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.     

对流扩散方程QUICK格式的数值摄动高精度重构格式

利用高智提出的数值摄动算法, 把对流扩散方程的常用QUICK格式(黏性和对流项分别用二阶中心和QUICK格式离散)进行了高精度重构, 包括利用离散单元内所有结点的全域重构和分别利用离散单元内上下游结点的上下游重构, 得到两类新的更高阶精度的数值摄动重构格式, 称为高的QUICK格式(G-QUICK格式). G-QUICK格式与QUICK格式相比简单性相当, 但精度更高; 全域重构G-QUICK格式和QUICK格式均为条件稳定, 上下游重构得到一些绝对稳定的G-QUICK格式. 解析分析和数值算例均证实了G-QUICK格式的优良性能, 上下游重构的G-QUICK格式为在对流扩散方程的QUICK格式中避免使用人工黏性提供了新途径.      

对流扩散方程的摄动有限体积(PFV)方法及讨论

在有限体积(FV)方法的重构近似中,引入数值摄动处理,即把界面数值通量摄动展开成网格间距的幂级数,并利用积分方程自身的性质求出幂级数的系数,同时获得高精度迎风和中心型摄动有限体积(PFV)格式.对标量输运方程给出积分近似为二阶、重构近似为二、三和四阶迎风和中心型PFV格式,这些PFV格式的结构形式及使用基点数与一阶迎风格式完全一致,迎风PFV格式满足对流有界准则;二阶和四阶中心PFV格式对网格Peclet数的任意值均为正型格式,比常用的二阶中心格式优越.用一维标量输运和方腔流动算例说明PFV格式的优良性能,并把PFV方法与性质相近的摄动有限差分(PFD)方法及相关的高精度方法作了对比分析.     

数值摄动算法及其CFD格式

作者提出的数值摄动算法把流体动力学效应耦合进NS方程组和对流扩散（CD）方程离散的数学基本格式（MBS）,特别是耦合进最简单的MBS即一阶迎风和二阶中心格式之中,由此构建成一系列新格式,称呼方便和强调耦合流体动力学起见,称它们为流体力学基本格式（FMBS）。构建FMBS的主要步骤是把MBS中的通量摄动重构为步长的幂级数,利用空间分裂和导出的高阶流体动力学关系式,把结点变量展开成Taylor级数,通过消除重构格式修正微分方程的截断误差诸项求出幂级数的待定系数,由此获得非线性FMBS。FMBS的公式是MBS与 （及 ）之简单多项式的乘积, 和 分别是网格Reynolds数和网格CFL数。FMBS和MBS使用相同结点,简单性彼此相当,但FMBS精度高稳定范围大,例如FMBS包含了许多绝对稳定和绝对正型、高阶迎风和中心有限差分（FD）格式和有限体积（FV）格式,这些格式对网格Reynolds数的任意值均为不振荡格式。可见对不振荡CFD格式的构建,数值摄动算法提供了不同于调节数值耗散等常见的人为构建方法,而利用流体力学自身关系以及把迎风机制通过上、下游摄动重构引入中心MBS的解析构建方法,FMBS除了直接应用于流体计算外;对于通过调节数值耗散、色散和数值群速度特性重构高分辨率格式的研究,最简单FMBS提供了比最简单MBS更精确、但同样简单的基础和起步格式。FMBS用于计算不可压缩流,可压缩流,液滴萃取传质,微通道两相流等,均获得良好数值结果或与已有Benchmark解一致的数值结果。已有文献称数值摄动算法为新型高精度格式和高的算法和高的格式;本文FMBS比数值摄动格式的称呼可更好反映FMBS的物理内容。文中也讨论了值得进一步研究的一些课题,该法亦可用于其它一些数学物理方程（例如,简化Boltzmann方程、磁流体方程、KdV-Burgers方程等）MBS耦合物理动力学效应的重构。      

