Boundary Treatment Research Articles

Direct imposition of the wall boundary condition for simulating free surface flows in SPH

In this study, a new method for treating the wall boundary in smoothed particle hydrodynamics (SPH) is proposed to simulate free surface flows effectively. Unlike conventional methods of wall boundary treatment through boundary particles, in the proposed method, the wall boundary condition is directly imposed by adding boundary truncation terms to the mass and momentum conservation equations. Thus, boundary particles are not used in boundary modeling. Doing so, the wall boundary condition is accurately imposed, boundary modeling is simplified, and computation is made efficient without losing stability in SPH. Performance of the proposed method is demonstrated through several numerical examples: dam break, dam break with a wedge, sloshing, inclined bed, cross-lever rotation, pulsating tank and sloshing with a flexible baffle. These results are compared with available experimental results, analytical solutions, and results obtained using the boundary particle method.

STUDY ON EARTHQUAKE DAMAGE MECHANISM OF AQUEDUCT STRUCTURE BASED ON DIFFERENT BOUNDARY

Numerically simulating an infinite domain foundation is an important method for solving structural dynamics problems. This paper introduces several artificial dynamic boundaries commonly used in the study of structural dynamics, and elaborates the theory and methods of the dynamic infinite element method boundary (IEMB) and viscous–spring artificial boundary (VSAB). The capacity of different boundary effects on seismic waves energy absorption is verified by establishing a layered half-space model. An irrigation aqueduct is taken as a research object. The IEMB, VSAB, and fixed boundary (FB) models are established and the Concrete Damaged Plasticity (CDP) constitutive is introduced, which is aimed at studying the dynamic failure mechanism and the rules of damage development to the aqueduct structure during the seismic duration. The results for the IEMB and VSAB show better energy absorption for the incident waves and a better simulation result for the damping effect of the far field foundation than that of the FB. Comparing the maximum displacement response rules of the three boundaries, it is seen that the maximum displacement response values of the VSAB and dynamic IEMB increased by 6%–48% and 9%–35%, respectively, over the FB. The calculation results of the VSAB are similar to that of the IEMB. The difference between the maximum acceleration response values is 2%–17% whereas the difference between the maximum displacement response values is 0.4%–19%. The IEMB studied in this paper provides a theoretical reference for large–scale building boundary treatment in structural dynamics calculations.

An integrated smoothed particle hydrodynamics method for numerical simulation of the droplet impacting with heat transfer

According to the complex flow dynamics of droplets impacting on the wall, an integrated smoothed particle hydrodynamics (SPH) method is developed by a series of improved techniques including artificial viscosity, kernel gradient correction (KGC) technique, boundary treatment and so on, combined with the surface tension based on the continuous surface force (CSF) model in this paper. This method is validated by simulations of the square cavity flow and natural variations of a square droplet, and effects of particle diameter, kernel function, and artificial viscosity coefficient are also analyzed. On this basis, dynamic behavior of the droplet impacting on the wall with heat transfer is studied, and the results agree well with the experimental results. It is observed that the wall characteristics have a significant influence on the droplet deformation, and the droplet tends to spread on the hydrophilic wall. In the investigated temperature difference, the influence of temperature difference could be ignored, but there is energy exchange between the droplet and the wall. In general, the spreading law of droplet is consistent with the physical phenomena, which shows that the integrated SPH method can effectively deal with engineering problems involving a droplet impacting on the wall with heat transfer.

Seven-velocity three-dimensional vectorial lattice Boltzmann method including various types of approximations to the pressure and two-parameterized second-order boundary treatments

In this paper we present a seven-velocity three-dimensional (D3N7) vectorial lattice Boltzmann method (LBM) including various types of approximations to the pressure and propose a family of two-parameterized second-order boundary schemes with accuracy independent of the boundary location. In order to show the numerical stability of the D3N7 model, we construct a symmetrizer to handle the nonlinear approximations to the pressure. In the meantime, we relate the stability based on the vectorial model to that based on the conventional scalar model through an orthogonal similarity transformation. Finally, two 3-D examples with straight and curved boundaries numerically validate the D3N7 model, including linear and nonlinear approximations to the pressure, together with the proposed boundary schemes.

