Implicit schemes for unsteady Euler equations on triangular meshes

An implicit finite element method is presented for the solution of steady and unsteady inviscid compressible flows on triangular meshes under transonic conditions. The method involves a first-order time-stepping scheme with a finite element discretization that reduces to central differencing on a rectangular mesh. On a solid wall the slip condition is prescribed and the pressure is obtained from an approximation of the normal momentum equation. With this solver no artificial viscosity is added to ensure the success of the calculation. Numerical examples are given for steady and unsteady cases.     

用隐式方法和WENO格式计算悬停旋翼跨声速无粘流场

寻找一种能够准确计算以涡为主要特征的复杂流场和克服尾迹耗散问题的数值方法,一直是旋翼空气动力学研究的热点和难点。本文发展了一种基于高阶迎风格式计算悬停旋翼无粘流场的隐式数值方法。无粘通量采用Roe通量差分分裂格式,为提高精度,使用五阶WENO格式进行左右状态插值,并与MUSCL插值进行比较。为提高收敛到定常解的效率,时间推进采用LU-SGS隐式方法。用该方法对一跨声速悬停旋翼无粘流场进行了数值计算,数值结果表明WENO-Roe的激波分辨率高于MUSCL-Roe,体现出了格式精度的提高对计算结果的改善,LU-SGS隐式方法的计算效率比5步Runge-Kutta显式方法的高。     

Domain Decomposition Methods for Sensitivity Analysis of a Nonlinear Aeroelasticity Problem

Abstract

We consider the nonlinear aeroelasticity problem of the interaction between a viscous, incompressible fluid and Lin elastic solid undergoing large displacement. The non-linearities of the problem formulation include the solid and fluid governing equations. as well as thc dependence of the How geometry on the solid deformation. The resulting coupling is thus two-way. We develop domain-decomposition methods for solution and sensitivity analysis of the coupled problem. The domain decomposition is in the form of a block-Gauss-Seidel-like prcconditioncr that decomposes ihc coupled-domain problem into distinct nonovcrlapping fluid and solid subdotnain problems. The preconditioner thus enables exploitation or single-domain algorithms for solid and fluid mechanics discretization and solution. On the other hand, two-way fluid-solid coupling is retained within the residuals, which is essential for correct sensitivities. Sensitivities of field quantities can be found with little additional work beyond that required for solving the coupled fluid-solid system. The methodology developed here is illustrated by the solution of a problem of viscous incompressible flow about an infinite clastic cylinder. Sensitivities of the resulting velocity and displacement fields with respect to elastic modulus and fluid viscosity are computed.     

A stabilized finite element formulation for high‐speed inviscid compressible flows using finite calculus

This paper aims at the development of a new stabilization formulation based on the finite calculus (FIC) scheme for solving the Euler equations using the Galerkin FEM on unstructured triangular grids. The FIC method is based on expressing the balance of fluxes in a space–time domain of finite size. It is used to prevent the creation of instabilities typically present in numerical solutions due to the high convective terms and sharp gradients. Two stabilization terms, respectively called streamline term and transverse term, are added via the FIC formulation to the original conservative equations in the space–time domain. An explicit fourth‐order Runge–Kutta scheme is implemented to advance the solution in time. The presented numerical test examples for inviscid flows prove the ability of the proposed stabilization technique for providing appropriate solutions especially near shock waves. Although the derived methodology delivers precise results with a nearly coarse mesh, a mesh refinement technique is coupled to the solution process for obtaining a suitable mesh particularly in the high‐gradient zones. Copyright © 2014 John Wiley & Sons, Ltd.     

A numerical method for simulating discontinuous shallow flow over an infiltrating surface

A numerical method based on the MacCormack finite difference scheme is presented. The method was developed for simulating two‐dimensional overland flow with spatially variable infiltration and microtopography using the hydrodynamic flow equations. The basic MacCormack scheme is enhanced by using the method of fractional steps to simplify application; treating the friction slope, a stiff source term, point‐implicitly, plus, for numerical oscillation control and stability, upwinding the convective acceleration term. A higher‐order smoothing operator is added to aid oscillation control when simulating flow over highly variable surfaces. Infiltration is simulated with the Green–Ampt model coupled to the surface water component in a manner that allows dynamic interaction. The developed method will also be useful for simulating irrigation, tidal flat and wetland circulation, and floods. Copyright © 2000 John Wiley & Sons, Ltd.     

