Proposed Boundary Treatment Research Articles

Conserved method for specified heat flux boundary in UGKS simulation of microscale gas flow

The unified gas kinetic scheme (UGKS) is a powerful method to simulate microscale gas flows. However, due to the lack of corresponding specified heat flux boundary (SHFB) treatment, there is great difficulty for UGKS to simulate problem with SHFB. In this paper, a SHFB treatment for UGKS is proposed based on gas kinetic theory. In the proposed boundary treatment, the heat flux term is directly added into the gas kinetic conservation equation of heat flux on boundary. In this conservation equation, the heat flux transferring from gas to the boundary is calculated using an extrapolated distribution function of inside gas, and the heat flux transferring form boundary to the gas is calculated using an equilibrium rebounding distribution function. Using the conservation equation of mass on boundary to close conservation equations, the gas density and temperature terms for rebounding gas distribution function can be calculated simultaneously. Distribution function on SHFB is reconstructed using rebounding gas distribution function and extrapolated distribution function of inside gas. Fluxes of gas distribution function and macro variables on SHFB is calculated with the reconstructed distribution function. Typical benchmark cases including steady and unsteady Fourier flow, steady Couette flow, pressure-driven flow, cavity flow and supersonic flow around square cylinder are tested to validate the proposed boundary treatment. The validations prove the accuracy, reliability and universality of the proposed method.

Boundary treatment of high order Runge-Kutta methods for hyperbolic conservation laws

In [4], we developed a boundary treatment method for implicit-explicit (IMEX) Runge-Kutta (RK) methods for solving hyperbolic systems with source terms. Since IMEX RK methods include explicit ones as special cases, this boundary treatment method naturally applies to explicit methods as well. In this paper, we examine this boundary treatment method for the case of explicit RK schemes of arbitrary order applied to hyperbolic conservation laws. We show that the method not only preserves the accuracy of explicit RK schemes but also possesses good stability. This compares favorably to the inverse Lax-Wendroff method in [5,6] where analysis and numerical experiments have previously verified the presence of order reduction [5,6]. In addition, we demonstrate that our method performs well for strong-stability-preserving (SSP) RK schemes involving negative coefficients and downwind spatial discretizations. It is numerically shown that when boundary conditions are present and the proposed boundary treatment is used, that SSP RK schemes with negative coefficients still allow for larger time steps than schemes with all nonnegative coefficients. In this regard, our boundary treatment method is an effective supplement to SSP RK schemes with/without negative coefficients for initial-boundary value problems for hyperbolic conservation laws.

Pore-scale simulation of gas flow in microscopic permeable media with complex geometries

Lattice Boltzmann method (LBM) is an efficient tool to perform direct numerical simulation of gas flow in micro-size pores. Despite great advances in LBM and its use in describing gas flow in micro-size channels in slip and transitional flow regimes, exploring formulations that are able to capture flow behavior in domains with complex boundary geometries remains a challenging task. As a result, the impact of complexities in pore structures on gas flow may not be fully explored using LBM. In this study, we propose to use the combined bounce-back and Maxwellian diffusive reflection scheme to capture the slip velocity with a new set of slip coefficients. Optimal values of slip coefficients are determined using a gradient based method, where the results from the linearized Boltzmann equation is used as the reference solution. The proposed formulation is further validated against predictions from molecular dynamics simulation in the presence of complex geometries by introducing obstacles of different shapes in straight channels. LBM with the proposed boundary treatment is then used to investigate gas flow in a synthetic microscopic permeable medium. Results indicate that gas exhibits different flow configurations as Knudsen number varies, and apparent gas permeability appears to have up to a second-order dependency on the reciprocal mean pressure in slip and transitional flow regimes.

