一种非等阶积分的求积元孔压单元（英文）
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摘要
目的:在对土体固结问题进行求解时,如果对孔压和位移采用等阶积分,当时间步长取值很小时,得到的孔压结果会出现数值振荡,导致精度降低。本文提出一种非等阶积分的求积元孔压单元,致力于减小孔压振荡,提高单元稳定性和计算精度。创新点:1.建立了一种对于孔压和位移采用不同阶积分的求积元复合单元;2.有效降低了小时间步长引起的孔压振荡。方法:1.通过求积元法对弱形式控制方程进行数值积分,并对比奥固结方程中的位移项和孔压项采用不等阶积分;2.通过拉格朗日插值获得位移点上的孔压和孔压点上的位移值;3.通过一维及二维问题的数值算例,验证所建立方法的有效性。结论:1.通过采用复合求积元单元,大大降低了孔压的数值振荡;2.应用复合求积单元时,采用二阶精度的Crank-Nicolson积分格式不会产生随时间的孔压振荡;3.本文所建立的单元可以大大提高固结问题求解的数值计算精度和计算效率。
The weak form quadrature element method is a high-order algorithm which has been applied successfully to consolidation analysis of saturated and unsaturated soils. Its superiority over the conventional finite element method has been verified. However, in consolidation analysis, pore pressure oscillations will appear when small time increments are used due to the very small permeability and near-incompressibility of the pore water. This can produce almost zero values of main diagonal elements in the coefficient matrix. To overcome the pressure oscillations, we propose a coupled composite quadrature element in which different orders of integration are employed for the pore pressure and the displacement. Its performance is evaluated and compared with that of the standard element through 1 D and 2 D numerical tests. Our results show that pressure oscillations can be effectively alleviated and stability and accuracy can be significantly enhanced by using the proposed element.
The weak form quadrature element method is a high-order algorithm which has been applied successfully to consolidation analysis of saturated and unsaturated soils. Its superiority over the conventional finite element method has been verified. However, in consolidation analysis, pore pressure oscillations will appear when small time increments are used due to the very small permeability and near-incompressibility of the pore water. This can produce almost zero values of main diagonal elements in the coefficient matrix. To overcome the pressure oscillations, we propose a coupled composite quadrature element in which different orders of integration are employed for the pore pressure and the displacement. Its performance is evaluated and compared with that of the standard element through 1 D and 2 D numerical tests. Our results show that pressure oscillations can be effectively alleviated and stability and accuracy can be significantly enhanced by using the proposed element.
引文
Aguilar G,Gaspar F,Lisbona F,et al.,2008.Numerical stabilization of Biot’s consolidation model by a perturbation on the flow equation.International Journal for Numerical Methods in Engineering,75(11):1282-1300.https://doi.org/10.1002/nme.2295
Azari B,Fatahi B,Khabbaz H,2015.Numerical analysis of vertical drains accelerated consolidation considering combined soil disturbance and visco-plastic behaviour.Geomechanics&Engineering,8(2):187-220.https://doi.org/10.12989/gae.2015.8.2.187
Bellman R,Casti J,1971.Differential quadrature and longterm integration.Journal of Mathematical Analysis and Applications,34(2):235-238.https://doi.org/10.1016/0022-247X(71)90110-7
Biot MA,1941.General theory of three-dimensional consolidation.Journal of Applied Physics,12(2):155-164.https://doi.org/10.1063/1.1712886
Boal N,Gaspar FJ,Lisbona FJ,et al.,2011.Finite-difference analysis of fully dynamic problems for saturated porous media.Journal of Computational and Applied Mathematics,236(6):1090-1102.https://doi.org/10.1016/j.cam.2011.07.032
Chen R,Zhou W,Wang H,et al.,2005.One-dimensional nonlinear consolidation of multi-layered soil by differential quadrature method.Computers and Geotechnics,32(5):358-369.https://doi.org/10.1016/j.compgeo.2005.05.003
Darkshanamurthy V,Fredlund DG,Rahardjo H,1984.Coupled three-dimensional consolidation theory of unsaturated porous media.Fifth International Conference on Expansive Soils,p.99.
Fei K,Liu HL,2009.Implementation and application of bounding surface model in ABAQUS.Journal of PLAUniversity of Science and Technology(Natural Science Edition),10(5):447-451(in Chinese).
