Spectral accuracy in fast Ewald-based methods for particle simulations

A spectrally accurate fast method for electrostatic calculations under periodic boundary conditions is presented. We follow the established framework of FFT-based Ewald summation, but obtain a method with an important decoupling of errors: it is shown, for the proposed method, that the error due to frequency domain truncation can be separated from the approximation error added by the fast method. This has the significance that the truncation of the underlying Ewald sum prescribes the size of the grid used in the FFT-based fast method, which clearly is the minimal grid. Both errors are of exponential-squared order, and the latter can be controlled independently of the grid size. We compare numerically to the established SPME method by Essmann et al. and see that the memory required can be reduced by orders of magnitude. We also benchmark efficiency (i.e. error as a function of computing time) against the SPME method, which indicates that our method is competitive. Analytical error estimates are proven and used to select parameters with a great degree of reliability and ease.     

Lekner summations and Ewald summations for quasi-two-dimensional systems

The purpose of this work is to carry out a comparison between the Ewald quasi-2D and Lekner summation methods. These methods were derived to treat the long-range electrostatic interactions in systems periodic in two directions, but bound in the third. The comparison is performed by Monte Carlo simulations on a very simple system, a bilayer of point ions; samplings of the phase space, average energies and structure functions are compared. When correctly implemented, the Lekner summation method is found to be in close agreement with the Ewald quasi-2D method; otherwise, a very complicated bias may plague computations.     

A numerical source of small-scale number-density fluctuations in Eulerian–Lagrangian simulations of multiphase flows

Eulerian–Lagrangian simulations of multiphase flow are known to suffer from two errors that can introduce small-scale fluctuations in the number-density of the dispersed phase. These errors can be reduced by increasing the number of particles in the simulation. Here, we present results to demonstrate that a third error exists that can also generate small-scale number-density fluctuations. In contrast to the two known errors, this error cannot be lowered by increasing the number of particles. Analysis shows that this error is caused by spatial variation at the subgrid scale in the interpolation error of the fluid velocity to the particle location. If the particle velocity divergence is zero, the particle concentration error remains at the subgrid scale. However, if particles preferentially accumulate either due to their inertia or due to divergence of the underlying fluid-velocity field, this error manifests as number-density fluctuations on the grid scale. The only mechanism of reducing these errors is through higher-order accurate interpolation. By studying two model problems, estimates for the errors are derived. These estimates are shown to be quite accurate for simulations of shock and expansion waves interacting with particles.     

Effect of electrostatics techniques on the estimation of thermal conductivity via equilibrium molecular dynamics simulation: application to methane hydrate

Equilibrium molecular dynamics (MD) simulations for three system sizes of fully occupied methane hydrate have been performed at around 265 K to estimate the thermal conductivity using the Ewald, Lekner, reaction field, shifted-force and undamped Fennell–Gezelter methods. The TIP4P water model was used in conjunction with a fully atomistic methane potential with which it had been parameterized from quantum simulation. The thermal conductivity was evaluated by integration of the heat flux autocorrelation function (ACF) derived from the Green–Kubo formalism; this approach vas validated by estimation of the average phonon mean free path. The thermal conductivities predicted by non-periodic techniques were in reasonable agreement with the experimental results of 0.62 and 0.68 W/m K, although it was found that the estimates by the non-periodic techniques were up to 25% larger than those of Lekner and Ewald estimates, particularly for larger systems. The results for the Lekner method exhibited the least variation with respect to system size. A decomposition of the heat flux vector into its respective contributions revealed the importance of electrostatic interactions, and how different electrostatic treatments affect the contribution to the thermal conductivity.     

快速多极边界元法用于扩散光学断层成像研究

在求解扩散光学断层成像中的正向问题时, 目前普遍采用有限元法, 但是随着实际模型规模的增大, 有限元法的计算量问题日益显著, 而边界元法则由于可以降低计算维度使计算量减少而备受关注. 本文以均匀的高散射介质为模型, 研究了将快速多极边界元法用于扩散光学断层成像的正向问题. 快速多极边界元法利用核函数的多极展开, 将常规边界元法中系数矩阵和迭代矢量的乘积项等价为相应四叉树结构的一次递归, 再结合广义最小残量法进行迭代求解. 将计算结果和蒙特卡罗法的模拟结果进行了比较, 表明利用快速多极边界元法的模拟结果和蒙特卡罗法的结果有很好的一致性. 研究结果验证了快速多极边界元法可以用于扩散光学断层成像, 为其大规模和实时成像带来可观的前景. 关键词： 扩散光学断层成像 边界元法 快速多极边界元法     

