一种基于等效线性化的非平稳随机动力响应分析的显式迭代算法

提出了一种基于等效线性化法的非平稳随机动力响应分析的显式迭代算法.首先根据等效线性化法把非线性系统转化为离散的线性系统,然后应用Newmark-β积分方法,推导出各个离散时刻的时域显式迭代公式,进而可以快速得到非线性系统的随机动力响应,最后用一个非线性的范德波尔系统和一个杜芬系统受非平稳随机荷载的算例验证了该算法的计算精度和计算效率.     

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.     

GPU通用计算平台上中心差分格式显式有限元并行计算

显式有限元是解决平面非线性动态问题的有效方法.由于显式有限元算法的条件稳定性,对于大规模的有限元问题的求解需要很长的计算时间.图形处理器(GPU)作为一种高度并行化的通用计算处理器,可以很好解决大规模科学计算的速度问题.统一计算架构(CUDA)为实现GPU通用计算提供了高效、简便的方法.因此,建立了基于GPU通用计算平台的中心差分格式的显式有限元并行计算方法.该方法针对GPU计算的特点,对串行算法的流程进行了优化和调整,通过采用线程与单元或节点的一一映射策略,实现了迭代过程的完全并行化.通过数值算例表明,在保证计算精度一致的前提下,采用NVIDIA GTX 460显卡,该方法能够大幅度提高计算效率,是求解平面非线性动态问题的一种高效简便的数值计算方法.     

一类非线性方程组奇异解的计算方法及其应用

针对一类特殊的非线性方程组雅克比矩阵奇异的问题,提出了一种基于对偶空间的牛顿迭代方法。给出了一个显式的计算对偶空间的公式,在此基础上利用对偶空间作用于原方程组构造新的方程,使扩充后的方程组在近似值点的雅可比矩阵满秩,从而恢复牛顿迭代算法的二次收敛性。实验结果表明,改进后的算法一般迭代3次计算精度就可以达到10^(-15)。所提算法丰富了代数几何中关于理想的对偶空间理论,也为工程应用中的数值计算提供了一种新方法。     

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.     

使用局部支撑径向基函数的隐式曲线曲面几何迭代算法

由散乱数据稳定重构曲线曲面,在变分拟插值方法的基础之上,提出了使用局部支撑径向基函数的隐式几何迭代算法.首先,根据给定数据点的法向构造隐式函数的非零约束,构造计算隐函数系数的迭代格式,并讨论其收敛性;然后,在此基础上引入加速因子,对隐式迭代算法进行加速,同时讨论了加速算法的收敛性;最后,为了降低迭代过程空间和时间的复杂度,给出了一种加速算法的改进版本.数值实验表明,使用局部支撑径向基函数的隐式几何迭代算法对曲线曲面重构是有效的,并对部分信息缺失、非均匀分布、带噪声采样数据的重构也达到了较好的效果,且实现简单,易于并行.     

一种新型基于3D-MUSIC的迭代算法

在信号传播的速度与方向有关时,传统的MUSIC、ESPRIT算法和联合位置偏差DOA估计方法计算量太大,且估计精度下降,因此有必要改进算法提高精度。本文提出一种新型的迭代MUSIC优化方法,该方法精确度高,计算量没有增加,仿真试验证明该方法的有效性,正确性。     

一种泊松-玻尔兹曼方程稳定算法的高效有限元并行实现

泊松-玻尔兹曼方程(Poisson-Boltzmann Equation,PBE)是广泛应用于溶剂化生物分子静电分析的隐式溶剂化模型.本文在原有有限元软件基础上对近来提出的基于高阶有限元求解PBE的无条件稳定方法~([9])设计并实现了一种高效的并行计算方法.无条件稳定方法对PBE拟时间迭代求解,避开了强非线性导致的不稳定性.基于非结构化四面体网格本文设计实现了基于代数分解的求解稀疏线性方程组的高效并行模型.规模可扩展至6400 CPU核,并行效率达到近86%.大规模并行迭代求解线性方程组是计算科学领域的共性问题,它的高效并行实现不仅对实际生物分子静电分析提供了很好的基础,也可扩展至其他各应用领域.     

