An Alternating Direction Multiplier Method for 4th-Order Variational Inequalities With Curvature Obstacle
-
摘要: 对于重调和算子和曲率障碍表示的变分不等式，提出了自适应交替方向乘子数值解法(SADMM). 对问题引入一个辅助变量表示曲率函数的增广Lagrange函数，导出一个约束极小值问题，并且该问题等价于一个鞍点问题. 然后采用交替方向乘子法(ADMM)求解这个鞍点问题. 通过采用平衡原理和迭代函数，得到了自动调整罚参数的自适应法则，从而提高了计算效率. 证明了该方法的收敛性，并给出了利用迭代函数近似罚参数的具体方法. 最后，用数值计算结果验证了该方法的有效性.Abstract: A self-adaptive alternating direction method of multipliers was proposed for the approximation solution of variational inequalities with biharmonic operators and curvature obstacle. An augmented Lagrange functional was introduced with an auxiliary variable to express the curvature function, and a constrained minimization problem equivalent to a saddle-point one was deduced. Then the alternating direction method of multipliers was applied to solve the saddle-point problem. By means of the balance principle and iterative functions, a self-adaptive rule was obtained to adjust the penalty parameter automatically, and improve the computation efficiency. The convergence of this method was proved and the penalty parameter approximation was given in detail with the iterative functions. The numerical results illustrate the effectiveness of the proposed method.
-
表 1 算法随步长变化所需迭代次数的情况
Table 1. The numbers of iterations required for the algorithm to change with the step size
ρ algorithm 1 (ADMM) algorithm 2 (SADMM) h=1/10 h=1/20 h=1/40 h=1/80 h=1/10 h=1/20 h=1/40 h=1/80 10－2 * * * * 25 28 24 28 10－1 * * * * 26 34 29 34 100 47 59 93 101 30 39 33 39 101 * * * * 32 41 35 41 102 * * * * 33 42 36 42 103 * * * * 34 43 37 43 104 * * * * 35 44 38 44 
下载: 导出CSV
表 2 算法随步长变化所需CPU时间情况
Table 2. CPU times required for the algorithm to change with the step size
ρ algorithm 1 (ADMM) algorithm 2 (SADMM) h=1/10 h=1/20 h=1/40 h=1/80 h=1/10 h=1/20 h=1/40 h=1/80 10－2 * * * * 0.117 5 0.255 1 1.911 9 80.095 4 10－1 * * * * 0.020 0 0.174 4 2.126 9 95.011 4 100 0.035 5 0.276 9 6.770 4 280.995 9 0.021 8 0.178 1 2.521 3 109.381 1 101 * * * * 0.024 8 0.168 2 2.647 1 114.965 5 102 * * * * 0.024 2 0.181 6 2.620 6 117.895 8 103 * * * * 0.024 3 0.172 3 2.791 4 120.384 8 104 * * * * 0.025 6 0.183 3 2.875 5 123.236 0
下载: 导出CSV
-
[1] CUI J T, ZHANG Y. A new analysis of discontinuous Galerkin methods for a fourth order variational inequality[J]. Computer Methods in Applied Mechanics and Engineering, 2019, 351(1): 531-547. [2] 郭楠馨, 张守贵. 自由边界问题的自适应Uzawa块松弛算法[J]. 应用数学和力学, 2019, 40(6): 682-693. doi: 10.21656/1000-0887.390347GUO Nanxin, ZHANG Shougui. Self-adaptive Uzawa block relaxation method for the free boundary problem[J]. Applied Mathematics and Mechanics, 2019, 40(6): 682-693. (in Chinese) doi: 10.21656/1000-0887.390347 [3] GUSTAFSSON T, STENBERG R, VIDEMAN J. A stabilized finite element method for the plate obstacle problem[J]. BIT Numerical Mathematics, 2019, 59(1): 97-124. doi: 10.1007/s10543-018-0728-7 [4] LI M, GUAN X, MAO S. New error estimates of the Morley element for the plate bending problems[J]. Journal of Computational and Applied Mathematics, 2014, 26(3): 405-416. http://www.cugb.edu.cn/uploadCms/file/20600/papers_upload/20140930105957094469.pdf [5] 饶玲. 单调迭代结合虚拟区域法求解非线性障碍问题[J]. 应用数学和力学, 2018, 39(4): 485-492. doi: 10.21656/1000-0887.380109RAO Ling. Monotone iterations combined with fictitious domain methods for numerical solution of nonlinear obstacle problems[J]. Applied Mathematics and Mechanics, 2018, 39(4): 485-492. (in Chinese) doi: 10.21656/1000-0887.380109 [6] 王霄婷, 龙宪军, 彭再云. 