Godunov-type upwind flux schemes of the two-dimensional finite volume discrete Boltzmann method

A simple unified Godunov-type upwind approach that does not need Riemann solvers for the flux calculation is developed for the finite volume discrete Boltzmann method (FVDBM) on an unstructured cell-centered triangular mesh. With piecewise-constant (PC), piecewise-linear (PL) and piecewise-parabolic (PP) reconstructions, three Godunov-type upwind flux schemes with different orders of accuracy are subsequently derived. After developing both a semi-implicit time marching scheme tailored for the developed flux schemes, and a versatile boundary treatment that is compatible with all of the flux schemes presented in this paper, numerical tests are conducted on spatial accuracy for several single-phase flow problems. Four major conclusions can be made. First, the Godunov-type schemes display higher spatial accuracy than the non-Godunov ones as the result of a more advanced treatment of the advection. Second, the PL and PP schemes are much more accurate than the PC scheme for velocity solutions. Third, there exists a threshold spatial resolution below which the PL scheme is better than the PP scheme and above which the PP scheme becomes more accurate. Fourth, besides increasing spatial resolution, increasing temporal resolution can also improve the accuracy in space for the PL and PP schemes.     

Normal,high order discrete velocity models of the Boltzmann equation

This paper aims at approximations of the collision operator in the Boltzmann equation. The developed framework guarantees the “normality” of the approximation, which means correct collision invariants, H-Theorem, and equilibrium solutions. It fits into the discrete velocity model framework, is given in such a way that it is understandable with undergraduate level mathematics and can be used to construct approximations with arbitrary high convergence orders. At last we give an example alongside a numerical verification. Here the convergence orders range up to $3$ ($2$) and the time complexity is given by $3+\frac{1}{2}$ ($4+\frac{2}{3}$) in $2$ ($3$) dimensions.     

A lattice Boltzmann modeling and analysis of the thermal convection in a lithium-ion battery

Thermal convection is a critical problem in the design of thermal management system, and is widely encountered in electric and hybrid electric vehicles. In the present work, the lattice Boltzmann method is adopted to investigate the thermal convection in the $\mathrm{LiN}{\mathrm{i}}_{\mathrm{x}}\mathrm{C}{\mathrm{o}}_{\mathrm{y}}\mathrm{M}{\mathrm{n}}_{\mathrm{z}}{\mathrm{O}}_{\mathrm{2}}$ (NCM) lithium-ion battery. The numerical results reveal that the thermal convection model considered in the current study can clearly depict the temperature evolution in the case of the thermal runaway. Additionally, it is found that as the adiabatic boundary condition is adopted, the maximum temperature inside the battery can reach 320°C at 240s, which in turn affects the surrounding batteries. To prevent the thermal runaway propagation in such a case, we also analyzed the forced convective heat transfer in this situation, and the numerical results indicate that thermal runaway can be effectively decreased if the value of the surface heat transfer coefficient for battery cell increases up to 200Wm^?2K^?1. Moreover, it is noted that when the temperature inside the battery reaches 110°C, the subsequent temperature distributions inside the battery have little influence on the surrounding batteries, which suggests that the thermal management of battery pack in both normal charge and discharge process should be considered.     

A discrete Boltzmann equation model for two-phase shallow granular flows

In this paper a Discrete Boltzmann Equation model (hereinafter DBE) is proposed as solution method of the two-phase shallow granular flow equations, a complex nonlinear partial differential system, resulting from the depth-averaging procedure of mass and momentum equations of granular flows. The latter, as e.g. a debris flow, are flows of mixtures of solid particles dispersed in an ambient fluid.The reason to use a DBE, instead of a more conventional numerical model (e.g. based on Riemann solvers), is that the DBE is a set of linear advection equations, which replaces the original complex nonlinear partial differential system, while preserving the features of its solutions. The interphase drag function, an essential characteristic of any two-phase model, is accounted for easily in the DBE by adding a physically based term. In order to show the validity of the proposed approach, the following relevant benchmark tests have been considered: the 1D simple Riemann problem, the dam break problem with the wet–dry transition of the liquid phase, the dry bed generation and the perturbation of a state at rest in 2D. Results are satisfactory and show how the DBE is able to reproduce the dynamics of the two-phase shallow granular flow.     

