隐-显混合格式在不稳定导热问题中的应用

一、引言 众所周知,在用有限元法计算不稳定导热问题时,由于对时间的离散方式不同,所得的计算格式也不同,通常考虑两种格式,隐式格式和显式格式,同一种格式,由于采用的公式不同,其数值稳定性的条件也不同。本文,我们只考虑无条件稳定的隐式格式及条件稳定的显式格式以及由它们构成的隐-显混合格式(详见第二节)。     

时间分数阶四阶扩散方程的显-隐和隐-显差分格式

时间分数阶四阶扩散方程是一类重要的发展型偏微分方程,其数值解的研究有重要的科学意义和工程实际价值.本文针对时间分数阶四阶扩散方程,研究一类显-隐(E-I)差分格式和隐-显(I-E)差分格式解法,该方法基于经典隐式和经典显式格式相结合构造而成,分析E-I和I-E两种差分格式解的存在唯一性、稳定性和收敛性.理论分析和数值试验结果证实本文E-I差分格式和I-E差分格式无条件稳定,具有空间2阶精度,时间2-α阶精度.在计算精度一致的要求下,E-I和I-E差分格式较经典隐式差分格式具有省时性,其计算时间相比古典隐格式减少约70%,研究表明本文格式求解时间分数阶四阶扩散方程是有效的.     

求解色散方程的两类阶梯形半显格式

一、引言 在求解色散方程u_t=αu_(xxx)的差分格式中,显格式的稳定性条件限制大,隐格式的计算量大(文[1—5])。曾文平同志提出了一类绝对稳定的半显格式,它们需要用另外的方法提供一个左端点(或右端点)。本文提出了另外两类绝对稳定的半显格式,只须给出左边界条件(或右边界条件),它们就可以显式计算。精确度与曾文平的格式相同,而计     

求解对流-扩散方程的交替分段显-隐式方法

1．引言求解下述对流一扩散方程已有很多显式和隐式差分方法．显式方法很适合于并行计算，但由于稳定性条件的限制，必须采用非常小的时间步长进行计算．隐式格式一般无稳定性条件，但在每一时间层上要求解线性代数方程组，实现并行计算有一定困难．Evans和Abdullah［‘，‘l巧妙地利用Saul’yev非对称格式构造了交替分组显式（xGz）方法来求解扩散方程，其后又将方法推广到求解对流一扩散方程＊．张宝琳＊提出了求解扩散方程的交替分段显一隐式方法，在一定意义上推广了Evans和Abdullah的方法，并在计算上更为精确．本文根据＊中关于扩…     

热传导方程基于界面修正的迭代并行计算方法

在许多实际计算中,由于对时间步长稳定性的要求,辐射热传导方程的计算通常采用隐式格式．隐式格式难以直接在并行机上实施,显式差分格式尽管易于在并行机上实施,但它的稳定性条件苛刻．在计算问题规模相当大时,例如需要具有数百、数千甚至上万台处理器的大型并行计算机进行计算时,数据的强相关与全局通讯等问题成为制约实现高性能计算的突出的瓶颈问题．因此,改造现有的隐式格式,研究适应于大型并行计算机的并行计算方法是目前大型科学与工程计算中迫切需要解决的具有挑战性的问题．本文简要介绍基于界面修正的迭代并行计算格式的构造及基本性质．所提出的并行格式的构造方法是将预测-校正技术应用于分区子区域的内边界,且与子区域内部的迭代求解相结合,讨论了这些并行格式的稳定性、收敛性与并行度等性质．     

粘性Burgers方程的高阶隐式WCNS格式

JFNK (Jacobian-free Newton-Krylov)方法是由外层Newton迭代法和内层Krylov子空间迭代法构成的嵌套迭代方法.本文提出了一种基于JFNK方法的高阶隐式WCNS (weighted compact nonlinear scheme)格式,并用于求解一维、二维粘性Burgers方程.外层迭代法采用含参数的多步Newton迭代法,给出了收敛性分析,内层迭代法采用无矩阵GMRES迭代法.粘性Burgers方程的非线性对流项采用五阶WCNS格式计算.为提高方法精度和计算效率,时间离散采用三阶隐式的DIRK (diagonal implicit Runge-Kutta)方法.数值结果表明基于JFNK方法的隐式WCNS格式在时间上能达到三阶精度,与显式TVD Runge-Kutta WCNS方法相比,计算效率更高.此外,基于JFNK方法的隐式WCNS格式稳定性好,且具有良好的激波捕捉能力.     

