无网格介点法求解Helmholtz方程

作为一种配点型无网格法，无网格介点MIP法具有数值实施简单、计算精度高、运算高效和适用范围广等优点。Helmholtz方程是科学与工程问题中广泛应用的一类特殊方程，因此对MIP法求解此类方程的适用性进行了验证。利用MIP法的d适应性，给出了MIP法求解该方程的两种计算格式。在数值算例中，分别对平面规则域和不规则域上的一般Helmholtz方程，以及轴对称Helmholtz方程进行了数值分析。结果表明，MIP法完全适用于求解Helmholtz方程。而且，MIP法的计算精度和收敛性都优于普通配点法。此外，MIP法的两种计算格式中，L2C0型通常具有更好的计算效果，故建议将该计算格式作为MIP法求解该类方程的标准形式。     

基于精细积分技术的非线性动力学方程的同伦摄动法

将精细积分技术（PIM）和同伦摄动方法（HPM）相结合,给出了一种求解非线性动力学方程的新的渐近数值方法。采用精细积分法求解非线性问题时,需要将非线性项对时间参数按Taylor级数展开,在展开项少时,计算精度对时间步长敏感;随着展开项的增加,计算格式会变得越来越复杂。采用同伦摄动法,则具有相对筒单的计算格式,但计算精度较差,应用范围也限于低维非线性微分方程。将这两种方法相结合得到的新的渐近数值方法则同时具备了两者的优点,既使同伦摄动方法的应用范围推广到高维非线性动力学方程的求解,又使精细积分方法在求解非线性问题时具有较简单的计算格式。数值算例表明,该方法具有较高的数值精度和计算效率。     

非稳态Hamilton-Jacobi方程的7阶加权紧致非线性格式

Hamilton-Jacobi (HJ) 方程是一类重要的非线性偏微分方程, 在物理学、流体力学、图像处理、微分几何、金融数学、最优化控制理论等方面有着广泛的应用. 由于HJ方程的弱解存在但不唯一, 且解的导数可能出现间断, 导致其数值求解具有一定的难度. 本文提出了非稳态HJ方程的7阶精度加权紧致非线性格式 (WCNS). 该格式结合了Hamilton函数的Lax-Friedrichs型通量分裂方法和一阶空间导数左、右极限值的高阶精度混合节点和半节点型中心差分格式. 基于7点全局模板和4个4点子模板推导了半节点函数值的高阶线性逼近和4个低阶线性逼近, 以及全局模板和子模板的光滑度量指标. 为避免间断附近数值解产生非物理振荡以及提高格式稳定性, 采用WENO型非线性插值方法计算半节点函数值. 时间离散采用3阶TVD型Runge-Kutta方法. 通过理论分析验证了WCNS格式对于光滑解具有最佳的7阶精度. 为方便比较, 经典的7阶WENO格式也被推广用于求解HJ方程. 数值结果表明, 本文提出的WCNS格式能够很好地模拟HJ方程的精确解, 且在光滑区域能够达到7阶精度; 与经典的同阶WENO格式相比, WCNS格式在精度、收敛性和分辨率方面更优, 计算效率略高.      

TVD格式在超音速喷管三维粘性流动求解中的应用

详细给出了任意三维曲线坐标系中Ｎｏｖｉｅｒ－Ｓｔｏｋｅｓ方程的对流项ＴＶＤ格式的构造过程，建立了数值求解三维粘性流动的计算方法，应用该方法对三维超音速喷管中有激波及无激波情况下的两种工况的层流流场进行了数值求解，并与实验做了对比。结果表明本文建立的计算方法具有较高的精度，同时也证明ＴＶＤ格式具有分辩率高，稳定收敛等优点，为进一步开展叶栅流场及紊流的研究打下了基础。     

无网格介点法求解Helmholtz方程

作为一种配点型无网格法,无网格介点MIP法具有数值实施简单、计算精度高、运算高效和适用范围广等优点。Helmholtz方程是科学与工程问题中广泛应用的一类特殊方程,因此对MIP法求解此类方程的适用性进行了验证。利用MIP法的d适应性,给出了MIP法求解该方程的两种计算格式。在数值算例中,分别对平面规则域和不规则域上的一般Helmholtz方程,以及轴对称Helmholtz方程进行了数值分析。结果表明,MIP法完全适用于求解Helmholtz方程。而且,MIP法的计算精度和收敛性都优于普通配点法。此外,MIP法的两种计算格式中,L2C0型通常具有更好的计算效果,故建议将该计算格式作为MIP法求解该类方程的标准形式。     

