

The novel studies of the dense plasma on transition properties and level structures for H-like Carbon, Nitrogen and Oxygen ions are presented. Plasma effect is included through the confined ion sphere plasma model. We have also included relativistic effects through mass-correction, Darwin and spin-orbit terms during the calculation. The energy level shift, oscillator strengths, transition rates, plasma shifts and probability densities are calculated by using Ritz’s variation method for different ion sphere plasma concentrations. The variation parameters are calculated by minimizing the expectation value of the energy. Eigenstate, probability and average position are calculated with plasma concentration for the internal structure of the system. We have compared our calculated data with experimental National Institute of Science and Technology (NIST) database and other theoretical results in the literature and it is showing a good quantitative agreement. The study may be very useful in astrophysical and fusion plasma.

Energy Levels and Oscillator Strengths of Ca V

We report an extensive and elaborate theoretical study of excitation energies and radiative data for the lowest 148 fine-structure levels of Sulphur-like Ca using Configuration Interaction Method (CIV3). We have included the relativistic effects using Breit-Pauli approximation by adding mass-correction, Darwin and Spin-Orbit terms to the non-relativistic Hamiltonian. Dipole allowed transitions are fined tuned by adjusting diagonal elements of the Hamiltonian Matrix. The exact identification of many levels becomes tedious due to the strong mixing between several fine-structure levels. We have compared our calculated values of energy with the data compiled by National Institute of standard and Technology (NIST) and found good agreement. We have reported data for many new levels which are not in the NIST database.

Effective non-adiabatic Hamiltonians for the quantum nuclear motion over coupled electronic states.

The quantum mechanical motion of the atomic nuclei is considered over a single- or a multidimensional subspace of electronic states which is separated by a gap from the rest of the electronic spectrum over the relevant range of nuclear configurations. The electron-nucleus Hamiltonian is block-diagonalized up to O(εn+1) through a unitary transformation of the electronic subspace, and the corresponding nth-order effective Hamiltonian is derived for the quantum nuclear motion. Explicit but general formulas are given for the second- and the third-order corrections. As a special case, the second-order Hamiltonian corresponding to an isolated electronic state is recovered which contains the coordinate-dependent mass-correction terms in the nuclear kinetic energy operator. For a multidimensional, explicitly coupled electronic band, the second-order Hamiltonian contains the usual Born-Oppenheimer terms and nonadiabatic corrections, but generalized mass-correction terms appear as well. These, earlier neglected terms, perturbatively account for the outlying (discrete and continuous) electronic states not included in the explicitly coupled electronic subspace.

A stable FSI algorithm for light rigid bodies in compressible flow

In this article we describe a stable partitioned algorithm that overcomes the added mass instability arising in fluid–structure interactions of light rigid bodies and inviscid compressible flow. The new algorithm is stable even for bodies with zero mass and zero moments of inertia. The approach is based on a local characteristic projection of the force on the rigid body and is a natural extension of the recently developed algorithm for coupling compressible flow and deformable bodies [1–3]. The new algorithm advances the solution in the fluid domain with a standard upwind scheme and explicit time-stepping. The Newton–Euler system of ordinary differential equations governing the motion of the rigid body is augmented by added mass correction terms. This system, which is very stiff for light bodies, is solved with an A-stable diagonally implicit Runge–Kutta scheme. The implicit system (there is one independent system for each body) consists of only 3d+d2 scalar unknowns in d=2 or d=3 space dimensions and is fast to solve. The overall cost of the scheme is thus dominated by the cost of the explicit fluid solver. Normal mode analysis is used to prove the stability of the approximation for a one-dimensional model problem and numerical computations confirm these results. In multiple space dimensions the approach naturally reveals the form of the added mass tensors in the equations governing the motion of the rigid body. These tensors, which depend on certain surface integrals of the fluid impedance, couple the translational and angular velocities of the body. Numerical results in two space dimensions, based on the use of moving overlapping grids and adaptive mesh refinement, demonstrate the behavior and efficacy of the new scheme. These results include the simulation of the difficult problems of shock impingement on an ellipse and a more complex body with appendages, both with zero mass.

