颗粒群碰撞搜索及CFD-DEM耦合分域求解的推进算法研究

在采用计算流体力学-离散元耦合方法(computational fluiddynamics-discrete element method, CFD-DEM)进行固液两相耦合分析时, 颗粒计算时间步的选取直接影响到耦合计算精度和计算效率. 为此, 本文选取每个目标颗粒为研究对象, 引入插值函数计算时间步的运动位移, 构建可变空间搜索网格; 通过筛选可能碰撞颗粒建立搜索列表, 采用逆向搜索方式判断碰撞颗粒, 从而提出一种改进的DEM方法(modified discreteelement method, MDEM). 该算法在颗粒群与流体耦合计算中, 颗粒计算初始时间步选取不受颗粒碰撞时间限制, 通过自动调整和修正实现大步长, 由颗粒和流体耦合条件实时更新流体计算时间步, 使颗粒计算时间步选取过小导致计算效率低、选取过大导致颗粒碰撞漏判的问题得以解决, 为颗粒与流体耦合的数值模拟提供了行之有效的计算方法. 通过两个颗粒和多个颗粒的数值模拟, 得到的颗粒间碰撞力、碰撞位置及次数, 与理论计算结果的相对误差均低于2%, 与传统的DEM碰撞搜索算法相比, 在选取的3种计算时间步均不会影响计算精度, 且有较高的计算效率. 通过多个颗粒与流体的耦合数值模拟, 采用传统的CFD-DEM方法, 只有颗粒计算时间步选取10$^{-6}$ s或更小才能得到精确解, 而采用本文方法取10$^{-4}$ s也能够得到精确解, 避免了颗粒碰撞随时间步增大而出现的漏判问题, 且计算耗时降低了16.7%.      

无条件稳定的更新精细积分方法

提出将Pade逼近与精细积分方法中的指数矩阵运算技巧结合起来,建立了精细积分法的更新形式及计算过程,对该更新精细积分方法的稳定性进行了论证与探讨.结果表明,该更新精细积分方法是无条件稳定的,整个积分方法的精度取决于所取Pade逼近的阶数与高斯积分点的数量.数值例题也显示了该方法的高效率及其可行性.     

结构动力方程的更新精细积分方法

将高斯积分方法与精细积分方法中的指数矩阵运算技巧结合起来,建立了精细积分法的更新形式及计算过程,对该更新精细积分方法的稳定性进行了论证与探讨。在实施精细积分过程中不必进行矩阵求逆,整个积分方法的精度取决于所选高斯积分点的数量。这种方法理论上可实现任意高精度,计算效率较高,其稳定性条件极易满足。数值例题也显示了这种方法的有效性。     

高黏性流动有限元模拟的迭代稳定分步算法

分步算法已被广泛应用于数值求解不可压缩N-S方程. Guermond等认为时间步长必须大于 某个临界值方能使算法稳定. 然而在高黏性流动模拟中,已有的显式和半隐式分步算法由于 其显式本质,必须采用小时间步长计算,不但降低了计算效率,同时也常与为使分步算法稳 分步算法已被广泛应用于数值求解不可压缩N-S方程. Guermond等认为时间步长必须大于 某个临界值方能使算法稳定. 然而在高黏性流动模拟中,已有的显式和半隐式分步算法由于 其显式本质,必须采用小时间步长计算,不但降低了计算效率,同时也常与为使分步算法稳 定必须满足的最小时间步长要求冲突. 本文目的是构造一种含迭代格式的分步算法,它能在 保证精度的前提下大幅度地增大时间步长. 方腔流和平面Poisseuille流数值计算结果证实 了此特点,该方法被有效应用于充填流动过程的数值模拟.     

Sizing optimization of truss structures by method of centers and force formulation

This paper describes a new approach to optimum weight design of truss structures. The force method is incorporated in an optimization algorithm based on the method of center points. Design variables are the member cross-sectional areas and the redundant forces evaluated for each independent loading condition acting on the structure. The optimization method utilizes the largest hyperspheres inscribed within the feasible space. The method of hyperspheres has been enhanced here to handle the compatibility equality constraints as well. By including the analysis step in the optimization cycle there is no longer the need to perform separate structural analyses thus saving computation time. The viability and efficiency of the proposed method are demonstrated for truss structures subject to multiple loading conditions and constraints on member stresses, nodal displacement and minimum gage. Numerical results are compared with those reported in the literature.     

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%.     

EFFICIENT COMPUTATIONS USING UPWIND BIASED SCHEMES

A new multiblock unfactored implicit upwind scheme for inviscid two-dimensional flow calculations is presented. Spatial discretization is carried out by means of an upwind first-order method; an original extension to higher accuracy is also presented. The integration algorithm is constructed in a ‘δ’ form that provides a direct derivation of the scheme and leads to an efficient computational method. Fast solutions of the linear systems arising at each time step are obtained by means of the bi-conjugate gradient stabilized technique. The computational results for super/hypersonic steady state flows illustrate the efficiency and accuracy of the algorithm.     

