二维平板通道中流动与传热的格子-Boltzmann模拟

格子-Boltzmann数值模拟方法(LBM),在最近十几年来得到迅速发展。本文发展了LBM的流动与传热模型,并对二维平板通道中的流动与传热进行了模拟,采用密度分布函数得到速度场,用单独的内能分布函数得到温度场,并与传统FVM方法所得到的多个特征量结果进行了比较,模拟结果与FVM解均吻合很好。鉴于LBM边界条件处理简单和易于实施等特点,该方法可望成为求解流动与传热的一种有效数值模拟手段。     

A Method of Lines Based on Immersed Finite Elements for Parabolic Moving Interface Problems

This article extends the finite element method of lines to a parabolic initial boundary value problem whose diffusion coefficient is discontinuous across an interface that changes with respect to time. The method presented here uses immersed finite element (IFE) functions for the discretization in spatial variables that can be carried out over a fixed mesh (such as a Cartesian mesh if desired), and this feature makes it possible to reduce the parabolic equation to a system of ordinary differential equations (ODE) through the usual semi-discretization procedure. Therefore, with a suitable choice of the ODE solver, this method can reliably and efficiently solve a parabolic moving interface problem over a fixed structured (Cartesian) mesh. Numerical examples are presented to demonstrate features of this new method.     

Link-Wise Artificial Compressibility Method for attached and separated flows past airfoils

The performance of the recently developed Link-Wise Artificial Compressibility Method (LW-ACM) is evaluated for aerodynamic applications and then applied for both attached and separated flows. Numerical flow simulations are performed around NACA-0012 and TSAGI-12 shaped airfoils at low speed and high angle of attack. Results of aerodynamic characteristics of the airfoils were found in good agreements with previous investigations including LBM (PowerFLOW) and FVM (CFL3D). Results also showed that LW-ACM is efficient and rapidly convergent when used to solve both attached and separated flows.     

A momentum exchange-based immersed boundary-lattice Boltzmann method for simulating incompressible viscous flows

A momentum exchange-based immersed boundary-lattice Boltzmann method is presented in this Letter for simulating incompressible viscous flows. This method combines the good features of the lattice Boltzmann method (LBM) and the immersed boundary method (IBM) by using two unrelated computational meshes, an Eulerian mesh for the flow domain and a Lagrangian mesh for the solid boundaries in the flow. In this method, the non-slip boundary condition is enforced by introducing a forcing term into the lattice Boltzmann equation (LBE). Unlike the conventional IBM using the penalty method with a user-defined parameter or the direct forcing scheme based on the Navier–Stokes (NS) equations, the forcing term is simply calculated by the momentum exchange of the boundary particle density distribution functions, which are interpolated by the Lagrangian polynomials from the underlying Eulerian mesh. Numerical examples show that the present method can provide very accurate numerical results.     

An improved immersed boundary-lattice Boltzmann method for simulating three-dimensional incompressible flows

The recently proposed boundary condition-enforced immersed boundary-lattice Boltzmann method (IB-LBM) [14] is improved in this work to simulate three-dimensional incompressible viscous flows. In the conventional IB-LBM, the restoring force is pre-calculated, and the non-slip boundary condition is not enforced as compared to body-fitted solvers. As a result, there is a flow penetration to the solid boundary. This drawback was removed by the new version of IB-LBM [14], in which the restoring force is considered as unknown and is determined in such a way that the non-slip boundary condition is enforced. Since Eulerian points are also defined inside the solid boundary, the computational domain is usually regular and the Cartesian mesh is used. On the other hand, to well capture the boundary layer and in the meantime, to save the computational effort, we often use non-uniform mesh in IB-LBM applications. In our previous two-dimensional simulations [14], the Taylor series expansion and least squares-based lattice Boltzmann method (TLLBM) was used on the non-uniform Cartesian mesh to get the flow field. The final expression of TLLBM is an algebraic formulation with some weighting coefficients. These coefficients could be computed in advance and stored for the following computations. However, this way may become impractical for 3D cases as the memory requirement often exceeds the machine capacity. The other way is to calculate the coefficients at every time step. As a result, extra time is consumed significantly. To overcome this drawback, in this study, we propose a more efficient approach to solve lattice Boltzmann equation on the non-uniform Cartesian mesh. As compared to TLLBM, the proposed approach needs much less computational time and virtual storage. Its good accuracy and efficiency are well demonstrated by its application to simulate the 3D lid-driven cubic cavity flow. To valid the combination of proposed approach with the new version of IBM [14] for 3D flows with curved boundaries, the flows over a sphere and torus are simulated. The obtained numerical results compare very well with available data in the literature.     

