基于Drucker-Prager准则的高阶单元自适应上限有限元研究

孙锐; 阳军生; 赵乙丁; 杨峰

doi:10.11779/CJGE202002022

基于Drucker-Prager准则的高阶单元自适应上限有限元研究 English Version

中南大学土木工程学院,湖南　长沙410075

基金项目:

国家自然科学基金项目 51878669

中南大学中央高校基本科研业务费专项资金项目 2019zzts292

详细信息

作者简介:
孙锐（1993— ）,男,博士研究生,主要从事隧道与地下工程方面的研究工作。E-mail：sunruilight@163.com

通讯作者:
杨峰, E-mail：yf5754@csu.edu.cn

中图分类号: TU43
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出版历程
- 收稿日期: 2019-03-01
- 网络出版日期: 2022-12-07
- 刊出日期: 2020-01-31

Upper bound adaptive finite element method with higher-order element based on Drucker-Prager yield criterion

School of Civil Engineering, Central South University, Changsha 410075, China

摘要

摘要: 引入Drucker-Prager屈服准则,建立了基于高阶单元和二阶锥规划的自适应上限有限元方法,并编制了计算程序。采用单元内部耗散能为控制指标的自适应加密策略,以一分为二的方式对能量耗散率较大的单元进行剖分加密,形成多次往复计算完成上限有限元的自适应加密过程。通过隧道稳定性及条形基础地基承载力算例,分析了系列Drucker-Prager屈服准则对极限荷载上限解的影响,揭示该屈服准则与Mohr-Coulomb屈服准则的差异所在,以及对上限解和破坏模式的影响,并进一步验证了基于高阶单元和二阶锥规划的自适应上限有限元法具有计算精度和求解效率高以及可搜索获取精细化破坏模式的特点。
- Drucker-Prager屈服准则 /
- 上限有限元 /
- 二阶锥规划 /
- 三角形高阶单元 /
- 自适应 /
- 破坏模式
Abstract: An upper bound adaptive finite element method with six-node triangular high-order element, which is based on Drucker-Prager yield criterion, is established. Based on the upper bound theory, the corresponding calculation program is compiled. The element dissipative energy is used as the control index in the adaptive refine strategy. Based on the calculated results of element dissipative energy, the mesh is refined by dividing the element with high dissipative energy into two parts, and the upper bound finite element adaptive calculation is completed through repeated calculation based on the refined mesh. The influences of a series of Drucker-Prager yield criteria on the upper limit solution are analyzed depending on the calculated results of stability of tunnels and bearing capacity of strip footings. The calculated results also show that the proposed upper bound finite element method can achieve high accuracy, and the failure modes can be obtained by the mesh distribution.
- Drucker-Prager yield criterion /
- upper bound finite element method /
- second-order cone programming /
- higher-order triangular element /
- self-adaptation /
- failure mode

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图 1 三角形六节点高阶单元

Figure 1. Six-node strain element

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图 2 基础初始网格及边界条件

Figure 2. Initial mesh and boundary conditions of strip footing

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图 3 地基承载力相对误差

Figure 3. Relative errors of bearing capacity

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图 4 地基承载力自适应上限有限元加密后网格

Figure 4. Adaptive refined meshes of bearing capacity using finite element upper bound solution

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图 5 隧道初始网格及边界条件

Figure 5. Initial mesh and boundary conditions of tunnel

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图 6 隧道稳定性自适应上限有限元加密后网格( $C / D = 1,$ $φ =$ 20°)

Figure 6. Adaptive refined meshes of bearing capacity using finite element upper bound solution

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表 1 各准则参数换算表^[21]

Table 1 Relationship among different yield criteria^[21]

编号	准则种类	$α$	$k$
DP1	MC外角点外接圆	$\frac{2 \sin φ}{\sqrt{3} (3 - \sin φ)}$	$\frac{6 c \sin φ}{\sqrt{3} (3 - \sin φ)}$
DP2	MC内角点外接圆	$\frac{2 \sin φ}{\sqrt{3} (3 + \sin φ)}$	$\frac{6 c \sin φ}{\sqrt{3} (3 + \sin φ)}$
DP3	M-C内切圆	$\frac{\sin φ}{\sqrt{3} \sqrt{(3 + \sin^{2} φ)}}$	$\frac{\sqrt{3} \sin φ}{\sqrt{3} \sqrt{(3 + \sin^{2} φ)}}$
DP4	MC等面积圆	$\frac{2 \sqrt{3} \sin φ}{\sqrt{2 \sqrt{3} π (9 - \sin^{2} φ)}}$	$\frac{2 \sqrt{3} \sin φ}{\sqrt{2 \sqrt{3} π (9 - \sin^{2} φ)}}$
DP5	平面应变MC匹配	$\frac{\sin φ}{3}$	$c \cos φ$

