Improved preconditioned conjugate gradient method and its application in F. E. A. for engineering

In this paper two theorems with theoretical and practical significance are given in respect to the preconditioned conjugate gradient method(PCCG).The theorems discuss respectively the qualitative property of the iterative solution and the construction principle of the iterative matrix.The authors put forward a new incompletely LU factorizing technique for non-M-matrix and the method of constructing the iterative matrix.This improved PCCG is used to calculate the ill-conditioned problems and large-scale three-dimensional finite element problems,and simultaneously contrasted with other methods.The abnormal phenomenon is analyzed when PCCG is used to solve the system of ill-conditioned equations,It is shown that the method proposed in this paper is quite effective in solving the system of large-scale finite element equations and the system of ill-conditioned equations.     

Preconditioned Krylov subspace methods used in solving two‐dimensional transient two‐phase flows

This paper investigates the performance of preconditioned Krylov subspace methods used in a previously presented two‐fluid model developed for the simulation of separated and intermittent gas–liquid flows. The two‐fluid model has momentum and mass balances for each phase. The equations comprising this model are solved numerically by applying a two‐step semi‐implicit time integration procedure. A finite difference numerical scheme with a staggered mesh is used. Previously, the resulting linear algebraic equations were solved by a Gaussian band solver. In this study, these algebraic equations are also solved using the generalized minimum residual (GMRES) and the biconjugate gradient stabilized (Bi‐CGSTAB) Krylov subspace iterative methods preconditioned with incomplete LU factorization using the ILUT(p, τ) algorithm. The decrease in the computational time using the iterative solvers instead of the Gaussian band solver is shown to be considerable. Copyright © 1999 John Wiley & Sons, Ltd.     

A 3D incompressible Navier–Stokes velocity–vorticity weak form finite element algorithm

The velocity–vorticity formulation is selected to develop a time‐accurate CFD finite element algorithm for the incompressible Navier–Stokes equations in three dimensions.The finite element implementation uses equal order trilinear finite elements on a non‐staggered hexahedral mesh. A second order vorticity kinematic boundary condition is derived for the no slip wall boundary condition which also enforces the incompressibility constraint. A biconjugate gradient stabilized (BiCGSTAB) sparse iterative solver is utilized to solve the fully coupled system of equations as a Newton algorithm. The solver yields an efficient parallel solution algorithm on distributed‐memory machines, such as the IBM SP2. Three dimensional laminar flow solutions for a square channel, a lid‐driven cavity, and a thermal cavity are established and compared with available benchmark solutions. Copyright © 2002 John Wiley & Sons, Ltd.     

Analysis of iterative methods for the viscous/inviscid coupled problem via a spectral element approximation

Based on a new global variational formulation, a spectral element approximation of the incompressible Navier–Stokes/Euler coupled problem gives rise to a global discrete saddle problem. The classical Uzawa algorithm decouples the original saddle problem into two positive definite symmetric systems. Iterative solutions of such systems are feasible and attractive for large problems. It is shown that, provided an appropriate pre‐conditioner is chosen for the pressure system, the nested conjugate gradient methods can be applied to obtain rapid convergence rates. Detailed numerical examples are given to prove the quality of the pre‐conditioner. Thanks to the rapid iterative convergence, the global Uzawa algorithm takes advantage of this as compared with the classical iteration by sub‐domain procedures. Furthermore, a generalization of the pre‐conditioned iterative algorithm to flow simulation is carried out. Comparisons of computational complexity between the Navier–Stokes/Euler coupled solution and the full Navier–Stokes solution are made. It is shown that the gain obtained by using the Navier–Stokes/Euler coupled solution is generally considerable. Copyright © 2000 John Wiley & Sons, Ltd.     

A new positive‐definite regularization of incompressible Navier–Stokes equations discretized with Q1/P0 finite element

A new regularization method is proposed for the Galerkin approximation of the incompressible Navier–Stokes equations with Q1/P0 element, by newly introducing a square‐type linear form into the variational divergence‐free constraint regularized with the global pressure jump (GPJ) method. The addition of the square‐type linear form is intended to eliminate the hydrostatic pressure mode appearing in confined flows, and to make the discretized matrix positive definite and then non‐singular without the pressure pegging trick. Effects of the free parameters for the regularization on the solutions are numerically examined with a 2‐D driven cavity flow problem. Furthermore, the convergences in the conjugate gradient iteration for the solution of the pressure Poisson equation are compared among the mixed method, the GPJ method and the present method for both leaky and non‐leaky 3‐D driven cavity flows. Finally, the non‐leaky 3‐D cavity flows at different Re numbers are solved to compare with the literature data and to demonstrate the accuracy of the proposed method. Copyright © 2003 John Wiley & Sons, Ltd.     