Parallel domain connectivity algorithm for unsteady flow computations using overlapping and adaptive grids

This paper describes the algorithms and functionality of a new module developed to support overset grid assembly associated with performing time-dependent and adaptive moving body calculations of external aerodynamic flows using a multi-solver paradigm (i.e. different CFD solvers in different parts of the computational domain). We use the term “domain connectivity” in this paper to denote all the procedures that are involved in an overset grid assembly, and the module developed is referred henceforth as the domain-connectivity module. The domain-connectivity module coordinates the data transfer between different solvers applied in different parts of the computational domain – body fitted structured or unstructured to capture viscous near-wall effects, and Cartesian adaptive mesh refinement to capture effects away from the wall. The execution of the CFD solvers and the domain-connectivity module are orchestrated by a Python-based computational infrastructure. The domain-connectivity module is fully parallel and performs all its operations (identification of grid overlaps and determination of data interpolation strategy) on the partitioned grid data. In addition, the domain connectivity procedures are completely automated such that no user intervention or manual input is necessary. The capabilities and performance of the package are presented for several test problems, including flow over a NACA 0015 wing and an AGARD A2 slotted airfoil, hover simulation of a scaled V-22 rotor, and dynamic simulation of a UH-60A rotor in forward flight. A modification to the algorithm for improved domain connectivity solutions in problems with tight tolerances as well as heterogeneous grid clustering is also presented.     

Compact integration factor methods for complex domains and adaptive mesh refinement

Implicit integration factor (IIF) method, a class of efficient semi-implicit temporal scheme, was introduced recently for stiff reaction–diffusion equations. To reduce cost of IIF, compact implicit integration factor (cIIF) method was later developed for efficient storage and calculation of exponential matrices associated with the diffusion operators in two and three spatial dimensions for Cartesian coordinates with regular meshes. Unlike IIF, cIIF cannot be directly extended to other curvilinear coordinates, such as polar and spherical coordinates, due to the compact representation for the diffusion terms in cIIF. In this paper, we present a method to generalize cIIF for other curvilinear coordinates through examples of polar and spherical coordinates. The new cIIF method in polar and spherical coordinates has similar computational efficiency and stability properties as the cIIF in Cartesian coordinate. In addition, we present a method for integrating cIIF with adaptive mesh refinement (AMR) to take advantage of the excellent stability condition for cIIF. Because the second order cIIF is unconditionally stable, it allows large time steps for AMR, unlike a typical explicit temporal scheme whose time step is severely restricted by the smallest mesh size in the entire spatial domain. Finally, we apply those methods to simulating a cell signaling system described by a system of stiff reaction–diffusion equations in both two and three spatial dimensions using AMR, curvilinear and Cartesian coordinates. Excellent performance of the new methods is observed.     

NS方程的各向异性笛卡尔网格法研究

将近年发展起来的用于Euler方程求解的具有局部均匀网格总体非结构特性的笛卡尔网格法推广到NS方程的求解。为了与流场的各向异性相适应、减少网格点数量,提出了一种各向异性网格加密法。另外还研究了分级笛卡尔网格对内点格式稳定性的影响和插值固体边界条件的稳定性。数值结果表明各向异性笛卡尔网格法相对于传统的各向同性网格方法能大量节省网格点数量而且与后者具有同样的精度。     

On using moving windows in finite element time domain simulation for long accelerator structures

A finite element moving window technique is developed to simulate the propagation of electromagnetic waves induced by the transit of a charged particle beam inside large and long structures. The window moving along with the beam in the computational domain adopts high-order finite element basis functions through p refinement and/or a high-resolution mesh through h refinement so that a sufficient accuracy is attained with substantially reduced computational costs. Algorithms to transfer discretized fields from one mesh to another, which are the keys to implementing a moving window in a finite element unstructured mesh, are presented. Numerical experiments are carried out using the moving window technique to compute short-range wakefields in long accelerator structures. The results are compared with those obtained from the normal finite element time domain (FETD) method and the advantages of using the moving window technique are discussed.     

Inverse Lax-Wendroff procedure for numerical boundary conditions of conservation laws

We develop a high order finite difference numerical boundary condition for solving hyperbolic conservation laws on a Cartesian mesh. The challenge results from the wide stencil of the interior high order scheme and the fact that the boundary intersects the grids in an arbitrary fashion. Our method is based on an inverse Lax-Wendroff procedure for the inflow boundary conditions. We repeatedly use the partial differential equation to write the normal derivatives to the inflow boundary in terms of the time derivatives and the tangential derivatives. With these normal derivatives, we can then impose accurate values of ghost points near the boundary by a Taylor expansion. At outflow boundaries, we use Lagrange extrapolation or least squares extrapolation if the solution is smooth, or a weighted essentially non-oscillatory (WENO) type extrapolation if a shock is close to the boundary. Extensive numerical examples are provided to illustrate that our method is high order accurate and has good performance when applied to one and two-dimensional scalar or system cases with the physical boundary not aligned with the grids and with various boundary conditions including the solid wall boundary condition. Additional numerical cost due to our boundary treatment is discussed in some of the examples.     

