基于高阶和高效格式的悬停旋翼可压缩无粘绕流的计算

发展了一种基于高精度和高效格式计算悬停旋翼复杂绕流的隐式有限体积方法。控制方程为Euler方程,其中对流项通量的左右状态量采用五阶加权基本无振荡(WENO)格式重构,时间推进应用隐式LU-SGS算法,为进一步加速收敛,采用三层V循环多重网格松弛方法。考虑到多重网格方法的思想以及五阶WENO格式涉及6个网格单元,建议仅在最细网格上使用WENO格式。计算结果表明本文方法能有效捕捉激波,对尾迹也有较高分辨率,基于隐式LU-SGS算法的多重网格方法能有效提高计算效率。     

三维对流扩散方程的高精度全隐式多重网格方法

提出了数值求解三维非定常变系数对流扩散方程的一种高精度全隐紧致差分格式,该格式在空间上具有四阶精度,时间具有二阶精度。为了克服传统迭代法在每一个时间步上迭代求解隐格式时收敛速度慢的缺点,采用多重网格加速技术,建立了适用于本文高精度全隐紧致格式的多重网格算法,从而大大加快了迭代收敛速度。数值实验结果验证了本文方法的精确性、稳定性和对高网格雷诺数问题的强适应性。     

二维对流扩散方程的高精度全隐式多重网格方法

提出了数值求解二维非定常变系数对流扩散方程的一种时间二阶、空间四阶精度的三层全隐紧致差分格式。为了加快迭代求解隐格式时在每一个时间步上的收敛速度，采用多重网格加速技术，建立了适用于本文高精度金隐紧致格式的多重网格算法。数值实验结果验证了本文方法的精确性、稳定性和对高网格雷诺数问题的强适应性。     

全机绕流Euler方程多重网格分区计算方法

全机三维复杂形状绕流数值求解只能采用分区求解的方法,本文采用可压缩Euler方程有限体积方法以及多重网格分区方法对流场进行分区计算。数值方法采用改进的van Leer迎风型矢通量分裂格式和MUSCL方法,基于有限体积方法和迎风型矢通量分裂方法,建立一套处理子区域内分界面的耦合条件。各个子区域之间采用显式耦合条件,区域内部采用隐式格式和局部时间步长等,以加快收敛速度。计算结果飞机表面压力分布等气动力特性与实验值进行了比较,二者基本吻合。计算结果表明采用分析“V”型多重网格方法,能提高计算效率,加快收敛速度达到接近一个量级。根据全机数值计算结果和可视化结果讨论了流场背风区域旋涡的形成过程。     

三维非结构聚合多重网格法数值模拟研究

在三维非结构网格上应用聚合式多重网格技术来加速Euler方程的收敛过程．自行设计了一种高效率的网格聚合方法．采用四重三维非结构网格，在每一层网格上采用有限体积法进行计算．通过对M6翼型的数值求解验证了多重网格加速收敛的高效性．     

一种有效的广义特征值分析方法

提出了一种适合于自适应有限元分析中求解广义特征值问题的多重网格方法．这种方法充分利用了初始网格下的结果，通过插值或最小二乘拟合技术来得到网格变化后的新的近似特征向量，然后由多重网格迭代过程实现对结构广义特征值问题的求解．在多重网格迭代的光滑步中，选择了收敛梯度法以提高其收敛率；在粗网格校正步中，则导出了一种近似求解特征向量误差的方程．这种方法将网格离散过程和数值求解过程很好地相结合，建立了一个网格细分后广义特征值问题的快速重分析方法，与传统有限元方法相比较，具有计算简便、计算量少等特点，可以作为结构动力问题自适应有限元分析的一种十分有效的工具．     

用差分法与超松弛迭代法求高维平稳FPK方程的解

用不同精度的差分格式将高维平稳FPK方程离散化为线性代数方程组,然后用超松弛迭代法求解该线性代数方程组得到平稳FPK方程的近似解。讨论了不同的差分格式、网格密度及超松弛因子对解精度及收敛速度的影响,并与其他方法的计算精度进行比较,提出用多重网格算法提高计算效率。研究了典型的二维及四维随机系统的稳态响应,算例表明,该算法具有简洁、节省存储量且精度高的特点,是求解高维平稳FPK方程解的有效算法。     

Walsh函数有限体积法的多重网格特征研究

Walsh函数有限体积法(FVM-WBF)是一种能够在网格内部捕捉间断的新型数值方法. 持续增加Walsh基函数数目能够稳步提高FVM-WBF方法的求解分辨率, 但计算量暴发式增长和收敛速度下降的问题也会同步出现. 针对Walsh基函数数目增加而引起的计算效率问题, 本文分析了Walsh基函数及其系数所能影响的网格单元局部均值区域尺度, 发现其中隐含类似多重网格的尺度特征, 据此提出一种结合多重网格策略的FVM-WBF方法. 在定常流场计算中根据各级Walsh基函数影响尺度的不同, 对每级Walsh基函数设置满足其稳定性约束的时间步长, 在时间推进求解的过程中快速消除不同波长的数值误差, 实现多重网格的加速收敛效果. 选取NACA0012翼型和二维圆柱的定常无黏绕流问题作为算例, 对引入多重网格策略的FVM-WBF方法和不考虑多重网格策略的FVM-WBF方法进行对比测试. 数值结果证实: 新发展的FVM-WBF方法具备多重网格的关键特征, 在不增加任何特殊处理和计算量的情况下, 只需通过时间步长的调整, 就能够达到多重网格的加速效果, 显著提升计算效率.      