同位网格摄动有限体积格式求解浮力驱动方腔流

利用对流扩散方程的摄动有限体积格式,在Rayleigh数从10$^{3}$ 到10$^{8}$的范围内对浮力驱动方腔流动问题作了数值模拟. 对流扩散方程的摄动 有限体积格式具有一阶迎风格式的简洁形式,使用相同的基点,重构近似精度高,特别是两 相邻控制体中心到公共界面的距离相等或不相等,PFV格式公式相同等优点. 在数值模拟中, 无论均匀网格还是非均匀网格均获得与DSC方法、自适应有限元法、多重网格法等Benchmark 解相符较好的数值结果,证明UPFV格式对高Rayleigh数对流传热问题的适用性和有效性.     

三维对流扩散方程的高精度全隐式多重网格方法

提出了数值求解三维非定常变系数对流扩散方程的一种高精度全隐紧致差分格式,该格式在空间上具有四阶精度,时间具有二阶精度。为了克服传统迭代法在每一个时间步上迭代求解隐格式时收敛速度慢的缺点,采用多重网格加速技术,建立了适用于本文高精度全隐紧致格式的多重网格算法,从而大大加快了迭代收敛速度。数值实验结果验证了本文方法的精确性、稳定性和对高网格雷诺数问题的强适应性。     

二维对流扩散方程的高精度全隐式多重网格方法

提出了数值求解二维非定常变系数对流扩散方程的一种时间二阶、空间四阶精度的三层全隐紧致差分格式。为了加快迭代求解隐格式时在每一个时间步上的收敛速度，采用多重网格加速技术，建立了适用于本文高精度金隐紧致格式的多重网格算法。数值实验结果验证了本文方法的精确性、稳定性和对高网格雷诺数问题的强适应性。     

对流扩散方程的迎风变换及相应有限差分方法

本文提出所谓迎风变换,将对流扩散方程分解为对流迎风函数和扩散方程,并构造相应的有限差分格式。对流迎风函数以简明的指数解析形式反映对流扩散现象的迎风效应,原则上消除了源于不对称对流算子的困难,能够便利对流扩散方程的数值求解。有限差分格式具有二阶精度和无条件稳定性,算例表明其准确性、收敛速度及对边界层效应的适应能力均明显优于中心差分格式和迎风差分格式。     

基于高阶和高效格式的悬停旋翼可压缩无粘绕流的计算

发展了一种基于高精度和高效格式计算悬停旋翼复杂绕流的隐式有限体积方法。控制方程为Euler方程,其中对流项通量的左右状态量采用五阶加权基本无振荡(WENO)格式重构,时间推进应用隐式LU-SGS算法,为进一步加速收敛,采用三层V循环多重网格松弛方法。考虑到多重网格方法的思想以及五阶WENO格式涉及6个网格单元,建议仅在最细网格上使用WENO格式。计算结果表明本文方法能有效捕捉激波,对尾迹也有较高分辨率,基于隐式LU-SGS算法的多重网格方法能有效提高计算效率。     

二维对流扩散方程的欧拉—拉格朗日分裂格式

本文在[1]基础上发展了一种有效的处理大P_e(R_e)数、非定常二维对流扩散方程的欧拉-拉格朗日(E-L)分裂格式,由于方法本质上与区域形状无关,且不需再分网格,因此是一种无网格的E-L方法,特别对于定常流动,E.-L.分裂格式可以导致比一阶迎风格式更精确的单调、无振荡格式,文中对于常系数、变系数和非线性的二维非定常和定常对流扩散方程的(初)边值问题进行了数值计算,数值结果与精确解的比较表明,本方法具有很好的精度,解是单调无振荡的,比通常一阶迎风格式具有较少的数值扩散,最大计算网格P-e(R-e)数可达100—500。     

Connections between the discontinuous Galerkin method and high‐order flux reconstruction schemes

