Numerical Integration Procedure Research Articles

Численное интегрирование краевых задач для нелинейных обыкновенных дифференциальных уравнений второго порядка произвольной структуры с использованием итерационных процедур

Предложена итерационная процедура численного интегрирования краевых задач для нелинейных обыкновенных дифференциальных уравнений второго порядка произвольной структуры. Исходное дифференциальное уравнение алгебраическими преобразованиями приведено к линейному неоднородному дифференциальному уравнению второго порядка с постоянными коэффициентами, правая часть которого представлена в виде линейной комбинации производных искомой функции вплоть до второй степени и исследуемого дифференциального уравнения произвольной структуры. При построении разностной краевой задачи были использованы многочлены Тейлора, что позволило отказаться от аппроксимации производных конечными разностями. Степень многочленов Тейлора может быть выбрана равной любому натуральному числу, большему или равному двум. Построенное линейное неоднородное дифференциальное уравнение имеет три произвольных коэффициента. Показано, что коэффициент при исходном дифференциальном уравнении произвольной структуры в правой части полученного неоднородного линейного дифференциального уравнения связан со сходимостью итерационной процедуры, а коэффициенты при производных искомой функции влияют на устойчивость разностной краевой задачи на каждой итерации. Теоретически установлены значения коэффициентов при производных искомой функции, обеспечивающие устойчивость разностной краевой задачи независимо от вида исходного уравнения. При выполнении численного эксперимента выявлено, что коэффициент, обеспечивающий сходимость итерационной процедуры, зависит от вида исходного дифференциального уравнения. Численный эксперимент показал, что увеличение степени используемого многочлена Тейлора приводит к уменьшению погрешности между точным и найденным численным решениями.

Acoustic radiation force on a rigid elliptical cylinder in plane (quasi)standing waves

The acoustic radiation force on a 2D elliptical (non-circular) cylinder centered on the axis of wave propagation of plane quasi-standing and standing waves is derived, based on the partial-wave series expansion (PWSE) method in cylindrical coordinates. A non-dimensional acoustic radiation force function, which is the radiation force per unit length, per characteristic energy density and per unit cross-sectional surface of the ellipse, is defined in terms of the scattering coefficients that are determined by applying the Neumann boundary condition for an immovable surface. A system of linear equations involving a single numerical integration procedure is solved by matrix inversion. Numerical simulations showing the transition from the quasi-standing to the (equi-amplitude) standing wave behaviour are performed with particular emphasis on the aspect ratio a/b, where a and b are the ellipse semi-axes, as well as the dimensionless size parameter kb (where k is the wavenumber), without the restriction to a particular range of frequencies. It is found that at high kb values > 1, the radiation force per length with broadside incidence is larger, whereas the opposite situation occurs in the long-wavelength limit (i.e., kb < 1). The results are particularly relevant in acoustic levitation of elliptical cylinders, the acoustic stabilization of liquid columns in a host medium, acousto-fluidics devices, and other particle dynamics applications to name a few. Moreover, the formalism presented here may be effectively applied to compute the acoustic radiation force on other 2D surfaces of arbitrary shape such as super-ellipses, Chebyshev cylindrical particles, or other non-circular geometries.

Orbit determination of trans-Neptunian objects and Centaurs for the prediction of stellar occultations

The prediction of stellar occultations by Transneptunian objects and Centaurs is a difficult challenge that requires accuracy both in the occulted star position as for the object ephemeris. Until now, the most used method of prediction involving tens of TNOs/Centaurs was to consider a constant offset for the right ascension and for the declination with respect to a reference ephemeris. This offset is determined as the difference between the most recent observations of the TNO and the reference ephemeris. This method can be successfully applied when the offset remains constant with time. This paper presents an alternative method of prediction based on a new accurate orbit determination procedure, which uses all the available positions of the TNO from the Minor Planet Center database plus sets of new astrometric positions from unpublished observations. The orbit determination is performed through a numerical integration procedure (NIMA), in which we develop a specific weighting scheme. The NIMA method was applied for 51 selected TNOs/Centaurs. For this purpose, we have performed about 2900 new observations during 2007-2014. Using NIMA, we succeed in predicting the stellar occultations of 10 TNOs and 3 Centaurs between 2013 and 2015. By comparing the NIMA and JPL ephemerides, we highlighted the variation of the offset between them with time. Giving examples, we show that the constant offset method could not accurately predict 6 out of the 13 observed positive occultations successfully predicted by NIMA. The results indicate that NIMA is capable of efficiently refine the orbits of these bodies. Finally, we show that the astrometric positions given by positive occultations can help to further refine the orbit of the TNO and consequently the future predictions. We also provide the unpublished observations of the 51 selected TNOs and their ephemeris in a usable format by the SPICE library.

