A First-Passage Kinetic Monte Carlo algorithm for complex diffusion–reaction systems

We develop an asynchronous event-driven First-Passage Kinetic Monte Carlo (FPKMC) algorithm for continuous time and space systems involving multiple diffusing and reacting species of spherical particles in two and three dimensions. The FPKMC algorithm presented here is based on the method introduced in Oppelstrup et al. [10] and is implemented in a robust and flexible framework. Unlike standard KMC algorithms such as the n-fold algorithm, FPKMC is most efficient at low densities where it replaces the many small hops needed for reactants to find each other with large first-passage hops sampled from exact time-dependent Green’s functions, without sacrificing accuracy. We describe in detail the key components of the algorithm, including the event-loop and the sampling of first-passage probability distributions, and demonstrate the accuracy of the new method. We apply the FPKMC algorithm to the challenging problem of simulation of long-term irradiation of metals, relevant to the performance and aging of nuclear materials in current and future nuclear power plants. The problem of radiation damage spans many decades of time-scales, from picosecond spikes caused by primary cascades, to years of slow damage annealing and microstructure evolution. Our implementation of the FPKMC algorithm has been able to simulate the irradiation of a metal sample for durations that are orders of magnitude longer than any previous simulations using the standard Object KMC or more recent asynchronous algorithms.     

Parametric study of horizontally vibrated grain packings: comparison between Discrete Element Method and experimental results

Numerical and experimental studies have been undertaken to analyze three parameters controlling the compaction of granular media submitted to sinusoidal horizontal vibrations. We have characterized the influence of the dimensionless acceleration Γ, the geometry of the container and the friction coefficients on the grain velocities and on the packing densities. Above a critical acceleration Γ, the velocities increases with Γ. For low values of Γ, the surface layers are compacted, whereas the bottom layers remain at their initial density. For high values of Γ, the bottom layers get compacted, the surface layers are fluidized so that the bulk dynamic and relaxed densities decreased. In the same way, the effect of the dimensions of the container and of the friction coefficients on the packing properties has been studied for given heights of sand, acceleration and frequency. It has been shown that the influence of the two last parameters is similar to that of acceleration. The numerical results given by the Discrete Element Method appear to be in good agreement with experimental results.     

Brownian dynamics simulations of colloidal hard spheres. Effects of sample dimensionality on self-diffusion

The self-diffusion coefficients of colloidal hard spheres were determined by Brownian dynamics (BD) computer simulations using a new efficient algorithm for treatment of the hard-sphere interactions. Calculations were done on an Apple PC type MacIIcx and on a Micro VAX 3000, considering samples in two and three dimensions at varying particle concentrations. Our results in three dimensions are compared with experimental results from our own group which were obtained by forced Rayleigh scattering (FRS), and with numerical results from a dynamical Monte Carlo simulation by Cichocki and Hinsen. Good agreement with the latter was found for particle volume fractions up to 0.40. Differences in the dynamical behavior of our numerically treated 2D and 3D samples are discussed using a simple geometrical model to enable comparison of particle concentrations in samples with different dimensionality.     

A method for resummation of perturbative series based on the stochastic solution of Schwinger-Dyson equations

We propose a numerical method for resummation of perturbative series, which is based on the stochastic perturbative solution of Schwinger-Dyson equations. The method stochastically estimates the coefficients of perturbative series, and incorporates Borel resummation in a natural way. Similarly to the “worm” algorithm, the method samples open Feynman diagrams, but with an arbitrary number of external legs. As a test of our numerical algorithm, we study the scale dependence of the renormalized coupling constant in a theory of one-component scalar field with quartic interaction. We confirm the triviality of this theory in four and five space-time dimensions, and the instability of the trivial fixed point in three dimensions.     

Three-dimensional weight-accumulation algorithm for generating multiple excitation spots in fast optical stimulation

We report here an algorithm for calculating a hologram to be employed in a high-access speed microscope for observing sensory-driven synaptic activity across all inputs to single living neurons in an intact cerebral cortex. The system is based on holographic multi-beam generation using a two-dimensional phase-only spatial light modulator to excite multiple locations in three dimensions with a single hologram. The hologram was calculated with a three-dimensional weighted iterative Fourier transform method using the Ewald sphere restriction to increase the calculation speed. Our algorithm achieved good uniformity of three dimensionally generated excitation spots; the standard deviation of the spot intensities was reduced by a factor of two compared with a conventional algorithm.     

