Application of the particle-in-cell method for numerical simulation of sheath plasma

We give a survey of papers on the numerical simulation of the sheath plasma using the particle-in-cell method. We study the problem of the behavior of a plasma bounded in the longitudinal direction of an absorbing wall. The model studied contains charged particles (electrons and ions) that move subject to a self-consistent electrostatic field. New pairs of particles are generated in the region of a distributed source. As a numerical model we use the electrostatic “particle-in-cell” method supplemented with the Emmert model for a bulk source and the algorithm of binary Coulomb collisions using a Monte Carlo method. Only electron-ion collisions are taken into account. Translated fromMetody Matematicheskogo Modelirovaniya, 1998, pp. 101–131.     

Numerical simulation of pollution and oil spill spreading by the stochastic discrete particle method

The features of the stochastic discrete particle method are discussed as applied to the simulation of pollutant advection and diffusion in a turbulent flow and to the spread of a thin film of a viscous substance (oil) on the surface of water. The diffusion tensor in the former problem depends on the scale of the pollution cloud, and the diffusivity in the latter problem depends nonlinearly on the desired function. For pollution dispersion by a turbulent flow, a stochastic discrete particle algorithm is constructed in the case when the diffusion tensor corresponds to the Richardson 4/3 law. The numerical and analytical results are shown to agree well. The problem of oil film spreading is described by a quasilinear advection-diffusion equation. For this problem, a random walking algorithm is constructed in which the variance of the walking particle step depends on the desired function. For both instantaneous and time-continuous sources of pollutants, the solution produced by the stochastic discrete particle method agrees well with the analytical and/or numerical solutions to the test problems under consideration.     

Method for taking into account gravity in free-surface flow simulation

A numerical algorithm that correctly takes into account the force of gravity in the presence of density discontinuities is constructed using unstructured collocated grids and splitting algorithms based on SIMPLE-type methods. A correct hydrostatic pressure field is obtained by explicitly extracting the gravity force contribution to the pressure equation and computing it using the solution of the gravity equilibrium problem for a two-phase medium. To ensure that the force of gravity is balanced by the pressure gradient in the case of a medium at rest, an algorithm is proposed according to which the pressure gradient in the equations of motion is replaced by a modification allowing for the force of gravity. Well-known free-surface problems are used to show that, in contrast to previously known algorithms, the proposed ones on unstructured meshes correctly predict hydrostatic pressure fields and do not yield velocity oscillations or free-surface distortions.     

Numerical simulation of seismic activity by the grid-characteristic method

Seismic activity in homogeneous and layered enclosing rock masses is studied. A numerical mechanical-mathematical model of a hypocenter is proposed that describes the whole range of elastic perturbations propagating from the hypocenter. Synthetic beachball plots computed for various fault plane orientations are compared with the analytical solution in the case of homogeneous rock. A detailed analysis of wave patterns and synthetic seismograms is performed to compare seismic activities in homogeneous and layered enclosing rock masses. The influence exerted by individual components of a seismic perturbation on the stability of quarry walls is analyzed. The grid-characteristic method is used on three-dimensional parallelepipedal and curvilinear structured grids with boundary conditions set on the boundaries of the integration domain and with well-defined contact conditions specified in explicit form.     

Numerical simulation two phase flows of casting filling process using SOLA particle level set method

Mould filling process is a typical gas–liquid metal two phase flow phenomenon. Numerical simulation of the two phase flows of mould filling process can be used to properly predicate the back pressure effect, the gas entrapment defects, and better understand the complex motions of the gas phase and the liquid phase. In this paper, a novel sharp interface incompressible two phase numerical model for mould filling process is presented. A simple ghost fluid method like discretization method and a density evaluation method at face centers of finite difference staggered grid are proposed to overcome the difficulties when solving two phase Navier–Stokes equations with large-density ratio and large-viscosity ratio. A new mass conservation particle level set method is developed to capture the gas–liquid metal phase interface. The classical pressure-correction based SOLA algorithm is modified to solve the two phase Navier–Stokes equations. Two numerical tests including the Zalesak disk problem and the broken dam problem are used to demonstrate the accuracy of the present method. The numerical method is then adopted to simulate three mould filling examples including two high speed CCD camera imaging water filling experiments and an in situ X-ray imaging experiment of pure aluminum filling. The simulation results are in good agreement with the experiments.     

Modeling and simulation of plasma jet by lattice Boltzmann method

In order to find a simple and efficient simulation for plasma spray process, an attempt of modeling was made to calculate velocity and temperature field of the plasma jet by hexagonal 7-bit lattice Boltzmann method (LBM) in this paper. Utilizing the methods of Chapman–Enskog expansion and multi-scale expansion, the authors derived the macro equations of the plasma jet from the lattice Boltzmann evolution equations on the basis of selecting two opportune equilibrium distribution functions. The present model proved to be valid when the predictions of the current model were compared with both experimental and previous model results. It is found that the LBM is simpler and more efficient than the finite difference method (FDM). There is no big variation of the flow characteristics, and the isotherm distribution of the turbulent plasma jet is compared with the changed quantity of the inlet velocity. Compared with the velocity at the inlet, the temperature at the inlet has a less influence on the characteristics of plasma jet.     

Numerical simulation of nonlinear consolidation problems by models based on the network method

The nonlinear consolidation problem in saturated soils, for any type of constitutive dependences of the hydraulic permeability and the void ratio on the effective pressure, has been numerically simulated by the network method. Three different network models, based on logarithmic and/or potential constitutive dependences, called the Davis and Raymond, Juárez-Badillo and Cornetti and Battaglio models, as well as a fourth one with dependences in tabulated form, are solved. In addition, new network models that delete the two restrictive hypotheses assumed by these authors are presented. These hypotheses are the influence of the void ratio changes in the term of contraction of the governing equation and the influence of the thickness change of the volume element as consolidation progresses. Only a few rules based on elementary theory of circuits are required for the design of the models, whose solution is reliable with relatively small grids and computational times. After verifying the results of the network method with the solutions of the classic authors, the extended models have been used to address a real case of consolidation.     

