二维双曲型守恒律的一种五阶松弛格式

松弛格式是Jin和Xin提出的无振荡有限差分方法,其主要思想是将守恒律转化为松弛方程组进行求解.本文用逐维五阶WENO重构和显隐式Runge-Kutta方法对松弛方程组的空间和时间进行离散,得到了一种求解二维双曲型守恒律五阶松弛格式.所得格式保持了松弛格式简单的优点,不用求解Riemann问题和计算通量函数的雅可比矩阵.通过二维Burgers方程和二维浅水方程的数值算例验证了格式的有效性.     

高阶紧致格式求解二维粘性不可压缩复杂流场

提出了一种求解二维不可压缩复杂流场的高精度算法．控制方程为原始变量、压力Ｐｏｉｓｓｏｎ方程提法．在任意曲线坐标下，采用四阶紧致格式求解Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程组，时间推进采用交替方向隐式（ＡＤＩ）格式，在非交错网格上用松弛法求解压力Ｐｏｉｓｓｏｎ方程．对于复杂的流场，采用了区域分解方法，并在每一时间步对各子域实施松弛迭代使之能精确地反映非定常流场．利用该算法计算了二维受驱空腔流动，弯管流动和垂直平板的突然起动问题．计算结果与实验结果和其他研究者的计算结果相比较吻合良好．对于平板起动流动，成功地模拟了流场中旋涡的生成以及Ｋａｒｍａｎ涡街的形成     

加性非平稳激励下结构速度响应的FPK方程降维

结构在随机激励下的非线性响应分析是具有高度挑战性的困难问题.对于白噪声或过滤白噪声激励,求解FPK方程将获得结构响应的精确解.遗憾的是,对于非线性多自由度系统,FPK方程难以直接求解.事实上,其数值解法严重受限于方程维度,而解析求解则仅适用于少数特定的系统,且多是稳态解.因此,将FPK方程进行降维,是求解高维随机动力响应分析问题的重要途径.本文针对幅值调制的加性白噪声激励下多自由度非线性结构的非平稳随机响应分析问题,将联合概率密度函数满足的高维FPK方程进行降维.针对结构速度响应概率密度函数求解,通过引入等价漂移系数,原FPK方程可转化为一维FPK型方程.建议了构造等价漂移系数的条件均值函数方法.进而,采用路径积分方法求解降维FPK型方程,得到速度概率密度函数的数值解答.结合单自由度Rayleigh振子、十层线性剪切型框架和非线性剪切型框架结构在幅值调制的加性白噪声激励下的非平稳速度响应求解,讨论了本文方法的精度和效率,验证了其有效性.     

多介质流体非守恒律欧拉方程组的数值计算方法

对多介质流体在界面处满足的Euler方程进行了探讨,方程组中增加了描述材料参数间断性质的对流形式非守恒律方程组 .以波传播算法为基础,通过Roe方程近似求解Riemann问题,同时采用相同的数值差分格式求解流体动力学Euler方程组和界面方程组.该方法可以有效消除多介质流体在界面处压力、速度可能出现的非物理振荡.给出了部分典型一维和二维数值计算结果.     

求大规模非线性随机动力系统概率密度函数解的子空间法

论文给出了一个求大规模非线性随机动力系统响应概率密度函数解的新方法,称之为子空间法.考察了此方法在求解大规模带位移项参数激励非线性随机动力系统响应概率密度函数的有效性.这里的概率密度函数解由Fokker-Planck-Kolmogorov(FPK)方程控制.该方法是基于将非线性随机动力系统状态变量空间分成两个子空间,然后在其中一个子空间上对FPK方程进行积分,采取一定措施后得到低维的FPK方程.该低维的FPK方程的维数可以人为确定,也可以取为二维,从而可以用现有的求解低维FPK方程的方法求得所需的概率密度函数解.文中给出了算例,用数值结果验证了子空间法的有效性.论文是采用作者曾提出的指数多项式闭合(EPC)法求解由子空间法降维的FPK方程.     

