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Immersed Finite Element method for Interface Problems with algebraic multigrid Solver

Communications in Computational Physics 2014年第4期15卷 1045-1067页

作者： Wenqiang Feng Xiaoming He Yanping Lin Xu Zhang Department of Mathematics and Statistics Missouri University of Science and TechnologyRollaMO 65409USA Department of Applied Mathematics Hong Kong Polytechnic UniversityHung HomHong Kong Department of Mathematical and Statistics Science University of AlbertaEdmontonABT6G 2G1Canada Department of Mathematics Virginia TechBlacksburgVA 24061USA

This article is to discuss the bilinear and linear immersed finite element(IFE)solutions generated from the algebraic multigrid solver for both stationary and moving interface problems.For the numerical methods based on finite difference formulation and a structured mesh independent of the interface,the stiffness matrix of the linear system is usually not symmetric positive-definite,which demands extra efforts to design efficient multigrid methods.On the other hand,the stiffness matrix arising from the IFE methods are naturally symmetric positive-definite.Hence the IFE-AMG algorithm is proposed to solve the linear systems of the bilinear and linear IFE methods for both stationary and moving interface problems.The numerical examples demonstrate the features of the proposed algorithms,including the optimal convergence in both L 2 and semi-H1 norms of the IFE-AMG solutions,the high efficiency with proper choice of the components and parameters of AMG,the influence of the tolerance and the smoother type of AMG on the convergence of the IFE solutions for the interface problems,and the relationship between the cost and the moving interface location.

关键词： Interface problems immersed finite elements algebraic multigrid method

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Finite Element Resistivity Forward Modeling using algebraic multigrid Preconditioned Conjugate Gradient method

Finite Element Resistivity Forward Modeling using Algebraic ...

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2012 3rd International Conference on Applied Mechanics and Mechanical Engineering(ICAMME 2012)

作者： Guihong Zou Huaqing Liang Dept.of Electronic Eng.China University of Petroleum

An algebraic multigrid by smoothed aggregation preconditioned conjugate gradient method is developed to solve the liner system arising from 3-D direct current finite element resistivity forward modeling. The algorithm combines the efficiency of algebraic multigrid method and the stability of conjugate gradient method. algebraic multigrid by smoothed aggregation keep in high-efficiency while simulation using local quasi-uniform mesh and its convergence effect will reduce while numerical modeling using anisotropic stretched grids. However tensor product non-equidistant mesh, a kind of anisotropic stretched grids, is often used in 3-D direct current resistivity forward modeling. In order to improve this situation, a factor is added to guide correct aggregation. Consequently, a typical example is used to prove that the improvement is the right. Finally, it is natural to conclude that the algorithm suggested in this paper is efficient and robust whether simulation using local quasi-uniform mesh or tensor product non-equidistant mesh

关键词： Preconditioned conjugate gradient method algebraic multigrid method smooothed aggregation 3-D resistivity forward finite element method

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A Robust Algorithm for Blind Total Variation Restoration

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Acta Mathematicae Applicatae Sinica 2008年第4期24卷 681-690页

作者： Jing Xu Qian-shun Chang School of Statistics and Mathematics Zhejiang Gongshang University Hangzhou 310018 China Academy of Mathematics and Systems Science Chinese Academy of Sciences Deijing 100190 China

Image restoration is a fundamental problem in image processing. Blind image restoration has a great value in its practical application. However, it is not an easy problem to solve due to its complexity and difficulty. In this paper, we combine our robust algorithm for known blur operator with an alternating minimization implicit iterative scheme to deal with blind deconvolution problem, recover the image and identify the point spread function（PSF）. The only assumption needed is satisfy the practical physical sense. Numerical experiments demonstrate that this minimization algorithm is efficient and robust over a wide range of PSF and have almost the same results compared with known PSF algorithm.

关键词： Blind deconvolution total variation algebraic multigrid method Krylov acceleration

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Numerical Solution of 3D Poisson-Nernst-Planck Equations Coupled with Classical Density Functional Theory for Modeling Ion and Electron Transport in a Confined Environment

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Communications in Computational Physics 2014年第10期16卷 1298-1322页

作者： Da Meng Bin Zheng Guang Lin Maria L.Sushko Pacific Northwest National Laboratory RichlandWA 99352USA. Department of Mathematics School of Mechanical EngineeringPurdue UniversityWest LafayetteIN 47907USA.

We have developed efficient numerical algorithms for solving 3D steadystate Poisson-Nernst-Planck(PNP)equations with excess chemical potentials described by the classical density functional theory(cDFT).The coupled PNP equations are discretized by a finite difference scheme and solved iteratively using the Gummel method with relaxation.The Nernst-Planck equations are transformed into Laplace equations through the Slotboom transformation.Then,the algebraic multigrid method is applied to efficiently solve the Poisson equation and the transformed Nernst-Planck equations.A novel strategy for calculating excess chemical potentials through fast Fourier transforms is proposed,which reduces computational complexity from O(N2)to O(NlogN),where N is the number of grid points.Integrals involving the Dirac delta function are evaluated directly by coordinate transformation,which yields more accurate results compared to applying numerical quadrature to an approximated delta function.Numerical results for ion and electron transport in solid electrolyte for lithiumion(Li-ion)batteries are shown to be in good agreement with the experimental data and the results from previous studies.