Acoustic scattering of a cylindrical quasi-Gaussian beam with arbitrary incidence focused on a rigid elliptical cylinder

Using the partial-wave series expansion method in cylindrical coordinates, a formal analytical solution for the acoustical scattering of a 2D cylindrical quasi-Gaussian beam with an arbitrary angle of incidence θi, focused on a rigid elliptical cylinder in a non-viscous fluid, is developed. The cylindrical focused beam expression is an exact solution of the Helmholtz equation. The scattering coefficients for the elliptical cylinder are determined by forcing the expression of the total (incident + scattered) field to satisfy the Neumann boundary condition for a rigid immovable surface, and performing the product of matrices involving an inversion procedure. Computations for the matrices elements require a single numerical integration procedure for each partial-wave mode. Numerical results are performed with particular emphasis on the focusing properties of the incident beam and its angle of incidence with respect to the major axis a of the ellipse as well as the aspect ratio a/b where b is the minor axis (assuming a > b). The method is validated and verified against previous results obtained via the T-matrix for plane waves. The present analysis is the first to consider an acoustical beam on an elliptic cylinder of variable cross-section as opposed to plane waves of infinite extent. Other 2D non-spherical and Chebyshev surfaces are mentioned that may be examined throughout this analytical formalism assuming a small deformation parameter ε.

Rheological Study of the Performance and the Vulnerability of an Element of RC Structure during and after a Seismic Signal

If the dimensioning methods of new structures are well controlled, the forecast of the seismic levels for which an existing building resists is a problem much more difficult to define. Their behavior and their comportment are completely unknown. The modeling of deteriorating hysteretic behavior is becoming increasingly important. With numerical integration procedures and the spectacular progress of computers, it is possible to predict the nonlinear behavior of reinforced concrete structures by using material rheological models during and after an earthquake. The objective of this research work is to develop a rheological model which allows digital nonlinear simulations (material) of structural systems excited by cyclic loads taken from seismic sources such as they are presented in reality, meaning in the form of digitalized accelerogram. This procedure requires a calibration dictated by particular transformations for the case of RC structures under this type of excitation in all points of the system and at every moment during the lasting of the signal. Besides, macrofibers models offer a satisfactory approach for analyzing all RC structures by taking into account various generated phenomena.

Resonance scattering and radiation force calculations for an elastic cylinder using the translational addition theorem for cylindrical wave functions

The standard Resonance Scattering Theory (RST) of plane waves is extended for the case of any two-dimensional (2D) arbitrarily-shaped monochromatic beam incident upon an elastic cylinder with arbitrary location using an exact methodology based on Graf’s translational addition theorem for the cylindrical wave functions. The analysis is exact as it does not require numerical integration procedures. The formulation is valid for any cylinder of finite size and material that is immersed in a nonviscous fluid. Partial-wave series expansions (PWSEs) for the incident, internal and scattered linear pressure fields are derived, and the analysis is further extended to obtain generalized expressions for the on-axis and off-axis acoustic radiation force components. The wave-fields are expressed using generalized PWSEs involving the beam-shape coefficients (BSCs) and the scattering coefficients of the cylinder. The off-axial BSCs are expressed analytically in terms of an infinite PWSE with emphasis on the translational offset distance d. Numerical computations are considered for a zeroth-order quasi-Gaussian beam chosen as an example to illustrate the analysis. Acoustic resonance scattering directivity diagrams are calculated by subtracting an appropriate background from the expression of the scattered pressure field. In addition, computations for the radiation force exerted on an elastic cylinder centered on the axis of wave propagation of the beam, and shifted off-axially are analyzed and discussed.

