Simulation Of Diffusion Processes Research Articles

Simulation of diffusion processes during electrothermal treatment of reaction crucibles of the Fe-Sn system

Simulation of diffusion processes during electrothermal treatment of reaction crucibles of the Fe-Sn system

A General Framework to Simulate Diffusions with Discontinuous Coefficients and Local Times

In this article, we propose an efficient general simulation method for diffusions that are solutions to stochastic differential equations with discontinuous coefficients and local time terms. The proposed method is based on sampling from the corresponding continuous-time Markov chain approximation. In contrast to existing time discretization schemes, the Markov chain approximation method corresponds to a spatial discretization scheme and is demonstrated to be particularly suited for simulating diffusion processes with discontinuities in their state space. We establish the theoretical convergence order and also demonstrate the accuracy and robustness of the method in numerical examples by comparing it to the known benchmarks in terms of root mean squared error, runtime, and the parameter sensitivity.

Investigation on the fusion reaction rate of deuterium and tritium under heterogeneous mixing

Mixing between shell material and gas fuel, caused by hydrodynamic instability, isolated defects, or kinetic effects, is the key to understand the degradation of implosion performance in the research of inertial confinement fusion. Understanding the mixing mechanism and reducing its impact is of extreme importance to achieve the ignition and high gain. The impact of mixing morphology on thermonuclear reaction rate in sub grid level has gradually attracted people’s attention in recent years due to its direct influence on burn rate and fusion process, the study on physical model of thermonuclear reaction rate in different mix morphology has important scientific significance and application value. In the paper, the dependence of thermonuclear reaction rate on mass distribution of different fuel concentrations at sub grid scale is derived. Based on thermodynamic equilibrium and ideal gas equation of state, the physical law of the evolution of the thermonuclear reaction rate with mix morphology under the dominance of diffusion mixing is revealed through analytical formula and numerical solution of diffusion equation in one-dimensional spherical geometry. It is convinced that the mixing amount directly affects the thermonuclear reaction rate by mainly affecting the volume fraction of the fuel, and the mixing diffusion time determined by heterogeneous mixing scale and diffusion coefficient directly affects the evolution behavior of the thermonuclear reaction rate. Furthermore, based on mutual diffusion coefficient obtained from direct simulation of diffusion process by Monte Carlo method, the difference of impact to thermonuclear reaction rate for low-<i>Z</i> Carbon and high-<i>Z</i> gold mixing is quantitatively investigated. Heterogeneous mix size with 0.1 μm, 0.01 μm respectively for the low-<i>Z</i> and high-<i>Z</i> mixing can be treated as atomic mix in burn rate aspect, and heterogeneous mix size with 10 μm, 1 μm respectively for the low-<i>Z</i> and high-<i>Z</i> mixing can be treated as ideal chunk mix in burn rate aspect, and heterogeneous mix size in the middle state needs to be evaluated by using the heterogeneous mixing model of thermonuclear reaction rate in the paper. Finally, the physical model is compared with 3D simulation results of the heterogeneous mixing effect experiment called “MARBLE Campaign” carried out on OMEGA laser facility, which is designed as a separated reactant experiments and capsules are filled with deuterated foam and HT gas pores of different size, covering typical mix morphology from atomic mix to chunk mix, which validate the reliability of the theoretical evaluation about the evolution of mixing morphology and its impact to thermonuclear reaction rate. This work is significant for the design and improvement of inertial confinement fusion mixing effect experiment in China.

The N-body interatomic potential for molecular dynamics simulations of diffusion in tungsten

Tungsten, as the most refractory metal, is applied in fusion reactor in parts subjected to high temperatures and strong neutron irradiation. These factors lead to intense diffusion processes causing degradation of the material. Experimental investigations under such conditions are usually highly complicated and cannot provide a comprehensive understanding of the occurring phenomena. Therefore, their combination with theoretical approaches is required. One of the most robust approaches to simulate diffusion processes is molecular dynamics simulations based on classical interatomic potentials. It allows modeling relatively large samples consisting of several grains, grain boundaries, dislocations, and other types of defects for a reasonable computational time. The reliable simulations of the diffusion process require interatomic potentials satisfying the following criteria: prediction of melting point and thermal expansion as close as possible to the experimental values because the diffusion coefficient strongly depends on the homologous temperature and size factor. In the present paper, we present the new interatomic potential for tungsten, developed within the N-body approach, which reproduces the experimental value of melting temperature (3695 K) and thermal expansion at temperatures up to a melting point. The calculated diffusion coefficient demonstrates adequate agreement with experimental results. The constructed potential is applicable for simulation of processes involving diffusion, one of which is the irradiation damage.

