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为了减少解在较小的局部区域内有着很强的奇异性、剧烈变化等的偏微分方程求解问题的计算量,提出了一种基于方程求解的移动网格方法,并将其应用于二维不可压缩Navier-Stokes方程的求解.与已有的大部分移动网格方法不同,网格节点的移动距离是通过求解一个变系数扩散方程得到的,避免了做区域映射,也不需要对控制函数进行磨光处理,所以算法很容易编程实现.数值算例表明所提算法能够在解梯度较大的位置加密网格,从而在保证提高数值解的分辨率的前提下,可以很好地节省了计算量.由于Navier-Stokes 的典型性,所得算法能够推广到求解很大一类偏微分方程数值问题. 相似文献
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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。 相似文献
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复线性方程组在科学与工程计算的诸多领域中有着重要的应用价值,如何高效的求解复线性方程组,一直是人们所关心的问题.目前对于复线性方程组,常用的处理方式有以下两种:一种是直接对方程组迭代求解,另外一种是将其转化为实线性方程组后进行求解.本文主要从两种处理方式讨论了共轭梯度法(CG法),并理论上证明了两种处理方式下的CG法具有相同的收敛性.之后基于变形共轭梯度法(MCG法)收敛速度的本质与CG法类似,只需将MCG法推广到复线性方程组进行研究,并且为了提高MCG法的收敛速度,提出了一种预处理MCG法.最后,通过数值算例验证了算法与理论分析的一致性,以及预处理算法的有效性. 相似文献
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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。 相似文献
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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。 相似文献
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等效夹杂方法是求解含杂质材料弹性应力场的一种有效方法,但是其解析求解只适用于椭球/椭圆类杂质问题。本文提出一种基于等效夹杂方法的数值化计算方法,介绍了其基本理论,并引入共轭梯度法求解该方法的一致性条件线性方程组。该方法通过计算区域的数值离散,能够实现对二维任意形状杂质弹性场的求解。将该方法得到的结果与解析解进行比较,验证了该方法的有效性。讨论了数值化等效夹杂方法在效率以及收敛性上的表现。通过对比证明,利用共轭梯度法实现该方法,能在保持精度的同时,相较于高斯消元法具有较大的效率优势。最后通过半椭圆杂质和氧化锆/氧化铝共挤复合材料算例验证了该方法处理任意形状杂质的能力。 相似文献
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概述了国内外油藏数值模拟的网格技术进展,介绍了结构化网格中的混合网格、快速自适应组合网格和非结构化网格中的角点几何网格、PEBI和CVFE网格、三角网格、Voronoi网格以及无网格方法。给出了它们的思路、适应范围以及优缺点。指出一个好的网格技术既要尽可能使模型的求解简单,又要尽可能全面、准确地模拟油藏静态和动态特征。同时,对未来的网格技术进行了展望,提出了应采用灵活的网格以及解决网格取向效应和网格细化的与计算稳定性、准确性、复杂程度和效率之间的矛盾以精确模拟我国的复杂油藏。 相似文献
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Denis
Davydov Jean-Paul Pelteret Daniel Arndt Martin Kronbichler Paul Steinmann 《International journal for numerical methods in engineering》2020,121(13):2874-2895
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. 相似文献
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Paul T. Lin John N. Shadid Paul H. Tsuji 《International journal for numerical methods in engineering》2019,120(12):1297-1309
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. 相似文献
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V. E. BULGAKOV M. V. BELYI K. M. MATHISEN 《International journal for numerical methods in engineering》1997,40(3):453-471
In this paper a novel iterative method of multilevel type for solving large-scale generalized eigenvalue problems encountered in structural dynamics is presented. A preconditioned iterative technique, which can be viewed as a modification of the Subspace Iteration method, is used for simultaneous calculation of a group of lowest modes and frequencies. The paper demonstrates that a coarse aggregation model can be employed in the hierarchical structure of the preconditioner in order to provide a good resemblance of the latter to the stiffness matrix of the finite element approximation with respect to low-frequency modes. This leads to a fast convergent procedure of subspace iterations. As opposed to the coarse grid used in methods of multigrid type, this model allows for solving problems with different finite elements including reticulated structures in the framework of large comprehensive finite element software systems. Numerical experiments performed for three-dimensional truss, frame and solid structures demonstrate an excellent performance of the method. © 1997 by John Wiley & Sons, Ltd. 相似文献
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V. E. Bulgakov G. Kuhn 《International journal for numerical methods in engineering》1995,38(20):3529-3544
The present study is aimed to overcome difficulties faced with industrial applications of multilevel iterative methods to arbitrary finite element (FE) structural analysis problems. The coarse grid concept, used in multigrid methods, is substituted with an aggregation coarse model based on the mechanical principle. On the base of this approach together with previously developed multilevel preconditioner, an efficient iterative equation solver FEAGS was developed for using in standard comprehensive finite element software systems. Numerical examples for analyses of three-dimensional (3-D) frame structures demonstrate the efficiency of the method. Comparison with incomplete Cholesky conjugate gradient (ICCG) method and direct methods is presented. 相似文献
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Charbel Farhat Nahil Sobh 《International journal for numerical methods in engineering》1989,28(7):1715-1723
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. 相似文献
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M. Rasquin H. Deconinck G. Degrez 《International journal for numerical methods in engineering》2010,82(12):1510-1536
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. 相似文献