Path-integral method for the source apportionment of photochemical pollutants

Abstract. A new, path-integral method is presented for apportioning the concentrations of pollutants predicted by a photochemical model to emissions from different sources. A novel feature of the method is that it can apportion the difference in a species concentration between two simulations. For example, the anthropogenic ozone increment, which is the difference between a simulation with all emissions present and another simulation with only the background (e.g., biogenic) emissions included, can be allocated to the anthropogenic emission sources. The method is based on an existing, exact mathematical equation. This equation is applied to relate the concentration difference between simulations to line or path integrals of first-order sensitivity coefficients. The sensitivities describe the effects of changing the emissions and are accurately calculated by the decoupled direct method. The path represents a continuous variation of emissions between the two simulations, and each path can be viewed as a separate emission-control strategy. The method does not require auxiliary assumptions, e.g., whether ozone formation is limited by the availability of volatile organic compounds (VOCs) or nitrogen oxides (NOx), and can be used for all the species predicted by the model. A simplified configuration of the Comprehensive Air Quality Model with Extensions (CAMx) is used to evaluate the accuracy of different numerical integration procedures and the dependence of the source contributions on the path. A Gauss–Legendre formula using three or four points along the path gives good accuracy for apportioning the anthropogenic increments of ozone, nitrogen dioxide, formaldehyde, and nitric acid. Source contributions to these increments were obtained for paths representing proportional control of all anthropogenic emissions together, control of NOx emissions before VOC emissions, and control of VOC emissions before NOx emissions. There are similarities in the source contributions from the three paths but also differences due to the different chemical regimes resulting from the emission-control strategies.

Mixed‐mode delamination in 2D layered beam finite elements

SummaryIn this work, a 2D finite element (FE) formulation for a multi‐layer beam with arbitrary number of layers with interconnection that allows for mixed‐mode delamination is presented. The layers are modelled as linear beams, while interface elements with embedded cohesive‐zone model are used for the interconnection. Because the interface elements are sandwiched between beam FEs and attached to their nodes, the only basic unknown functions of the system are two components of the displacement vector and a cross‐sectional rotation per layer. Damage in the interface is modelled via a bi‐linear constitutive law for a single delamination mode and a mixed‐mode damage evolution law. Because in a numerical integration procedure, the damage occurs only in discrete integration points (i.e. not continuously), the solution procedure experiences sharp snap backs in the force‐displacements diagram. A modified arc‐length method is used to solve this problem. The present model is verified against commonly used models, which use 2D plane‐strain FEs for the bulk material. Various numerical examples show that the multi‐layer beam model presented gives accurate results using significantly less degrees of freedom in comparison with standard models from the literature. Copyright © 2015 John Wiley & Sons, Ltd.

A non-associated plasticity model with anisotropic and nonlinear kinematic hardening for simulation of sheet metal forming

A material model for more thorough analysis of plastic deformation of sheet materials is presented in this paper. This model considers the following aspects of plastic deformation behavior of sheet materials: (1) the anisotropy in yield stresses and in work hardening by using Hill’s 1948 quadratic yield function and non-constant stress ratios which leads to different flow stress hardening in different directions, (2) the anisotropy in plastic strains by using a quadratic plastic potential function and non-associated flow rule, also based on Hill’s 1948 model and r-values, and (3) the cyclic hardening phenomena such as the Bauschinger effect, permanent softening and transient behavior for reverse loading by using a coupled nonlinear kinematic hardening model. Plasticity fundamentals of the model were derived in a general framework and the model calibration procedure was presented for the plasticity formulations. Also, a generic numerical stress integration procedure was developed based on backward-Euler method, so-called multi-stage return mapping algorithm. The model was implemented in the framework of the finite element method to evaluate the simulation results of sheet metal forming processes. Different aspects of the model were verified for two sheet metals, namely DP600 steel and AA6022 aluminum alloy. Results show that the new model is able to accurately predict the sheet material behavior for both anisotropic hardening and cyclic hardening conditions. The drawing of channel sections and the subsequent springback were also simulated with this model for different drawbead configurations. Simulation results show that the current non-associated anisotropic hardening model is able to accurately predict the sidewall curl in the drawn channel sections.