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表 2 $N_{c}$ 计算结果与文献结果对比

Table 2 Comparison between present results and those available in literatures

内摩擦角	0°		10°		15°		20°		25°		30°
内摩擦角	本文方法	文献[21]	本文方法	文献[21]	本文方法	文献[21]	本文方法	文献[21]	本文方法	文献[21]	本文方法	文献[21]
DP1	59.69	60.23	116.89	121.50	181.65	192.81	316.96	362.25	664.67	891.16	2350.02	2373.60
DP2	59.69	60.23	96.26	98.59	124.48	129.91	163.26	175.00	216.34	243.13	291.25	351.88
DP3	51.53	52.19	83.08	84.98	107.88	111.90	142.48	151.75	190.84	212.08	259.96	310.00
DP4	54.10	54.81	90.70	92.94	122.35	127.51	171.65	184.90	252.06	289.28	394.98	508.83
DP5	51.53	52.19	83.72	86.20	110.22	115.51	149.25	159.75	208.38	201.19	303.23	370.48
Prandtl	51.42		83.45		109.77		148.35		207.21		301.4
注：DP1代表M-C外角点外接圆,DP2代表M-C内角点外接圆,DP3代表M-C内切圆,DP4代表M-C等面积圆,DP5代表M-C匹配DP圆。

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表 3 自重作用下圆形隧道临界破坏系数计算结果(DP5)

Table 3 Calculated results of critical failure coefficient of circular tunnel under gravity (DP5)

$C / D$	$φ$ /(°)	Yang等^[26]	Sahoo等^[25]	Yamamoto等^[24]	本文方法
1	5	2.34	—	2.33	2.29
	10	2.69	2.63	2.61	2.62
	20	3.63	3.67	—	3.52
2	5	1.8	—	1.76	1.77
	10	2.14	2.2	2.13	2.11
	20	3.13	3.28	—	3.11

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表 4 系列DP准则下隧道临界破坏系数计算结果

Table 4 Calculated results of critical failure coefficient of circular tunnel under gravity based on Drucker-Prager yield criteria

$C / D$	$φ$ /(°)	DP1	DP2	DP3	DP4	DP5
1	0	2.33	2.33	2.02	2.12	2.02
	5	2.80	2.62	2.29	2.42	2.29
	10	3.45	2.94	2.60	2.80	2.62
	20	6.00	3.76	3.40	3.90	3.52
	25	8.76	4.28	3.92	4.75	4.17
	30	13.78	4.85	4.51	5.89	4.98
	35	25.18	5.48	5.17	7.41	5.98
2	0	1.74	1.74	1.51	1.58	1.51
	5	2.18	2.03	1.77	1.88	1.77
	10	2.83	2.39	2.09	2.26	2.11
	20	5.63	3.35	3.00	3.49	3.11
	25	8.65	3.97	3.60	4.46	3.86
	30	13.76	4.67	4.31	5.75	4.80
	35	25.19	5.40	5.07	7.38	5.94

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参考文献(28)

