Characteristic-based finite volume scheme for 1D Euler equations

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。     

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。     

THREE HIGH-ORDER SPLITTING SCHEMES FOR 3D TRANSPORT EQUATION

IntroductionWith the development of modern industry, various pollutants discharge into the air,rivers, lakes and oceans, which makes the environmental qualities worse and has bad effectson the mankind’s health and the sustained development of industry an…     

用DQM求解成层地基一维非线性固结问题

由于多层地基的一维非线性固结问题求解的复杂性,其解析解很难求得。本文基于Davis和Raymond一维非线性固结理论,利用DQM(Differential Quadrature Method)导了初始有效应力沿深度变化、任意边界条件、任意荷载作用下成层地基一维非线性固结的统一表达式,求得了孔压、有效应力和平均固结度的解答。通过解的收敛性分析讨论了DQM解的有效性。由于DQM解对于固结间题各种复杂条件具有统一的矩阵表达式,更便于编程计算和工程应用。最后,用本文解答对三层地基一维非线性固结问题进行了讨论。     

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。     

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。     

A CLASS OF COMPACT UPWIND TVD DIFFERENCE SCHEMES

A new method was proposed for constructing total variation diminishing (TVD) upwind schemes in conservation forms. Two limiters were used to prevent non-physical oscillations across discontinuity. Both limiters can ensure the nonlinear compact schemes TVD property. Two compact TVD (CTVD) schemes were tested, one is third-order accuracy, and the other is fifth-order. The performance of the numerical algorithms was assessed by one-dimensional complex waves and Riemann problems, as well as a two-dimensional shock-vortex interaction and a shock-boundary flow interaction. Numerical results show their high-order accuracy and high resolution, and low oscillations across discontinuities.     

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).     

Three-dimensional free vibration analysis of functionally graded piezoelectric annular plates on elastic foundations

Three-dimensional free vibration analysis of functionally graded piezoelectric (FGPM) annular plates resting on Pasternak foundations with different boundary conditions is presented. The material properties are assumed to have an exponent-law variation along the thickness. A semi-analytical approach which makes use of state-space method in thickness direction and one-dimensional differential quadrature method in radial direction is utilized to obtain the influences of the Winkler and shearing layer elastic coefficients of the foundations on the non-dimensional natural frequencies of functionally graded piezoelectric annular plates. The analytical solution in the thickness direction can be acquired using the state-space method and approximate solution in the radial direction can be obtained using the one-dimensional differential quadrature method. Numerical results are given to demonstrate the convergency and accuracy of the present method. The influences of the material property graded index, circumferential wave number and thickness of the annular plate on the dynamic behavior are also investigated. Since three-dimensional free vibration analysis of FGPM annular plates on elastic foundations has not been implemented before, the new results can be used as benchmark solutions for future researches.     

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。     

基于交叉降阶取样的改进响应面方法

针对经典响应面法和加权响应面法存在的计算精度不高和稳定性不强的问题,提出一种新的样本点选择策略,即交叉降阶思想,并基于此发展出一种新的改进响应面方法。该方法以均值点加上0.5倍的标准差作为初始迭代点。在Bucher试验设计的基础之上,将n维坐标系降阶为n个一维坐标系。然后,依据样本点的重要性,筛选出n+1个靠近真实失效面的优秀样本点,并用这些样本点去拟合线性响应面函数。最后依据样本中心间的距离与上一次迭代得到的设计点的范数之商作为其收敛准则。通过5个数值算例与工程算例,验证了交叉降阶响应面法具有较高的精度、效率、稳定性和一定的工程适用性。     

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.     

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.     

Harten-Lax-van Leer-contact (HLLC) approximation Riemann solver with elastic waves for one-dimensional elastic-plastic problems

A Harten-Lax-van Leer-contact (HLLC) approximate Riemann solver is built with elastic waves (HLLCE) for one-dimensional elastic-plastic flows with a hypoelastic constitutive model and the von Mises’ yielding criterion. Based on the HLLCE, a third-order cell-centered Lagrangian scheme is built for one-dimensional elastic-plastic problems. A number of numerical experiments are carried out. The numerical results show that the proposed third-order scheme achieves the desired order of accuracy. The third-order scheme is used to the numerical solution of the problems with elastic shock waves and elastic rarefaction waves. The numerical results are compared with a reference solution and the results obtained by other authors. The comparison shows that the presented high-order scheme is convergent, stable, and essentially non-oscillatory. Moreover, the HLLCE is more efficient than the two-rarefaction Riemann solver with elastic waves (TRRSE).     

基于改进位移模式的一维C1有限元超收敛算法

提出了基于改进位移模式的一维C1有限元超收敛算法。利用单元内部需满足平衡方程的条件,推导了超收敛计算解析公式的显式,即将高阶有限元解的位移模式用常规有限元解的位移模式表示。用常规有限元解的位移模式与高阶有限元解的位移模式之和构造新的位移模式。采用积分形式推导了单元刚度矩阵。该算法在前处理阶段使用了超收敛计算公式,在常规试函数的基础上,增加了高阶试函数,使得单元内平衡方程的残差减少,从而达到提高精度的目标。对于Hermite单元,本文的结点和单元的位移、导数都达到了h4阶的超收敛精度。     

THE INFLUENCE OF GLOBAL-DIRECTION STENCIL ON GRADIENT AND HIGH-ORDER DERIVATIVES RECONSTRUCTION OF UNSTRUCTURED FINITE VOLUME METHODS 1)

模板选择方式对非结构有限体积方法的计算准确性会产生显著影响. 在之前的工作中, 基于局部方向模板存在的问题, 我们探索了一种更加简单有效的全局方向模板选择方法, 并将其应用于二阶精度非结构有限体积求解器. 基于该方法找到的模板单元均沿着壁面法向与流向, 可有效捕捉流场变化, 反映流动的各向异性, 并且模板选择过程脱离了对网格拓扑的依赖, 避免了局部方向模板选择方法中复杂的阵面推进与方向判断过程, 克服了在大压缩比三角形网格上模板单元偏离壁面法向的现象, 同时在二阶精度求解器上得到了较高的计算精度与计算准确性. 为了进一步验证全局方向模板在高阶精度非结构有限体积方法中应用的可行性, 本文初步测试了该模板对变量梯度及高阶导数重构的影响. 经检验, 在不同类型的网格上, 采用全局方向模板得到的变量梯度与高阶导数误差明显低于局部方向模板, 同时也低于共点模板的计算误差. 此外, 在高斯积分点处由全局方向模板得到的变量点值与导数误差同样在三种模板中最低. 因此该模板选择方法在非结构有限体积梯度与高阶导数重构方面具有较好的数值表现, 具备在高阶精度非结构有限体积求解器中应用并推广的可行性.     

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.     

基于核重构思想的配点型无网格方法的研究--一维问题

无网格方法按其离散原理可分为Galerkin型、配点型等。其中Galerkin型无网格方法的实施需要背景网格，不属于真正的无网格法；配点型无网格方法的实施不需要背景网格，是真正的无网格法。本文首先介绍了重构核点法的基本原理，然后基于核重构思想，与配点法相结合，以一维问题为例，研究了配点型无网格方法，对该方法构造过程中的近似函数及其导数的计算、修正函数的计算及方法的实现等问题进行了探讨。并结合若干典型算例，检验了其计算精度与收敛姓。