体积约束的非局部扩散问题的加罚有限元方法

doi:10.19596/j.cnki.1001-246x.7730

计算物理 ›› 2018, Vol. 35 ›› Issue (2): 161-168.DOI: 10.19596/j.cnki.1001-246x.7730

体积约束的非局部扩散问题的加罚有限元方法

葛志昊, 吴慧丽

河南大学数学与统计学院, 河南开封 475004

收稿日期:2017-07-19 修回日期:2017-09-29 出版日期:2018-03-25 发布日期:2018-03-25
作者简介:葛志昊(1980-),男,河南淮阳,副教授,从事偏微方程理论与计算研究,Email:zhihaoge@vip.henu.edu.cn
基金资助:
河南省自然科学基金（162300410031）和河南大学优秀青年（YQPY20140039）资助项目

A Penalized Finite Element Method of Nonlocal Diffusion Problem with Volume Constraints

GE Zhihao, WU Huili

School of Mathematics and Statistics, Henan University, Kaifeng 475004, China

Received:2017-07-19 Revised:2017-09-29 Online:2018-03-25 Published:2018-03-25

摘要/Abstract

摘要： 针对非齐次和齐次体积约束的非局部扩散问题设计了新的有限元方法——加罚有限元方法，并给出该方法的误差分析.数值算例验证了加罚有限元方法的稳定性和有效性.

关键词: 非局部扩散问题, 误差分析, 有限元方法

Abstract: We propose a finite element method, penalized finite element method, to solve a homogenous and inhomogenous nonlocal diffusion problem with volume constraints, and give error estimate of the method. Finally, we show numerical experiments to verify the theoretical results.

Key words: nonlocal diffusion problem, error analysis, finite element method

中图分类号:

O241.82

葛志昊, 吴慧丽. 体积约束的非局部扩散问题的加罚有限元方法[J]. 计算物理, 2018, 35(2): 161-168.

GE Zhihao, WU Huili. A Penalized Finite Element Method of Nonlocal Diffusion Problem with Volume Constraints[J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2018, 35(2): 161-168.

参考文献

[1] KRONER E. Elasticity theory of materials with long range forces[J]. International Journal of Solids and Structures, 1967, 3(5):731-742.
[2] ROGULA D. Nonlocal theory of material media[M]. Berlin:Springer, 1982.
[3] ALTAN B. Uniqueness of initial-boundary value problems in nonlocal elasticity[J]. International Journal of Solids and Structures, 1989, 25(11):1271-1278.
[4] ALTAN B. Uniqueness in nonlocal thermoelasticity[J]. Journal of Thermal Stresses, 1991, 14(2):121-128.
[5] WANG J, DHALIWAL R. Uniqueness in generalized nonlocal thermoelasticity[J]. Journal of Thermal Stresses, 1993, 16(1):71-77.
[6] WANG J, DHALIWAL R. On some theorems in the nonlocal theory of micropolar elasticity[J]. International Journal of Solids and Structures, 1993, 30(10):1331-1338.
[7] SILLING S. Reformulation of elasticity theory for discontinuities and long-range forces[J]. Journal of the Mechanics and Physics of Solids, 2000, 48(1):175-209.
[8] BOBARU F, SILLING S. Peridynamic 3D problems of nanofiber networks and carbon nanotube-reinforced composites[J]//Proceedings of the International Conference on Numerical Methods in Industrial Forming Processes, American Institute of Physics, 2004:1565-1570.
[9] BOBARU F, SILLING S, JIANG H. Peridynamic frature and damage modeling of membranes and nanofiber networks[J]. Proceedings of the XI International Conference on Fracture, Turin, 2005, 5748:1-6.
[10] SILLING S. Dynamic frature modeling with a meshfree peridynamic code[J]. Computational Fluid and Solid Mechanics, 2003, 641-644.
[11] GILBOA G, OSHER S.Nonlocal operators with applications to image processing[J]. Siam Journal on Multiscale Modeling and Simulation, 2008,7(3):1005-1028.
[12] LOU Y, ZHANG X, OSHER S, BERTOZZI A. Image recovery via nonlocal operators[J]. Journal of Scientific Computing, 2010, 42(2):185-197.
[13] SILLING S A, LEHOUCQ R B. Convergence of peridynamics to classical elasticity theory[J]. Journal of Elasticity, 2008, 93(1):13-37.
[14] DU Q, ZHUO K. Mathematical analysis for the peridynamics nonlocal continuum theory[J]. Esaim Mathematical Modelling and Numerical Analysis, 2011, 45(2):217-234.
[15] EMMRICH E, WECKNER O. On the well-posedness of the linear peridynamic model and its convergence towards the Navier equation of linear elasticity[J]. Communications in Mathematical Sciences, 20075(4):851-864.
[16] EMMRICH E, WECKNER O. Analysis and numerical approximation of an integro-differential equation modeling nonlocal effects in linear elasticity[J]. Mathematics and Mechanics of Solids, 2007, 4(4):363-384.
[17] LEHOUCQ R B, SILLING S. Force flux and the peridynamic stress tensor[J]. Journal of the Mechanics and Physics of Solids, 2008, 56(4):1566-1577.
[18] GUNZBURGER M, LEHOUCQ R B. A nonlocal vector calculus with application to nonlocal boundary value problems[J]. SIAM Journal on Multiscale Modeling and Simulation, 2010, 8(5):1581-1598.
[19] DU Q, GUNZBUEGER M, LEHOUCQ R B, et al. A nonlocal vector caculus, nonlocal volume-constrained problems, and the nonlocal balance laws[J]. Mathematical Models and Methods in Applied Sciences, 2013, 23(3):493-540.
[20] AKSOYLU B, PARKS M. Variational theory and domain decomposition for nonlocal problems[J]. Applied Mathematics and Computation, 2011, 217(14):6498-6515.
[21] ZHU X G, NIE Y F, WANG J G, et al. A characteristic finite element method for fractional convection-diffusion equations[J]. Chinese Journal of Computational Physics, 2017, 34(4):417-424.
[22] CHEN X, GUNZBURGER M. Continuous and discontinuous finite element methods for a peridynamics model of mechanics[J]. Computer Methods in Applied Mechanics and Engineering, 2011, 200(9-12):1237-1250.
[23] MACEK R W, SILLING S. Peridynamics via finite element analysis[J]. Finite Elements in Analysis and Design, 2007, 43(15):1169-1178.
[24] SILLING S, ASKARI E. A meshfree method based on the peridynamic model of solid mechanics[J]. Computers and Structures, 2005, 83(17-18):1526-1535.
[25] ZHOU K, DU Q. Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions[J]. SIAM Journal on Numerical Analysis, 2010, 48(5):1759-1780.
[26] RIVIERE B. Discontinuous Galerkin methods for solving elliptic and parabolic equations:Theory and implementation[M]. SIAM, Philadelphia, 2008.

