Stochastic Boundary Research Articles

Asymptotic solutions by the successive disordering method

We develop the periodic componentmethod [1] and represent the solution of a stochastic boundary value elasticity problem for a random quasiperiodic structure with a given disordering degree of inclusions in the matrix via the deviations from the corresponding solution for a random structure with a smaller disordering degree. An example in which the tensor of elastic properties of a composite is calculated is used to illustrate the asymptotic and differential approaches of the successive disordering method. The asymptotic approach permits representing the quasiperiodic structure with a given chaos coefficient and the desired tensor of effective elastic properties as a result of small successive disordering of an originally ideally periodic structure of a composite with known tensor of elastic properties. In the differential approach, a differential equation is obtained for the tensor of effective elastic properties as a function of the chaos coefficient. Its solution coincides with the solution provided by the asymptotic approach. The solution is generalized to the case of piezoactive composites, and a numerical analysis of the effective properties is performed for a PVF (polyvinylidene fluoride) piezoelectric with various quasiperiodic structures on the basis of the cubic structure with spherical inclusions of a high-module elastic material.

A FRACTIONAL POISSON EQUATION: EXISTENCE, REGULARITY AND APPROXIMATIONS OF THE SOLUTION

We consider a stochastic boundary value elliptic problem on a bounded domain D ⊂ ℝk, driven by a fractional Brownian field with Hurst parameter H = (H1,…,Hk) ∈ [½, 1[k. First, we define the stochastic convolution derived from the Green kernel and prove some properties. Using monotonicity methods, we prove the existence and uniqueness of solution along with regularity of the sample paths. Finally, we propose a sequence of lattice approximations and prove its convergence to the solution of the SPDE at a given rate.

Resonant Pedestal Pressure Reduction Induced by a Thermal Transport Enhancement due to Stochastic Magnetic Boundary Layers in High Temperature Plasmas

Good alignment of the magnetic field line pitch angle with the mode structure of an external resonant magnetic perturbation (RMP) field is shown to induce modulation of the pedestal electron pressure p(e) in high confinement high rotation plasmas at the DIII-D tokamak with a shape similar to ITER, the next step tokamak experiment. This is caused by an edge safety factor q95 resonant enhancement of the thermal transport, while in contrast, the RMP induced particle pump out does not show a significant resonance. The measured p(e) reduction correlates to an increase in the modeled stochastic layer width during pitch angle variations matching results from resistive low rotation plasmas at the TEXTOR tokamak. These findings suggest a field line pitch angle resonant formation of a stochastic magnetic edge layer as an explanation for the q95 resonant character of type-I edge localized mode suppression by RMPs.

Analytical results of the Nagel-Schreckenberg model with stochastic open boundary conditions

The deterministic Nagel-Schreckenberg model with stochastic open boundary conditions is investigated in a mostly analytical way. By means of the Markov Chain, we model the working process of the stochastic open boundaries. First, the analytical expression of the free-flow density profiles is derived. Then, we discuss theoretically how the right boundary determines the traffic capacity, global density, and density profiles. For these features, the analytical and numerical results agree well. This paper implies that the deterministic Nagel-Schreckenberg model with stochastic open boundaries is almost totally analyzable.

Thermal noise and stochastic strings in AdS/CFT

We clarify the structure of thermal noise in AdS/CFT by studying the dynamics of an equilibrated heavy quark string. Using the Kruskal extension of the correspondence to generate the dynamics of the field theory on the Keldysh contour, we show that the motion of the string is described by the classical equations of motion with a stochastic boundary condition on the stretched horizon. The form of the stochastic boundary condition is consistent with the dissipation on this surface and is found by integrating out the fluctuations inside of the stretched horizon. Solving the equations of motion for the fluctuating string we determine the full frequency dependence of the random force on the boundary quark and show that it is consistent with the frequency dependent dissipation. We show further that the stochastic motion reproduces the bulk to bulk two point functions of the Kruskal formalism. These turn out to be related to the usual retarded bulk to bulk propagator by KMS relations. Finally we analyze the stochastic equations and give a bulk picture of the random boundary force as a flip-flopping trailing string solution. The basic formalism can be applied to the fluctuations of gravitons, dilatons, and other fields.

