Characteristic-based finite volume scheme for 1D Euler eouations

In this paper, a high-order finite-volume scheme is presented for the one-dimensional scalar and inviscid Euler conservation laws. The Simpson's quadrature rule is used to achieve high-order accuracy in time. To get the point value of the Simpson's quadrature, the characteristic theory is used to obtain the positions of the grid points at each sub-time stage along the characteristic curves, and the third-order and fifth-order central weighted essentially non-oscillatory (CWENO) reconstruction is adopted to estimate the cell point values. Several standard one-dimensional examples are used to verify the high-order accuracy, convergence and capability of capturing shock.     

微分求积单元法在结构工程中的应用

微分求积法（Differential Quadrature Method）是求鳃偏微分方程和积分-微分方程的一种数值方法,该法具有计算简便、精度较高和易于实现等优点。微分求积单元法（Differential Quadrature Element Method）是在微分求积法的基础上结合区域分割和集成规则而形成的一种新的数值计算方法,能通过自适应地选取微分求积网点数目正确模拟构件的刚度和荷载性质,其精度可通过细分单元或增加离散点数目加以提高。微分求积单元法是一种可供选择的、性能优越的数值计算方法。本文将详细论述这一数值方法的基本原理,并通过数值算例说明该方法的应用过程及其优越性,为这一方法在结构工程中的推广应用提供参考。     

Non-singular direct formulation of boundary integral equations for potential flows

This paper presents a general direct integral formulation for potential flows. The singularities of Green's functions are desingularized theoretically, using a subtracting and adding back technique, so that Gaussian quadrature or any other numerical integration methods can be applied directly to evaluate all the integrals without any difficulty. When high-order quadrature formulas are applied globally, the number of unknowns can be reduced. Interpolation functions are not necessary for unknown variables in the present paper. Therefore, the present method is much simpler and more efficient than the conventional one. Several numerical examples are calculated and compared satisfactorily with analytical solutions or published results. © 1998 John Wiley & Sons, Ltd.     

轴向均布载荷下压杆稳定问题的DQ解

叙述了微分求积法(differential quadrature method)的一般方法，研究用微分求积法求解在均布轴向载荷下细长杆的稳定问题．通过Newton-Raphson法求解非线性方程组，以及对问题进行线性假设后求解广义特征值方程，得到了精度很高的后屈曲挠度数值和临界载荷数值．与解析解和其他近似解相比，微分求积法具有较高的精度和简便性．     

角度非均匀材料平面V形切口应力奇性分析

提出了一种确定角度非均匀材料平面V形切口尖端应力奇性指数的有效方法。首先,在弹性力学基本方程中引入V形切口尖端位移场的级数渐近展开,建立以位移为特征函数的变系数和非线性微分方程组。然后,采用微分求积法(DQM)求解微分方程组,可得到多阶应力奇性指数及其相对应的特征函数,该法具有公式简单、编程方便、计算量少和精度高等优点,可处理任意开口角度和任意材料组合的V形切口。典型算例验证了微分求积法的有效性和精确性。     

Verification of higher-order discontinuous Galerkin method for hexahedral elements

A high-order implementation of the Discontinuous Galerkin (dg) method is presented for solving the three-dimensional Linearized Euler Equations on an unstructured hexahedral grid. The method is based on a quadrature free implementation and the high-order accuracy is obtained by employing higher-degree polynomials as basis functions. The present implementation is up to fourth-order accurate in space. For the time discretization a four-stage Runge–Kutta scheme is used which is fourth-order accurate. Non-reflecting boundary conditions are implemented at the boundaries of the computational domain.The method is verified for the case of the convection of a 1D compact acoustic disturbance. The numerical results show that the rate of convergence of the method is of order $p+1$ in the mesh size, with p the order of the basis functions. This observation is in agreement with analysis presented in the literature. To cite this article: H. Özdemir et al., C. R. Mecanique 333 (2005).     

微分求积法的特性及其改进

微分求积法已在科学和工程计算中得到了广泛应用。然而,有关时域微分求积法的数值稳定性、计算精度即阶数等基本特性,仍缺乏系统性的分析结论。依据微分求积法的基本原理,推导证明了微分求积法的权系数矩阵满足V-变换这一重要特性;利用微分求积法和隐式Runge-Kutta法的等值性,证明了时域微分求积法是A-稳定、s级s阶的数值方法。在此基础上,为进一步提高传统微分求积法的计算精度,利用待定系数法和Padé逼近,推导出了一类新的s级2s阶的微分求积法。数值计算对比结果验证了所提出的新微分求积法比传统的微分求积法具有更高的计算精度。     

Mechanical quadrature methods and extrapolation for solving nonlinear boundary Helmholtz integral equations

This paper presents mechanical quadrature methods (MQMs) for solving nonlinear boundary Helmholtz integral equations. The methods have high accuracy of order O(h ³) and low computation complexity. Moreover, the mechanical quadrature methods are simple without computing any singular integration. A nonlinear system is constructed by discretizing the nonlinear boundary integral equations. The stability and convergence of the system are proved based on an asymptotical compact theory and the Stepleman theorem. Using the h ³-Richardson extrapolation algorithms (EAs), the accuracy to the order of O(h ⁵) is improved. To slove the nonlinear system, the Newton iteration is discussed extensively by using the Ostrowski fixed point theorem. The efficiency of the algorithms is illustrated by numerical examples.     