Improved coupling of time integration and hydrodynamic interaction in particle suspensions using the lattice Boltzmann and discrete element methods

This paper introduces improvements to the simulation of particle suspensions using the lattice Boltzmann method (LBM) and the discrete element method (DEM). First, the benefit of using a two-relaxation-time (TRT) collision operator, instead of the popular Bhatnagar–Gross–Krook (BGK) collision operator, is demonstrated. Second, a modified solid weighting function for the partially saturated method (PSM) for fluid–solid interaction is defined and tested. Results are presented for a range of flow configurations, including sphere packs, duct flows, and settling spheres, with good accuracy and convergence observed. Past research has shown that the drag, and consequently permeability, predictions of the LBM exhibit viscosity-dependence when used with certain boundary conditions such as bounce-back or interpolated bounce-back, and this is most pronounced when the BGK collision operator is employed. The improvements presented here result in a range of computational viscosities, and therefore relaxation parameters, within which drag and permeability predictions remain invariant. This allows for greater flexibility in using the relaxation parameter to adjust the LBM timestep, which can subsequently improve synchronisation with the time integration of the DEM. This has significant implications for the simulation of large-scale suspension phenomena, where the limits of computational hardware persistently constrain the resolution of the LBM lattice.     

A conservative discrete ordinate method for model Boltzmann equations

A class of conservative discrete ordinate method (C-DOM) for the Bhatnagar–Gross–Krook (BGK) model Boltzmann equation is presented. The C-DOM is the extension of the discrete ordinate method in my previous study (Yang and Huang, 1995 [3]). In the C-DOM, a conservative molecular collision process is employed and thus the conservation properties of the collision integral are maintained at the molecular level. For a broad range of Knudsen number, several test problems, including unsteady shock-tube problem and supersonic/hypersonic flows over circular cylinder, are utilized to demonstrate the performance and validity of the DOM and C-DOM. Results show that the C-DOM can greatly reduce the computer time and memory requirements in hypersonic rarefied gas flow computations.     

Lattice Boltzmann equation for axisymmetric thermal flows

A simple lattice Boltzmann equation (LBE) model for axisymmetric thermal flow is proposed in this paper. The flow field is solved by a quasi-two-dimensional nine-speed (D2Q9) LBE, while the temperature field is solved by another four-speed (D2Q4) LBE. The model is validated by a thermal flow in a pipe and some nontrivial thermal buoyancy-driven flows in vertical cylinders, including Rayleigh-Bénard convection, natural convection, and heat transfer of swirling flows. It is found that the numerical results agree excellently with analytical solution or other numerical results.     

A Galerkin finite element method for time-fractional stochastic heat equation

In this study, a Galerkin finite element method is presented for time-fractional stochastic heat equation driven by multiplicative noise, which arises from the consideration of heat transport in porous media with thermal memory with random effects. The spatial and temporal regularity properties of mild solution to the given problem under certain sufficient conditions are obtained. Numerical techniques are developed by the standard Galerkin finite element method in spatial direction, and Gorenflo–Mainardi–Moretti–Paradisi scheme is applied in temporal direction. The convergence error estimates for both semi-discrete and fully discrete schemes are established. Finally, numerical example is provided to verify the theoretical results.     

A discrete layer beam finite element for the dynamic analysis of composite sandwich beams with integral damping layers

A discrete layer finite element is presented for the dynamic analysis of laminated beams. The element uses C⁰ continuous linear and quadratic polynominals to interpolate the in-plane and transverse displacement field, respectively, and is free from the effects of shear locking. Modal frequencies and damping are estimated using both the modal strain energy method and the complex modulus method. A forced response version of the model is also presented. The model predictions are compared with experimental data for composite sandwich beams with integral damping layers. Four damping configurations are considered, a constrained layer treatment, a segmented constrained layer treatment and two internal treatments.     

Two-grid method for the two-dimensional time-dependent Schrödinger equation by the finite element method

In this paper, we construct a backward Euler full-discrete two-grid finite element scheme for the two-dimensional time-dependent Schrödinger equation. With this method, the solution of the original problem on the fine grid is reduced to the solution of same problem on a much coarser grid together with the solution of two Poisson equations on the same fine grid. We analyze the error estimate of the standard finite element solution and the two-grid solution in the $H^1$ norm. It is shown that the two-grid algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy $H=O\left(h^{\frac{k}{k+1}}\right)$. Finally, a numerical experiment indicates that our two-grid algorithm is more efficient than the standard finite element method.     