Implicit solvers using stabilized mixed approximation

This paper concerns a new class of robust and efficient methods for solving the Navier–Stokes equations for unsteady incompressible flow. In previous work we established the effectiveness of an implicit time integrator using a stabilized trapezoid rule with an explicit Adams–Bashforth method for error control. The role of the stability of the spatial approximation on the overall accuracy of the implicit solution algorithm is the primary focus here. In particular, the relationship between spatial stabilization and temporal solution accuracy is assessed computationally for the case of the lowest order conforming mixed approximation. Copyright © 2012 John Wiley & Sons, Ltd.     

非稳态流动的隐式最小二乘等几何方法

对非稳态粘性不可压流动问题提出了一种隐式最小二乘等几何计算方法。该方法先用隐式的向后多步差分格式对Navier-Stokes方程进行时间离散,再用Newton法线性化对流项,最后在每个时间步上用最小二乘等几何方法进行求解。根据该算法编制了计算程序,通过构造解析解的方法验证了程序的正确性,用该程序求解了雷诺数为5000时的非稳态二维顶盖驱动流问题,计算结果捕捉到了流动过程中涡的演化过程,表明本文方法可用于非稳态流动的求解。     

有限谱ENO格式及其应用

首先对王健平提出的有限谱法^[1-3]做了进一步的理论研究，发现了一些新的有限谱法的插值基函数组，并互将有限谱法应用在ENO格式中，构造了有限谱ENO计算格式，然后通过对一维Euler方程的几个经典的模型问题和二维湍流与弱激波相互作用问题的数值计算，并且与理论解或准精确解进行比较分析，从而表明了此格式对于激波和其他间断具有较高的分辨率，在激波附近基本上没有明显的数值振荡，而且对于流场中的细致结构也具有相当高的精度。     

循环硬化材料本构模型的隐式应力积分和有限元实现

针对新发展的、能够描述循环硬化行为应变幅值依赖性的粘塑性本构模型,讨论了它的数值实现方法。首先,为了能够对材料的循环棘轮行为（Ratcheting）和循环应力松弛现象进行描述,对已有的本构模型进行了改进;然后,在改进模型的基础上,建立了一个新的、全隐式应力积分算法,进而推导了相应的一致切线刚度（Consistent Tangent Modulus）矩阵的表达式;最后,通过ABAQUS用户材料子程序UMAT将上述本构模型进行了有限元实现,并通过一些算例对一些构件的循环变形行为进行了有限元数值模拟,讨论了该类本构模型有限元实现的必要性和合理性。     

Parallel finite element methods for large-scale computation of storm surges and tidal flows

Massively parallel finite element methods for large-scale computation of storm surges and tidal flows are discussed here. The finite element computations, carried out using unstructured grids, are based on a three-step explicit formulation and on an implicit space–time formulation. Parallel implementations of these unstructured grid-based formulations are carried out on the Fujitsu Highly Parallel Computer AP1000 and on the Thinking Machines CM-5. Simulations of the storm surge accompanying the Ise-Bay typhoon in 1959 and of the tidal flow in Tokyo Bay serve as numerical examples. The impact of parallelization on this type of simulation is also investigated. The present methods are shown to be useful and powerful tools for the analysis of storm surges and tidal flows. © 1997 John Wiley & Sons, Ltd.     

伽辽金有限元素法对旋翼气弹稳定性的应用

采用伽辽金加权余数有限元素法发展了一种悬停状态下无铰旋翼桨叶气弹稳定性的分析方法。分析模型包括预锥角、下垂角、预掠角、总距角、桨根预安装角、桨叶预扭角、变距轴偏置、根部外伸量和操纵线系刚度等结构参数，对无铰旋翼桨叶气弹稳定性研究有普遍适用意义。试验证明该理论可行并能用于研究无铰旋翼结构参数对桨叶气弹稳定性的影响，也能用于直升机旋翼的型号设计。     

Effects of Entropic Wave in Vibroacoustic Problem Using Boundary Element Analysis

A variational formulation for a vibroacoustic problem of a membrane and a viscothermal fluid is investigated in this paper. The formulation combines a variational formulation by integral equations of the fluid, that takes into account the acoustic and entropic waves coupling, with a variational formulation of the membrane. The formulation has been implemented numerically for the problems with axisymmetric geometry. The numerical results are compared to the analytical solution for a circular membrane coupled to a cylindrical cavity filled with air. These results show the validity of numerical implementation and illustrate the thermal effects of air on the membrane-cavity system modes in the micro cavities cases.     

