A three-dimensional two-level gradient smoothing meshfree method for rainfall induced landslide simulations
Dongdong WANG1,2(), Jiarui WANG1, Junchao WU1, Junjun DENG1, Ming SUN1
1. Department of Civil Engineering and Xiamen Engineering Technology Center for Intelligent Maintenance of Infrastructures, Xiamen University, Xiamen 361005, China 2. Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computation, Xiamen University, Xiamen 361005, China
A three-dimensional two-level gradient smoothing meshfree method is presented for rainfall induced landslide simulations. The two-level gradient smoothing for meshfree shape function is elaborated in the three-dimensional Lagrangian setting with detailed implementation procedure. It is shown that due to the successive gradient smoothing operation without the requirement of derivative computation in the present formulation, the two-level smoothed gradient of meshfree shape function is capable of achieving a given influence domain more efficiently than the standard gradient of meshfree shape function. Subsequently, the two-level smoothed gradient of meshfree shape function is employed to discretize the weak form of coupled rainfall seepage and soil motion equations in a nodal integration format, as provides an efficient three-dimensional regularized meshfree formulation for large deformation rainfall induced landslide simulations. The exponential damage and pressure dependent plasticity relationships are utilized to describe the failure evolution in landslides. The plastic response of soil is characterized by the true effective stress measure, which is updated according to the rotationally neutralized objective integration algorithm. The effectiveness of the present three-dimensional two-level gradient smoothing meshfree method is demonstrated through numerical examples.
O CZienkiewicz, R LTaylor, D DFox. The Finite Element Method for Solid and Structural Mechanics. 7th ed. Oxford: Butterworth-Heinemann, 2013
2
L BLucy. A numerical approach to the testing of the fission hypothesis. Astronomical Journal, 1977, 82: 1013–1024 https://doi.org/10.1086/112164
3
R AGingold, J J Monaghan. Smoothed particle hydrodynamics: Theory and application to non-spherical stars. Monthly Notices of the Royal Astronomical Society, 1977, 181(3): 375–389 https://doi.org/10.1093/mnras/181.3.375
4
M BLiu, G R Liu. Smoothed particle hydrodynamics (SPH): An overview and recent developments. Archives of Computational Methods in Engineering, 2010, 17(1): 25–76 https://doi.org/10.1007/s11831-010-9040-7
5
TBelytschko, Y Y Lu, L Gu. Element-free Galerkin methods. International Journal for Numerical Methods in Engineering, 1994, 37(2): 229–256 https://doi.org/10.1002/nme.1620370205
6
W KLiu, S Jun, Y FZhang. Reproducing kernel particle methods. International Journal for Numerical Methods in Fluids, 1995, 20(8–9): 1081–1106 https://doi.org/10.1002/fld.1650200824
7
TBelytschko, Y Y Lu, L Gu. Crack propagation by element-free Galerkin methods. Engineering Fracture Mechanics, 1995, 51(2): 295–315 https://doi.org/10.1016/0013-7944(94)00153-9
8
J SChen, C Pan, C TWu, W KLiu. Reproducing kernel particle methods for large deformation analysis of non-linear structures. Computer Methods in Applied Mechanics and Engineering, 1996, 139(1–4): 195–227 https://doi.org/10.1016/S0045-7825(96)01083-3
9
L DLibersky, P W Randles, T C Carney, D L Dickinson. Recent improvements in SPH modeling of hypervelocity impact. International Journal of Impact Engineering, 1997, 20(6–10): 525–532 https://doi.org/10.1016/S0734-743X(97)87441-6
TRabczuk, T Belytschko. A three-dimensional large deformation meshfree method for arbitrary evolving cracks. Computer Methods in Applied Mechanics and Engineering, 2007, 196(29–30): 2777–2799 https://doi.org/10.1016/j.cma.2006.06.020
12
YVidal, J Bonet, AHuerta. Stabilized updated Lagrangian corrected SPH for explicit dynamic problems. International Journal for Numerical Methods in Engineering, 2007, 69(13): 2687–2710 https://doi.org/10.1002/nme.1859
13
DWang, Z Li, LLi, YWu. Three dimensional efficient meshfree simulation of large deformation failure evolution in soil medium. Science China. Technological Sciences, 2011, 54(3): 573–580 https://doi.org/10.1007/s11431-010-4287-7
14