Improved treatment of wall boundary conditions for a particle method with consistent spatial discretization

In recent years, consistent spatial discretization schemes for meshfree particle methods to numerically simulate incompressible flow have been studied by many researchers. This study is focused on the treatment of solid wall boundary conditions for one of those schemes, namely the least squares MPS (LSMPS) scheme, and proposes a new technique to deal with no-slip and free-slip wall boundary conditions. With the proposed treatment, wall geometries are expressed by surface meshes, i.e. polygons in 3D and line segments in 2D. Thus, complicated geometries can be handled easily. Based on a Taylor series expansion, wall boundary conditions are incorporated into the differential operators acting on fluid particles located in the vicinity of a wall, through a least squares approach. As a consequence, Neumann boundary conditions can be treated quite efficiently. To verify consistency of the proposed discretization scheme, a convergence study was carried out. As numerical examples, Couette flow, plane Poiseuille flow, gravity-driven flow in a 3D square duct, a rigid rotation problem, Taylor–Green vortices and lid-driven cavity flow have been calculated using the proposed boundary treatment, with both no-slip and free-slip conditions applied. As a result, the present method agreed well with the reference solutions, which verified its computational accuracy.

A Meshless WCSPH Boundary Treatment for Open-Channel Flow over Small-Scale Rough Bed

A weakly compressible smoothed particle hydrodynamics (WCSPH) method was developed to model open-channel flow over rough bed. An improved boundary treatment is proposed to quantitatively characterize bed roughness based on the ghost boundary particles (GBPs). In this model, the velocities of GBPs are explicitly calculated by using evolutionary polynomial regression with a multiobjective genetic algorithm. The simulation results show that the proposed boundary treatment can well reflect the influence of wall roughness on the vertical flow structure. A fully developed open channel is established, and its flume length, processing time, and turbulent model are discussed. The mixed-length-based subparticle scale (SPS) turbulence model is adopted to simulate uniform flow in open channel, and this model is compared with the Smagorinsky-based one. For the modified WCSPH model, the results show that the calculated vertical velocity and turbulent shear stress distribution are in good agreement with experimental data and fit better than the calculations obtained from the traditional Smagorinsky-based model.

Relay-Zone Technique for Numerical Boundary Treatments in Simulating Dark Solitons

To simulate dark solitons in the defocusing nonlinear Schrodinger equation, we introduce a relay-zone technique, by alternately using a Robin boundary condition to treat the nonzero far field, and a derivative boundary condition to match the dark soliton. Numerical tests and comparisons demonstrate the effectiveness of the proposed boundary treatment. Stability and interaction of dark solitons are also studied.

General curved boundary treatment for two- and three-dimensional stationary and moving walls in flow and nonflow lattice Boltzmann simulations.

The aim of this study is to introduce a general approach to implement curved boundaries in lattice Boltzmann simulations. The main idea is to determine boundary values by extrapolating macroscopic properties from some reference points inside the computational domain. The introduced approach is based on a unified extrapolation equation that can be employed for any macroscopic value (flow and nonflow properties) in arbitrary two- and three-dimensional geometries. In the case of nonflow simulations, the present treatment can easily apply Dirichlet and Neumann boundary conditions. By introducing a point cloud description of geometry, the new treatment can handle any complex geometry that is modeled by a CAD program. The application of the new treatment is also extended to moving boundaries, by developing a novel force calculation method. The proposed boundary treatment is tested against several well-established problems and the order of accuracy of solutions is evaluated. Numerical results show that the present treatment is of second-order accuracy with respect to the grid spacing in flow simulations, and it leads to a significant enhancement in nonflow simulations as well.

On a third order CWENO boundary treatment with application to networks of hyperbolic conservation laws

High order numerical methods for networks of hyperbolic conservation laws have recently gained increasing popularity. Here, the crucial part is to treat the boundaries of the single (one-dimensional) computational domains in such a way that the desired convergence rate is achieved in the smooth case but also stability criterions are fulfilled, in particular in the presence of discontinuities. Most of the recently proposed methods rely on a WENO extrapolation technique introduced by Tan and Shu (2010). Within this work, we refine and in a sense generalize these results for the case of a third order scheme. Numerical evidence for the analytically found parameter bounds is given as well as results for a complete third order scheme based on the proposed boundary treatment.

