• 论文 • 上一篇    

科恩-沈方程与含时科恩-沈方程的一个自适应有限元数值方法

况阳1, 申烨丹2, 胡光辉2,3   

  1. 1. 新加坡国立大学数学系, 新加坡;
    2. 澳门大学数学系, 澳门;
    3. 珠海澳大科技研究院, 珠海
  • 收稿日期:2020-10-20 发布日期:2021-03-18
  • 通讯作者: 胡光辉,garyhu@um.edu.mo.
  • 基金资助:
    国家自然科学基金(11922120,11871489,11401608),澳门科技发展基金(FDCT-029/2016/A1),澳门大学(MYRG2017-00189-FST,MYRG2019-00154-FST)及新加坡国立大学(R-146-000-291-114)资助.

况阳, 申烨丹, 胡光辉. 科恩-沈方程与含时科恩-沈方程的一个自适应有限元数值方法[J]. 数值计算与计算机应用, 2021, 42(1): 33-55.

Kuang Yang, Shen Yedan, Hu Guanghui. AN H-ADAPTIVE FINITE ELEMENT METHOD FOR KOHN-SHAM AND TIME-DEPENDENT KOHN-SHAM EQUATIONS[J]. Journal on Numerica Methods and Computer Applications, 2021, 42(1): 33-55.

AN H-ADAPTIVE FINITE ELEMENT METHOD FOR KOHN-SHAM AND TIME-DEPENDENT KOHN-SHAM EQUATIONS

Kuang Yang1, Shen Yedan2, Hu Guanghui2,3   

  1. 1. Department of Mathematics, National University of Singapore, Singapore 119076;
    2. Department of Mathematics, University of Macau, Macao SAR, China;
    3. Zhuhai UM Science & Technology Research Institute, Guangdong Province, China
  • Received:2020-10-20 Published:2021-03-18
在材料分析、纳米光学等研究中,高质量数值模拟多体系统电子密度的随时间演化是一类重要研究内容.演化中产生的时间依赖偶极子等物理量,是更进一步研究的基础.此类数值模拟分为两个步骤,即多体系统的基态求解、及以基态为初值的系统的动态演化模拟.这两个步骤可以分别通过数值求解科恩-沈(Kohn——Sham)方程及含时科恩-沈(time-dependent Kohn——Sham)方程实现.本文中,我们提出一类基于有限元方法的数值求解框架,为这两个步骤提供一个统一的模拟实现.在基态求解中,我们利用一类自洽场迭代对方程进行线性化,采用局部最优块预处理共轭梯度法求解导出的广义特征值问题,并设计了一个基于多重网格方法的预优对求解进行有效加速.在动态演化模拟中,针对方程的结构,我们提出了一个基于隐式中点公式的数值方法,利用预估-校正方法对方程进行线性化处理,并设计了一个针对复值线性系统的代数多重网格求解器用于加速时间推进.特别地,我们基于提出的数值方法,分别针对科恩-沈及含时科恩-沈方程导出了残量型后验误差估计子,并实现了基于局部加密的网格自适应方法,用于进一步改善数值模拟效率.数值解展示了方法的有效性.
Quality numerical simulations of the dynamics of a given many-body electronic structure system is an important research area in material analysis and nano-optics, etc. Quantities such as the time-dependent dipole moment are essential for further study. There are two components in such simulations, i.e., the ground state calculation, and the following dynamic simulations with the ground state as an initial state. These two components can be obtained by solving Kohn-Sham and time-dependent Kohn-Sham equations, respectively. In this paper, based on the finite element method, a unified numerical framework is proposed for the whole simulation. For the ground state calculation, the classical self-consistent field iteration method is employed for the linearization of the equation, in which the derived generalized eigenvalue problem is solved by the locally optimal blocked preconditioned conjugate gradient method, and we also design an effective preconditioner based on the multigrid method for the acceleration of the iteration. For the simulation of the dynamics, an implicit midpoint scheme is used for the temporal discretization, while the linear finite element method is used for the spatial discretization. A predictor-corrector method is used for the linearization of the equation, and an algebraic multigrid solver is developed for the derived complexvalued system in order to accelerate the simulation. In particular, an h-adaptive finite element method is developed for further improving the efficiency, in which two residual type a posteriori error indicators are designed for the Kohn-Sham and time-dependent Kohn- Sham equations, respectively. A variety of numerical experiments verify the effectiveness of our method.