Simulation of Diffusion Processes in Chemical and Thermal Processing of Machine Parts

To solve a number of technological issues, it is advisable to use mathematical modeling, which will allow us to obtain the dependences of the influence of the technological parameters of chemical and thermal treatment processes on forming the depth of the diffusion layers of steels and alloys. The paper presents mathematical modeling of diffusion processes based on the existing chemical and thermal treatment of steel parts. Mathematical modeling is considered on the example of 38Cr2MoAl steel after gas nitriding. The gas nitriding technology was carried out at different temperatures for a duration of 20, 50, and 80 h in the SSHAM-12.12/7 electric furnace. When modeling the diffusion processes of surface hardening of parts in general, providing a specifically given distribution of nitrogen concentration over the diffusion layer’s depth from the product’s surface was solved. The model of the diffusion stage is used under the following assumptions: The diffusion coefficient of the saturating element primarily depends on temperature changes; the metal surface is instantly saturated to equilibrium concentrations with the saturating atmosphere; the surface layer and the entire product are heated unevenly, that is, the product temperature is a function of time and coordinates. Having satisfied the limit, initial, and boundary conditions, the temperature distribution equations over the diffusion layer’s depth were obtained. The final determination of the temperature was solved by an iterative method. Mathematical modeling allowed us to get functional dependencies for calculating the temperature distribution over the depth of the layer and studying the influence of various factors on the body’s temperature state of the body.

Simulating Stochastic Diffusion Processes and Processes with “Market” Time

The paper focuses on modelling, simulation techniques and numerical methods concerned stochastic processes in subject such as financial mathematics and financial engineering. The main result of this work is simulation of a stochastic process with new market active time using Monte Carlo techniques.The processes with market time is a new vision of how stock price behavior can be modeled so that the nature of the process is more real. The iterative scheme for computer modelling of this process was proposed.It includes the modeling of diffusion processes with a given marginal inverse gamma distribution. Graphs of simulation of the Ornstein-Uhlenbeck random walk for different parameters, a simulation of the diffusion process with a gamma-inverse distribution and simulation of the process with market active time are presented.To simulate stochastic processes, an iterative scheme was used:
 xk+1 = xk + a(xk, tk) ∆t + b(xk, tk) √ (∆t) εk,,
 where εk each time a new generation with a normal random number distribution.Next, the tools of programming languages for generating random numbers (evenly distributed, normally distributed) are investigated. Simulation (simulation) of stochastic diffusion processes is carried out; calculation errors and acceleration of convergence are calculated, Euler and Milstein schemes. At the next stage, diffusion processes with a given distribution function, namely with an inverse gamma distribution, were modelled. The final stage was the modelling of stock prices with a new "market" time, the growth of which is a diffusion process with inverse gamma distribution. In the proposed iterative scheme of stock prices, we use the modelling of market time gains as diffusion processes with a given marginal gamma-inverse distribution.The errors of calculations are evaluated using the Milstein scheme. The programmed model can be used to predict future values of time series and for option pricing.

Discrete-Event Modeling and Simulation of Diffusion Processes in Multiplex Networks

A variety of phenomena (such as the spread of diseases, pollution in rivers, etc.) can be studied as diffusion processes over networks (i.e., the diffusion of the phenomenon over a set of interconnected entities). This research introduces a method to study such diffusion processes in multiplex dynamic networks. We use a formal Modeling and Simulation methodology (in our case, DEVS, Discrete-Event System Specification). We use DEVS formal models to integrate models defined using Agent-Based Modeling and Network Theory. We present (1) an Architecture to study Diffusion Processes in Multiplex dynamic networks (ADPM) and (2) a systematic Process to define, implement, and simulate diffusion processes over such networks. We show a theoretical definition and a concrete implementation of ADPM. We show how to use ADPM and the process in a case study based on a real nuclear emergency plan; this illustrates the application of the process, the architecture, and the developed software. Different scenarios are studied as Diffusion Processes to demonstrate the usability of ADPM.

