Accurate Integration Method Research Articles

A family of noniterative integration methods with desired numerical dissipation

ABSTRACTA new family of unconditionally stable integration methods for structural dynamics has been developed, which possesses the favorable numerical dissipation properties that can be continuously controlled. In particular, it can have zero damping. This numerical damping is helpful to suppress or even eliminate the spurious participation of high frequency modes, whereas the low frequency modes are almost unaffected. The most important improvement of this family method is that it involves no nonlinear iterations for each time step, and thus it is very computationally efficient when compared with a general second‐order accurate integration method, such as the constant average acceleration method. Copyright © 2014 John Wiley & Sons, Ltd.

Ion fluence dependence of the total sputtering yield and differential angular sputtering yield of bismuth due to 50 keV argon ion irradiation

The dependences of the total sputtering yield of Bi and the differential angular distribution of these sputtered Bi atoms on the fluence of 50keV Ar+ ions at normal incidence have been experimentally measured. Polycrystalline Bi targets were used for these purposes. The collector technique and accurate current integration methods were adopted for the determination of angular distributions of sputtered Bi atoms. The ion fluence was varied from 1.9×1019 to 3.1×1020ions/cm2. The sputtered atoms were collected on high purity aluminum foils under ultra-high vacuum (∼5×10−9Torr). The collector foils were subsequently analyzed using heavy ion Rutherford backscattering spectroscopy. The shape of the angular distribution of sputtered atoms was found not to change significantly with the fluence, but the sputtering yield increased significantly from 2.2±0.2 to 9.6±0.6atoms/ion over the fluence range studied.

An accurate application of the integral method applied to the diffusion of oxygen in absorbing tissue

Accurate integral methods are applied to a one dimensional moving boundary problem describing the diffusion of oxygen in absorbing tissue. These methods have been well studied for classic Stefan problems but this situation is unusual because there is no condition which contains the velocity of the moving boundary explicitly. This paper begins by giving a short time solution and then discusses some of the previous integral methods found in the literature. The main drawbacks of these solutions are that they cannot be solved from t=0 and also cannot determine the end behaviour. This is due to the non-uniform initial profile which integral methods typically fail to capture. The use of a novel transformation removes this non-uniformity and, on applying optimal integral methods to the resulting system, leads to simple and yet very accurate approximate solutions that overcome the deficiencies of previous methods.

Multilayer integral method for simulation of eddy currents in thin volumes of arbitrary geometry produced by MRI gradient coils

This article aims to present a fast, efficient and accurate multi-layer integral method (MIM) for the evaluation of complex spatiotemporal eddy currents in nonmagnetic and thin volumes of irregular geometries induced by arbitrary arrangements of gradient coils. The volume of interest is divided into a number of layers, wherein the thickness of each layer is assumed to be smaller than the skin depth and where one of the linear dimensions is much smaller than the remaining two dimensions. The diffusion equation of the current density is solved both in time-harmonic and transient domain. The experimentally measured magnetic fields produced by the coil and the induced eddy currents as well as the corresponding time-decay constants were in close agreement with the results produced by the MIM. Relevant parameters such as power loss and force induced by the eddy currents in a split cryostat were simulated using the MIM. The proposed method is capable of accurately simulating the current diffusion process inside thin volumes, such as the magnet cryostat. The method permits the priori-calculation of optimal pre-emphasis parameters. The MIM enables unified designs of gradient coil-magnet structures for an optimal mitigation of deleterious eddy current effects.