On the resolution of the phase space density required to obtain a specified accuracy of the solar wind bulk velocity

AbstractPlasma measurements with significantly improved accuracy and time resolution are needed to advance knowledge and understanding of plasma physical process in the magnetosphere and the solar wind. In this study, it is shown how small unresolved features in particle distribution functions can have significant impacts on the measurement of plasma bulk parameters such as the plasma bulk velocity. Errors are caused by the numerical integration scheme used to compute the phase space integrals and by errors or experimental uncertainties in measurements of the phase space distribution. Here the errors in the bulk velocity caused by the numerical integration procedure are investigated for both Gaussian (Maxwellian) velocity distributions and Kappa distributions. The particle distribution functions are perturbed by a small “bump” represented by a relatively small amplitude Gaussian with a thermal width that is much smaller than the thermal width of the unperturbed distribution. The goal is to find the grid spacing Δv in phase space required to obtain a specified accuracy of the mean flow velocity . For parameters encountered in the solar wind near the orbit of the Earth, it is shown that to determine the radial component of the solar wind bulk velocity with an accuracy less than 0.1 km/s requires a grid spacing less than Δv/v = 0.8% or, equivalently, an energy spacing less than ΔE/E = 1.6%. To accurately resolve the velocity components transverse to the heliocentric radial direction, it is found that the spacing of the angle bins must be less than Δθ = 0.4∘. The required spacing of the energy and angle bins are approximately 1 order of magnitude less than those of any solar wind ion instrument ever flown in space.

Green’s functions and Eshelby tensors for an ellipsoidal inclusion in a non-centrosymmetric and anisotropic micropolar medium

We derive simple integral solutions for the Green’s tensors and Eshelby tensor for a generally anisotropic and non-centrosymmetric micropolar material. The material properties of a micropolar medium are characterized by three fourth-order tensors, C, B and D, in which C relates the stress to the strain, D connects the couple stress to the curvature tensor, and B is the coupling tensor, linking the stress to the curvature tensor or the couple stress to the strain tensor. Here we find that when C and B tensors possess minor symmetry conditions, the Green’s tensors for a general anisotropic micropolar medium can be derived in simple integral expressions. We note however that in our formulation there is no any restriction on the D tensor. Specifically, we will show that, under the conditions on the C and B tensors, the Green’s tensors can be exactly expressed as a simple line integral over a unit circle, and that the Eshelby’s tensors for a general ellipsoidal inclusion can be derived as surface integrals over a unit sphere, entirely analogous to those of the classical elasticity. The exact integral form of the tensors can be implemented with numerical integration procedures. We will demonstrate that our numerical solutions are in good accuracy compared with the existing solutions for simple situations. In the literature analytic Green’s tensors are existed only for isotropic and non-centrosymmetric micropolar medium. The solutions derived here can serve as benchmark solutions for certain classes of anisotropic micropolar medium, and also can be used as an approximate solution for a micropolar medium with general material properties without any constraint conditions.

Complete Analytical Solution of the Heston Model for Option Pricing and Value-at-Risk Problems: A Probability Density Function Approach