Application of artificial neural networks for rigid lattice kinetic Monte Carlo studies of Cu surface diffusion

Kinetic Monte Carlo (KMC) is a powerful method for simulation of diffusion processes in various systems. The accuracy of the method, however, relies on the extent of details used for the parameterization of the model. Migration barriers are often used to describe diffusion on atomic scale, but the full set of these barriers may become easily unmanageable in materials with increased chemical complexity or a large number of defects. This work is a feasibility study for applying a machine learning approach for Cu surface diffusion. We train an artificial neural network on a subset of the large set of 226 barriers needed to correctly describe the surface diffusion in Cu. Our KMC simulations using the obtained barrier predictor show sufficient accuracy in modelling processes on the low-index surfaces and display the correct thermodynamical stability of these surfaces.

Applying forces to generate cartograms: a fast and flexible transformation framework

ABSTRACT Automatic production of contiguous area cartograms has become practical with computer algorithms, particularly applying forces to rubber-sheets and simulating diffusion processes. Reformulating the existing force-based and rubber-sheet methods, this article presents a fast and flexible force-based computational framework for general space transformation with solid physical and mathematical foundations. When being applied to cartogram production, this framework guarantees topological integrity, allows flexible force generation, achieves fast convergence, and avoids extreme shape deformation. Benchmarked against the recently published fast flow-based diffusion method using five datasets of various volume, compactness, and complexity, this force-based framework is faster, reduces more shape deformation, and can produce cartograms with distinctive slim or inflated styles. Additionally, the force-based framework is robust and can handle complicated datasets, whereas the flow-based method produces errors from them. Although the flow-based diffusion method often reduces cartogram size error more than the force-based (typical weighted mean error is below 0.1% versus 1%), both are outstanding with such low levels of error and their cartograms generally register little visual differences. Overall, the force-based transformation framework provides a fast, flexible, and robust alternative to the diffusion method for cartogram production.

Assessment of the Environmental Consequences Caused by Endogenous and Anthropogenic Hazards for the Azov Recreation Area

The Russian sector of the Azov recreational area is under the influence of numerous pollution sources of anthropogenic or natural origin. To ensure a comfortable recreation and recovery of the population, it is necessary to apply methods of integrated system monitoring, which makes it possible to identify patterns of the pollution formation in heterogeneous areas of the territory and places of increased environmental risk. Mathematical and simulation modelling should be an integral part of such systems, as a tool for scenario analysis, forecasting and warning. The paper proposes cellular automaton modelling methods, which have the advantages of expressivity, which allows for expansion of the possibilities for studying complex physicochemical processes and phenomena associated with the migration and transformation of reagents in the atmosphere and in the aquatic environment. The method is implemented in the form of a software application. We also present the results of the simulation of diffusion process and two substances - products of combustion in a fire with a point localization of epicentres; independent dispersion and interaction of light fractions of emission products upon activation of a land volcano and heavy fractions of mud volcano breccias of an underwater mud volcano.

Diffusion process simulation and ventilation strategy for small-hole natural gas leakage in utility tunnels

The utility tunnel provides enough spaces, dry environment, better monitoring and maintenance for municipal pipelines. When gas pipelines are incorporated into the utility tunnel, the flammability and explosion property of the natural gas would threaten its safe operation. At present, the safety prevention technology and the safety design specifications for gas pipeline in utility tunnels are not mature. To investigate the gas diffusion characteristics in the utility tunnel, and to propose an emergency accident ventilation strategy, in this paper, the mechanism of small-hole gas leakage diffusion under natural and mechanical ventilation is studied, and the influential factors are simulated and analyzed in detail using the Fluent software. The results indicate that under the natural ventilation, the gas spreads to the whole compartment at the leakage time of 300 s, while under the normal ventilation scheme, the leaked natural gas fills the downstream half of the compartment at the time of 120 s. When the leak size and pressure increase, the alarm response time shortens, the concentration of test point increases, and the danger zone expands. To prevent alarm failure and gas gathering for the near-outlet leakage, it is recommended that the last detector and the air outlet should be placed as close as possible to the fireproof wall. In addition, when a leak occurs, increasing the air change rate can greatly reduce the overall gas concentration. When the leak size is no more than 5 mm, and the pipeline pressure is lower than 0.4 MPa, the minimum accident ventilation scheme could guarantee the safe operation of the utility tunnel. When the pipeline pressure grows to 0.8 MPa, the effective air change rates of the emergency accident ventilation are 15 times/h, 18 times/h, and 24 times/h, respectively for the leak sizes of 4 mm, 5 mm, and 6 mm. Moreover, for the pipeline pressure of 1.6 MPa, the effective air change rates are 21 times/h, 30 times/h, and 36 times/h, respectively for the leak sizes of 4 mm, 5 mm, and 6 mm.

