Application of Preconditioned GMRES Algorithm for Analysis of Millimeter Wave Ferrite Sphere Circulators

The generalized minimal residual (GMRES) iterative method is applied to solve such sparse large non-symmetric system of linear equations resulting from the use of edge-based finite element method. In order to speed up the convergence of GMRES, the symmetric successive overrelaxation (SSOR) preconditioning scheme is applied for the analysis of millimeter wave ferrite circulator. Consequently, this preconditioned GMRES (PGMRES) approach can reach convergence 19 times faster than GMRES. The isolation and insertion losses of millimeter wave waveguide circulator are compared with those obtained from literature.     

Application of Preconditioned GMRES Algorithm for Analysis of Millimeter Wave Ferrite Sphere Circulators

The generalized minimal residual (GMRES) iterative method is applied to solve such sparse large non-symmetric system of linear equations resulting from the use of edge-based finite element method. In order to speed up the convergence of GMRES, the symmetric successive overrelaxation (SSOR) preconditioning scheme is applied for the analysis of millimeter wave ferrite circulator. Consequently, this preconditioned GMRES (PGMRES) approach can reach convergence ten times faster than GMRES. The reflection and insertion losses of millimeter wave waveguide circulator are compared with those obtained from literature.     

An implicit immersed boundary method for three-dimensional fluid–membrane interactions

We present an implicit immersed boundary method for the incompressible Navier–Stokes equations capable of handling three-dimensional membrane–fluid flow interactions. The goal of our approach is to greatly improve the time step by using the Jacobian-free Newton–Krylov method (JFNK) to advance the location of the elastic membrane implicitly. The most attractive feature of this Jacobian-free approach is Newton-like nonlinear convergence without the cost of forming and storing the true Jacobian. The Generalized Minimal Residual method (GMRES), which is a widely used Krylov-subspace iterative method, is used to update the search direction required for each Newton iteration. Each GMRES iteration only requires the action of the Jacobian in the form of matrix–vector products and therefore avoids the need of forming and storing the Jacobian matrix explicitly. Once the location of the boundary is obtained, the elastic forces acting at the discrete nodes of the membrane are computed using a finite element model. We then use the immersed boundary method to calculate the hydrodynamic effects and fluid–structure interaction effects such as membrane deformation. The present scheme has been validated by several examples including an oscillatory membrane initially placed in a still fluid, capsule membranes in shear flows and large deformation of red blood cells subjected to stretching force.     

A matrix-free,implicit, incompressible fractional-step algorithm for fluid–structure interaction applications

In this paper we detail a fast, fully-coupled, partitioned fluid–structure interaction (FSI) scheme. For the incompressible fluid, new fractional-step algorithms are proposed which make possible the fully implicit, but matrix-free, parallel solution of the entire coupled fluid–solid system. These algorithms include artificial compressibility pressure-poisson solution in conjunction with upwind velocity stabilisation, as well as simplified pressure stabilisation for improved computational efficiency. A dual-timestepping approach is proposed where a Jacobi method is employed for the momentum equations while the pressures are concurrently solved via a matrix-free preconditioned GMRES methodology. This enables efficient sub-iteration level coupling between the fluid and solid domains. Parallelisation is effected for distributed-memory systems. The accuracy and efficiency of the developed technology is evaluated by application to benchmark problems from the literature. The new schemes are shown to be efficient and robust, with the developed preconditioned GMRES solver furnishing speed-ups ranging between 50 and 80.     

一种可压缩流动的高阶加权紧致非线性格式(WCNS)的加速收敛方法

以高超声速粘性绕流的数值模拟为例,研究LU-SGS、高斯-赛德尔点松弛、线松弛以及GMRES等隐式求解方法在空间项采用高阶精度格式WCNS离散时的收敛性,并对GMRES(generalized minimal residual)方法中的子迭代影响作了对比计算.结果表明,采用准确的解析雅克比矩阵的点、线松弛的收敛速度优于LU-SGS,以线松弛为预处理的GMRES算法具有良好的收敛特性.     