Large eddy simulation of Cambridge bluff-body coal (CCB2) flames with a flamelet progress variable model

In the present study, we report the first large eddy simulation (LES) study of the Cambridge CCB2 coal flames, one of the target flames in the Workshop on Measurement and Simulation of Coal and Biomass Conversion, with an extended flamelet progress variable (FPV) model. The extended FPV model is based on two mixture fractions considering the volatiles and char off-gases. The normalized total enthalpy is used for the interphase heat transfer modelling. Turbulence-chemistry interaction is treated with an assumed probability density function approach. The results show that the present LES can generally capture the flow field and particle distribution, while there are considerable deviations in the OH prediction due to the boundary treatment of using a mixture of volatiles and carrier gas to replace the methane-containing mixtures in the primary and pilot flow. It indicates that for such gas-assisted coal flames, the pilot fuel stream needs to be rigorously considered in the flamelet tabulation that could be resolved by extending the two-mixture-fraction model into a three-mixture-fraction model. The instantaneous Lagrangian particles histories show that the increasing of coal load has a negligible effect on devolatilization, but delays the char conversion.

An inverse Lax-Wendroff procedure for hyperbolic conservation laws with changing wind direction on the boundary

In this paper, we reconsider the inverse Lax-Wendroff (ILW) procedure, which is a numerical boundary treatment for solving hyperbolic conservation laws, and propose a new approach to evaluate the values on the ghost points. The ILW procedure was firstly proposed to deal with the “cut cell” problems, when the physical boundary intersects with the Cartesian mesh in an arbitrary fashion. The key idea of the ILW procedure is repeatedly utilizing the partial differential equations (PDEs) and inflow boundary conditions to obtain the normal derivatives of each order on the boundary. A simplified ILW procedure was proposed in [28] and used the ILW procedure for the evaluation of the first order normal derivatives only. The main difference between the simplified ILW procedure and the proposed ILW procedure here is that we define the unknown u and the flux f(u) on the ghost points separately. One advantage of this treatment is that it allows the eigenvalues of the Jacobian f′(u) to be close to zero on the boundary, which may appear in many physical problems. We also propose a new weighted essentially non-oscillatory (WENO) type extrapolation at the outflow boundaries, whose idea comes from the multi-resolution WENO schemes in [32]. The WENO type extrapolation maintains high order accuracy if the solution is smooth near the boundary and it becomes a low order extrapolation automatically if a shock is close to the boundary. This WENO type extrapolation preserves the property of self-similarity, thus it is more preferable in computing the hyperbolic conservation laws. We provide extensive numerical examples to demonstrate that our method is stable, high order accurate and has good performance for various problems with different kinds of boundary conditions including the solid wall boundary condition, when the physical boundary is not aligned with the grids.

A novel fluid-structure interaction algorithm for compressible flows and deformable structures

In this study, we propose a new immersed boundary method for compressible fluid-structure interaction simulations. We take a partitioned coupling strategy that comprises three parts: fluid solver, structure solver, and fluid-structure boundary treatment. The fluid solver is implemented by a second-order finite volume scheme, and the structure is solved by an implicit finite element solver. For the boundary treatment, this method employs a multilevel marking algorithm to determine the direction from the inside of the structure to the outside. Following the direction, fluids that permeate the structure are transported out of the structure boundaries to satisfy the boundary conditions. This method strictly maintains mass conservation and uses no cross-boundary fitting or interpolation; thus, it appears to be convenient when treating complex-shaped or deformable structures and diminishes the singularity problem. The 1D and 2D numerical experiments show that our method is capable of obtaining approximately first-order convergent results, agreeing well with the published data. The method's limitations are discussed in the conclusion.

Peridynamics boundary condition treatments via the pseudo-layer enrichment method and variable horizon approach

Peridynamics is a nonlocal theory that applies an integral term to represent the material response. Without a spatial differential term involved, peridynamics possesses certain advantages for solving discontinuity-involved problems. However, due to the reduction of stiffness, the deformation near the boundary region by peridynamics has a low accuracy compared to the local elastic deformation. Previous peridynamics boundary condition treatment enriches the stiffness on the boundary region by adding an artificial layer outside the domain boundary. However, this requires the deformation of the pseudo-layer to be pre-determined. In addition, the accuracy of the response on the physical domain depends largely on the accuracy of the pre-determined deformation on the pseudo-layer. Considering the fact that peridynamics reduces to local elasticity as the horizon size goes to zero, and previous researches indicate the potential advantage of boundary enrichment via the horizon varying approach, in this paper, peridynamics with a variable horizon is utilized as an efficient way to reduce the boundary-induced inaccuracy. Firstly, the pseudo-layer boundary condition treatments are discussed by using the symmetrical enrichment function and other extrapolation functions. Then, the variable horizon boundary condition treatment is introduced and a robust improvement of deformation accuracy on the boundary region is observed compared to other boundary condition treatments for both one- and two-dimensional examples. The variable horizon approach requires no additional pseudo-layer and the boundary conditions are applied directly on the physical boundary. Thus, the variable horizon approach is easy to implement and its computational cost is reduced.

