An MPI parallel level-set algorithm for propagating front curvature dependent detonation shock fronts in complex geometries

We present a parallel, two-dimensional, grid-based algorithm for solving a level-set function PDE that arises in Detonation Shock Dynamics (DSD). In the DSD limit, the detonation shock propagates at a speed that is a function of the curvature of the shock surface, subject to a set of boundary conditions applied along the boundaries of the detonating explosive. Our method solves for the full level-set function field, φ(x, y, t), that locates the detonation shock with a modified level-set function PDE that continuously renormalises the level-set function to a distance function based off of the locus of the shock surface, φ(x, y, t)=0. The boundary conditions are applied with ghost nodes that are sorted according to their connectivity to the interior explosive nodes. This allows the boundary conditions to be applied via a local, direct evaluation procedure. We give an extension of this boundary condition application method to three dimensions. Our parallel algorithm is based on a domain-decomposition model which uses the Message-Passing Interface (MPI) paradigm. The computational order of the full level-set algorithm, which is O(N ⁴), where N is the number of grid points along a coordinate line, makes an MPI-based algorithm an attractive alternative. This parallel model partitions the overall explosive domain into smaller sub-domains which in turn get mapped onto processors that are topologically arranged into a two-dimensional rectangular grid. A comparison of our numerical solution with an exact solution to the problem of a detonation rate stick shows that our numerical solution converges at better than first-order accuracy as measured by an L1-norm. This represents an improvement over the convergence properties of narrow-band level-set function solvers, whose convergence is limited to a floor set by the width of the narrow band. The efficiency of the narrow-band method is recovered by using our parallel model.     

FCT有限元法数值求解三维可压缩高速无粘流问题

从Euler方程出发,利用流量修正有限元法(FEM-FCT)求解三维无粘流动的高速流场。通过对圆球、简化航天飞机和双子星座飞船返回舱外形的数值模拟证明,这种方法对激波有较高的分辨能力,是一种有效的方法。     

用非结构网格与欧拉方程计算复杂区域的二维流动

提出用Delaunay三角化方法生成非结构网格的一种过程。所生成的网格可用于复杂多连通域内的可压流计算。采用Euler方程和格心有限体积法,研制出程序,给出了算例。     

求解Euler方程的区域分解方法与并行算法

将复杂形状区域划分成多块子区域,研究发展了一种多块区域之间迎风守恒型的内边界耦合方法,实现相邻子区域解的光滑过渡,使多区耦合得到总体流场的数值解。对二维翼型跨音速流动和圆弧形隆起物超音速流动等进行了分区数值计算,并将计算结果与单区计算结果和实验结果作了比较。并行分区计算引入\"先进先出\"的同步控制等待机制,实现了高效率并行计算,还分析了影响并行效率的主要因素。     

应用随机模型方法预测汽车风噪声

采用随机模型方法,对简化汽车头部外形进行风噪声数值模拟.将计算区域分为声源区域和传播区域,在声源区域采用随机模型构造湍流脉动速度场,传播区域通过求解带源项的线化欧拉方程实现声波向外传播得到声场解.同直接模拟方法相比,该方法具有计算量小、计算所需内存少等优点.数值模拟结果与实验数据吻合较好,验证了该方法预测汽车风噪声的可行性,为研究实际汽车外形的风噪声问题打下基础.     

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。     

Solutions of Doubly and Higher Order Iterated Equations

Michael Fisher once studied the solution of the equation f(f(x))=x ²–2. We offer solutions to the general equation f(f(x))=h(x) in the form f(x)=g(ag ^–1(x)) where a is in general a complex number. This leads to solving duplication formulas for g(x). For the case h(x)=x ²–2, the solution is readily found, while the h(x)=x ²+2 case is challenging. The solution to these types of equations can be related to differential equations.     

多分量流计算的高分辨KFVS有限体积方法

论及高分辨分子动力学通向量分裂（ＫＦＶＳ）有限体积方法的推广。在方法中提出了适当修改Ｍａｘｗｅｌｌ平衡分布用以修复Ｅｕｌｅｒ方程。基于熟知的Ｅｕｌｅｒ方程与Ｂｏｌｔｚｍａｎｎ方程的关系提出了一类求解多分量Ｅｕｌｅｒ方程的高分辨分子动力爱向量分裂（ＫＦＶＳ）有限体积方法,应用该方法不需要求解任何Ｒｉｅｍａｎｎ问题或求附加的非守恒压力方程也需要任何非守恒修正。数值计算表明,数值解在物质界面附近无振荡,     

特形脉冲的欧拉角

本文提出了用量子力学的空间转动变换算符描述特形脉冲的方法。它把任意的特形脉冲用三个欧拉角来表示,并且使得在特形脉冲下的相干演化可以很容易地利用多极NMR理论,张量算符理论或者积算符理论来分析,作为例子,用数值方法计算了高斯脉冲的三个参数。     

The discontinuous Petrov-Galerkin method for one-dimensional compressible Euler equations in the Lagrangian coordinate

In this paper, a Petrov-Galerkin scheme named the Runge-Kutta control volume (RKCV) discontinuous finite element method is constructed to solve the one-dimensional compressible Euler equations in the Lagrangian coordinate. Its advantages include preservation of the local conservation and a high resolution. Compared with the Runge-Kutta discontinuous Galerkin (RKDG) method, the RKCV method is easier to implement. Moreover, the advantages of the RKCV and the Lagrangian methods are combined in the new method. Several numerical examples are given to illustrate the accuracy and the reliability of the algorithm.     