With high‐order methods becoming more widely adopted throughout the field of computational fluid dynamics, the development of new computationally efficient algorithms has increased tremendously in recent years. One of the most recent methods to be developed is the flux reconstruction approach, which allows various well‐known high‐order schemes to be cast within a single unifying framework. Whilst a connection between flux reconstruction and the more widely adopted discontinuous Galerkin method has been established elsewhere, it still remains to fully investigate the explicit connections between the many popular variants of the discontinuous Galerkin method and the flux reconstruction approach. In this work, we closely examine the connections between three nodal versions of tensor‐product discontinuous Galerkin spectral element approximations and two types of flux reconstruction schemes for solving systems of conservation laws on quadrilateral meshes. The different types of discontinuous Galerkin approximations arise from the choice of the solution nodes of the Lagrange basis representing the solution and from the quadrature approximation used to integrate the mass matrix and the other terms of the discretization. By considering both linear and nonlinear advection equations on a regular grid, we examine the mathematical properties that connect these discretizations. These arguments are further confirmed by the results of an empirical numerical study. Copyright © 2014 John Wiley & Sons, Ltd.     

A CFD Euler solver from a physical acoustics–convection flux Jacobian decomposition

This paper introduces a continuum, i.e. non‐discrete, upstream‐bias formulation that rests on the physics and mathematics of acoustics and convection. The formulation induces the upstream‐bias at the differential equation level, within a characteristics‐bias system associated with the Euler equations with general equilibrium equations of state. For low subsonic Mach numbers, this formulation returns a consistent upstream‐bias approximation for the non‐linear acoustics equations. For supersonic Mach numbers, the formulation smoothly becomes an upstream‐bias approximation of the entire Euler flux. With the objective of minimizing induced artificial diffusion, the formulation non‐linearly induces upstream‐bias, essentially locally, in regions of solution discontinuities, whereas it decreases the upstream‐bias in regions of solution smoothness. The discrete equations originate from a finite element discretization of the characteristic‐bias system and are integrated in time within a compact block tridiagonal matrix statement by way of an implicit non‐linearly stable Runge–Kutta algorithm for stiff systems. As documented by several computational results that reflect available exact solutions, the acoustics–convection solver induces low artificial diffusion and generates essentially non‐oscillatory solutions that automatically preserve a constant enthalpy, as well as smoothness of both enthalpy and mass flux across normal shocks. Copyright © 1999 John Wiley & Sons, Ltd.     

两套节点格林元嵌入式离散裂缝模型数值模拟方法

对于原始嵌入式离散裂缝模型(EDFM),在计算包含裂缝单元的基质网格内的压力分布时采用了线性分布假设,这导致了油藏开发早期对非稳态窜流量的计算精度不足.因此,本文提出了一种两套节点格林元法的EDFM数值模拟方法.两套节点格林元法的核心思想是将压力节点与流量节点区分开,一套压力节点设置在单元顶点,另一套流量节点设置在网格边的中点,满足局部物质守恒、具有二阶精度的同时,可适用于任意网格类型.本文将两套节点格林元法与EDFM耦合,采用了非稳态渗流控制方程的边界积分形式推导了基质网格与裂缝网格之间传质量的新格式,代替了线性分布假设以提高模拟精度;此外,修正后的EDFM能适应任意形态的基质网格剖分,拓展了原始EDFM仅适用于矩形基质网格、难以考虑复杂油藏边界的局限性.研究表明:通过对比商业模拟软件tNavigator?LGR模块与原始EDFM,验证了本文模型具有较高的早期计算精度;以复杂油藏边界-缝网-SRV分区模型为例,通过对比SFEM-COMSOL商业模拟软件,验证了本文模型处理复杂问题的适应性.本文研究可用于裂缝性油藏开发动态的精确模拟.     