Exact Stationary States of a Two-Dimensional Transport Model

Previous calculations of transport coefficients of particle systems in two or more dimensions have used computer simulations or various approximations, like those implicit in the Boltzmann equation or in the standard probability theory of diffusion. Here an exactly solved model of crystal growth in three dimensions is transformed into a 2D model of steady particle transport. Then the exact, many-body stationary distribution and exact transport coefficients can be found as well as correlation functions. The model is highly asymmetric. In the transverse direction, particles are strongly attracted, so that long transverse rows of particles tend to form at low temperature. Kinks in these rows are rare, so the transverse positions of kinks effectively move continuously on a large distance scale.     

Parametric study of horizontally vibrated grain packings Comparison between Discrete Element Method and experimental results

Numerical and experimental studies have been undertaken to analyze three parameters controlling the compaction of granular media submitted to sinusoidal horizontal vibrations. We have characterized the influence of the dimensionless acceleration G \Gamma , the geometry of the container and the friction coefficients on the grain velocities and on the packing densities. Above a critical acceleration G_crit \Gamma_{{\rm crit}}^{} , the velocities increases with G \Gamma . For low values of G \Gamma , the surface layers are compacted, whereas the bottom layers remain at their initial density. For high values of G \Gamma , the bottom layers get compacted, the surface layers are fluidized so that the bulk dynamic and relaxed densities decreased. In the same way, the effect of the dimensions of the container and of the friction coefficients on the packing properties has been studied for given heights of sand, acceleration and frequency. It has been shown that the influence of the two last parameters is similar to that of acceleration. The numerical results given by the Discrete Element Method appear to be in good agreement with experimental results.     

High-order quadratures for the solution of scattering problems in two dimensions

We construct an iterative algorithm for the solution of forward scattering problems in two dimensions. The scheme is based on the combination of high-order quadrature formulae, fast application of integral operators in Lippmann–Schwinger equations, and the stabilized bi-conjugate gradient method (BI-CGSTAB). While the FFT-based fast application of integral operators and the BI-CGSTAB for the solution of linear systems are fairly standard, a large part of this paper is devoted to constructing a class of high-order quadrature formulae applicable to a wide range of singular functions in two and three dimensions; these are used to obtain rapidly convergent discretizations of Lippmann–Schwinger equations. The performance of the algorithm is illustrated with several numerical examples.     

Comment on "A geometric representation of spectral and temporal vowel features: quantification of vowel overlap in three linguistic varieties" [J. Acoust. Soc. Am. 119, 2334-2350 (2006)

In a recent paper by Wassink [J. Acoust Soc. Am. 119, 2334-2350 (2006)] the spectral overlap assessment metric (SOAM) was proposed for quantifying the degree of acoustic overlap between vowels. The SOAM does not fully take account of probability densities. An alternative metric is proposed which is based on quadratic discriminant analysis and takes account of probability densities in the form of a posteriori probabilities. Unlike the SOAM, the a posteriori probability-based metric allows for a direct comparison of vowel overlaps calculated using different numbers of dimensions, e.g., three dimensions (Fl, F2, and duration) versus two dimensions (Fl and F2).     

A nonlinear controller for three-dimensional tracking of a fluorescent particle in a confocal microscope

We describe an algorithm for using a confocal microscope for tracking single fluorescent particles diffusing in three dimensions. The algorithm uses a standard confocal setup and directly translates each fluorescence measurement into an actuator command. Through physical simulations, we illustrate 3-D tracking in both stage scanning and beam scanning confocal systems. The simulated stage scanning system achieved tracking of particles diffusing in 3-D with coefficients up to 0.2 μm²/s when the average fluorescence intensities was less than 1.84 counts per measurement cycle (corresponding to less than 18,400 counts per second) in the presence of background fluorescence with a rate of 5,000 counts per second. Increasing the fluorescence intensity to approximately 193 counts per measurement cycle (1,930,000 counts per second) allowed the system to track up to particles diffusing with coefficients as large as 0.7 μm²/s. The beam steering system allowed for faster motion of the focal volume of the microscope and successfully tracked particles diffusing with coefficients up to 0.7 μm²/s with fluorescence measurement intensities of approximately 0.189 counts per measurement cycle (37,570 counts per second) and with coefficients up to 90 μm²/s when the fluorescence intensity was increased to 19 counts per measurement cycle (3,807,500 counts/sec).     