[1]	孙聪, 李春光, 郑宏, 等. 基于单元速度泰勒展开的上限原理有限元法[J]. 岩土力学, 2016, 37(4): 1153-1160. https://www.cnki.com.cn/Article/CJFDTOTAL-YTLX201604031.htm SUN Cong, LI Chun-guang, ZHENG Hong, et al. Upper bound limit analysis based on Taylor expansion form of element velocity[J]. Rock and Soil Mechanics, 2016, 37(4): 1153-1160. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YTLX201604031.htm
[2]	赵明华, 胡啸, 张锐. 临坡地基承载力极限分析上限有限元数值模拟[J]. 岩土力学, 2016, 37(4): 1137-1143. https://www.cnki.com.cn/Article/CJFDTOTAL-YTLX201604029.htm ZHAO Ming-hua, HU Xiao, ZHANG Rui. Numerical simulation of the bearing capacity of a foundation near slope using the upper bound finite element method[J]. Rock and Soil Mechanics, 2016, 37(4): 1137-1143. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YTLX201604029.htm
[3]	ZHANG J, YANG F, YANG J, et al. Upper-bound stability analysis of dual unlined elliptical tunnels in cohesive- frictional soils[J]. Computers and Geotechnics, 2016, 80: 283-289. doi: 10.1016/j.compgeo.2016.08.023
[4]	BOTTERO A, NEGRE R, PASTOR J, et al. Finite element method and limit analysis theory for soil mechanics problems[J]. Computer Methods in Applied Mechanics & Engineering, 1980, 22(1): 131-149.
[5]	SLOAN S W, KLEEMAN P W. Upper bound limit analysis with discontinuous velocity fields[J]. Computer Methods in Applied Mechanics & Engineering, 1995, 127(1): 293-314.
[6]	杨小礼, 李亮, 刘宝琛. 大规模优化及其在上限定理有限元中的应用[J]. 岩土工程学报, 2001, 23(5): 602-605. https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC200105017.htm YANG Xiao-li, LI Liang, LIU Bao-chen. Large-scale optimization and its application to upper bound theorem using kinematical element method[J]. Chinese Journal of Geotechnical Engineering, 2001, 23(5): 602-605. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC200105017.htm
[7]	杨峰, 阳军生, 李昌友, 等. 基于六节点三角形单元和线性规划模型的上限有限元研究[J]. 岩石力学与工程学报, 2012, 31(12): 2556-2563. https://www.cnki.com.cn/Article/CJFDTOTAL-YSLX201212021.htm YANG Feng, YANG Jun-sheng, LI Chang-you, et al. Investigation of finite element upper bound solution based on six nodal triangular elements and linear programming model[J]. Chinese Journal of Rock Mechanics and Engineering, 2012, 31(12): 2556-2563. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YSLX201212021.htm
[8]	YU H S, SLOAN S W, KLEEMAN P W. A quadratic element for upper bound limit analysis[J]. Engineering Computations, 1994, 11(3): 195-212. doi: 10.1108/02644409410799281
[9]	SLOAN S W. A steepest edge active set algorithm for solving sparse linear programming problems[J]. International Journal for Numerical Methods in Engineering, 1988, 26: 2671-2685. doi: 10.1002/nme.1620261207
[10]	SLOAN S W. Upper bound limit analysis using finite element and linear programming[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 1989, 13: 263-282. doi: 10.1002/nag.1610130304
[11]	MAKRODIMOPOULOS A, MARTIN C M. Upper bound limit analysis using simplex strain elements and secondorder-cone programming[J]. International Journal for Numerical & Analytical Methods in Geomechanics, 2007, 31(6): 835-865.
[12]	NGUYEN-THOI T, PHUNG-VAN P, NGUYEN-THOI M H, et al. An upper-bound limit analysis of Mindlin plates using CS-DSG3 method and second-order cone programming[J]. Journal of Computational & Applied Mathematics, 2015, 281(C): 32-48.
[13]	杨昕光, 周密, 张伟, 等. 基于二阶锥规划的边坡稳定上限有限元分析[J]. 长江科学院院报, 2016, 33(12): 61-67. https://www.cnki.com.cn/Article/CJFDTOTAL-CJKB201612013.htm YANG Xin-guang, ZHOU Mi, ZHANG Wei, et al. Upper bound finite element limit analysis of slope stability using second-order cone programming[J]. Journal of Yangtze River Scientific Research Institute, 2016, 33(12): 61-67. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-CJKB201612013.htm
[14]	SUÁREZ C, HÉCTOR. Computation of Upper and Lower Bounds in Limit Analysis Using Second-order Cone Programming and Mesh Adaptivity[R]. Massachusetts: Massachusetts Institute of Technology, 2002.
[15]	赵明华, 张锐. 有限元上限分析网格自适应方法及其工程应用[J]. 岩土工程学报, 2016, 38(3): 537-545. https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC201603021.htm ZHAO Ming-hua, ZHANG Rui. Adaptive mesh refinement of upper bound finite element method and its applications in geotechnical engineering[J]. Chinese Journal of Geotechnical Engineering, 2016, 38(3): 537-545. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC201603021.htm
