Reflecting Boundary Conditions Research Articles

Boundary conditions for two-sided fractional diffusion

This paper develops appropriate boundary conditions for the two-sided fractional diffusion equation, where the usual second derivative in space is replaced by a weighted average of positive (left) and negative (right) fractional derivatives. Mass preserving, reflecting boundary conditions for two-sided fractional diffusion involve a balance of left and right fractional derivatives at the boundary. Stable, consistent explicit and implicit Euler methods are detailed, and steady state solutions are derived. Steady state solutions for two-sided fractional diffusion equations using both Riemann–Liouville and Caputo flux are computed. For Riemann–Liouville flux and reflecting boundary conditions, the steady-state solution is singular at one or both of the end-points. For Caputo flux and reflecting boundary conditions, the steady-state solution is a constant function. Numerical experiments illustrate the convergence of these numerical methods. Finally, the influence of the reflecting boundary on the steady-state behavior subject to both the Riemann–Liouville and Caputo fluxes is discussed.

Thermoacoustic Modes of Quasi-One-Dimensional Combustors in the Region of Marginal Stability

It may be generally believed that the thermoacoustic eigenfrequencies of a combustor with fully acoustically reflecting boundary conditions depend on both flame dynamics and geometry of the system. In this work, we show that there are situations where this understanding does not strictly apply. The purpose of this study is twofold. In the first part, we show that the resonance frequencies of two premixed combustors with fully acoustically reflecting boundary conditions in the region of marginal stability depend only on the parameters of the flame dynamics but do not depend on the combustor's geometry. This is shown by means of a parametric study, where the time delay and the interaction index of the flame response are varied and the resulting complex eigenfrequency locus is shown. Assuming longitudinal acoustics and a low Mach number, a quasi-1D Helmholtz solver is utilized. The time delay and interaction index of the flame response are parametrically varied to calculate the complex eigenfrequency locus. It is found that all the eigenfrequency trajectories cross the real axis at a resonance frequency that depends only on the time delay. Such marginally stable frequencies are independent of the resonant cavity modes of the two combustors, i.e., the passive thermoacoustic modes. In the second part, we exploit the aforementioned observation to evaluate the critical flame gain required for the systems to become unstable at four eigenfrequencies located in the marginally stable region. A computationally efficient method is proposed. The key ingredient is to consider both direct and adjoint eigenvectors associated with the four eigenfrequencies. Hence, the sensitivity of the eigenfrequencies to changes in the gain at the region of marginal stability is evaluated with cheap and accurate calculations. This work contributes to the understanding of thermoacoustic stability of combustors. In the same manner, the understanding of the nature of distinct resonance frequencies in unstable combustors may be enhanced by employing the analysis of the eigenfrequency locus here reported.

Underestimation of statistical uncertainty of local tallies in Monte Carlo eigenvalue calculation for simple and LWR lattice geometries

ABSTRACT A prediction method of the true variance of local tally in a Monte Carlo (MC) critical calculation is developed. In the MC calculation, the effective multiplication factor () and the fission rate distribution are estimated by simulating fission chain reactions. The statistical uncertainty of calculation result is commonly estimated by the standard error based on the central limit theorem. However, the evaluated statistical uncertainty would be underestimated when the inter-cycle correlation is not appropriately taken into account. In this study, a theoretical formula of the underestimation ratio (UR) of the statistical uncertainty for a local tally is derived using the eigenfunction expansion and the Autoregressive model. Note that the UR is defined by the ratio of uncertainty estimated by a MC calculation to the true statistical uncertainty. The proposed method is applied to one-dimensional slab and multi-assembly geometries with reflective boundary conditions. In the one-dimensional slab geometry, the prediction results of UR show reasonable agreement with the reference. In the multi-assembly geometry, on the other hand, the prediction results of UR are agreement with the reference regarding the relative spatial shape but there are considerable differences in the absolute values of UR.