An accurate and efficient method for treating aerodynamic interactions of cloud droplets

Motivated by a need to improve the representation of short-range interaction forces in hybrid direct numerical simulation of interacting cloud droplets, an efficient method for treating the aerodynamic interaction of two spherical particles settling under gravity is developed. An effort is made to ensure the accuracy of our method for any inter-particle separation by considering three separation ranges. The first is the long-range interaction where a multipole method is applied. After a decomposition into six simple configurations, explicit formulae for drag forces and torques are derived from an approximate Force–Torque–Stresslet (FTS) formulation. The FTS formulation is found to be accurate when the separation distance normalized by the average radius is larger than 5. The second range concerns the short-range interaction where the interaction force could be very large. Leading-order lubrication expansions are employed for this range and are found to be accurate when the normalized separation is less than about 0.01. Finally, for the intermediate range where no simple method is available, a third-order polynomial fitting is proposed to bridge the treatments for long-range and short-range interactions. After optimizing the precise form of polynomial fitting and matching locations, the force representation is found to be highly accurate when compared with the exact solution for Stokes flows. Using this method, collision efficiencies of cloud droplets sedimenting under gravity have been calculated. It is shown that the results of collision efficiency are in excellent agreement with results based on the exact Stokes flow solution. Collision efficiency results are also compared to previous results to further illustrate the accuracy of our calculations. The effects of particle rotation and the attractive van der Waals force on the collision efficiency are also studied. The efficient force representation developed here is more general than the usual lubrication expansion and thus can serve as a better approach to correct unresolved short-range interactions in particle-resolved simulations.     

High Conductivity Mediated by Thermal Excitation of Solectrons

We study the quantum dynamics and statistics of electrons interacting with nonlinear excitations of a classical thermal lattice of atoms on a semi‐phenomenological basis. By theoretical estimates based on tight‐binding approximations, Wigner distributions and computer simulations we show the existence of fast and nearly loss‐free motions of electrons along crystallographic axes of a nonlinear lattice. Using mass‐action relations we estimate the density of moving bound states between electrons and lattice solitons and estimate analytically and by simulations Wigner momentum distributions which are non‐Maxwellian. Calculating the currents from these bimodal distributions we show that thermally excited solectrons in nonlinear media may lead to a considerable transport enhancement. Our estimates and simulations demonstrate that in a temperature window, where solectrons are excited and are relatively stable, conductivity and diffusion may be enhanced by up to two orders of magnitude. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Estimation of Particle Size Distributions Using Pipet-Withdrawal Centrifuges

The mathematical treatments used in the evaluation of particle size distribution from pipet-withdrawal centrifuge data are reviewed. An error in the published procedure for multiple-sample withdrawal systems is pointed out and a corrected formulation is presented. An alternative treatment, based on a piece-wise linearization of the size distribution is described. Computer simulations of the analytical and data evaluation procedures indicate that the errors in the original formulation are not generally serious and that the corrected and alternative procedures lead to very similar estimates. Except for some cases involving very narrow size distributions, the alternative approach appears to give slightly closer estimates of the size distribution. Comparisons based on actual experimental data are also described.     

Estimation and application of spatially variable noise fields in diffusion tensor imaging

Optimal interpretation of magnetic resonance image content often requires an estimate of the underlying image noise, which is typically realized as a spatially invariant estimate of the noise distribution. This is not an ideal practice in diffusion tensor imaging because the noise distribution is usually spatially varying due to the use of fast imaging and noise suppression techniques. A new estimation approach for spatially varying noise fields (NFs) is proposed in this article. The approach is based on a noise invariance property in scenarios in which more than one image, each with potentially different signal levels, is acquired on each slice, as in diffusion-weighted MRI. This technique leads to improved NF estimates in simulations, phantom experiments and in vivo studies when compared to traditional NF estimators that use regional variability or background intensity histograms. The proposed method reduces the NF estimation error by a factor of 100 in simulations, shows a strong linear correlation (R²=0.99) between theoretical and estimated noise changes in phantoms and demonstrates consistent (<5% variability) NF estimates in vivo. The advantages of spatially varying NF estimation are demonstrated for power analysis, outlier detection and tensor estimation.     

Molecular dynamics simulation of ultrafast laser ablation of fused silica film

Ultrafast laser ablation of fused silica is studied using molecular dynamics simulations. Ionization and generation of free electrons, absorption of the laser energy by free electrons and energy coupling between free electrons and ions are considered. The BKS potential is applied and modified to describe molecular interactions and the effect of free electrons. Smooth particle mesh of the Ewald method (SPME) is adopted to calculate the Coulomb force. It is found that the electrostatic Coulomb force, which is caused by the ionization, plays an important role in the laser ablation process.     