MR(2010)主题分类: 

()
[1] Genovese L, Videau B, Ospici M, Deutsch T, Goedecker S, Méhaut J F. Daubechies wavelets for high performance electronic structure calculations:The BigDFT project[J]. Comptes Rendus Mécanique, 2011, 339(2):149-164. High Performance Computing.
[2] Tancogne-Dejean N, Oliveira M J T, Andrade X, Appel H, Borca C H, Breton G L, Buchholz F, Castro A, Corni S, Correa A A, Giovannini U D, Delgado A, Eich F G, Flick J, Gil G, Gomez A, Helbig N, Hübener H, Jestädt R, Jornet-Somoza J, Larsen A H, Lebedeva I V, Lüders M, Marques M A L, Ohlmann S T, Pipolo S, Rampp M, Rozzi C A, Strubbe D A, Sato S A, Schäfer C, Theophilou I, Welden A, Rubio A. Octopus, a computational framework for exploring light-driven phenomena and quantum dynamics in extended and finite systems[J]. The Journal of Chemical Physics, 2020, 152(12):124119.
[3] Bao G, Hu G, Liu D. Real-time adaptive finite element solution of time-dependent Kohn-Sham equation[J]. Journal of Computational Physics, 2015, 281:743-758.
[4] Chen H, Dai X, Gong X, He L, Zhou A. Adaptive finite element approximations for Kohn-Sham models[J]. Multiscale Modeling & Simulation, 2014, 12(4):1828-1869.
[5] 戴小英, 周爱辉. 电子结构计算的有限元方法[J]. 中国科学:化学, 2015, 45(8):800-811.
[6] 戴小英. 电子结构计算的数值方法与理论[J].计算数学, 2020, 42(2):131-158.
[7] Lin L, Lu J, Ying L, Weinan E. Adaptive local basis set for Kohn-Sham density functional theory in a discontinuous Galerkin framework I:Total energy calculation[J]. Journal of Computational Physics, 2012, 231(4):2140-2154.
[8] Dai X, Gong X, Yang Z, Zhang D, Zhou A. Finite volume discretizations for eigenvalue problems with applications to electronic structure calculations[J]. Multiscale Modeling & Simulation, 2011, 9(1):208-240.
[9] Motamarri P, Gavini V. Subquadratic-scaling subspace projection method for large-scale Kohn-Sham density functional theory calculations using spectral finite-element discretization[J]. Physical Review B, 2014, 90(11):115127.
[10] Gaussian. https://gaussian.com/.
[11] COMSOL. https://www.comsol.com/.
[12] VASP. http://www.vasp.at.
[13] DFT-FE. https://sites.google.com/umich.edu/dftfe.
[14] Kosloff R, Tal-Ezer H. A direct relaxation method for calculating eigenfunctions and eigenvalues of the Schrödinger equation on a grid[J]. Chemical Physics Letters, 1986, 127(3):223-230.
[15] Hu G, Xie H, Xu F. A multilevel correction adaptive finite element method for Kohn-Sham equation[J]. Journal of Computational Physics, 2018, 355:436-449.
[16] Gao B, Hu G, Kuang Y, Liu X. An orthogonalization-free parallelizable framework for all-electron calculations in density functional theory[J]. arXiv preprint arXiv:2007.14228, 2020.
