基于二阶锥规划理论的有限元强度折减法及应用
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摘要
针对岩土体稳定性问题,常用的方法有极限平衡法和有限元强度折减法等。传统的有限元强度折减法通常需设置很大的最大允许非线性迭代次数(如200或500),计算耗时严重,此外,采用的平衡迭代和应力积分算法可能导致岩土体塑性区计算不够准确,进而影响稳定性分析结果。提出一种二阶锥规划有限元强度折减法,该方法基于Hellinger–Reissner混合变分原理和有限元法,将岩土体弹塑性问题构造成基于有限元框架的二阶锥规划问题,结合强度折减技术来分析岩土体稳定性。将该新方法应用于平面应变岩土体稳定性分析,结果表明:与传统的有限元强度折减法相比,新方法结果可靠,但其计算效率更高,所获得的塑性区更加平滑。
For geotechnical stability problems, the limit equilibrium method(LEM) and shear strength reduction finite element method(SSRFEM) have been commonly used. In the traditional elasto-plastic finite element method, a large maximum allowable number of nonlinear iterations(such as 200 or 500) are often set in the SSRFEM, so that the calculation is generally time-consuming; besides, the equilibrium iteration and stress integration algorithm may probably lead to inaccurate calculation of plastic zone and stability. Based on the Hellinger-Reissner mixed variational principle and finite element method, a new shear strength reduction finite element method is proposed based on the finite element method of second-order cone programming(FEM-SOCP). In the mathematical programming finite element framework, the elasto-plastic finite element problem can be cast into a form of second-order cone programming(SOCP), and when being utilized in conjunction with the strength reduction technique, the resultant approach named SSRFEM-SOCP can be applied to geotechnical stability analysis.When being applied to plane strain problems, it is observed that SSRFEM-SOCP is reliable and efficient, and particularly the plastic zone attained by the SSRFEM-SOCP is generally smoother than that by the conventional SSRFEM method.
For geotechnical stability problems, the limit equilibrium method(LEM) and shear strength reduction finite element method(SSRFEM) have been commonly used. In the traditional elasto-plastic finite element method, a large maximum allowable number of nonlinear iterations(such as 200 or 500) are often set in the SSRFEM, so that the calculation is generally time-consuming; besides, the equilibrium iteration and stress integration algorithm may probably lead to inaccurate calculation of plastic zone and stability. Based on the Hellinger-Reissner mixed variational principle and finite element method, a new shear strength reduction finite element method is proposed based on the finite element method of second-order cone programming(FEM-SOCP). In the mathematical programming finite element framework, the elasto-plastic finite element problem can be cast into a form of second-order cone programming(SOCP), and when being utilized in conjunction with the strength reduction technique, the resultant approach named SSRFEM-SOCP can be applied to geotechnical stability analysis.When being applied to plane strain problems, it is observed that SSRFEM-SOCP is reliable and efficient, and particularly the plastic zone attained by the SSRFEM-SOCP is generally smoother than that by the conventional SSRFEM method.
引文
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[16]张雪,盛岱超.一种模拟土体流动的连续体数值方法[J].岩土工程学报,2016,38(3):562-569.(ZHANG Xue,SHENG Dai-chao.Continuum approach for modelling soil flow in geotechnical engineering[J].Chinese Journal of Geotechnical Engineering,2016,38(3):562-569.(in Chinese))
[17]ZHOU T.Stabilized hybrid finite element methods based on the combination of saddle point principles of elasticity problems[J].Mathematics of Computation,2003,72(244):1655-1673.
[18]KRABBENH?FT K,LYAMIN A V,SLOAN S W.Formulation and solution of some plasticity problems as conic programs[J].International Journal of Solids and Structures,2007,44(5):1533-1549.
[19]MOSEK A S.The MOSEK C optimizer API manual,version8.0[OL].http://docs.mosek.com/8.0/capi/index.html,2016-12-25.
[20]DONALD I B,GIAM P S K.The ACADS Slope Stability Programs Review[C]//6th International Symposium on Landslides.Christchurch,1992.
[21]DONALD I B,GIAM P S K.Soil slope stability programs review[C]//ACADS Publication No.U255.Melbourne,1989.
[2]GRIFFITHS D V,LANE P A.Slope stability analysis by finite elements[J].Géotechnique,1999,49(3):387-403.