Numerical simulations for two‐dimensional incompressible Navier‐Stokes equations with stochastic boundary conditions

AbstractThe article mainly concerns modeling the stochastic input and its propagation in incompressible Navier‐Stokes(N‐S) flow simulations. The stochastic input is represented spectrally by employing orthogonal polynomial functionals from the Askey scheme as trial basis to represent the random space. A standard Galerkin projection is applied in the random dimension to derive the equations in the weak form. The resulting set of deterministic equations is then solved with standard methods to obtain the mean solution and variance of the stochastic velocity. In this article, the main method employs the Hermite polynomial as the basis in random space. Cavity problems are given to demonstrate the process of numerical simulation. Furthermore, Monte‐Carlo simulation method is applied to illustrate the accurate numerical results. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010

A hybrid collocation/Galerkin scheme for convective heat transfer problems with stochastic boundary conditions

AbstractWe present a numerical method to study convective heat transfer in a high Reynolds number incompressible flow around a cylinder subject to uncertain boundary conditions. We exploit the one‐way coupling of the energy and momentum transport to derive a semi‐intrusive uncertainty propagation scheme, which combines Galerkin and collocation approaches for computing approximate statistics of the stochastic temperature field. The hybrid scheme converges rapidly and dramatically reduces the overall computational cost compared with the conventional uncertainty propagation schemes. Copyright © 2009 John Wiley & Sons, Ltd.

A third generation erosion model: The combination of probabilistic and deterministic components

A combination of the classical deterministic mechanics of soil erosion and stochastic description of controlling erosion factors (driving and resistance forces) gives significant advantages and leads to the development of a new (the third) generation of erosion models. In contrast to existing models, the STOSEM model allows determination of the relationship between the rate of erosion and controlling factors inside the model theoretically, by a solution of the governing deterministic equations with stochastic initial and boundary conditions. It is possible to calculate the mean erosion rate explicitly from the probability density functions (PDFs) of flow velocity, soil aggregate size, cohesion, integrity and other basic controlling factors. Variability of the PDFs of controlling factors through time and space due to the changes in flow regimen and/or soil composition produces differences in the structure of relationship between the mean erosion rate and controlling factors, as well as in resulting mean rate of erosion. The model is still far from application due to the lack of specific experimental data for validation. Nevertheless, its theoretical basement, transparency and flexibility show new direction in soil erosion science.

Model prediction of impurity retention in stochastic magnetic boundary and comparison with edge carbon emission in LHD

Impurity retention (screening) potential in the edge region with stochastic magnetic field in LHD (large helical device) is studied numerically using the three-dimensional edge transport code, EMC3-EIRENE, and experimentally applying VUV and EUV spectrometers. The code predicts an impurity retention regime at high densities, where the friction force dominates over the thermal force. The carbon radiation intensities from each charge state are measured in experiments, and compared with the modelling results. The density dependencies of the calculated and measured radiation intensities agree well with each other, confirming the impurity retention in the edge region of LHD. Geometrical advantages of the stochastic field line structure in this concern are also discussed.

Carbon transport in the stochastic magnetic boundary of TEXTOR

For given conditions, significant change in main particle and carbon transport is observed in TEXTOR under the action of the Dynamic Ergodic Divertor, DED, operating both in 6/2 and 3/1 basic modes. In particular, the stochastic layer created at the plasma edge by the applied perturbing field, is responsible for the so called pump out, PO, effect (observed in DIII-D and on JET during experiments of ELM mitigation) characterized by a decrease in the plasma density and the reduction in the intrinsic carbon concentration in the plasma core. For a sufficiently high value of the applied field strength, in DED 3/1 basic operational mode a 3/1 island connects to the wall, with related further enhancement of particle transport. With respect to carbon, this phenomenon simply leads to a further carbon de-contamination from the central plasma.

Solution of the plane stochastic creep boundary value problem

The solution of the non-linear stochastic boundary-value problem of the creep of a thin plate in a plane stress state when the elastic strains are small and can be neglected is presented. The plate material is stochastically inhomogeneous so that the stress and strain tensors are random functions of the coordinates. The constitutive creep relation, taken as in non-linear viscous flow theory, is formulated in a stochastic form. Using the perturbation method, the non-linear stochastic problem is reduced to a system of three linear partial differential equations in the fluctuations of the stress tensor and, then, changing by implementing the stress function, to a differential equation, the solution of which is represented in the form of the sum of two series. The first series is the solution far from the boundary of the plate, ignoring edge effects, and the second is the solution in the boundary layer, and its terms rapidly decay as the distance from the boundary of the plate increases. The stretching of a stochastically inhomogeneous half-plane in the direction of two mutually orthogonal axes is considered as an example. The stress concentration in the boundary of the half-plane is investigated. It is shown that the spread of the stresses in the surface layer, the width of which depends on the degree of non-linearity of the material, can be much greater than in the deep layers.