Time fractional derivative model with Mittag-Leffler function kernel for describing anomalous diffusion: Analytical solution in bounded-domain and model comparison

Non-Fickian or anomalous diffusion had been well documented in material transport through heterogeneous systems at all scales, whose dynamics can be quantified by the time fractional derivative equations (fDEs). While analytical or numerical solutions have been developed for the standard time fDE in bounded domains, the standard time fDE suffers from the singularity issue due to its power-law function kernel. This study aimed at deriving the analytical solutions for the time fDE models with a modified kernel in bounded domains. The Mittag-Leffler function was selected as the alternate kernel to improve the standard power-law function in defining the time fractional derivative, which was known to be able to overcome the singularity issue of the standard fractional derivative. Results showed that the method of variable separation can be applied to derive the analytical solution for various time fDEs with absorbing and/or reflecting boundary conditions. Finally, numerical examples with detailed comparison for fDEs with different kernels showed that the models and solutions obtained by this study can capture anomalous diffusion in bounded domains.

Prediction of optimum design variables for maximum heat transfer through a rectangular porous fin using particle swarm optimization

The current study deals with the optimization of significant variables which govern the heat transfer from a porous fin in convective medium using particle swarm optimization (PSO). The temperature distribution of the fin is obtained analytically by a perturbation technique called homotopy perturbation method (HPM). To validate the temperature distribution obtained by HPM, finite difference method is employed. Next a significance analysis has been carried out to identify important variables that play a vital role in transferring heat from the porous fin. The set of variables thus obtained was then optimized by PSO to enhance the heat transfer rate. Reflective boundary condition is incorporated in the PSO to prevent particles from wandering in the infeasible region. The convergence plots of the variables show the effectiveness of PSO in solving non linear problems of this magnitude which are often encountered in the analysis of heat transfer through porous fins.

Improvements of the Embedded Self-Shielding Method with Pseudo-Resonant Isotope Model on the Multifuel Lattice System

The deviations brought by the embedded self-shielding method with the pseudo-resonant isotope model is investigated. Numerical results show that error sources mainly come from the inconsistency in the heterogeneous resonance integral (RI) generated in the two-dimensional square pin–cell case with reflective boundary conditions. The high-order resonance interference effect also contributes to the deviation. The black assumption on the macroscopic cross section of the fuel is proposed to enhance the consistency in the generation of the heterogeneous RI table. Numerical results show that the modification on the original embedded self-shielding method improves the accuracy of the cross-section prediction in the multifuel lattice systems.

Application of Krylov Acceleration Technique in Method of Characteristics–Based Neutron Transport Code

In our earlier work, a computer code based on Method of Characteristics (MOC) was developed to solve the neutron transport equation for mainly assembly-level lattice calculations with reflective and periodic boundary conditions and to some extent core-level calculation with a vacuum boundary condition. Performance of the MOC code was also demonstrated with flat and linear flux approximations. Since neutron transport calculations involve extensive computation, an attempt is made to develop an efficient numerical recipe that will reduce the computation time. First, a conventional MOC solution of the neutron transport equation is transformed into a matrix equation to apply the Krylov subspace iteration method for accelerating the solution. It is found that even in the most sophisticated and compact formats, forming the matrix equation explicitly by storing its nonzero elements requires extremely large computer memory. Hence, an alternate way to apply the Krylov iteration is demonstrated by incorporating the effect of the matrix-based approach into the solution without storing the matrix elements. This computationally viable and novel acceleration technique is used in combination with the existing formalism of flat as well as linear flux approximation to solve a number of benchmark problems. Results show significant improvement in terms of faster convergence of the solution over the conventional inner-outer iteration without compromising accuracy.

Charged black hole bombs in a Minkowski cavity

Press and Teukolsky famously introduced the concept of a black hole bomb system: a scalar field scattering a Kerr black hole confined inside a mirror undergoes superradiant amplification that keeps repeating due to the reflecting boundary conditions at the mirror. A similar charged black hole bomb system exists if we have a charged scalar field propagating in a Reissner–Nordström black hole confined inside a box. We point out that scalar fields propagating in such a background are unstable not only to superradiance but also to a mechanism known as the near-horizon scalar condensation instability. The two instabilities are typically entangled but we identify regimes in the phase space where one of them is suppressed but the other is present, and vice-versa (we do this explicitly for the charged but non-rotating black hole bomb). These ‘corners’ in the phase space, together with a numerical study of the instabilities allow us to identify accurately the onset of the instabilities. Our results should thus be useful to make educated choices of initial data for the Cauchy problem that follows the time evolution and endpoint of the instabilities. Finally, we use a simple thermodynamic model (that makes no use of the equations of motion) to find the leading order thermodynamic properties of hairy black holes and solitons that should exist as a consequence (and that should be the endpoint) of these instabilities. To find the properties of these hairy solutions at higher order in perturbation theory, the Einstein–Maxwell-scalar equations of motion would have to be solved.