Size-velocity correlations in hybrid high order moment/multi-fluid methods for polydisperse evaporating sprays: Modeling and numerical issues

Kah et al. (2010) [30,33] recently developed the Eulerian multi-size moment model (EMSM) which tackles the modeling and numerical simulation of polydisperse multiphase flows. Using a high order moment method in a compact interval, they suggested to reconstruct the number density function (NDF) by entropy maximization, which leads to a unique and realizable NDF, potentially in several size intervals, thus leading to an hybrid method between Multifluid and high order. This reconstruction is used to simulate the evaporation process, by an evaluation of the flux of droplet disappearance at zero size, the fluxes of droplets between size intervals, and an accurate description of the size shift induced by evaporation Massot et al. (2010) [15]. Although this method demonstrated its potential for evaporating polydisperse flows, two issues remain to be addressed. First, the EMSM only considers one velocity for all droplets, thus decoupling size from velocity, which is too restrictive for distributions with a large size spectrum. In most applications size-conditioned dynamics have to be accounted for. Second, the possibility to have separated dynamics for each size can lead to quasi-monodisperse distributions, which corresponds to a hard limiting case for the EM algorithm. So the behavior of the algorithm needs to be investigated, in order to reproduce the entire moment space with a reasonable accuracy. The aim of this paper is thus twofold. The EM and its related algorithm are enhanced by using a more accurate integration method in order to handle NDF close to the frontier of the moment space associated with an adaptive number of parameters to reconstruct the NDF accurately and efficiently, as well as tabulated initial guess to optimize the computational time. Then, a new model called CSVM (coupled size-velocity moments model) is introduced. Size-velocity correlations are addressed either in the evaporation and drag processes, or in the convective transport. To reach this goal, a velocity reconstruction for each size is suggested, using only one additional moment per dimension, and which can be directly applied to several size intervals. Thus, this method is a direct generalization of EMSM. To handle the convective transport, a flux splitting scheme is proposed, based on the underlying kinetic description of the disperse phase. Comparing to existing approaches, a main novelty of the CSVM is that our kinetic approach ensures built-in realizability conditions, no additional corrections of the moments being needed at each time step. The full strategy is first evaluated in 0D and 1D cases, which either demonstrates the ability to reproduce both evaporation, drag force and convection with size-velocity correlations, or the possible extension to several size intervals. Finally, the method is applied on 2D cases with only one section, showing the ability of the CSVM and its related algorithms to capture the main physics of polydisperse evaporating sprays with a minimal number of moments.

Structure-dependent difference equations for time integration

A structure-dependent explicit method with enhanced stability properties is proposed in this study. In general, the method offers unconditional stability for structural systems except those with a particular instantaneous stiffness hardening behavior. In addition, it is second-order accurate and displays no overshooting in high frequency responses. Numerical experiments reveal that the proposed method saves a substantial amount of computational effort in solving inertial problems where only the low frequency responses are of interest, when compared to a general second-order accurate integration method.

Fast multi-particle scattering: A hybrid solver for the Maxwell equations in microstructured materials

A variety of problems in device and materials design require the rapid forward modeling of Maxwell’s equations in complex micro-structured materials. By combining high-order accurate integral equation methods with classical multiple scattering theory, we have created an effective simulation tool for materials consisting of an isotropic background in which are dispersed a large number of micro- or nano-scale metallic or dielectric substructures.

Stable and accurate integral equation methods for scattering problems with multiple material interfaces in two dimensions

Stable and accurate integral equation methods for scattering problems with multiple material interfaces in two dimensions

Singularities in the complex physical plane for deep water waves

AbstractDeep water waves in two-dimensional flow can have curvature singularities on the surface profile; for example, the limiting Stokes wave has a corner of $2\lrm{\pi} / 3$ radians and the limiting standing wave momentarily forms a corner of $\lrm{\pi} / 2$ radians. Much less is known about the possible formation of curvature singularities in general. A novel way of exploring this possibility is to consider the curvature as a complex function of the complex arclength variable and to seek the existence and nature of any singularities in the complex arclength plane. Highly accurate boundary integral methods produce a Fourier spectrum of the curvature that allows the identification of the nearest singularity to the real axis of the complex arclength plane. This singularity is in general a pole singularity that moves about the complex arclength plane. It approaches the real axis very closely when waves break and is associated with the high curvature at the tip of the breaking wave. The behaviour of these singularities is more complex for standing waves, where two singularities can be identified that may collide and separate. One of them approaches the real axis very closely when a standing wave forms a very narrow collapsing column of water almost under free fall. In studies so far, no singularity reaches the real axis in finite time. On the other hand, the surface elevation $y(x)$ has square-root singularities in the complex $x$ plane that do reach the real axis in finite time, the moment when a wave first starts to break. These singularities have a profound effect on the wave spectra.

