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。     

Adomian修正分解法的后处理方法比较

Adomian修正分解法在求解非线性微分方程中得到广泛应用。Adomian修正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。但是Adomian修正分解法的计算精度取决于其收敛域。为了扩大Adomian修正分解法的收敛域,需要对所得解进行后处理,目前常见的后处理方法包括Padé近似、LaplacePadé近似和多步迭代方法。本文首先简要回顾了Adomian修正分解法,然后讨论了这三种后处理方法,最后通过Duffing振子为例对这些后处理方法的优缺点进行讨论和分析。数值计算结果表明,多步迭代方法能够加速Adomian修正分解法解的收敛,并扩大其收敛域。     

弯扭耦合刚度对薄壁梁弯扭耦合振动的影响研究

通过Adomian修正分解法对包含弯扭耦合刚度的等截面弯扭耦合薄壁梁进行自由振动分析。通过Adomian修正分解法可以把弯扭耦合梁的特征微分方程组变换成为一组递归代数公式,随后通过边界条件即可得到该弯扭耦合梁的固有频率及相应的振形函数解析表达式。Adomian修正分解法的主要优点在于计算简单快速,并且不需要进行离散化或线性化。通过与前人的计算结果比较,本文方法的最大误差小于0.09%,从而验证了本文方法的有效性,并指出如果不考虑弯扭耦合刚度,第1阶和第3阶固有频率会高估30%。     

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。     

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。     

轴向力作用下双层梁的振动与稳定性分析

基于Adomian修正分解法研究轴向力作用下双层梁的自由振动和稳定性.通过Euler-Bernoulli梁振动理论建立轴向力作用下、具有Winkler弹性联系的双层梁自由振动微分方程组.并通过Adomian修正分解法把该特征微分方程组转换成递归代数公式,然后利用边界条件推导得到该双层梁的固有频率及相应的振形函数解析表达式.通过与前人的计算结果比较,验证了本文方法的有效性.并讨论了双层梁的厚度比以及作用在双层梁上的轴向力之比等参数对其固有频率和稳定性的影响.     

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。     

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.     

Chebyshev谱元方法结合并行算法求解三维区域的Helmholtz方程

本文探讨了采用Chebyshev谱元方法结合并行计算求解三维区域的Helmholtz方程问题。首先应用变分方法,得到了带有第一类边界条件的三维区域Helmholtz方程的弱形式。然后在三维的标准单元内,采用Chebyshev正交多项式展开函数u和试函数v,并且将其带入弱形式方程,通过积分,得到单元刚度矩阵;通过合成单元刚度矩阵,得到总体矩阵。最后通过基于MPI的并行计算,求解了以总体矩阵为系数的方程组,得到了Helmholtz方程的数值解,和解析解对比表明了数值解的正确性,并且数值解具有8阶精度。在并行求解方程组过程中,充分利用矩阵的对称性和矢量存储来获取上三角元素,这大幅的节约了存储量和计算进程间的通讯量,获得的并行效率可达76.6%。     

奇异边界法求解多连通域的瞬态热传导问题

提出一种基于奇异边界法结合双重互易法的数值模型来求解瞬态热传导问题。奇异边界法属于配点型边界无网格方法,相对于网格方法,其具有无需划分网格,只需边界配点的优势。运用差分格式来处理热传导方程中的时间变量,将原热传导方程化为非齐次修正Helmholtz方程。修正Helmholtz方程的解由齐次解和特解两部分组成,齐次解通过奇异边界法求出,特解由双重互易法求出,源项由径向基函数近似。通过数值算例检验了本文数值模型的精度及有效性;算例结果表明,该数值模型计算精度较高,误差基本都在1%以内,具有很好的稳定性,能有效地应用于求解多连通域的瞬态热传导问题。     

边界节点法的计算稳定性研究

边界节点法利用满足控制方程的非奇异通解作为基函数,半解析边界数值离散偏微分方程,具有精度高、收敛快、易编程等优点,是一种纯无网格配点方法.但是在求解具体问题时,随着节点数的增加,边界节点法经常得到严重病态的插值矩阵.本文利用有效条件数评价边界节点法求解Helmholtz问题线性方程组的计算稳定性;然后利用三种正则化方法处理其病态的线性方程组,并与高斯消元法比较计算精度和收敛性.通过数值实验,本文研究了有效条件数、误差和正则化方法之间的关系.     

Fourth‐order accurate compact‐difference discretization method for Helmholtz and incompressible Navier–Stokes equations

A fourth‐order accurate solution method for the three‐dimensional Helmholtz equations is described that is based on a compact finite‐difference stencil for the Laplace operator. Similar discretization methods for the Poisson equation have been presented by various researchers for Dirichlet boundary conditions. Here, the complicated issue of imposing Neumann boundary conditions is described in detail. The method is then applied to model Helmholtz problems to verify the accuracy of the discretization method. The implementation of the solution method is also described. The Helmholtz solver is used as the basis for a fourth‐order accurate solver for the incompressible Navier–Stokes equations. Numerical results obtained with this Navier–Stokes solver for the temporal evolution of a three‐dimensional instability in a counter‐rotating vortex pair are discussed. The time‐accurate Navier–Stokes simulations show the resolving properties of the developed discretization method and the correct prediction of the initial growth rate of the three‐dimensional instability in the vortex pair. Copyright © 2004 John Wiley & Sons, Ltd.     

