组合杂交四边形元的多重网格预处理共轭梯度方法

组合杂交元方法是一种求解弹性力学问题的稳定化有限元方法.为了快速求解组合杂交元离散得到的大型、稀疏、对称正定系统,本文研究了多重网格预处理共轭梯度方法.首先,通过选用合适的网格转移算子和光滑策略,得到了有效的多重网格预处理器.其次,通过分析数值试验结果证明所得到的多重网格预处理共轭梯度方法是有效可行的,利用该预处理方法大大降低了系数矩阵的条件数,提高了计算效率.此外,对于一类高性能的组合杂交元,多重网格预处理共轭梯度方法在网格畸变时依然收敛.     

预条件共轭梯度法的实现以及一些改进

本文讨论了工程有限元分析中预条件共轭梯度法的实现，并分析了此方法的优缺点，为了提高整体的效率和改进ＬＤＬ预优矩阵的稳定性，本文提出了双参数松弛方案和按元素的绝对值确定矩阵Ｌ的分解方案，双参数松弛方案基本上解决了ＬＤＬ预优矩阵的稳定性问题，按元素的绝对值大确定矩阵Ｌ的分解方案可以改善ＬＤＬ预优矩阵的稳定性，并提高预条件共轭梯度法在整体效率。     

散射问题中复线性系统的扰动预条件技术

利用稀疏策略可以控制不完全分解因子的稀疏度,对角扰动技术则通过对原系数矩阵的对角元的轻微扰动,提高不完全分解预条件方法的效率.本文结合稀疏策略和对角扰动技术的修正的不完全LLT分解预条件技术,用来加速共轭垂直共轭梯度法(COCG)求解离散散射问题得到的大型、稀疏的复对称线性系统的求解速率,数值试验验证了基于扰动的不完全分解预条件方法,对迭代求解散射问题有着很好的提速效果.     

预处理共轭梯度法求解线性方程组Ax=b

针对共轭梯度法求解线性方程组Ax=b,提出一种预处理思想。基于次思想,首先给出预处理矩阵,然后求解预处理线性方程组,再使用共轭梯度法求解。最后通过几个数值试验,与直接使用共轭梯度法求解线性方程组相比较,本文的方法提高了收敛速度。     

基于PVM的网络并行子结构共轭梯度法

网络并行环境是近年来国际上并行环境的一个重要方向,ＰＶＭ是当前最流行的支持异构或同构型网络并行计算的软件平台之一。本文采用子结构共轭梯度法研究了基于ＰＶＭ的网络并行有限元,该方法将有限元网格划分为ｎ个子结构,再将ｎ个子结构的数据分送给网上ｎ台可用微机,ｎ台微机并行形成和组集ｎ个子结构的劲度矩阵和荷载列阵,然后采用预条件共轭梯度法并行求解结点位移,最后ｎ台微机并行对ｎ个子结构进行应变和应力分析。该方法不需形成结构的总体劲度矩阵和荷载列阵,可同时迭代求出所有结点位移,且比一般的迭代法收敛要快。算例表明此种并行子结构共轭梯度法在网络上能获得较高的并行加速比。     

求解非线性等式约束优化问题的共轭梯度投影算法

利用投影矩阵,对求解无约束规划的共轭梯度算法中的参数βk给一限制条件确定βk的取值范围,以保证得到目标函数的共轭梯度投影下降方向,建立了求解非线性等式约束优化问题的共轭梯度投影算法,并证明了算法的收敛性。数值例子表明算法是有效的。     

一个充分下降的修正 PRP 型谱共轭梯度法

谱共轭梯度法是共轭梯度法的一种重要延拓,可以通过共轭参数和谱参数二维度调整,使得所设计算法的搜索方向满足某一预设条件,比如充分下降条件或共轭条件等。谱参数和共轭参数的设计是谱共轭梯度法的两大核心工作,决定方法的收敛性和数值效果。基于 PRP 方法,构造了一个修正的 PRP 型共轭参数,该共轭参数不仅保持了 PRP 公式的结构和性能,而且具有 FR 方法的收敛性质。利用充分下降条件取定一个谱参数,与修正的 PRP 型共轭参数结合,建立一个新的谱共轭梯度算法。该算法不依赖于任何线搜索就可以满足充分下降条件。常规假设条件下,采用强 Wolfe 线搜索准则产生步长,证明了新算法的全局敛性。通过 100 个算例对该算法进行数值测试并与其他五个算法进行比较,同时采用性能图对数值结果进行直观展示,结果表明该算法是有效的。     

一类Armijo搜索下的混合HS-PRP共轭梯度法?