体积约束的非局部扩散问题的加罚有限元方法

A Penalized Finite Element Method of Nonlocal Diffusion Problem with Volume Constraints

RichHTML

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

编辑推荐

Metrics

作者中心

审稿中心

期刊浏览

期刊介绍

[1]	朱家莉, 尚月强. 不可压缩流的并行两水平稳定有限元算法[J]. 计算物理, 2022, 39(3): 309-317.
[2]	丁琪, 尚月强. 非定常Navier-Stokes方程基于两重网格离散的有限元并行算法[J]. 计算物理, 2020, 37(1): 10-18.
[3]	张庆福, 姚军, 黄朝琴, 李阳, 王月英. 裂缝性介质多尺度深度学习模型[J]. 计算物理, 2019, 36(6): 665-672.
[4]	温小静, 屈瑜, 陈城钊. 双层开口环的圆二色性数值研究[J]. 计算物理, 2019, 36(3): 357-362.
[5]	杨晓成, 尚月强. 大雷诺数Navier-Stokes方程的两水平亚格子模型稳定化方法[J]. 计算物理, 2017, 34(6): 657-665.
[6]	朱晓钢, 聂玉峰, 王俊刚, 袁占斌. 分数阶对流扩散方程的特征有限元方法[J]. 计算物理, 2017, 34(4): 417-424.
[7]	赵国忠, 蔚喜军, 郭怀民. 二维Lagrangian坐标系下可压气动方程组的间断Petrov-Galerkin方法[J]. 计算物理, 2017, 34(3): 294-308.
[8]	叶珍宝, 朱剑, 周海京. 基于曲四面体单元剖分的CN E-H TDFEM方法[J]. 计算物理, 2016, 33(6): 652-660.
[9]	叶珍宝, 周海京. 基于高阶叠层矢量基函数的E-H时域有限元方法分析谐振腔和波导结构[J]. 计算物理, 2016, 33(3): 333-340.
[10]	叶珍宝, 周海京. 高阶间断伽辽金时域有限元方法分析三维谐振腔[J]. 计算物理, 2015, 32(4): 449-454.
[11]	赵国忠, 蔚喜军, 李珍珍. 多介质流模拟的Runge-Kutta控制体积间断有限元方法[J]. 计算物理, 2014, 31(3): 271-284.
[12]	李珍珍, 蔚喜军, 赵国忠, 冯涛. RKDG有限元法求解一维拉格朗日形式的Euler方程[J]. 计算物理, 2014, 31(1): 1-10.
[13]	冯涛, 蔚喜军, 安恒斌, 崔霞, 吴迪, 李珍珍. Lax-Wendroff时间离散的自适应间断有限元方法求解三维可压缩欧拉方程[J]. 计算物理, 2013, 30(6): 791-798.
[14]	高永峰, 张相炎, 刘宁. 任意不可压多介质流的粒子有限元方法模拟[J]. 计算物理, 2013, 30(1): 35-43.
[15]	赵国忠, 蔚喜军, 张荣培. Lagrange坐标系下二维气动方程组的RKDG有限元方法[J]. 计算物理, 2012, 29(2): 166-174.