Gaspar FJ,Lisbona FJ,Vabishchevich PN,2003.A finite difference analysis of Biot’s consolidation model.Applied Numerical Mathematics,44(4):487-506.https://doi.org/10.1016/S0168-9274(02)00190-3
Gaspar FJ,Lisbona FJ,Vabishchevich PN,2006.Staggered grid discretizations for the quasi-static Biot’s consolidation problem.Applied Numerical Mathematics,56(6):888-898.https://doi.org/10.1016/j.apnum.2005.07.002
Gaspar FJ,Lisbona FJ,Matus P,et al.,2016.Numerical methods for a one-dimensional non-linear Biot’s model.Journal of Computational and Applied Mathematics,293:62-72.https://doi.org/10.1016/j.cam.2015.03.039
Ho L,Fatahi B,2015.Analytical solution for the twodimensional plane strain consolidation of an unsaturated soil stratum subjected to time-dependent loading.Computers&Geotechnics,67:1-16.https://doi.org/10.1016/j.compgeo.2015.02.011
Ho L,Fatahi B,Khabbaz H,2016.Analytical solution to axisymmetric consolidation in unsaturated soils with linearly depth-dependent initial conditions.Computers&Geotechnics,74:102-121.https://doi.org/10.1016/j.compgeo.2015.12.019
Hua L,2011.Stable element-free Galerkin solution procedures for the coupled soil-pore fluid problem.International Journal for Numerical Methods in Engineering,86(8):1000-1026.https://doi.org/10.1002/nme.3087
Le TM,Fatahi B,2016.Trust-region reflective optimisation to obtain soil visco-plastic properties.Engineering Computations,33(2):410-442.https://doi.org/10.1108/EC-11-2014-0236
Le TM,Fatahi B,Khabbaz H,et al.,2016.Numerical optimisation applying trust-region reflective least squares algorithm with constraints to optimise the non-linear creep parameters of soft soil.Applied Mathematical Modelling,16:1-21.
Manzari MT,Dafalias YF,1997.A critical state two-surface plasticity model for sands.Geotechnique,47(2):255-272.https://doi.org/10.1680/geot.1997.47.2.255
Mira P,Pastor M,Li T,et al.,2003.A new stabilized enhanced strain element with equal order of interpolation for soil consolidation problems.Computer Methods in Applied Mechanics and Engineering,192(37-38):4257-4277.https://doi.org/10.1016/S0045-7825(03)00416-X
Preisig M,Prévost JH,2011.Stabilization procedures in coupled poromechanics problems:a critical assessment.International Journal for Numerical and Analytical Methods in Geomechanics,35(11):1207-1225.https://doi.org/10.1002/nag.951
Reed M,1984.An investigation of numerical errors in the analysis of consolidation by finite elements.International Journal for Numerical and Analytical Methods in Geomechanics,8(3):243-257.https://doi.org/10.1002/nag.1610080304
Rodrigo C,Gaspar F,Hu X,et al.,2016.Stability and monotonicity for some discretizations of the Biot’s consolidation model.Computer Methods in Applied Mechanics and Engineering,298:183-204.https://doi.org/10.1016/j.cma.2015.09.019
Samimi S,Pak A,2012.Three-dimensional simulation of fully coupled hydro-mechanical behavior of saturated porous media using element free Galerkin(EFG)method.Computers&Geotechnics,46(4):75-83.https://doi.org/10.1016/j.compgeo.2012.06.004
Sandhu RS,Liu H,Singh KJ,1977.Numerical performance of some finite element schemes for analysis of seepage in porous elastic media.International Journal for Numerical and Analytical Methods in Geomechanics,1(2):177-194.https://doi.org/10.1002/nag.1610010205
Simo JC,Rifai M,1990.A class of mixed assumed strain methods and the method of incompatible modes.International Journal for Numerical Methods in Engineering,29(8):1595-1638.https://doi.org/10.1002/nme.1620290802
Wang W,Wang J,Wang Z,et al.,2007.An unequal-order radial interpolation meshless method for Biot’s consolidation theory.Computers and Geotechnics,34(2):61-70.https://doi.org/10.1016/j.compgeo.2006.10.010
Wang X,Gu H,1997.Static analysis of frame structures by the differential quadrature element method.International Journal for Numerical Methods in Engineering,40(4):759-772.https://doi.org/10.1002/(SICI)1097-0207(19970228)40:4<759::AID-NME87>3.0.CO;2-9
