Reflecting Boundary Conditions Research Articles

On the Application of the Nyström Method to Two-Dimensional Neutral Particle Transport Problems in Heterogeneous Media

Over time, several methods were developed to deal with neutral particle transport problems. The interest in these problems is related to their wide range of applications, from neutron transport and heat transfer in nuclear reactors to radiative transfer in atmospheric clouds. Unlike the discrete ordinates or discrete ordinates–like methods, integral methods do not require discretization of angular variables. Instead, angular variables are completely eliminated by an integration procedure over the solid angle, which allows elimination of the ray effect. That said, this paper presents a new approach to estimate the scalar flux in two-dimensional fixed-source neutron transport problems in a heterogeneous medium, considering isotropic scattering and vacuum and reflective boundary conditions. Here, the Nyström method is combined with the singularity-subtraction technique to present an integral formulation for the scalar flux in a mesh grid over all regions of the domain. The iterative method of the Neumann series is used as an alternative to direct methods to solve the resulting system of equations generated from the domain discretization. Numerical results are given to verify the offered method.

Numerical analysis of a gas flow in a square cavity driven by spanwise lid motion on the basis of kinetic theory: Behavior of the gas near a sharp corner

A gas flow in a square cavity driven by a lid sliding in the direction of its line of contact with the cavity wall is considered. The steady behavior of the gas is numerically investigated based on the linearized Bhatnagar–Gross–Krook kinetic equation and the diffuse reflection boundary condition. When one applies the Stokes equation and the no-slip boundary condition to the system considered here, the flow velocity becomes multivalued at the corner between the lid and the cavity wall, and the shear stress diverges at the corner inversely proportionally to the distance from there, which is known as the so-called corner singularity. In the present work, the behavior of the gas near the corner is examined based on numerical results obtained from the kinetic theory. Although the range of the flow velocity value in the kinetic solution is limited due to the significant velocity slip near the corner, the flow velocity is, nevertheless, multivalued at the corner. The shear stress varies inversely proportionally to the distance from the corner up to the position that is a few tens of mean free paths away from there. The increase in the stress is suppressed at positions closer to the corner and its magnitude remains bounded. Thus, the total forces acting on the lid and the side cavity walls are bounded as well. Due to the distinctive behavior of the stress near the corner, the resulting nondimensional total forces behave with an unconventional rate Kn ln Kn for small Knudsen numbers Kn.

Flame-acoustic interactions in high-pressure H2/air diffusion flames

Understanding flame-acoustics interaction (FAI) that can potentially trigger thermoacoustic instabilities is of critical importance to mitigate the serious side effects of pressure oscillations. While FAI has been studied extensively in premixed flames, comparatively less analysis has been conducted to explore FAI mechanisms in non-premixed flames, especially under high-pressure conditions. This study aims to numerically investigate FAI in high-pressure hydrogen/air counterflow diffusion flames, with special attention on the impact of pressure on FAI. To this end, a fully compressible unsteady counterflow solver combined with Navier-Stokes Characteristic Boundary Conditions (NSCBC) is employed to capture the reflection of the acoustic waves at the boundaries. The results show that for all ambient pressures ( 1 ≤ P a ≤ 60 bar) considered here, real-gas effects on flame structures are negligible and that increasing P a would lead to a power-law dependence of density gradient on P a with an exponent of 1.5. Furthermore, all tested cases show acoustic growth with time under fully reflecting boundary conditions, which is found to be pressure dependent. The flames become less resilient to FAI when the pressure (or density gradient) is increased. The results from the spectral analysis show that for all ambient pressures, there is only a single dominant frequency at the low-density fuel side and two dominant frequencies at the high-density oxidiser side. This indicates that the pressure oscillation is more sinusoidal at the H 2 side than that at the air side. Moreover, the phase space portraits of pressure signals indicate that the dynamics of the system are becoming periodic and approaching a limit cycle. It is found that the integrated heat release rate and the pressure fluctuations are partially in phase, which is responsible for the growth of high-frequency acoustic waves at both fuel and oxidiser sides. On the other hand, the significant density gradient contributes to the growth of low-frequency acoustic waves at the oxidiser side due to the high degree of acoustic reflectivity at the density boundary.