Extremely Accurate Symplectic Integration Method with Frequency Dependent Coefficients for Wave Equations

Extremely Accurate Symplectic Integration Method with Frequency Dependent Coefficients for Wave Equations

Measurement of the eddy diffusion term in chromatographic columns. I. Application to the first generation of 4.6 mm I.D. monolithic columns

Abstract The corrected heights equivalent to a theoretical plate (HETP) of three 4.6 mm I.D. monolithic Onyx-C 18 columns (Onyx, Phenomenex, Torrance, CA) of different lengths (2.5, 5, and 10 cm) are reported for retained (toluene, naphthalene) and non-retained (uracil, caffeine) small molecules. The moments of the peak profiles were measured according to the accurate numerical integration method. Correction for the extra-column contributions was systematically applied. The peak parking method was used in order to measure the bulk diffusion coefficients of the sample molecules, their longitudinal diffusion terms, and the eddy diffusion term of the three monolithic columns. The experimental results demonstrate that the maximum efficiency was 60 000 plates/m for retained compounds. The column length has a large impact on the plate height of non-retained species. These observations were unambiguously explained by a large trans-column eddy diffusion term in the van Deemter HETP equation. This large trans-rod eddy diffusion term is due to the combination of a large trans-rod velocity bias (≃3%), a small radial dispersion coefficient in silica monolithic columns, and a poorly designed distribution and collection of the sample streamlets at the inlet and outlet of the monolithic rod. Improving the performance of large I.D. monolithic columns will require (1) a detailed knowledge of the actual flow distribution across and along these monolithic rod and (2) the design of appropriate inlet and outlet distributors designed to minimize the nefarious impact of the radial flow heterogeneity on band broadening.

Integration of MR images with electroanatomical maps: feasibility and utility in guiding left ventricular substrate mapping

The purpose of the study is to evaluate the feasibility and utility of magnetic resonance (MR) image and electroanatomic (EA) maps integration in guiding detailed left ventricle (LV) anatomical and substrate mapping, identifying the most accurate registration strategy. Twenty-five patients with dilated ischemic or non-ischemic cardiomyopathy were enrolled. We first verified the feasibility and accuracy of EA mapping and MR image integration using four different strategies (15 patients). Different EA maps were performed according to the strategy in exam: aortic map, collected from the descending portion of the arch to the ascending one; partial or complete LV map, reconstructed with a minimum of 40 widely distributed points or 200 points, respectively. We then evaluated the utility in LV substrate mapping of the most accurate integration method identified (ten patients). Strategy III, based on aortic map and a partial LV map, allowed us to obtain an accurate integration with MR images of aorta and LV with a lower number of EA LV points; we therefore used this strategy during phase II of the study. Both mean LV end diastolic volume and long- and short-axis LV end diastolic diameters obtained by MR were not significantly different compared with Carto measurements. Eighty-eight percent of the segments with transmural/subendocardial scar detected by delayed enhanced MR were localized on bipolar voltage maps projected on MR-integrated images. This study shows that integration strategy III represents the optimal registration method. Its clinical utility consists on guiding the catheter roving inside the chamber, mapping all areas of the LV and optimizing scar reconstruction.