A Hybrid Monte Carlo and Finite Difference Method for Option Pricing

We propose an accurate, efficient, and robust hybrid finite difference method, with a Monte Carlo boundary condition, for solving the Black–Scholes equations. The proposed method uses a far-field boundary value obtained from a Monte Carlo simulation, and can be applied to problems with non-linear payoffs at the boundary location. Numerical tests on power, powered, and two-asset European call option pricing problems are presented. Through these numerical simulations, we show that the proposed boundary treatment yields better accuracy and robustness than the most commonly used linear boundary condition. Furthermore, the proposed hybrid method is general, which means it can be applied to other types of option pricing problems. In particular, the proposed Monte Carlo boundary condition algorithm can be implemented easily in the code of the existing finite difference method, with a small modification.

A new curved boundary treatment in lattice Boltzmann method for micro gas flow in the slip regime

A new curved boundary treatment in lattice Boltzmann method is developed for micro gas flow in the slip regime. The proposed treatment is a combination of the nonequilibrium extrapolation scheme for curved boundary with no-slip velocity condition and the counter-extrapolation method for the velocity and its normal gradient on the curved boundary. Taking into consideration the effect of the offset between the physical boundary and the closest grid line, the new treatment is proved to be more accurate than the traditional half-way diffusive bounce-back (DBB) scheme. The present treatment is also more applicable than the modified DBB scheme because the specific gas-wall interaction parameters need to be determined to ensure the validation of the modified DBB scheme.The proposed boundary treatment is implemented to simulate the benchmark problems, which include a Poiseuille flow in the aligned/inclined micro-channel, a flow past a microcylinder and a microcylindrical Couette flow. The results and conclusions are summarized as follows.1) The force-driven Poiseuille flow in an aligned microchannel is simulated separately with different values of wall-grid offset qδx (q=0.25, 0.5, 0.75, 1.0). With the consideration of the wall-grid offset, the numerical results with the new boundary treatment show good agreement with the analytical solutions. However, the results obtained by using the half-way DBB scheme only accord well with the analytical solutions under the condition of a fixed wall-grid offset (q=0.5).2) To demonstrate the capability of the present treatment in dealing with gas flow in a more complex geometry, the force-driven Poiseuille flow in a micro-channel is investigated separately with different inclined angles. The present numerical results fit well with the analytical solutions. However, the discrepancy between the results obtained with the half-way DBB scheme and the analytical solutions can be clearly observed near the inclined boundaries.3) The gas flow past a microcylinder is simulated to prove that the present treatment can deal with the curved boundary. The slip velocity profile along the micro cylinder periphery obtained with the present treatment accords well with the available data in the published literature. However, the results obtained with the half-way DBB scheme show lower values than the results from the published work.4) In the simulations of the microcylindrical Couette flow between two coaxial rotating cylinders for different Knudsen numbers the results obtained by using the present treatment show excellent agreement with the analytical solutions. However, the results obtained with the half-way DBB scheme and the modified DBB scheme deviate obviously from the analytical solutions near the inner and outer cylindrical walls, respectively.In summary, the new boundary treatment proposed in this work is capable of dealing with the complex gas-solid boundary in the slip regime. The new treatment has a higher accuracy than the half-way DBB scheme and shows a better applicability than the modified DBB scheme.

Exact Boundary Condition for Semi-discretized Schrödinger Equation and Heat Equation in a Rectangular Domain

A convolution type exact/transparent boundary condition is proposed for simulating a semi-discretized linear Schrodinger equation on a rectangular computational domain. We calculate the kernel functions for a single source problem, and subsequently those over the rectangular domain. Approximate kernel functions are pre-computed numerically from discrete convolutionary equations. With a Crank---Nicolson scheme for time integration, the resulting approximate boundary conditions effectively suppress boundary reflections, and resolve the corner effect. The proposed boundary treatment, with a parameter modified, applies readily to a semi-discretized heat equation.