• The first ever explicit formulation of the concept of an option’s probability density functions has been introduced in our publications Breakthrough in Understanding Derivatives and Option Based Hedging - Marginal and Joint Probability Density Functions of Vanilla Options -- True Value-at-Risk and Option Based Hedging Strategies and Complete Analytical Solution of the Asian Option Pricing and Asian Option Value-at-Risk Problems. A Probability Density Function Approach. (See links http://ssrn.com/abstract=2489601 and http://ssrn.com/abstract=2546430). • In this paper we report similar unique results for pricing options in the presence of stochastic volatility (Heston model), enabling complete analytical resolution of all problems associated with options considered within the Heston Model. • Our discovery of the probability density function for options with stochastic volatility enables exact closed-form analytical results for their expected values (prices) for the first time without depending on approximate numerical methods and a Fourier transform that only abbreviates complex numerical integration procedure. • Expected value is the first moment. All higher moments are as easily represented in closed form based on our probability density function, but are not calculable by extensions of other numerical methods, such as a Fourier transform, now used to represent the first moment. • Our formulation of the density function for options with stochastic volatility within the Heston model is expressive enough to enable derivation for the first time ever of corollary closed-form analytical results for such Value-At-Risk characteristics as the probabilities that options with stochastic volatility will be below or above any set of thresholds at termination. Such assessments are absolutely out of reach of current published methods for treating options within the Heston model.• All numerical evaluations based on our analytical results are practically instantaneous and absolutely accurate.

Complete Analytical Solution of the Asian Option Pricing and Asian Option Value-at-Risk Problems: A Probability Density Function Approach

•The first ever explicit formulation of the concept of an option’s probability density functions has been introduced in our publication “Breakthrough in Understanding Derivatives and Option Based Hedging - Marginal and Joint Probability Density Functions of Vanilla Options - True Value-at-Risk and Option Based Hedging Strategies” (see link http://ssrn.com/abstract=2489601). •In this paper we report similar unique results for Asian Options, enabling complete analytical resolution of all problems associated with Asian Options. •Our discovery of the Asian Option probability density function enables exact closed-form analytical results for its expected value (price) for the first time without depending on Inverse Laplace or Fourier transforms that only abbreviate complex numerical integration procedures. •Expected value is the first moment. All higher moments are as easily represented in closed form based on our probability density function, but are not calculable by extensions of other numerical methods, such as Inverse Laplace or Fourier transforms, now used to represent the first moment. •Our formulation of the Asian Option probability density function is general enough to cover interesting and practical cases that are not addressed in the literature at all: for example, cases for which the averaging period is a terminal subset of the contract period, e.g. the last 3 months of the option lifetime. •Our formulation of the Asian Option probability density function is expressive enough to enable derivation for the first time ever of corollary closed-form analytical results for such Value-At-Risk characteristics as the probabilities that an Asian Option will be below or above any set of thresholds at any future time before or at termination. Such assessments are absolutely out of reach of current published methods for treating Asian Options. •Resolution of the Asian Option Pricing problem enables knowledge of their linear correlation with Vanilla Options and, therefore, hedging in terms of Vanilla Options. Moreover, an opportunity to construct “synthetic” Asian Options from Vanilla Options for a given termination becomes a trivial exercise. •All numerical evaluations based on our analytical results are practically instantaneous and absolutely accurate.

Evaluation of the Effective Temperature of Sunspots Using Molecular Parameters of AlF

The physical conditions of celestial objects can be analyzed using the spectrum of atoms or molecules present in the object. The present work focuses on the spectroscopic analysis of astrophysically significant molecule AlF. The evaluation of Franck-Condon (FC) factors andr-centroids is done by a numerical integration procedure using the suitable potential energy curves forC1Σ+-A1Σ+,b3Σ+-a3Πr,c3Σ-a3Πr, andf3Π-a3Πrband systems of AlF molecule. The intensity of various bands is discussed with the help of derived FC factors. The band degradation and the nature of potential energy curves are studied usingr-centroid values. The vibrational temperature of sunspot is estimated to be around 1220 ± 130 K which falls in the reported temperature range of cold sunspots.

Dynamic characteristics of hybrid tower of cable-stayed bridges

The dynamic characterization is important in making accurate predictions of the seismic response of the hybrid structures dominated by different damping mechanisms. Different damping characteristics arise from the construction of the tower with different materials: steel for the upper part; reinforced concrete for the lower main part and interaction with supporting soil. The process of modeling damping matrices and experimental verification is challenging because damping cannot be determined via static tests as can mass and stiffness. The assumption of classical damping is not appropriate if the system to be analyzed consists of two or more parts with significantly different levels of damping, such as steel/ concrete mixed structure - supporting soil coupled system. The dynamic response of structures is critically determined by the damping mechanisms, and its value is very important for the design and analysis of vibrating structures. An analytical approach capable of evaluating the equivalent modal damping ratio from structural components is desirable for improving seismic design. Two approaches are considered to define and investigate dynamic characteristics of hybrid tower of cable-stayed bridges: The first approach makes use of a simplified approximation of two lumped masses to investigate the structure irregularity effects including damping of different material, mass ratio, frequency ratio on dynamic characteristics and modal damping; the second approach employs a detailed numerical step-by step integration procedure in which the damping matrices of the upper and the lower substructures are modeled with the Rayleigh damping formulation.