[16]	阳军生, 张箭, 杨峰. 浅埋隧道掌子面稳定性二维自适应上限有限元分析[J]. 岩土力学, 2015, 36(1): 257-264. https://www.cnki.com.cn/Article/CJFDTOTAL-YTLX201501035.htm YANG Jun-sheng, ZHANG Jian, YANG Feng. Stability analysis of shallow tunnel face using two-dimensional finite element upper bound solution with mesh adaptation[J]. Rock and Soil Mechanics, 2015, 36(1): 257-264. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YTLX201501035.htm
[17]	杨峰, 阳军生. 用于上限有限元的非结构网格重划加密方法研究[J]. 中南大学学报(自然科学版), 2014(10): 3571-3577. https://www.cnki.com.cn/Article/CJFDTOTAL-ZNGD201410033.htm YANG Feng, YANG Jun-sheng. Investigation of unstructured mesh regeneration and Refinement method for finite element upper bound solution[J]. Journal of Central South University (Science and Technology), 2014(10): 3571-3577. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-ZNGD201410033.htm
[18]	MUÑOZ J J, BONET J, HUERTA A, et al. Upper and lower bounds in limit analysis: Adaptive meshing strategies and discontinuous loading[J]. International Journal for Numerical Methods in Engineering, 2009, 77(4): 471-501.
[19]	杨雪强, 凌平平, 向胜华. 基于系列Drucker-Prager破坏准则评述土坡的稳定性[J]. 岩土力学, 2009, 30(4): 865-870. https://www.cnki.com.cn/Article/CJFDTOTAL-YTLX200904002.htm YANG Xue-qiang, LIN Ping-ping, XIANG Sheng-hua. Comments on slope stability based on a series of Drucker-Prager failure criteria[J]. Rock and Soil Mechanics, 2009, 30(4): 865-870. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YTLX200904002.htm
[20]	王渭明, 赵增辉, 王磊. 不同强度准则下软岩巷道底板破坏安全性比较分析[J]. 岩石力学与工程学报, 2012, 31(增刊2): 3920-3927. https://www.cnki.com.cn/Article/CJFDTOTAL-YSLX2012S2063.htm WANG Wei-ming, ZHAO Zeng-hui, WANG Lei. Safety analysis for soft rock tunnel floor destruction based on different yield criterions[J]. Chinese Journal of Rock Mechanics and Engineering, 2012, 31(S2): 3920-3927. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YSLX2012S2063.htm
[21]	邓楚键, 何国杰, 郑颖人. 基于M-C准则的D-P系列准则在岩土工程中的应用研究[J]. 岩土工程学报, 2006, 28(6): 735-739. https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC200606011.htm DENG Chu-jian, HE Guo-jie, ZHENG Ying-ren. Studies on Drucker-Prager yield criterions based on M-C yield criterion and application in geotechnical engineering[J]. Chinese Journal of Geotechnical Engineering, 2012, 28(6): 735-739. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC200606011.htm
[22]	王先军, 陈明祥, 常晓林, 等. Drucker-Prager系列屈服准则在稳定分析中的应用研究[J]. 岩土力学, 2009, 30(12): 3733-3738. https://www.cnki.com.cn/Article/CJFDTOTAL-YTLX200912032.htm WANG Xian-jun, CHEN Ming-xiang, CHANG Xiao-lin, et al. Studies of application of Drucker- Prager yield criteria to stability analysis[J]. Rock and Soil Mechanics, 2009, 30(12): 3734-3738. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YTLX200912032.htm
[23]	MAKRODIMOPOULOS A, MARTIN C M. Upper bound limit analysis using discontinuous quadratic displacement fields[J]. Communications in Numerical Methods in Engineering, 2008, 24: 911-927.
[24]	YAMAMOTO K, LYAMIN A V, WILSON D W, et al. Stability of a circular tunnel incohesive-frictional soil subjected to surcharge loading[J]. Computers and Geotechnics, 2011, 38: 504-514.
[25]	SAHOO J P, KUMAR J. Stability of long unsupported twin circular tunnels insoils[J]. Tunnelling and Underground Space Technology, 2013, 38: 326-335.
[26]	YANG F, ZHANG J, YANG J, et al. Stability analysis of unlined elliptical tunnel using finite element upper-bound method with rigid translatory moving elements[J]. Tunnelling and Underground Space Technology, 2015, 50: 13-22.
[27]	赵明华, 张锐, 雷勇, 等. 基于可行弧内点算法的上限有限单元法优化求解[J]. 岩土工程学报, 2014, 36(4): 604-611. https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC201404003.htm ZHAO Ming-hua, ZHANG Rui, LEI Yong, et al. Optimization of upper bound finite element method based on feasible arc interior point algorithm[J]. Chinese Journal of Geotechnical Engineering, 2014, 36(4): 604-611. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC201404003.htm
[28]	赵明华, 张锐, 刘猛. 下限分析有限单元法的非线性规划求解[J]. 岩土力学, 2015, 36(12): 3589-3597. https://www.cnki.com.cn/Article/CJFDTOTAL-YTLX201512032.htm ZHAO Ming-hua, ZHANG Rui, LIU Meng. Nonlinear programming of lower bound finite element method[J]. Rock and Soil Mechanics, 2015, 36(12): 3589-3597. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YTLX201512032.htm