Coupled finite-volume method and smoothed-particle hydrodynamics method for numerical simulation of interactions between inviscid shock waves and structures

In this work, a novel coupled finite-volume method (FVM) and a smoothed-particle-hydrodynamics (SPH) method were developed for the simulation of interactions between inviscid shock waves and structures. In this approach, which considers the particles of a meshless method immersed in an FVM grid, the FVM grid cells are classified into either pure or mixed FVM cells, the latter of which contain SPH particles. A finite-element-method shape function is applied to map the variables from the SPH particles to the FVM cells, and the nodal and cell velocities are then obtained. The interaction of the fluid with the structure is computed using moving reflection boundary conditions at cell interfaces with SPH particles. The interactions of the structure with the fluid are computed from the pressure differences around the SPH particles. The processes for computing the coupled FVM–SPH method are described in detail herein. The validity of the presented coupled FVM–SPH method was verified using a theoretical model of a piston, and the numerical results were found to agree well with the theoretical approximations, indicating the accuracy of the proposed coupled method. The results of the method were then compared with the results of an experiment involving a blast-driven steel plate. Good agreement between the experimental and numerical results was obtained, and the maximum difference was 3.44%, demonstrating the effectiveness of the proposed coupled FVM–SPH method when applied to the interaction of a shock wave with a structure.

Assessing Nonlinear Polarization in Electrochemical Cells using AC Impedance Spectroscopy

AC impedance spectroscopy is widely used to evaluate performance limitations in energy storage and conversion devices (e.g., batteries, supercapacitors, and fuel cells). This work shows that integrating the resistive elements in an equivalent circuit as functions of steady-state current enables one to recover overpotentials associated with different processes (e.g., ion migration, charge transfer, and diffusion) in nonlinear electrochemical power supplies. Closed form expressions for diffusion overpotentials are derived using this method for transmissive and reflective boundary conditions and three electrode symmetries (planar, cylindrical, and spherical). Discussion is also extended to macroscopically homogenous porous electrodes which are relevant for most real-world devices. Overall, the approach described herein is a powerful tool to identify rate-limiting steps and guide material/component design.

The Landau and non-cutoff Boltzmann equations in union of cubes

The existence and stability of collisional kinetic equations, especially non-cutoff Boltzmann equation, in a bounded domain with physical boundary condition is a longstanding open problem. This work proves the global stability of the Landau equation and non-cutoff Boltzmann equation in union of cubes with the specular reflection boundary condition when an initial datum is near Maxwellian. Moreover, the solution enjoys exponential large-time decay in the union of cubes. Our method is based on the fact that normal derivatives in cubes are also derivatives along the axis, which allows us to obtain high-order derivative estimates.

The scattering map on collapsing charged spherically symmetric spacetimes

In this paper we generalise our previous results (Alford 2020 Ann. Inst. Henri Poincare 21 2031–92) concerning scattering on the exterior of collapsing dust clouds to the charged case, including in particular the extremal case. We analyse the energy boundedness of solutions φ to the wave equation on the exterior of collapsing spherically symmetric charged matter clouds. We then proceed to define the scattering map on this spacetime, and look at the implications of our boundedness results on this map. More specifically, we first construct a class of spherically symmetric charged collapsing matter cloud exteriors, and then consider solutions to the wave equation with Dirichlet (reflective) boundary conditions on the surface of these clouds. We then show that the energy of φ remains uniformly bounded going forwards or backwards in time, and that the scattering map is bounded going forwards but not backwards. Therefore, the scattering map is not surjective onto the space of finite energy on I+∪H+ . Thus there does not exist a backwards scattering map from finite energy radiation fields on I+∪H+ to finite energy radiation fields on I− for these models. These results will be used to give a treatment of Hawking radiation in a companion paper (Alford 2023 arXiv:2309.03022).

Numerical approximation of kinetic Fokker–Planck equations with specular reflection boundary conditions

This work is devoted to the analysis of a numerical approximation to a general multi-dimensional kinetic Fokker–Planck (FP) equation with reaction and source terms and subject to specular reflection boundary conditions. This numerical approximation is based on splitting the kinetic FP model into a transport equation in space and a FP diffusive model in the velocity coordinates. The former is discretized by a Kurganov-Tadmor finite-volume scheme, while the latter is approximated by a generalized Chang & Cooper finite-volume method. Time integration is performed by a strong stability-preserving Runge-Kutta method where the reaction and source terms are accommodated with a Strang splitting technique and the use of a Magnus integrator. It is proved that the resulting numerical solution method is conservative and positive preserving, in the case where the continuous model has these properties, and it is second-order accurate in time and in phase space in the L1-norm, subject to a CFL condition. Results of numerical experiments are reported that validate these theoretical results.

