基于紧致差分的Grad-Shafranov方程快速解法

在等离子体平衡重建迭代计算过程中,需要快速求解Grad-Shafranov方程(G-S方程)。构造了具有四阶精度紧致差分格式的离散方程,采用离散正弦变换技术对其进行快速求解并采用CUDATM实现GPU并行加速,将其应用到EAST等离子体平衡重建PEFIT代码中,实现基于紧致差分格式的快速G-S方程求解。结果表明,在65×65的网格下,给定方程右端项电流分布的前提下,使用GPU求解G-S方程所需时间为大约34μs。     

基于大规模并行计算的三维多群中子扩散方程有限差分方法

三维多群中子扩散方程的精确、高效求解是核动力堆芯设计及燃料管理的基础。应用有限差分方法求解该方程具有简便、精确、成熟的优点;然而,该方法的计算量和存储量均较大,极大地限制了它的计算规模和应用范围。本文基于大规模并行计算,研究三维多群中子扩散方程有限差分方法：采用中心有限差分格式离散中子扩散方程;基于MPI并行编程模型,采用空间区域分解的方式实现大规模并行计算;采用多群多区域耦合PGMRES算法进行并行加速。在集群服务器上开发了ParaFiDi程序,并采用IAEA3D,PHWR等多个基准题对该程序进行验证。数值结果表明,ParaFiDi程序具有较高的计算精度和计算效率。     

A simple multigrid scheme for solving the Poisson equation with arbitrary domain boundaries

We present a new multigrid scheme for solving the Poisson equation with Dirichlet boundary conditions on a Cartesian grid with irregular domain boundaries. This scheme was developed in the context of the Adaptive Mesh Refinement (AMR) schemes based on a graded-octree data structure. The Poisson equation is solved on a level-by-level basis, using a “one-way interface” scheme in which boundary conditions are interpolated from the previous coarser level solution. Such a scheme is particularly well suited for self-gravitating astrophysical flows requiring an adaptive time stepping strategy. By constructing a multigrid hierarchy covering the active cells of each AMR level, we have designed a memory-efficient algorithm that can benefit fully from the multigrid acceleration. We present a simple method for capturing the boundary conditions across the multigrid hierarchy, based on a second-order accurate reconstruction of the boundaries of the multigrid levels. In case of very complex boundaries, small scale features become smaller than the discretization cell size of coarse multigrid levels and convergence problems arise. We propose a simple solution to address these issues. Using our scheme, the convergence rate usually depends on the grid size for complex grids, but good linear convergence is maintained. The proposed method was successfully implemented on distributed memory architectures in the RAMSES code, for which we present and discuss convergence and accuracy properties as well as timing performances.     

Sixth order compact scheme combined with multigrid method and extrapolation technique for 2D poisson equation

We develop a sixth order finite difference discretization strategy to solve the two dimensional Poisson equation, which is based on the fourth order compact discretization, multigrid method, Richardson extrapolation technique, and an operator based interpolation scheme. We use multigrid V-Cycle procedure to build our multiscale multigrid algorithm, which is similar to the full multigrid method (FMG). The multigrid computation yields fourth order accurate solution on both the fine grid and the coarse grid. A sixth order accurate coarse grid solution is computed by using the Richardson extrapolation technique. Then we apply our operator based interpolation scheme to compute sixth order accurate solution on the fine grid. Numerical experiments are conducted to show the solution accuracy and the computational efficiency of our new method, compared to Sun–Zhang’s sixth order Richardson extrapolation compact (REC) discretization strategy using Alternating Direction Implicit (ADI) method and the standard fourth order compact difference (FOC) scheme using a multigrid method.     