Monte Carlo Simulation of Diffusion Processes in Three-Component Alloys

The methods of molecular dynamics applied to a representative volume of a substance make it possible to calculate atomic trajectories within the time intervals on the order of 1 ns, which ensures an investigation of such slow processes as diffusion. This problem can be solved using the Monte Carlo method successfully applied to investigation of such diffusion-controlled processes as order – disorder transitions in the alloys or diffusion welding of heterogeneous metals through a backing plate. The majority of studies have been made for binary alloys, while the alloys in practical use contain a larger number of components. A theoretical model is presented, which allows investigating diffusion processes in three-component alloys via the vacancy mechanism in a solid-sphere approximation. A relation is derived for calculating the potential energy of an alloy, which is specified for the case of a completely disordered alloy. The difference between these energies is expressed via the ordering energies and order parameter. The proposed model is applicable to crystal lattices of any dimensionality. An example of its use for a three-component alloy of the A2BC stoichiometry is given, whose atoms occupy the sites of a two-dimensional square lattice is given.

Towards modelling diffusion mechanisms for sustainable off-grid electricity planning

The electrification-based literature reports a limited knowledge about the mechanisms of evolution of electricity demand in off-grid settings, especially in remote contexts of developing countries, due to the lack of robust and appropriate modelling frameworks. Such lack of understanding and modelling endeavour contributes to an inefficient allocation of resources for electrification projects and inappropriate off-grid sizing processes. As a first step towards the development of a more appropriate electricity demand model, we present a comparative study of two approaches for modelling different diffusion mechanisms of electricity connections: system-dynamics and agent-based models. The latter includes the modelling of social network archetypes in the simulation of diffusion processes. We model different scenarios of diffusion and we use them for evaluating the impact on the sizing process of an off-grid hydroelectric system. The results suggest that the structure of the social network can represent a crucial parameter that can impact on timing needed to complete the diffusion of electricity access – from few months to even >10 years. This affects the sizing process and the long-term sustainability of the power system, leading to variation of the hydroelectric capacity and the battery size up to around 55% and 100%, respectively. Our results indicate that the agent-based approach allows a more diversified representation of diffusion processes, but the limitations and scarcity of data can be an obstacle to their prompt application for energy application in unelectrified areas. On the contrary, system-dynamics can represent a more appropriate method since it requires less quantitative data and it provides a more structural and holistic modelling framework for conceptualising and formulating in a the determinants and complexities affecting the evolution of electricity demand in unelectrified areas.

An exponential timestepping algorithm for diffusion with discontinuous coefficients

We present a new Monte Carlo algorithm to simulate diffusion processes in presence of discontinuous convective and diffusive terms. The algorithm is based on the knowledge of close form analytic expressions of the resolvents of the diffusion processes which are usually easier to obtain than close form analytic expressions of the density. In the case of diffusion processes with piecewise constant coefficients, known as Skew Diffusions, such close form expressions for the resolvent are available. Then we apply our algorithm to this particular case and we show that the approximate densities of the particles given by the algorithm replicate the particularities of the true densities well (discontinuities, bimodality, …). Besides, numerical experiments show a quick convergence.

Discovery of rare event testing for stochastic simulations of diffusion processes

Stochastic modeling of diffusion processes in various spatial domains and with boundary conditions is widely applicable especially in combination with non-linear kinetics of chemical reactions. However, the comparison with exact solution is possible in simple cases e.g. linear diffusion models. The novel testing method introduced in this article enables to study the behavior of linear stochastic diffusion models with possible low particle concentration in target sub-domains. The proposed method is demonstrated on various diffusion models with the known Green function. The novel method is recommended whenever the χ2 goodness-of-fit test fails or the localization of critical domains is required.

An adaptive simulation of nonlinear heat and moisture transfer as a boundary value problem

This work presents an alternative view on the numerical simulation of diffusion processes applied to the heat and moisture transfer through porous building materials. Traditionally, by using the finite-difference approach, the discretization follows the Method Of Lines (MOL), when the problem is first discretized in space to obtain a large system of coupled Ordinary Differential Equations (ODEs). Thus, this paper proposes to change this viewpoint. First, we discretize in time to obtain a small system of coupled ODEs, which means instead of having a Cauchy (Initial Value) Problem (IVP), we have a Boundary Value Problem (BVP). Fortunately, BVPs can be solved efficiently today using adaptive collocation methods of high order. To demonstrate the benefits of this new approach, three case studies are presented, in which one of them is compared with experimental data. The first one considers nonlinear heat and moisture transfer through one material layer while the second one considers two material layers. Results show how the nonlinearities and the interface between materials are easily treated, by reasonably using a fourth-order adaptive method. Finally, the last case study compares numerical results with experimental measurements, showing a good agreement.

