Random collision model for random genetic drift and stochastic difference equation

Summary At first we introduce a simple stochastic difference equation, to simulate random sampling drift in population genetics, which is naturally obtained from a random collision model. Next, we introduce a random collision model to simulate overdominance model in population genetics. We assume in a time interval °t, a random collision of four particles, which represents overdominant selection, takes place at a certain probability, where a particle corresponds to a gene. We assume that mutation takes place by some rate and assume that every new mutation is different from extant alleles. We estimate mean heterozygosity by our simulation method and compare it with the result obtained by using a stochastic difference equation for overdominance model. The Institute of Statistical Mathematics     

Gas-dynamic equations for spatially inhomogeneous gas mixtures with internal degrees of freedom. I. General theory

A general algorithm for building a uniform asymptotic solution of the kinetic equations for spatially inhomogeneous reactive gas mixtures is proposed. It solves the problem of irregular asymptotic solution arising in the ordinary Chapman–Enskog method, providing expressions for chemical reaction rates that agree with the mono-molecular reaction theory. We study a quasi-stationary behavior of the system, characterized by the slowly varying gas-dynamic variables which number is greater than the number of integral invariants of the collision operator. The gas-dynamic equations for reacting and relaxing gas mixtures are derived in general form. It is shown that accurate treatment of non-equilibrium processes gives rise to additional terms caused by the strong influence of small perturbations of quasi-equilibrium distribution functions on the kinetics of high-threshold physical and chemical processes. These terms are describing the influence of inelastic collisions, expansion/compression processes and spatial non-uniformity of gas-dynamic variables.     

Optimization-based simulation of nonsmooth rigid multibody dynamics

We present a time-stepping method to simulate rigid multibody dynamics with inelastic collision, contact, and friction. The method progresses with fixed time step without backtracking for collision and solves at every step a strictly convex quadratic program. We prove that a solution sequence of the method converges to the solution of a measure differential inclusion. We present numerical results for a few examples, and we illustrate the difference between the results from our scheme and previous, linear-complementarity-based time-stepping schemes.     

Direct particle motion and interaction modeling method applied to simulate propellant burn

A direct particle motion and particle interaction modeling method was developed to provide an alternative means of capturing the fundamental phenomena occurring during the burning of propellant grains. Individual propellant grains and other moving components are directly incorporated into the computational domain, removing the need for correlations for particle drag and interaction effects. The motion of the individual particles is calculated from the locally acting fluid induced and collision effect forces and moments. Particle/object interactions are handled through a soft particle collision algorithm. Localized mass and energy sources, accompanied by a shrinking particle size, simulate the effects of the combustion process.     

Rapid-equilibrium Approximation applied to mathematical models of Tracer dynamics in biochemical reaction systems

A method is presented which allows to reduce the size of the differential equation system describing the dynamics of radioactive tracers in biochemical reaction systems endowed with time hierarchy, i.e., encompassing fast and slow reaction rates which may differ by several orders of magnitude. The basic idea of this approach is to apply the rapid-equilibrium approximation commonly used in the mathematical modelling of metabolic systems (cf. [1] for lumping into single “pool variables” all those reactive groups which can be labeled and which are connected by fast reversible reactions. The set of remaining slow reactions governing the dynamics of the reduced system of pool variables depends on the special way of labeling chosen. Thus, the reduction procedure permits to judge the maximal possible number of flux rates which can be estimated from stationary or time-dependent tracer experiments by regression analysis (fitting). Moreover, the method can be employed for identifying near-equilibrium and nonequilibrium reactions. An illustration of the reduction algorithm is provided by considering the distribution of ¹⁴C-tracers in the pentose phosphate pathway.     

An improved dissipative particle dynamics scheme

Dissipative particle dynamics (DPD) and smoothed dissipative particle dynamics (sDPD) have become most popular numerical techniques for simulating mesoscopic flow phenomena in fluid systems. Several DPD/sDPD simulations in the literature indicate that the model fluids should be designed with their dynamic response, measured by the Schmidt number, in a relevant range in order to reach a good agreement with the experimental results. In this paper, we propose a new dissipative weighting function (or a new kernel) for the DPD (or the sDPD) formulation, which allows both the viscosity and the Schmidt number to be independently specified as input parameters. We also show that some existing dissipative functions/kernels are special cases of the proposed one, and the imposed viscosity of the present DPD/sDPD system has a lower and upper limit. Numerical verification of the proposed function/kernel is conducted in viscometric flows.     