On numerical techniques in CFD

Numerical techniques play an important role in CFD. Some of them are reviewed in this paper. The necessity of using high order difference scheme is demonstrated for the study of high Reynolds number viscous flow. Physical guide lines are provided for the construction of these high order schemes. To avoid unduly ad hoc tremtment in the boundary region the use of compact scheme is recommended because it has a small stencil size compared with the traditional finite difference scheme. Besides preliminary Fourier analysis shows the compact scheme can also yield better space resolution which makes it more suitable to study flow with multiscales e.g. turbulence. Other approaches such as perturbation method and finite spectral method are also emphasized. Typical numerical simulations were carried out. The first deals with Euler equations to show its capabilities to capture flow discontinuity. The second deals with Navier-Stokes equations studying the evolution of a mixing layer, the pertinent structures at different times are shown. Asymmetric break down occurs and also the appearance of small vortices. The project supported by the National Natural Science Foundation of China (No.19393100) The Lecture in Memory of Prof. Pei-Yuan Chou, Presented at the Eighth Asian Congress of Fluid Mechanics, Shenzhen, China, December 6–10, 1999     

A second-order positivity-preserving well-balanced finite volume scheme for Euler equations with gravity for arbitrary hydrostatic equilibria

We present a well-balanced finite volume scheme for the compressible Euler equations with gravity, where the approximate Riemann solver is derived using a Suliciu relaxation approach. Besides the well-balanced property, the scheme is robust with respect to the physical admissible states. General hydrostatic solutions are captured up to machine precision by deriving, for a given initial value problem, suitable time-independent functions and using them in the discretization of the source term. The first-order scheme is extended to a second-order scheme by reconstructing in equilibrium variables while preserving the well-balanced and robustness properties. Numerical examples are performed to demonstrate the accuracy, well-balanced, and robustness properties of the presented scheme for up to three space dimensions.     

NND格式在非结构网格中的推广

在张涵信提出的无波动、无自由参数的差分格式（ＮＮＤ格式）的基础上，构造了适用于非结构网格的二阶精度ＮＮＤ有限体积格式，解决了现有非结构网格方法中为抑制激波附近的波动而必须引入含自由参数的人工粘性项的困难，并采用网格自适应技术以提高效率．通过对二维平板激波反射和前台阶在管道内的流动问题的计算，表明本方法可有效地用于Ｅｕｌｅｒ方程的求解．     

水槽涡的变化规律数值模拟

建立了不可压缩Navier-Stokes方程的Crank-Nicolson有限差分方法,数值模拟水槽晃动中流场及其涡流的数值变化规律。将数值解与解析解和前人的数值解进行比较,数值验证了不可压缩Naver-Stokes方程有限差分方法的有效性。通过数值模拟得到水槽在不同程度的倾斜激励晃动下流场及涡流的数值变换规律,当倾斜激励晃动的频率接近或远离共振频率时,水槽涡场的变化逐步由双涡变成单涡,再到不规则的涡场。当倾斜激励晃动的频率靠近共振频率ω_p=0.95ω_1附近时,水槽流场上部形成一个小涡,然后小涡扩大成整个水槽中的大涡,大涡下沉分裂成两个单涡,最后在底部消失;当倾斜激励晃动的频率在ω_p=0.75ω_1附近时,水槽底部形成一个小涡,然后扩大成大的单涡,最后在自由面消失;当倾斜激励晃动的频率在ω_p=0.55ω_1附近时,水槽底部出现小涡,然后扩大成大的单涡,大涡在自由面消失,继而出现不规则的大涡和不规则的小涡。     

A streamline diffusion nonconforming finite element method for the time-dependent linearized Navier-Stokes equations

A nonconforming finite element method of finite difference streamline diffusion type is proposed to solve the time-dependent linearized Navier-Stokes equations. The backward Euler scheme is used for time discretization. Crouzeix-Raviart nonconforming finite element approximation, namely, nonconforming （P1）2 - P0 element, is used for the velocity and pressure fields with the streamline diffusion technique to cope with usual instabilities caused by the convection and time terms. Stability and error estimates are derived with suitable norms.     

The dynamic plastic response characteristics of circular beam

In this paper the problem of a circular beam subjected to radial impact by a rigid mass at its tip in its own plane is investigated on the basis of rigid-perfectly plastic assumption. The analytical solution of the particle velocities is obtained as the function of travelling plastic hinge location. By analysing the solution, some special properties of circular beam problem are found.     

岩土弹塑性有限元分析的隐式积分弹性刚度法

本文提出了用于岩土弹塑性有限元分析的隐式积分弹性刚度算法。该算法既具有隐式积分法精度好、效率高、无条件稳定等优点，也具有弹性刚度法中刚度矩阵正定、对称的特点，更重要的是它避免了传统切线刚度法在处理岩土非相关联塑性流动和屈服面“角点”所遇到的非对称性和奇异性计算问题。通过算例分析了该算法的精度、效率