Comparison of capability of time integration methods in capturing dynamic loading

Numerical properties of the time integration method proposed by the first author of this paper in 2007 are the same as those of the constant average acceleration method (AAM) for linear elastic systems, except that the capability to capture dynamic loading was not explored. It was found that there were different quadrature equations to predict the next step displacement increment. A modified quadrature equation of this method was derived so that the equation to determine the next step displacement was numerically equivalent to the equation used in the constant AAM. It was verified that the original form of this method, in general, had a better capability to capture dynamic loadings than the constant AAM. This excellent property, in addition to computational efficiency, will help to make this method competitive with general secondorder accurate integration methods.

A sector-integration method for dose/MU calculation in a uniform scanning proton beam

An accurate, simple and time-saving sector integration method for calculating the proton output (dose/monitor unit, MU) is presented based on the following treatment field parameters: aperture shape, aperture size, measuring position, beam range and beam modulation. The model is validated with dose/MU values for 431 fields previously measured at our center. The measurements were obtained in a uniform scanning proton beam with a parallel plate ionization chamber in a water phantom. For beam penetration depths of clinical interest (6–27 cm water), dose/MU values were measured as a function of spread-out Bragg peak (SOBP) extent and aperture diameter. First, 90 randomly selected fields were used to derive the model parameters, which were used to compute the dose/MU values for the remaining 341 fields. The min, max, average and the standard deviation of the difference between the calculated and the measured dose/MU values of the 341 fields were used to evaluate the accuracy and stability, for different energy ranges, aperture sizes, measurement positions and SOBP values. The experimental results of the five different functional sets showed that the calculation model is accurate with calculation errors ranging from −2.4% to 3.3%, and 99% of the errors are less than ±2%. The accuracy increases with higher energy, larger SOBP and bigger aperture size. The average error in the dose/MU calculation for small fields (field size <25 cm2) is 0.31 ± 0.96 (%).

Explicit Pseudodynamic Algorithm with Improved Stability Properties

An explicit pseudodynamic algorithm with an improved stability property is proposed herein. This algorithm is shown to be unconditionally stable for any linear elastic systems and any instantaneous stiffness softening systems. The most attracting stability property is that it can have unconditional stability for the instantaneous hardening systems with the instantaneous degree of nonlinearity less than or equal to 2. This property has never been found among the currently available explicit algorithms. Hence, it may be applied to perform a general pseudodynamic test without considering the stability problem since it is rare for a civil engineering structure whose instantaneous degree of nonlinearity is greater than 2. This explicit algorithm can be implemented as a common explicit pseudodynamic algorithm, such as the use of the Newmark explicit method, since it does not involve any iteration procedure. In addition, it possesses comparable accuracy as that of a general second-order accurate integration method such as the Newmark explicit method. Both numerical and error propagation properties are analytically studied and numerical experiments are used to confirm these properties. Actual pseudodynamic tests attested to the feasibility of this proposed explicit pseudodynamic algorithm.

New approximate formula for the generalized temperature integral

AbstractThe generalized temperature integral $ \int\nolimits_0^T {T^m \,\exp ( - E/RT)dT} $ frequently occurs in nonisothermal kinetic analysis. This article has proposed a new approximate formula for the generalized temperature integral, which is in the following form: For commonly used values of m in kinetic analysis, the deviation of the new approximation from the numerical values of the integral is within 0.4%. More importantly, the new formula represents the exponential approximation, which is not found earlier, and it can result in a new and very accurate integral method in kinetic analysis. © 2009 American Institute of Chemical Engineers AIChE J, 2009

A Gaussian quadrature method for total energy analysis in electronic state calculations

