间断有限元方法在弹尾超音速喷流计算中的应用

采用间断有限元方法对超音速无粘喷流流动进行数值模拟.将二维双曲守恒方程的间断有限元方法发展到轴对称Euler方程,并就某导弹尾部超音速伴随射流进行数值计算.计算结果与实验照片反映的流动特征吻合较好,与高精度、高分辨率TVD格式的计算结果相比,间断有限元方法的计算结果在轴线反射点附近具有较高的分辨率,表明该方法对激波具有较强的捕捉能力,在激波阵面上不会产生振荡或抹平间断现象.     

隐式间断有限元方法的参数化边界修正

本文采用并行任意阶精度隐式格式的间断有限元方法,求解来流马赫数为0.01的低速可压缩无黏流动,为增强间断有限元对网格的适应性,采用参数化的方法同时对曲线边界上的积分点位置及外法向向量进行修正,计算结果表明在不做预处理的情况下,隐式间断有限元方法也能较好的计算低速流动,本文所做的参数化边界修正方法有助于间断有限元方法使用通用非结构网格进行高精度的计算.     

有限球中14MeV中子点源问题的数值模拟

点源问题的物理实验与数值计算是研究反应堆防护和辐射效应,考查中子参数和计算方法的重要手段。本工作解决了有限元节点上的加源问题,并对几个典型的14MeV中子点源问题的物理实验,用双向间断有限元法进行了数值模拟,并和其它方法的计算结果及物理实验的测量结果进行了比较,取得了较为满意的结果。     

解二维中子输运问题的三角网格配置法

本文研究了二维轴对称中子输运(包括各向同性散射和各向异性散射)问题的一种有限元数值方法。给出了一系列的数值计算结果,并与国内外的SN、间断有限元方法的数值结果及实验结果进行比较,取得了令人满意的结果。     

高分辨率间断有限元方法

间断有限元方法是集高分辨率有限差分方法和有限体积方法的优点发展起来的一种数值方法,在计算流体动力学问题上显示了优良的效能.利用守恒问题给出间断有限元方法的基本概念和过程,利用简单算例给出该方法的精度分析和限制器对精度的影响,并给出浅水波问题、交通流问题和波传播问题的数值模拟结果,进一步,综合评介该方法在椭圆、抛物、对流扩散、Hamilton-Jacobi方程、Navier-Stokes方程等的实际应用进展.     

一维双曲守恒律的龙格-库塔控制体积间断有限元方法

给出数值求解一维双曲守恒律方程的新方法——龙格-库塔控制体积间断有限元方法(RKCVDFEM),其中空间离散基于控制体积有限元方法,时间离散基于二阶TVB Runge-Kutta技术,有限元空间选取为分段线性函数空间.理论分析表明,格式具有总变差有界(TVB)的性质,而且空间和时间离散形式上具有二阶精度.数值算例表明,数值解收敛到熵解并且对光滑解的收敛阶是最优的,优于龙格-库塔间断Galerkin方法(RKDGM)的计算结果.     

非规则形状介质内辐射-导热耦合传热的间断有限元求解

采用间断有限元法(discontinuous finite element method,DFEM)求解非规则形状介质内的辐射导热耦合传热问题,得到了典型非规则形状介质内辐射导热耦合传热问题的高精度数值结果.和传统连续型有限元方法不同,DFEM将计算区域划分成相互独立的离散单元,形函数的构造、未知量的加权近似以及控制方程的求解均在每一个离散单元上进行.通过在单元之间施加迎风格式的数值通量,DFEM保证了整个计算区域的连续性,因此这种方法兼具良好的几何灵活性和局部守恒性.推导了辐射传输方程和能量扩散方程的射导热耦合传热问题,得到了典型非规则形状介质内辐射导热耦合传热的高精度数值结果.     

非结构网格上Level Set方程的间断有限元解法

针对间断有限元弱形式难于求解可压缩流场中Level Set方程的问题提出间断有限元强形式,从而在统-框架下解决Level Set方程在可压缩与不可压缩流场中的求解问题.通过非结构网格上采用Legendre-Gauss-Lobatto节点构造基函数,在复杂区域上可以达到任意高阶的精度.将若干-、二、三维算例与已有文献或解析解比较,验证方法追踪自由界面的有效性.结果表明,该方法适合各种情形下Level Set方程求解,易于在复杂区域的非结构网格上实施,精度高、分辨率高且具有高质量守恒性,既能避免重新初始化过程又方便向高维扩展.     