平面射流中纳米粒子积聚的矩方法

应用大涡模拟方法求解平面湍射流场,矩方法求解纳米粒子的一般动力学方程．通过对每种情况3 000个时间步的平均,得到了Schmidt数和Damkohler数对纳米粒子动力学特性的影响．结果发现, 当气体参数不变时,Schmidt数的变化只对直径小于1 nm的颗粒数密度的分布产生影响．直径小的颗粒其颗粒数密度沿流动方向下降迅速,而具有大Schmidt数的颗粒,沿横向的分布较窄．较小的颗粒容易发生积聚和扩散,并且体积增长较快,因而颗粒多分散性较为明显．小的颗粒积聚时间尺度能增强颗粒的碰撞和积聚频率,导致颗粒尺寸迅速增大．Damkohler数越大,颗粒的多分散也越明显．     

Orbital analysis in the planar circular Copenhagen problem using polar coordinates

We examine the orbital dynamics of the planar Copenhagen problem, by classifying sets of starting conditions of orbits, expressed in polar coordinates. Specifically, we examine how the Jacobi constant influences several aspects of the overall dynamics of the test particle, such as its final state, as well as the time of escape/collision. By using modern diagrams with color codes, we manage to present the different types of basins. It is shown that both the character of the orbits and the fractal degree of the system are highly dependent on the Jacobi constant.     

Closure method for spatially averaged dynamics of particle chains

We study the closure problem for continuum balance equations that model the mesoscale dynamics of large ODE systems. The underlying microscale model consists of classical Newton equations of particle dynamics. As a mesoscale model we use the balance equations for spatial averages obtained earlier by a number of authors: Murdoch and Bedeaux, Hardy, Noll and others. The momentum balance equation contains a flux (stress), which is given by an exact function of particle positions and velocities. We propose a method for approximating this function by a sequence of operators applied to the average density and momentum. The resulting approximate mesoscopic models are systems in closed form. The closed form property allows one to work directly with the mesoscale equations without the need to calculate the underlying particle trajectories, which is useful for the modeling and simulation of large particle systems. The proposed closure method utilizes the theory of ill-posed problems, in particular iterative regularization methods for solving first order linear integral equations. The closed form approximations are obtained in two steps. First, we use Landweber regularization to (approximately) reconstruct the interpolants of the relevant microscale quantities from the average density and momentum. Second, these reconstructions are substituted into the exact formulas for stress. The developed general theory is then applied to non-linear oscillator chains. We conduct a detailed study of the simplest zero-order approximation, and show numerically that it works well as long as the fluctuations of velocity are nearly constant.     

双尺度二阶矩颗粒相湍流模型经验系数的影响

基于将颗粒脉动分成湍流引起的大尺度脉动和颗粒间碰撞产生的小尺度脉动的概念,建立了双尺度二阶矩两相湍流模型．用该模型对下行床内两相流动进行了数值模拟,颗粒体积浓度、平均速度的计算结果和实验数据吻合较好．分析了双尺度二阶矩两相湍流模型经验系数变化对预报结果的影响:在经验系数的一定变化范围内,预报结果并无明显的影响,但是变化范围增大,预报结果会产生较大变化．     

Effective models for reactive flow under a dominant Péclet number and order one Damköhler number: Numerical simulations

The paper is devoted to the longitudinal dispersion of a soluble substance released in a steady laminar flow through a slit channel with heterogeneous reaction at the outer wall. The reactive transport happens in the presence of a dominant Péclet number and order one Damköhler number. In particular, these Péclet numbers correspond to Taylor’s dispersion regime. An effective model for the enhanced diffusion in this context was derived recently. It contains memory effects and contributions to the effective diffusion and effective advection velocity, due to the flow and chemistry reaction regime. In the present paper, we show through numerical simulations the efficiency of this new model. In particular, using Taylor’s ‘historical’ parameters, we illustrate that our derived contributions are important and that using them is necessary in order to simulate correctly the reactive flows. We emphasize three main points. First, we show how the effective diffusion is enhanced by chemical effects at dispersive times. Second, our model captures an intermediate regime where the diffusion is anomalous and the distribution is asymmetric. Third, we show how the chemical effects also slow down the average speed of the front.     