AbstractThis article reports studies by Fukushima and coworkers since 1980 concerning their highly accurate numerical integral method using Gaussian quadratures to evaluate the total energy in electronic state calculations. Gauss‐Legendre and Gauss‐Laguerre quadratures were used for integrals in the finite and infinite regions, respectively. Our previous article showed that, for diatomic molecules such as CO and FeO, elliptic coordinates efficiently achieved high numerical integral accuracy even with a numerical basis set including transition metal atomic orbitals. This article will generalize straightforward details for multiatomic systems with direct integrals in each decomposed elliptic coordinate determined from the nuclear positions of picked‐up atom pairs. Sample calculations were performed for the molecules O3 and H2O. This article will also try to present, in another coordinate, a numerical integral by partially using the Becke's decomposition published in 1988, but without the Becke's fuzzy cell generated by the polynomials of internuclear distance between the pair atoms. Instead, simple nuclear weights comprising exponential functions around nuclei are used. The one‐center integral is performed with a Gaussian quadrature pack in a spherical coordinate, included in the author's original program in around 1980. As for this decomposition into one‐center integrals, sample calculations are carried out for Li2. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

A hybrid time/Laplace integration method based on numerical Green’s functions in conduction heat transfer

The present paper describes an efficient time/Laplace domain approach to analyze numerically heat conduction problems. An efficient recurrence relationship for the temperature in the time-domain, based on the Green’s functions of the model, is presented. Primarily, Green’s functions in nodal coordinates are explicitly calculated by the finite element method in the Laplace domain and subsequently, the Stehfest and the Zakian Laplace inversion schemes are employed to compute numerically Green’s functions that transfer solution at time 0 to time Δ t. As a result, a new family of highly accurate time integration methods called ExGA-Stehfest and ExGA-Zakian is obtained. Finally, numerical examples are presented in order to illustrate the high accuracy and potentialities of these novel approaches.

Growth of Cloud Droplets by Turbulent Collision–Coalescence

Abstract An open question in cloud physics is how rain forms in warm cumulus as rapidly as it is sometimes observed. In particular, the growth of cloud droplets across the size gap from 10 to 50 μm in radius has not been fully explained. In this paper, the authors investigate the growth of cloud droplets by collision–coalescence, taking into account both the gravitational mechanism and several enhancements of the collision–coalescence rate due to air turbulence. The kinetic collection equation (KCE) is solved with an accurate bin integral method and a newly developed parameterization of turbulent collection kernel derived from direct numerical simulation of droplet-laden turbulent flows. Three other formulations of the turbulent collection kernel are also considered so as to assess the dependence of the rain initiation time on the nature of the collection kernel. The results are compared to the base case using the Hall hydrodynamical–gravitational collection kernel. Under liquid water content and eddy dissipation rate values typical of small cumulus clouds, it is found that air turbulence has a significant impact on the collection kernel and thus on the time required to form drizzle drops. With the most realistic turbulent kernel, the air turbulence can shorten the time for the formation of drizzle drops by about 40% relative to the base case, applying measures based on either the radar reflectivity or the mass-weighted drop size. A methodology is also developed to unambiguously identify the three phases of droplet growth, namely, the autoconversion phase, the accretion phase, and the larger hydrometeor self-collection phase. The important observation is that even a moderate enhancement of collection kernel by turbulence can have a significant impact on the autoconversion phase of the growth.

An accurate time integration method for simplified overland flow models

An accurate time integration method for the diffusion-wave and kinematic-wave approximated models for the overland flow is proposed. The discretization of the first- and second-order spatial derivatives in the basic equation is obtained by using the second-order Lax–Wendroff and the three-point centred finite difference schemes, respectively. For the solution in time, the system of ordinary differential equations, obtained by the linearization of the celerity and of the hydraulic diffusivity by Taylor series expansions, is integrated analytically. The stability and the numerical dissipation and dispersion are investigated by the Fourier analysis. A proper Courant number, and the corresponding time step for the numerical simulations can be established. In addition, the proposed diffusion-wave and kinematic-wave models are straightforwardly extended to the two-dimensional flow. Test cases for both one- and two-dimensional problems, compare the solutions of the diffusion-wave and kinematic-wave models with analytical solutions, with experimental results and with numerical solutions obtained by the Saint–Venant equations. These simulations show that the proposed numerical–analytical models accurately predict the overland flow for several situations, in particular for unsteady rainfall rate and for spatial variations of the surface roughness.