Periodic travelling wave solutions for a reaction-diffusion system on landscape fitted domains

In this article, we propose a new landscape fitted domain construction and its boundary treatment of periodic travelling wave solutions for a diffusive predator-prey system with landscape features. The proposed method uses the distance function based on an obstacle. The landscape fitted domain is defined as a region whose distance from the obstacle is positive and less than a pre-defined distance. At the exterior boundary of the domain, we use the zero-Neumann boundary condition and define the boundary value from the bilinearly interpolated value in the normal direction of the distance function. At the interior boundary, we use the homogeneous Dirichlet boundary condition. Typically, reaction-diffusion systems are numerically solved on rectangular domains. However, in the case of periodic travelling wave solutions, the boundary treatment is critical because it may result in unexpected chaotic pattern. To avoid this unwanted chaotic behavior, we need to use sufficiently large computational domain to minimize the boundary treatment effect. Using the proposed method, we can get accurate results even though we use relatively small domain sizes.

Elastic wave propagation in anisotropic solids using energy-stable finite differences with weakly enforced boundary and interface conditions

Summation-by-parts (SBP) finite difference methods have several desirable properties for second-order wave equations. They combine the computational efficiency of narrow-stencil finite difference operators with provable stability on curvilinear multiblock grids. While several techniques for boundary and interface conditions exist, weak imposition via simultaneous approximation terms (SATs) is perhaps the most flexible one. Although SBP methods have been applied to elastic wave equations many times, an SBP-SAT method for general anisotropic elastic wave equations has not yet been presented in the literature. We fill this gap by deriving energy-stable self-adjoint SBP-SAT methods for general anisotropic materials on curvilinear multiblock grids. The methods are based on fully compatible SBP operators. Although this paper focuses on classical SBP finite difference operators, the presented boundary and interface treatments are general and apply to a range of methods that satisfy an SBP property. We demonstrate the stability and accuracy properties of a particular set of fully compatible SBP-SAT schemes using the method of manufactured solutions. We also demonstrate the utility of the new method in elastodynamic cloaking and seismic imaging in mountainous regions.

Boundary treatment of implicit-explicit Runge-Kutta method for hyperbolic systems with source terms

In this paper, we develop a high order finite difference boundary treatment method for the implicit-explicit (IMEX) Runge-Kutta (RK) schemes solving hyperbolic systems with possibly stiff source terms on a Cartesian mesh. The main challenge is how to obtain the solutions at ghost points resulting from the wide stencil of the interior high order scheme. We address this problem by combining the idea of using the RK schemes at the boundary and an inverse Lax-Wendroff (ILW) procedure in [29,30]. Specifically, we only apply the ILW procedure in the starting stage of the RK method. In the intermediate stages, we solve out the intermediate solutions as well as their first-order spatial derivatives at the boundary by using the RK schemes, which are then used to compute solutions at ghost points by Taylor expansions. Our method is different from the widely used approach for the explicit RK schemes by imposing boundary conditions at intermediate stages [14,29,30] which does not apply to IMEX schemes. In addition, the intermediate boundary conditions are available for explicit RK schemes only up to third order while our method applies to IMEX and explicit RK schemes of arbitrary order. The stability and the desired accuracy of our boundary treatment with IMEX RK method up to fourth order are demonstrated numerically through 1D examples and 2D reactive Euler equations.

Steady-state two-relaxation-time lattice Boltzmann formulation for transport and flow, closed with the compact multi-reflection boundary and interface-conjugate schemes

We introduce the steady-state two-relaxation-time (TRT) Lattice Boltzmann method. Owing to the symmetry argument, the bulk system and the closure equations are all expressed in terms of the equilibrium and non-equilibrium unknowns with the half discrete velocity set. The local mass-conservation solvability condition is adjusted to match the stationary, but also the quasi-stationary, solutions of the standard TRT solver. Additionally, the developed compact, boundary and interface-conjugate, multi-reflection (MR) concept preserves the efficient directional bulk structure and shares its parametrization properties. The method is exemplified in grid-inclined stratified slabs for two-phase Stokes flow and the linear advection-diffusion equation featuring the discontinuous coefficients and sources. The piece-wise parabolic benchmark solutions are matched exactly with the novel Dirichlet, pressure-stress, Neumann flux and Robin MR schemes. The popular, anti-bounce-back and shape-fitted Dirichlet continuity schemes are improved in the presence of both interface-parallel and perpendicular advection velocity fields. The steady-state method brings numerous advantages: it skips transient numerical instability, overpasses severe von Neumann parameter range limitations, tolerates high physical contrasts and arbitrary MR coefficients. The method is promising for faster computation of Stokes/Brinkman/Darcy linear flows in heterogeneous soil, but also heat and mass transfer problems governed by an accurate boundary and interface treatment.