Neutronic examination of the U–Mo accident tolerant fuel for VVER-1200 reactors

In this study, we investigated the possibility of employing accident tolerant fuel (ATF) in VVER-1200/V491 assembly without gadolinium-containing fuel rods using the Monte Carlo code Serpent 1.1.7 with ENDF/B-VII cross-section library. The analysis involves assembly design with reflective boundary conditions. To compare the neutronic performances, U–5Mo, U-7.5Mo, U–10Mo, and U–15Mo fuels were chosen in addition to ordinary UO2 fuel. The concentration of 135Xe, 149Sm, fissile and fertile isotopes with burnup, reactivity feedback with fuel temperature variation, and βeff values were calculated. The results indicate that the fuel cycle length increases by 54.27% for U–5Mo, 32.6% for U-7.5Mo, and 13.8% for U–10Mo, while it decreases by 16.4% for U–15Mo fuel. Additionally, the effect of 95Mo content in natural Mo was investigated by reducing the 95Mo concentration. According to the results, each proposed fuel's fuel cycle length extended when the depletion ratio of 95Mo increased. Additionally, the calculations for reactivity feedback guarantee safe operating conditions for all U-xMo fuels.

The Fokker–Planck–Boltzmann equation in the finite channel

In this paper, we establish the existence of small-amplitude unique solutions near the Maxwellian for the Fokker–Planck–Boltzmann equation in a finite channel with specular reflection boundary conditions. The solution space we consider is denoted as Lk̄1LT∞Lx1,v2, introduced in Duan et al. [Commun. Pure Appl. Math. 74(5), 932–1020 (2021)]. In addition, we investigate the long-time behavior of solutions for both hard and soft potentials.

PWR core calculation based on pin-cell homogenization in three-dimensional pin-by-pin geometry

For the pressurized water reactor two-step calculation, the traditional assembly homogenization and two-group neutron diffusion calculation have been widely used. When it comes to the core pin-by-pin simulation, many models and techniques are different and unsettled. In this paper, the homogenization methods based on the pin discontinuity factors and super homogenization factors are used to get the pin-cell homogenized parameters. The heterogeneous leakage model is applied to modify the infinite flux spectrum of the single assembly with reflective boundary condition and to determine the diffusion coefficients for the SP3 solver which is used in the core simulation. To reduce the environment effect of the single-assembly reflective boundary condition, the on-line method for the SPH factors updating is applied in this paper, and the functionalization of SPH factors based on the least-squares method will be pre-made alone with the table of the group constants. The fitting function will be used to update the thermal-group SPH factors with a whole-core pin-by-pin homogeneous solution on-line. The three-dimensional Watts Bar Nuclear Unit 1 (WBN1) problem was utilized to test the performance of pin-by-pin calculation. And numerical results have demonstrated that PWR pin-by-pin core calculation has more accurate results compared with the traditional assembly-homogenization scheme.

Reflective Boundary Conditions Coupled With the SPH Method for the Three-Dimensional Simulation of Fluid-Structure Interaction With Solid Boundaries

Reflective Boundary Conditions Coupled With the SPH Method for the Three-Dimensional Simulation of Fluid-Structure Interaction With Solid Boundaries

The Slab Critical Thickness Problem with Reflecting Boundary Condition for the Anlı-Güngör Scattering Function

The criticality equation, which defines the relation between the secondary neutron number and the thickness of the slab, and the numerical solutions of this equation are investigated with reflecting boundary condition for the recently studied the Anlı-Güngör (AG) scattering function. The analytical calculations are performed by HN method. The numerical results are calculated with Wolfram Mathematica software for the varying secondary neutron number, the varying scattering parameter, and the varying reflection coefficient. The critical slab thickness values decrease for increasing reflection coefficient as expected.

A probabilistic algorithm for optimising the steady-state diffusional flux into a partially absorbing body

Cells in an aqueous environment absorb diffusing nutrient molecules through nanoscale protein channels in their outer membranes. Assuming that there are constraints on the number of such channels a cell can produce, we ask the question: given a nondepleting source of nutrients, what is the optimal distribution of these channels over the cell surface? We coarse-grain this problem, phrasing it as a diffusion problem with position-dependent Robin boundary conditions on the surface. The aim is to maximize the steady-state total flux through the partially absorbing surface under an integral constraint on the local reactivities. We develop an algorithm to tackle this problem that uses the stored and processed results of a particle-based simulation with reflective boundary conditions to a posteriori estimate absorption flux at essentially negligible additional computational cost. We validate the algorithm against a few cases for which analytical or semi-analytical results are available. We apply it to two examples: a spherical cell in the presence of a point source and a spheroidal cell with an isotropic source at infinity. In the former case, there is a significant gain relative to the homogeneous case, while in the latter case the gain is only 1%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\%$$\\end{document}.