FPU方程的多尺度保辛摄动积分

FPU问题是一个经典非线性问题,其计算涉及多尺度分析。本文针对FPU问题,提出多尺度保辛摄动算法,该方法具有多尺度效应,可以按不同尺度显示计算结果,长时间计算保真,可以克服刚性问题,采用较大的积分步长,可以克服数值共振现象。数值算例显示了本文算法的有效性。     

一种广义精细积分法

提出了求解非齐次动力方程特解的一种精细数值积分法,该方法与通解 精细积分法具有相同精度. 首先选取一个积分形式的非齐次方程特解,将积分区域划分为 2$^{N}$份,并对之进行精细的数值积分;然后针对载荷为多项式、指数函数及三角函数的情 况,将积分求和转化为一个递推过程,按此只需$n$次矩阵乘法就能计算出积分和,从而得到 非齐次方程的特解. 该方法的优点是能与通解的精细积分过程有机地结合起来,具有极高的 精度和效率,同时还具有较广泛的适用范围. 算例结果证明了该方法的有效性.     

Towards consistent hybrid overset mesh methods for rotorcraft CFD

The overset mesh method chimera is popular within the rotorcraft research community, because the use of multiple, non‐matching grids make the CFD simulations of bodies in relative motion much simpler. Consequently, the relative motion between the helicopter blades and fuselage can be accurately accounted for. In this paper, the method for treating overset grids within CFD codes is presented. It is compatible with multi‐block, structured‐grid solvers. The proposed method is based on hierarchy of overset, non‐matching grids, whose cells are automatically identified as computational or non‐computational and localised with respect to all grids they overlap with. The efficiency of the method relies on the hierarchical, multi‐step approach, for the overset mesh localisation and the use of a tree search. Because of the high efficiency of the algorithm, the search for overlapping cells can be carried out on‐the‐fly, during time‐marching of the unsteady, implicit CFD solver. In addition, the algorithm is suitable for parallel execution. The method has been demonstrated for several flows, ranging from simple aerofoils to rotor‐body interaction. The paper presents and demonstrates the method and shows that it has a low CPU overhead. It also highlights the limitations of the method and suggests remedies for improvement. Copyright © 2013 John Wiley & Sons, Ltd.     

An adaptive algorithm of precise integration for transient analysis

This paper presents an improved precise integration algorithm for transient analysis of heat transfer and some other problems. The original precise integration method is improved by means of the inverse accuracy analysis so that the parameterN, which has been taken as a constant and an independent parameter without consideration of the problems in the original method, can be generated automatically by the algorithm itself. Thus, the improved algorithm is adaptive and the accuracy of the algorithm is not dependent on the length of the time step in the integration process. It is shown that the numerical results obtained by the method proposed are more accurate than those obtained by the conventional time integration methods such as the difference method and others. Four examples are given to demonstrate the validity, accuracy and efficiency of the new method. Project supported by the National Natural Science Foundation of China (No. 19872016, 19872017), the National Key Basic Research Special Foundation (G1999032805) and the Foundation for University Key Teachers by the Ministry of Education of China.     

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。     

一种改进的LLL模糊度规约算法

整周模糊度的高效解算是GNSS高精度数据处理中的关键,基于格论进行GNSS模糊度估计时需要通过格基规约来实现最优整周模糊度向量的快速搜索。针对高维情况下常规LLL规约算法辅助整周模糊度解算存在规约耗时较长和规约性能有限的问题,引入最小列旋转QR分解技术对基向量进行预排序,采用延后尺度规约和部分尺度规约来减少规约过程中的冗余尺度规约,以改善LLL算法的执行效果。分别通过模拟和实测数据进行实验,结果表明:改进后的LLL算法可以明显降低格基规约耗时,实测环境下其规约效率相比于传统方法提高了约10倍,且能够保证较好的规约性能,从而有效提升高维模糊度的解算效率。     

A fractional two‐step implicit algorithm using curvilinear collected grids

An efficient fractional two‐step implicit algorithm is reported to simulate incompressible fluid flows in a boundary‐fitted curvilinear collocated grid system. Using the finite volume method, the convection terms are discretized by the high‐accuracy Roe's scheme to minimize numerical diffusion. An implicitness coefficient Π is introduced to accelerate the rate of convergence. It is demonstrated that the proposed algorithm links the fractional step method to the pressure correction procedure, and the SIMPLEC method could be considered as a special case of the fractional two‐step implicit algorithm (when Π=1). The proposed algorithm is applicable to unsteady flows and steady flows. Three benchmark two‐dimensional laminar flows are tested to evaluate the performance of the proposed algorithm. Performance is measured by sensitivity analyses of the efficiency, accuracy, grid density, grid skewness and Reynolds number on the solutions. Results show that the model is efficient and robust. Copyright © 2006 John Wiley & Sons, Ltd.     

