A parallel two-level finite element method for the Navier-Stokes equations

Based on domain decomposition, a parallel two-level finite element method for the stationary Navier-Stokes equations is proposed and analyzed. The basic idea of the method is first to solve the Navier-Stokes equations on a coarse grid, then to solve the resulted residual equations in parallel on a fine grid. This method has low communication complexity. It can be implemented easily. By local a priori error estimate for finite element discretizations, error bounds of the approximate solution are derived. Numerical results are also given to illustrate the high efficiency of the method.     

Local and parallel finite element algorithms for time-dependent convection-diffusion equations

Local and parallel finite element algorithms based on two-grid discretization for the time-dependent convection-diffusion equations are presented. These algorithms are motivated by the observation that, for a solution to the convection-diffusion problem, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel proce- dures. Hence, these local and parallel algorithms only involve one small original problem on the coarse mesh and some correction problems on the local fine grid. One technical tool for the analysis is the local a priori estimates that are also obtained. Some numerical examples are given to support our theoretical analvsis.     

Parallel finite element computation of incompressible magnetohydrodynamics based on three iterations

Based on local algorithms, some parallel finite element(FE) iterative methods for stationary incompressible magnetohydrodynamics(MHD) are presented. These approaches are on account of two-grid skill include two major phases: find the FE solution by solving the nonlinear system on a globally coarse mesh to seize the low frequency component of the solution, and then locally solve linearized residual subproblems by one of three iterations(Stokes-type, Newton, and Oseen-type) on subdomains with fine...     

一种新的并行自适应网格有限元算法

给出了一种新的适用于流体力学问题的并行自适应有限元算法。首先，基于初始稀网格上获得的事后误差估算值，应用反复谱对剖分方法对初网格进行划分，使各子域上总体误差近似相等，从而解决并行自适应计算中的负载平衡问题。然后在各处理器上独立地求解整体问题，并进行指定子域上的网格自适应处理。最后将各子域上的自适应网格组合成一个整体网格，应用基于粘接元技术的区域分裂法在该网格上获得最终解。文末给出了数值实验结果。     

TWO-GRID ERROR ESTIMATES FOR THE STREAM FUNCTION FORM OF NAVIER-STOKES EQUATIONS

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Two-grid methods for semi-linear elliptic interface problems by immersed finite element methods

In this paper, two-grid immersed finite element(IFE) algorithms are proposed and analyzed for semi-linear interface problems with discontinuous diffusion coefficients in two dimension. Because of the advantages of finite element(FE) formulation and the simple structure of Cartesian grids, the IFE discretization is used in this paper. Two-grid schemes are formulated to linearize the FE equations. It is theoretically and numerically illustrated that the coarse space can be selected as coarse asH= O(h~(1/4))(orH=O(h~(1/8))), and the asymptotically optimal approximation can be achieved as the nonlinear schemes. As a result, we can settle a great majority of nonlinear equations as easy as linearized problems. In order to estimate the present two-grid algorithms, we derive the optimal error estimates of the IFE solution in theL pnorm. Numerical experiments are given to verify the theorems and indicate that the present two-grid algorithms can greatly improve the computing efficiency.     

应变局部化分析的嵌入强间断多尺度有限元法

固体材料的应变局部化行为是导致结构破坏失效的重要因素之一,开展相关数值模拟分析对于结构安全性评估具有重要意义.然而由于材料的非均质和多尺度特性,采用传统数值方法进行求解时通常需要从最小特征尺度离散求解的结构,这将大幅度增加计算规模和成本.针对这一问题,本文提出了一种基于嵌入强间断模型的多尺度有限元方法.该方法从粗细两个尺度离散求解模型,首先在细尺度单元上引入嵌入强间断模型来描述单元间断特性,所附加的跳跃位移自由度则通过凝聚技术进行消除,从而保持细尺度单元刚度阵维度不变.其次,提出了一种增强多节点粗单元技术,其可根据局部化带与粗单元边界相交情况自适应动态地增加粗节点,新构造的增强数值基函数可以捕捉细尺度间断特性,完成物理信息从细单元到粗单元的准确传递以及宏观响应的快速分析;再次,在细尺度解的计算中,将细尺度解分解为降尺度解与单胞局部摄动解,从而消除弹塑性分析时单胞内部的不平衡力.最后,通过两个典型算例分析,并与完全采用细单元的嵌入有限元结果进行对比,验证了所提出算法的正确性与有效性.     

Parallel‐in‐time simulation of the unsteady Navier–Stokes equations for incompressible flow

In this paper, we present an application of a parallel‐in‐time algorithm for the solution of the unsteady Navier–Stokes model equations that are of parabolic–elliptic type. This method is based on the alternated use of a coarse global sequential solver and a fine local parallel one. A standard finite volume/finite differences first‐order approach is used for discretization of the unsteady two‐dimensional Navier–Stokes equations. The Taylor vortex decay problem and the confined flow around a square cylinder were selected as unsteady flow examples to illustrate and analyse the properties of the parallel‐in‐time method through numerical experiments. The influence of several parameters on the computing time required to perform a parallel‐in‐time calculation on a PC cluster was verified. Among them we have analysed the influence of the number of processors, the number of iterations for convergence, the resolution of the spatial domain and the influence of the time‐step sizes ratio between the coarse and fine grids. Significant computer time saving was achieved when compared with the single processor computing time, particularly when the spatial dimension of the problem is low and the temporal scale is large. Copyright © 2004 John Wiley & Sons, Ltd.     