Generalized Langevin equation for solids. II. Stochastic boundary conditions for nonequilibrium molecular dynamics simulations

The generalized Langevin equation (GLE) [L. Kantorovich, Phys. Rev. B78, 094304 (2008)], which describes dynamics of a finite and possibly highly anharmonic subsystem surrounded by an extended harmonic solid, is simplified here assuming short-range interactions between atoms. We show that in this case quite naturally the GLE can be worked into a form which corresponds to considering central atoms of the finite region as governed by usual Newtonian mechanics, while the boundary atoms are treated as Langevin atoms, i.e., they experience friction and random forces [the so-called stochastic boundary conditions (SBCs)]. We show that the random forces constitute a stationary Gaussian random process with the dispersion directly related to the friction coefficient in a usual way. Next, we rigorously demonstrate that, even though not all atoms are stochastic within the SBC model, the system should still arrive at canonical distribution at long times. Since the SBC model follows directly from the general GLE description and can perform as a correct $NVT$ thermostat, we propose that the SBC is a method of choice if one wants to do nonequilibrium molecular dynamics simulations correctly. Our derivation of SBC is physically more acceptable than given previously when the central region atoms were assumed to be much heavier than those in the surrounding lattice (the zero-frequency approximation).

Algebra of Generalized Stochastic Processes and the Stochastic Dirichlet Problem

Following the embedding idea of generalized functions into the Colombeau algebra of generalized functions, we construct a new algebra of generalized stochastic processes and denote it 𝒢(W 2,2;(S)−1). This is done by using the chaos expansion form of generalized stochastic processes regarded as linear continuous mappings from the Sobolev space into the Kondratiev space (S)−1. As an application, we prove existence and uniqueness of the solution of the equation Lu = h with given stochastic boundary condition. The operator L is assumed to be strictly elliptic in divergence form Lu = ∇·(A·∇ u) + c·∇ u + du. Its coefficients: the elements of the matrix A and of the vectors b, c and d are assumed to be deterministic Colombeau generalized functions.

Scale-Dependent Homogenization of Inelastic Random Polycrystals

Rigorous scale-dependent bounds on the constitutive response of random polycrystalline aggregates are obtained by setting up two stochastic boundary value problems (Dirichlet and Neumann type) consistent with the Hill condition. This methodology enables one to estimate the size of the representative volume element (RVE), the cornerstone of the separation of scales in continuum mechanics. The method is illustrated on the single-phase and multiphase aggregates, and, generally, it turns out that the RVE is attained with about eight crystals in a 3D system. From a thermodynamic perspective, one can also estimate the scale dependencies of the dissipation potential in the velocity space and its complementary potential in the force space. The viscoplastic material, being a purely dissipative material, is ideally suited for this purpose.

Probability distribution of sea surface wind stresses

[1] The probability density function (pdf) of sea surface wind stresses is considered. Observed sea surface wind stresses are highly non-Gaussian, with characteristic relationships between moments. A consideration of empirical models of sea surface wind stress pdfs built from models of sea surface vector winds demonstrates that non-Gaussian structure in the vector winds must be accounted for to accurately characterise the observed relationships between wind stress moments. An idealised stochastic boundary layer model is shown to provide a good qualitative characterisation of these pdfs.

3D numerical simulations of energy transport in the stochastic boundary of TEXTOR-DED with a finite difference method

The effect of magnetic field line ergodization that eliminates magnetic surfaces (either by a resonant magnetic perturbation like in TEXTOR-DED or by intrinsic plasma effects like in W7-X) imposes the need for plasma transport models being able to describe this properly. To handle the ergodicity the concept of local magnetic coordinates allowing a correct discretization of the transport equations with minimized numerical errors is used. For the simulation of plasma transport in perturbed volume, a numerical method based on the finite difference concept has been developed, using a custom-tailored unstructured grid in local magnetic coordinates. This grid is generated by field line tracing to guarantee complete separation of the large parallel transport along B and that perpendicular to B and the ergodicity of the magnetic field does not limit applicability of the method in contrast to the methods based on finite volume ansatz. Perpendicular and parallel fluxes can be effectively separated in our approach and treated independently in the numerical method which has been implemented in the FINDIF code.The finite difference code FINDIF is used to investigate the energy transport in the complex 3D TEXTOR-DED tokamak geometry, where the plasma structures and transport are closely related to the structure of the magnetic field lines. Numerical grids have been prepared in order to simulate 12/4 and 6/2 modes of the DED operation, respectively. In particular, the question, what is the role of long and short magnetic field lines in the heat transfer from the core plasma to divertor surface, is addressed. Simulation results are compared with experimentally determined temperature profiles and heat fluxes at the target.