The role of multiple microscopic mechanisms in cluster interface evolution

In this paper we discuss mesoscopic models describing pattern formation mechanisms for a prototypical model of surface processes that involves multiple microscopic mechanisms. We focus on a mean field partial differential equation, which contains qualitatively microscopic information on particle-particle interactions and multiple particle dynamics, and we rigorously derive the macroscopic cluster evolution laws and transport structure. We show that the motion by mean curvature is given by V=μσκ, where k is the mean curvature, σ is the surface tension and μ is an effective mobility that depends on the presence of the multiple mechanisms and speeds up the cluster evolution. This is in contrast with the Allen-Cahn equation where V=κ.     

Application of the finite-element method to the diffusion and reaction of chemical species in multilayered polymeric bodies

Application of the finite-element method (FEM) to chemical species diffusion and reaction in polymers by Fickian mass transport is described. The method is developed by analogy to heat conduction and is extended to include multiple, reactive chemical species dissolved in multilayered polymeric materials. Because of the analogy to conductive heat transfer, existing FEM thermal codes can be readily adapted to solve chemical diffusion problems. The method described is limited to Fickian diffusion at a constant material temperature.     

Adaptive Model Reduction for Chemical Kinetics

An adaptive model reduction algorithm is proposed for systems of ODEs from chemical kinetics. Its goal is to provide an accurate approximation to the solution of these systems faster than could be obtained through straightforward numerical integration. The algorithm approximates a system with a sequence of reduced models, each one appropriate to the dynamics of the system during a period of the trajectory. Reduced models are identical to the original system except for the deletion of some chemical reactions. This saves the cost of computing unimportant reaction coefficients. Both the reduced models and the durations for which they are used are selected adaptively in order to efficiently yield an accurate approximate solution. The performance of the algorithm is assessed through numerical experiments.     

Simulation of a reacting gas–liquid bubbly flow with CFD and PBM: Validation with experiments

In this work we use computational fluid dynamics (CFD) to simulate a reactive gas–liquid bubbly system in a rectangular bubble column, operating at low superficial velocities (i.e. homogeneous regime). The gas bubbles, injected in the column through a sparger, contain one of the reactants, namely CO₂, that via mass transfer moves to the continuous liquid phase, where it reacts with NaOH. A key role is played by the bubble size distribution (BSD) and the specific surface area that define the overall mass transfer rate in the CFD model. In order to correctly predict the BSD and the polydispersity of the bubbly system the population balance equation is solved by the quadrature method of moments (QMOM), within the OpenFOAM (v. 2.2.x) two-fluid solver compressibleTwoPhaseEulerFoam. To reduce the computational time and increase stability, a second-order operator-splitting technique for the solution of the chemically reactive species is also implemented, allowing to solve the different processes involved with their own time-scale. To our knowledge this is the first time that QMOM is employed for the simulation of a real reactive bubbly system and predictions are validated against experiments.     

Mesoscopic supply chain simulation

This paper reviews and compares existing approaches for supply chain modeling and simulation and applies the mesoscopic modeling and simulation approach using the simulation software MesoSim, an own development. A simplified real-world supply chain example is modeled with discrete event, mesoscopic and system dynamics simulation. The objective of the study is to compare the process of model creation and its validity using each approach. The study examines advantages of the mesoscopic approach for the simulation. Major benefits of the mesoscopic approach are that modeling efforts are balanced with the necessary level of detail and facilitate quick and simple model creation and simulation.     

Hybrid systems: Modelling and analysis using emergent dynamics

In this paper, we discuss modelling and analysis of hybrid systems with physical interaction dynamics. Such systems are typically considered complex and they are modelled using abstractions. Abstractions may, however, unintentionally exclude critical details, leading to partial or false results. Therefore, we study here use of a particle system in modelling and analysis. The novelty of the particle system is that it is designed to reveal interaction dynamics as emergent dynamics; thus, supporting analysis of complex and intricate interaction dynamics with acceptable modelling effort. As the main contribution, we formalize the particle system, and use it to model and analyze hybrid systems, both mechanical and biological, with nontrivial interaction dynamics.     

Unique and multiple PHAM series solutions of a class of nonlinear reactive transport model

The purpose of this paper is to visit a class of nonlinear reactive transport model in the case including advective and diffusive transport with the Michaelis-Menten reaction term. We apply the method so-called predictor homotopy analysis method (PHAM) which has been recently proposed to predict multiplicity of solutions of nonlinear BVPs. Consequently two consequential matters are indicated which confirms the authority of PHAM to identify multiple solutions: (i) The Talylor series solutions are improved by the so-called convergence-controller parameter (ii) The possibility of existence of multiple solutions is discovered in some cases for the model.