三维Euler方程的隐式间断有限元算法

为了求解三维欧拉方程,对隐式时间离散格式间断有限元方法进行了研究。根据间断Galerkin有限元方法思想,构造内迭代SOR-LU-SGS隐式时间离散格式,结合当地时间步长技术、多重网格方法,实现了三维流场的计算。数值计算了ONERAM6机翼、大攻角尖前缘三角翼以及DLR-F4翼身组合体的亚声速绕流问题。结果表明,加入SOR内迭代步的LU-SGS隐式算法具有较大的优势,相较于GMRES算法所占用的内存少且收敛速度相当,是LU-SGS算法的三倍以上。针对三维算例,具有较好的稳定性和较高的收敛速度,能够给出准确的流场信息。与原方法相比,SOR-LU-SGS方法无论是在迭代步数上还是在CPU时间上,效率均有明显提高,适合于三维复杂流场计算。     

CAE软件操作小百科 (46)

Abaqus提供显式和隐式2种求解类型，其中：显式计算方法是“有条件收敛的”，只要增量步小于限值，大多数情况均能顺利完成计算；而隐式计算方法在非线性情况下极易出现不收敛的情况，比如欠约束、接触、材料塑性或失效、断裂、屈曲失稳等，都可能导致多次迭代不收敛，增量步大小一降再降，直到满足终止条件而退出计算。Abaqus的隐式求解就是计算出一个很大的刚度矩阵方程的解。这个方程能否通过迭代到最后达到系统默认的收敛准则标准范围之内，就决定这一次计算能否收敛。下面总结解决不收敛问题的几种方法。     

Parallel implementation of finite-element/newton method for solution of steady-state and transient nonlinear partial differential equations

Domain decomposition by nested dissection for concurrent factorization and storage (CFS) of asymmetric matrices is coupled with finite element and spectral element discretizations and with Newton's method to yield an algorithm for parallel solution of nonlinear initial-and boundary-value problem. The efficiency of the CFS algorithm implemented on a MIMD computer is demonstrated by analysis of the solution of the two-dimensional, Poisson equation discretized using both finite and spectral elements. Computation rates and speedups for the LU-decomposition algorithm, which is the most time consuming portion of the solution algorithm, scale with the number of processors. The spectral element discretization with high-order interpolating polynomials yields especially high speedups because the ratio of communication to computation is lower than for low-order finite element discretizations. The robustness of the parallel implementation of the finite-element/Newton algorithm is demonstrated by solution of steady and transient natural convection in a two-dimensional cavity, a standard test problem for low Prandtl number convection. Time integration is performed using a fully implicit algorithm with a modified Newton's method for solution of nonlinear equations at each time step. The efficiency of the CFS version of the finite-element/Newton algorithm compares well with a spectral element algorithm implemented on a MIMD computer using iterative matrix methods.Submitted toJ. Scientific Computing, August 25, 1994.     

Comparison of numerical integration methods in the analysis of impulsively loaded elasto-plastic and viscoplastic structures

Direct integration methods for ordinary differential equations encountered in the finite element analysis of dynamic problems of structural mechanics are studied and compared. Examples considered are a single-degree-of-freedom oscillator and a transversal impact on a beam. The material behavior is assumed to be elastic, elasto-plastic or viscoplastic. Both geometrically linear and nonlinear cases are treated. The time-integrators used include both explicit and implicit, single- and multistep methods, e.g. methods of Runge-Kutta, Euler, Newmark, Houbolt, Wilson, the central difference method and stiffly stable methods by Gear. Methods are compared in terms of computation time and accuracy and ease of formulation.     

An explicit-implicit water-level model for tidal computations of rivers

The combined use of explicit and implicit time integration methods leads to a numerical model, based on the finite element method, that overcomes the often claimed advantages of finite difference methods. The implicit part of the model shows the whole range of flexibility the finite element method is famous for, whereas the explicit part reduces to simple difference quotients for regular and irregular discretizations. The stability of the numerical solution is ensured and high numerical damping is avoided by means of a generalized collocation method for the integrations in space.     

A high-order Lagrangian-decoupling method for the incompressible Navier-Stokes equations

In this paper we present a high-order Lagrangian-decoupling method for the unsteady convection diffusion and incompressible Navier-Stokes equations. The method is based upon Lagrangian variational forms that reduce the convection-diffusion equation to a symmetric initial value problem, implicit high-order backward-differentiation finite difference schemes for integration along characteristics, finite element or spectral element spatial discretizations and mesh-invariance procedures and high-order explicit time-stepping schemes for deducing function values at convected space-time points. The method improves upon previous finite element characteristic methods through the systematic and efficient extension to high-order accuracy and the introduction of a simple structure-preserving characteristic-foot calculation procedure which is readily implemented on modern architectures. The new method is significantly more efficient than explicit-convection schemes for the Navier-Stokes equations due to the decoupling of the convection and Stokes operators and the attendant increase in temporal stability. Numerous numerical examples are given for the convection-diffusion and Navier-Stokes equations for the particular case of a spectral element spatial discretization.     