BRen, S Li, JQian, XZeng. Meshfree simulations of spall fracture. Computer Methods in Applied Mechanics and Engineering, 2011, 200(5–8): 797–811 https://doi.org/10.1016/j.cma.2010.10.003
15
YWu, D Wang, C TWu. Three dimensional fragmentation simulation of concrete structures with a nodally regularized meshfree method. Theoretical and Applied Fracture Mechanics, 2014, 72: 89–99 https://doi.org/10.1016/j.tafmec.2014.04.006
16
RDrathi, A J M Das, A Rangarajan. Meshfree simulation of concrete structures and impact loading. International Journal of Impact Engineering, 2016, 91: 194–199 https://doi.org/10.1016/j.ijimpeng.2015.10.013
17
C TWu, Y Wu, J ECrawford, J MMagallanes. Three-dimensional concrete impact and penetration simulations using the smoothed particle Galerkin method. International Journal of Impact Engineering, 2017, 106: 1–17 https://doi.org/10.1016/j.ijimpeng.2017.03.005
18
S NAtluri, S P Shen. The Meshless Local Petrov-Galerkin (MLPG) Method. Henderson: Tech Science Press, 2002
19
S FLi, W K Liu. Meshfree Particle Methods. New York: Springer, 2004
20
XZhang, Y Liu. Meshless Methods. Beijing: Tsinghua University Press, 2004 (in Chinese)
21
V PNguyen, T Rabczuk, SBordas, MDuflot. Meshless methods: A review and computer implementation aspects. Mathematics and Computers in Simulation, 2008, 79(3): 763–813 https://doi.org/10.1016/j.matcom.2008.01.003
22
G RLiu. Meshfree Methods: Moving Beyond the Finite Element Method. 2nd ed. Boca Raton: CRC Press, 2009
H HBui, R Fukagawa, KSako, J CWells. Slope stability analysis and discontinuous slope failure simulation by elasto-plastic smoothed particle hydrodynamics (SPH). Geotechnique, 2011, 61(7): 565–574 https://doi.org/10.1680/geot.9.P.046
25
MPastor, T Blanc, BHaddad, SPetrone, MSanchez Morles, VDrempetic, DIssler, G BCrosta, LCascini, GSorbino, SCuomo. Application of a SPH depth-integrated model to landslide run-out analysis. Landslides, 2014, 11(5): 793–812 https://doi.org/10.1007/s10346-014-0484-y
26
MHu, M B Liu, M W Xie, G R Liu. Three-dimensional run-out analysis and prediction of flow-like landslides using smoothed particle hydrodynamics. Environmental Earth Sciences, 2015, 73(4): 1629–1640 https://doi.org/10.1007/s12665-014-3513-1
27
ZDai, Y Huang. A three-dimensional model for flow slides in municipal solid waste landfills using smoothed particle hydrodynamics. Environmental Earth Sciences, 2016, 75(2): 132 https://doi.org/10.1007/s12665-015-4923-4
28
TRabczuk, P M A Areias. A new approach for modelling slip lines in geological materials with cohesive models. International Journal for Numerical and Analytical Methods in Geomechanics, 2006, 30(11): 1159–1172 https://doi.org/10.1002/nag.522
29
WZheng, X Zhuang, DTannant, YCai, S Nunoo. Unified continuum/discontinuum modeling framework for slope stability assessment. Engineering Geology, 2014, 179: 90–101 https://doi.org/10.1016/j.enggeo.2014.06.014
30
GLiu, X Zhuang, ZCui. Three-dimensional slope stability analysis using independent cover based numerical manifold and vector method. Engineering Geology, 2017, 225: 83–95 https://doi.org/10.1016/j.enggeo.2017.02.022
31
JDolbow, T Belytschko. Numerical integration of the Galerkin weak form in meshfree methods. Computational Mechanics, 1999, 23(3): 219–230 https://doi.org/10.1007/s004660050403
32
J SChen, M Hillman, MRüter. An arbitrary order variationally consistent integration for Galerkin meshfree methods. International Journal for Numerical Methods in Engineering, 2013, 95(5): 387–418 https://doi.org/10.1002/nme.4512
33
QDuan, X Gao, BWang, XLi, H Zhang, TBelytschko, YShao. Consistent element free Galerkin method. International Journal for Numerical Methods in Engineering, 2014, 99(2): 79–101 https://doi.org/10.1002/nme.4661
34
M RHematiyan, AKhosravifard, G RLiu. A background decomposition method for domain integration in weak-form meshfree methods. Computers & Structures, 2014, 142: 64–78 https://doi.org/10.1016/j.compstruc.2014.07.001
35
G RJoldes, A Wittek, KMiller. Adaptive numerical integration in element-free Galerkin methods for elliptic boundary value problems. Engineering Analysis with Boundary Elements, 2015, 51: 52–63 https://doi.org/10.1016/j.enganabound.2014.10.007
36