Efficient Analysis of Millimeter Wave Ferrite Circulators by the GMRES Iterative Algorithm

The finite element method (FEM) combined with the perfectly matched layers (PML) is given for simulation of waveguide ferrite circulators. The generalized minimal residual (GMRES) iterative method is applied to solve such sparse large non-symmetric system of linear equations resulting from the use of edge-based finite element method. The formulation of FEM and the algorithm of GMRES method are described in detail. The reflection and insertion losses of millimeter wave waveguide circulator are analyzed and the results are compared with those obtained from literature.     

A kernel-free boundary integral method for elliptic boundary value problems

This paper presents a class of kernel-free boundary integral (KFBI) methods for general elliptic boundary value problems (BVPs). The boundary integral equations reformulated from the BVPs are solved iteratively with the GMRES method. During the iteration, the boundary and volume integrals involving Green’s functions are approximated by structured grid-based numerical solutions, which avoids the need to know the analytical expressions of Green’s functions. The KFBI method assumes that the larger regular domain, which embeds the original complex domain, can be easily partitioned into a hierarchy of structured grids so that fast elliptic solvers such as the fast Fourier transform (FFT) based Poisson/Helmholtz solvers or those based on geometric multigrid iterations are applicable. The structured grid-based solutions are obtained with standard finite difference method (FDM) or finite element method (FEM), where the right hand side of the resulting linear system is appropriately modified at irregular grid nodes to recover the formal accuracy of the underlying numerical scheme. Numerical results demonstrating the efficiency and accuracy of the KFBI methods are presented. It is observed that the number of GMRES iterations used by the method for solving isotropic and moderately anisotropic BVPs is independent of the sizes of the grids that are employed to approximate the boundary and volume integrals. With the standard second-order FEMs and FDMs, the KFBI method shows a second-order convergence rate in accuracy for all of the tested Dirichlet/Neumann BVPs when the anisotropy of the diffusion tensor is not too strong.     

A kernel-free boundary integral method for elliptic boundary value problems

This paper presents a class of kernel-free boundary integral (KFBI) methods for general elliptic boundary value problems (BVPs). The boundary integral equations reformulated from the BVPs are solved iteratively with the GMRES method. During the iteration, the boundary and volume integrals involving Green’s functions are approximated by structured grid-based numerical solutions, which avoids the need to know the analytical expressions of Green’s functions. The KFBI method assumes that the larger regular domain, which embeds the original complex domain, can be easily partitioned into a hierarchy of structured grids so that fast elliptic solvers such as the fast Fourier transform (FFT) based Poisson/Helmholtz solvers or those based on geometric multigrid iterations are applicable. The structured grid-based solutions are obtained with standard finite difference method (FDM) or finite element method (FEM), where the right hand side of the resulting linear system is appropriately modified at irregular grid nodes to recover the formal accuracy of the underlying numerical scheme. Numerical results demonstrating the efficiency and accuracy of the KFBI methods are presented. It is observed that the number of GMRES iterations used by the method for solving isotropic and moderately anisotropic BVPs is independent of the sizes of the grids that are employed to approximate the boundary and volume integrals. With the standard second-order FEMs and FDMs, the KFBI method shows a second-order convergence rate in accuracy for all of the tested Dirichlet/Neumann BVPs when the anisotropy of the diffusion tensor is not too strong.     

Krylov子空间法在SIMPLER算法中的求解性能分析

本文开发了Krylov子空间法中的Bi-CGSTAB、GMRES(m)、CGS、TFQMR及QMR方法的计算程序,并将其实施于SIMPLER算法作为其内迭代方法,针对CFD/NHT领域的问题,研究了它们的求解特性;发现:Bi-CGSTAB方法有着高效的收敛速度和良好的稳定性;N-S方程求解中不同方程不同m值的协调选取是GMRES(m)方法在CFD/NHT领域推广应用的关键;CGS和QMR方法易于中断;TFQMR方法收敛速度慢于其他方法,但能适用于更广泛问题的求解.     