On numerical stabilization in the solution of Saint-Venant equations using the finite element method

Solving the Saint-Venant equations by using numerical schemes like finite difference and finite element methods leads to some unwanted oscillations in the water surface elevation. The reason for these oscillations lies in the method used for the approximation of the nonlinear terms. One of the ways of smoothing these oscillations is by adding artificial viscosity into the scheme. In this paper, by using a suitable discretization, we first solve the one-dimensional Saint-Venant equations by a finite element method and eliminate the unwanted oscillations without using an artificial viscosity. Second, our main discussion is concentrated on numerical stabilization of the solution in detail. In fact, we first convert the systems resulting from the discretization to systems relating to just water surface elevation. Then, by using M-matrix properties, the stability of the solution is shown. Finally, two numerical examples of critical and subcritical flows are given to support our results.     

A direct-forcing immersed boundary method for the thermal lattice Boltzmann method

In this study, a direct-forcing immersed boundary method (IBM) for thermal lattice Boltzmann method (TLBM) is proposed to simulate the non-isothermal flows. The direct-forcing IBM formulas for thermal equations are derived based on two TLBM models: a double-population model with a simplified thermal lattice Boltzmann equation (Model 1) and a hybrid model with an advection–diffusion equation of temperature (Model 2). As an interface scheme, which is required due to a mismatch between boundary and computational grids in the IBM, the sharp interface scheme based on second-order bilinear and linear interpolations (instead of the diffuse interface scheme, which uses discrete delta functions) is adopted to obtain the more accurate results. The proposed methods are validated through convective heat transfer problems with not only stationary but also moving boundaries – the natural convection in a square cavity with an eccentrically located cylinder and a cold particle sedimentation in an infinite channel. In terms of accuracy, the results from the IBM based on both models are comparable and show a good agreement with those from other numerical methods. In contrast, the IBM based on Model 2 is more numerically efficient than the IBM based on Model 1.     

A finite difference/finite element technique with error estimate for space fractional tempered diffusion-wave equation

An efficient numerical technique is proposed to solve one- and two-dimensional space fractional tempered fractional diffusion-wave equations. The space fractional is based on the Riemann–Liouville fractional derivative. At first, the temporal direction is discretized using a second-order accurate difference scheme. Then a classic Galerkin finite element is employed to obtain a full-discrete scheme. Furthermore, for the time-discrete and the full-discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method. Finally, two test problems have been illustrated to verify the efficiency and simplicity of the proposed technique.     

Development of finite element computer code for thermal analysis of roller compacted concrete dams

Thermal analysis of roller compacted concrete (RCC) dams plays an important role in their design and construction. This paper deals with the development of a finite element based computer code for the determination of temperatures within the dam body. The finite element code is then applied to the real full-scale problem to determine the impact of the placement schedule on the thermal response of roller compacted concrete dam. Based on the results obtained, it could be concluded that for a given roller compacted concrete dam, changing the placing schedule can optimize the locations of maximum temperature zones.     

Numerical analysis and testing of a stable and convergent finite element scheme for approximate deconvolution turbulence models

This report presents a stable and convergent finite element scheme for the approximate deconvolution turbulence models (ADM). The ADM is a popular turbulence model intensely studied lately but the computation of its numerical solution raises issues in terms of efficiency and accuracy. This report addresses this question. The proposed scheme presented herein is based on a new interpretation of the ADM model recently introduced by the author. Following this interpretation, the solution of the ADM is viewed as the average of a perturbed Navier–Stokes system. The scheme uses the Crank–Nicolson time discretization and the finite element spatial discretization and is proved to be stable and convergent provided a moderate choice of the time step is made. Numerical tests to verify the convergence rates and performance on a benchmark problem are also provided and they prove the correctness of this approach to numerically solve the ADM.     

Fully discrete potential-based finite element methods for a transient eddy current problem

Fully discrete potential-based finite element methods called methods are used to solve a transient eddy current problem in a three-dimensional convex bounded polyhedron. Using methods, fully discrete coupled and decoupled numerical schemes are developed. The existence and uniqueness of solutions for these schemes together with the energy-norm error estimates are provided. To verify the validity of both schemes, some computer simulations are performed for the model from TEAM Workshop Problem 7. This work was supported by Postech BSRI Research Fund-2009, National Basic Research Program of China (2008CB425701), NSFC under the grant 10671025 and the Key Project of Chinese Ministry of Education (No. 107018).