基于小波压缩的马铃薯全粉还原糖近红外光谱检测研究

为探讨小波压缩算法结合近红外光谱技术在马铃薯全粉还原糖含量检测中的可行性,采用傅里叶变换近红外光谱仪采集了250份马铃薯全粉样品的近红外光谱。分别优化了消失矩、小波系数和主成分因子数,优化结果为10,100和20。基于db小波函数将1 501个马铃薯全粉的近红外光谱变量压缩成100个小波系数。分别以1 501个光谱变量和100个小波系数为变量分别建立了偏最小二乘(PLS)校正模型。以62个未参与建模的样品作为预测集,考察模型的预测能力。经比较,小波压缩结合PLS的校正模型预测结果最优,模型预测相关系数为0.98,预测均方根误差为0.181%。实验结果表明小波压缩算法结合近红外光谱技术有效地保留了有效光谱信息,实现了光谱数据降维,简化了马铃薯全粉还原糖PLS校正模型,提高了模型的预测能力。     

A Faster Implementation of the Pivot Algorithm for Self-Avoiding Walks

The pivot algorithm is a Markov Chain Monte Carlo algorithm for simulating the self-avoiding walk. At each iteration a pivot which produces a global change in the walk is proposed. If the resulting walk is self-avoiding, the new walk is accepted; otherwise, it is rejected. Past implementations of the algorithm required a time O(N) per accepted pivot, where N is the number of steps in the walk. We show how to implement the algorithm so that the time required per accepted pivot is O(N ^q ) with q<1. We estimate that q is less than 0.57 in two dimensions, and less than 0.85 in three dimensions. Corrections to the O(N ^q ) make an accurate estimate of q impossible. They also imply that the asymptotic behavior of O(N ^q ) cannot be seen for walk lengths which can be simulated. In simulations the effective q is around 0.7 in two dimensions and 0.9 in three dimensions. Comparisons with simulations that use the standard implementation of the pivot algorithm using a hash table indicate that our implementation is faster by as much as a factor of 80 in two dimensions and as much as a factor of 7 in three dimensions. Our method does not require the use of a hash table and should also be applicable to the pivot algorithm for off-lattice models.     

Third Order Lagrangians, Weyl Invariants and Classical Trace Anomaly in Six Dimensions

We have proceeded the analogy (represented in our previous works) of the Einstein tensor and the alternative form of the Einstein field equations for the generic coefficients of the eight terms in the third order of the Lovelock Lagrangian. We have found the constraint between the coefficients into two forms, an independent and a dimensional dependent versions. Each form has three degrees of freedom, and not only the exact coefficients of the third order Lovelock Lagrangian do satisfy the two forms of the constraints, but also the two independent cubic of the Weyl tensor satisfy the independent constraint in six dimensions and yield the dimensional dependent version identically independent of the dimension. Then, we have introduced the most general effective expression for a total third order type Lagrangian with the homogeneity degree number three which includes the previous eight terms plus the new three ones among the all seventeen independent terms. We have proceeded the analogy for this combination, and have achieved the relevant constraint. We have shown that the expressions given in the literature as the third Weyl-invariant combination in six dimensions do satisfy this constraint. Thus, we suggest that these constraint relations to be considered as the necessary consistency conditions on the numerical coefficients that a Weyl-invariant in six dimensions should satisfy. Finally, we have calculated the “classical” trace anomaly (an approach that was presented in our previous works) for the introduced total third order type Lagrangian and have achieved a general expression with four degrees of freedom in more than six dimensions (three degrees in six dimensions). Then, we have demonstrated that the resulted expression contains exactly the relevant coefficient of the Schwinger–DeWitt proper time method (that linked with the relevant heat kernel coefficient) in six dimensions, as a particular case. Of course, this result is a necessary consistency check, nevertheless our approach can be regarded as an alternative (perhaps simpler, and classical) derivation of the trace anomaly which also gives a general expression with the relevant degrees of freedom.     

Quantum binding of the BCS interaction model

The Schrödinger (as opposed to the Cooper or BCS-gap) equation is solved without approximation in momentum space for the BCS interaction model to obtain the quantum bound-state spectrum of an isolated pair of fermions in one, two, and three dimensions. Regardless of dimensionality, there is never more than a single bound state (in analogy with the nucleon-nucleon interaction), but a threshold value of the potential strength is needed to support this state in any dimension. For very low densities one recovers previously known formulas for two and three dimensions which are consistent in this limit with the more familiar properties of quantum binding for simple, purely attractive wells. Results are illustrated for typical conventional, cuprate, and superconducting semiconductors having controversially low carrier densities.     

