影响无网格方法求解精度的因素分析

基于移动最小二乘法的无网格方法的计算精度除受到节点的分布密度和基底函数的阶次影响外，还受到其它因素的影响，其中权函数的选取、权函数影响域的大小及位移边界条件的引入对计算精度影响较大。本文分析了几种常用权函数在数值计算时的特点，包括计算精度、收敛情况、计算效率等，同时分析了影响域大小及边界条件的引入对计算精度的影响。通过分析给出了确定权函数及其影响域大小的方法。当受约束的自由度较多时，通过配点法引入位移边界条件会引起计算结果的振荡，通过施加稳定项可以消除振荡现象，通过对带孔方板的受力分析证明了其可行性。应用以上结论对J23—10曲柄压力机机身进行了受力分析，应力集中部位的计算结果得到了较高的精度。     

应力高梯度问题的无网格分析

基于移动最小二乘法的无网格计算，采用线性基函数即可得到C^1连续位移场，使得应力，应变场在整个求解域内保持连续；节点之间脱离了单元的约束，对求解域进行离散和加密节点时变得十分灵活，因此适合分析应力高梯度问题。本文简要介绍了无网格方法的基本原理，给出了确定节点影响域大小的方法，应用无网格方法对带有V型缺口的受拉方板J23-10曲柄压力机机身进行了受力分析，得到的应力集中部位的计算结果与实际值更为接近。     

求解二维Euler方程的时一空守恒格式

将作者原来得出的一维时－空守恒格式推广到了二维情形,得到了二维Ｅｕｌｅｒ方程的时．空守恒格式,并用几个典型算例进行了检验计算,结果表明：得到的二维时一空守恒格式保留了一维格式所有的优点,格式简单,通用性强,对微波等间断具有很高的分辨率．     

全机绕流Euler方程多重网格分区计算方法

全机三维复杂形状绕流数值求解只能采用分区求解的方法,本文采用可压缩Euler方程有限体积方法以及多重网格分区方法对流场进行分区计算。数值方法采用改进的van Leer迎风型矢通量分裂格式和MUSCL方法,基于有限体积方法和迎风型矢通量分裂方法,建立一套处理子区域内分界面的耦合条件。各个子区域之间采用显式耦合条件,区域内部采用隐式格式和局部时间步长等,以加快收敛速度。计算结果飞机表面压力分布等气动力特性与实验值进行了比较,二者基本吻合。计算结果表明采用分析“V”型多重网格方法,能提高计算效率,加快收敛速度达到接近一个量级。根据全机数值计算结果和可视化结果讨论了流场背风区域旋涡的形成过程。     

带源参数的二维热传导反问题的无网格方法

利用无网格有限点法求解带源参数的二维热传导反问题,推导了相应的离散方程. 与 其它基于网格的方法相比,有限点法采用移动最小二乘法构造形函数,只需要节点信息,不 需要划分网格,用配点法离散控制方程,可以直接施加边界条件,不需要在区域内部求积分. 用有限点法求解二维热传导反问题具有数值实现简单、计算量小、可以任意布置节点等优点. 最后通过算例验证了该方法的有效性.     

高雷诺数下求解NS方程的无网格算法

提出了一种适合高雷诺数NS方程求解的隐式无网格算法。针对高雷诺数粘性流动的特点,在附面层内的粘性影响区域采用法向层次推进布点的方法形成离散点云,在附面层外的计算区域内实行填充式布点的方法形成离散点云。根据附面层内外点云的不同构造特点,推导出运用格林公式和最小二乘曲面拟合方法求取空间导数的统一形式,在此基础上运用AUSM _up格式求得数值通量,并引入BL湍流模型对雷诺平均NS方程的湍流应力项进行封闭。时间推进格式方面,采用了计算效率较高的隐式高斯-赛德尔迭代算法。为了验证本文方法的计算精度和鲁棒性,对NACA0012翼型低速流动、RAE2822翼型跨音速绕流和二维圆柱的分离流动进行了数值模拟。     

附面层修正应用于无网格算法的研究

研究了无网格算法中的附面层修正方法,在一种布置点云方法的基础上,发展一种曲面拟合的重构方式构造流场物理量;找出了无网格算法与网格算法之间的联系,成功将AUSM＋-up格式移植到无网格算法当中,并应用于计算欧拉方程的数值通量;计算中采用了一种改进的隐式时间推进,并引入当地时间步长和残值光顺等加速收敛措施,成功的将附面层修...     