Curious Convergence Properties of Lattice Boltzmann Schemes for Diffusion with Acoustic Scaling

We consider the D1Q3 lattice Boltzmann scheme with an acoustic scale for the simulation of diffusive processes. When the mesh is refined while holding the diffusivity constant, we first obtain asymptotic convergence. When the mesh size tends to zero, however, this convergence breaks down in a curious fashion, and we observe qualitative discrepancies from analytical solutions of the heat equation. In this work, a new asymptotic analysis is derived to explain this phenomenon using the Taylor expansion method, and a partial differential equation of acoustic type is obtained in the asymptotic limit. We show that the error between the D1Q3 numerical solution and a finite-difference approximation of this acoustic-type partial differential equation tends to zero in the asymptotic limit. In addition, a wave vector analysis of this asymptotic regime demonstrates that the dispersion equation has nontrivial complex eigenvalues, a sign of underlying propagation phenomena, and a portent of the unusual convergence properties mentioned above.

Diffusion in Condensed Matter by Finite Element Method

In this Chapter, the finite element simulations of diffusion processes in homogeneous and polycrystalline materials are presented as well as some analytical solutions and implementations of basic diffusion relations. For the homogeneous materials the presented examples show the changes in time of the concentration of diffusing matter within the semi-infinite system and simulation of anisotropic nature of diffusion processes.The polycrystalline materials have been analysed for three cases, namely influence of average grain size and the homogeneity of grain size on the macroscopic diffusivity as well as simulation of the diffusion strains. The homogenisation technique has been used to estimate the diffusion property of grains aggregates.

Multiscale models and stochastic simulation methods for computing rare but key binding events in cell biology

The main difficulty in simulating diffusion processes at a molecular level in cell microdomains is due to the multiple scales involving nano- to micrometers. Few to many particles have to be simulated and simultaneously tracked while there are exploring a large portion of the space for binding small targets, such as buffers or active sites. Bridging the small and large spatial scales is achieved by rare events representing Brownian particles finding small targets and characterized by long-time distribution. These rare events are the bottleneck of numerical simulations. A naive stochastic simulation requires running many Brownian particles together, which is computationally greedy and inefficient. Solving the associated partial differential equations is also difficult due to the time dependent boundary conditions, narrow passages and mixed boundary conditions at small windows. We present here two reduced modeling approaches for a fast computation of diffusing fluxes in microdomains. The first approach is based on a Markov mass-action law equations coupled to a Markov chain. The second is a Gillespie's method based on the narrow escape theory for coarse-graining the geometry of the domain into Poissonian rates. The main application concerns diffusion in cellular biology, where we compute as an example the distribution of arrival times of calcium ions to small hidden targets to trigger vesicular release.

Modifying continuous-time random walks to model finite-size particle diffusion in granular porous media

The continuous-time random walk (CTRW) model is useful for alleviating the computational burden of simulating diffusion in actual media. In principle, isotropic CTRW only requires knowledge of the step-size, P_l, and waiting-time, P_t, distributions of the random walk in the medium and it then generates presumably equivalent walks in free space, which are much faster. Here we test the usefulness of CTRW to modelling diffusion of finite-size particles in porous medium generated by loose granular packs. This is done by first simulating the diffusion process in a model porous medium of mean coordination number, which corresponds to marginal rigidity (the loosest possible structure), computing the resulting distributions P_l and P_t as functions of the particle size, and then using these as input for a free space CTRW. The CTRW walks are then compared to the ones simulated in the actual media. In particular, we study the normal-to-anomalous transition of the diffusion as a function of increasing particle size. We find that, given the same P_l and P_t for the simulation and the CTRW, the latter predicts incorrectly the size at which the transition occurs. We show that the discrepancy is related to the dependence of the effective connectivity of the porous media on the diffusing particle size, which is not captured simply by these distributions. We propose a correcting modification to the CTRW model—adding anisotropy—and show that it yields good agreement with the simulated diffusion process. We also present a method to obtain P_l and P_t directly from the porous sample, without having to simulate an actual diffusion process. This extends the use of CTRW, with all its advantages, to modelling diffusion processes of finite-size particles in such confined geometries.