求解非单调变分不等式的一种二次投影算法[J]. 应用数学和力学, 2022, 43(8): 927-934. doi: 10.21656/1000-0887.420414WANG Xiaoting, LONG Xianjun, PENG Zaiyun. A double projection algorithm for solving non-monotone variational inequalities[J]. Applied Mathematics and Mechanics, 2022, 43(8): 927-934. (in Chinese) doi: 10.21656/1000-0887.420414 [7] AL-SAID E A, NOOR M A, KAYA D. Finite difference method for solving fourth-order obstacle problems[J]. International Journal of Computer Mathematics, 2004, 81(6): 741-748. doi: 10.1080/00207160410001661654 [8] GLOWINSKI R, MARINI L D, VIDRASCU M. Finite-element approximations and iterative solutions of a fourth-order elliptic variational inequality[J]. IMA Journal of Numerical Analysis, 1984, 4(2): 127-167. doi: 10.1093/imanum/4.2.127 [9] SHI D Y, CHEN S C, HAGIWARA I. Highly nonconforming finite element approximations for a fourth order variational inequality with curvature obstacle[J]. Journal of Systems Science and Complexity, 2005, 18(1): 136-142. http://kns.cnki.net/KCMS/detail/detail.aspx?dbcode=CJFD&filename=XTYW200501015 [10] ZHANG S G, GUO N X. Uzawa block relaxation method for free boundary problem with unilateral obstacle[J]. International Journal of Computer Mathematics, 2021, 98(4): 671-689. doi: 10.1080/00207160.2020.1777402 [11] ESSOUFI E H, KOKO J, ZAFRAR A. Alternating direction method of multiplier for a unilateral contact problem in electro-elastostatics[J]. Computers & Mathematics With Applications, 2017, 73(8): 1789-1802. http://www.sciencedirect.com/science?_ob=ShoppingCartURL&_method=add&_eid=1-s2.0-S0898122117301062&originContentFamily=serial&_origin=article&_ts=1491406629&md5=807865dba836e8cb2a960afd9226280c [12] KOKO J. Uzawa block relaxation method for the unilateral contact problem[J]. Journal of Computational and Applied Mathematics, 2011, 235(8): 2343-2356. http://www.onacademic.com/detail/journal_1000034579755210_10ce.html [13] 张茂林, 冉静, 张守贵. 具有滑动边界条件Stokes问题的自适应Uzawa块松弛算法[J]. 应用数学和力学, 2021, 42(2): 188-198. doi: 10.21656/1000-0887.410170ZHANG Maolin, RAN Jing, ZHANG Shougui. A self-adaptive Uzawa block relaxation method for Stokes problems with slip boundary conditions[J]. Applied Mathematics and Mechanics, 2021, 42(2): 188-198. (in Chinese) doi: 10.21656/1000-0887.410170 [14] ICHIRO H. Highly nonconforming finite element approximations for a fourth order variational inequality with curvature obstacle[J]. Journal of Systems Science and Complexity, 2005, 18(1): 136-142. http://kns.cnki.net/KCMS/detail/detail.aspx?dbcode=CJFD&filename=XTYW200501015 [15] CAO W, YANG D. Adaptive optimal control approximation for solving a fourth-order elliptic variational inequality[J]. Computers & Mathematics With Applications, 2014, 66(12): 2517-2531. http://www.sciencedirect.com/science?_ob=ShoppingCartURL&_method=add&_eid=1-s2.0-S0898122113005853&originContentFamily=serial&_origin=article&_ts=1437831935&md5=6cbc7dab618e6d86736c8e5eec00aa0e [16] GLOWINSKI R. Numerical Methods for Nonlinear Variational Problems[M]. Berlin: Spring-Verlag, 2008. [17] SCHOLZ R. Mixed finite element approximation of a fourth order variational inequality by the penalty method[J]. Numerical Functional Analysis and Optimization, 2007, 9(3/4): 233-247. http://www.ams.org/mathscinet-getitem?mr=887070 -
点击查看大图
图(4) / 表(2)
计量
- 文章访问数: 272
- HTML全文浏览量: 79
- PDF下载量: 45
- 被引次数: 0