网格与高精度差分计算问题

研究NS方程差分求解时来流雷诺数、计算格式精度和计算网格之间的关系．给出了判定空间三个方向上的粘性贡献在给定雷诺数、格式精度和网格下是否能够正确计入的估计方法．指出在NS方程的二阶差分方法的数值模拟中,由于物面法向采用了压缩网格技术,物面附近的网格间距很小,该方向上的粘性贡献可被计入．但是如果流向和周向的网格较粗,相应的差分方程中的粘性贡献可能落入截断误差相同的量级,因此在精度上等于仍是求解略去流向和周向粘性项的薄层近似方程．指出,高阶精度的差分计算格式,可以避免对网格要求苛刻的困难．并进一步讨论了建立高阶精度格式的问题,提出了建立高阶精度格式应该满足的原则：耗散控制原则以及色散控制原则．为了避免激波附近可能出现的微小非物理振荡,建议发展混合高阶精度格式,即在激波区,采用网格自适应的NND格式,在激波以外的区域,采用按上述原则发展的高阶格式．     

满足"抑制波动原则"的广义紧致格式的特性及应用

详细分析了满足“抑制波动原则”和“稳定性原则”的五阶精度三点广义紧致格式的数值特性。该格式可以无波动地捕捉激波，并五具有类似于谱方法的高分辨率，以及很好的各向同性性质。另外还采用了时间相关的边界处理方法，以保证边界点上也满足“抑制波动原则”和“稳定性原则”。典型数值实验表明，本文格式及其边界处理方法能够有效求解Euler方程与N—S方程。     

用物理黏性构建高阶不振荡对流扩散差分格式

利用数值摄动算法, 通过扩散格式数值摄动重构把对流扩散方程的2阶中心差分格式(2-CDS)重构为高精度高分辨率格式, 解析分析和模型方程计算证实了新格式的高精度不振荡性质. 新格式是把物理黏性使流动光滑化的扩散运动规律引入2-CDS 中的结果. 该法显然与构建高级离散格式的常见方法不同. 证实: 数值摄动重构中引入扩散运动规律的结果格式与引入对流运动规律(下游不影响上游的规律)的结果格式一致, 说明对离散方程的数值摄动运算, 在维持原格式结构形式不动的条件下, 不仅能提高格式精度和稳健性, 且可揭示对流离散运动规律与扩散离散运动规律之间的内在关联;同时证实, 文中提出和使用的上、下游分裂方法是构建高精度不振荡离散格式的一个有效方法.     

硬球加强模型在CFD-DEM耦合计算中的验证与分析

针对CFD-DEM耦合计算中，颗粒计算时间步的选取影响颗粒碰撞计算精度和效率的问题。本文引入插值算法，将动量定理求解颗粒碰撞前后速度进行加权平均；根据弹性理论计算得到颗粒碰撞力，进行动力学方程求解；通过速度收敛准则修正初值速度并自动调整迭代求解次数，提出一种计算精度不受计算时间步长影响，无需对碰撞过程进行精细描述的高效率和高精度的加强硬球模型。对两个颗粒匀和变速碰撞算例进行数值模拟，碰撞后速度、碰撞力和碰撞时间与理论计算误差小于4%,与采用软球碰撞模型的DEM方法相比，颗粒碰撞计算精度不受计算时间步长影响，计算效率提高36.3%和36.8%。对单个颗粒在静水中沉降进行数值模拟，计算步长取10 s～5 s,颗粒与壁面即可得到精确解，计算效率提高33.5%。通过压力损失实验验证了该模型能够准确计算颗粒体积分数小于12%条件下两相流的压力损失。     

污染扩散方程高精度紧致格式及其稳定性分析

针对污染扩散方程提出了时间任意阶精度的显式格式,并对该格式的稳定性和精度进行了分析,理论结果表明:一阶精度的计算格式是传统的显格式,其稳定条件为:s≤1/2(s=D.Δt/Δx2,D为扩散系数,Δt为时间步长,Δx为空间步长),随着保留精度阶数的增加,稳定性范围也会随之增大;当保留无穷阶精度时,格式是无条件稳定的。这也就从一个侧面揭示了稳定性与时间精度之间的关系,为高性能数值计算格式的构思提供了可以借鉴的原则。数值算例的结果表明,本文格式具有一定的实用性。     

Numerical calculations for two-phase jet flow of a mixture of pulverized-coal and gas

In this paper, Euler-Lagrange type equations are used to describe the jet flow of a mixture of pulverized-coal and gas, which is an unsteady axisymmetric two-phase flow. By means of the finite-difference method, the coal particle's distribution, velocity and trajectory in the flow field are obtained. The coal particles are represented by a finite number of computational particles. Each particle's diameter is randomly assigned according to a given distribution. The states of the computational particles are different from each other. Turbulence is accounted for in a stochastic model. Explicit time-splitting scheme is used to calculate the strongly coupling interphase term. The numerical results are reasonable. The comparison between the numerical results and the experiment data for the case of the oil droplet injection shows good agreement. This numerical technique can be extended to the calculation of other two-phase flows of dilute particles or a droplet system. Mr. Mei Renwei also participated in the work of this paper.     