无网格方法的研究进展与展望

目前正在发展的无网格方法采用基于点的近似,可以彻底或部分地消除网格,因此在处理不连续和大变形问题时可以完全抛开网格重构。无网格方法是目前科学和工程计算方法研究的热点,也是科学和工程计算发展的趋势。本文首先简单地阐述了无网格方法,然后详细叙述了目前提出的各种无网格方法的研究进展,最后对目前无网格方法存在的问题进行了探讨,提出了今后的研究方向。     

一般坐标系下的二维Euler方程时—空守恒格式

将文「１－３」中的一维时－空守恒格式推广到了二维情形,得到了一般坐标系下的二维Ｅｕｌｅｒ方程时－空守恒格式,并用几个典型算例进行了检验计算,结果表明：本文得到的二维时－空守恒格式保留了一维格式所有的优点,格式简单,通用性强,而且对激波等间断具有很高的分辨率。     

求解二维Euler方程的时一空守恒格式

将作者原来得出的一维时－空守恒格式推广到了二维情形,得到了二维Ｅｕｌｅｒ方程的时．空守恒格式,并用几个典型算例进行了检验计算,结果表明：得到的二维时一空守恒格式保留了一维格式所有的优点,格式简单,通用性强,对微波等间断具有很高的分辨率．     

A hybrid building‐block and gridless method for compressible flows

A hybrid building‐block Cartesian grid and gridless method is presented to compute unsteady compressible flows for complex geometries. In this method, a Cartesian mesh based on a building‐block grid is used as a baseline mesh to cover the computational domain, while the boundary surfaces are represented using a set of gridless points. This hybrid method combines the efficiency of a Cartesian grid method and the flexibility of a gridless method for the complex geometries. The developed method is used to compute a number of test cases to validate the accuracy and efficiency of the method. The numerical results obtained indicate that the use of this hybrid method leads to a significant improvement in performance over its unstructured grid counterpart for the time‐accurate solution of the compressible Euler equations. An overall speed‐up factor from six to more than one order of magnitude and a saving in storage requirements up to one order of magnitude for all test cases in comparison with the unstructured grid method are demonstrated. Copyright © 2008 John Wiley & Sons, Ltd.     

Symmetry techniques for the numerical solution of the 2D Euler equations at impermeable boundaries

The implementation of boundary conditions at rigid, fixed wall boundaries in inviscid Euler solutions by upwind, finite volume methods is considered. Some current methods are reviewed. Two new boundary condition procedures, denoted as the symmetry technique and the curvature-corrected symmetry technique are then presented. Their behaviour in relation to the problem of the subsonic flow about blunt and slender elliptic bodies is analysed. The subsonic flow inside the Stanitz elbow is then computed. The symmetry technique is proven to be as accurate as one of the current methods, second-order pressure extrapolation technique. Finally, for arbitrary curved geometries, dramatic advantages of the curvature-corrected symmetry technique over the other methods are shown. © 1998 John Wiley & Sons, Ltd.     

