Accurate Analytical Formula Research Articles

Path integral for the quartic oscillator: an accurate analytic formula for the partition function

Path integral for the quartic oscillator: an accurate analytic formula for the partition function

Quasinormal modes of the Mannheim–Kazanas black holes

Abstract A spherically symmetric black hole solution in the conformal Weyl gravity was found by Mannheim and Kazanas in 1988. While the quasinormal modes (QNMs) of these black holes have been considered in a few works, here we complement these studies by considering in detail the regime of vanishing cosmological constant and negative values of the Weyl parameter for which we find quasinormal frequencies for scalar, Dirac and electromagnetic perturbations with the time-domain integration and WKB methods. In particular, we derive the compact and remarkably accurate analytic formula for the frequencies in the form of expansion in terms of the inverse multipole number. Comparison with the time-domain integration shows that the 6th order WKB method with the Padé approximants is quite accurate, unless the black hole is in the near extreme state.

Nanoscale dynamic mechanical analysis of a viscoelastic matrix inclusion within an elastic substrate

Advanced composite materials incorporate viscoelastic matrix inclusions in designated locations within elastic substrates, which grant them specialized energy dissipation capabilities of dynamic loadings. Identifying the local nanomechanical properties of the matrix-substrate complex within the composite is critical to its design and adaptation toward a specific target function. Here, we conduct FE simulations of nanoscale dynamic mechanical analysis (DMA) testing on hemispherical matrix inclusions within elastic substrates and analyze the variations in its indentation storage loss moduli measures for different matrix-substrate configurations. Then, we describe the mechanical system by simple spring models, identify the dominating parameters at different contact states of the nanoscale DMA testing, and obtain highly accurate analytical formulae that link the indentation measures of the matrix-substrate complex to the individual mechanical properties of the matrix and substrate parts. Our analysis can be directly integrated into benchmark nanomechanical testing methodologies of composite materials and promotes the local dynamic-mechanical characterization of complex materials systems, including nanomaterials, micro-architected structures, and bio-inspired designs.

Formulas for Beam Shift and Beam Narrowing in 1-D Leaky-Wave Antennas Due to Element Pattern

The radiation pattern of one-dimensional (1-D) unidirectional leaky-wave antennas (LWAs) is typically derived from the analysis of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">space factor</i> (SF) only. However, when the antenna aperture is characterized by a longitudinal radiating current, a cosine-type <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">element pattern</i> term has to be included: this factor may considerably affect the beam properties. Specifically, the beam peak is shifted and the beamwidth is narrowed when the element pattern is included. These effects are more prominent as the antenna size decreases and the beam angle increases. Approximate but accurate analytical formulas are here provided and numerically validated to predict the beam angle and the half-power beamwidth for these kinds of 1-D unidirectional LWAs when the element pattern is included.

A modified electromagnetic force calculation method has high accuracy and applicability for EMS maglev vehicle dynamics simulation

The traditional electromagnetic force calculation method does not consider the non-linear magnetization characteristics of the ferromagnetic material or the magnetic resistance in full circuit, resulting in large calculation errors when the electromagnet operation state is far from the rated state, and causing the dynamics simulation results to diverge from the actual situation. A more accurate analytical formula for electromagnetic force is derived based on the full circuit magnetic resistance modification and considering the non-linear magnetization characteristics of ferromagnetic materials. Then combined with the finite element simulation analysis, the magnetic resistance modification (MRM) method is proposed for calculating electromagnetic levitation force and guiding force. This MRM method greatly improves the accuracy of electromagnetic force calculations and provides greater accuracy for EMS maglev vehicle dynamics simulations.