High accuracy eigensolution and its extrapolation for potential equations

From the potential theorem, the fundamental boundary eigenproblems can be converted into boundary integral equations （BIEs） with the logarithmic singularity. In this paper, mechanical quadrature methods （MQMs） are presented to obtain the eigensolutions that are used to solve Laplace＇s equations. The MQMs possess high accuracy and low computation complexity. The convergence and the stability are proved based on Anselone＇s collective and asymptotical compact theory. An asymptotic expansion with odd powers of the errors is presented. By the h3-Richardson extrapolation algorithm （EA）, the accuracy order of the approximation can be greatly improved, and an a posteriori error estimate can be obtained as the self-adaptive algorithms. The efficiency of the algorithm is illustrated by examples.     

集中载荷作用下梯度复合材料梁的小波-微分求积法分析

将梯度复合材料梁作为平面应力问题处理,采用小波和微分求积混合法,对集中荷载作用下结构的响应进行了分析.考虑材料特性参数沿高度方向呈梯度分布,在该方向上采用广义微分求积法进行离散;鉴于广义微分求积法求解集中荷载问题精度不高的缺点,在梁的长度方向上引入对突变信号敏感的小波插值函数.数值计算表明,小波-微分求积混合法不仅保留了广义微分求积法高效的优点,而且能够很好地模拟结构局部化特征.     

混合微分求积法在功能梯度材料平板柱形弯曲问题中的应用

利用混合微分求积法,对任意荷载作用下不同材料梯度分布的功能梯度材料平板柱形弯曲问题进行了分析。针对广义微分求积法求解集中荷载问题精度不高的缺点,本文利用小波微分求积法进行了改进。由于小波对突变信号具有良好的自适应描述能力,因此在平板宽度方向上,利用小波微分求积法可以有效地处理集中荷载;而在材料梯度变化的板厚方向上,则利用广义微分求积法计算量小且精度高的特点进行离散计算。计算表明,混合微分求积法不仅保留了广义微分求积法高效的特点,而且能有效地求解任意荷载作用的问题。通过算例,分析了在机械荷载作用下,材料不同梯度形式、平板上下表面材料性质差异对功能梯度平板结构响应的影响。     

Two-stage fourth-order gas-kinetic scheme for three-dimensional Euler and Navier-Stokes solutions

ABSTRACT

For the one-stage third-order gas-kinetic scheme (GKS), successful applications have been achieved for the three-dimensional compressible flows [Pan, L., K. Xu, Q. Li, and J. Li. 2016. “An Efficient and Accurate Two-stage Fourth-order Gas-kinetic Scheme for the Navier-Stokes Equations.” Journal of Computational Physics 326: 197–221]. The high-order accuracy of the scheme is obtained by integrating a multidimensional time-accurate gas distribution function over the cell interface within a time step without using Gaussian quadrature points and Runge-Kutta time-stepping technique. However, to the further increase of the order of the scheme, such as the fourth-order one, the one step formulation becomes very complicated for the multidimensional flow. Recently, a two-stage fourth-order GKS with high efficiency has been constructed for two-dimensional inviscid and viscous flow computations ([Li, J., and Z. Du. 2016. “A Two-stage Fourth Order Time-accurate Discretization for Lax-Wendroff Type Flow Solvers I. Hyperbolic Conservation Laws.” SIAM Journal on Scientific Computing 38: 3046–3069]; Pan et al. 2016), and the scheme uses the time accurate flux function and its time derivatives. In this paper, a fourth-order GKS is developed for the three-dimensional flows under the two-stage framework. Based on the three-dimensional WENO reconstruction and flux evaluation at Gaussian quadrature points on a cell interface, the high-order accuracy in space is achieved first. Then, the two-stage time stepping method provides the high accuracy in time. In comparison with the formal third-order GKS [Pan, L., and K. Xu. 2015. “A Third-order Gas-kinetic Scheme for Three-dimensional Inviscid and Viscous Flow Computations.” Computers & Fluids 119: 250–260], the current fourth-order method not only improves the accuracy of the scheme, but also reduces the complexity of the gas-kinetic flux solver greatly. More importantly, the fourth-order GKS has the same robustness as the second-order shock capturing scheme [Xu, K. 2001. “A Gas-kinetic BGK Scheme for the Navier-Stokes Equations and its Connection with Artificial Dissipation and Godunov Method.” Journal of Computational Physics 171: 289–335]. Numerical results validate the outstanding reliability and applicability of the scheme for three-dimensional flows, such as the cases related to turbulent simulations.     