GMRES算法在二维定常无粘流计算中的应用

发展了GMRES算法的两种不同预处理方法求解二维无粘流体动力学方程组。在保证计算效率的基础上,采用了一种减小内存需求的途径。用两个算例对GMRES算法以及两种不同的预处理方法进行分析,同时与DDADI方法进行比较。通过对NACA0012有攻角超临界流动以及GAMM通道超音流的计算,表明两种预处理下的GMRES算法都具有收敛速度快的优点,LUSGS预处理方法略优于ILU预处理方法。     

一种二阶混合有限体元格式的GAMG预条件子

针对一种含跳系数椭圆问题的二阶混合有限体元格式,讨论求解相应离散系统PGMRES法的预条件子构造问题.通过严格的理论分析,建立分层基下该二阶混合有限体元刚度矩阵和二次有限元刚度矩阵的谱等价关系,并利用关于二次有限元刚度矩阵的一种基于分层思想的GAMG预条件子,为二阶混合有限体元刚度矩阵设计一种高效GAMG预条件子.数值结果验证理论分析的正确性和新预条件子的高效性与稳定性.     

The immersed boundary method for advection–electrodiffusion with implicit timestepping and local mesh refinement

We describe an immersed boundary method for problems of fluid–solute-structure interaction. The numerical scheme employs linearly implicit timestepping, allowing for the stable use of timesteps that are substantially larger than those permitted by an explicit method, and local mesh refinement, making it feasible to resolve the steep gradients associated with the space charge layers as well as the chemical potential, which is used in our formulation to control the permeability of the membrane to the (possibly charged) solute. Low Reynolds number fluid dynamics are described by the time-dependent incompressible Stokes equations, which are solved by a cell-centered approximate projection method. The dynamics of the chemical species are governed by the advection–electrodiffusion equations, and our semi-implicit treatment of these equations results in a linear system which we solve by GMRES preconditioned via a fast adaptive composite-grid (FAC) solver. Numerical examples demonstrate the capabilities of this methodology, as well as its convergence properties.     

On the evaluation of layer potentials close to their sources

When solving elliptic boundary value problems using integral equation methods one may need to evaluate potentials represented by a convolution of discretized layer density sources against a kernel. Standard quadrature accelerated with a fast hierarchical method for potential field evaluation gives accurate results far away from the sources. Close to the sources this is not so. Cancellation and nearly singular kernels may cause serious degradation. This paper presents a new scheme based on a mix of composite polynomial quadrature, layer density interpolation, kernel approximation, rational quadrature, high polynomial order corrected interpolation and differentiation, temporary panel mergers and splits, and a particular implementation of the GMRES solver. Criteria for which mix is fastest and most accurate in various situations are also supplied. The paper focuses on the solution of the Dirichlet problem for Laplace’s equation in the plane. In a series of examples we demonstrate the efficiency of the new scheme for interior domains and domains exterior to up to 2000 close-to-touching contours. Densities are computed and potentials are evaluated, rapidly and accurate to almost machine precision, at points that lie arbitrarily close to the boundaries.     

An efficient flexible-order model for 3D nonlinear water waves

The flexible-order, finite difference based fully nonlinear potential flow model described in [H.B. Bingham, H. Zhang, On the accuracy of finite difference solutions for nonlinear water waves, J. Eng. Math. 58 (2007) 211–228] is extended to three dimensions (3D). In order to obtain an optimal scaling of the solution effort multigrid is employed to precondition a GMRES iterative solution of the discretized Laplace problem. A robust multigrid method based on Gauss–Seidel smoothing is found to require special treatment of the boundary conditions along solid boundaries, and in particular on the sea bottom. A new discretization scheme using one layer of grid points outside the fluid domain is presented and shown to provide convergent solutions over the full physical and discrete parameter space of interest. Linear analysis of the fundamental properties of the scheme with respect to accuracy, robustness and energy conservation are presented together with demonstrations of grid independent iteration count and optimal scaling of the solution effort. Calculations are made for 3D nonlinear wave problems for steep nonlinear waves and a shoaling problem which show good agreement with experimental measurements and other calculations from the literature.     

Computation of linear and nonlinear stationary states of photonic structures using modern iterative solvers

In this paper we describe efficient methods to obtain the stationary states of linear and nonlinear photonic systems, which have gained particular interest in the field of integrated and nonlinear optics. While the methods presented are directly applicable to optical physics, they are also general and should be of interest in a broad range of phenomena presently under study in other areas of physics and engineering. The strategy consists in combining the use of classical methods, such as inverse iteration or the Newton method, together with modern, nonstationary linear solvers, such as SYMMLQ or GMRES, in order to obtain efficient numerical computations to problems involving large matrices. We have selected several example problems in order to discuss the practical implementation details, not normally described in the present literature. Moreover, the problems we have selected provide a backdrop to contrast and motivate the use of different methods for systems which are symmetric and non-symmetric, single and multi-component, and also real and complex. Information relative to numerical performance of the different algorithms, including a survey for a nonsymmetric problem, which requires the adjustment of a restarting parameter for the GMRES algorithm, is also presented.     