Improved Analytical Triple-2D Leakage Inductance Model of Cone Winding Matrix Transformers

The transformer leakage inductance is one of the limiting factors for pulse shape quality in high voltage pulsed power (HVPP) converters that are essential in applications such as cancer treatment, particle accelerators, and free electron lasers. Cone winding matrix (CWM) transformers are commonly used in HVPP converters as they offer low leakage inductance and high insulation distance. This paper proposes an improved Triple-2D (iT2D) leakage inductance model for CWM transformers. The iT2D model identifies three basic cross sections of CWM transformers and calculates their associated leakage inductance contributions. The paper proposes an accurate and compact analytical formula of the leakage inductance contribution resulting from the magnetic energy stored in between the magnetic cores. For cross sections with cone windings, an expression is presented that computes the leakage inductance per unit length of non-parallel windings. The iT2D model is verified with measurements and finite element simulations of three transformer prototypes. The analytical solutions used in the iT2D model ensure applicability to arbitrary geometric aspect ratios and designs. Furthermore, the iT2D model is rapidly executable enabling its time-efficient integration in converter optimisations.

Quantum interference in gravitational particle production

Previous numerical investigations of gravitational particle production during the coherent oscillation period of inflation displayed unexplained fluctuations in the spectral density of the produced particles. We argue that these features are due to the quantum interference of the coherent scattering reactions that produce the particles. We provide accurate analytic formulae to compute the particle production amplitude for a conformally- coupled scalar field, including the interference effect in the kinematic region where the production can be interpreted as inflaton scattering into scalar final states via graviton exchange.

A formula for estimating the buckling capacity of corroded tubular members

PurposeSeveral experimental and numerical studies were performed in the past to estimate buckling capacity of corroded tubular members. However, the effect of initial imperfections has not been properly considered in most of these earlier proposed formulas. Therefore, the main objective of this paper is to propose an accurate analytical formula to determine the buckling capacity of patched corroded tubular members.Design/methodology/approachTubular members with initial geometrical imperfections can be regarded as beam-columns because of the combination of axial load and bending moment. The proposed formula is derived for a rectangular corrosion patch. The proposed formula is verified with results from finite element analysis of corroded tubular members and experimental results. The formula is also applied to an existing offshore jacket structure to highlight its significance and applicability. It is found that the buckling capacity of jacket members in splash zone reduces significantly with ageing. This reduction is around 29 and 14% for the selected brace and leg member respectively, during the design life. Finally, it is concluded that corrosion reduces the buckling capacity significantly and the proposed formula can be easily applied by practicing engineers to give an accurate and slightly conservative estimate the remaining buckling capacity.FindingsThe main finding is the new formula which accurately and conservatively estimate the buckling capacity of corroded tubular members. The proposed formula considers the secondary effect of both initial geometrical imperfections and shifting of centroid because of corrosion.Originality/valueThe proposed new formula is unique and original in that it considers both secondary effects from geometrical imperfections, reduction of cross-section from corrosion wastage and shifting of centroid because of corrosion. Finally, it is concluded that corrosion reduces the buckling capacity significantly and the proposed formula can be easily applied by practicing engineers to conservatively estimate the remaining buckling capacity and verify if further, more advanced estimations are needed.

Mechanical Properties of the Component “Column Web in Compression” in Steel Beam‐to‐column Joints

AbstractThe component method is widely used nowadays for the analytical prediction of the mechanical properties (rotational elastic stiffness, design resistance and ductility/deformation capacity) of steel and composite beam‐to‐column joints. Eurocode 3 Part 1‐8 (EN 1993‐1‐8) provides expressions for the evaluation of the properties of many different joint components. However, a quick examination of EN 1993‐1‐8 highlights a glaring disproportion between the quantity of information related to rotational stiffness and plastic resistance, and that related to ductility and deformation capacity. For the latter, only few recommendations are provided, sometimes in line, and sometimes even not with the component approach itself. Present paper concerns one of these components, the “column web in transverse compression” (CWC), for which no information is basically available in terms of ductility. An original procedure for the prediction of the plastic deformation capacity is presented. It relies on a preliminary evaluation of the four main component properties: elastic stiffness, plastic resistance, post‐plastic stiffness and ultimate resistance. Through this study, more accurate analytical formulae than those provided by EN 1993‐1‐8 for the evaluation of the plastic resistance are also suggested. All the developments are validated through extensive comparisons with available experimental tests.