计算物理前沿及其与计算技术的交叉

计算物理学是随着计算技术发展而形成的一门新兴的交叉学科。它是用现代计算技术武装起来的“实验的”理论物理学和“虚拟的”实验物理学。文章回顾了计算物理的发展历史，介绍了计算物理研究状况和前沿问题，并展望了新世纪计算物理面临的挑战。     

龙格库塔间断有限元方法在计算爆轰问题中的应用

构造求解带源项守恒律方程组的龙格库塔间断有限元(RKDG)方法,并分别结合源项的Strang分裂法和无分裂法数值求解模型守恒律方程和反应欧拉方程.为了和有限体积型WENO方法进行比较,设计计算源项的WENO重构格式.对一维带源项守恒律的计算表明,对于非刚性问题,RKDG方法比有限体积型WENO方法的误差更小;对于刚性问题,RKDG方法对于间断面位置的捕捉更为精确.对于一二维爆轰波问题的计算结果表明,RKDG方法对爆轰波结构的分辨和爆轰波位置的捕捉能力更强.     

RKDG有限元法求解一维拉格朗日形式的Euler方程

描述一种新的求解Euler方程的拉格朗日格式,该格式用Runge-Kutta Discontinuous Galerkin(RKDG)方法在拉格朗日坐标系求解Euler方程,剖分网格随流体运动.新格式不仅保证流体的质量、动量和能量守恒,而且能够在时间和空间上同时达到二阶精度.数值算例表明在一维情况,随着拉氏网格的移动和改变,格式在时间和空间上仍保持二阶精度,并且没有数值震荡.     

二维随机介质中的能量分布和频谱特性

研究了二维随机介质中能量的空间分布和一些特定区域的频谱特性.结果显示介质中能量呈局域化分布,其分布与介质中颗粒的随机分布有关,还与激发光的波长有密切的关系,介质中能量聚集的区域和介质的准封闭区域有一定关系但并非一一对应.所考察的各准封闭区域的频谱均随时间变化,在同一激发光的激励下各准封闭区域的频谱均不相同,而同一区域在不同的激发光的激励下频谱也不相同.当激发波长与准封闭区域的结构参数偏差较大时,频谱中谱峰会出现此消彼涨的模式竞争现象,而且能量衰减得较快;而当激发波长与该结构参数较接近时,相应的频谱中存在一关键词：激光物理有限时域差分法局域化频谱特性     

一类新的差分格式优化目标函数

对差分格式进行优化处理可以提高其谱精度。与高精度(指Taylor展开精度)格式相比,优化格式放大因子的误差随波数的变化不是单调的,而是必然会出现极值点,这样就存在临界距离R_cr,在此距离内优化格式描述的数值波的积累误差小于高精度格式,而超出此距离后优化格式的误差反而大,对于非定常流及气动声学计算来说,控制差分格式的临界距离是必要的。一般的优化目标函数以每个时间推进步的误差为基础(即放大因子法),R_cr不能在优化过程中确定。对此进行分析,指出积累误差的重要性并提出以此为基础的新的优化目标函数,这样在对格式进行优化时可以直接指定临界距离,从而为控制谱精度提供方便。     

双曲型守恒律的一种高精度TVD差分格式

构造了一维双曲型守恒律方程的一个高精度高分辨率的守恒型TVD差分格式.其主要思想是:首先将计算区域划分为互不重叠的小单元,且每个小单元再根据希望的精度阶数分为细小单元;其次,根据流动方向将通量分裂为正、负通量,并通过小单元上的高阶插值逼近得到了细小单元边界上的正、负数值通量,为避免由高阶插值产生的数值振荡,进一步根据流向对其进行TVD校正;再利用高阶Runge KuttaTVD离散方法对时间进行离散,得到了高阶全离散方法.进一步推广到一维方程组情形.最后对一维欧拉方程组计算了几个算例.     

A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids

Recently a new high-order formulation for 1D conservation laws was developed by Huynh using the idea of “flux reconstruction”. The formulation was capable of unifying several popular methods including the discontinuous Galerkin, staggered-grid multi-domain method, or the spectral difference/spectral volume methods into a single family. The extension of the method to quadrilateral and hexahedral elements is straightforward. In an attempt to extend the method to other element types such as triangular, tetrahedral or prismatic elements, the idea of “flux reconstruction” is generalized into a “lifting collocation penalty” approach. With a judicious selection of solution points and flux points, the approach can be made simple and efficient to implement for mixed grids. In addition, the formulation includes the discontinuous Galerkin, spectral volume and spectral difference methods as special cases. Several test problems are presented to demonstrate the capability of the method.     

The effect of shocks on second order sensitivities for the quasi-one-dimensional Euler equations

The effect of discontinuity in the state variables on optimization problems is investigated on the quasi-one-dimensional Euler equations in the discrete level. A pressure minimization problem and a pressure matching problem are considered. We find that the objective functional can be smooth in the continuous level and yet be non-smooth in the discrete level as a result of the shock crossing grid points. Higher resolution can exacerbate that effect making grid refinement counter productive for the purpose of computing the discrete sensitivities. First and second order sensitivities, as well as the adjoint solution, are computed exactly at the shock and its vicinity and are compared to the continuous solution. It is shown that in the discrete level the first order sensitivities contain a spike at the shock location that converges to a delta function with grid refinement, consistent with the continuous analysis. The numerical Hessian is computed and its consistency with the analytical Hessian is discussed for different flow conditions. It is demonstrated that consistency is not guaranteed for shocked flows. We also study the different terms composing the Hessian and propose some stable approximation to the continuous Hessian.