多介质流体非守恒律欧拉方程组的数值计算方法

对多介质流体在界面处满足的Euler方程进行了探讨 ,方程组中增加了描述材料参数间断性质的对流形式非守恒律方程组。以波传播算法为基础 ,通过Roe方程近似求解Riemann问题 ,同时采用相同的数值差分格式求解流体动力学Euler方程组和界面方程组。该方法可以有效消除多介质流体在界面处压力、速度可能出现的非物理振荡。给出了部分典型一维和二维数值计算结果。     

一种基于Associated Hermite正交函数求解对流扩散方程的算法

提出了一种基于AH(Associated Hermite)正交基函数求解对流扩散方程的无条件稳定算法。该算法将方程的时间项通过Hermite多项式作为正交基函数进行展开,利用Galerkin方法消除时间变量项,从而导出有限维AH域隐式差分方程,突破了传统显式差分格式稳定性条件的限制,最后通过对AH域展开系数的求解得到该对流扩散方程的数值解。在数值算例中,将该算法与传统显示差分法和交替方向隐式差分法进行对比分析,数值计算结果表明,算法无条件稳定且其计算精度与时间步长无关,对于具有精细结构的对流换热问题,该算法具有明显的效率优势,且保持了较高的精度。     

网格与高精度差分计算问题

研究NS方程差分求解时来流雷诺数、计算格式精度和计算网格之间的关系．给出了判定空间三个方向上的粘性贡献在给定雷诺数、格式精度和网格下是否能够正确计入的估计方法．指出在NS方程的二阶差分方法的数值模拟中,由于物面法向采用了压缩网格技术,物面附近的网格间距很小,该方向上的粘性贡献可被计入．但是如果流向和周向的网格较粗,相应的差分方程中的粘性贡献可能落入截断误差相同的量级,因此在精度上等于仍是求解略去流向和周向粘性项的薄层近似方程．指出,高阶精度的差分计算格式,可以避免对网格要求苛刻的困难．并进一步讨论了建立高阶精度格式的问题,提出了建立高阶精度格式应该满足的原则：耗散控制原则以及色散控制原则．为了避免激波附近可能出现的微小非物理振荡,建议发展混合高阶精度格式,即在激波区,采用网格自适应的NND格式,在激波以外的区域,采用按上述原则发展的高阶格式．     

用斜交坐标差分法计算叶型的弯曲中心

对于象叶型这样具有复杂曲线边界的截面,本文介绍用斜交坐标差分法数值求解应力函数和翘曲函数来计算弯曲中心.首先选取能逼近边界的网格并建立斜交坐标系,然后对网格区域用回路积分导出差分格式,再用直接法求解所得的对称带状线性代数方程组,并对网格点上的函数值用Romberg 公式数值积分,从而得到弯曲中心值.文末以椭圆和半圆为例考察计算精确度,所得计算值与精确值非常接近.     

非结构混合网格高超声速绕流与磁场干扰数值模拟

对均匀磁场干扰下的二维钝头体无粘高超声速流场进行了基于非结构混合网格的数值模拟.受磁流体力学方程组高度非线性的影响及考虑到数值模拟格式的精度,目前在此类流场的数值模拟中大多使用结构网格及有限差分方法,因而在三维复杂外形及复杂流场方面的研究受到限制.本文主要探索使用非结构网格(含混合网格)技术时的数值模拟方法.控制方程为耦合了Maxwell方程及无粘流体力学方程的磁流体力学方程组,数值离散格式采用Jameson有限体积格心格式,5步Runge-Kutta显式时间推进.计算模型为二维钝头体,初始磁场均匀分布.对不同磁感应强度影响下的高超声速流场进行了数值模拟,并与有限的资料进行了对比,得到了较符合的结果.     