A solution-adaptive lattice Boltzmann method for two-dimensional incompressible viscous flows

A stencil adaptive lattice Boltzmann method (LBM) is developed in this paper. It incorporates the stencil adaptive algorithm developed by Ding and Shu [26] for the solution of Navier–Stokes (N–S) equations into the LBM calculation. Based on the uniform mesh, the stencil adaptive algorithm refines the mesh by two types of 5-points symmetric stencils, which are used in an alternating sequence for increased refinement levels. The two types of symmetric stencils can be easily combined to form a 9-points symmetric structure. Using the one-dimensional second-order interpolation recently developed by Wu and Shu [27] along the straight line and the D2Q9 model, the adaptive LBM calculation can be effectively carried out. Note that the interpolation coefficients are only related to the lattice velocity and stencil size. Hence, the simplicity of LBM is not broken down and the accuracy is maintained. Due to the use of adaptive technique, much less mesh points are required in the simulation as compared to the standard LBM. As a consequence, the computational efficiency is greatly enhanced. The numerical simulation of two dimensional lid-driven cavity flows is carried out. Accurate results and improved efficiency are reached. In addition, the steady and unsteady flows over a circular cylinder are simulated to demonstrate the capability of proposed method for handling problems with curved boundaries. The obtained results compare well with data in the literature.     

一种非结构/结构多层混合网格方法及其应用

对于粘性绕流的数值模拟,在自适应直角网格基础上,结合三角形非结构网格和结构化网格,利用其各自的优势和特点,提出一种生成混合杂交网格的思路和方法.在物面附近生成适合粘性流计算的大长宽比结构化网格,在远场分布自适应直角网格,快速离散计算空间.对于复杂的多体问题,采用三角形网格来连接各体网格,并运用网格合并的方法,保证各网格之间的光滑过渡与连接,提高网格质量.针对一些二维、三维外形的绕流问题,在上述网格基础上,采用B-L代数湍流模型和中心有限体积法,完成Navier-Stokes和Euler方程数值模拟的对比计算,结果表明网格生成和流场计算是正确的.     

十三点格子Boltzmann模型仿真

格子气和格子Boltzmann方法的迅速发展提供了一类求解流体力学问题的新方法。格子Boltzmann方法在保留了格子气模型优点的同时,克服了它的不足之处。本文讨论了一种三迭加HPP十三点模型,通过选择适当的平衡分布及参数,并用Chapman－Enskog展开和多尺度技术导出了Navier－Stokes方程。在微机上模拟了空腔流的流动问题,并与传统方法的计算结果进行了比较,结果表明该模型能较好的模拟复杂流动现象,并具有较好的工程应用背景。     

A variational adiabatic hyperspherical finite element R matrix methodology: general formalism and application to H + H2 reaction

The aim of this paper is to present an efficient numerical procedure for the theoretical study of bimolecular reactions. It is based on the R matrix variational formalism and the p-version of the finite element method (p-FEM) for expanding the wave function in a finite basis set, and facilitates the development of an efficient algorithm to invert matrices that significantly reduces the computational time in R matrix calculations. We also utilise the self-consistent finite element method to optimise the elements mesh and provide faster convergence of results. We apply our methodology to the study of the collinear H + H₂ process and evaluate its efficiency by comparing our results with several results previously published in the literature.     

A priori mesh quality metric error analysis applied to a high-order finite element method

Characterization of computational mesh’s quality prior to performing a numerical simulation is an important step in insuring that the result is valid. A highly distorted mesh can result in significant errors. It is therefore desirable to predict solution accuracy on a given mesh. The HiFi/SEL high-order finite element code is used to study the effects of various mesh distortions on solution quality of known analytic problems for spatial discretizations with different order of finite elements. The measured global error norms are compared to several mesh quality metrics by independently varying both the degree of the distortions and the order of the finite elements. It is found that the spatial spectral convergence rates are preserved for all considered distortion types, while the total error increases with the degree of distortion. For each distortion type, correlations between the measured solution error and the different mesh metrics are quantified, identifying the most appropriate overall mesh metric. The results show promise for future a priori computational mesh quality determination and improvement.     