Accurate boundary treatment for transient Schrödinger equation under polar coordinates

A new local boundary condition is designed for the two dimensional Schrödinger equation under polar coordinates. Based on an approximate linear relation among the kernel functions for a free one-dimensional Schrödinger equation of a new variable, it takes a simple form of ordinary differential equation that relates the neighboring grid points. Numerical tests and comparisons demonstrate the effectiveness of the proposed boundary treatment.

Robust absorbing boundary conditions for shallow water flow models

This paper investigates the cause of the spurious waves generated at the domain boundaries and proposes robust absorbing conditions in 2D shallow water simulations. The spurious waves are caused by the missing information when imposing boundary conditions for subcritical flows over irregular topography adjacent to the boundaries. The resulting boundary disturbances are likely to affect the flow solution inside the domain. To prevent the generation of such boundary spurious waves, an approach is proposed by extending the computational domain in the normal direction by a small number of cells. The number of extended cells is properly determined through numerical experiments by taking into account the effects of wave height, water depth, boundary topographic features (i.e., bump height, bump slope and bump type) and grid resolution. Five extended cells are found to be adequate in eliminating the spurious waves and maintaining stable and accurate numerical solutions. The robustness of the proposed boundary treatment is tested and confirmed through applications to reproduce two field-scale tsunami events.

Entropy stable wall boundary conditions for the three-dimensional compressible Navier–Stokes equations

Non-linear entropy stability and a summation-by-parts framework are used to derive entropy stable wall boundary conditions for the three-dimensional compressible Navier–Stokes equations. A semi-discrete entropy estimate for the entire domain is achieved when the new boundary conditions are coupled with an entropy stable discrete interior operator. The data at the boundary are weakly imposed using a penalty flux approach and a simultaneous-approximation-term penalty technique. Although discontinuous spectral collocation operators on unstructured grids are used herein for the purpose of demonstrating their robustness and efficacy, the new boundary conditions are compatible with any diagonal norm summation-by-parts spatial operator, including finite element, finite difference, finite volume, discontinuous Galerkin, and flux reconstruction/correction procedure via reconstruction schemes. The proposed boundary treatment is tested for three-dimensional subsonic and supersonic flows. The numerical computations corroborate the non-linear stability (entropy stability) and accuracy of the boundary conditions.

An improved particle method for simulation of the non-isothermal viscoelastic fluid mold filling process

A non-isothermal molding filling process for a non-Newtonian viscoelastic fluid is investigated in this work. An extended smoothed particle hydrodynamics with diffusive term and kernel gradient correction (extended SPH_DTKGC) method is first presented to simulate the non-isothermal molding filling for the viscoelastic fluid based on the coupled eXtended Pom-Pom (XPP) and thermal power-law model. Meanwhile, a suitable discrete temperature scheme in the SPH frame is proposed. And a new way to treat the zig-zag boundary is given and expected to deal with arbitrary shaped rigid wall. Subsequently, the merits of the proposed boundary treatment are demonstrated by a spin-down problem and the filling process in a C-shaped mould. The ability of the extended SPH_DTKGC to simulate the viscoelastic free surface flow is verified by the several classical problems. Two benchmark problems are presented to display the reliability of the proposed temperature scheme. The non-isothermal filling process for a power-law fluid is performed to show the feasibility of the extended SPH_DTKGC to simulate the non-isothermal mold filling process. Then the extended SPH_DTKGC method combined with other improvements is applied to the non-isothermal filling processes based on the coupled XPP and power-law model. Especially, the influences of the parameters of the XPP model on the physical quantities, such as the first normal stress difference and the temperature field, are numerically predicted.