高精度有限差分法模拟Kelvin-Helmholtz不稳定性

 WENO有限差分格式有较高的分辨精度,适合复杂流场的计算,在国际上被广泛采用。本文利用WENO有限差分格式求解2维守恒型欧拉方程,实现了对无粘流体中Kelvin-Helmholtz不稳定性的数值模拟。速度剪切方向采用周期边界条件;扰动增长方向采用嵌边出流边界条件,一个不稳定波长分布64个网格。数值模拟给出的扰动幅值线性增长率与线性稳定性分析给出的结果很好符合,显示了该格式的有效性和精度。数值模拟给出了清晰的密度等值线,表明该方法还具有较好的界面变形捕捉能力。     

Multigrid method based on the transformation-free HOC scheme on nonuniform grids for 2D convection diffusion problems

In this paper, a multigrid method based on the high order compact (HOC) difference scheme on nonuniform grids, which has been proposed by Kalita et al. [J.C. Kalita, A.K. Dass, D.C. Dalal, A transformation-free HOC scheme for steady convection–diffusion on non-uniform grids, Int. J. Numer. Methods Fluids 44 (2004) 33–53], is proposed to solve the two-dimensional (2D) convection diffusion equation. The HOC scheme is not involved in any grid transformation to map the nonuniform grids to uniform grids, consequently, the multigrid method is brand-new for solving the discrete system arising from the difference equation on nonuniform grids. The corresponding multigrid projection and interpolation operators are constructed by the area ratio. Some boundary layer and local singularity problems are used to demonstrate the superiority of the present method. Numerical results show that the multigrid method with the HOC scheme on nonuniform grids almost gets as equally efficient convergence rate as on uniform grids and the computed solution on nonuniform grids retains fourth order accuracy while on uniform grids just gets very poor solution for very steep boundary layer or high local singularity problems. The present method is also applied to solve the 2D incompressible Navier–Stokes equations using the stream function–vorticity formulation and the numerical solutions of the lid-driven cavity flow problem are obtained and compared with solutions available in the literature.     

An accurate and robust numerical method for micromagnetics simulations

We propose a new robust, accurate, and fast numerical method for solving the Landau–Lifshitz equation which describes the relaxation process of the magnetization distribution in ferromagnetic material. The proposed numerical method is second-order accurate in both space and time. The approach uses the nonlinear multigrid method for handling the nonlinearities at each time step. We perform numerical experiments to show the efficiency and accuracy of the new algorithm on two- and three-dimensional space. The numerical results show excellent agreements with exact analytical solutions, the second-order accuracy in both space and time, and the energy conservation or dissipation property.     

Modified finite-volume method based on a cell vertex scheme for the solution of radiative transfer problems in complex 3D geometries

A modified finite-volume method based on a cell vertex scheme was applied to solve radiative transfer problems within a participating medium of complex three-dimensional shaped domain. The computational spatial domain of interest was divided into four-node tetrahedron elements with unstructured meshes while the adopted formulation was combined with a closure relation based on an exponential scheme. The studied medium was assumed to be grey, non-scattering and was bounded by black surfaces. Our results were then compared with those found in other articles on the subject. The approach shows a very good level of performance for wall heat transfer evaluation. Accurate results were obtained on coarse computational meshes and solution errors were found to decrease with grid refinement.     

Numerical simulation of the soliton solutions for a complex modified Korteweg—de Vries equation by a finite difference method

In this paper,a Crank-Nicolson-type finite difference method is proposed for computing the soliton solutions of a complex modifed Korteweg de Vries(MKdV)equation(which is equivalent to the Sasa-Satsuma equation)with the vanishing boundary condition.It is proved that such a numerical scheme has the second order accuracy both in space and time,and conserves the mass in the discrete level.Meanwhile,the resuling scheme is shown to be unconditionally stable via the von Nuemann analysis.In addition,an iterative method and the Thomas algorithm are used together to enhance the computational efficiency.In numerical experiments,this method is used to simulate the single-soliton propagation and two-soliton collisions in the complex MKdV equation.The numerical accuracy,mass conservation and linear stability are tested to assess the scheme's performance.     