为有效求解大规模无约束优化问题,本文基于HS方法和PRP方法,提出了一类新的混合共轭梯度法。该方法在每步迭代中都不依赖于函数的凸性和搜索条件而自行产生充分下降方向。在精确搜索下,本文算法将还原为标准的PRP方法。在适当的条件下,获证了该法在Armijo搜索下,即使求解非凸函数极小化的问题,算法也具有全局收敛性。同时,数值实验表明本文算法可以有效求解优化测试问题。     

一族共轭梯度法的全局收敛性

提出了求解无约束优化问题的一族共轭梯度法，这族方法包古Fkcher提出的共轭下降法．文中证明了一种非精确线性搜索条件能够保证这族方法的下降性和全局收敛性，其收敛结果与Dai和Yuan 1996年给出的关于共轭下降法的相一致．     

桥梁移动荷载识别的不适定性及其试验研究

对桥梁移动荷载识别方程不适定问题进行研究，提出采用预处理共轭梯度法（PCGM）求解超定方程组，通过选择不同的预优矩阵，改善和解决超定方程组的欠秩和病态问题。为验证基于PCGM方法的现场实用性，设计制作了车桥试验模型，通过试验采集到的桥梁弯矩响应数据识别桥面移动荷载。比较桥梁模态数、预处理共轭梯度法迭代次数、桥面粗糙度、车辆重量以及测点选择对识别结果精度的影响后，研究结果表明：基于PCGM方法能够很好地识别车辆荷载，收敛较快且能较好改善荷载识别方程的不适定性。     

Two-step preconditioning technique for hierarchical time-domain finite-element method

A two-step factorised sparse approximation inverse and symmetric successive over relaxation preconditioned conjugate gradient (CG) algorithm is proposed to solve the large system of linear equations resulted from the hierarchical implicit time-domain finite-element method (TDFEM). Convergence properties and CPU time of the proposed algorithm are compared with those of other preconditioned CG schemes. Numerical results demonstrate that the present approach is efficient for solving the large sparse system from hierarchical implicit TDFEM.     

Normalized explicit approximate inverse preconditioning for solving 3D boundary value problems on uniprocessor and distributed systems

Normalized explicit approximate inverse matrix techniques, based on normalized approximate factorization procedures, for solving sparse linear systems resulting from the finite difference discretization of partial differential equations in three space variables are introduced. Normalized explicit preconditioned conjugate gradient schemes in conjunction with normalized approximate inverse matrix techniques are presented for solving sparse linear systems. The convergence analysis with theoretical estimates on the rate of convergence and computational complexity of the normalized explicit preconditioned conjugate gradient method are also derived. A Parallel Normalized Explicit Preconditioned Conjugate Gradient method for distributed memory systems, using message passing interface (MPI) communication library, is also given along with theoretical estimates on speedups, efficiency and computational complexity. Application of the proposed method on a three‐dimensional boundary value problem is discussed and numerical results are given for uniprocessor and multicomputer systems. Copyright © 2005 John Wiley & Sons, Ltd.     

Algorithms for construction of preconditioners based on incomplete block-factorizations of the matrix

When applying an incomplete block-factorization technique one needs sparse approximate inverses of the successive Schur complements computed throughout the factorization. Here we propose a method for the construction of such sparse approximate inverses. The method has an advantage over earlier versions, in that such approximate inverses of block-tridiagonal matrices can be computed in parallel. Comparative numerical experiments for solving a number of discretized diffusion equations by this preconditioning matrix in a preconditioned conjugate gradient method and earlier versions of incomplete block-factorization preconditioners are presented.     

The effect of ordering on preconditioned GMRES algorithm,for solving the compressible Navier-Stokes equations

We investigate the effect of the ordering of the blocks of unknowns on the rate of convergence of a preconditioned non-linear GMRES algorithm, for solving the Navier-Stokes equations for compressible flows, using finite element methods on unstructured grids. The GMRES algorithm is preconditioned by an incomplete LDU block factorization of the Jacobian matrix associated with the non-linear problem to solve. We examine a wide range of ordering methods including minimum degree, (reverse) Cuthill-McKee and snake, and consider preconditionings without fill-in. We show empirically that there can be a significant difference in the number of iterations required by the preconditioned non-linear GMRES method and suggest a criterion for choosing a good ordering algorithm, according to the problem to solve. We also consider the effect of orderings when an incomplete factorization which allows some fill-in is performed. We consider the effect of automatically controlling the sparsity of the incomplete factorization through the level of fill-in. Finally, following the principal ideas of non-linear GMRES algorithm, we suggest other inexact Newton methods.     

A unified dual‐primal finite element tearing and interconnecting approach for incompressible Stokes equations

A unified framework of dual‐primal finite element tearing and interconnecting (FETI‐DP) algorithms is proposed for solving the system of linear equations arising from the mixed finite element approximation of incompressible Stokes equations. A distinctive feature of this framework is that it allows using both continuous and discontinuous pressures in the algorithm, whereas previous FETI‐DP methods only apply to discontinuous pressures. A preconditioned conjugate gradient method is used in the algorithm with either a lumped or a Dirichlet preconditioner, and scalable convergence rates are proved. This framework is also used to describe several previously developed FETI‐DP algorithms and greatly simplifies their analysis. Numerical experiments of solving a two‐dimensional incompressible Stokes problem demonstrate the performances of the discussed FETI‐DP algorithms represented under the same framework.Copyright © 2012 John Wiley & Sons, Ltd.     