包含集总模型的ADI-FDTD扩展算法的数值稳定性

本文利用Von-Neumann技术系统地研究了包含集总模型的ADI-FDTD扩展算法的稳定性问题,其中集总模型包括电阻、电容和电感.集总模型伏安特性方程按显式、半隐式和隐式三种差分格式进行差分.分析结果表明:集总模型按显式格式差分时,ADI-FDTD扩展算法不是无条件稳定的,稳定性条件受集总元件值的影响,而在半隐式和隐式差分格式下,扩展算法是无条件稳定的.最后通过分析两个简单的包含集总元件的微带电路,验证了理论结果的正确性.     

有限元法在板料二次成形中的应用

本文建立了板料成形动力显式有限元模拟，采用四节点退化壳单元对板料进行离散化，利用中心差分法离散时间域，建立显式计算格式，采用罚函数法和修正库仑定理计算接触力和摩擦力。对二次成形过程，建立了有限元分析计算模型。通过Dynaform对板料成形进行仿真得出最后结论。     

双曲型守恒律的一种三阶松弛格式

对一维双曲型守恒律,给出了一种形式更简单、计算量更小的三阶松弛格式.该格式以三阶WENO重构和三阶显隐式Runge-Kutta方法为基础.由于不用求解Riemann问题和计算非线性通量函数的雅可比矩阵,所以本文格式保持了松弛格式简单的优点.数值试验表明:该方法具有较高的分辨率.     

复合隐式时间积分格式的非线性问题仿真研究

梯形时间格式在求解某些非线性问题时会出现失稳现象,无法获得收敛解,为了验证复合隐式时间积分格式在求解此类非线性问题时的优越性,通过比较分析数值耗散的大小确定复合隐式时间积分格式中γ的合理取值。通过Updated La-grangian方法建立非线性运动增量方程,应用9节点四边形单元划分计算域网格。分析比较了相同时间步长,γ取值对研究对象的位移、速度和加速度时程曲线的影响。悬臂梁承受恒定集中力和承受冲击载荷的算例均表明:γ取0.2时复合隐式积分格式引入的数值耗散较小,精度较高。     

An efficient algorithm for the discrete-time algebraic Riccatiequation

The authors develop an algorithm to solve the standard discrete-time algebraic Riccati equation by using a skew-Hamiltonian transformation and the square-root method. The algorithm is structure-preserving and efficient because the Hamiltonian structure is fully exploited and only orthogonal transformations are used. The efficiency and stability of the algorithm are analyzed, numerical examples are included     

极限数值演示

介绍了VB编写程序演示limn→∞（1＋1/n）^n的数值变化过程,分析了理论计算与实际计算的差异,并提出了解决方法;比较完整地反映了数学建模、修改模型、编制算法、优化算法的过程。     

Finite-time stability analysis of a class of nonlinear time-varying systems: a numerical algorithm

The paper investigates the finite-time stability (FTS) analysis of a very general class of nonlinear time-varying systems. The FTS of the considered system, whose vector field consists of a nonlinear part which can be sublinear or superlinear, and a linear part which can be time-varying, has not been fully studied before. By estimating the bound of the norm of the considered system's states with the generalised Gronwall–Bellman inequality, a sufficient criterion is established to guarantee the FTS of the considered system. To facilitate checking the criterion in practice, a novel numerical algorithm is proposed by numerically solving certain differential equations. Therefore, the FTS of the considered class of nonlinear time-varying systems can be easily analysed by the numerical algorithm. Further considering the numerical errors in the practical numerical computation, we strictly prove the credibility and programmability of the numerical algorithm in theory. Finally, three numerical examples are provided to illustrate the effectiveness the proposed results.     