一种基于块雅可比迭代的高阶FR格式隐式方法

最近, 基于非结构网格的高阶通量重构格式(flux reconstruction, FR)因其构造简单且通用性强而受到越来越多人的关注. 但将FR格式应用于大规模复杂流动的模拟时仍面临计算开销大、求解时间长等问题. 因此, 亟需发展与之相适应的高效隐式求解方法和并行计算技术. 本文提出了一种基于块Jacobi迭代的高阶FR格式求解定常二维欧拉方程的单GPU隐式时间推进方法. 由于直接求解FR格式空间和隐式时间离散后的全局线性方程组效率低下并且内存占用很大. 而通过块雅可比迭代的方式, 能够改变全局线性方程组左端矩阵的特征, 克服影响求解并行性的相邻单元依赖问题, 使得只需要存储和计算对角块矩阵. 最终将求解全局线性方程组转化为求解一系列局部单元线性方程组, 进而又可利用LU分解法在GPU上并行求解这些小型局部线性方程组. 通过二维无黏Bump流动和NACA0012无黏绕流两个数值实验表明, 该隐式方法计算收敛所用的迭代步数和计算时间均远小于使用多重网格加速的显式Runge-Kutta格式, 且在计算效率方面至少有一个量级的提升.      

一种改进的无单元方法

使用 1阶或 1阶以上最小滑动二乘法 ( MLS)形函数的无网格伽辽金法 ( EFGM) ,它们的主要缺点是形函数构造复杂、计算费用十分昂贵。本文提出了一种改进的无单元方法 ( IEFM) ,它通过采用 Shepard形函数 ( 0阶 MLS形函数 )对结点的覆盖位移函数加权求和来简化整体近似位移函数的构造 ,且能够避免 EFGM里求解结点形函数时矩阵的求逆及相乘计算。文中的数值算例表明 ,这种改进的 IEFM法收敛快、精度高 ,与标准的EFGM相比其计算时间得到了大幅度的减少     

A discontinuous Galerkin‐based sharp‐interface method to simulate three‐dimensional compressible two‐phase flow

A numerical method for the simulation of compressible two‐phase flows is presented in this paper. The sharp‐interface approach consists of several components: a discontinuous Galerkin solver for compressible fluid flow, a level‐set tracking algorithm to follow the movement of the interface and a coupling of both by a ghost‐fluid approach with use of a local Riemann solver at the interface. There are several novel techniques used: the discontinuous Galerkin scheme allows locally a subcell resolution to enhance the interface resolution and an interior finite volume Total Variation Diminishing (TVD) approximation at the interface. The level‐set equation is solved by the same discontinuous Galerkin scheme. To obtain a very good approximation of the interface curvature, the accuracy of the level‐set field is improved and smoothed by an additional P_NP_M‐reconstruction. The capabilities of the method for the simulation of compressible two‐phase flow are demonstrated for a droplet at equilibrium, an oscillating ellipsoidal droplet, and a shock‐droplet interaction problem at Mach 3. Copyright © 2015 John Wiley & Sons, Ltd.     

H 1 space-time discontinuous finite element method for convection-diffusion equations

An H~1 space-time discontinuous Galerkin (STDG) scheme for convectiondiffusion equations in one spatial dimension is constructed and analyzed. This method is formulated by combining the H~1 Galerkin method and the space-time discontinuous finite element method that is discontinuous in time and continuous in space. The existence and the uniqueness of the approximate solution are proved. The convergence of the scheme is analyzed by using the techniques in the finite difference and finite element methods. An optimal a-priori error estimate in the L~∞ (H~1 ) norm is derived. The numerical exper- iments are presented to verify the theoretical results.     

Spectral p‐multigrid discontinuous Galerkin solution of the Navier–Stokes equations

Discontinuous Galerkin (DG) methods are very well suited for the construction of very high‐order approximations of the Euler and Navier–Stokes equations on unstructured and possibly nonconforming grids, but are rather demanding in terms of computational resources. In order to improve the computational efficiency of this class of methods, a high‐order spectral element DG approximation of the Navier–Stokes equations coupled with a p‐multigrid solution strategy based on a semi‐implicit Runge–Kutta smoother is considered here. The effectiveness of the proposed approach in the solution of compressible shockless flow problems is demonstrated on 2D inviscid and viscous test cases by comparison with both a p‐multigrid scheme with non‐spectral elements and a spectral element DG approach with an implicit time integration scheme. Copyright © 2010 John Wiley & Sons, Ltd.     