Adaptive variational multiscale method for the Stokes equations

An adaptive variational multiscale method for the Stokes equations is presented in this paper. We solve the coarse scale problem on the coarse mesh and approximate the fine scale solution by solving a series of local residual equations defined on some local fine grids, which can be implemented in parallel. In addition, we also propose a reliable local a posteriori error estimator and construct an adaptive algorithm based on the corresponding a posterior error estimate. Finally, numerical examples are presented to verify the algorithm.Copyright © 2012 John Wiley & Sons, Ltd.     

Design of an Optimal Grid for Finite Element Methods

ABSTRACT Finite element solutions of improved quality are obtained by optimizing the location of nodes of the finite element grid, while keeping the number of degrees of freedom fixed. The formulation of the grid optimization problem is based on the reduction of error associated with interpolation of the exact solution, using functions from the finite element space. Element sizes are selected as design variables: length in R1 and area in R2. Analytically derived optimality conditions are presented and an approximation to these conditions is introduced to obtain a set of operationally useful equations that can be used as guidelines for construction of improved grids. Example problems are given for illustration.     

Two-level stabilized finite element method for Stokes eigenvalue problem

A two-level stabilized finite element method for the Stokes eigenvalue problem based on the local Gauss integration is considered.This method involves solving a Stokes eigenvalue problem on a coarse mesh with mesh size H and a Stokes problem on a fine mesh with mesh size h = O(H 2),which can still maintain the asymptotically optimal accuracy.It provides an approximate solution with the convergence rate of the same order as the usual stabilized finite element solution,which involves solving a Stokes eigenvalue problem on a fine mesh with mesh size h.Hence,the two-level stabilized finite element method can save a large amount of computational time.Moreover,numerical tests confirm the theoretical results of the present method.     

时域自适应算法求解弹性地基薄板的动力问题

为更精确地求解弹性地基薄板的动力响应,发展了一种分段时域自适应算法,通过变量在离散时段内的展开,将时空耦合的初边值问题转化为一系列递推的基于有限元(FEM)的空间问题求解,通过自适应计算保持稳定的计算精度。数值算例表明:本文解与解析解相比最大相对误差不超过3.59%;当步长较大时四阶Runge-Kutta法和Newmark法均失效,本文所提算法仍可得到满意的计算结果。     

离散颗粒集合体-Cosserat连续体模型的连接尺度方法

基于针对分子动力学-Cauchy连续体模型提出的连接尺度方法(BSM)[1,2],发展了耦合细尺度上基于离散颗粒集合体模型的离散单元法(DEM)和粗尺度上基于Cosserat连续体模型的有限元法(FEM)的BSM。仅在有限局部区域内采用DEM以从细观层次模拟非连续破坏现象,而在全域则采用花费计算时间和存储空间较少的FEM。通过连接尺度位移(包括平移和转动)分解,和基于作用于Cosserat连续体有限元节点和颗粒集合体颗粒形心的离散系统虚功原理,得到了具有解耦特征的粗细尺度耦合系统运动方程。讨论和提出了在准静态载荷条件下粗细尺度域的界面条件,以及动态载荷条件下可以有效消除粗细尺度域界面上虚假反射波的非反射界面条件(NRBC)。本文二维数值算例结果说明了所提出的颗粒材料BSM的可应用性和优越性,及所实施界面条件对模拟颗粒材料动力学响应的有效性。     

PRECISE INTEGRAL ALGORITHM BASED SOLUTION FOR TRANSIENT INVERSE HEAT CONDUCTION PROBLEMS WITH MULTI-VARIABLES

IntroductionIHCPs (InverseHeatConductionProblems)arecloselyassociatedwithmanyengineeringaspects,andwelldocumentedintheliteratures,coveringtheidentificationsofthermalparameters[1,2 ],boundaryshapes[3],boundaryconditions[4 ]andsource_relatedterms[5 ,6 ]etc .Howeveritseemsthatonlylittleworkisdirectlyconcernedwithmulti_variablesidentificationsbyauthors’knowledge.Tsengetal.presentedanapproachtodeterminingtwokindsofvariables[7],butonlygavefewnumericalexamplestodeterminethemsimultaneously .Thesol…     

无穷域势流问题的有限元／差分线法混合求解

提出了一种将有限元和差分线法相结合求解无穷域势流问题的算法。用两同心圆将求解域划分为存在重叠的有限和无限两个区域,在有限和无限域上分别用有限元和差分线法求解Laplace方程边值问题。用差分线法推导出的关系式修正有限元方程,求解该方程组从而得到原问题的解。本算法将求解无穷域问题转化为代数特征值问题和有限域内线性方程组的...     