Yuan S,Zhong HZ,2014.Consolidation analysis of nonhomogeneous soil by the weak form quadrature element method.Computers and Geotechnics,62:1-10.https://doi.org/10.1016/j.compgeo.2014.06.012
Yuan S,Zhong HZ,2016a.Three dimensional analysis of unconfined seepage in earth dams by the weak form quadrature element method.Journal of Hydrology,533:403-411.https://doi.org/10.1016/j.jhydrol.2015.12.034
Yuan S,Zhong HZ,2016b.A weak form quadrature element formulation for coupled analysis of unsaturated soils.Computers and Geotechnics,76:1-11.https://doi.org/10.1016/j.compgeo.2016.02.006
Yuan S,Zhong HZ,2017.Finite deformation elasto-plastic consolidation analysis of soft clay by the weak form quadrature element method.Journal of Zhejiang University-SCIENCE A(Applied Physics&Engineering),18(12):942-957.https://doi.org/10.1631/jzus.A1600671
Zhong HZ,Yu T,2009.A weak form quadrature element method for plane elasticity problems.Applied Mathematical Modelling,33(10):3801-3814.https://doi.org/10.1016/j.apm.2008.12.007
Azari B,Fatahi B,Khabbaz H,2015.Numerical analysis of vertical drains accelerated consolidation considering combined soil disturbance and visco-plastic behaviour.Geomechanics&Engineering,8(2):187-220.https://doi.org/10.12989/gae.2015.8.2.187
Bellman R,Casti J,1971.Differential quadrature and longterm integration.Journal of Mathematical Analysis and Applications,34(2):235-238.https://doi.org/10.1016/0022-247X(71)90110-7
Biot MA,1941.General theory of three-dimensional consolidation.Journal of Applied Physics,12(2):155-164.https://doi.org/10.1063/1.1712886
Boal N,Gaspar FJ,Lisbona FJ,et al.,2011.Finite-difference analysis of fully dynamic problems for saturated porous media.Journal of Computational and Applied Mathematics,236(6):1090-1102.https://doi.org/10.1016/j.cam.2011.07.032
Chen R,Zhou W,Wang H,et al.,2005.One-dimensional nonlinear consolidation of multi-layered soil by differential quadrature method.Computers and Geotechnics,32(5):358-369.https://doi.org/10.1016/j.compgeo.2005.05.003
Darkshanamurthy V,Fredlund DG,Rahardjo H,1984.Coupled three-dimensional consolidation theory of unsaturated porous media.Fifth International Conference on Expansive Soils,p.99.
Fei K,Liu HL,2009.Implementation and application of bounding surface model in ABAQUS.Journal of PLAUniversity of Science and Technology(Natural Science Edition),10(5):447-451(in Chinese).
Gaspar FJ,Lisbona FJ,Vabishchevich PN,2003.A finite difference analysis of Biot’s consolidation model.Applied Numerical Mathematics,44(4):487-506.https://doi.org/10.1016/S0168-9274(02)00190-3
Gaspar FJ,Lisbona FJ,Vabishchevich PN,2006.Staggered grid discretizations for the quasi-static Biot’s consolidation problem.Applied Numerical Mathematics,56(6):888-898.https://doi.org/10.1016/j.apnum.2005.07.002
Gaspar FJ,Lisbona FJ,Matus P,et al.,2016.Numerical methods for a one-dimensional non-linear Biot’s model.Journal of Computational and Applied Mathematics,293:62-72.https://doi.org/10.1016/j.cam.2015.03.039
Ho L,Fatahi B,2015.Analytical solution for the twodimensional plane strain consolidation of an unsaturated soil stratum subjected to time-dependent loading.Computers&Geotechnics,67:1-16.https://doi.org/10.1016/j.compgeo.2015.02.011
Ho L,Fatahi B,Khabbaz H,2016.Analytical solution to axisymmetric consolidation in unsaturated soils with linearly depth-dependent initial conditions.Computers&Geotechnics,74:102-121.https://doi.org/10.1016/j.compgeo.2015.12.019
Hua L,2011.Stable element-free Galerkin solution procedures for the coupled soil-pore fluid problem.International Journal for Numerical Methods in Engineering,86(8):1000-1026.https://doi.org/10.1002/nme.3087
Le TM,Fatahi B,2016.Trust-region reflective optimisation to obtain soil visco-plastic properties.Engineering Computations,33(2):410-442.https://doi.org/10.1108/EC-11-2014-0236
Le TM,Fatahi B,Khabbaz H,et al.,2016.Numerical optimisation applying trust-region reflective least squares algorithm with constraints to optimise the non-linear creep parameters of soft soil.Applied Mathematical Modelling,16:1-21.