基于分区加速和总体共轭梯度法的耦合界面数据传递问题研究

对于耦合动力学问题的分析过程,在界面上需频繁进行数据交换。为此,基于紧支径向基函数和多项式基函数推导了界面数据传递的插值算法,给出了传递矩阵的具体形式。通过分析时间复杂度,找出该算法在大节点量时效率不高的原因在于径向基矩阵的构造和传递矩阵的计算。为加快径向基矩阵的构造速度,提出分区加速处理以提高相关节点的搜索效率;为避免传递矩阵求解过程中的求逆运算,将其转化为多右端项的大型稀疏对称线性方程组问题,引入多右端项的总体共轭梯度迭代方法求解,并讨论了初始估计矩阵的选取方法。数值算例结果表明,结合使用分区加速原理和总体共轭梯度迭代方法,可在不损失插值精度的前提下显著提高求解效率。     

Parallel conjugate gradient performance for least-squares finite elements and transport problems

In this study we consider parallel conjugate gradient solution of sparse systems arising from the least-squares mixed finite element method. Of particular interest are transport problems involving convection. The least-squares approach leads to a symmetric positive system and the conjugate gradient scheme is directly applicable. The scheme is applied to both the convection–diffusion equation and to the stationary Navier–Stokes equations. Here we demonstrate parallel solution and performance studies for a representative MIMD parallel computer with hypercube architecture. © 1998 John Wiley & Sons, Ltd.     

A momentum coupling method for the unsteady incompressible Navier–Stokes equations on the staggered grid

A new numerical method is developed to efficiently solve the unsteady incompressible Navier–Stokes equations with second-order accuracy in time and space. In contrast to the SIMPLE algorithms, the present formulation directly solves the discrete x- and y-momentum equations in a coupled form. It is found that the present implicit formulation retrieves some cross convection terms overlooked by the conventional iterative methods, which contribute to accuracy and fast convergence. The finite volume method is applied on the fully staggered grid to solve the vector-form momentum equations. The preconditioned conjugate gradient squared method (PCGS) has proved very efficient in solving the associate linearized large, sparse block-matrix system. Comparison with the SIMPLE algorithm has indicated that the present momentum coupling method is fast and robust in solving unsteady as well as steady viscous flow problems. © 1998 John Wiley & Sons, Ltd.     

Numerical study on the vortex motion patterns around a rotating circular cylinder and their critical characters

A hybrid finite difference and vortex method (HFDV), based on the domain decomposition method (DDM), is used for calculating the flow around a rotating circular cylinder at Reynolds number Re=1000, 200 and the angular‐to‐rectilinear speed ratio α∈(0.5, 3.25) respectively. A fully implicit third‐order eccentric finite difference scheme is adopted in the finite difference method, and the deduced large broad band sparse matrix equations are solved by a highly efficient modified incomplete LU decomposition conjugate gradient method (MILU‐CG). The long‐time, fully developed features about the variations of the vortex patterns in the wake, as well as the drag and lift forces on the cylinder, are given. The calculated streamline contours are in good agreement with the experimentally visualized flow pictures. The existence of the critical state is confirmed again, and the single side shed vortex pattern at the critical state is shown for the first time. Also, the optimized lift‐to‐drag force ratio is obtained near the critical state. Copyright © 1999 John Wiley & Sons, Ltd.     

Transient solutions for three-dimensional lid-driven cavity flows by a least-squares finite element method

A time-accurate least-squares finite element method is used to simulate three-dimensional flows in a cubic cavity with a uniform moving top. The time- accurate solutions are obtained by the Crank-Nicolson method for time integration and Newton linearization for the convective terms with extensive linearization steps. A matrix-free algorithm of the Jacobi conjugate gradient method is used to solve the symmetric, positive definite linear system of equations. To show that the least-squares finite element method with the Jacobi conjugate gradient technique has promising potential to provide implicit, fully coupled and time-accurate solutions to large-scale three-dimensional fluid flows, we present results for three-dimensional lid-driven flows in a cubic cavity for Reynolds numbers up to 3200.     