Fluctuation relation in continuous-time random walks driven by an external field

We study a fluctuation relation representing a non-equilibrium equality indicating that the ratio between the distribution of trajectories obtained by exchanging the initial and final positions is characterized by free energy differences for the duration of the trajectories. We examine the fluctuation relation for noninteracting charge carriers driven by an external electric field by using a continuous-time lattice random walk model with a general waiting-time distribution of transitions. The fluctuation relation is obtained regardless of the lattice structure factor or the form of the waiting-time distribution. However, the fluctuation relation is satisfied only after taking the continuum limit in the presence of a reflecting boundary. Moreover, in free space without boundary conditions, exchanging the initial and final positions is equivalent to exchanging the field (or drift) directions. However, we show that the exchanging field (or drift) directions is not relevant for studying the fluctuation relation under the reflecting boundary condition.

The Boltzmann Equation with a Class of Large-Amplitude Initial Data and Specular Reflection Boundary Condition

The Boltzmann Equation with a Class of Large-Amplitude Initial Data and Specular Reflection Boundary Condition

A First collision source method using spherical harmonics method for arbitrary shape and order finite element mesh

The discrete ordinates (SN) method is a highly effective and accurate angular discretization method for the neutron transport equation, but suffers from the inherent ray effect in low-scattering regions. The first collision source (FCS) method employing the methods without ray effect to obtain the uncollided flux is a common ray effect mitigation technique. In this paper, a novel FCS method using the spherical harmonics (PN) method to calculate the uncollided flux is proposed. The PN for the first collision source (PN-FCS) method possesses distinctive advantages in several aspects: (1) it can be applied in the problem containing reflective boundary conditions that are difficult to handle with traditional FCS method, (2) it uses the discontinuous Galerkin method to perform spatial discretization, and thus can be easily applied in arbitrary shape and order finite element meshes, (3) it can be effectively calculated in parallel on distributed meshes, (4) the accuracy and computational cost can be flexibly adjusted. Besides, the filter technique is adopted to improve the performance of the PN method in stream-dominated regions. The PN-FCS method has been implemented in two- and three-dimensional neutron transport code and tested by various testing problems. Numerical results show that the PN-FCS method can effectively mitigate the ray effect.

Overview of Method of ChAracteristics based Neutron TRAnsport Code (MANTRA) and its validation

An in-house computer code MANTRA is developed to solve the neutron transport equation at lattice or assembly level (in 2D) applying Method of Characteristics (MOC) for reactor design calculation. The code has the capability to divide the lattice into triangular meshes. By default, the code assumes constant neutron source inside the meshes. If required, it is possible to use linear source assumption to reduce the number of meshes for speeding up the solution. Using MOC, the solution of transport equation is obtained along a number of parallel lines or rays traced through the meshes in different directions and connect them through reflective or periodic boundary condition. In the code, there is option to choose either Power or Krylov subspace iteration technique to converge the solution with desired accuracy. The code is coupled with multigroup cross section library WIMSD. Performance of MANTRA is tested against a number of benchmark problems and overall good agreement is observed with the results obtained from internationally recognized codes like WIMSD, DRAGON, MCNP etc.

Hölder regularity of the Boltzmann equation past an obstacle

AbstractRegularity and singularity of the solutions according to the shape of domains is a challenging research theme in the Boltzmann theory. In this paper, we prove an Hölder regularity in for the Boltzmann equation of the hard‐sphere molecule, which undergoes the elastic reflection in the intermolecular collision and the contact with the boundary of a convex obstacle. In particular, this Hölder regularity result is a stark contrast to the case of other physical boundary conditions (such as the diffuse reflection boundary condition and in‐flow boundary condition), for which the solutions of the Boltzmann equation develop discontinuity in a codimension 1 subset (Kim [Comm. Math. Phys. 308 (2011)]), and therefore the best possible regularity is BV, which has been proved by Guo et al. [Arch. Rational Mech. Anal. 220 (2016)].

Reflective conditions for radiative transfer in integral form with H-matrices

In a recent article the authors showed that the radiative Transfer equations with multiple frequencies and scattering can be formulated as a nonlinear integral system. In the present article, the formulation is extended to handle reflective boundary conditions. The fixed point method to solve the system is shown to be monotone. The discretization is done with a P1 Finite Element Method. The convolution integrals are precomputed at every vertex of the mesh and stored in compressed hierarchical matrices, using Partially Pivoted Adaptive Cross-Approximation. Then the fixed point iterations involve only matrix vector products. The method is O(NN3ln⁡N), with respect to the number of vertices, when everything is smooth. A numerical implementation is proposed and tested on two examples. As there are some analogies with ray tracing the programming is complex.