Solution of the incompressible Navier–Stokes equations by the method of lines

The numerical method of lines (NUMOL) is a numerical technique used to solve efficiently partial differential equations. In this paper, the NUMOL is applied to the solution of the two‐dimensional unsteady Navier–Stokes equations for incompressible laminar flows in Cartesian coordinates. The Navier–Stokes equations are first discretized (in space) on a staggered grid as in the Marker and Cell scheme. The discretized Navier–Stokes equations form an index 2 system of differential algebraic equations, which are afterwards reduced to a system of ordinary differential equations (ODEs), using the discretized form of the continuity equation. The pressure field is computed solving a discrete pressure Poisson equation. Finally, the resulting ODEs are solved using the backward differentiation formulas. The proposed method is illustrated with Dirichlet boundary conditions through applications to the driven cavity flow and to the backward facing step flow. Copyright © 2015 John Wiley & Sons, Ltd.     

Helmholtz边界积分方程中奇异积分间接求解方法

提出了间接求解传统Helmholtz边界积分方程CBIE的强奇异积分和自由项系数,以及Burton-Miller边界积分方程BMBIE中的超强奇异积分的特解法。对于声场的内域问题,给出了满足Helmholtz控制方程的特解,间接求出了CBIE中的强奇异积分和自由项系数。对于声场外域对应的BMBIE中的超强奇异积分,按Guiggiani方法计算其柯西主值积分需要进行泰勒级数展开的高阶近似,公式繁复,实施困难。本文给出了满足Helmholtz控制方程和Sommerfeld散射条件的特解,提出了间接求出超强奇异积分的方法。推导了轴对称结构外场问题的强奇异积分中的柯西主值积分表达式,并通过轴对称问题算例证明了本文方法的高效性。数值结果表明,对于内域问题,采用本文特解法的计算结果优于直接求解强奇异积分和自由项系数的结果,且本文的特解法可避免针对具体几何信息计算自由项系数,因而具有更好的适用性。对于外域问题,两者精度相当,但本文的特解法可避免对核函数进行高阶泰勒级数展开,更易于数值实施。     

A complete boundary integral formulation for compressible Navier–Stokes equations

A complete boundary integral formulation for compressible Navier–Stokes equations with time discretization by operator splitting is developed using the fundamental solutions of the Helmholtz operator equation with different order. The numerical results for wall pressure and wall skin friction of two‐dimensional compressible laminar viscous flow around airfoils are in good agreement with field numerical methods. Copyright © 2004 John Wiley & Sons, Ltd.     

Complex variable function method for the scattering of plane waves by an arbitrary hole in a porous medium

Based on the complex variable function method, a new approach for solving the scattering of plane elastic waves by a hole with an arbitrary configuration embedded in an infinite poroelastic medium is developed in the paper. The poroelastic medium is described by Biot's theory. By introducing three potentials, the governing equations for Biot's theory are reduced to three Helmholtz equations for the three potentials. The series solutions of the Helmholtz equations are obtained by the wave function expansion method. Through the conformal mapping method, the arbitrary hole in the physical plane is mapped into a unit circle in the image plane. Integration of the boundary conditions along the unit circle in the image plane yields the algebraic equations for the coefficients of the series solutions. Numerical solution of the resulting algebraic equations yields the displacements, the stresses and the pore pressure for the porous medium. In order to demonstrate the proposed approach, some numerical results are given in the paper.     

Green quasifunction method for vibration of simply-supported thin polygonic plates on Pasternak foundation

A new numerical method—Green quasifunction is proposed.The idea of Green quasifunction method is clarified in detail by considering a vibration problem of simply-supported thin polygonic plates on Pasternak foundation.A Green quasifunction is established by using the fundamental solution and boundary equation of the problem. This function satisfies the homogeneous boundary condition of the problem.The mode shape differential equation of the vibration problem of simply-supported thin plates on Pasternak foundation is reduced to two simultaneous Fredholm integral equations of the second kind by Green formula.There are multiple choices for the normalized boundary equation.Based on a chosen normalized boundary equation,a new normalized boundary equation can be established such that the irregularity of the kernel of integral equations is overcome.Finally,natural frequency is obtained by the condition that there exists a nontrivial solution in the numerically discrete algebraic equations derived from the integral equations.Numerical results show high accuracy of the Green quasifunction method.     

A Newton's method scheme for solving free-surface flow problems

A Newton's method scheme is described for solving the system of non-linear algebraic equations arising when finite difference approximations are applied to the Navier–Stokes equations and their associated boundary conditions. The problem studied here is the steady, buoyancy-driven motion of a deformable bubble, assumed to consist of an inviscid, incompressible gas. The linear Newton system is solved using both direct and iterative equation solvers. The numerical results are in excellent agreement with previous work, and the method achieves quadratic convergence.     

Solution to transient Navier–Stokes equations by the coupling of differential quadrature time integration scheme with dual reciprocity boundary element method

The two‐dimensional time‐dependent Navier–Stokes equations in terms of the vorticity and the stream function are solved numerically by using the coupling of the dual reciprocity boundary element method (DRBEM) in space with the differential quadrature method (DQM) in time. In DRBEM application, the convective and the time derivative terms in the vorticity transport equation are considered as the nonhomogeneity in the equation and are approximated by radial basis functions. The solution to the Poisson equation, which links stream function and vorticity with an initial vorticity guess, produces velocity components in turn for the solution to vorticity transport equation. The DRBEM formulation of the vorticity transport equation results in an initial value problem represented by a system of first‐order ordinary differential equations in time. When the DQM discretizes this system in time direction, we obtain a system of linear algebraic equations, which gives the solution vector for vorticity at any required time level. The procedure outlined here is also applied to solve the problem of two‐dimensional natural convection in a cavity by utilizing an iteration among the stream function, the vorticity transport and the energy equations as well. The test problems include two‐dimensional flow in a cavity when a force is present, the lid‐driven cavity and the natural convection in a square cavity. The numerical results are visualized in terms of stream function, vorticity and temperature contours for several values of Reynolds (Re) and Rayleigh (Ra) numbers. Copyright © 2008 John Wiley & Sons, Ltd.