Multiphase smoothed particle hydrodynamics modeling of forced liquid sloshing

AbstractIn the present paper, an improved multiphase weakly compressible smoothed particle hydrodynamics model for balancing the accuracy and stability of the long‐term simulations is proposed to model the forced liquid sloshing in a tank. The governing equations of the multiphase flow are discretized by considering the density discontinuity over the interface. To suppress the pressure oscillation, a previous density correction term suitable only for single‐phase problems is modified and incorporated into the discrete continuity equation to suit multiphase problems. The modified density reinitialization algorithm is implemented to calculate the pressure of the boundary particles, and the coupled dynamic solid boundary treatment (SBT) is employed to determine the rigid wall condition. For convenience, a numerical probe algorithm is also proposed to accurately measure the wave height. The present model exhibits a better numerical stability than the previous multiphase smoothed particle hydrodynamics model, and its results well confirm with the experimental data of the forced sloshing of liquid excited by swaying or rolling.

A curved boundary treatment for discontinuous Galerkin method applied to Euler equations on triangular and tetrahedral grids

For Euler equations defined in a domain Ω with boundary conditions specified on a curved boundary Γ, we consider a discontinuous Galerkin (DG) method with high order polynomial basis functions on a geometry fitting triangular and tetrahedral grids. We derive a modification to the DG scheme defined on a boundary triangle and tetrahedron based on the general curvilinear element DG method. In the modified DG scheme, even though integration along the curved boundary is still necessary, integrals over any curvilinear element are avoided. To further simplify the calculation of integration along the curved boundary, the modified DG scheme is constructed by the relationship between normal vector of geometric boundary and surface Jacobian. In the new modified DG scheme, not only integrals over any curvilinear element are avoided, but also the line and face integrations along the curved boundary are not required. Numerical tests suggest that such a new modified DG scheme is as accurate as the basic modified DG scheme and full curvilinear DG scheme and more efficient than latter, and is stable on reasonably coarse, as well as more refined, grids. It is shown that this new modified DG scheme can be extremely useful to solve the problems defined in a domain with boundary conditions specified on a curved boundary.

IQ-MPM

We propose a novel scheme for simulating two-way coupled interactions between nonlinear elastic solids and incompressible fluids. The key ingredient of this approach is a ghost matrix operator-splitting scheme for strongly coupled nonlinear elastica and incompressible fluids through the weak form of their governing equations. This leads to a stable and efficient method handling large time steps under the CFL limit while using a single monolithic solve for the coupled pressure fields, even in the case with highly nonlinear elastic solids. The use of the Material Point Method (MPM) is essential in the designing of the scheme, it not only preserves discretization consistency with the hybrid Lagrangian-Eulerian fluid solver, but also works naturally with our novel interface quadrature (IQ) discretization for free-slip boundary conditions. While traditional MPM suffers from sticky numerical artifacts, our framework naturally supports discontinuous tangential velocities at the solid-fluid interface. Our IQ discretization results in an easy-to-implement, fully particle-based treatment of the interfacial boundary, avoiding the additional complexities associated with intermediate level set or explicit mesh representations. The efficacy of the proposed scheme is verified by various challenging simulations with fluid-elastica interactions.

Lattice Boltzmann simulation for unsteady shock wave/boundary layer interaction in a shock tube