多介质流模拟的Runge-Kutta控制体积间断有限元方法（英文）

构造可用于多介质流数值模拟的Runge-Kutta控制体积(RKCV)间断有限元方法.对于多介质流模拟,使用线性和非线性的Riemann问题解法器计算界面处的数值流通量.该方法是一种高精度的数值方法且可以保证流体的局部守恒.数值结果表明,即使是利用线性Riemann问题解法器的计算格式也可获得较好的数值结果.与Runge-kutta间断Galerkin方法的比较展示了本文构造算法的优势.     

自适应间断有限元方法求解三维欧拉方程

将龙格库塔间断有限元方法(RDDG)与自适应方法相结合,求解三维欧拉方程.区域剖分采用非结构四面体网格,依据数值解的变化采用自适应技术对网格进行局部加密或粗化,减少总体网格数目,提高计算效率.给出四种自适应策略并分析不同自适应策略的优缺点.数值算例表明方法的有效性.     

Discontinuous finite element method for radiative heat transfer in semitransparent graded index medium

To avoid the complicated and time-consuming computation of curved ray trajectories, a discontinuous finite element method based on discrete ordinate equation is extended to solve the radiative transfer problem in a multi-dimensional semitransparent graded index medium. Two cases of radiative heat transfer in two-dimensional rectangular gray semitransparent graded index medium enclosed by opaque boundary are examined to verify this discontinuous finite element method. Special layered and radial graded index distributions are considered. The predicted dimensionless net radiative heat fluxes and dimensionless temperature distributions are determined by the discontinuous finite element method and compared with the results obtained by the curved Monte Carlo method in references. The results show that the discontinuous finite element method has a good accuracy in solving the multi-dimensional radiative transfer problem in a semitransparent graded index medium.     

局部时间步长间断有限元方法求解三维欧拉方程

使用间断有限元方法求解三维流体力学方程.空间剖分采用非结构四面体网格,为了克服显格式在单元网格尺寸差别较大时计算效率低下的问题,在格式中采用局部时间步长技术(LTS),即控制方程在空间、时间上积分得到一种单步格式,既可以局部计算每个单元又避免了Runge-Kutta高精度格式处理三维问题时存储量过大的问题.为了提高流体力学方程计算精度,在计算单元边界的数值流通量时使用任意高阶精度方法(ADER).数值算例表明格式稳定有效.     

Artificial boundary conditions for the numerical solution of the Euler equations by the discontinuous galerkin method

We present artificial boundary conditions for the numerical simulation of compressible flows using high-order accurate discretizations with the discontinuous Galerkin (DG) finite element method. The construction of the proposed boundary conditions is based on characteristic analysis and applied for boundaries with arbitrary shape and orientation. Numerical experiments demonstrate that the proposed boundary treatment enables to convect out of the computational domain complex flow features with little distortion. In addition, it is shown that small-amplitude acoustic disturbances could be convected out of the computational domain, with no significant deterioration of the overall accuracy of the method. Furthermore, it was found that application of the proposed boundary treatment for viscous flow over a cylinder yields superior performance compared to simple extrapolation methods.     

Computational simulation of wave propagation problems in infinite domains

This paper deals with the computational simulation of both scalar wave and vector wave propagation problems in infinite domains. Due to its advantages in simulating complicated geometry and complex material properties, the finite element method is used to simulate the near field of a wave propagation problem involving an infinite domain. To avoid wave reflection and refraction at the common boundary between the near field and the far field of an infinite domain, we have to use some special treatments to this boundary. For a wave radiation problem, a wave absorbing boundary can be applied to the common boundary between the near field and the far field of an infinite domain, while for a wave scattering problem, the dynamic infinite element can be used to propagate the incident wave from the near field to the far field of the infinite domain. For the sake of illustrating how these two different approaches are used to simulate the effect of the far field, a mathematical expression for a wave absorbing boundary of high-order accuracy is derived from a two-dimensional scalar wave radiation problem in an infinite domain, while the detailed mathematical formulation of the dynamic infinite element is derived from a two-dimensional vector wave scattering problem in an infinite domain. Finally, the coupled method of finite elements and dynamic infinite elements is used to investigate the effects of topographical conditions on the free field motion along the surface of a canyon.     

