谱元方法求解二维不可压缩Navier-Stokes方程

利用有限元的思想并结合谱方法的精度提出求解偏微分方程的谱元方法,在元素内插值函数使用伪谱Chebyshev逼近,并将此方法应用于求解不可压Navier-Stokes方程,具体求解了二维方腔顶盖驱动流,与公认基准解对比获得了较好的结果。     

基本解方法与边界节点法求解Helmholtz方程的比较研究

基本解方法和边界节点法是基于径向基函数的两种重要无网格边界离散数值技术。针对Helmholtz方程,本文比较研究这两种数值方法在不同计算区域问题上的计算精度、插值矩阵对称性、病态性及计算成本。数值试验结果表明,两种方法都可以有效求解边界数据准确的Helmholtz问题。在数值离散过程中,两种方法都可以通过调整配置点的位...     

关于Orr—Sommerfeld方程的Chebyshev谱方法的讨论

本文讨论了Ｏｒｒ－Ｓｏｍｍｅｒｅｌｄ方程的各种Ｃｈｅｂｙｓｈｅｖ谱离散方法，数值证明了Ｃｈｅｂｙｓｈｅｖ配置法离散Ｏｒｒ－Ｓｏｍｍｅｒｆｅｌｄ方程没有伪谱，并以此构造了适于任意平面平行速度剖面情形，对时间和时空稳定性模式一致有效的无伪谱的离离散方法，其中，对时空稳定性问题本文给出了一种新的迭代法可以快速有效地求出复频率的鞍点，对平面Ｐｏｉｓｅｕｉｌｌｅ流，Ｂｌａｓｉｕｓ边界层流和Ｇａｕｓｓ模型尾迹     

通过Adomian分解法求解二维Helmholtz方程

提出基于Adomian分解法求解二维Helmholtz方程。通过Adomian分解法可以把Helmholtz微分方程和边界条件分别转换成递归代数公式和适用符号计算的简单代数公式。利用边界条件可以很容易得到方程的解析解表达式。Adomian分解法的主要特点在于计算简单快速,并且不需要进行线性化或离散化。最后给出数值实例以验证Adomian分解法求解二维Helmholtz方程的有效性。通过数值计算可以发现,基于Adomian分解法的计算结果非常接近精确解,并且该方法具有良好的收敛性。这表明Adomian分解法能够快速有效求解Helmholtz方程。     

通过Adomian分解法求解二维Helmholtz方程

提出基于Adomian分解法求解二维Helmholtz方程。通过Adomian分解法可以把Helmholtz微分方程和边界条件分别转换成递归代数公式和适用符号计算的简单代数公式。利用边界条件可以很容易得到方程的解析解表达式。Adomian分解法的主要特点在于计算简单快速,并且不需要进行线性化或离散化。最后给出数值实例以验证Adomian分解法求解二维Helmholtz方程的有效性。通过数值计算可以发现,基于Adomian分解法的计算结果非常接近精确解,并且该方法具有良好的收敛性。这表明Adomian分解法能够快速有效求解Helmholtz方程。     

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。     

谱元方法求解环形空间内自然对流换热

提出了将谱元方法应用到极坐标系下，利用极坐标系下的谱元方法求解环形空间内自然对流问题。具体求解了原始变量速度和压力的不可压缩Navier-Stokes方程和能量方程，通过在时间方向采用时间分裂方法和空间采用谱元方法对方程进行离散求解，取得了与基准解较一致的计算结果。     

Helmholtz方程的微分容积解法

用一种新型的数值技术－－微分容积法（Differential Cubature Method）求解二维Helmholtz方程的边值问题，几个数值算例表明，该方法稳定收敛，并具有较好的数值精度，本文方法适用于求解具有较小波数的Helmholtz方程。     