Characterization of unstiffened column webs in transverse compression in steel beam-to-column joints

The component method is widely used nowadays for the analytical prediction of the mechanical properties (rotational elastic stiffness, design resistance and ductility/deformation capacity) of steel and composite beam-to-column joints. The Part 1-8 of Eurocode 3 (EN 1993-1-8) provides expressions for the evaluation of the properties of many different joint components. However, a quick examination of EN 1993-1-8 highlights a glaring disproportion between the quantity of information related to rotational stiffness and plastic resistance, and that related to ductility and deformation capacity. For the latter, only few recommendations are provided, sometimes in line, and sometimes even not with the component approach itself. Present paper concerns one of these components, the “column web in transverse compression” (CWC), for which no information is basically available in terms of ductility. An original procedure for the prediction of the plastic deformation capacity is presented. It relies on a preliminary evaluation of the four main component properties: elastic stiffness, plastic resistance, post-plastic stiffness and ultimate resistance. Through this study, more accurate analytical formulae than those provided by EN 1993-1-8 for the evaluation of the plastic resistance are also suggested. All the developments are validated through extensive comparisons with already available experimental tests performed in various countries.

Onset of Electron Captures and Shallow Heating in Magnetars

The loss of magnetic pressure accompanying the decay of the magnetic field in a magnetar may trigger exothermic electron captures by nuclei in the shallow layers of the stellar crust. Very accurate analytical formulas are obtained for the threshold density and pressure, as well as for the maximum amount of heat that can be possibly released, taking into account the Landau–Rabi quantization of electron motion. These formulas are valid for arbitrary magnetic field strengths, from the weakly quantizing regime to the most extreme situation in which electrons are all confined to the lowest level. Numerical results are also presented based on experimental nuclear data supplemented with predictions from the Brussels-Montreal model HFB-24. This same nuclear model has been already employed to calculate the equation of state in all regions of magnetars.

Kinetic specific features of singlet fission in highly anisotropic organic semiconductors.

Kinetics of singlet fission (SF) in molecular semiconductors, i.e., spontaneous splitting of the excited singlet state into a pair of triplet (T) excitons, is known to be strongly affected by geminate annihilation of created TT-pairs. In our work, we analyze in detail the specific properties of SF-kinetics in highly anisotropic molecular crystals (in which T-excitons undergo strongly anisotropic hopping migration) within the earlier proposed two-state model (TSM). This model allows for accurate treatment of the characteristic effects of anisotropic relative migration of T-excitons and TT-interaction on SF-kinetics, describing these effects within the approximation, that assumes kinetic coupling of two states: the [TT]-state of interacting TT-pairs and the [T + T]-state of freely migrating T-excitons. The TSM makes it possible to represent the TT-migration and interaction effects in terms of lattice-migration Green's functions, accurate analytical formulas that are obtained in this work. The TSM is applied to the analysis of SF-kinetics in rubrene single crystals, recently measured in a wide range of times (0.1ns < t < 104ns). The analysis enables one to obtain important information on specific properties of SF-kinetics in highly anisotropic crystals. In particular, the observed specific "hump" of SF-kinetics at intermediate times can be treated as a manifestation of the TT-coupling in the [TT]-state. It is also found that the characteristic asymptotic time-dependence of SF-kinetics (∼t-3/2) can markedly be distorted by spin relaxation in TT-pairs.