On the boundary conditions in Lagrangian particle methods and the physical foundations of continuum mechanics

This paper aims to discuss the boundary conditions techniques employed in the solution of physical and engineering problems using the Lagrangian particle approach in continuum mechanics. Simulations using different boundary treatment techniques have been performed, and in addition, a review of the literature on the techniques commonly employed in SPH simulations was also performed. The fictitious (virtual/ghost), the dynamic particles and the artificial repulsive forces are widely employed in problem-solving, mixing concepts of molecular and continuum scales, being inadequate for the use in continuum scale (considering the disrespect of the classical physics laws). On the other hand, the reflective boundary conditions, based on fundamentals of Physics (Newton’s restitution law) and analytic geometry, are in accordance with the continuum hypothesis and their use is recommended in continuum mechanics problems, out of the molecular scale. In many boundary conditions employed in particle methods, models of repulsive molecular forces in conjunction with virtual particles are being used without theoretical foundation in continuum mechanics and classical Physics. Purely computational solutions employing fictitious particles and artificial molecular forces should be avoided, and methods that respect the continuum theory, such as the reflective boundary conditions, should be employed on the macroscopic scale. Both hydrostatic and hydrodynamics analysis have been performed. An excellent agreement with the analytical solution or experimental results have been achieved. From the results found, the applicability of the reflexive boundary technique in the continuum domain, discretized by particles, was verified.

Conservative fourth-order finite-volume Vlasov–Poisson solver for axisymmetric plasmas in cylindrical (r,vr,vθ) phase space coordinates

A fourth-order finite-volume Vlasov–Poisson algorithm is developed for simulating axisymmetric plasma configurations in (r,vr,vθ) phase space coordinates. The Vlasov equation for cylindrical phase space coordinates is cast into conservation-law form and is discretized on a structured grid. The conservative finite-volume discretization is based on fifth-order upwind reconstructions of the distribution function and a fourth-order quadrature rule that accounts for transverse variations of fluxes along control volume surfaces. High-order specular reflection boundary conditions enable high-fidelity treatment of plasma distribution functions at the axis and at wall boundaries. The numerical method is applied to simulate a confined uniform neutral gas to assess convergence properties for an equilibrium system in which effects of finite-temperature and acceleration due to centrifugal and Coriolis forces are present. The discretization is also applied to a Z-pinch configuration to study electrostatic ion confinement and the dynamics of the associated breathing mode. Simulations show that the ion distribution function exhibits non-Maxwellian features and that a temperature anisotropy develops and is sustained. The finite-volume implementation is demonstrated to converge at fourth-order for both applications.

A morphology-independent data analysis method for detecting and characterizing gravitational wave echoes

The ability to directly detect gravitational waves has enabled us to empirically probe the nature of ultra-compact relativistic objects. Several alternatives to the black holes of classical general relativity have been proposed which do not have a horizon, in which case a newly formed object (e.g. as a result of binary merger) may emit echoes: bursts of gravitational radiation with varying amplitude and duration, but arriving at regular time intervals. Unlike in previous template-based approaches, we present a morphology-independent search method to find echoes in the data from gravitational wave detectors, based on a decomposition of the signal in terms of generalized wavelets consisting of multiple sine-Gaussians. The ability of the method to discriminate between echoes and instrumental noise is assessed by inserting into the noise two different signals: a train of sine-Gaussians, and an echoing signal from an extreme mass-ratio inspiral of a particle into a Schwarzschild vacuum spacetime, with reflective boundary conditions close to the horizon. We find that both types of signals are detectable for plausible signal-to-noise ratios in existing detectors and their near-future upgrades. Finally, we show how the algorithm can provide a characterization of the echoes in terms of the time between successive bursts, and damping and widening from one echo to the next.