Manzari MT,Dafalias YF,1997.A critical state two-surface plasticity model for sands.Geotechnique,47(2):255-272.https://doi.org/10.1680/geot.1997.47.2.255
Mira P,Pastor M,Li T,et al.,2003.A new stabilized enhanced strain element with equal order of interpolation for soil consolidation problems.Computer Methods in Applied Mechanics and Engineering,192(37-38):4257-4277.https://doi.org/10.1016/S0045-7825(03)00416-X
Preisig M,Prévost JH,2011.Stabilization procedures in coupled poromechanics problems:a critical assessment.International Journal for Numerical and Analytical Methods in Geomechanics,35(11):1207-1225.https://doi.org/10.1002/nag.951
Reed M,1984.An investigation of numerical errors in the analysis of consolidation by finite elements.International Journal for Numerical and Analytical Methods in Geomechanics,8(3):243-257.https://doi.org/10.1002/nag.1610080304
Rodrigo C,Gaspar F,Hu X,et al.,2016.Stability and monotonicity for some discretizations of the Biot’s consolidation model.Computer Methods in Applied Mechanics and Engineering,298:183-204.https://doi.org/10.1016/j.cma.2015.09.019
Samimi S,Pak A,2012.Three-dimensional simulation of fully coupled hydro-mechanical behavior of saturated porous media using element free Galerkin(EFG)method.Computers&Geotechnics,46(4):75-83.https://doi.org/10.1016/j.compgeo.2012.06.004
Sandhu RS,Liu H,Singh KJ,1977.Numerical performance of some finite element schemes for analysis of seepage in porous elastic media.International Journal for Numerical and Analytical Methods in Geomechanics,1(2):177-194.https://doi.org/10.1002/nag.1610010205
Simo JC,Rifai M,1990.A class of mixed assumed strain methods and the method of incompatible modes.International Journal for Numerical Methods in Engineering,29(8):1595-1638.https://doi.org/10.1002/nme.1620290802
Wang W,Wang J,Wang Z,et al.,2007.An unequal-order radial interpolation meshless method for Biot’s consolidation theory.Computers and Geotechnics,34(2):61-70.https://doi.org/10.1016/j.compgeo.2006.10.010
Wang X,Gu H,1997.Static analysis of frame structures by the differential quadrature element method.International Journal for Numerical Methods in Engineering,40(4):759-772.https://doi.org/10.1002/(SICI)1097-0207(19970228)40:4<759::AID-NME87>3.0.CO;2-9
Yuan S,Zhong HZ,2014.Consolidation analysis of nonhomogeneous soil by the weak form quadrature element method.Computers and Geotechnics,62:1-10.https://doi.org/10.1016/j.compgeo.2014.06.012
Yuan S,Zhong HZ,2016a.Three dimensional analysis of unconfined seepage in earth dams by the weak form quadrature element method.Journal of Hydrology,533:403-411.https://doi.org/10.1016/j.jhydrol.2015.12.034
Yuan S,Zhong HZ,2016b.A weak form quadrature element formulation for coupled analysis of unsaturated soils.Computers and Geotechnics,76:1-11.https://doi.org/10.1016/j.compgeo.2016.02.006
Yuan S,Zhong HZ,2017.Finite deformation elasto-plastic consolidation analysis of soft clay by the weak form quadrature element method.Journal of Zhejiang University-SCIENCE A(Applied Physics&Engineering),18(12):942-957.https://doi.org/10.1631/jzus.A1600671
Zhong HZ,Yu T,2009.A weak form quadrature element method for plane elasticity problems.Applied Mathematical Modelling,33(10):3801-3814.https://doi.org/10.1016/j.apm.2008.12.007