High accuracy measurement of unsteady flows using digital particle image velocimetry

The analysis of digital PIV data, either derived from CCD technology or through film and then scanned, typically involves two quantization steps: spatial and intensity quantization. The all-optical systems do not introduce these sources of error. For systems which make use of digital technology however, it is of crucial importance to have reliable error bounds and a sufficiently accurate estimate of particle position, taking into consideration both types of quantization. The accuracy demanded by aerodynamicists from PIV has been a major barrier to its practical application in the past. The more recent approach of using the Gaussian profile of the particle images to yield sub-pixel accurate position estimates has resulted in robust measurements being taken to an accuracy of 1/10th pixel and 1% in velocity for the in-plane velocity, in hostile industrial environments. A major problem for 3D PIV estimation has historically been that the out-of-plane velocity error was of the order of 3–4 times larger than in-plane. The out-of-plane velocity estimate can be derived from the change in the ratio of amplitude to variance—known as the depth factor—of the Gaussian form, as a particle traverses the beam profile. However, such measurements are crucially dependent not only on an accurate position estimate but also on an equally accurate estimate of the amplitude and variance. The accuracy of the Gaussian profile fit using a Nelder–Meade optimisation method as developed until now however, is not capable of providing the required accuracies. Therefore, this paper presents a development of the “locales” approach to position estimation to achieve the desired objective of high accuracy PIV measurements. This approach makes use of the fact that by considering all the possible digital representations of the Gaussian particle profile, regions of indistinguishable position can be derived. These positions are referred to as “locales”. By considering the density, distribution, and shape of these locales, the available precision can be estimated and an accurate (no worse than 0.5% error for a typical PIV image) in-plane velocity, accuracy can be obtained; while at the same time providing estimates of the depth factor with an error of approximately 0.8%. This paper describes the implementation of an efficient algorithm to provide velocity estimates to an accuracy of at least 0.5% in-plane, together with a discussion of the required constraints imposed on the imaging. The method was validated by creating a synthetic PIV image with CCD-type noise. The flow being analysed is that of flow past the near wake of a cylinder at a Reynolds Number of 140,000. This image was then analysed with the new method and the velocity estimates compared to the CFD data for a range of signal-to-noise ratios (SNR). For a realistic SNR of 5, the accuracy of the method is confirmed as being at least 0.5% in-plane. Finally, the algorithm was used to map an experimental transonic flow field of the stator trailing edge region of a full-size annular cascade with an estimated error of 0.5%. The experimental results are found to be in good agreement with a previously reported steady state viscous calculation and PIV mapping.     

Molecular dynamics simulations of liquid water using various long-range electrostatics techniques

Water is one of the most extensively studied molecules, owing to its crucial role in biological processes. The water molecule is both highly polar and highly polarizable. Properties of water computed from molecular simulations are therefore critically dependent on both the intermolecular potential and the method for computing long-range electrostatic corrections. In this paper, the effects of the potential and the long-range electrostatic corrections are quantified for liquid water from 260 to 400?K. Simulations were carried out for a system of 256 molecules in the NVT ensemble. Thermodynamic, structural, dynamical, hydrogen bonding and dielectric properties have been computed for the flexible SPC and rigid SPC, SPC/E, TIP4P, TIP4P-Ew and TIP4P-FQ potentials, using the Lekner, Ewald and reaction field techniques to handle long-range electrostatics. The Lekner method gave the best overall agreement with experimental data, while the reaction field approach produced poorer results. Some measurable differences were found between the Lekner and Ewald techniques. For dielectric properties, the performance of the TIP4P-FQ model was superior relative to other potentials. For 256 molecules, the computational speeds of the Ewald and reaction field methods were found to be 2.5 to 3 times and 3.5 to 5 times faster than the Lekner technique, respectively.     

A faster aggregation for 3D fast evanescent wave solvers using rotations

A novel technique to accelerate the aggregation and disaggregation stages in evanescent plane wave methods is presented. The new method calculates the six plane wave radiation patterns from a multipole expansion (aggregation) and calculates the multipole expansion of an incoming field from the six plane wave incoming field patterns. It is faster than the direct approach for multipole orders larger than one, and becomes six times faster for large multipole orders. The method relies on a connection between the discretizations of the six integral representations, and on the fact that the Wigner D-matrices become diagonal for rotations around the z-axis. The proposed technique can also be extended to the vectorial case in two different ways, one of which is very similar to the scalar case. The other method relies on a Beltrami decomposition of the fields and is faster than the direct approach for any multipole order. This decomposition is also not limited to evanescent wave solvers, but can be used in any vectorial multilevel fast multipole algorithm.     