注塑流动与传热分析的四边形单元控制体积法

阐述了注塑分析的基于Hele-Shaw假设四边形单元控制体积法。由于四边形控制体积法会产生不对称矩阵的方程组,求解压力时把方程不对称的部分移到方程右边,这样方程转化为对称的线性方程组。在求解转化后的线性方程组时,只在压力求解区域改变时更新方程的右边。注塑分析的时间步长进行自动调整,并大致保证流动前锋在两个时间步内向前流动大略一单元层。与现有文献的计算结果比较表明四边形单元控制体积法可以有效地模拟中面模型的注射过程,该方法并可以与三角形单元混合使用,且可应用于基于表面的注塑分析方法。     

FAST ALGORITHMS FOR DETERMINING POSITIVE DEFINITENESS OF LARGE SCALE MATRICES IN SOLID STRUCTURE ANALYSIS

对于结构稳定性分析中超大规模矩阵正定性判定,必须采用并行计算方法,传统方法如计算特征值、主子式行列式及LDLT等直接方法难以实现.本文提出了一些可适用于并行的迭代判定算法.借鉴力学系统中能量下降的思想,发展了一种判定矩阵正定性的新思路,即将矩阵的正定性判定问题转化为一个优化问题,并基于优化算法来判定矩阵的正定性.提出了基于最速下降法和共轭梯度法来进行矩阵正定性判定的算法.然后考虑到力学系统刚度矩阵的稀疏性和结构刚失稳状态的弱非正定性,提出可以先截超平面后解方程求驻值点的方法来判定弱非正定矩阵的正定性.为了保证对强非正定矩阵判定的准确性,本文提出可以高效混杂使用截平面法和共轭梯度法.数值实验结果表明,本文提出的算法具有准确性和高效性.对于强非正定矩阵而言,共轭梯度算法更加高效;而对于弱非正定矩阵,则是截平面法和混杂算法更加高效.这些算法都容易在机群上实现并行计算,能够快速判定大规模矩阵的正定性.     

A domain decomposition method for modelling Stokes flow in porous materials

An algorithm is presented for solving the Stokes equation in large disordered two‐dimensional porous domains. In this work, it is applied to random packings of discs, but the geometry can be essentially arbitrary. The approach includes the subdivision of the domain and a subsequent application of boundary integral equations to the subdomains. This gives a block diagonal matrix with sparse off‐block components that arise from shared variables on internal subdomain boundaries. The global problem is solved using a biconjugate gradient routine with preconditioning. Results show that the effectiveness of the preconditioner is strongly affected by the subdomain structure, from which a methodology is proposed for the domain decomposition step. A minimum is observed in the solution time versus subdomain size, which is governed by the time required for preconditioning, the time for vector multiplications in the biconjugate gradient routine, the iterative convergence rate and issues related to memory allocation. The method is demonstrated on various domains including a random 1000‐particle domain. The solution can be used for efficient recovery of point velocities, which is discussed in the context of stochastic modelling of solute transport. Copyright © 2002 John Wiley & Sons, Ltd.     

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.     

Iterative solution of panel method discretizations for potential flow problems. The modal multipolar preconditioning

The iterative solution of linear systems arising from panel method discretization of three‐dimensional (3D) exterior potential problems coming mainly from aero‐hydrodynamic engineering problems, is discussed. An original preconditioning based on an approximate eigenspace decomposition is proposed, which corrects bad conditioning arising from a pair of surfaces that are very close to each other, which is a very common situation in slender wings and other aerodynamic profiles. This preconditioning has been tested with the standard Bi‐conjugate gradient (Bi‐CG) and conjugate gradient squared (CGS) iterative methods. Copyright © 2000 John Wiley & Sons, Ltd.     

MULTIGRID AND KRYLOV SUBSPACE METHODS FOR THE DISCRETE STOKES EQUATIONS

Discretization of the Stokes equations produces a symmetric indefinite system of linear equations. For stable discretizations a variety of numerical methods have been proposed that have rates of convergence independent of the mesh size used in the discretization. In this paper we compare the performance of four such methods, namely variants of the Uzawa, preconditioned conjugate gradient, preconditioned conjugate residual and multigrid methods, for solving several two-dimensional model problems. The results indicate that multigrid with smoothing based on incomplete factorization is more efficient than the other methods, but typically by no more than a factor of two. The conjugate residual method has the advantage of being independent of iteration parameters.     