Mathematical modeling of the electric field in the planar and coaxial dielectric barrier discharge reactor in water treatment application

PurposeTo design a pulse power water treatment system, it is necessary to design a reactor optimally. One of the most essential types of reactors used in water treatment is the dielectric barrier discharge (DBD) reactor. The purpose of this paper is to model the electric field in the two types of planar and coaxial reactors to have an accurate analytical formula for using in the optimal design according to the required electric field of the treatment.Design/methodology/approachThe method proposed in this paper focuses on the voltage of different areas in the reactor and different boundary conditions to obtain the surface charge density. In this regard, parameters of the dielectric and treated material, as well as the reactor dimension, have been affected in the equations. To confirm the analytical results, the finite element method simulation has been performed, and it shows the accuracy of this method.FindingsThe exact analytical equation of the electric field is found within the discharge zone of the planar and coaxial DBD reactors. These equations can predict the values of different parameters of the reactor required to purify the material before each design and it does not even require simulation.Originality/valueThe electric field formula presented in this paper can allow the manufacturers of pulse power water treatment systems to optimize their design easily, cost-effectively and in less time. Also, the formulas provided are completely general and remain effective for all materials.

Medium-induced radiative kernel with the Improved Opacity Expansion

We calculate the fully differential medium-induced radiative spectrum at next-to-leading order (NLO) accuracy within the Improved Opacity Expansion (IOE) framework. This scheme allows us to gain analytical control of the radiative spectrum at low and high gluon frequencies simultaneously. The high frequency regime can be obtained in the standard opacity expansion framework in which the resulting power series diverges at the characteristic frequency ωc ∼ hat{q} L2. In the IOE, all orders in opacity are resumed systematically below ωc yielding an asymptotic series controlled by logarithmically suppressed remainders down to the thermal scale T « ωc, while matching the opacity expansion at high frequency. Furthermore, we demonstrate that the IOE at NLO accuracy reproduces the characteristic Coulomb tail of the single hard scattering contribution as well as the Gaussian distribution resulting from multiple soft momentum exchanges. Finally, we compare our analytic scheme with a recent numerical solution, that includes a full resummation of multiple scatterings, for LHC-inspired medium parameters. We find a very good agreement both at low and high frequencies showcasing the performance of the IOE which provides for the first time accurate analytic formulas for radiative energy loss in the relevant perturbative kinematic regimes for dense media.

Polyisobutylene and Silicone in Warm Edge Glazing Systems—Evaluation of Long-Term Performance

This article describes the characteristics of one type of sealing system used in warm edge glazing units and analyses the possible causes of damage. Attention was focused on the performance of the dual seal, PIB/silicone system. This type of glazing is widely used for modern curtain walls and roofs of office buildings and shopping centres. Study was focused on PIB displacement defects, which affects both the appearance and thermal performance of the curtain wall system. Wide-ranging field surveys were conducted to examine the problems identified in some office buildings. The information gathered in this way was used to identify the critical areas and causes of seal displacement in the analysed insulating glass units (IGUs). Laboratory tests were conducted on PIB and silicone seals retrieved from the removed defective units. The properties of these materials were determined and used to evaluate the applied edge sealing system and build a representative numerical model. Due to the problems encountered in deriving accurate analytical formulas, finite element (FE) approximation was used as a problem solving tool. The generated FE model and strain analysis were the key parts to obtaining a true representation of the actual behaviour of IGUs subjected to various environmental loads, taking into account the influence of the air cavity. Results of computer simulations and laboratory tests were compared for model validation. The effect of changes in ambient pressure was examined, showing the development of tensile strains in the silicone and PIB, which can lead to debonding. The greatest principal strains occur at the silicone/butyl rubber interface and this location should be considered to be the most susceptible to failure. The observations are summarised in the final conclusions. Additionally, as field study showed, after ten years in service, the percentage of damaged units is considerable. More frequent IGUs inspection should cover both appearance and thermal imaging to detect unsealed panels. From the standpoint of both durability and appearance, dual silicone/PIB should be phased out in favour of modern seal systems.