CIP Method of Characteristics for the Solution of Tide Wave Equations

The CIP-MOC (Constrained Interpolation Profile/Method of Characteristics) is proposed to solve the tide wave equations with large time step size. The bottom topography and bottom friction, which are very important factor for the tidal wave model, are included to the equation of Riemann invariants as the source term. Numerical experiments demonstrate the good performance of the scheme. Compared to traditional semi-implicit (SI) finite difference scheme which is widely used in tidal wave simulation, CIP-MOC has better stability in simulating large gradient water surface change and has the ability to use much longer time step size under the premise of maintaining accuracy. Besides, numerical tests with reflective boundary conditions are carried out by CIP-MOC with large time step size and good results are obtained.

Sweetest taboo processes

Brownian dynamics play a key role in understanding the diffusive transport of micro particles in a bounded environment. In geometries containing confining walls, physical laws determine the behavior of the random trajectories at the boundaries. For impenetrable walls, imposing reflecting boundary conditions to the Brownian particles leads to dynamics described by reflecting stochastic differential equations. In practice, these stochastic differential equations as well as their refinements are quite challenging to handle, and more importantly, many physical processes are better modeled by processes conditioned to stay in a prescribed bounded region. In the mathematical literature, these processes are known as taboo processes, and despite their simplicity, at least compared to the reflecting stochastic differential equations approach, are surprisingly not much exploited in physics. This paper explores some aspect of taboo processes and other constrained processes in simple geometries: interval in one dimension, circular annulus in two dimensions, hollow sphere in three dimensions, and more. In particular, for the two-dimensional (2D) taboo process in a circular annulus, the Gaussian behavior of the stochastic angle is established.

Permeability correlation with porosity and Knudsen number for rarefied gas flow in Sierpinski carpets

In recent years, application of porous media is highlighted among researchers due to their wide range of usability in micro-scale problems, such as gas reservoirs, micro-filtering, heat exchangers, etc. With this respect, the accurate description of flow behavior using governing equations based on the continuum assumption is not valid since the mean free path is comparable to the characteristics length of the problem. For this purpose, a simple methodology for diffusion reflection boundary condition is developed and validated for two valuable benchmarks, namely micro-channel flow and fractal porous media, where the results were in good agreement with literature. Then, pore-scale simulation of the rarefied gas flow inside the square- and circular-based Sierpinski carpets in the slip and transition flow regimes (based on the developed boundary condition) for different porosities are carried out, using lattice Boltzmann method (LBM). In each case, the effects of different key parameters such as pressure gradient in the x-direction (body force), porosity, and Knudsen number on the velocity contours, velocity profiles, flow structures, and permeability are investigated. It is found that the velocity magnitude is reduced as Knudsen number increases, where the reduction rate of velocity in the slip flow regime is greater than that of the transition one. Also, a correlation between the permeability with porosity and Knudsen number in the Darcy flow regime (Re<<1) is presented. For both geometries, it is observed that the permeability varies logarithmically with Knudsen number and exponentially with the porosity.

Theoretical analysis of extreme wave oscillation in Paradip Port using a 3-D boundary element method

A mathematical model is developed to estimate the amplification of multidirectional random waves in the Paradip port, Odisha, India under the resonance conditions. A 3-D boundary element method with the consideration of variable bathymetry is utilized for solving Laplace equation in an irregular domain including the partial reflection boundary condition. The current numerical scheme is validated through the comparison of simulation results with previous well-defined studies along with measurement data for the rectangular harbor. Six key recorder stations are chosen as port location of moored ship to analyze the wave response in the Paradip port for multidirectional random waves. Further, wave spectrum is also estimated based on Fast Fourier Transformation (FFT) with respect to wave period at six recorder stations. Actual bathymetry of Paradip port is utilized to estimate the wave height ratio at each record station for the incident waves with different directions. Further, the resonance frequencies are determined for each directional incident wave and safe locations in Paradip port is also identified based on the simulation results. Thus, the proposed numerical scheme can be utilized to foster the prediction of wave profile in the realistic ports or harbors with complex geometries for the safe navigation.

