等代数结构面网格剖分下三维弹性问题的代数多重网格法

在一种等代数结构面网格剖分下,建立了求解三维弹性问题有限元方程的代数多重网格法及相应的预处理共轭梯度法,详细描述了代数多重网格方法中网格粗化技术与插值算子的构造,并将所构造的代数多重网格法应用于某些实际问题如非均匀介质、高应力梯度问题的数值求解。结果表明,建立的代数多重网格法对求解三维弹性问题是十分有效的,具有很好的鲁棒性,较直接解法和其它常用迭代方法具有明显的优越性。     

几类典型网格下三维弹性问题的代数多层网格法

有限元方法是数值求解三维弹性问题的一类重要的离散化方法.在有限元分析中,网格的几何形状及网格质量会对有限元离散代数系统的求解产生很大影响.该文系统研究了几类典型网格对几种常用AMG法计算效率的影响,并进行了详细的性能测试与比较.利用容易获知的部分几何与分析信息(如方程类型,节点自由度信息),再结合经典AMG法中的网格粗...     

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.     

多重网格--特征有限元方法计算地下水溶质运移问题

本文用多重网格特征有限元方法求解溶质运移问题,有效的克服了通常数值方法中数值弥散、计算速度慢、计算量大等缺点。     

弹性力学问题自适应有限元及其局部多重网格法

针对平面弹性力学问题,利用最新顶点二分法,设计了一种不需要标记振荡项和加密单元不需要满足“内节点”性质的自适应有限元法;利用自适应加密过程中每层网格上只有局部单元需要加密这一特性,设计了一种基于局部松弛的多重网格法.数值实验结果表明:该文设计的自适应有限元法具有一致收敛性和拟最优计算复杂度,基于局部松弛的多重网格法对求解弹性力学问题自适应网格下的有限元方程具有很好的计算效率和鲁棒性.     

一种新的无网格方法与有限元耦合法

本文分析了Belytschko和Huerta提出的无网格方法和有限元耦合法各自存在的问题,提出了一种新的无网格方法与有限元耦合法。Belytschko提出的方法的缺点是,无网格方法子域和有限元法子域的界面必须是规则的,交界域内有限元不能随意划分,交界域内无网格方法的节点也不能随意分布。Huerta提出的方法的缺点是对交界域内无网格方法的节点影响域可能无法覆盖交界域。本文提出的无网格方法与有限元耦合法解决了以上两种方法存在的问题,并保留了无网格方法随意配点的优点、交界面可以不规则、提高了无网格子域内的求解精度,从而提高问题的整体求解精度。然后,建立了弹性力学的无网格方法与有限元法的耦合法。最后给出了数值算例。     

节理性质对岩体爆破破坏模式影响的数值分析

针对目前常用的有限元和离散元等数值方法难以客观反映岩体中存在的大量断续节理和在外力作用下岩体破碎及块体运动的不足,提出了采用数值流形方法以解决目前岩体爆破模拟中存在的上述问题.数值流形方法采用数学网格与物理网格以形成求解流形单元,因而很容易反映岩体中存在的众多初始节理,采用断裂力学准则以模拟节理、裂纹扩展,采用DDA中的块体运动学理论以模拟块体运动.最后通过算例对比分析了完整岩体和节理岩体爆破破坏模式的差异,说明了节理存在对岩体爆破破坏模式有着重要影响,且其影响程度与节理的几何分布及物理力学性质有着密切关系.     

三维弹性问题高次有限元离散线性系统的块对角逆预条件PCG法

高次有限元由于对问题具有更好的逼近效果及某些特殊的优点,如能解决弹性问题的闭锁现象(Poisson’s ratio locking),使得它们在实际计算中被广泛使用。但与线性元相比,它具有更高的计算复杂性。该文基于标量椭圆问题高次有限元离散化系统的代数多层网格(AMG)法,针对三维弹性问题高次有限元离散化线性系统的求解,设计了一种以块对角逆为预条件子的共轭梯度法(AMG-BPCG)。数值实验表明,该文设计的AMG-BPCG法较标准的ILU-型PCG法具有更好的计算效率和鲁棒性。     

用微分求积法求解梁的弹塑性问题

根据梁塑性弯曲的工程理论,采用微分求积法进行了梁的弹塑性平面弯曲分析。微分求积法是一种直接求解微分方程(组)的数值方法,不依赖于变分原理,且能以较少的网格点求得微分方程的高精度数值解。与有限元分析结果的比较,表明了微分求积法求解梁的弹塑性问题的计算效率和精度。微分求积法的计算结果不受荷载步长的限制,也不需要迭代求解,特别对于承受非线性分布荷载作用的梁的弹塑性分析具有很大的优越性。通过选用不同的网格点数目,分析了微分求积法的稳定性和收敛性。     

有限元概率多重网格法

本文提出了一种求解椭园问题有限元方程的概率算法-有限元概率多重网格法.使用这种方法可以在不需要形成总刚矩阵的情况下计算出一个或少数几个网格点上的函数值,并证明此算法有二次元逼近精度。     