Highly parallel time integration of viscoelastic flows

A highly parallel time integration method is presented for calculating viscoelastic flows with the DEVSS-G/DG finite element discretization. The method is a synthesis of an operator splitting time integration method that decouples the calculation of the polymeric stress by solution of a hyperbolic constitutive equation from the evolution of the velocity and pressure fields by solution of a generalized Stokes problem. Both steps are performed in parallel. The discontinuous finite element discretization of the hyperbolic constitutive equation leads to highly-parallel element-by-element calculation of the stress at each time step. The Stokes-like problem is solved by using the BiCGStab Krylov iterative method implemented with the block complement and additive levels method (BCALM) preconditioner. The solution method is demonstrated for the calculation of two-dimensional (2D) flow of an Oldroyd-B fluid around an isolated cylinder confined between two parallel plates. These calculations use extremely fine finite elements and expose new features of the solution structure.     

Fast Fourier homogenization for elastic wave propagation in complex media

In the context of acoustic or elastic wave propagation, the non-periodic asymptotic homogenization method allows one to determine a smooth effective medium and equations associated with the wave propagation in a given complex elastic or acoustic medium down to a given minimum wavelength. By smoothing all discontinuities and fine scales of the original medium, the homogenization technique considerably reduces meshing difficulties as well as the numerical cost associated with the wave equation solver, while producing the same waveform as for the original medium (up to the desired accuracy). Nevertheless, finding the effective medium requires one to solve the so-called “cell problem”, which corresponds to an elasto-static equation with a finite set of distinct loadings. For general elastic or acoustic media, the cell problem is a large problem that has to be solved on the whole domain and its resolution implies the use of a finite element solver and a mesh of the fine scale medium. Even if solving the cell problem is simpler than solving the wave equation in the original medium (because it is time and source independent, based on simple tetrahedral meshes and embarrassingly parallel) it is still a challenge. In this work, we present an alternative method to the finite element approach for solving the cell problem. It is based on a well-known method designed by H. Moulinec and P. Suquet in 1998 in structural mechanics. This iterative technique relies on Green functions of a simple reference medium and extensively uses Fast Fourier Transforms. It is easy to implement, very efficient and relies on a simple regular gridding of the medium. Through examples we show that the method gives excellent results, even, under some conditions, for discontinuous media.     

岩质圆形隧洞围岩应力场弹塑性新解

针对动态接触问题的有限元并行计算，提出了一种新的接触算法. 新算法引入局部拉氏 乘子技术来计算接触力. 由于同时考虑了无穿透的接触约束条件和相邻接触对的相互影响， 较之广泛使用的罚参数法，新算法使接触约束条件和系统平衡方程得到更充分的满足. 虽然 为提高接触计算精度而在局部采用了迭代技术，但算法仍然具有较高的效率，且与显式时间 积分方案完全相容. 此外，通过构造专门的区域分解方案，实现了将现有为串行程序开发的 搜索算法平滑移植到并行环境的目标. 数值算例表明，所提出的接触算法具有很好的并行性， 在保证了接触问题并行计算精度的同时，取得了满意的并行效率.     

颗粒材料多尺度分析的连接尺度方法

本文提出了耦合细尺度上基于离散颗粒集合体模型的离散单元法（DEM）和粗尺度上基于Cosserat连续体模型的有限元法（FEM）的连接尺度方法（BSM）以研究颗粒材料的力学行为。采用Cosserat连续体模型和FEM模拟的粗尺度域覆盖全域,而采用离散颗粒集合体模型的DEM模拟的细尺度域仅限于需特别关注材料微结构演变和非连续变形行为的局部区域。对这两个区域间的界面提出了适当的界面条件及其实施方案。通过采用适当的连接尺度投影算子,空间离散的粗、细尺度耦合系统多尺度运动方程具有解耦和允许分别求解、因而也允许分别采用不同时间步长对粗、细尺度计算的特点,可极大地提高BSM的计算效率。文中二维地基数值算例结果说明了所陈述方法的可应用性,以及相对基于Cosserat连续体模型的FEM和基于离散颗粒集合体模型的DEM的优越性。      

A Bridging Scale Method for Multi-scale Analysis of Granular Materials

本文提出了耦合细尺度上基于离散颗粒集合体模型的离散单元法（DEM）和粗尺度上基于Cosserat连续体模型的有限元法（FEM）的连接尺度方法（BSM）以研究颗粒材料的力学行为。采用Cosserat连续体模型和FEM模拟的粗尺度域覆盖全域,而采用离散颗粒集合体模型的DEM模拟的细尺度域仅限于需特别关注材料微结构演变和非连续变形行为的局部区域。对这两个区域间的界面提出了适当的界面条件及其实施方案。通过采用适当的连接尺度投影算子,空间离散的粗、细尺度耦合系统多尺度运动方程具有解耦和允许分别求解、因而也允许分别采用不同时间步长对粗、细尺度计算的特点,可极大地提高BSM的计算效率。文中二维地基数值算例结果说明了所陈述方法的可应用性,以及相对基于Cosserat连续体模型的FEM和基于离散颗粒集合体模型的DEM的优越性。