Massive scalar perturbations on Myers-Perry\u2013de Sitter black holes with a single rotation

This study investigates the stability of higher-dimensional singly rotating Myers-Perry–de Sitter (MP–dS) black holes against scalar field perturbations. The phase spaces of MP-dS black holes with one spin parameter are discussed. Additionally, the quasinormal modes (QNMs) of MP-dS black holes are calculated via the asymptotic iteration method and sixth-order Wentzel–Kramers–Brillouin approximation. For near-extremal MP-dS black holes, the event horizon may be considerably close to the cosmological horizon. In such cases, the Pöschl–Teller technique yields an accurate analytic formula for the QNMs. It is found that when the spin parameter of a black hole increases, the scalar perturbation modes oscillate at higher frequencies and decay faster. Furthermore, the MP-dS black hole with a single rotation is found to be stable under perturbation.

Prediction of Structural Performance of Vinyl Ester Polymer Concrete Using FEM Elasto-Plastic Model.

This paper presents the methodology for predicting the mechanical performance of structural elements made of polymer concrete (PC). A vinyl ester polymer concrete composition and the results of experimental studies to determine the basic mechanical properties of the material are presented. Following the strategy for sustainable development in the building industry, the material cost of polymer concrete was lowered by reducing the consumption of raw materials and the partial replacing of the microfiller fraction with recycled waste products—calcium fly ash. An accurate computational model enabling stress analysis is a convenient way to verify the suitability of PC as a construction material in structural applications. Due to difficulty in deriving an accurate analytical formula, numerical approximation (finite element method) was used as a method for solving the problem. Constitutive modeling of PC is a very important aspect of the strength calculations and here it was done within the framework of elasto-plasticity. Numerical evaluation of the static bearing capacity of PC manhole covers is shown as an example of the proposed FEM methodology. The results of computer simulations were compared with laboratory tests. Finally, the adequacy of the numerical modeling for testing new construction and material improvements is discussed. The study showed that the concrete damaged plasticity material model can be effectively used for the description of PC mechanical behavior.

Accurate analytic formula for light bending in Schwarzschild metric

We propose new analytic formulae describing light bending in the Schwarzschild metric. For an emission radii above the photon orbit at the 1.5 Schwarzschild radius, the formulae have an accuracy of better than 0.2% for the bending angle and 3% for the lensing factor for any trajectories that turn around a compact object by less than about 160°. In principle, they can be applied to any emission point above the horizon of the black hole. The proposed approximation can be useful for problems involving emission from neutron stars and accretion discs around compact objects when fast accurate calculations of light bending are required. It can also be used to test the codes that compute light bending using exact expressions via elliptical integrals.

Analytical determination of the structure of the outer crust of a cold nonaccreted neutron star

A very fast iterative method is presented to calculate the internal constitution of the outer crust of a cold nonaccreted neutron star, making use of very accurate analytical formulas for the transition pressures between adjacent crustal layers and their density. In addition to the composition of the different crustal layers, their depth and their baryonic mass content can be simultaneously estimated using an approximate solution of Einstein's general relativistic equations. The overall computing time is drastically reduced compared to the traditional approach, thus opening the door to large-scale statistical studies and sensitivity analyses.

KT and azimuth sub‐region deramp‐based high‐squint SAR imaging algorithm mounted on manoeuvring platforms

Owing to the existence of high-order motion parameters and vertical velocity, severe range–azimuth coupling and two-dimensional spatial variability of the imaging parameters degrade the image quality of high-squint synthetic aperture radar (SAR) mounted on manoeuvring platforms. To accommodate these issues, a novel sub-aperture SAR imaging algorithm based on keystone transform (KT) and azimuth sub-region deramp is proposed. In the range dimension processing, a correction function is constructed based on the scene centre to realise non-spatial-variant range cell migration (RCM) correction and range–azimuth decoupling, and then the residual spatial-variant RCM is removed by the KT. In the azimuth dimension processing, considering the impact of the range distortion on the azimuth modulated phase, accurate analytical formula of the signal in the azimuth time domain is deduced. On this basis, a fast azimuth compression method based on the azimuth sub-region deramp is proposed, and the theoretical effective imaging area is analysed. The theoretical analysis and the simulations show that the proposed imaging algorithm has high computational efficiency and can effectively expand the imaging area of high-squint SAR mounted on manoeuvring platforms.