关键词： Poisson-Nernst-Planck equations classical density functional theory algebraic multigrid method fast Fourier transform Li-ion battery

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Application of splitting scheme and multigrid method for TV-Stokes denoising

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Science China(Information Sciences) 2011年第4期54卷 745-756页

作者： CHANG QianShun 1,TAI XueCheng 2 & XING LiLi 3 1 Institute of Applied Mathematics,Academy of Mathematics and Systems Sciences,Chinese Academy of Sciences,Beijing 100190,China 2 Department of Mathematics,University of Bergen,Bergen 5007,Norway School of Physical and Mathematical Sciences,Nanyang Technological University,639798,Singapore 3 School of Mathematics and Computational Science,China University of Petroleum (East China),Dongying 257061,China Institute of Applied Mathematics Academy of Mathematics and Systems Sciences Chinese Academy of Sciences Beijing China Department of Mathematics University of Bergen Bergen Norway School of Mathematics and Computational Science China University of Petroleum (East China) Dongying China

Based on some previous work on the connection between image restoration and fluid dynamics,we apply a two-step algorithm for image denoising.In the first step,using a splitting scheme to study a nonlinear Stokes equation,tangent vectors are obtained.In the second step,an image is restored to fit the constructed tangent directions.We apply a fixed point iteration to solve the total variation-based image denoising problem,and use algebraic multigrid method to solve the corresponding linear equations.Numerical results demonstrate that our algorithm is efficient and robust,and boundary conditions are satisfactory for image denoising.

关键词： image denoising total variation Stokes equation algebraic multigrid method

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Two-Level Hierarchical PCG methods for the Quadratic FEM Discretizations of 2D Concrete Aggregate Models

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Advances in Applied Mathematics and Mechanics 2019年第6期11卷 1376-1397页

作者： Yingxiong Xiao Heng Chen Lingjie Xie Civil Engineering and Mechanics College Xiangtan UniversityXiangtan 411105HunanChina

The concrete aggregate model is considered as a type of weakly discontinuous problem consisting of three phases:aggregates which randomly distributed in different shapes,cement paste and internal transition zone(ITZ).Because of different shapes of aggregate and thin ITZs,a huge number of elements are often used in the finite element(FEM)analysis.In order to ensure the accuracy of the numerical solutions near the interfaces,we need to use higher-order elements.The widely used FEM softwares such as ANSYS and ABAQUS all provide the option of quadratic elements.However,they have much higher computational complexity than the linear elements.The corresponding coefficient matrix of the system of equations is a highly ill-conditioned matrix due to the large difference between three phase materials,and the convergence rate of the commonly used solving methods will deteriorate.In this paper,two types of simple and efficient preconditioners are proposed for the system of equations of the concrete aggregate models on unstructured triangle meshes by using the resulting hierarchical structure and the properties of the diagonal block matrices.The main computational cost of these preconditioners is how to efficiently solve the system of equations by using linear elements,and thus we can provide some efficient and robust solvers by calling the existing geometric-based algebraic multigrid(GAMG)methods.Since the hierarchical basis functions are used,we need not present those algebraic criterions to judge the relationships between the unknown variables and the geometric node types,and the grid transfer operators are also trivial.This makes it easy to find the linear element matrix derived directly from the fine level matrix,and thus the overall efficiency is greatly improved.The numerical results have verified the efficiency of the resulting preconditioned conjugate gradient(PCG)methods which are applied to the solution of several typical aggregate models.

关键词： Concrete aggregate model hierarchical quadratic elements ill-conditioned matrix algebraic multigrid method preconditioner

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非结构四边形二次Lagrangian有限元方程的代数多重网格法及收敛性分析

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高等学校计算数学学报 2005年第S1期 215-222页

作者：刘轩舒适湘潭大学数学与计算科学学院湘潭大学数学与计算科学学院湘潭 411105

1 引言代数多重网格(AMG)法是求解有结构、非结构冈格下的PDEs离散化系统,以及其它大型稀疏矩阵方程的最为有效的方法之一,是当前多重网格法研究领域的热点．它主要侧重于粗化技术的研究,目前这方面已有大量的工作(见文[1～4]),其中基... 详细信息

1 引言代数多重网格(AMG)法是求解有结构、非结构冈格下的PDEs离散化系统,以及其它大型稀疏矩阵方程的最为有效的方法之一,是当前多重网格法研究领域的热点．它主要侧重于粗化技术的研究,目前这方面已有大量的工作(见文[1～4]),其中基于部分几何信息和分析信息的AMG法,是当前AMG法发展的一个新趋势(见文[5～7])．

关键词： two-level grid method algebraic multigrid method quadrilateral Lagrangian finite element.

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