The impact of numerical viscosity in SPH simulations of galaxy clusters

A SPH code employing a time-dependent artificial viscosity scheme is used to construct a large set of N-body/SPH cluster simulations for studying the impact of artificial viscosity on the thermodynamics of the ICM and its velocity field statistical properties. Spectral properties of the gas velocity field are investigated by measuring for the simulated clusters the velocity power spectrum E(k). The longitudinal component E_c(k) exhibits over a limited range a Kolgomorov-like scaling k^{-5/3}, whilst the solenoidal power spectrum component E_s(k) is strongly influenced by numerical resolution effects. The dependence of the spectra E(k) on dissipative effects is found to be significant at length scales 100-300Kpc, with viscous damping of the velocities being less pronounced in those runs with the lowest artificial viscosity. The turbulent energy density radial profile E_{turb}(r) is strongly affected by the numerical viscosity scheme adopted in the simulations, with the turbulent-to-total energy density ratios being higher in the runs with the lowest artificial viscosity settings and lying in the range between a few percent and ~10%. These values are in accord with the corresponding ratios extracted from previous cluster simulations realized using mesh-based codes. At large cluster radii, the mass correction terms to the hydrostatic equilibrium equation are little affected by the numerical viscosity of the simulations, showing that the X-ray mass bias is already estimated well in standard SPH simulations. Finally, simulations in which the gas can cool radiatively are characterized by the presence in the cluster inner regions of high levels of turbulence, generated by the interaction of the compact cool gas core with the ambient medium.

KINETIC ENERGY FUNCTIONALS: EXACT ONES FROM ANALYTIC MODEL WAVE FUNCTIONS AND APPROXIMATE ONES IN ORBITAL-FREE MOLECULAR DYNAMICS

We consider the problem of constructing kinetic energy functionals in density functional theory. We first discuss the functional generated through the application of local-scaling transformations to the exact analytic wavefunctions for the Hookean model of 4He (a finite mass three-particle system), and contrast this result with a previous one for ∞He (infinite mass system). It is shown that an exact non-Born-Oppenheimer treatment not only leads to mass-correction terms in the kinetic energy, but to a basically different functional expression. In addition, we report and comment on some recently advanced approximate kinetic energy functionals generated in the context of the constraint-based approach to orbital-free molecular dynamics. The positivity and non-singularity of a new family of kinetic energy functionals specifically designed for orbital-free molecular dynamics is discussed. Finally, we present some conclusions.

Calculating electron paramagnetic resonance g-matrices for triplet state molecules from multireference spin-orbit configuration interaction wave functions

We present a way to calculate electron paramagnetic resonance (EPR) g-matrices from variationally optimized spin-orbit coupled wave functions. Our method constructs a triangular g-matrix from the matrix representation of the total electron magnetic moment in the basis of the spin-orbit coupled wave functions by means of a projection technique. Principal g-values are obtained in the standard fashion by forming from the triangular matrix g the tensor G=gg(t) and diagonalizing it. In principle, the scheme allows to calculate the spin-orbit orbital Zeeman cross term which usually gives the dominating contribution to the EPR g-shifts for any multiplicity. We have implemented this approach into a multireference spin-orbit configuration interaction (MRSOCI) program [M. Kleinschmidt et al., J. Chem. Phys. 124, 124101 (2006)]. Test applications are carried out for various triplet state sytems. The g-shifts of several of main group diatomics with X (3)Sigma(g)(-) ground state are investigated at the level of ab initio MRSOCI. We obtain perpendicular g-shifts which underestimate experimental Delta g(perpendicular) values from literature by approximately 13% on the average. For a set of organic triplet state molecules we employ the combined density functional theory/multireference configuration interaction (DFT/MRCI) technique [S. Grimme and M. Waletzke, J. Chem. Phys. 111, 5645 (1999)] to reduce the computational costs of the spin-free correlation problem. This approach yields principal g-values that match experiment well in many cases. Due to the small absolute g-shifts, a rigorous comparison will require the inclusion of first-order contributions such as the relativistic mass correction and gauge correction terms which have not been included here. For the triplet state dication trans-(CNSSS)(2)(2+) the principal g-shifts Delta g(a)=-0.3 ppt, Delta g(b)=17.5 ppt, and Delta g(c)=26.6 ppt are significantly larger and compare rather well to the experimental values Delta g(1)=-0.1+/-0.2 ppt, Delta g(2)=14.8+/-0.2 ppt, and Delta g(3)=24.8+/-0.1 ppt [A. Berces et al., Magn. Reson. Chem. 37, 353 (1999)]. In comparison to conventional truncated sum-over state techniques based on Rayleigh-Schrodinger perturbation theory, our new variational approach shows, in practice, robust and advantageous convergence characteristics with respect to the size of the many-particle basis set. We demonstrate that the DFT/MRSOCI technology is a very feasible means to compute reliable g-shifts for large organic triplet systems at low computational cost.