格子Boltzmann方法求解Burgers方程

众所周知,格子方法（包括格子气和格子Ｂｏｌｔｚｍａｎｎ方法）在计算物理领域取得巨大进展。与之形成鲜明对比,格子方法的数学理论始终处于停滞前的状况。为求解Ｂｕｒｇｅｒｓ方程,一类带有ＢＧＫ模型格子方法被构造出来,经过变量替换,发现他们属于三层非性差分方法。使用极值原理,给出此类格式稳定性的严格证明,最后,从数值实验中可以看出,使用ＬＢＭ得到的结果,与经典二阶守恒差分方法的结果符合得非常好。     

Modeling of wall pressure fluctuations for finite element structural analysis

This paper investigates a modeling technique of wall pressure fluctuations (WPF) due to turbulent boundary layer flows on a surface for finite element structural analysis. This study is motivated by critical issues of structural vibrations due to turbulent WPF over the surface of a body. The WPFs are characterized with random behavior in time and space. The temporal and spatial random behavior of the WPFs is mathematically expressed in the form of the auto- and cross-power spectral density functions (PSDF) in the frequency domain (e.g., Corcos Model). For finite element modeling of the random distributed fluctuations, the cross-PSDF is properly modeled over the finite element structural mesh. The quality of modeling of the cross-PSDF is directly affected by the structural mesh size. We first examine the maximum mesh size required for reliable finite element analysis. The reliability of the FEA results is confirmed by the theoretical results. It is found that the maximum mesh size should be determined under consideration of the spatial distribution of the cross-PSDF in addition to the representation of dynamic behavior of the structure in the frequency range of interest. It is also recognized that the requirement of the maximum mesh size is unrealistic in many practical cases. We then investigate practical modeling schemes under a realistic mesh size condition. We found that the WPF can be modeled without the exact consideration of cross-correlation if the power due to the wall pressure fluctuation can be properly compensated. This is owing to the feature of decreasing the cross-correlation of WPFs at high frequencies and the fact that the WPF does weakly couple into the structural modes at high wavenumbers such that 2πf/U_c<k_max. The wall pressure fluctuations can therefore be modeled as uncorrelated loadings with power compensations.     

Analysis of transport of collimated radiation in a participating media using the lattice Boltzmann method

Application of the lattice Boltzmann method (LBM) recently proposed by Asinari et al. [Asinari P, Mishra SC, Borchiellini R. A lattice Boltzmann formulation to the analysis of radiative heat transfer problems in a participating medium. Numer Heat Transfer B 2010; 57:126–146] is extended to the analysis of transport of collimated radiation in a planar participating medium. To deal with azimuthally symmetric radiation in planar medium, a new lattice structure for the LBM is used. The transport of the collimated component in the medium is analysed by two different, viz., flux splitting and direct approaches. For different angles of incidence of the collimated radiation, the LBM formulation is tested for the effects of the extinction coefficient, the anisotropy factor, and the boundary emissivities on heat flux and emissive power distributions. Results are compared with the benchmark results obtained using the finite volume method. Both the approaches in LBM provide accurate results.     

Adaptation and performance of the Cartesian coordinates fast multipole method for nanomagnetic simulations

An implementation of the fast multiple method (FMM) is performed for magnetic systems with long-ranged dipolar interactions. Expansion in spherical harmonics of the original FMM is replaced by expansion of polynomials in Cartesian coordinates, which is considerably simpler. Under open boundary conditions, an expression for multipole moments of point dipoles in a cell is derived. These make the program appropriate for nanomagnetic simulations, including magnetic nanoparticles and ferrofluids. The performance is optimized in terms of cell size and parameter set (expansion order and opening angle) and the trade off between computing time and accuracy is quantitatively studied. A rule of thumb is proposed to decide the appropriate average number of dipoles in the smallest cells, and an optimal choice of parameter set is suggested. Finally, the superiority of Cartesian coordinate FMM is demonstrated by comparison to spherical harmonics FMM and FFT.     

A Cartesian method for fitting the bathymetry and tracking the dynamic position of the shoreline in a three-dimensional, hydrodynamic model

This paper presents a Cartesian method for the simultaneous fitting of the bathymetry and shorelines in a three-dimensional, hydrodynamic model for free-surface flows. The model, named LESS3D (Lake & Estuarine Simulation System in Three Dimensions), solves flux-based finite difference equations in the Cartesian-coordinate system (x,y,z). It uses a bilinear bottom to fit the bottom topography and keeps track the dynamic position of the shoreline. The resulting computational cells are hybrid: interior cells are regular Cartesian grid cells with six rectangular faces, and boundary/bottom cells (at least one face is the water–solid interface) are unstructured cells whose faces are generally not rectangular. With the bilinear interpolation, the shape of a boundary/bottom cell can be determined at each time step. This allows the Cartesian coordinate model to accurately track the dynamic position of the shorelines. The method was tested with a laboratory experiment of a Tsunami runup case on a circular island. It was also tested for an estuary in Florida, USA. Both model applications demonstrated that the Cartesian method is quite robust. Because the present method does not require any coordinate transformation, it can be an attractive alternative to curvilinear grid model.