[17] Pueyo A G, Marques M A L, Rubio A, Castro A. Propagators for the Time-Dependent Kohn-Sham Equations:Multistep, Runge-Kutta, Exponential Runge-Kutta, and Commutator Free Magnus Methods[J]. Journal of Chemical Theory and Computation, 2018, 14(6):3040-3052. PMID:29672048.
[18] Andrade X, Strubbe D, Giovannini U D, Larsen A H, Oliveira M J T, Alberdi-Rodriguez J, Varas A, Theophilou I, Helbig N, Verstraete M J, Stella L, Nogueira F, Aspuru-Guzik A, Castro A, Marques M A L, Rubio A. Real-space grids and the Octopus code as tools for the development of new simulation approaches for electronic systems[J]. Phys. Chem. Chem. Phys., 2015, 17:31371-31396.
[19] Kanungo B, Gavini V. Real time time-dependent density functional theory using higher order finite-element methods[J]. Phys. Rev. B, Sep 2019, 100:115148.
[20] Chen L, Chen Y. Multigrid Method for Poroelasticity Problem by Finite Element Method[J]. Advances in Applied Mathematics and Mechanics, 2019, 11(6):1339-1357.
[21] Lin L, Lu J, Ying L, E W. Adaptive local basis set for Kohn-Sham density functional theory in a discontinuous Galerkin framework I:Total energy calculation[J]. Journal of Computational Physics, 2012, 231(4):2140-2154.
[22] Bao G, Hu G, Liu D. An h-adaptive finite element solver for the calculations of the electronic structures[J]. Journal of Computational Physics, 2012, 231(14):4967-4979.
[23] Kuang Y, Hu G. On stabilizing and accelerating SCF using ITP in solving Kohn-Sham equation[J]. Commun. Comput. Phys., 2020, 28(3):999-1018.
[24] Yang L, Shen Y, Hu Z, Hu G. An Implicit Solver for the Time-Dependent Kohn-Sham Equation[J]. Numerical Mathematics:Theory, Methods and Applications, 2021, 14(1):261-284.
[25] Day D, Heroux M A. Solving Complex-Valued Linear Systems via Equivalent Real Formulations[J]. SIAM Journal on Scientific Computing, 2001, 23(2):480-498.
[26] Li R, Tang T, Zhang P. Moving mesh methods in multiple dimensions based on harmonic maps[J]. Journal of Computational Physics, 2001, 170(2):562-588.
[27] Marques M A L, Oliveira M J T, Burnus T. Libxc:A library of exchange and correlation functionals for density functional theory[J]. Computer physics communications, 2012, 183(10):2272-2281.
[28] Lin L, Yang C. Elliptic preconditioner for accelerating the self-consistent field iteration in Kohn-Sham density functional theory[J]. SIAM Journal on Scientific Computing, 2013, 35(5):S277-S298.
[29] Liu X, Wang X, Wen Z, Yuan Y. On the convergence of the self-consistent field iteration in Kohn-Sham density functional theory[J]. SIAM J. Matrix Anal. Appl., 2014, 35(2):546-558.
[30] Broyden C G. A class of methods for solving nonlinear simultaneous equations[J]. Mathematics of computation, 1965, 19(92):577-593.
[31] Pulay P. Convergence acceleration of iterative sequences. The case of SCF iteration[J]. Chemical Physics Letters, 1980, 73(2):393-398.