Fine-structure energy levels, oscillator strengths and lifetimes in Co XV

Excitation energies from ground state for 97 fine-structure levels as well as oscillator strengths and radiative decay rates for all electric-dipole-allowed and intercombination transitions among the fine-structure levels of the terms belonging to the (1s 22s 22p 6)3s 23p, 3s 3 p 2, 3s 23d, 3p 3, 3s3p3d, 3p 23d, 3s3d 2, 3s 24s, 3s 24p, 3s 24d, 3s 24f, and 3s3p4s configurations of Co XV are calculated, using extensive configuration-interaction (CI) wave functions, obtained with the CIV3 computer code of Hibbert. The important relativistic effects in intermediate coupling are included through the Breit-Pauli approximation via spin-orbit, spin-other-orbit, spin-spin, Darwin and mass correction terms. Small adjustments to the diagonal elements of the Hamiltonian matrices have been made. Our calculated excitation energies, including their ordering, are in excellent agreement with the experimental results and the experimentally compiled energy values of the National Institute for Standards and Technology (NIST) wherever available. The mixing among several fine-structure levels is found to be very strong, with most of the strongly mixed levels belonging to the (1s 22s 22p 6)3p 23d and 3s3d 2 configurations. The strong mixing among several fine-structure levels makes it very difficult to identify them uniquely. Perhaps, that may be the reason for the lack of both experimental and theoretical results for these levels. We believe that our extensive calculated values can guide experimentalists in identifying the fine-structure levels in their future work. From our radiative decay rates we have also calculated radiative lifetimes of some fine-structure levels. In this calculation we also predict new data for several fine-structure levels where no other theoretical and/or experimental results are available.

Fine-structure energy levels, oscillator strengths and lifetimes in Mg-like chromium

Excitation energies from ground state for 86 fine-structure levels as well as oscillator strengths and radiative decay rates for all fine-structure transitions among the levels of the terms (1s22s22p6)3s2(1S), 3s3p(1,3Po), 3s3d(1,3D), 3s4s(1,3S), 3s4p(1,3Po), 3s4d(1,3D), 3s4f(1,3Fo), 3p2(1S, 3P, 1D), 3p3d(1,3Po, 1,3Do, 1,3Fo), 3p4s(1,3Po), 3p4p(1,3S, 1,3P, 1,3D), 3p4d(1,3Po, 1,3Do, 1,3Fo), 3p4f(1,3D, 1,3F, 1,3G) and 3d2(1S, 3P, 1D, 3F,1G) of Cr XIII are calculated using extensive configuration-interaction (CI) wave functions obtained with the CIV3 computer code of Hibbert. The important relativistic effects in intermediate coupling are included through the Breit-Pauli approximation via spin-orbit, spin-other-orbit, spin-spin, Darwin and mass correction terms. Small adjustments to the diagonal elements of the Hamiltonian matrices have been made. The mixing among several fine-structure levels is found to be very strong. Our excitation energies, including their ordering, are in excellent agreement (better than 0.5%) with the available experimental results. From our transition probabilities, we have also calculated radiative lifetimes of some fine-structure levels. Our calculated lifetime for the longer-lived level 3s3p(3P1) is found to be in excellent agreement with the experimental result of Curtis compared to other theoretical calculations.