[3]连镇营,韩国城,孔宪京.强度折减有限元法研究开挖边坡的稳定性[J].岩土工程学报,2001,23(4):406-411.(LIAN Zhen-ying,HAN Guo-cheng,KONG Xian-jing.Stability analysis of excavation by strength reduction FEM[J].Chinese Journal of Geotechnical Engineering,2001,23(4):406-411.(in Chinese))
[4]赵尚毅,郑颖人,时卫民,等.用有限元强度折减法求边坡稳定安全系数[J].岩土工程学报,2002,24(3):343-346.(ZHAO Shang-yi,ZHENG Ying-ren,SHI Wei-min,et al.Analysis on safety factor of slope by strength reduction FEM[J].Chinese Journal of Geotechnical Engineering,2002,24(3):343-346.(in Chinese))
[5]林鸿州,于玉贞,李广信,等.强度折减有限元法在滑坡特性预测的应用探讨[J].岩土工程学报,2009,31(2):229-233.(LIN Hong-chou,YU Yu-zhen,LI Guang-xin,et al.Finite element method with consideration shear strength reduction for prediction of landslide[J].Chinese Journal of Geotechnical Engineering,2009,31(2):229-233.(in Chinese))
[6]陈曦,张训维,苗姜龙,等.土体剪胀特性对土质边坡体系稳定性的影响[J].重庆交通大学学报(自然科学版),2017,36(1):52-57.(CHEN Xi,ZHANG Xun-wei,MIAOJiang-long,et al.Stability analysis and evaluation of soil slope system considering the shear dilatancy effect[J].Journal of Chongqing Jiaotong University(Natural Science),2017,36(1):52-57.(in Chinese))
[7]DAVIS E H.Theories of plasticity and failure of soil masses[M].London:Butterworths,1968:341-380.
[8]TSCHUCHNIGG F,SCHWEIGER H F,SLOAN S W.Slope stability analysis by means of finite element limit analysis and finite element strength reduction techniques:part Inumerical studies considering non-associated plasticity[J].Computers and Geotechnics,2015,70:169-177.
[9]陈曦,刘春杰.有限元强度折减法中安全系数的搜索算法[J].岩土工程学报,2010,28(9):1443-1447.(CHEN Xi,LIU Chun-jie.Search algorithms for safety factor in finite element shear strength reduction method[J].Chinese Journal of Geotechnical Engineering,2010,28(9):1443-1447.(in Chinese))
[10]CHEN X,WU Y,YU Y,et al.A two-grid search scheme for large-scale 3-D finite element analyses of slope stability[J].Computers and Geotechnics,2014,62:203-215.
[11]CHEN X,JIE Y,LIU J.Robust partitioned block preconditioners for large-scale geotechnical applications with soil-structure interactions[J].International Journal for Numerical and Analytical Methods in Geomechanics,2014,38(1):72-91.
[12]杨小礼.线性与非线性破坏准则下岩土极限分析方法及其应用[D].长沙:中南大学,2002.(YANG Xiao-li.Limit analysis method and its application to geotechnical engineering with linear and nonlinear failure criteria[D].Changsha:Central South University,2002.(in Chinese))
[13]殷建华,陈健,李焯芬.岩土边坡稳定性的刚体有限元上限分析法[J].岩石力学与工程学报,2004,23(6):898-905.(YIN Jian-hua,CHEN Jian,LEE C F.Ultimate limit analysis of stability of rock and soil slopes using rigid finite elements[J].Chinese Journal of Rock Mechanics and Engineering,2004,23(6):898-905.(in Chinese))
[14]WANG D,CHEN X,FAN Q,et al.Comparison of discontinuity layout optimization and finite element optimization for plane plasticity problems[C]//The 15th International Conference of the International Association for Computer Methods and Advances in Geomechanics.Wuhan,2017.
[15]TANG C,PHOON K K,TOH K C.Lower-bound limit analysis of seismic passive earth pressure on rigid walls[J].International Journal of Geomechanics,2014,14(5):04014022.
[16]张雪,盛岱超.一种模拟土体流动的连续体数值方法[J].岩土工程学报,2016,38(3):562-569.(ZHANG Xue,SHENG Dai-chao.Continuum approach for modelling soil flow in geotechnical engineering[J].Chinese Journal of Geotechnical Engineering,2016,38(3):562-569.(in Chinese))
[17]ZHOU T.Stabilized hybrid finite element methods based on the combination of saddle point principles of elasticity problems[J].Mathematics of Computation,2003,72(244):1655-1673.
[18]KRABBENH?FT K,LYAMIN A V,SLOAN S W.Formulation and solution of some plasticity problems as conic programs[J].International Journal of Solids and Structures,2007,44(5):1533-1549.
[19]MOSEK A S.The MOSEK C optimizer API manual,version8.0[OL].http://docs.mosek.com/8.0/capi/index.html,2016-12-25.
[20]DONALD I B,GIAM P S K.The ACADS Slope Stability Programs Review[C]//6th International Symposium on Landslides.Christchurch,1992.
[21]DONALD I B,GIAM P S K.Soil slope stability programs review[C]//ACADS Publication No.U255.Melbourne,1989.
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