Equation-oriented system: an efficient programming approach to solve multilinear and polynomial equations by the conjugate gradient algorithm

The factor analysis problem can be conceptualized as an expansion of polynomial equations that are solvable using least-squares methods. The equation-oriented system (EOS) is introduced as a method for solving polynomial equations using a preconditioned conjugate gradient (CG) algorithm for the normal equations. EOS is a fast, easy to program, low computer memory requirement method for accomplishing this task. EOS can be used to solve multilinear and PARAFAC problems. The practical aspects of implementing EOS in MATLAB are discussed.     

Nonlinear three-dimensional magnetic field computations usingLagrange finite-element functions and algebraic multigrid

The paper proposes an efficient solution strategy for nonlinear three-dimensional (3-D) magnetic field problems. The spatial discretization of Maxwell's equations uses Lagrange finite-element functions. The paper shows that this discretization is appropriate for the problem class. The nonlinear equation is linearized by the standard fixed-point scheme. The arising sequence of symmetric positive definite matrices is solved by a preconditioned conjugate gradient method, preconditioned by an algebraic multigrid technique. Because of the relatively high setup time of algebraic multigrid, the preconditioner is kept constant as long as possible in order to minimize the overall CPU time. A practical control mechanism keeps the condition number of the overall preconditioned system as small as possible and reduces the total computational costs in terms of CPU time. Numerical studies involving the TEAM 20 and the TEAM 27 problem demonstrate the efficiency of the proposed technique. For comparison, the standard incomplete Cholesky preconditioner is used     

The hybrid boundary node method accelerated by fast multipole expansion technique for 3D potential problems

This paper presents a fast formulation of the hybrid boundary node method (Hybrid BNM) for solving problems governed by Laplace's equation in 3D. The preconditioned GMRES is employed for solving the resulting system of equations. At each iteration step of the GMRES, the matrix–vector multiplication is accelerated by the fast multipole method. Green's kernel function is expanded in terms of spherical harmonic series. An oct‐tree data structure is used to hierarchically subdivide the computational domain into well‐separated cells and to invoke the multipole expansion approximation. Formulations for the local and multipole expansions, and also conversion of multipole to local expansion are given. And a binary tree data structure is applied to accelerate the moving least square approximation on surfaces. All the formulations are implemented in a computer code written in C++. Numerical examples demonstrate the accuracy and efficiency of the proposed approach. Copyright © 2005 John Wiley & Sons, Ltd.     

An implicit–explicit solution method for electro‐osmotic flow through three‐dimensional micro‐channels

This paper presents a comprehensive finite‐element modelling approach to electro‐osmotic flows on unstructured meshes. The non‐linear equation governing the electric potential is solved using an iterative algorithm. The employed algorithm is based on a preconditioned GMRES scheme. The linear Laplace equation governing the external electric potential is solved using a standard pre‐conditioned conjugate gradient solver. The coupled fluid dynamics equations are solved using a fractional step‐based, fully explicit, artificial compressibility scheme. This combination of an implicit approach to the electric potential equations and an explicit discretization to the Navier–Stokes equations is one of the best ways of solving the coupled equations in a memory‐efficient manner. The local time‐stepping approach used in the solution of the fluid flow equations accelerates the solution to a steady state faster than by using a global time‐stepping approach. The fully explicit form and the fractional stages of the fluid dynamics equations make the system memory efficient and free of pressure instability. In addition to these advantages, the proposed method is suitable for use on both structured and unstructured meshes with a highly non‐uniform distribution of element sizes. The accuracy of the proposed procedure is demonstrated by solving a basic micro‐channel flow problem and comparing the results against an analytical solution. The comparisons show excellent agreement between the numerical and analytical data. In addition to the benchmark solution, we have also presented results for flow through a fully three‐dimensional rectangular channel to further demonstrate the application of the presented method. Copyright © 2007 John Wiley & Sons, Ltd.     

动态载荷时域识别的联合去噪修正和正则化预优迭代方法

系统响应可表示为单位脉冲响应函数与激励载荷的卷积,将其离散化一组线性方程组,则载荷识别问题即转化为求解线性方程组的反问题。针对响应中带有噪音时载荷识别的困难,提出了联合奇异熵去噪修正和正则化预优的共轭梯度迭代识别方法。一方面对含噪信号进行基于奇异熵的去噪处理,提高反问题求解中输入数据的精度。另一方面利用正则化方法对共轭梯度迭代算法进行预优,改善反问题的非适定性。由于从输入的响应数据去噪和正则化算法两方面同时改善动态载荷识别反问题的求解,因此可以有效地抑制噪声,提高识别精度。通过数值算例分析,表明在不同的噪声水平干扰下,其识别精度均优于常规的正则化方法,能够实现有效稳定地识别动态载荷。最后通过实验研究进一步验证了该方法的正确性和有效性。