Large deformation measurement scheme for 3D digital image correlation method

Difficulties often arise for digital image correlation (DIC) technique when serious de-correlation occurs between the reference image and the deformed image due to large deformation. An updating reference image scheme could be employed to deal with large deformation situation, however that will introduce accumulated errors. A large deformation measurement scheme, combining improved coarse search method and updating reference image scheme, is proposed in this paper. For a series of deformation images, the correlation calculation begins with a seed point and spreads out. An improved coarse search method is developed to calculate the initial correlation parameters for the seed point, which guarantees that the correlation calculation can be carried out successfully even in large deformation situation. Only for extremely large deformation, the reference image is updated. Using this method, not only extremely large deformation can be measured successfully but also the accumulated error could be controlled. A polymer material tensile test and a foam compression test are used to verify the proposed scheme. Experimental results show that up to 450% tensile deformation and 83% compression deformation can be measured successfully.     

耦合不可压流场输运方程的格子Boltzmann方法研究

基于格子Boltzmann方法,提出了求解耦合不可压缩流场输运方程的一种改进数值方法. 该方法使用格子Boltzmann方法求解流场方程,并根据流场格子模型的密度分布函数构建了输运方程的二阶离散格式. 通过二维平板通道流场输运系统验证了该方法的有效性.数值结果表明,该方法可以有效地减少计算过程中出现的非物理耗散, 并克服了传统模型所需巨大存储量的缺点.     

三维对流扩散方程非等距网格上的四阶紧致格式及其多重网格方法

提出了数值求解三维变系数对流扩散方程非等距网格上的四阶精度19点紧致差分格式,为了提高求解效率,采用多重网格方法求解高精度格式所形成的大型代数方程组。数值实验结果表明本文方法对于不同的网格雷诺数问题,在精确性、稳定性和减少计算工作量方面均明显优于7点中心差分格式。     

基于JASMIN框架的抛物方程有限差分解法并行计算及其应用

抛物方程有限差分解法的网格步长严格受波长限制,在求解城市小区电波传播问题时,计算速度明显变慢,为此,基于JASMIN框架研究了抛物方程有限差分解法的并行方法,通过将同一步进面划分成多个网格片,并分配到不同的处理器进行运算,实现了抛物方程有限差分解法的并行计算。与解析解的对比验证了并行程序的正确性,同时通过实例分析了并行程序的高效性,算例表明,抛物方程有限差分解法的求解效率得到了有效的提高。最后,模拟和分析了某一电信基站天线在包含9栋规则建筑物的城市小区环境中的电磁特性,结果表明,该方法能够得到基站在空间各处的信号覆盖强弱,可以为基站选址提供参考。     

A Monte Carlo method for 3D thermal infrared radiative transfer

A 3D Monte Carlo model for specific application to the broadband thermal radiative transfer has been developed in which the emissivities for gases and cloud particles are parameterized by using a single cubic element as the building block in 3D space. For spectral integration in the thermal infrared, the correlated k-distribution method has been used for the sorting of gaseous absorption lines in multiple-scattering atmospheres involving 3D clouds. To check the Monte-Carlo simulation, we compare a variety of 1D broadband atmospheric fluxes and heating rates to those computed from the conventional plane-parallel (PP) model and demonstrate excellent agreement between the two. Comparisons of the Monte Carlo results for broadband thermal cooling rates in 3D clouds to those computed from the delta-diffusion approximation for 3D radiative transfer and the independent pixel-by-pixel approximation are subsequently carried out to understand the relative merits of these approaches.     

High-order compact schemes for incompressible flows: A simple and efficient method with quasi-spectral accuracy

In this paper, a finite difference code for Direct and Large Eddy Simulation (DNS/LES) of incompressible flows is presented. This code is an intermediate tool between fully spectral Navier–Stokes solvers (limited to academic geometry through Fourier or Chebyshev representation) and more versatile codes based on standard numerical schemes (typically only second-order accurate). The interest of high-order schemes is discussed in terms of implementation easiness, computational efficiency and accuracy improvement considered through simplified benchmark problems and practical calculations. The equivalence rules between operations in physical and spectral spaces are efficiently used to solve the Poisson equation introduced by the projection method. It is shown that for the pressure treatment, an accurate Fourier representation can be used for more flexible boundary conditions than periodicity or free-slip. Using the concept of the modified wave number, the incompressibility can be enforced up to the machine accuracy. The benefit offered by this alternative method is found to be very satisfactory, even when a formal second-order error is introduced locally by boundary conditions that are neither periodic nor symmetric. The usefulness of high-order schemes combined with an immersed boundary method (IBM) is also demonstrated despite the second-order accuracy introduced by this wall modelling strategy. In particular, the interest of a partially staggered mesh is exhibited in this specific context. Three-dimensional calculations of transitional and turbulent channel flows emphasize the ability of present high-order schemes to reduce the computational cost for a given accuracy. The main conclusion of this paper is that finite difference schemes with quasi-spectral accuracy can be very efficient for DNS/LES of incompressible flows, while allowing flexibility for the boundary conditions and easiness in the code development. Therefore, this compromise fits particularly well for very high-resolution simulations of turbulent flows with relatively complex geometries without requiring heavy numerical developments.     