Steady‐state solution of incompressible Navier–Stokes equations using discrete least‐squares meshless method

A fractional step method for the solution of the steady state incompressible Navier–Stokes equations is proposed in this paper in conjunction with a meshless method, named discrete least‐squares meshless (DLSM). The proposed fractional step method is a first‐order accurate scheme, named semi‐incremental fractional step method, which is a general form of the previous first‐order fractional step methods, i.e. non‐incremental and incremental schemes. One of the most important advantages of the proposed scheme is its capability to use large time step sizes for the solution of incompressible Navier–Stokes equations. DLSM method uses moving least‐squares shape functions for function approximation and discrete least‐squares technique for discretization of the governing differential equations and their boundary conditions. As there is no need for a background mesh, the DLSM method can be called a truly meshless method and enjoys symmetric and positive‐definite properties. Several numerical examples are used to demonstrate the ability and the efficiency of the proposed scheme and the discrete least‐squares meshless method. The results are shown to compare favorably with those of the previously published works. Copyright © 2010 John Wiley & Sons, Ltd.     

A Hybrid Domain Decomposition and Multigrid Method for the Acceleration of Compressible Viscous Flow Calculations on Unstructured Triangular Meshes

This paper is concerned with the formulation and the evaluation of a hybrid solution method that makes use of domain decomposition and multigrid principles for the calculation of two-dimensional compressible viscous flows on unstructured triangular meshes. More precisely, a non-overlapping additive domain decomposition method is used to coordinate concurrent subdomain solutions with a multigrid method. This hybrid method is developed in the context of a flow solver for the Navier-Stokes equations which is based on a combined finite element/finite volume formulation on unstructured triangular meshes. Time integration of the resulting semi-discrete equations is performed using a linearized backward Euler implicit scheme. As a result, each pseudo time step requires the solution of a sparse linear system. In this study, a non-overlapping domain decomposition algorithm is used for advancing the solution at each implicit time step. Algebraically, the Schwarz algorithm is equivalent to a Jacobi iteration on a linear system whose matrix has a block structure. A substructuring technique can be applied to this matrix in order to obtain a fully implicit scheme in terms of interface unknowns. In the present approach, the interface unknowns are numerical fluxes. The interface system is solved by means of a full GMRES method. Here, the local system solves that are induced by matrix-vector products with the interface operator, are performed using a multigrid by volume agglomeration method. The resulting hybrid domain decomposition and multigrid solver is applied to the computation of several steady flows around a geometry of NACA0012 airfoil.     

Implementing a high‐order accurate implicit operator scheme for solving steady incompressible viscous flows using artificial compressibility method

This paper uses a fourth‐order compact finite‐difference scheme for solving steady incompressible flows. The high‐order compact method applied is an alternating direction implicit operator scheme, which has been used by Ekaterinaris for computing two‐dimensional compressible flows. Herein, this numerical scheme is efficiently implemented to solve the incompressible Navier–Stokes equations in the primitive variables formulation using the artificial compressibility method. For space discretizing the convective fluxes, fourth‐order centered spatial accuracy of the implicit operators is efficiently obtained by performing compact space differentiation in which the method uses block‐tridiagonal matrix inversions. To stabilize the numerical solution, numerical dissipation terms and/or filters are used. In this study, the high‐order compact implicit operator scheme is also extended for computing three‐dimensional incompressible flows. The accuracy and efficiency of this high‐order compact method are demonstrated for different incompressible flow problems. A sensitivity study is also conducted to evaluate the effects of grid resolution and pseudocompressibility parameter on accuracy and convergence rate of the solution. The effects of filtering and numerical dissipation on the solution are also investigated. Test cases considered herein for validating the results are incompressible flows in a 2‐D backward facing step, a 2‐D cavity and a 3‐D cavity at different flow conditions. Results obtained for these cases are in good agreement with the available numerical and experimental results. The study shows that the scheme is robust, efficient and accurate for solving incompressible flow problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Multigrid flow solutions in complex two-dimensional geometries

A finite difference solution algorithm is described for use on two-dimensional curvilinear meshes generated by the solution of the transformed Laplace equation. The efficiency of the algorithm is improved through the use of a full approximation scheme (FAS) multigrid algorithm using an extended pressure correction scheme as smoother. The multigrid algorithm is implemented as a fixed V-cycle through the grid levels with a constant number of sweeps being performed at each grid level. The accuracy and efficiency of the numerical code are validated using comparisons of the flow over two backward step configurations. Results show close agreement with previous numerical predictions and experimental data. Using a standard Cartesian co-ordinate flow solver, the multigrid efficiency obtainable in a rectangular system is shown to be reproducible in two-dimensional body-fitted curvilinear co-ordinates. Comparisons with a standard one-grid method show the multigrid method, on curvilinear meshes, to give reductions in CPU time of up to 93%.     