Electron collisional excitation of argon-like Ni XI using the Breit-Pauli R-matrix method

Electron excitation collision strengths for the transitions from the ground state to the fine-structure levels of the 3s23p53d and 3s23p63d configurations in Ni XI are calculated using the Breit-Pauli R-matrix method. Configuration interaction wavefunctions have been used to represent the target states. The relativistic effects are incorporated in the Breit-Pauli approximation by including one-body mass correction, Darwin, and spin-orbit interaction terms in the scattering equations. Collision strengths are tabulated at selected energies in the range 10 to 75 Rydberg. Effective collision strengths are determined by integrating collision strengths by assuming a Maxwellian distribution of electron energies. The effective collision strengths are listed over a wide temperature range (2×104-1×107 K) applicable to astrophysical plasmas. Our results are the only collision strengths and effective collision strengths available for this ion. We believe that the data calculated in this work will be useful in solar, astrophysical and laser applications.

Breit-Pauli energy levels and radiative lifetimes in neutral chlorine

Breit-Pauli Energy levels, oscillator strengths and transition probabilities for all the transitions in Cl I between the fine structure levels of 3s2 3p5, 3s2 3p43d, 3s2 3p44s and 3s23p44p states are calculated using extensive configuration interaction (CI) wave functions. We have also determined the lifetimes of 3s2 3p43d, 3s2 3p44s and 3s2 3p44p levels. The relativistic effects are included through Breit-Pauli approximation via spin-orbit, Darwin and mass correction terms. Prior to the calculations of the oscillator strengths and transition probabilities, we fine-tune the CI coefficient using experimental energies. Our results are compared with experimental and other available theoretical data. The calculated energy levels are in close agreement with most of the NIST compiled data. We predict new lifetime data for several levels where no other theoretical and/or experimental results are available, which will form the basis for the future experimental work.

Excitation Energies, Oscillator Strengths and Lifetimes in Ca IX

We have calculated the excitation energies for the lowest 46 LS and 86 fine-structure levels as well as oscillator strengths and radiative decay rates for transitions among the (1s22s22p6)3s2(1S), 3s3p(1,3Po), 3s3d(1,3D), 3s4s(1,3S), 3s4p(1,3Po), 3s4d(1,3D), 3s4f(1,3Fo), 3p2(1S, 3P, 1D), 3p3d(1,3Po, 1,3Do, 1,3Fo), 3p4s(1,3Po), 3p4p(1,3S, 1,3P, 1,3D), 3p4d(1,3Po, 1,3Do, 1,3Fo), 3p4f(1,3D, 1,3F, 1,3G) and 3d2(1S, 3P, 1D, 3F, 1G) states in Ca IX. These states are represented by extensive configuration-interaction (CI) wavefunctions obtained with the CIV3 computer code of Hibbert. From our transition probabilities we have also calculated the radiative lifetimes of singlet and triplet states of Ca IX. Our results are compared with other available theoretical calculations and experimental data. To assess the importance of relativistic effects on our calculated values, we have also carried out calculations in the intermediate-coupling scheme. These effects are incorporated through the Breit-Pauli approximation via spin-orbit, spin-other-orbit, spin-spin, Darwin and mass correction terms. In order to keep our calculated energy splittings as close as possible to the experimental values, we have made small adjustments to the diagonal elements of the Hamiltonian matrices. The energy splittings of 87 fine-structure levels, the oscillator strengths and transition probabilities for some strong dipole-allowed and intercombination transitions and the lifetimes of some fine-structure levels are presented and compared with available experimental and other theoretical values. Our calculated lifetimes of the relatively long-lived 3p3d(3FJ) levels show remarkable improvement over the theoretical values of Fawcett, compared to the experimental results of Trabert et al. Also, our lifetime for the 3p2(1D2) level calculated in intermediate-coupling scheme, while differing significantly from our LS value, shows good agreement with the experimental value and thus confirms the need to include relativistic effects in calculations. In this calculation we also predict new data for several levels where no other theoretical and experimental results are available.

Collisional excitation of neon-like Ni XIX using the Breit-PauliR-matrix method

Collision strength for the transition within the first five fine-structure levels in Ni XIX are calculated using the Breit-PauliR-matrix method. Configuration interaction wave functions are used to represent the target states included in theR-matrix expansion. The relativistic effects are incorporated in the Breit-Pauli approximation by including the one-body mass correction, Darwin and spin-orbit interaction terms in scattering equations.