High-efficiency improved symmetric successive over-relaxation preconditioned conjugate gradient method for solving large-scale finite element linear equations

Fast solving large-scale linear equations in the finite element analysis is a classical subject in computational mechanics. It is a key technique in computer aided engineering （CAE） and computer aided manufacturing （CAM）. This paper presents a high-efficiency improved symmetric successive over-relaxation （ISSOR） preconditioned conjugate gradient （PCG） method, which maintains lelism consistent with the original form. Ideally, the by 50% as compared with the original algorithm. the convergence and inherent paralcomputation can It is suitable for be reduced nearly high-performance computing with its inherent basic high-efficiency operations. By comparing with the numerical results, it is shown that the proposed method has the best performance.     

极坐标系下Fourier-Legendre谱元方法与有限差分法数值扩散的比较

提出一种Fourier-Legendre谱元方法用于求解极坐标系下的Navier-Stokes方程,其中极点所在单元的径向采用Gauss-Radau积分点,避免了r=0处的1/r坐标奇异性。时间离散采用时间分裂法,引入数值同位素模型跟踪同位素的输运过程验证数值模拟的精度,分别利用谱元法和有限差分法的迎风差分格式求解匀速和加速坩埚旋转流动中的同位素方程。计算结果表明,有限差分法中的一阶迎风差分格式存在严重的数值假扩散,二阶迎风差分格式的数值结果较精确,增加节点可以有效地缓解数值扩散。然而,谱元法具有以较少节点得到高精度解的优势。     

An all-at-once algebraic multigrid method for finite element discretizations of Stokes problem

We consider numerical solution of finite element discretizations of the Stokes problem. We focus on the transform-then-solve approach, which amounts to first apply a specific algebraic transformation to the linear system of equations arising from the discretization, and then solve the transformed system with an algebraic multigrid method. The approach has recently been applied to finite difference discretizations of the Stokes problem with constant viscosity, and has recommended itself as a robust and competitive solution method. In this work, we examine the extension of the approach to standard finite element discretizations of the Stokes problem, including problems with variable viscosity. The extension relies, on one hand, on the use of the successive over-relaxation method as a multigrid smoother for some finite element schemes. On the other hand, we present strategies that allow us to limit the complexity increase induced by the transformation. Numerical experiments show that for stationary problems our method is competitive compared to a reference solver based on a block diagonal preconditioner and MINRES, and suggest that the transform-then-solve approach is also more robust. In particular, for problems with variable viscosity, the transform-then-solve approach demonstrates significant speed-up with respect to the block diagonal preconditioner. The method is also particularly robust for time-dependent problems whatever the time step size.     

一种处理弹性动力边界元法中奇异积分的方法

利用边界元法求解瞬态弹性动力学问题时，时域基本解函数的分段连续性和奇异性为该问题的求解带来很大的困难。为了解决时域基本解中的奇异性问题，本文依据柯西主值的定义，对经过时间解析积分之后的时域基本解进行奇异值分解，将其分成奇异和正则积分两部分；其中正则部分可通过采用常规高斯积分方法来计算，而奇异部分具有简单的形式，可以利用解析积分计算。经过上述操作之后，就可以达到直接消除时域基本解中奇异积分的目的。和传统方法相比，本文方法并不依赖静力学基本解来消除奇异性，是一种直接求解方法。最后给定两个数值算例来验证本文提出方法的正确性和可行性，结果表明使用本文算法可以解决弹性动力学边界积分方程中的奇异性问题。     

Boundary element method to solve torsion problem of cracked cylinder

Separating the discontinuous solution by use of the single crack solution, together with the regular solution of harmonic function, the torsion problem of a cracked cylinder is reduced to solving a set of mixed-type integral equations and its numerical technique is then proposed by combining the numerical method of singular integral equation with the boundary element method. Several numerical examples are calculated which will be useful to engineering practice. The method proposed is characterized by its fine accuracy and convenience for using, which can be extended to the cases of multiple crack.The project supported by National Natural Science Foundation of China.     