Effects of long-range interactions on the dynamics of ions in aqueous solution

Molecular dynamics simulations have been carried out for a Br⁻ ion in aqueous solution in order to establish the effect of truncation of long-range interactions on the dynamical properties of the ion. Simulations using smooth truncation of the potential at different cutoff radii were carried out and compared to results using the Ewald summation method. It is shown that when small cutoffs are applied (i.e. R_c = 8Å), the calculations yield low ionic diffusion coefficients relative to experiment, as well as short-time dynamical behavior which is inconsistent with the Ewald calculations. As the cutoff is increased, the results approach both the Ewald and the experimental results. In contrast to the results with the truncated potential, the short-time dynamical behavior of the hydrated bromide ion obtained with the Ewald method can be described by simple Langevin dynamics.     

A fast Fourier transform on multipoles (FFTM) algorithm for solving Helmholtz equation in acoustics analysis

This article presents a fast algorithm for the efficient solution of the Helmholtz equation. The method is based on the translation theory of the multipole expansions. Here, the speedup comes from the convolution nature of the translation operators, which can be evaluated rapidly using fast Fourier transform algorithms. Also, the computations of the translation operators are accelerated by using the recursive formulas developed recently by Gumerov and Duraiswami [SIAM J. Sci. Comput. 25, 1344-1381(2003)]. It is demonstrated that the algorithm can produce good accuracy with a relatively low order of expansion. Efficiency analyses of the algorithm reveal that it has computational complexities of O(Na), where a ranges from 1.05 to 1.24. However, this method requires substantially more memory to store the translation operators as compared to the fast multipole method. Hence, despite its simplicity in implementation, this memory requirement issue may limit the application of this algorithm to solving very large-scale problems.     

Resolving statistical uncertainty in correlation dimension estimation

In this paper we propose a novel method for obtaining standard errors and confidence intervals for the correlation dimension estimated on an observed chaotic time series. This method is based on the U-Statistics theory and an ingenious combination of the moving block and parametric bootstrap procedures. We test the method on the basis of computer simulations for both clean and noisy series. We show that the distribution of the correlation dimension estimate obtained by our method agrees very well with the "true" distribution obtained by the Monte Carlo simulation. One of the main advantage of our method is the ability to estimate the distribution (and hence, the standard error) of the correlation dimension estimate using only one observed time series.     

Yukawa potentials in systems with partial periodic boundary conditions. II. Lekner sums for quasi-two-dimensional systems

Yukawa potentials may be long-ranged when the Debye screening length is large. In computer simulations, such long-ranged potentials have to be taken into account with convenient algorithms to avoid systematic bias in the sampling of the phase space. Recently, we provided Ewald sums for quasi-two-dimensional systems with Yukawa interaction potentials [J. Chem. Phys. 126, 056101 (2007); Molec. Phys. paper I of this series]. Sometimes, Lekner sums are used as an alternative to Ewald sums for Coulomb systems. In the present work, we derive the Lekner sums for quasi-two-dimensional systems with Yukawa interaction potentials and we give numerical tests for practical implementations. The main result of this paper is to emphasize that Lekner sums cannot be considered as an alternative to Ewald sums for Yukawa potentials. As a conclusion to this work: Lekner sums should not be used for quasi-two-dimensional systems with Yukawa interaction potentials.     

Optimization of Nanofluidic Devices for Geometry-Induced Electrostatic Trapping

Nanoparticle trapping in a nanofluidic device utilizing geometry-induced electrostatic (GIE) potential trap is an efficient and robust way to perform nano‑object confinement and single particle studies. The GIE‑trapping is a passive method that solely depends on the device geometry and device-particle surface interaction. Therefore, optimization of a nanofluidic device based on experimental requirements, helps to achieve stiffer single-particle trapping. The efficiency of a GIE‑trapping device is defined in terms of residence time and trapping stiffness of the nanoparticle inside a potential trap. The present study reveals all crucial parameters that affect the device efficiency, particle trapping stiffness, and particle residence time. Furthermore, the trends of particle trapping stiffness are presented as a function of crucial parameters and demonstrate two variants of simulations to estimate the particle trapping efficiency: (a) using charged particle, and (b) using point charge approximation. Simulations with charged particle give more realistic values related to particle trapping whereas simulations with point charge approximation is a faster approach which gives approximate values and a guideline for more rigorous simulations. The results demonstrate a good agreement with experimental observations and hold the key for future developments in this field, wherein a device geometry can be very precisely optimized.