Calculated Energy Levels, Oscillator Strengths and Lifetimes in Al-like Argon

Excitation energies, oscillator strengths and transition probabilities for electric-dipole-allowed and inter-combination transitions among the 25 LS levels belonging to the (1s22s22p6)3s23p, 3s3p2, 3s23d, 3p3, 3s3p3d, 3s24s, 3s24p, 3s24d, 3s24f and 3s3p4s configurations of Ar VI are calculated using extensive configuration-interaction (CI) wave functions. From our transition probabilities we have also calculated the radiative lifetimes of doublet and quartet states of Ar VI. Our results are compared with other available theoretical calculations and the experimental data. To assess the importance of relativistic effects on our calculated values, we have also carried out calculations in the intermediate-coupling scheme. These effects are incorporated through the Breit–Pauli approximation via spin–orbit, spin–other–orbit, spin–spin, Darwin and mass correction terms. Small adjustments to the diagonal elements of the Hamiltonian matrices have been made so that the energy splittings are as close as possible to the experimental values. The energy splitting of 54 fine-structure levels, the oscillator strengths and transition probabilities for some strong dipole-allowed and intercombination transitions and the lifetimes of some fine-structure levels are presented and compared with available experimental and other theoretical values. Our lifetime for the 3s3p(1Po)3d(2Po) level calculated in intermediate-coupling scheme, while differing significantly from our LS value, shows excellent agreement with the experimental result of Pinnington et al. In this calculation we also predict new data for several levels where no other theoretical and experimental results are available.

Excitation Energies, Oscillator Strengths and Radiative Rates in Triply Ionised Phosphorous

We have calculated the excitation energies for the lowest 23 LS and 43 fine-structure levels as well as oscillator strengths and radiative decay rates for the transitions among (1s22s22p6) 3s2 (1S), 3s3p (1,3Po), 3p2 (1S, 3P, 1D), 3s3d (1,3D), 3s4s (1,3S), 3p3d (1,3Po, 1,3Do, 1,3Fo), 3s4p (1,3Po), 3s4d (1,3D), 3s4f (1,3Fo) and 3p4s (1,3Po) states in P IV. These states are represented by extensive configuration-interaction (CI) wavefunctions obtained with the CIV3 computer code of Hibbert. The calculations have been carried out in both LS and intermediate-coupling schemes. Relativistic effects are included through the Breit-Pauli approximation via spin-orbit, spin-other-orbit, spin-spin, Darwin and mass correction terms. We have also calculated the radiative lifetimes from our radiative decay rates. Our results are compared with other theoretical calculations and the available experimental data.

Fine-Structure Energy Levels, Oscillator Strengths and Transition Probabilities in Ni XVI

Fine-structure energy levels relative to the ground state, oscillator strengths and transition probabilities for transitions among the lowest 40 fine-structure levels belonging to the configurations 3s23p, 3s3p2, 3s23d, 3p3 and 3s3p3d of Ni XVI are calculated using a large scale CI in Program CIV3 of Hibbert. Relativistic effects are included through the Breit–Pauli approximation via spin-orbit, spin-other-orbit, spin-spin, Darwin and mass correction terms. The existing discrepancies between the calculated and measured values for many of the relative energy positions are resolved in the present calculation which yields excellent agreement with measurement. Also, many of our oscillator strengths for allowed and intercombination transitions are in very good agreement with the recommended data by the National Institute of Standard and Technology (NIST).

Effective Collision Strengths of Fine‐Structure Transitions in Fex, Fexi, and Fexiii

Electron excitation collision strengths for fine-structure transitions in Fe X, Fe XI, and Fe XIII are calculated using full Breit-Pauli R-matrix approach. We included 49 fine-structure levels of Fe X, 38 fine-structure levels of Fe XI, and 26 fine-structure levels of Fe XIII in our scattering calculations. These levels are represented by configuration-interaction wave functions. The relativistic effects are included in the Breit-Pauli approximation via one-body mass correction, Darwin, and spin-orbit interaction terms in the scattering equations. Rydberg series of resonances converging to the excited state thresholds are explicitly included in the calculations. The effective collision strengths are determined by integrating collision strengths and assuming a Maxwellian distribution of electron energies.