基于径向基点插值法的旋转Mindlin板高次刚柔耦合动力学模型

将无网格径向基点插值法(radial point interpolation method,RPIM)用于中心刚体?旋转柔性板的动力学分析.基于浮动坐标系方法和一阶剪切变形理论即Mindlin板理论,考虑剪切变形的影响,并计入板面内变形的非线性耦合变形项,采用径向基点插值法描述板的变形场,保留动能中有关非线性耦合变形项...     

Efficient high-order immersed interface methods for heat equations with interfaces

An efficient high-order immersed interface method （IIM） is proposed to solve two-dimensional （2D） heat problems with fixed interfaces on Cartesian grids, which has the fourth-order accuracy in the maximum norm in both time and space directions. The space variable is discretized by a high-order compact （HOC） difference scheme with correction terms added at the irregular points. The time derivative is integrated by a Crank-Nicolson and alternative direction implicit （ADI） scheme. In this case, the time accuracy is just second-order. The Richardson extrapolation method is used to improve the time accuracy to fourth-order. The numerical results confirm the convergence order and the efficiency of the method.     

Galerkin meshless methods based on partition of unity quadrature

Introduction Meshlessmethodsarenewmethodsofnumericalcomputationwhichhavebeendeveloped rapidlyinrecentyears.Inthesemethods,onlynodesareneeded,meshinformationistotally unnecessary.Thiscanavoidorpartlyavoidthedifficultyofmeshgeneration.Duetohigh accuracyandstability,Galerkinmeshlessmethodsareappliedbroadly,butitisunavoidable tocomputetheintegrationoverthewholephysicaldomaininGalerkinweakform,whichisa greatchallengeforGalerkinmeshlessmethodsbecauseoftheabsenceofmesh.TocarryouttheintegrationinGal…     

非线性双曲型守恒律的高精度MmB差分格式

构造了一维非线性双曲型守恒律方程的一个高精度、高分辨率的广义G odunov型差分格式。其构造思想是:首先将计算区间划分为若干个互不相交的小区间,再根据精度要求等分小区间,通过各细小区间上的单元平均状态变量,重构各等分小区间交界面上的状态变量,并加以校正;其次,利用近似R iem ann解算子求解细小区间交界面上的数值通量,并结合高阶R unge-K u tta TVD方法进行时间离散,得到了高精度的全离散方法。证明了该格式的Mm B特性。然后,将格式推广到一、二维双曲型守恒方程组情形。最后给出了一、二维Eu ler方程组的几个典型的数值算例,验证了格式的高效性。     

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。     

高阶数值流形方法的初应力公式

高阶数值流形方法和高阶DDA方法可以显著提高结构变形的计算精度,但目前涉及几何非线性问题的研究成果大都计算精度差甚至不收敛,这是由高阶初应力公式的不准确或不正确引起的。本文介绍数值流形方法的大变形计算格式,基于平面三角形数学网格和多项式覆盖函数,提出高阶流形法的两种初应力处理方法,首次导出了高阶初应力的准确公式。该公式在分步计算的初应力累加中考虑了大变形结构的构形变化,并将初应力表示成多项式函数形式以满足单纯形积分的要求。文中给出的悬臂梁大变形数值算例与理论解的对比结果证明了方法的正确性。本文的方法和公式也适用于三维四面体数学网格,稍加修改后将可应用于高阶DDA方法和常规的有限元方法。     

NUMERICAL ANALYSIS OF MINDLIN SHELL BY MESHLESS LOCAL PETROV-GALERKIN METHOD

The objectives of this study are to employ the meshless local Petrov-Galerkin method （MLPGM） to solve three-dimensional shell problems. The computational accuracy of MLPGM for shell problems is affected by many factors, including the dimension of compact support domain, the dimension of quadrture domain, the number of integral cells and the number of Gauss points. These factors＇ sensitivity analysis is to adopt the Taguchi experimental design technology and point out the dimension of the quadrature domain with the largest influence on the computational accuracy of the present MLPGM for shells and give out the optimum combination of these factors. A few examples are given to verify the reliability and good convergence of MLPGM for shell problems compared to the theoretical or the finite element results.     

Holberg's optimisation for high-order compact finite difference staggered schemes

Two optimised high-order compact finite difference (FD) staggered schemes are presented in this communication. Following Holberg's optimisation strategy, the least squares problem to minimising the group velocity (MGV) error, for the fourth- and sixth-order pentadiagonal schemes, is formulated. For a fixed level of group velocity accuracy, the optimised spectrum of wave number and the optimised coefficients for the schemes, are analytically evaluated. The spectral accuracy of these schemes has been verified by several comparisons with the FD staggered schemes obtained following Kim and Lee's (1996) optimisation procedure. Fewer group and phase velocity errors, greater resolution in terms of absolute error and resolving efficiency have been achieved by the optimised schemes proposed. High-order accuracy in time is obtained by marching the solution with an optimised Runge–Kutta scheme. Next, the comparison in terms of the number of grid points per wavelength required to achieve a standard accuracy for distances expressed in terms of the number of wavelengths travelled is presented. Numerical results from benchmark tests for the one-dimensional shallow water equations are presented.