Two‐objective optimization strategies using the adjoint method and game theory in inverse natural convection problems

This paper considers the problem of estimating the strengths of two time‐varying heat sources simultaneously, from measurements of the temperature inside the square domain in a porous medium, when prior knowledge of the source functions is not available. This problem is an inverse natural convection problem. In order to circumvent this problem, we define several optimization criteria (objective functionals) that measure discrepancies between model and measured data, where objective functionals depend on two heat sources and use multi‐criteria optimization to identify Nash equilibria, which are solutions to the non‐cooperative game according to game theory. Two non‐cooperative game strategies are considered: competitive (Nash) game and hierarchical (modified Stackelberg) game. The methodology that we employ relies on a combination of mixed finite element space approximations, finite difference time discretizations, adjoint equation and sensitivity equation techniques, and nonlinear conjugate gradient algorithms for the solutions of estimating two heat sources. Applying the Sobolev gradient for the noise removal is investigated. The performance of the present technique of inverse analysis is evaluated, by means of several numerical experiments, and is found to be very accurate as well as efficient. Copyright © 2012 John Wiley & Sons, Ltd.     

Multi‐scale time integration for transient conjugate heat transfer

The quasi‐steady assumption is commonly adopted in existing transient fluid–solid‐coupled convection–conduction (conjugate) heat transfer simulations, which may cause non‐negligible errors in certain cases of practical interest. In the present work, we adopt a new multi‐scale framework for the fluid domain formulated in a triple‐timing form. The slow‐varying temporal gradient corresponding to the time scales in the solid domain has been effectively included in the fluid equations as a source term, whilst short‐scale unsteadiness of the fluid domain is captured by a local time integration at a given ‘frozen’ large scale time instant. For concept proof, validation and demonstration purposes, the proposed methodology has been implemented in a loosely coupled procedure in conjunction with a hybrid interfacing treatment for coupling efficiency and accuracy. The present results indicate that a much enhanced applicability can be achieved with relatively small modifications of existing transient conjugate heat transfer methods at little extra cost. Copyright © 2016 John Wiley & Sons, Ltd.     

On the approximate computation of extreme eigenvalues and the condition number of nonsingular matrices

From the formulas of the conjugate gradient, a similarity between a symmetric positive definite (SPD) matrix A and a tridiagonal matrix B is obtained. The elements of the matrix B are determinedby the parameters of the conjugate gradient. The computation of eigenvalues of A is then reduced to the case of the tridiagonal matrix B. The approximation of extreme eigenvalues of A can be obtained as a by-product in the computation of the conjugate gradient ifa computational cost of O(s) arithmetic operations is added, where s is the number of iterations This computational cost is negligible compared with the conjugate gradient. If the matrix A is not SPD, the approximation of the condition number of A can be obtained from the computation of the conjugate gradient on A~T A. Numerical results show that this is a convenient and highly efficient method for computing extreme eigenvalues and the condition number of nonsingular matrices.     

Large-scale computational fluid dynamics by the finite element method

Solution methods are presented for the large systems of linear equations resulting from the implicit, coupled solution of the Navier-Stokes equations in three dimensions. Two classes of methods for such solution have been studied: direct and iterative methods. For direct methods, sparse matrix algorithms have been investigated and a Gauss elimination, optimized for vector-parallel processing, has been developed. Sparse matrix results indicate that reordering algorithms deteriorate for rectangular, i.e. M × M × N, grids in three dimensions as N gets larger than M. A new local nested dissection reordering scheme that does not suffer from these difficulties, at least in two dimensions, is presented. The vector-parallel Gauss elimination is very efficient for processing on today's supercomputers, achieving execution rates exceeding 2.3 Gflops the Cray YMP-8 and 9.2 Gflops on the NEC on SX3. For iterative methods, two approaches are developed. First, conjugate-gradient-like methods are studied and good results are achieved with a preconditioned conjugate gradient squared algorithm. Convergence of such a method being sensitive to the preconditioning, a hybrid viscosity method is adopted whereby the preconditioner has an artificial viscosity that is gradually lowered, but frozen at a level higher than the dissipation introduced in the physical equations. The second approach is a domain decomposition one in which overlapping domain and side-by-side methods are tested. For the latter, a Lagrange multiplier technique achieves reasonable rates of convergence.