Radiative GRMHD simulations of accretion and outflow in non-magnetized neutron stars and ultraluminous X-ray sources

We run two GRRMHD simulations of super-Eddington accretion disks around a black hole and a non-magnetized, non-rotating neutron star. The neutron star was modeled using a reflective inner boundary condition. We observe the formation of a transition layer in the inner region of the disk in the neutron star simulation which leads to a larger mass outflow rate and a lower radiative luminosity over the black hole case. Sphereization of the flow leads to an observable luminosity at infinity around the Eddington value when viewed from all directions for the neutron star case, contrasting to the black hole case where collimation of the emission leads to observable luminosities about an order of magnitude higher when observed along the disk axis. We find the outflow to be optically thick to scattering, which would lead to the obscuring of any neutron star pulsations observed in corresponding ULXs.

Discrete spherical harmonics method for radiative transfer in scalar planar inhomogeneous atmosphere.

The radiative transfer problems in a participating inhomogeneous scalar planar atmosphere, subjected to diffuse or collimated incidence, are investigated using the discrete spherical harmonics method. In developing the method, the radiative intensity is expanded in a finite series of Legendre polynomials and the resulting first-order coupled differential equations of radiance moments are expressed in a set of discrete polar directions. The method is applied to homogeneous/inhomogeneous atmospheres of various anisotropic scattering degrees and thicknesses, and reflective boundary conditions. The discrete spherical harmonics method albedo, transmittance, and radiative intensity predictions agree well with benchmark literature results. Additionally, numerical predictions show that the discrete spherical harmonics method using Mark boundary conditions are more efficient than using Marshak boundary conditions.

Characteristics of ion-cyclotron surface waves in semi-bounded (r, q) distribution dusty plasmas

The influence of magnetic field strength, ion mass, and the non-thermal character on the dispersion properties of ion-cyclotron surface wave is investigated in a semi-bounded (r, q) distribution dusty plasma. In the limit of short wave number, the dispersion relation is derived by adopting the specular reflection boundary condition and the effective screening distance in (r, q) distribution dusty plasma. It is found that the stronger magnetic field strength suppresses the wave speed, but the heavier ions will enhance the wave propagation. To investigate the wave propagation in the non-Maxwellian plasma, the typical values of r and q are chosen and the dispersion relation is plotted to obtain the general character of wave propagation. The result would reduce to the case of Maxwellian plasma for r→0 and q→∞.

Non-self-averaging behaviors and ergodicity in quenched trap models with finite system sizes.

Tracking tracer particles in heterogeneous environments plays an important role in unraveling material properties. These heterogeneous structures are often static and depend on the sample realizations. Sample-to-sample fluctuations of such disorder realizations sometimes become considerably large. When we investigate the sample-to-sample fluctuations, fundamental averaging procedures are a thermal average for a single disorder realization and the disorder average for different disorder realizations. Here we report on non-self-averaging phenomena in quenched trap models with finite system sizes, where we consider the periodic and the reflecting boundary conditions. Sample-to-sample fluctuations of diffusivity greatly exceed trajectory-to-trajectory fluctuations of diffusivity in the corresponding annealed model. For a single disorder realization, the time-averaged mean square displacement and position-dependent observables converge to constants because of the existence of the equilibrium distribution. This is a manifestation of ergodicity. As a result, the time-averaged quantities depend neither on the initial condition nor on the thermal histories but depend crucially on the disorder realization.

Reaction time for trimolecular reactions in compartment-based reaction-diffusion models.

Trimolecular reaction models are investigated in the compartment-based (lattice-based) framework for stochastic reaction-diffusion modeling. The formulae for the first collision time and the mean reaction time are derived for the case where three molecules are present in the solution under periodic boundary conditions. For the case of reflecting boundary conditions, similar formulae are obtained using a computer-assisted approach. The accuracy of these formulae is further verified through comparison with numerical results. The presented derivation is based on the first passage time analysis of Montroll [J. Math. Phys. 10, 753 (1969)]. Montroll's results for two-dimensional lattice-based random walks are adapted and applied to compartment-based models of trimolecular reactions, which are studied in one-dimensional or pseudo one-dimensional domains.