三次相位板波前编码系统彩色图像恢复的迭代算法

采用三次相位板进行景深延拓的波前编码系统得到非对称的点扩展函数.为了获得最终清晰的彩色图像.研究了一种基于广义极小残差法(GMREs)的迭代算法,结合Tikhonov规整化方法,并利用多通道处理过程对中间图像进行左卷积恢复.为了消除恢复图像边界的振铃效应,推导了新的光学成像过程数学模型,该模型采用反镜像边界条件并利用直积近似对卷积核进行处理.模拟数据的分析表明,采用多通道处理过程对彩色图片进行恢复时,新的算法在给出精确的反巷积结果的同时能有效地抑制噪声的放大;实验结果显示,较之经典的维纳滤波恢复结果,新算法能够更好的消除边界的振铃和图像边缘的振动波纹.     

PCCSAP-3D程序压力场算法改进

介绍大型先进压水堆安全壳专用分析程序PCCSAP-3D计算采用的方法,引入GMRES(Generalized Minimal RESidual)方法改进该程序的压力场算法.使用GMRES算法的实用变形,并采用合适的预处理技术,比较GMRES算法和ML-ADI算法在求解压力方程时的收敛速度.结果表明,利用压力矩阵结构化和稀疏性的特点,采用预处理GMRES算法能够更快速地求解压力方程.当压力矩阵规模变大时,GMRES算法相对于ML-ADI方法能够节省更多的时间.     

不可压缩流基于块预处理的并行有限元计算

发展了一个模拟非定常不可压缩粘性流的并行有限元求解器,时间离散使用具有二阶精度的隐式中点格式,基于三维非结构四面体网格剖分,使用高阶混合有限元离散速度场（P2）和压力场（P1）.全离散格式产生的代数方程组是大型、稀疏、非对称和病态的,基于修正的压力对流扩散预处理（PCD）和精心设计的子问题迭代执行策略,采用预处理的GMRES迭代法来高效求解线性方程组.利用相同的子问题迭代策略,同时给出基于最小二乘交换子（LSC）预处理的并行效率对比.大量数值算例验证了算法的精度、可扩展性和可靠性.三维驱动方腔流模拟结果（Re=3200.0）清晰地显示了方腔流中主涡（PE）、下游二次涡（DSE）、上游二次涡（USE）、侧壁涡（EWV）和TGL涡的存在.     

JFNK方法在S N输运计算中的应用

JFNK(Jacobian-free Newton-Krylov)方法是一种求解非线性方程的高效迭代算法.传统输运计算中的负通量修正与k-特征值迭代使得原本线性的输运计算转变为非线性问题数值求解.为提高非线性输运问题的计算效率,将这两类非线性问题离散成残差形式的非线性方程组,并采用JFNK方法对其进行迭代求解.分析不同...     

A numerical scheme for the integration of the Vlasov–Poisson system of equations, in the magnetized case

We present a numerical algorithm for the solution of the Vlasov–Poisson system of equations, in the magnetized case. The numerical integration is performed using the well-known “splitting” method in the electrostatic approximation, coupled with a finite difference upwind scheme; finally the algorithm provides second order accuracy in space and time. The cylindrical geometry is used in the velocity space, in order to describe the rotation of the particles around the direction of the external uniform magnetic field.Using polar coordinates, the integration of the Vlasov equation is very simplified in the velocity space with respect to the cartesian geometry, because the rotation in the velocity cartesian space corresponds to a translation along the azimuthal angle in the cylindrical reference frame. The scheme is intrinsically symplectic and significatively simpler to implement, with respect to a cartesian one. The numerical integration is shown in detail and several conservation tests are presented, in order to control the numerical accuracy of the code and the time evolution of the entropy, strictly related to the filamentation problem for a kinetic model, is discussed.