[32] Simon B. Schrödinger operators in the twentieth century[J]. Journal of Mathematical Physics, 2000, 41(6):3523-3555.
[33] Genovese L, Neelov A, Goedecker S, Deutsch T, Ghasemi S A, Willand A, Caliste D, Zilberberg O, Rayson M, Bergman A,. Daubechies wavelets as a basis set for density functional pseudopotential calculations[J]. The Journal of chemical physics, 2008, 129(1):014109.
[34] Bai Z, Demmel J, Dongarra J, Ruhe A, van der Vorst H. Templates for the solution of algebraic eigenvalue problems:a practical guide[M]. SIAM, 2000.
[35] Beck T L. Real-space mesh techniques in density-functional theory[J]. Reviews of Modern Physics, 2000, 72(4):1041.
[36] Saad Y. Numerical Methods for Large Eigenvalue Problems:Revised Edition[M]. SIAM, 2011.
[37] 李大潜, 秦铁虎. 物理学与偏微分方程(上, 下册)[M]. 教育出版社, 1997.
[38] Ciarlet P G. The Finite Element Method for Elliptic Problems[M]. SIAM, 2002.
[39] Ullrich C A. Time-Dependent Density-Functional Theory:Concepts and Applications[M]. OUP Oxford, 2011.
[40] Cleary A J, Falgout R D, Henson V E, Jones J E, Manteuffel T A, McCormick S F, Miranda G N, Ruge J W. Robustness and scalability of algebraic multigrid[J]. SIAM Journal on Scientific Computing, 2000, 21(5):1886-1908.
[41] Li R. On multi-mesh h-adaptive methods[J]. Journal of Scientific Computing, 2005, 24(3):321-341.
[42] Russell D. Johnson III (Ed.). NIST Computational Chemistry Comparison and Benchmark Database, NIST Standard Reference Database Number 101[J]. Release 19, April 2018. http://cccbdb.nist.gov/,DOI:10.18434/T47C7Z.
[43] Oliveira M J T, Nogueira F. Generating relativistic pseudo-potentials with explicit incorporation of semi-core states using APE, the Atomic Pseudo-potentials Engine[J]. Computer Physics Communications, 2008, 178(7):524-534.
[1] 谢和虎. 子空间扩展算法及其应用[J]. 数值计算与计算机应用, 2020, 41(3): 169-191.
[2] 王芹, 马召灿, 白石阳, 张林波, 卢本卓, 李鸿亮. 三维半导体器件漂移扩散模型的并行有限元方法研究[J]. 数值计算与计算机应用, 2020, 41(2): 85-104.
[3] 葛志昊, 李婷婷, 王慧芳. 双资产欧式期权定价问题的特征有限元方法[J]. 数值计算与计算机应用, 2020, 41(1): 27-41.
[4] 李瑜, 谢和虎. 基于特征线法的群体平衡系统的数值模拟[J]. 数值计算与计算机应用, 2019, 40(4): 261-278.
[5] 邓维山, 徐进. 一种泊松-玻尔兹曼方程稳定算法的高效有限元并行实现[J]. 数值计算与计算机应用, 2018, 39(2): 91-110.
[6] 余涛, 张镭. 线性弹性问题的局部正交分解方法[J]. 数值计算与计算机应用, 2018, 39(1): 10-19.
[7] 周宇, 李秋齐. 基于降基多尺度有限元的PGD方法及其在含参数椭圆方程中的应用[J]. 数值计算与计算机应用, 2017, 38(2): 105-122.
[8] 杨建宏. 定常Navier-Stokes问题低次等阶稳定有限体积元算法研究[J]. 数值计算与计算机应用, 2017, 38(2): 91-104.
[9] 曹济伟, 葛志昊, 刘鸣放. Stokes方程基于多尺度函数的稳定化有限元方法[J]. 数值计算与计算机应用, 2017, 38(1): 68-80.
[10] 周志强, 吴红英. 分数阶对流-弥散方程的移动网格有限元方法[J]. 数值计算与计算机应用, 2014, 35(1): 1-7.
[11] 杨建宏. 定常Navier-Stokes方程的三种两层稳定有限元算法计算效率分析[J]. 数值计算与计算机应用, 2011, 32(2): 117-124.
[12] 程俊霞, 任健. 含曲率的水平集方程在非结构四边形网格上的数值离散方法[J]. 数值计算与计算机应用, 2011, 32(1): 33-40.
阅读次数
全文


摘要