密度矩阵重正化群的异构并行优化

密度矩阵重正化群方法（DMRG）在求解一维强关联格点模型的基态时可以获得较高的精度,在应用于二维或准二维问题时,要达到类似的精度通常需要较大的计算量与存储空间.本文提出一种新的DMRG异构并行策略,可以同时发挥计算机中央处理器（CPU）和图形处理器（GPU）的计算性能.针对最耗时的哈密顿量对角化部分,实现了数据的分布式存储,并且给出了CPU和GPU之间的负载平衡策略.以费米Hubbard模型为例,测试了异构并行程序在不同DMRG保留状态数下的运行表现,并给出了相应的性能基准.应用于4腿梯子时,观测到了高温超导中常见的电荷密度条纹,此时保留状态数达到10⁴,使用的GPU显存小于12 GB.     

The space–time CESE method for solving special relativistic hydrodynamic equations

The special relativistic hydrodynamic equations are more complicated than the classical ones due to the nonlinear and implicit relations that exist between conservative and primitive variables. In this article, a space–time conservation element and solution element (CESE) method is proposed for solving these equations in one and two space dimensions. The CESE method has capability to capture sharp propagating wavefront of the relativistic fluids without excessive numerical diffusion or spurious oscillations. In contrast to the existing upwind finite volume schemes, the Riemann solver and reconstruction procedure are not the building blocks of the suggested method. The method differs from previous techniques because of global and local flux conservation in a space–time domain without resorting to interpolation or extrapolation. The scheme is efficient, robust, and gives results comparable to those obtained with more sophisticated algorithms, even in highly relativistic two-dimensional test problems.     

An approximation for the boundary optimal control problem of a heat equation defined in a variable domain

In this paper, we consider a numerical approximation for the boundary optimal control problem with the control constraint governed by a heat equation defined in a variable domain. For this variable domain problem, the boundary of the domain is moving and the shape of theboundary is defined by a known time-dependent function. By making use of the Galerkin finite element method, we first project the original optimal control problem into a semi-discrete optimal control problem governed by a system of ordinary differential equations. Then, based on the aforementioned semi-discrete problem, we apply the control parameterization method to obtain an optimal parameter selection problem governed by a lumped parameter system, which can be solved as a nonlinear optimization problem by a Sequential Quadratic Programming （SQP） algorithm. The numerical simulation is given to illustrate the effectiveness of our numerical approximation for the variable domain problem with the finite element method and the control parameterization method.     

A New Numerical Method for Solving the Acoustic Radiation Problem

A numerical method of solving the problem of acoustic wave radiation in the presence of a rigid scatterer is described. It combines the finite element method and the boundary algebraic equation one. In the proposed method, the exterior domain around the scatterer is discretized, so that there appear an infinite domain with regular discretization and a relatively small layer with irregular mesh. For the infinite regular mesh, the boundary algebraic equation method is used with spurious resonance suppression according to Burton and Miller. In the thin layer with irregular mesh, the finite element method is used. The proposed method is characterized by simple implementation, fair accuracy, and absence of spurious resonances.     

Discontinuous Galerkin methods in nanophotonics

This article reviews the state of the recently developed discontinuous Galerkin finite element method for the efficient numerical treatment of nanophotonic systems. This approach combines the accurate and flexible spatial discretisation of classical finite elements with efficient time stepping capabilities. We describe in detail the underlying principles of the discontinuous Galerkin technique and its application to the simulation of complex nanophotonic structures. In addition, formulations for both time‐ and frequency‐domain solvers are provided and specific advantages and limitations of the technique are discussed. The potential of the discontinuous Galerkin approach is illustrated by modelling and simulating several experimentally relevant systems.     

三维电磁扩散场数值模拟及磁化效应的影响

采用矢量有限元法实现了三维电磁扩散场数值模拟,并成功将其应用在大地电磁的正演研究中.为灵活精确地拟合起伏地形和地下不规则构造,采用由不规则四面体单元组成的非结构化网格,可根据模型设计的需要调整网格的大小.引入了基于二次场理论,将解析的一次场从总场中扣除,直接计算二次场,使得误差仅局限于相对较小的二次场,以提高总场计算精度.常规的节点有限元法不满足电性分界面上法向电场不连续和无源区单元内电流密度无散,违反麦克斯韦方程组.为克服节点有限元法的弊端,使用矢量有限元法求解基于二次电场的偏微分方程.另外,在算法设计中,考虑了磁导率参数的变化,可以模拟磁导率不均匀的模型.通过与COMMEMI模型已发表的结果对比,证明了本文算法的正确性和精确性.为突显非结构网格优势,计算了椭球异常体模型和任意地形模型的MT响应,并详细讨论了地形和磁化效应对三维数值模拟结果的影响.