一种空时联合的自适应波束形成算法

为了进一步提高传统最优空时滤波算法的性能,提出了一种新的空时联合的自适应波束形成算法。在给出算法的信号模型和算法原理的基础上,详细地阐述了该算法实现空时联合自适应滤波的机理,然后利用计算机仿真较全面地分析和比较了算法的性能。理论分析和仿真结果表明,该算法在提高输出信干噪比和削弱指向误差影响等方面明显优于传统的最优空时滤波方法。     

Parallel least-squares finite element method for time-dependent convection–diffusion system

On the basis of overlapping domain decomposition, we construct a parallel least-square finite element algorithm (PLS) for solving the first-order time-dependent convection–diffusion system. The algorithm is fully parallel. At each time step, only one or two iterations are needed to reach to given accuracy. Some numerical results are reported to confirm the theoretical results.     

Real‐time Shading with Filtered Importance Sampling

We propose an analysis of numerical integration based on sampling theory, whereby the integration error caused by aliasing is suppressed by pre‐filtering. We derive a pre‐filter for evaluating the illumination integral yielding filtered importance sampling, a simple GPU‐based rendering algorithm for image‐based lighting. Furthermore, we extend the algorithm with real‐time visibility computation. Free from any pre‐computation, the algorithm supports fully dynamic scenes and, above all, is simple to implement.     

A new model-dependent time integration scheme with effective numerical damping for dynamic analysis

This paper presents a new semi-explicit dissipative model-dependent time integration algorithm for solving structural dynamics problems. Motivated by the superior properties of the composite time-stepping scheme, the proposed method is designed, so that it fully inherits the numerical characteristics of its parent algorithm, namely the Bathe method. The algorithm design procedure is carried out by assuming unknown integration parameters for the proposed method. Afterwards, by time discretization of an SDOF model equation, the unknown parameters can be obtained explicitly by solving nonlinear system of equations. Some numerical examples are analyzed by the presented technique and comparisons are also made with two other dissipative model-dependent time integration algorithms as well as the Bathe method. Results demonstrate that the suggested technique can effectively damp out the spurious oscillations of the high-frequency modes, while the other schemes exhibit significant overshoot in the calculated responses. Furthermore, it is also observed that numerical results of the presented method totally coincide with the parent algorithm. While the Bathe method subdivides each time increment into two sub-steps, the proposed algorithm is single-step, non-iterative and does not involve any time-step subdividing.

     

Numerical analysis of coupled thermomechanical frictional contact problems. Computational model and applications

Summary In this paper a numerical model for the analysis of coupled thermomechanical multi-body frictional contact problems at finite deformations is presented. The multi-body frictional contact formulation is fully developed on the continuum setting and then a spatial (Galerkin projection) and temporal (time-stepping algorithm) discretization is applied. A contact pressure and temperature dependent thermal contact model has been used. A fractional step method arising from an operator split of the governing equations has been used to solve the coupled nonlinear system of equations, leading to a staggered solution algorithm. The numerical model has been implemented into an enhanced version of the computational finite element program FEAP. Numerical examples and simulation of industrial metal forming processes show the performance of the numerical model in the analysis of coupled thermomechanical frictional contact problems.     

基于国产十亿亿次超算系统的近连续过渡流区N-S/DSMC耦合算法并行优化研究

过渡流区气动问题的数值模拟一直是空气动力学领域的难点。首先介绍了在已有 N-S解算器和 DSMC方法研究基础上,采用 MPC耦合技术建立N-S/DSMC 耦合算法,把 DSMC 方法和 N-S 方法的应用范围拓展到近连续过渡流区。然后详述了基于国家超级计算无锡中心的国产十亿亿次超级计算机开展的耦合算法多级并行优化技术,并首次实现了耦合算法的众核并行。测试表明,本文的进程级优化技术取得了超线性加速比;众核级优化受制于原算法特点和系统特点没有取得预期效果,但进行了探讨和分析,为 N-S/DSMC 耦合算法的众核并行提供了研究和分析依据,为过渡流区高超声速气动特性数值模拟研究提供了有效的途径。     

对带有有限缓冲区和可修机器的串行生产线的生产方差研究

把生产与市场结合起来并考虑供需关系时,系统的生产方差研究显得非常必要.讨 论的模型为两级的串行生产线,利用结构随机矩阵理论和马氏过程的协方差结构,给出了在 一个给定时间区间内系统生产成品数方差的有效算法.基于正态分布,这个算法对订单满足 的程度进行了概率刻划.文中的数值分析充分体现了这个算法的有效性和实用性.