A unified and preserved Dirichlet boundary treatment for the cell-centered finite volume discrete Boltzmann method

A new boundary treatment is proposed for the finite volume discrete Boltzmann method (FVDBM) that can be used for accurate simulations of curved boundaries and complicated flow conditions. First, a brief review of different boundary treatments for the general Boltzmann simulations is made in order to primarily explain what type of boundary treatment will be developed in this paper for the cell-centered FVDBM. After that, the new boundary treatment along with the cell-centered FVDBM model is developed in detail. Next, the proposed boundary treatment is applied to a series of numerical tests with a detailed discussion of its qualitative and quantitative properties. From the results, it can be concluded that the new boundary treatment is at least first-order accurate for a variety of Dirichlet boundary conditions (BCs). It can handle both the velocity and density Dirichlet BCs in a unified way and further realize some BCs that the conventional lattice Boltzmann model fails to simulate. In addition, such a boundary treatment can incorporate different lattice models without changing its framework, and it can preserve the Dirichlet BCs up to machine accuracy in different situations.

An extrapolation-based boundary treatment for using the lattice Boltzmann method to simulate fluid-particle interaction near a wall

Two important issues need to be addressed when using lattice Boltzmann methods to simulate particulate flows: complex moving boundaries and near contact of two solid objects. This study introduces a new boundary treatment method for using lattice Boltzmann methods to study fluid-particle interaction problems in which solid objects are close to a wall or two solid objects are near contact. The new method does not need to use any additional lubrication correction to total hydrodynamic force, instead implementing the no-slip boundary condition by introducing an appropriate fictitious flow field outside the fluid domain. The new method was verified by simulating a spherical particle falling down in fluids with and without rebound. It is concluded that the proposed boundary treatment method can accurately simulate the fluid-particle interaction near a rigid wall and collision in fluid.

Entropy-Stable Schemes for the Euler Equations with Far-Field and Wall Boundary Conditions

In this paper entropy-stable numerical schemes for the Euler equations in one space dimension subject to far-field and wall boundary conditions are derived. Furthermore, a stable numerical treatment of interfaces between different grid domains is proposed. Numerical computations with second- and fourth-order accurate schemes corroborate the stability and accuracy of the proposed boundary treatment.

A high-order compact local integrated-RBF scheme for steady-state incompressible viscous flows in the primitive variables

This study is concerned with the development of integrated radialbasis- function (IRBF) method for the simulation of two-dimensional steady-state incompressible viscous flows governed by the pressure-velocity formulation on Cartesian grids. Instead of using low-order polynomial interpolants, a high-order compact local IRBF scheme is employed to represent the convection and diffusion terms. Furthermore, an effective boundary treatment for the pressure variable, where Neumann boundary conditions are transformed into Dirichlet ones, is proposed. This transformation is based on global 1D-IRBF approximators using values of the pressure at interior nodes along a grid line and first-order derivative values of the pressure at the two extreme nodes of that grid line. The performance of the proposed scheme is investigated numerically through the solution of several linear (analytic tests including Stokes flows) and non-linear (recirculating cavity flow driven by combined shear & body forces and lid-driven cavity flow) problems. Unlike the global 1D-IRBF scheme, the proposed method leads to a sparse system matrix. Numerical results indicate that (i) the present solutions are more accurate and converge faster with grid refinement in comparison with standard finite-difference results; and (ii) the proposed boundary treatment for the pressure is more effective than conventional direct application of the Neumann boundary condition.

A New Boundary Closure Scheme for the Multiresolution Time-Domain (MRTD) Method

This paper introduces a novel boundary closure treatment for the wavelet based multiresolution time-domain (MRTD) solution of Maxwell's equations. Accommodating non-trivial boundary conditions, such as the Robin condition or time dependent condition, has been a challenging issue in the MRTD analysis of wave scattering, radiation, and propagation. A matched interface and boundary (MIB) method is introduced to overcome this difficulty. Several numerical benchmark tests are carried out to validate the MIB boundary scheme. The proposed boundary treatment can also be applied to other high order finite-difference time-domain (FDTD) approaches, such as the dispersion-relation-preserving (DRP) method. The MIB boundary scheme greatly enhances the feasibility for applying the MRTD methods to more complicated electromagnetic structures.