Numerical Simulation of Acoustic Waves Interacting with a Shock Wave in a Quasi-ID Convergent-Divergent Nozzle Using an Unstructured Finite Volume Algorithm

Numerical simulations of a very small amplitude acoustic wave interacting with a shock wave in a quasi-ID convergent-divergent nozzle is performed using an unstructured finite volume algorithm with piece-wise linear, least square reconstruction, Roe flux difference splitting, and second-order MacCormack time marching. First, the spatial accuracy of the algorithm is evaluated for steady flows with and without the normal shock by running the simulation with a sequence of successively finer meshes. Then the accuracy of the Roe flux difference splitting near the sonic transition point is examined for different reconstruction schemes. Finally, the unsteady numerical solutions with the acoustic perturbation are presented and compared with linear theory results.     

Implicit FEM‐FCT algorithms and discrete Newton methods for transient convection problems

A new generalization of the flux‐corrected transport (FCT) methodology to implicit finite element discretizations is proposed. The underlying high‐order scheme is supposed to be unconditionally stable and produce time‐accurate solutions to evolutionary convection problems. Its nonoscillatory low‐order counterpart is constructed by means of mass lumping followed by elimination of negative off‐diagonal entries from the discrete transport operator. The raw antidiffusive fluxes, which represent the difference between the high‐ and low‐order schemes, are updated and limited within an outer fixed‐point iteration. The upper bound for the magnitude of each antidiffusive flux is evaluated using a single sweep of the multidimensional FCT limiter at the first outer iteration. This semi‐implicit limiting strategy makes it possible to enforce the positivity constraint in a very robust and efficient manner. Moreover, the computation of an intermediate low‐order solution can be avoided. The nonlinear algebraic systems are solved either by a standard defect correction scheme or by means of a discrete Newton approach, whereby the approximate Jacobian matrix is assembled edge by edge. Numerical examples are presented for two‐dimensional benchmark problems discretized by the standard Galerkin finite element method combined with the Crank–Nicolson time stepping. Copyright © 2007 John Wiley & Sons, Ltd.     

自然单元法在线弹性断裂力学中的应用研究

采用Laplace插值构造近似场函数,代入弹性力学边值问题的变分弱形式,由变分原理推导出自然单元法的离散控制方程。运用线弹性断裂力学理论,分析了有限板单边Ⅰ型裂纹的应力强度因子和有限板拉剪复合裂纹的扩展,给出了分析线弹性断裂力学问题的自然单元法。数值计算结果表明了方法的正确性和有效性。     

The Edge-Based Face Element Method for 3D-Stream Function and Flux Calculations in Porous Media Flow

We present a velocity-oriented discrete analog of the partial differential equations governing porous media flow: the edge-based face element method. Conventional finite element techniques calculate pressures in the nodes of the grid. However, such methods do not satisfy the requirement of flux continuity at the faces. In contrast, the edge-based method calculates vector potentials along the edges, leading to continuity of fluxes. The method is algebraically equivalent with the popular block-centered finite difference method and with the mixed-hybrid finite element method, but is algorithmically different and has the same robustness as the more conventional node-based velocity-oriented method. The numerical examples are computed analytically and may, therefore, be considered as an 'heuristic proof' of the theory and its practical applicability for reservoir engineering and geohydrology.     

复合材料层合板的摄动随机Cell-Based光滑有限元

随机性是实际工程结构的固有特性,如何更真实地描述含随机参数结构的随机响应及统计特性,对工程结构的可靠性设计具有非常重要的意义。本文基于Cell-Based光滑有限元,采用四边形单元,推导了基于一阶剪切变形理论的复合材料层合板的光滑有限元公式,降低了网格划分要求,适应不规则网格,并采用离散剪切间隙有效地消除了剪切自锁;结合摄动法和随机场理论,导出了复合材料层合板的摄动随机光滑有限元平衡方程,并给出了结构随机响应数字特征的计算公式,求解了材料属性含随机性的复合材料层合板的随机响应问题,数值算例结果表明了本方法的有效性和准确性。