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在三维三温辐射流体力学方程组求解过程中,为解决复杂的等离子体流场和难度大的三维网格计算,提出了用物理量迁移法求解方程组,并对三温能量方程采用了一种改进的显示算法。由此获得了较好的数值结果。     

基于编码标志点的数码相机三维测量与重构

介绍了利用数码相机作为图像传感器,在物体表面设置编码标志点,通过对编码标志点进行识别和检测,从而确定标志点的空间位置,实现了对物体表面某些特定点三维信息的无接触检测。介绍了其中的关键技术,如标志点的设计与检测、基础矩阵的求解、数码相机参数的标定以及内外参数的捆绑调整等,给出了检测实例并进行了精度验证。实验证明,该方法结构简单,成本低廉,数据采集快,移动方便,检测范围大,可满足三维物体特别是大面积物体空间检测的需求。     

用于腔体内电磁脉冲模拟的三维并行全电磁粒子方法

X射线辐照飞行器等腔体在其内部产生的腔体内电磁脉冲,会干扰其内部电子系统的正常工作,进而影响飞行器的运行和生存。介绍一种三维并行全电磁粒子方法,用于模拟X射线辐照腔体在其内部产生的瞬态电磁脉冲响应。在这一数值方法中,时域有限差分方法和Particle-in-Cell方法用来求解瞬态电磁场的产生和带电粒子运动之间的耦合关系,有效电流分配方法用来计算瞬态电磁场产生的源项。该方法基于JASMIN并行框架实现,可模拟含数亿网格和数亿粒子的三维腔体结构的内电磁脉冲响应,且具备大规模并行的优势。用这一方法来模拟圆柱腔体在X射线辐照下的腔体内电磁脉冲响应,其计算结果与文献结果吻合较好,验证了算法的有效性和正确性。     

Modeling of radiative heat transfer in 3D complex boiler with non-gray sooting media

The radiative heat transfer problem is solved for 3D complex industrial boiler with five baffles containing a mixture of carbon dioxide and water vapor for non-uniform temperature fields. A numerical formulation using the FTn finite volume method coupled with the bounded high-order resolution CLAM scheme, the blocked-off-region procedure and the narrow-band based weighted-sum-of-gray-gases (WSGG) [Kim OJ, Song T-H. Data base of WSGGM-based spectral model for radiation properties of combustion products, JQSRT 2000; 64: 379-94] model is adapted. The effect of soot volumetric fraction, particle temperature and uniform particle concentration on the radiative heat flux and radiative heat source is investigated and discussed. Also the advantages, in non-gray media, of the FTnFVM compared to the classical FVM are highlighted.     

An efficient and robust reconstruction method for optical tomography with the time-domain radiative transfer equation

An efficient and robust method based on the complex-variable-differentiation method (CVDM) is proposed to reconstruct the distribution of optical parameters in two-dimensional participating media. An upwind-difference discrete-ordinate formulation of the time-domain radiative transfer equation is well established and used as forward model. The regularization term using generalized Gaussian Markov random field model is added in the objective function to overcome the ill-posed nature of the radiative inverse problem. The multi-start conjugate gradient method was utilized to accelerate the convergence speed of the inverse procedure. To obtain an accurate result and avoid the cumbersome formula of adjoint differentiation model, the CVDM was employed to calculate the gradient of objective function with respect to the optical parameters. All the simulation results show that the CVDM is efficient and robust for the reconstruction of optical parameters.