基于步长因子改进的导重法求解拓扑优化问题

结合固体各向同性惩罚微结构模型SIMP（Solid Isotropic Microstructures with Penalization）,将导重法用于求解拓扑优化问题。针对导重法迭代公式中步长因子的取值问题,提出两种变步长因子的控制策略,以结构最优性指标为参考,自动确定每步迭代的步长因子。同时引入密度补偿方法,以结构最优性指标为依据自动判断加入密度补偿的时机。利用经典拓扑优化算例,验证两种步长因子控制策略的优越性;通过细长梁算例,比较优化准则法OC（Optimality Criteria）和导重法的差异,分析对比两种步长因子控制策略施加密度补偿方法前后的计算结果。研究结果表明,两种步长因子控制策略能够取得更优解,有效提升求解效率;对于细长梁的拓扑优化问题,导重法求得的解较OC算法更具有全局性,优化效果更佳;密度补偿方法可进一步提升导重法的求解效率。     

一种结构动力时程分析的积分求微方法

传统采用微分求积(differential quadrature,DQ)法求解动力问题时都是以位移响应作为基本未知量,而将速度响应和加速度响应表示为位移响应的加权和的形式.如此做法需要处理线性方程组或者矩阵方程(Sylvester方程)才能求得动力响应,导出的算法一般为有条件稳定算法.本文利用动力响应的Duhamel积分解,逆用DQ原理,提出了一种计算卷积的高精度显式算法.该算法可以逐时段地求解出动力时程响应,当各时段内DQ节点分布完全一致时,仅须进行一次Vandermonde矩阵求逆计算即可应用于各个时段,一次性获得时段内多个时刻的位移响应值,因而具有计算效率高的优点.通过分析动力方程积分格式,证明本文动力算法传递矩阵的谱半径恒等于1,因而该算法具有无条件稳定特性,且计算过程中不会产生数值耗散. 本文算法的数值精度取决于分析时段内布置的DQ节点数量$N$,具有$N-1$阶代数精度.实际操作时可以取10个甚至更多的DQ节点数,从而获得比较高的数值精度.      

粘性流基于特征线的四阶Runge-Kutta有限元法

对于二维不可压缩粘性流,通过沿流线方向的坐标变换,推导了无对流项的二维N-S（Navier-Stokes）方程。采用四阶Runge-Kutta法对N-S方程进行时间离散,并沿流线进行Taylor展开,得到显式的时间离散格式,然后利用Galerkin法对其进行空间离散,得到了高精度的有限元算法。利用本文算法对方腔驱动流和圆柱绕流进行了数值计算,通过对时间步长、网格尺寸和流场区域的计算分析,进一步验证了本文算法相比经典CBS法在时间步长、收敛性、耗散性和计算精度方面更具有优势。     

Semi‐Lagrangian semi‐implicit time‐splitting scheme for the shallow water equations

Time‐splitting technique applied in the context of the semi‐Lagrangian semi‐implicit method allows the use of extended time steps mainly based on physical considerations and reduces the number of numerical operations at each time step such that it is approximately proportional to the number of the points of spatial grid. To control time growth of the additional truncation errors, the standard stabilizing correction method is modified with no penalty for accuracy and efficiency of the algorithm. A linear analysis shows that constructed scheme is stable for time steps up to 2h. Numerical integrations with actual atmospheric fields of pressure and wind confirm computational efficiency, extended stability and accuracy of the proposed scheme. Copyright © 2006 John Wiley & Sons, Ltd.     

A parallelized multi-degrees-of-freedom cell mapping method

Cell mapping methods, in general, provide a computationally efficient way to analyze the long-term global dynamics of lower-dimensional systems. The multi-degrees-of-freedom cell mapping (MDCM) method, in particular, overcomes the scaling limitations of other cell mapping methods, allowing efficiency benefits to be realized for higher-dimensional systems. Unfortunately, the sequential structure of the MDCM algorithm limits the ability to utilize the parallel processing capabilities of modern computers. In this paper, the parallelized multi-degrees-of-freedom cell mapping (PMDCM) method is introduced. The PMDCM method features a restructured algorithm that employs parallel computation to streamline one of the most time-consuming elements: numerical integration. The PMDCM algorithm is described in detail and is demonstrated by comparing results produced by the PMDCM method to those produced by MDCM and the grid-of-starts. By using the PMDCM method on a quad-core processor with 100 simultaneous integrations per mapping step, the total computation time is reduced by 93 %, as compared with the MDCM method. With PMDCM, the global integrity measure also agrees more closely with the computationally intensive grid-of-starts method when compared with the MDCM method.     

树形多体系统动力学的隐式数值算法

研究了树形多体系统动力学的隐式算法．用矩阵形式给出了多体系统的正则方程及其右端函数的Ｊａｃｏｂｉ矩阵，并给出该矩阵的分块算法和对角隐式Ｒｕｎｇｅ－Ｋｕｔａ法（ＤＩＲＫＭ）以及隐式辛Ｒｕｎｇｅ－Ｋｕｔａ法（ＩＳＲＫＭ）．该算法便于编程计算，能提高计算效率，保持长期计算的稳定性．并用算例说明该算法的有效性