Improved CVP scheme for laminar incompressible flows

An improved scheme of the continuity vorticity pressure (CVP) variational equations method is presented. The changes from the original version of the CVP method concern the velocity and the pressure correction equations that are used in the solution procedure and the topology of the grid where the method is applied. The improved CVP scheme is faster, simpler and more stable than the original version of the method. The efficiency and the accuracy of the new scheme are tested and validated through comparison of predictions and of computational time, with numerical results obtained with the SIMPLE method. Moreover, we present extensive comparisons of the results of the improved CVP scheme with numerical and experimental data from various researchers that show excellent agreement for a wide range of benchmark 2D and 3D laminar internal flow problems such as flow over a backward facing step, flow in square, circular and elliptical curved ducts and pulsating flow. Copyright © 2010 John Wiley & Sons, Ltd.     

基于特征分裂有限元准隐格式的共轭传热整体耦合数值模拟方法

共轭传热现象在科学和工程领域中大量存在. 随着计算能力的发展, 对共轭传热现象进行准确有效的数值模拟, 成为科学研究和工程设计上的重要挑战.共轭传热数值模拟的方法可以分为两大类: 分区耦合和整体耦合.本文采用有限元法对共轭传热问题进行整体耦合模拟. 固体传热求解采用标准的伽辽金有限元方法.流动求解采用基于特征分裂的有限元方法. 该方法是一种重要的求解流动问题的有限元方法, 可以使用等阶有限元. 该方法的准隐格式与其他格式相比, 具有时间步长大的特点. 将稳定项中的时间步长与全局时间步长分开, 改进了准隐格式的稳定性. 基于改进的特征分裂有限元方法的准隐格式, 发展了一种层流共轭传热数值模拟的整体耦合方法. 采用这种方法可以将流体计算域和固体计算域作为一个整体划分有限元网格, 并且所有变量都可以采用相同的插值函数, 从而有利于程序的实现. 通过对典型问题的模拟, 验证了这种方法的准确性. 本工作还研究了固体区域时间步长对定常共轭传热问题数值模拟收敛性的影响.      

群桩作用机理的数值试验研究

 在分析桩土支撑体系及其相互作用关系的基础上，利用有限元分析软件ANSYS建立了群桩 体系计算模型. 通过对均质土体例题计算结果分析，可以看出利用该建模方法与传统解答有 很好的一致性. 根据辽宁工程技术大学实验馆场地土实测资料，利用该建模方法，求解出了承台与桩的 荷载分担比，对安全、经济地进行群桩基础设计具有重要指导作用和应用价值.     

A finite element analysis of a free surface drainage problem of two immiscible fluids

A sharp interface problem arising in the flow of two immiscible fluids, slag and molten metal in a blast furnace, is formulated using a two-dimensional model and solved numerically. This problem is a transient two-phase free or moving boundary problem, the slag surface and the slag–metal interface being the free boundaries. At each time step the hydraulic potential of each fluid satisfies the Laplace equation which is solved by the finite element method. The ordinary differential equations determining the motion of the free boundaries are treated using an implicit time-stepping scheme. The systems of linear equations obtained by discretization of the Laplace equations and the equations of motion of the free boundaries are incorporated into a large system of linear equations. At each time step the hydraulic potential in the interior domain and its derivatives on the free boundaries are obtained simultaneously by solving this linear system of equations. In addition, this solution directly gives the shape of the free boundaries at the next time step. The implicit scheme mentioned above enables us to get the solution without handling normal derivatives, which results in a good numerical solution of the present problem. A numerical example that simulates the flow in a blast furnace is given.     

High‐order compact numerical schemes for non‐hydrostatic free surface flows

The development of a numerical scheme for non‐hydrostatic free surface flows is described with the objective of improving the resolution characteristics of existing solution methods. The model uses a high‐order compact finite difference method for spatial discretization on a collocated grid and the standard, explicit, single step, four‐stage, fourth‐order Runge–Kutta method for temporal discretization. The Cartesian coordinate system was used. The model requires the solution of two Poisson equations at each time‐step and tridiagonal matrices for each derivative at each of the four stages in a time‐step. Third‐ and fourth‐order accurate boundaries for the flow variables have been developed including the top non‐hydrostatic pressure boundary. The results demonstrate that numerical dissipation which has been a problem with many similar models that are second‐order accurate is practically eliminated. A high accuracy is obtained for the flow variables including the non‐hydrostatic pressure. The accuracy of the model has been tested in numerical experiments. In all cases where analytical solutions are available, both phase errors and amplitude errors are very small. Copyright © 2006 John Wiley & Sons, Ltd.