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Hybrid parallel multigrid preconditioner based on automatic mesh coarsening for 3D metal forming simulations
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Frederic Vi Katia Mocellin Hugues Digonnet Etienne Perchat Lionel Fourment 《International journal for numerical methods in engineering》2018,114(6):598-618
A parallel multigrid (MG) method is developed to reduce the large computational costs involved by the finite element simulation of highly viscous fluid flows, especially those resulting from metal forming applications, which are characterized by using a mixed velocity/pressure implicit formulation, unstructured meshes of tetrahedra, and frequent remeshings. The developed MG method follows a hybrid approach where the different levels of nonnested meshes are geometrically constructed by mesh coarsening, while the linear systems of the intermediate levels result from the Galerkin algebraic approach. A linear O(N) convergence rate is expected (with N being the number of unknowns), while keeping software parallel efficiency. These objectives lead to selecting unusual MG smoothers (iterative solvers) for the upper grid levels and to developing parallel mesh coarsening algorithms along with parallel transfer operators between the different levels of partitioned meshes. Within the utilized PETSc library, the developed MG method is employed as a preconditioner for the usual conjugate residual algorithm because of the symmetric undefinite matrix of the system to solve. It shows a convergence rate close to optimal, an excellent parallel efficiency, and the ability to handle the complex forming problems encountered in 3‐dimensional hot forging, which involve large material deformations and frequent remeshings. 相似文献
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Paul T. Lin 《International journal for numerical methods in engineering》2012,91(9):971-989
This study considers the scaling of three algebraic multigrid aggregation schemes for a finite element discretization of a drift–diffusion system, specifically the drift–diffusion model for semiconductor devices. The approach is more general and can be applied to other systems of partial differential equations. After discretization on unstructured meshes, a fully coupled multigrid preconditioned Newton–Krylov solution method is employed. The choice of aggregation scheme for generating coarser levels has a significant impact on the performance and scalability of the multigrid preconditioner. For the test cases considered, the uncoupled aggregation scheme, which aggregates/combines the immediate neighbors, followed by repartitioning and data redistribution for the coarser level matrices on a subset of the Message Passing Interface (MPI) processes, outperformed the two other approaches, including the baseline aggressive coarsening scheme. Scaling results are presented up to 147,456 cores on an IBM Blue Gene/P platform. A comparison of the scaling of a multigrid V‐cycle and W‐cycle is provided. Results for 65,536 cores demonstrate that a factor of 3.5 × reduction in time between the uncoupled aggregation and baseline aggressive coarsening scheme can be obtained by significantly reducing the iteration count due to the increased number of multigrid levels and the generation of better quality aggregates. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献
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Rongliang Chen Xiao‐Chuan Cai 《International journal for numerical methods in engineering》2014,99(13):945-965
A two‐level domain decomposition method is introduced for general shape optimization problems constrained by the incompressible Navier–Stokes equations. The optimization problem is first discretized with a finite element method on an unstructured moving mesh that is implicitly defined without assuming that the computational domain is known and then solved by some one‐shot Lagrange–Newton–Krylov–Schwarz algorithms. In this approach, the shape of the domain, its corresponding finite element mesh, the flow fields and their corresponding Lagrange multipliers are all obtained computationally in a single solve of a nonlinear system of equations. Highly scalable parallel algorithms are absolutely necessary to solve such an expensive system. The one‐level domain decomposition method works reasonably well when the number of processors is not large. Aiming for machines with a large number of processors and robust nonlinear convergence, we introduce a two‐level inexact Newton method with a hybrid two‐level overlapping Schwarz preconditioner. As applications, we consider the shape optimization of a cannula problem and an artery bypass problem in 2D. Numerical experiments show that our algorithm performs well on a supercomputer with over 1000 processors for problems with millions of unknowns. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献