Algorithm for the Fast Calculation of Magnetic Fields Generated by Arc-Shaped Conductors With Rectangular Cross Section

This paper presents a novel procedure for the fast numerical integration of the magnetic field produced by arc-shaped conductors, which are characterized by a rectangular cross section. Available procedures are based on the analytical integration of Biot-Savart's law, but most of them exploit the analytical integration along one coordinate and then perform a 2-D numerical integration. In the proposed procedure, the analytical integration was performed along two coordinates, obtaining a 1-D integrand (thus avoiding the use of elliptic integrals), which is very easy to process using a state-of-the-art numerical quadrature library. The result is very satisfactory in terms of high speed and precision, particularly on the conductor surface, and when its cross dimensions are very uneven.

To theory of asymptotically stable accelerating Universein Riemann-Cartan spacetime

Homogeneous isotropic cosmological models built in the framework of the Poincar'e gauge theory of gravity based on general expression of gravitational Lagrangian with indefinite parameters are analyzed. Special points of cosmological solutions for flat cosmological models at asymptotics and conditions of their stability in dependence of indefinite parameters are found. Procedure of numerical integration of the system of gravitational equations at asymptotics is considered. Numerical solution for accelerating Universe without dark energy is obtained.

Comparación de la metodología utilizada para el cálculo de los parámetros orbitales en los Centros de Análisis del Servicio GNSS Internacional (IGS)

The orbital information given in the form of Ultra-Rapid Orbits (IGA and IGU), Rapid Orbits (IGR), and Final Orbits (IGS), as well as the Earth rotational parameters and the clock offset provided by the International GNSS Service plays an important role in supporting the Global Navigation Satellite Systems GPS and GLONASS. In this paper, we compare the observables, the force models, the adjustment methods and the numerical integration procedures applied by the different International GNSS Service Analysis Centers for the orbital parameter determination.

Optimum dynamic analysis of 2D frames using free-scaled wavelet functions

This paper presents a wavelet-based scheme for dynamic analysis of 2-dimensional (2D) frames. In the proposed approach, free-scaled wavelet functions are developed for Multi-Degrees-of-Freedom (MDOF) structures, particularly, complex Chebyshev and simple Haar wavelets are implemented. A simple step-by-step and explicit algorithm is presented to calculate the time history response of 2D frames. The validity of the proposed procedure is demonstrated with two examples compared with several common numerical integration procedures such as family ofNewmark-β, Wilson-θ and central difference method. Finally, it is shown that dynamic analysis of 2D frames is optimally accomplished by lesser computational time and high accuracy of results.

Intensity distribution in the A1 Σ+−X1 Σ+ and A′ 1 Π−X1 Σ+ systems of barium monoxide

The integrated intensities of a few bands in the vibrational structure of the astrophysically significant A1Σ+−X1Σ+ and A′ 1Π−X1Σ+ systems of barium monoxide (BaO) have been measured experimentally using band spectra. The radiative transition parameters such as Franck-Condon (FC) factor, r-centroid, electronic transition moment, Einstein coefficient, band oscillator strength, radiative life time and effective vibrational temperature have been computed for A1Σ+−X1Σ+ and A′ 1Π−X1Σ+ systems of BaO molecule by the more reliable numerical integration procedure for the experimentally known vibrational levels using Rydberg-Klein-Rees (RKR) potential energy curves. The effective vibrational temperatures of these band systems of BaO molecule are found to be nearly 3100 °K. Hence, the radiative transition parameters help us to ascertain the presence of BaO molecule in the interstellar medium and S-stars.