饱和两相介质动力固结问题时域求解的精细时程积分方法

针对u-p形式的饱和两相介质波动方程,采用精细时程积分方法计算固相位移u,采用向后差分算法求解流体压力p,建立了饱和两相介质动力固结问题时域求解的精细时程积分方法。针对标准算例,对该方法的计算精度进行了校核。开展了该方法相关算法特性的研究,对采用不同数值积分方法计算非齐次波动方程特解项计算精度的差异进行了对比研究,并对采用不同积分点数目的高斯积分法计算特解项条件下计算精度的差异进行了对比研究。研究结果表明,(1)该方法具有良好的计算精度。(2)计算非齐次波动方程特解项的数值积分方法中,梯形积分法的计算精度最差,高斯积分法、辛普生积分法和科茨积分法都具有较好的计算精度。(3)增加高斯积分点数目对于提高计算精度的作用并不显著。     

Numerical simulation of chemically reactive Powell-Eyring liquid flow with double diffusive Cattaneo-Christov heat and mass flux theories

numerical study is reported for two-dimensional flow of an incompressible Powell-Eyring fluid by stretching the surface with the Cattaneo-Christov model of heat diffusion. Impacts of heat generation/absorption and destructive/generative chemical reactions are considered. Use of proper variables leads to a system of non-linear dimensionless expressions. Velocity, temperature and concentration profiles are achieved through a finite difference based algorithm with a successive over-relaxation (SOR) method. Emerging dimensionless quantities are described with graphs and tables. The temperature and concentration profiles decay due to enhancement in fluid parameters and Deborah numbers.     

An analysis of overlapping Schwarz method for a weakly coupled system of singularly perturbed convection-diffusion equations

In this article, we have developed an overlapping Schwarz method for a weakly coupled system of convection-diffusion equations. The method splits the original domain into two overlapping subdomains. A hybrid difference scheme is proposed in which on the boundary layer region, we use the central finite difference scheme on a uniform mesh, whereas on the nonlayer region, we use the mid-point difference scheme on a uniform mesh. It is shown that the numerical approximations converge in the maximum norm to the exact solution. We have proved that, when appropriate subdomains are used, the method produces almost second-order convergence. Furthermore, it is shown that two iterations are sufficient to achieve the expected accuracy. Numerical examples are presented to support the theoretical results. The main advantage of this method used with the proposed scheme is that it reduces iteration counts very much and easily identifies in which iteration the Schwarz iterate terminates.     

An indirect shooting method based on the POD/DEIM technique for distributed optimal control of the wave equation

This paper presents a fast numerical method, based on the indirect shooting method and Proper Orthogonal Decomposition (POD) technique, for solving distributed optimal control of the wave equation. To solve this problem, we consider the first‐order optimality conditions and then by using finite element spatial discretization and shooting strategy, the solution of the optimality conditions is reduced to the solution of a series of initial value problems (IVPs). Generally, these IVPs are high‐order and thus their solution is time‐consuming. To overcome this drawback, we present a POD indirect shooting method, which uses the POD technique to approximate IVPs with smaller ones and faster run times. Moreover, in the presence of the nonlinear term, to reduce the order of the nonlinear calculations, a discrete empirical interpolation method (DEIM) is applied and a POD/DEIM indirect shooting method is developed. We investigate the performance and accuracy of the proposed methods by means of 4 numerical experiments. We show that the presented POD and POD/DEIM indirect shooting methods dramatically reduce the CPU time compared to the full indirect shooting method, whereas there is no significant difference between the accuracy of the reduced and full indirect shooting methods.     

一类非线性奇异积分方程及其数值方法研究

探讨了一类非线性奇异积分方程的数理性质以及在颗粒雷诺数Rep＜1时此类方程解的存在条件，然后详细研究了该方程的数值计算方法并构造称之为P（EC）^k多步法的差分格式，分析了该格式的收敛性和代数精度，得到时域离散步长的约束关系。运用该格式计算了静止流场和均匀振荡流中球形小颗粒的非恒定运动，将计算结果与其解析解及有关实验数据的比较表明，上述数值方法具有良好的计算精度。