Angular and axial evaluation of superficial defects on non-accessible pipes by wavelet transform and neural network-based classification

In this paper an effective procedure that allows evaluating the dimensions of corrosive flaws on non-accessible pipes is presented. The method is based on the propagation of ultrasound waves, analyzing the informative content of echoes reflected by defects. The approach exploits the properties of the wavelet transform to represent signals by a reduced form. The coefficients of this representation are selected properly by making use of a filter method followed by a genetic algorithm and, then, they feed a neural network classifier which evaluates the dimensions of defects on the pipe under test. Numerical results show low error rates in the evaluation of both angular and axial extension of each flaw. The main advantage offered by the method consists of analyzing long lines of non-accessible pipes, realizing an automatic evaluation of the dimensions of superficial flaws in pipelines.     

A ray-based algorithm for multi-dimensional linear conversion

A numerical algorithm is proposed for connecting the incoming and outgoing wave fields in studies of linear conversion. This is the first such ray-based algorithm for wave conversion in multiple spatial dimensions. It is demonstrated that, aside from the overall phase of the coupling, one can directly evaluate all quantities needed for the connection coefficients from the ray geometry. The ray dynamics is generated using the determinant of the dispersion matrix as the Hamiltonian. Using information available while following an incoming ray, the algorithm automatically detects that the ray has entered a conversion region, evaluates the transmission and conversion coefficients, and launches the transmitted ray. The algorithm does not require any prior knowledge of the geometry of the conversion region. The algorithm is illustrated using a two-dimensional toroidal model with resonant conversion from a magnetosonic to an ion-hybrid wave.     

A high-resolution mapped grid algorithm for compressible multiphase flow problems

We describe a simple mapped-grid approach for the efficient numerical simulation of compressible multiphase flow in general multi-dimensional geometries. The algorithm uses a curvilinear coordinate formulation of the equations that is derived for the Euler equations with the stiffened gas equation of state to ensure the correct fluid mixing when approximating the equations numerically with material interfaces. A γ-based and a α-based model have been described that is an easy extension of the Cartesian coordinates counterpart devised previously by the author [30]. A standard high-resolution mapped grid method in wave-propagation form is employed to solve the proposed multiphase models, giving the natural generalization of the previous one from single-phase to multiphase flow problems. We validate our algorithm by performing numerical tests in two and three dimensions that show second order accurate results for smooth flow problems and also free of spurious oscillations in the pressure for problems with interfaces. This includes also some tests where our quadrilateral-grid results in two dimensions are in direct comparisons with those obtained using a wave-propagation based Cartesian grid embedded boundary method.     

Simulating finger phenomena in porous media with a moving finite element method

The non-equilibrium Richards equation is solved using a moving finite element method in this paper. The governing equation is discretized spatially with a standard finite element method, and temporally with second-order Runge–Kutta schemes. A strategy of the mesh movement is based on the work by Li et al. [R.Li, T.Tang, P.W. Zhang, A moving mesh finite element algorithm for singular problems in two and three space dimensions, Journal of Computational Physics, 177 (2002) 365–393]. A Beckett and Mackenzie type monitor function is adopted. To obtain high quality meshes around the wetting front, a smoothing method which is based on the diffusive mechanism is used. With the moving mesh technique, high mesh quality and high numerical accuracy are obtained successfully. The numerical convergence and the advantage of the algorithm are demonstrated by a series of numerical experiments.     

Fast free-surface detection and level-set function definition in SPH solvers

The present paper proposes a novel algorithm to detect the free-surface in particle simulations, both in two and three dimensions. Since the proposed algorithms are based on SPH interpolations their implementation does not require complex geometrical procedures. Thus the free-surface detection can be easily embedded in SPH solvers, without a significant increase of the CPU time. Throughout this procedure accurate normal vectors to the free-surface are made available. Then it is possible to define a level-set function algorithm which is presented in detail. The latter allows in-depth analyses of three-dimensional free-surface simulations by using standard visualization tools, including internal features of the flow. The algorithms proposed for detecting free-surface particles and defining the level-set function are validated on simple and complex two- and three-dimensional flow simulations. The usefulness of the proposed procedures to post-process and analyze complex flows are illustrated on realistic examples.     

A particle method with adjustable transport properties—the generalized consistent Boltzmann algorithm

The consistent Boltzmann algorithm (CBA) for dense, hard-sphere gases is generalized to obtain the van der Waals equation of state and the corresponding exact viscosity at all densities except at the highest temperatures. A general scheme for adjusting any transport coefficients to higher values is presented.