无摩擦弹性接触问题的自适应交替方向乘子法

袁欣; 张守贵

doi:10.21656/1000-0887.440079

无摩擦弹性接触问题的自适应交替方向乘子法

doi: 10.21656/1000-0887.440079

袁欣,
张守贵^,

重庆师范大学数学科学学院, 重庆 401331

基金项目:

国家自然科学基金项目 11971085

重庆市自然科学基金项目 cstc2020jcyj-msxmX0066

重庆市研究生教育教学改革研究项目 yjg213071

详细信息

作者简介:
袁欣(1997—), 女, 硕士生(E-mail: 1577279037@qq.com)

通讯作者:
张守贵(1973—), 男, 教授, 博士, 硕士生导师(通讯作者. E-mail: shgzhnag@cqnu.edu.cn)

中图分类号: O221.6
计量
- 文章访问数: 270
- HTML全文浏览量: 90
- PDF下载量: 40
- 被引次数: 0
出版历程
- 收稿日期: 2023-03-24
- 修回日期: 2023-04-24
- 刊出日期: 2023-08-01

A Self-Adaptive Alternating Direction Multiplier Method for Frictionless Elastic Contact Problems

YUAN Xin,
ZHANG Shougui^,

School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, P.R.China

摘要

摘要: 对一类无摩擦的弹性接触问题，得到了求其数值解的自适应交替方向乘子法.由该问题导出相应的变分问题，引入辅助变量将原问题转化为一个基于增广Lagrange函数表示的鞍点问题，并采用交替方向乘子法求解；为了提高算法性能，提出了利用边界迭代函数自动选取合适罚参数的自适应法则. 该算法的优点是每次迭代只需计算一个线性变分问题，同时显式计算了辅助变量和Lagrange乘子. 对算法的收敛性进行了理论分析，最后用数值结果验证了该算法的可行性和有效性.
- 弹性接触问题 /
- 交替方向乘子法 /
- 自适应法则 /
- 增广Lagrange
Abstract: A self-adaptive alternating direction multiplier method was designed for frictionless elastic contact problems. An augmented Lagrange function was introduced for the variational formulation of the problem with an auxiliary variable, to deduce a minimization problem and an equivalent saddle-point problem. Then the alternating direction multiplier method was used to solve the problem. To enhance the performance of the algorithm, a self-adaptive rule based on the iterative function on the boundary was proposed to automatically select the proper penalty parameter. The advantage of this algorithm is that, each iteration only needs to solve a linear variational problem and explicitly calculate the auxiliary variable and the Lagrange multiplier. The convergence of the algorithm was analyzed theoretically. The numerical results illustrate the feasibility and effectiveness of the proposed method.
- elastic contact problem /
- alternating direction multiplier method /
- self-adaptive rule /
- augmented Lagrange function

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图 1 形变后的弹性体及障碍函数(算例1)

Figure 1. The deformed configuration and the obstacle function (example 1)

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图 2 g=0.01时的算法迭代次数

Figure 2. The number of iterations for each obstacle function (example 1) method with g=0.01

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图 3 形变后的弹性体及障碍函数(算例2)

Figure 3. The deformed configuration and the obstacle

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图 4 各算法的迭代次数

Figure 4. The number of iterations for each method function (example 2)

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表 1 g=0.01时3种算法的CPU运行时间

Table 1. CPU time for 3 methods with g=0.01

ρ	ADMM1				ADMM2				SADMM
ρ	$h=\frac{1}{10}$	$h=\frac{1}{20}$	$h=\frac{1}{30}$	$h=\frac{1}{40}$	$h=\frac{1}{10}$	$h=\frac{1}{20}$	$h=\frac{1}{30}$	$h=\frac{1}{40}$	$h=\frac{1}{10}$	$h=\frac{1}{20}$	$h=\frac{1}{30}$	$h=\frac{1}{40}$
1	3.080	9.305	29.344	37.788	5.213	12.865	24.283	41.639	0.562	1.171	2.617	4.214
10	2.670	8.665	34.726	37.629	5.137	12.331	23.814	40.909	0.454	1.135	2.342	3.891
10²	0.611	2.890	6.224	10.178	1.474	4.169	7.156	11.393	0.610	0.984	2.145	3.632
10³	0.229	0.896	1.842	3.765	0.577	1.223	1.978	3.943	0.425	0.650	1.321	2.516
10⁴	0.306	1.223	2.905	5.742	0.502	1.228	2.479	4.286	0.409	0.743	1.496	2.699

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表 2 3种算法的CPU运行时间

Table 2. CPU time for 3 methods

ρ	ADMM1				ADMM2				SADMM
ρ	$h=\frac{1}{10}$	$h=\frac{1}{20}$	$h=\frac{1}{30}$	$h=\frac{1}{40}$	$h=\frac{1}{10}$	$h=\frac{1}{20}$	$h=\frac{1}{30}$	$h=\frac{1}{40}$	$h=\frac{1}{10}$	$h=\frac{1}{20}$	$h=\frac{1}{30}$	$h=\frac{1}{40}$
1	4.114	13.733	35.051	54.791	4.008	13.227	30.753	53.383	0.378	1.313	3.129	7.029
10	3.827	13.505	39.201	65.841	3.792	12.91	29.465	52.564	0.381	1.227	2.982	6.649
10²	1.565	5.294	16.247	26.196	1.256	5.132	12.322	22.394	0.314	1.116	2.645	5.985
10³	0.933	2.552	7.530	12.051	0.872	2.386	5.276	8.389	0.238	0.786	1.932	5.017
10⁴	0.699	2.059	6.003	10.109	0.384	1.306	3.059	5.449	0.315	0.948	2.313	3.849

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