In this paper, lattice Boltzmann simulation for unsteady shock wave/boundary layer interaction in a shock tube is presented. This kind of problem, which contains the typical characteristics of supersonic viscous flows with shock wave and boundary layer, has rarely been well validated in the lattice Boltzmann community. This leads to few application of lattice Boltzmann method in supersonic flows with boundary layer. Some problems related to numerical dissipation and stability are exposed when simulating shock wave/boundary layer interaction using the conventional compressible lattice Boltzmann approach. The implementations of lattice Boltzmann for this kind of flow are investigated. As a novel problem for the lattice Boltzmann method, three main issues need to be addressed carefully, i.e., lattice Boltzmann model, solving scheme, and boundary treatment. For the first issue, a double-distribution-function compressible lattice Boltzmann model, in which the lattice velocity is studied for the stability in the simulation, is employed since we have demonstrated its feasibility in supersonic viscous flow previously. For the solving scheme, various space discretization methods are tested to enhance numerical performance. Meanwhile, the boundary treatment for the ghost layer is studied. The simulation result by the improved lattice Boltzmann implementation agrees well with the results through Navier–Stokes equations and gas-kinetic scheme reported in previous literatures. This work probes the numerical implementations of lattice Boltzmann method for unsteady shock wave/boundary layer interaction, providing important theoretical support for lattice Boltzmann application in supersonic viscous flows with shock wave and boundary layer.

Direct Numerical Simulations of Turbulent Channel Flow With Polymer Additives

ABSTRACTDirect numerical simulations have been applied to simulate flows with polymer additives. FENE-P (finite-extensible-nonlinear-elastic-Peterlin) dumbbell model solving for the conformation tensor is adopted to investigate the influence of the polymer on the flowfield. Boundary treatments of the conformation tensor on the flowfield are examined first, where boundary condition based on the linear extrapolation scheme provides more accurate results with second-order accurate error norms. Further simulations of the turbulent channel flow at different Weissenberg numbers are also conducted to investigate the influence on drag reduction. Drag reduction increases in tandem with the increase of Weissenberg number and the increase saturates at Weτ~200, where the drag reduction is close to the maximum drag reduction (MDR) limit. At the regime of y+ > 5, the viscous layer thickens with the increase of the Weissenberg number showing a departure from the traditional log-law profile, and the velocity profiles approach the MDR line at high Weissenberg number. The Reynolds stress decreases in tandem with the increase of Weτ, whereas the levels of laminar stress and polymer stress act adversely. However, as the Weissenberg number increases, the proportion of the laminar stress in the total stress increases, and this contributes to the drag reduction of the polymer flow.

Parametrized reduced order modeling for cracked solids

SummaryA parametrized reduced order modeling methodology for cracked two dimensional solids is presented, where the parameters correspond to geometric properties of the crack, such as location and size. The method follows the offline‐online paradigm, where in the offline, training phase, solutions are obtained for a set of parameter values, corresponding to specific crack configurations and a basis for a lower dimensional solution space is created. Then in the online phase, this basis is used to obtain solutions for configurations that do not lie in the training set. The use of the same basis for different crack geometries is rendered possible by defining a reference configuration and employing mesh morphing to map the reference to different target configurations. To enable the application to complex geometries, a mesh morphing technique is introduced, based on inverse distance weighting, which increases computational efficiency and allows for special treatment of boundaries. Applications in linear elastic fracture mechanics are considered, with the extended finite element method being used to represent discontinuous and asymptotic fields.

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.

A Sequel of Inverse Lax–Wendroff High Order Wall Boundary Treatment for Conservation Laws

When solving CFD problems, the solver, or the numerical code, plays an important role. Depending on the phenomena and problem domain, designing such numerical codes can be hard work. One strategy is to start with simple problems and construct the code as building blocks. The purpose of this work is to provide a detailed review of the theory to compute analytical and exact solutions, and recent numerical methods to construct a code to solve compressible and inviscid fluid flows with high-resolution, arbitrary domains, non-linear phenomena, and on rectangular meshes. We also propose a modification to the inverse Lax–Wendroff procedure solid wall treatment and two-dimensional WENO-type extrapolation stencil selection and weights to handle more generic situations. To test our modifications, we use the finite difference method, Lax–Friedrichs splitting, WENO-Z+ scheme, and third-order strong stability preserving Runge-Kutta time discretization. Our first problem is a simple one-dimensional transient problem with periodic boundary conditions, which is useful for constructing the core solver. Then, we move to the one-dimensional Rayleigh flow, which can handle flows with heat exchange and requires more detailed boundary treatment. The next problem is the quasi-one-dimensional nozzle flow with and without shock, where the boundary treatment needs a few adjustments. The first two-dimensional problem is the Ringleb flow, and despite being smooth, it has a curved wall as the left boundary. Finally, the last problem is a two-dimensional conical flow, which presents an oblique shock and an inclined straight line wall being the cone surface. We show that the designed accuracy is being reached for smooth problems, that high-resolution is being attained for non-smooth problems, and that our modifications produce similar results while providing a more generic way to treat solid walls.