极坐标系下的Legendre谱元方法求解Poisson-型方程

r=0处的坐标奇异性是求解极坐标下Poisson-型方程的关键。本文提出一种极坐标系下基于Galerkin变分的Legendre谱元方法用于求解圆形区域内的Poisson-型方程,物理区域的径向和周向划分若干单元,计算单元均采用Legendre多项式展开;圆心所在单元的径向使用LGR(Legendre Gauss Radau)积分点,其他单元径向使用LGL(Legendre Gauss Lobatto)积分点,从而避免了极点处1/r坐标奇异性,周向单元均采用LGL积分点。利用区域分解技术,可以避免节点在极点附近聚集;最后求解了多个Dirichlet或Neumann边界条件下的Poisson-型方程算例。数值结果表明,谱元方法具有很高的精度。     

拟谱-微分求积混合方法求解一类双曲电报方程

拟谱方法和微分求积法是两类重要的无网格法,二者都已在科学和工程计算中获得了广泛应用。采用拉格朗日插值多项式作为二者的试函数,且采用同一种网格点分布,指出了在空间域上,微分求积法是拟谱方法的一种特殊形式。在此基础上,结合二者各自的特点,提出了拟谱-微分求积混合方法用于求解一类双曲电报方程。理论分析和数值测试表明,新方法在空间域上具有谱精度收敛性,在时间域上是A-稳定的,比较适合于求解多维电报方程。     

Parallel domain decomposition method for finite element approximation of 3D steady state non‐Newtonian fluids

We introduce a stabilized finite element method for the 3D non‐Newtonian Navier–Stokes equations and a parallel domain decomposition method for solving the sparse system of nonlinear equations arising from the discretization. Non‐Newtonian flow problems are, generally speaking, more challenging than Newtonian flows because the nonlinearities are not only in the convection term but also in the viscosity term, which depends on the shear rate. Many good iterative methods and preconditioning techniques that work well for the Newtonian flows do not work well for the non‐Newtonian flows. We employ a Galerkin/least squares finite element method, with stabilization parameters adjusted to count the non‐Newtonian effect, to discretize the equations, and the resulting highly nonlinear system of equations is solved by a Newton–Krylov–Schwarz algorithm. In this study, we apply the proposed method to some inelastic power‐law fluid flows through the eccentric annuli with inner cylinder rotation and investigate the robustness of the method with respect to some physical parameters, including the power‐law index and the Reynolds number ratios. We then report the superlinear speedup achieved by the domain decomposition algorithm on a computer with up to 512 processors. Copyright © 2015 John Wiley & Sons, Ltd.     

Dynamic analysis of beam-cable coupled systems using Chebyshev spectral element method

The dynamic characteristics of a beam–cable coupled system are investigated using an improved Chebyshev spectral element method in order to observe the effects of adding cables on the beam. The system is modeled as a double Timoshenko beam system interconnected by discrete springs. Utilizing Chebyshev series expansion and meshing the system according to the locations of its connections,numerical results of the natural frequencies and mode shapes are obtained using only a few elements, and the results are validated by comparing them with the results of a finiteelement method. Then the effects of the cable parameters and layout of connections on the natural frequencies and mode shapes of a fixed-pinned beam are studied. The results show that the modes of a beam–cable coupled system can be classified into two types, beam mode and cable mode, according to the dominant deformation. To avoid undesirable vibrations of the cable, its parameters should be controlled in a reasonable range, or the layout of the connections should be optimized.     

An adaptive cell-based domain integration method for treatment of domain integrals in 3D boundary element method for potential and elasticity problems

An adaptive cell-based domain integration method(CDIM) is proposed for the treatment of domain integrals in 3D boundary element method(BEM). The domain integrals are computed in background cells rather than volume elements. The cells are created from the boundary elements based on an adaptive oct-tree structure and no other discretization is needed. Cells containing the boundary elements are subdivided into smaller sub-cells adaptively according to the sizes and levels of the boundary elements; and the sub-cells outside the domain are deleted to obtain the desired accuracy. The method is applied in the 3D potential and elasticity problems in this paper.     

