Axisymmetric adaptive lower bound finite element method based on Mohr-Coulomb yield criterion and second-order cone programming
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摘要: 轴对称Mohr-Coulomb准则屈服面的角点问题导致其在数值计算中存在困难，如何高效处理该屈服准则一直是极限分析下限有限元法的重要内容。首先，引入完全塑性假定将轴对称Mohr-Coulomb准则转化为1组不等式约束和3个线性等式约束；然后，将不等式约束直接转化为二阶锥约束，避免了对角点进行光滑近似处理；最后，将基于Mohr-Coulomb准则的轴对称极限分析下限有限元计算模型转化为具有较高计算效率的二阶锥规划数学优化模型。极限分析下限有限元法采用的线性应力单元难以精确模拟破坏区域的应力变化，单元的分布形式对计算精度存在较大影响。因此提出一种基于单元应力的网格自适应加密策略，通过判断单元内节点应力接近屈服的程度，自动识别破坏区域待加密的单元，实现对破坏区域应力分布的精确模拟，进而能够以较少单元获得高精度下限解。通过分析圆形基础承载力及竖向锚板极限抗拔力等典型轴对称岩土工程稳定性问题，表明了所提方法具有较高计算效率及计算精度，具有一定的理论价值和应用前景。
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关键词:
- 下限有限元 /
- 自适应加密 /
- 轴对称 /
- 二阶锥规划 /
- Mohr-Coulomb准则
Abstract: The axisymmetric Mohr Coulomb yield surfaces have edges and corners in the three-dimensional stress space, which leads to difficulties in numerical calculation. Therefore, how to deal with the axisymmetric Mohr-Coulomb criterion efficiently has always been an important research content of the lower bound finite element limit analysis (LB-FELA) method. Firstly, the axisymmetric Mohr-Coulomb criterion is transformed into a set of inequality constraints and three linear equality constraints by introducing the complete plasticity assumption. Then, the inequality constraints are directly transformed into the second-order cone ones, which avoids the smooth approximation of numerical singularities. Finally, the axisymmetric LB-FELA model based on the Mohr Coulomb criterion is transformed into a mathematical optimization one of the second-order cone programming. The linear stress element adopted by the axisymmetric LB-FELA cannot accurately simulate the stress change in the failure region. Therefore, the mesh distribution form has a great impact on the calculation accuracy of the LB-FELA. To solve this problem, an adaptive mesh refinement strategy based on the element yield residual is proposed. By judging the degree of the node stress in the element close to the yield, the elements can be automatically identified and refined in the failure area. By analyzing the stability problems of typical axisymmetric geotechnical projects such as the bearing capacity of circular foundation and the ultimate uplift capacity of vertical anchor plate, it is shown that the proposed method has high calculation efficiency and accuracy, and it has certain theoretical value and application prospect. -
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图 3 圆形基础地基承载力下限有限元加密网格(接触光滑)
Figure 3. Adaptive refined meshes of bearing capacity using lower bound finite element method with smooth interface
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图 4 圆形基础地基承载力下限有限元加密网格(接触粗糙)
Figure 4. Adaptive refined meshes of bearing capacity using lower bound finite element method with rough interface
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图 5 网格自适应加密次数对地基承载力系数的影响
Figure 5. Effects of mesh refinement times on bearing capacity coefficient
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图 6 圆形锚板竖向抗拔力问题初始网格及边界条件
Figure 6. Initial mesh and boundary conditions for vertical uplift of circular anchor plate
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图 7 圆形锚板竖向极限抗拔力问题加密网格（局部破坏区域）
Figure 7. Refinement meshes for vertical uplift of circular anchor plates
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图 8 计算结果与文献结果对比
Figure 8. Comparison between present results and those available in Reference[1]
表 1 计算效率对比(地基与基础直径接触粗糙)
Table 1 Comparison of calculation efficiencies between present method and those available in Reference [18]
方法 单元数 节点数 优化变量数 锥约束数量 线性约束数量 承载力系数 计算误差/% 计算时长/s 承载力系数 计算误差/% 计算时长/s 本文方法 22846 11592 479766 68538 593794 7.73 3.37 31 22.35 3.74 37 文献[18] 22846 11592 890994 205614 799408 7.74 3.25 68 22.37 3.67 87 
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表 2 Nc计算结果与文献结果对比
Table 2 Comparison between present results and those available in references
本文方法 Kumar等[5] Zhang等[25] Martin等[8] 5 8.03(7.41) 8.00(7.31) 8.00(7.47) 8.06(7.43) 10 11.04(9.94) 10.99(9.78) 11.16(10.05) 11.09(9.99) 15 15.76(13.86) 15.66(13.51) 16.03(13.99) 15.84(13.87) 20 23.53(19.94) 23.22(19.38) 24.02(20.32) 23.67(20.07) 25 36.98(30.25) 36.17(29.06) 37.96(30.97) 37.31(30.52) 30 61.92(48.78) 61.48(47.10) 64.27(50.48) 62.70(49.29) 35 112.07(84.59) 112.47(81.47) 117.99(89.01) 113.99(85.88) 注：括号外为接触粗糙时地基承载力系数，括号内为接触光滑时的承载力系数；自适应加密次数均为15次。
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