FLEXMG: A new library of multigrid preconditioners for a spectral/finite element incompressible flow solver

A new library called FLEX MG has been developed for a spectral/finite element incompressible flow solver called SFELES. FLEX MG allows the use of various types of iterative solvers preconditioned by algebraic multigrid methods. Two families of algebraic multigrid preconditioners have been implemented, namely smooth aggregation‐type and non‐nested finite element‐type. Unlike pure gridless multigrid, both of these families use the information contained in the initial fine mesh. A hierarchy of coarse meshes is also needed for the non‐nested finite element‐type multigrid so that our approaches can be considered as hybrid. Our aggregation‐type multigrid is smoothed with either a constant or a linear least‐square fitting function, whereas the non‐nested finite element‐type multigrid is already smooth by construction. All these multigrid preconditioners are tested as stand‐alone solvers or coupled with a GMRES method. After analyzing the accuracy of the solutions obtained with our solvers on a typical test case in fluid mechanics, their performance in terms of convergence rate, computational speed and memory consumption is compared with the performance of a direct sparse LU solver as a reference. Finally, the importance of using smooth interpolation operators is also underlined in the study. Copyright © 2010 John Wiley & Sons, Ltd.     

Parallel multigrid solvers for 3D unstructured finite element problems in large deformation elasticity and plasticity

Multigrid is a popular solution method for the set of linear algebraic equations that arise from PDEs discretized with the finite element method. The application of multigrid to unstructured grid problems, however, is not well developed. We discuss a method, that uses many of the same techniques as the finite element method itself, to apply standard multigrid algorithms to unstructured finite element problems. We use maximal independent sets (MISs) as a mechanism to automatically coarsen unstructured grids; the inherent flexibility in the selection of an MIS allows for the use of heuristics to improve their effectiveness for a multigrid solver. We present parallel algorithms, based on geometric heuristics, to optimize the quality of MISs and the meshes constructed from them, for use in multigrid solvers for 3D unstructured problems. We discuss parallel issues of our algorithms, multigrid solvers in general, and the parallel finite element application that we have developed to test our solver on challenging problems. We show that our solver, and parallel finite element architecture, does indeed scale well, with test problems in 3D large deformation elasticity and plasticity, with 40 million degree of freedom problem on 240 IBM four‐way SMP PowerPC nodes. Copyright © 2000 John Wiley & Sons, Ltd.     

On the performance of Krylov smoothing for fully coupled AMG preconditioners for VMS resistive MHD

This study explores the performance and scaling of a GMRES Krylov method employed as a smoother for an algebraic multigrid preconditioned Newton-Krylov solution approach applied to a fully implicit variational multiscale finite element resistive magnetohydrodynamics formulation. In this context, a Newton iteration is used for the nonlinear system and a parallel MPI-based Krylov method is employed for the linear subsystems. The efficiency of this approach is critically dependent on the scalability and performance of the parallel algebraic multigrid preconditioner for the linear solutions and the performance of the multigrid smoothers play a critical role. Krylov multigrid smoothers are considered in an attempt to reduce the time and memory requirements of existing robust smoothers based on additive Schwarz domain decomposition with incomplete LU factorization solves on each subdomain. Three time-dependent resistive magnetohydrodynamics test cases are considered to evaluate the method. Compared with a domain decomposition incomplete LU smoother, the GMRES smoother can reduce the solve time due to a significant decrease in the preconditioner setup time and often a reduction in outer Krylov solver iterations, and requires less memory, typically 35% less memory.     

A matrix-free approach for finite-strain hyperelastic problems using geometric multigrid

This work investigates matrix-free algorithms for problems in quasi-static finite-strain hyperelasticity. Iterative solvers with matrix-free operator evaluation have emerged as an attractive alternative to sparse matrices in the fluid dynamics and wave propagation communities because they significantly reduce the memory traffic, the limiting factor in classical finite element solvers. Specifically, we study different matrix-free realizations of the finite element tangent operator and determine whether generalized methods of incorporating complex constitutive behavior might be feasible. In order to improve the convergence behavior of iterative solvers, we also propose a method by which to construct level tangent operators and employ them to define a geometric multigrid preconditioner. The performance of the matrix-free operator and the geometric multigrid preconditioner is compared to the matrix-based implementation with an algebraic multigrid (AMG) preconditioner on a single node for a representative numerical example of a heterogeneous hyperelastic material in two and three dimensions. We find that matrix-free methods for finite-strain solid mechanics are very promising, outperforming linear matrix-based schemes by two to five times, and that it is possible to develop numerically efficient implementations that are independent of the hyperelastic constitutive law.     