Chebyshev spectral method and Chebyshev noise processing procedure for vorticity calculation in PIV post-processing

Chebyshev spectral method and Chebyshev noise processing procedure are proposed for the calculation of vorticity from PIV experimental data. The Chebyshev spectral method offers superior intrinsic accuracy of derivative calculations. To overcome its noise sensitivity, the Chebyshev noise processing procedure can be applied prior to the derivative calculation to remove the high-frequency noise in the Chebyshev transform space. We compare the Chebyshev spectral method against the least-squares approach and test their performance in the calculation of vorticity with an Oseen vortex and with PIV data of the wake of a trapezoidal mixing tab. It is found that for clean velocity data the Chebyshev spectral method is extremely accurate. However, the Chebyshev spectral method alone is found to be more sensitive to noise than the least-squares method. When the Chebyshev noise processing procedure is applied together with the Chebyshev spectral method it greatly reduces the error and makes the Chebyshev spectral method more accurate than the least-squares method for a wide range of vorticity values. A special requirement imposed by the Chebyshev spectral method is that the PIV velocity processing must be carried out on special grids such as Gauss–Lobatto points.     

Direct spectral domain decomposition method for 2D incompressible Navier-Stokes equations

An efficient direct spectral domain decomposition method is developed coupled with Chebyshev spectral approximation for the solution of 2D, unsteady and incompressible Navier-Stokes equations in complex geometries. In this numerical approach, the spatial domains of interest are decomposed into several non-overlapping rectangular sub-domains. In each sub-domain, an improved projection scheme with second-order accuracy is used to deal with the coupling of velocity and pressure, and the Chebyshev collocation spectral method (CSM) is adopted to execute the spatial discretization. The influence matrix technique is employed to enforce the continuities of both variables and their normal derivatives between the adjacent sub-domains. The imposing of the Neumann boundary conditions to the Poisson equations of pressure and intermediate variable will result in the indeterminate solution. A new strategy of assuming the Dirichlet boundary conditions on interface and using the first-order normal derivatives as transmission conditions to keep the continuities of variables is proposed to overcome this trouble. Three test cases are used to verify the accuracy and efficiency, and the detailed comparison between the numerical results and the available solutions is done. The results indicate that the present method is efficiency, stability, and accuracy.     

Laplace transform method in boundary value problems for the Helmholtz equation

The purpose of this present note is to obtain boundary integral equation formulations of boundary value problems for the two-dimensional Helmholtz equation. The intergral equations are derived using the Laplace transform method.     

Chebyshev finite spectral method with extended moving grids

A Chebyshev finite spectral method on non-uniform meshes is proposed. An equidistribution scheme for two types of extended moving grids is used to generate grids. One type is designed to provide better resolution for the wave surface, and the other type is for highly variable gradients. The method has high-order accuracy because of the use of the Chebyshev polynomial as the basis function. The polynomial is used to interpolate the values between the two non-uniform meshes from a previous time step to the current time step. To attain high accuracy in the time discretization, the fourth-order Adams-Bashforth-Moulton predictor and corrector scheme is used. To avoid numerical oscillations caused by the dispersion term in the Korteweg-de Vries (KdV) equation, a numerical technique on non-uniform meshes is introduced. The proposed numerical scheme is validated by the applications to the Burgers equation (nonlinear convectiondiffusion problems) and the KdV equation (single solitary and 2-solitary wave problems), where analytical solutions are available for comparisons. Numerical results agree very well with the corresponding analytical solutions in all cases.     

并行元胞单元法

在对元胞单元法的基本原理与特点进行简要介绍之后,探讨了元胞单元法的并行计算方法,给出了算法描述和解题步骤。以平面应力问题分析为例,进行了数值试验,给出了详细的计算步骤和结果。结果表明元胞单元法可形成一种高度并行的算法,适用于未来并行机的要求,有望在大型结构和纳米力学的大规模摸拟方面得到应用。