An hp‐adaptive discontinuous Galerkin method for shallow water flows

An adaptive spectral/hp discontinuous Galerkin method for the two‐dimensional shallow water equations is presented. The model uses an orthogonal modal basis of arbitrary polynomial order p defined on unstructured, possibly non‐conforming, triangular elements for the spatial discretization. Based on a simple error indicator constructed by the solutions of approximation order p and p?1, we allow both for the mesh size, h, and polynomial approximation order to dynamically change during the simulation. For the h‐type refinement, the parent element is subdivided into four similar sibling elements. The time‐stepping is performed using a third‐order Runge–Kutta scheme. The performance of the hp‐adaptivity is illustrated for several test cases. It is found that for the case of smooth flows, p‐adaptivity is more efficient than h‐adaptivity with respect to degrees of freedom and computational time. Copyright © 2010 John Wiley & Sons, Ltd.     

The Runge–Kutta control volume discontinuous finite element method for systems of hyperbolic conservation laws

In this paper, a new high‐order and high‐resolution method called the Runge–Kutta control volume discontinuous finite element method (RKCVDFEM) was proposed to solve 1D and 2D systems of hyperbolic conservation laws. Its main advantage lies in the local conservation, and it is simpler than the Runge–Kutta discontinuous Galerkin finite element method (RKDGM). The theoretical analysis showed that the RKCVDFEM has formally an optimal convergence order for 1D systems. Based on logically rectangular grids of irregular quadrilaterals, a scheme for 2D systems was constructed. Some classical problems were simulated and the validity of the method was presented. Copyright © 2010 John Wiley & Sons, Ltd.     

A 3D shell-like approach using element-free Galerkin method for analysis of thin and thick plate structures

A new efficient meshless method based on the element-free Galerkin method is proposed to analyze the static deformation of thin and thick plate structures in this paper. Using the new 3D shell-like kinematics in analogy to the solid-shell concept of the finite element method, discretization is carried out by the nodes located on the upper and lower surfaces of the structures. The approximation of all unknown field variables is carried out by using the moving least squares (MLS) approximation scheme in the in-plane directions, while the linear interpolation is applied through the thickness direction. Thus, different boundary conditions are defined only using displacements and penalty method is used to enforce the essential boundary conditions. The constrained Galerkin weak form, which incorporates only displacement degrees of freedom (d.o.f.s), is derived. A modified 3D constitutive relationship is adopted in order to avoid or eliminate some self-locking effects. The numeric efficiency of the proposed meshless formulation is illustrated by the numeric examples.     

Remarks on the links between low‐order DG methods and some finite‐difference schemes for the Stokes problem

In this paper we demonstrate that some well‐known finite‐difference schemes can be interpreted within the framework of the local discontinuous Galerkin (LDG) methods using the low‐order piecewise solenoidal discrete spaces introduced in (SIAM J. Numer. Anal. 1990; 27 (6): 1466–1485). In particular, it appears that it is possible to derive the well‐known MAC scheme using a first‐order Nédélec approximation on rectangular cells. It has been recently interpreted within the framework of the Raviart–Thomas approximation by Kanschat (Int. J. Numer. Meth. Fluids 2007; published online). The two approximations are algebraically equivalent to the MAC scheme, however, they have to be applied on grids that are staggered on a distance h/2 in each direction. This paper also demonstrates that both discretizations allow for the construction of a divergence‐free basis, which yields a linear system with a ‘biharmonic’ conditioning. Both this paper and Kanschat (Int. J. Numer. Meth. Fluids 2007; published online) demonstrate that the LDG framework can be used to generalize some popular finite‐difference schemes to grids that are not parallel to the coordinate axes or that are unstructured. Copyright © 2008 John Wiley & Sons, Ltd.     

A Bubble Element for the Compressible Euler Equations

In this paper, we present a new Galerkin finite element method with bubble function for the compressible Euler equations. This method is derived from the scaled bubble element for the advection-diffusion problems developed by Simo and his colleagues, which is based on the equivalence between the Galerkin method employing piecewise linear interpolation with bubble functions and the Streamline-Upwind/Petrov Galerkin (SUPG) finite element method using P₁ approximation in the steady advection-diffusion problem. Simo and this author have applied this approach to transient advection-diffusion problems by using a special scaled bubble function called P-scaled bubble, which is designed to work in the transient advection-diffusion problems for any Peclet number from 0 to ∞. The method presented in this paper is an application of this p-scaled bubble element to a pure hyperbolic system.     

ANALYSIS/DESIGN CALCULATIONS OF TRANSONIC FLOWS

SUMMARY

Analysis/design calculations of transonic flow are discussed and several improvements are made. The nonisentropic potential method is used to calculate the inviscid transonic flow analysis problem instead of the traditional potential method. An inverse integral 3D boundary layer method is used to calculate the boundary layer in the viscous transonic flow analysis problem. The viscous/inviscid interaction calculations are carried out by a semi-inverse coupling scheme. In design problem calculations, an improved residual-correction method is used. Three individual methods are combined in a global algorithm and computing code. The improvements speed up the convergence, increase applicability and computational efficiency. Some numerical results are given to illustrate that the present method provides an effective engineering tool of high accuracy and efficiency in three dimensional transonic analysis and design situations.     

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).