A distributed memory parallel implementation of the multigrid method for solving three‐dimensional implicit solid mechanics problems

We describe the parallel implementation of a multigrid method for unstructured finite element discretizations of solid mechanics problems. We focus on a distributed memory programming model and use the MPI library to perform the required interprocessor communications. We present an algebraic framework for our parallel computations, and describe an object‐based programming methodology using Fortran90. The performance of the implementation is measured by solving both fixed‐ and scaled‐size problems on three different parallel computers (an SGI Origin2000, an IBM SP2 and a Cray T3E). The code performs well in terms of speedup, parallel efficiency and scalability. However, the floating point performance is considerably below the peak values attributed to these machines. Lazy processors are documented on the Origin that produce reduced performance statistics. The solution of two problems on an SGI Origin2000, an IBM PowerPC SMP and a Linux cluster demonstrate that the algorithm performs well when applied to the unstructured meshes required for practical engineering analysis. Copyright © 2004 John Wiley & Sons, Ltd.     

A monolithic geometric multigrid solver for fluid‐structure interactions in ALE formulation

We present a monolithic geometric multigrid solver for fluid‐structure interaction problems in Arbitrary Lagrangian Eulerian coordinates. The coupled dynamics of an incompressible fluid with nonlinear hyperelastic solids gives rise to very large and ill‐conditioned systems of algebraic equations. Direct solvers usually are out of question because of memory limitations, and standard coupled iterative solvers are seriously affected by the bad condition number of the system matrices. The use of partitioned preconditioners in Krylov subspace iterations is an option, but the convergence will be limited by the outer partitioning. Our proposed solver is based on a Newton linearization of the fully monolithic system of equations, discretized by a Galerkin finite element method. Approximation of the linearized systems is based on a monolithic generalized minimal residual method iteration, preconditioned by a geometric multigrid solver. The special character of fluid‐structure interactions is accounted for by a partitioned scheme within the multigrid smoother only. Here, fluid and solid field are segregated as Dirichlet–Neumann coupling. We demonstrate the efficiency of the multigrid iteration by analyzing 2d and 3d benchmark problems. While 2d problems are well manageable with available direct solvers, challenging 3d problems highly benefit from the resulting multigrid solver. Copyright © 2015 John Wiley & Sons, Ltd.     

An algebraic multigrid approach to solve extended finite element method based fracture problems

This article proposes an algebraic multigrid (AMG) approach to solve linear systems arising from applications where strong discontinuities are modeled by the extended finite element method. The application of AMG methods promises optimal scalability for solving large linear systems. However, the straightforward (or ‘black‐box’) use of existing AMG techniques for extended finite element method problems is often problematic. In this paper, we highlight the reasons for this behavior and propose a relatively simple adaptation that allows one to leverage existing AMG software mostly unchanged. Numerical tests demonstrate that optimal iterative convergence rates can be attained that are comparable with AMG convergence rates associated with linear systems for standard finite element approximations without discontinuities. Published 2012. This article is a US Government work and is in the public domain in the USA.     

Algebraic Multigrid Methods for Direct Frequency Response Analyses in Solid Mechanics

Algebraic multigrid (AMG) methods have proven to be effective for solving the linear algebraic system of equations that arise from many classes of unstructured discretized elliptic PDEs. Standard AMG methods, however, are not suitable for shifted linear systems from elliptic PDEs, such as discretized Helmholtz operators, due to the indefiniteness of the system and the presence of low energy modes which are difficult for multigrid methods to resolve effectively. This paper investigates simple methods to adapt existing standard AMG methods to these shifted systems from direct frequency response analyses in solid mechanics.     

A coarse/fine preconditioner for very ill-conditioned finite element problems

We consider the problem of applying the conjugate gradient method to solve-ill-conditioned large algebraic systems of equations resulting from the finite element discretization of some three-dimensional boundary value problems. We present an effective preconditioner for such systems based on a multigrid technique. We assess its performance with examples borrowed from large flexible aerospace structures.     

Algebraic multigrid methods for dual mortar finite element formulations in contact mechanics

This paper proposes novel strategies to enable multigrid preconditioners within iterative solvers for linear systems arising from contact problems based on mortar finite element formulations. The so‐called dual mortar approach that is exclusively employed here allows for an easy condensation of the discrete Lagrange multipliers. Therefore, it has the advantage over standard mortar methods that linear systems with a saddle‐point structure are avoided, which generally require special preconditioning techniques. However, even with the dual mortar approach, the resulting linear systems turn out to be hard to solve using iterative linear solvers. A basic analysis of the mathematical properties of the linear operators reveals why the naive application of standard iterative solvers shows instabilities and provides new insights of how contact modeling affects the corresponding linear systems. This information is used to develop new strategies that make multigrid methods efficient preconditioners for the class of contact problems based on dual mortar methods. It is worth mentioning that these strategies primarily adapt the input of the multigrid preconditioners in a way that no contact‐specific enhancements are necessary in the multigrid algorithms. This makes the implementation comparably easy. With the proposed method, we are able to solve large contact problems, which is an important step toward the application of dual mortar–based contact formulations in the industry. Numerical results are presented illustrating the performance of the presented algebraic multigrid method.