Boundary function method to find the asymptotic solution of the plasma—sheath equation

We consider the asymptotic solution of the Tonks—Langmuir integro-different equation with an Emmert kernel, which describes the behavior of the potential both inside the main plasma volume and in a thin boundary layer. Equations of this type are singularly perturbed due to the small coefficient at the highest order (second) derivative. The asymptotic solution is obtained by the boundary function method. Equations are derived for the first two coefficients in the regular expansion series and in the boundary function expansion. The equation for the first coefficient of the regular series has only a trivial solution. Second-order differential equations are obtained for the first two boundary functions. The equation for the first boundary function is solved numerically on a discrete grid with locally uniform spacing. An approximate analytical expression for the first boundary function is obtained from the linearized equation. This solution adequately describes the behavior of the potential on small distances only. __________ Translated from Prikladnaya Matematika i Informatika, No. 19, pp. 21–40, 2004.     

NFFT-based extension of a particle simulation method using multigrid

In the calculation of electrostatic problems using multigrid methods, computationally the most expensive part is often the charge assignment to a grid. We show how to improve an existing multigrid method by using an NFFT-based scheme for this task. Numerical tests show that the new method outperforms the direct charge assignment schemes for the case where smooth charge distributions with large support are considered. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Computation of the lifetime of coverings taking account of corrosion phenomena

We propose a mathematical model for describing diffusion processes in piecewise homogeneous bodies, in particular in structural elements with coverings taking account of corrosion in the space of the bodies in contact and at the interface. The dependence of the stresses and the adhesion on internal and external parameters obtained from experimentally determined criteria make it possible to predict the lifetime of coverings.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 28, 1988, pp. 6–11.     

Numerical simulation of the modified regularized long wave equation by He's variational iteration method

The modified regularized long wave (MRLW) equation, with some initial conditions, is solved numerically by variational iteration method. This method is useful for obtaining numerical solutions with high degree of accuracy. The variational iteration solution for the MRLW equation converges to its exact solution. Moreover, the conservation laws properties of the MRLW equation are also studied. Finally, interaction of two and three solitary waves is shown. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Numerical simulation of reaction-diffusion systems by modified cubic B-spline differential quadrature method

In this paper, we have applied modified cubic B-spline based differential quadrature method to get numerical solutions of one dimensional reaction-diffusion systems such as linear reaction-diffusion system, Brusselator system, Isothermal system and Gray-Scott system. The models represented by these systems have important applications in different areas of science and engineering. The most striking and interesting part of the work is the solution patterns obtained for Gray Scott model, reminiscent of which are often seen in nature. We have used cubic B-spline functions for space discretization to get a system of ordinary differential equations. This system of ODE’s is solved by highly stable SSP-RK43 method to get solution at the knots. The computed results are very accurate and shown to be better than those available in the literature. Method is easy and simple to apply and gives solutions with less computational efforts.     

Probabilistic simulation of offshore collisions

Collisions between tankers and offshore structures may result in serious problems such as oil pollution. It is therefore of interest to estimate the probability of a collision. This is done here by means of a simulation model using the GASP IV simulation program. The new method is based on a statistical prediction of the ship's path after some kind of critical failure affecting the manoeuvrability of the tanker has taken place.     

Theory of plasticity and creep taking account of microstrains

A relationship between the theories of plasticity and creep of the type /1, 2/ and theories based on the concept of slip is set up. A most logical structure is proposed for the constitutive equations of the theory which is convenient for engineering calculations.

It has been shown /3/ that the theory of slip /4/ results from the theories /1, 2/. However, it remains unclear whether a deeper connection exists between these theories. Moreover, the connection between creep theories constructed using the approach in /1, 2/ and creep theories based on the slip concept was not generally examined. A survey of the development of polycrystalline strain theory /5/ yields a complete representation of the state of matters in plasticity and creep theories.     

Contact problems of thermoelasticity taking account of heat production

We give a survey of papers that discuss the mathematical statement and methods of solution of contact problems of thermoelasticity taking account of heat production due to frictional forces on the surface of contact.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 93–100.     

Numerical investigation of MHD plasma instability by the method of projections on the unstable manifold

Translated from Matematicheskie Modeli i Optimizatsiya Vychislitel'nykh Algoritmov, pp. 217–223, 1993.     

Numerical simulation of anomalous plasma transport in the presence of magnetic turbulence

The structure of plasma in the interplanetary space is briefly presented, and the problems related to the variability of solar activity are discussed. The features of magnetic turbulence in the solar wind are also described. Magnetic field fluctuations are one of the causes of enhanced transport both in laboratory and astrophysical plasmas. To a first approximation, the plasma particles follow the magnetic field lines, whose equations form a non-linear one and a half degrees of freedom system. Unless the fluctuation level is very low, numerical simulations are needed to study such a system. We review three-dimensional numerical simulations of field line transport in anisotropic magnetic turbulence. Several transport regimes are found: for low Kubo number, anomalous transport is obtained, featuring both subdiffusion, corresponding to trapping in cantori structures, and superdiffusion, corresponding to Levy flights in the stochastic layer. Increasing the Kubo number, and hence stochasticity, quasilinear, intermediate, and percolative regimes are found, in the order. An expression of the diffusion coefficient valid for generalized anisotropy is presented.