A constitutive operator splitting method for nonlinear transient analysis

A constitutive operator splitting method for the time integration of the nonlinear dynamic equations of motion which result from the consideration of nonlinear material behavior is studied. In the method, the material constitutive law is split into a constant, history independent relation (implicit portion) and a variable history dependent relation (explicit portion); the resulting constituents are then integrated by implicit and explicit methods, respectively. Analysis of stability and some example solutions indicate that for materials with decreasing stiffness (materials which soften with increasing strain) the method is unconditionally stable. Several examples are considered which compare the accuracy of the method to exact solutions and solutions obtained by explicit central difference time integration; in all cases there is good agreement.     

Approximate response sensitivities for nonlinear problems in explicit dynamic formulation

Explicit and implicit time integration schemes are discussed in the context of sensitivity analysis of dynamic problems. The application of the fully explicit central difference method (CDM) proves to be efficient for many nonlinear problems. In the case of the corresponding dynamic sensitivity problem the CDM is less advantageous both from efficiency and accuracy points of view. Approximate sensitivity expressions are derived in the paper for nonlinear path-dependent problems allowing the application of an unconditionally stable implicit time integration scheme with the time step much larger than the time step of the explicit CDM scheme of the direct problem. The method seems to be particularly suitable for problems of quasi-static nature in which the dynamic terms are artificially introduced to allow explicit CDM solution of highly nonlinear equations. Received January 21, 2000     

Structural dynamics methods for concurrent processing computers

In the area of crash impact, research is urgently required on the development and evaluation of parallel methods for crash dynamics analysis of complex nonlinear finite element and/or finite difference structural problems. An investigation of selected nonlinear dynamics algorithms appropriate for parallel computers is reported. Implicit methods such as those of the Newmark type which build on the Cholesky decomposition strategy and explicit methods such as the central difference time integration method are included. Both implicit and explicit dynamics algorithms are investigated on two significantly different parallel computers, the FLEX/32 shared memory multicomputer and the INTEL iPSC Hypercube local memory computer.     

An analysis of the time integration algorithms for the finite element solutions of incompressible Navier–Stokes equations based on a stabilised formulation

This work is concerned with the analysis of time integration procedures for the stabilised finite element formulation of unsteady incompressible fluid flows governed by the Navier–Stokes equations. The stabilisation technique is combined with several different implicit time integration procedures including both finite difference and finite element schemes. Particular attention is given to the generalised-α method and the linear discontinuous in time finite element scheme. The time integration schemes are first applied to two model problems, represented by a first order differential equation in time and the one dimensional advection–diffusion equation, and subjected to a detailed mathematical analysis based on the Fourier series expansion. In order to establish the accuracy and efficiency of the time integration schemes for the Navier–Stokes equations, a detailed computational study is performed of two standard numerical examples: unsteady flow around a cylinder and flow across a backward facing step. It is concluded that the semi-discrete generalised-α method provides a viable alternative to the more sophisticated and expensive space–time methods for simulations of unsteady flows of incompressible fluids governed by the Navier–Stokes equations.     

面向半物理仿真的陈述式模型求解方法研究

通过符号操作和数值计算相结合,提出了一种求解半物理仿真模型的新方法。为了满足半物理仿真对实时性的要求,在模型编译阶段将代表数值积分的隐式离散公式插入到仿真模型中,增广后的方程系统伴随着非线性方程的出现,需要在积分的每一步对这些非线性方程进行迭代求解,而求解非线性方程的时间复杂度随维度的变大成指数增加,因此引入代数环撕裂减小代数方程块耦合变量数,以满足实时求解对粒度的要求。最后通过实例对文中提出的方法进行了验证。     

利用弹簧质点模型和隐式方法的布料模拟研究

首先运用弹簧-质点模型建立布料的面模型,然后对质点进行力的分解以及受力分析并优化。提出逼近的隐式数值积分方法模拟质点的运动轨迹,这解决了显式数值积分方法的不稳定性和小时间间隔的缺点和其他隐式方法计算量大的缺点,这也是实现基于物理模型的布料仿真的关键技术。针对具体碰撞对象采用简单的包围盒方法进行碰撞检测,和利用二分法进行碰撞的处理,大大增加了碰撞处理的逼真效果。实验证明其模拟方法具有稳定性和实用性。