用拟压缩性方法求解不可压Euler方程

用拟压缩性方法和Ｊａｍｅｓｏｎ的有限体积算法求解了二维和三维定常可可压Ｅｕｌｅｒ方程。分别采用显、隐式时间离散推进求解;分析了人工粘性的阶数对定常解收敛性的影响,应用该方法计算了单个翼型和翼身组合体的低速绕流,结果与实验吻合较好。     

一般形式的二维时—空守恒格式

本文将经作者改进后的一维时－空守恒格式推广到了二维情形，得到了一个一般形式的二维Ｅｕｌｅｒ方程时－空守恒格式，该格式对各种不规则几何区域内的流动问题具有很强的适应性，同时它还保留了一维格式的优点。几个典型算例的计算结果表明，本文格式不仅精度高，通用性好，而且对激波等间断具有很高的分辨率。     

An adaptive least‐squares method for the compressible Euler equations

An adaptive least‐squares finite element method is used to solve the compressible Euler equations in two dimensions. Since the method is naturally diffusive, no explicit artificial viscosity is added to the formulation. The inherent artificial viscosity, however, is usually large and hence does not allow sharp resolution of discontinuities unless extremely fine grids are used. To remedy this, while retaining the advantages of the least‐squares method, a moving‐node grid adaptation technique is used. The outstanding feature of the adaptive method is its sensitivity to directional features like shock waves, leading to the automatic construction of adapted grids where the element edge(s) are strongly aligned with such flow phenomena. Using well‐known transonic and supersonic test cases, it has been demonstrated that by coupling the least‐squares method with a robust adaptive method shocks can be captured with high resolution despite using relatively coarse grids. Copyright © 1999 John Wiley & Sons, Ltd.     

A composite grid approach for the Euler equations

A zonal grid methodology has been developed for the calculation of compressible fluid flows. The domain subdivision is based on patched grid systems composed of zones or blocks within which a distinct curvilinear grid is generated. The flow simulation is then carried out with a modified scheme based on the Euler finite volume solver of Ni. This scheme uses a distribution procedure that provides an easy and accurate way for the transfer of information from one block to another. This method results in a naturally conservative computation at the interfaces. It is analysed and developed for the treatment of embedded grids with a grid point common to more than four blocks.     

Least-squares finite element methods for compressible Euler equations

A method based on backward finite differencing in time and a least-squares finite element scheme for first-order systems of partial differential equations in space is applied to the Euler equations for gas dynamics. The scheme minimizes the L²-norm of the residual within each time step. The method naturally generates numerical dissipation proportional to the time step size. An implicit method employing linear elements has been implemented and proves robust. For high-order elements, computed solutions based on the L²-method may have oscillations for calculations at similar time step sizes. To overcome this difficulty, a scheme which minimizes the weighted H¹-norm of the residual is proposed and leads to a successful scheme with high-degree elements. Finally, a conservative least-squares finite element method is also developed. Numerical results for two-dimensional problems are given to demonstrate the shock resolution of the methods and compare different approaches.     

Improved artificial dissipation schemes for the Euler equations

The influence of artificial dissipation schemes on the accuracy and stability of the numerical solution of compressible flow is extensively examined. Using an implicit central difference factored scheme, an improved form of artificial dissipation is introduced which highly reduces the errors due to numerical viscosity. A function of the local Mach number is used to scale the amount of numerical damping added into the solution according to the character of the flow in several flow regimes. The resulting scheme is validated through several inviscid flow test cases.     

Application of the method of lines to unsteady compressible Euler equations

In the sense of method of lines, numerical solution of the unsteady compressible Euler equations in 1D, 2D and 3D is split into three steps: First, space discretization is performed by the first‐order finite volume method using several approximate Riemann solvers. Second, smoothness and Lipschitz continuity of RHS of the arising system of ordinary dimensional equations (ODEs) is analysed and its solvability is discussed. Finally, the system of ODEs is integrated in time by means of implicit and explicit higher‐order adaptive schemes offered by ODE packages ODEPACK and DDASPK, by a backward Euler scheme based on the linearization of the RHS and by higher‐order explicit Runge–Kutta methods. Time integrators are compared from several points of view, their applicability to various types of problems is discussed, and 1D, 2D and 3D numerical examples are presented. Copyright © 2003 John Wiley & Sons, Ltd.