Identification and analysis of transport domains in the stochastic boundary of TEXTOR-DED for different mode spectra

At the TEXTOR tokamak an external resonant magnetic perturbation is applied with the dynamic ergodic divertor to control the edge transport properties. The approaches to analyse the impact of such a kind of edge stochastization on transport apply mostly a shell-like picture which includes a dependence of transport from magnetic field topology in the radial direction only. In this paper multiple experimental evidence is presented in contrast to these approaches the perturbation applied forms a poloidally heterogenous edge layer in which the transport characteristics are determined by the poloidally alternating field line behaviour. A thorough analysis of density and temperature profiles and their gradients for base mode spectra with poloidal/toroidal mode numbers of m/n = 12/4 and m/n = 6/2 is worked out in comparison with the modelled magnetic field topology and results from three dimensional transport modelling with EMC3/EIRENE. Hereby two poloidally adjacent transport domains are identified for the first time in such detail. A domain representing a helical scrape off layer is formed by field lines with short connection and therefore prevailing parallel transport to the wall elements. Here, the field lines are clustered into extended flux tubes embedded into a long connection length ergodic domain with diffusive transport characteristics and enhanced radial transport.

Stochastic FitzHugh-Nagumo equations on networks with impulsive noise

We consider a system of nonlinear partial differential equations with stochastic dynamical boundary conditions that arises in models of neurophysiology for the diffusion of electrical potentials through a finite network of neurons. Motivated by the discussion in the biological literature, we impose a general diffusion equation on each edge through a generalized version of the FitzHugh-Nagumo model, while the noise acting on the boundary is described by a generalized stochastic Kirchhoff law on the nodes. In the abstract framework of matrix operators theory, we rewrite this stochastic boundary value problem as a stochastic evolution equation in infinite dimensions with a power-type nonlinearity, driven by an additive Lévy noise. We prove global well-posedness in the mild sense for such stochastic partial differential equation by monotonicity methods.

Tensor-valued random fields for meso-scale stochastic model of anisotropic elastic microstructure and probabilistic analysis of representative volume element size

The main objective of this paper is to present a generic meso-scale probability model for a large class of random anisotropic elastic microstructures in order to perform a parametric analysis of the Representative Volume Element (RVE) size. This new approach can be useful for a direct experimental identification of random anisotropic elastic microstructures when the standard method cannot easily be applied to anisotropic elastic microstructures. Such a RVE is used to construct the macroscopic properties in the context of stochastic homogenization. The probability analysis is not performed as usual for a given particular random microstructure defined in terms of its constituents. Instead, it is performed for a large class of random anisotropic elastic microstructures. For this class, the probability distribution of the random effective stiffness tensor is explicitly constructed. This allows a full probability analysis of the RVE size to be carried out and its convergence to be studied. The procedure of homogenization is based on a homogeneous Dirichlet condition on the boundary of the RVE. The probability model used for the stiffness tensor-valued random field of the random anisotropic elastic microstructure is an extension of the model recently introduced by the author for elliptic stochastic partial differential operators. The stochastic boundary value problem is numerically solved by using the stochastic finite element method. The probability analysis of the RVE size is performed by studying the probability distribution of the random operator norm of the random effective stiffness tensor with respect to the spatial correlation length of the random microstructure.

An Impact of Stochastic Dynamic Boundary Conditions on the Evolution of the Cahn-Hilliard System

Nonlinear systems are often subject to random influences. Sometimes the noise enters the system through physical boundaries and this leads to stochastic dynamic boundary conditions. A dynamic, as opposed to static, boundary condition involves the time derivative as well as spatial derivatives for the system state variables on the boundary. Although stochastic static (Neumann or Dirichet type) boundary conditions have been applied for stochastic partial differential equations, not much is known about the dynamical impact of stochastic dynamic boundary conditions. The purpose of this article is to study possible impacts of stochastic dynamic boundary conditions on the long term dynamics of the Cahn-Hilliard equation arising in the materials science. We show that the dimension estimation of the random attractor increases as the coefficient for the dynamic term in the stochastic dynamic boundary condition decreases. However, the dimension of the random attractor is not affected by the corresponding stochastic static boundary condition.