DWang, J Wu. An efficient nesting sub-domain gradient smoothing integration algorithm with quadratic exactness for Galerkin meshfree methods. Computer Methods in Applied Mechanics and Engineering, 2016, 298: 485–519 https://doi.org/10.1016/j.cma.2015.10.008
37
HWei, J S Chen, M Hillman. A stabilized nodally integrated meshfree formulation for fully coupled hydro-mechanical analysis of fluid-saturated porous media. Computers & Fluids, 2016, 141: 105–115 https://doi.org/10.1016/j.compfluid.2015.11.002
38
C TWu, S W Chi, M Koishi, YWu. Strain gradient stabilization with dual stress points for the meshfree nodal integration method in inelastic analyses. International Journal for Numerical Methods in Engineering, 2016, 107(1): 3–30 https://doi.org/10.1002/nme.5147
39
JWu, J Deng, JWang, DWang. A review of numerical integration approaches for Galerkin meshfree methods. Chinese Journal of Solid Mechanics, 2016, 37: 208–233 (in Chinese)
40
SBeissel, T Belytschko. Nodal integration of the element-free Galerkin method. Computer Methods in Applied Mechanics and Engineering, 1996, 139(1–4): 49–74 https://doi.org/10.1016/S0045-7825(96)01079-1
J SChen, S P Yoon, C T Wu. Non-linear version of stabilized conforming nodal integration for Galerkin mesh-free methods. International Journal for Numerical Methods in Engineering, 2002, 53(12): 2587–2615 https://doi.org/10.1002/nme.338
43
O L AKwok, P CGuan, W PCheng, C TSun. Semi-Lagrangian reproducing kernel particle method for slope stability analysis and post-failure simulation. KSCE Journal of Civil Engineering, 2015, 19(1): 107–115 https://doi.org/10.1007/s12205-013-0550-3
44
P CGuan, J S Chen, Y Wu, HTeng, JGaidos, KHofstetter, MAlsaleh. Semi-Lagrangian reproducing kernel formulation and application to modeling earth moving operations. Mechanics of Materials, 2009, 41(6): 670–683 https://doi.org/10.1016/j.mechmat.2009.01.030
45
YLian, X Zhang, YLiu. Coupling between finite element method and material point method for problems with extreme deformation. Theoretical and Applied Mechanics Letters, 2012, 2(2): 021003 https://doi.org/10.1063/2.1202103
46
XZhang, K Krabbenhoft, DSheng, WLi. Numerical simulation of a flow-like landslide using the particle finite element method. Computational Mechanics, 2015, 55(1): 167–177 https://doi.org/10.1007/s00466-014-1088-z
47
TBelytschko, Z P Bažant, H Yul-Woong, CTa-Peng. Strain-softening materials and finite-element solutions. Computers & Structures, 1986, 23(2): 163–180 https://doi.org/10.1016/0045-7949(86)90210-5
J SChen, X Zhang, TBelytschko. An implicit gradient model by a reproducing kernel strain regularization in strain localization problems. Computer Methods in Applied Mechanics and Engineering, 2004, 193(27–29): 2827–2844 https://doi.org/10.1016/j.cma.2003.12.057
50
HAskes, J Pamin, Rde Borst. Dispersion analysis and element-free Galerkin solutions of second- and fourth-order gradient-enhanced damage models. International Journal for Numerical Methods in Engineering, 2000, 49(6): 811–832 https://doi.org/10.1002/1097-0207(20001030)49:6<811::AID-NME985>3.0.CO;2-9
51
DWang, Z Li. A two-level strain smoothing regularized meshfree approach with stabilized conforming nodal integration for elastic damage analysis. International Journal of Damage Mechanics, 2013, 22(3): 440–459 https://doi.org/10.1177/1056789512455938
52
DWang, L Li, ZLi. A regularized Lagrangian meshfree method for rainfall infiltration triggered slope failure analysis. Engineering Analysis with Boundary Elements, 2014, 42: 51–59 https://doi.org/10.1016/j.enganabound.2013.09.001
53
TRabczuk, T Belytschko, S PXiao. Stable particle methods based on Lagrangian kernels. Computer Methods in Applied Mechanics and Engineering, 2004, 193(12‒14): 1035–1063 https://doi.org/10.1016/j.cma.2003.12.005
54
JMaxars. Mechanical damage and fracture of concrete structures. In: Proceedings of the 5th International Conference of Fracture. Cannes, 1981, 4: 1499–1506
55
J CSimo, J W Ju. Strain- and stress-based continuum damage models—II. Computational aspects. International Journal of Solids and Structures, 1987, 23(7): 841–869 https://doi.org/10.1016/0020-7683(87)90084-9
56
J WJu. On energy-based coupled elastoplastic damage theories: Constitutive modeling and computational aspects. International Journal of Solids and Structures, 1989, 25(7): 803–833 https://doi.org/10.1016/0020-7683(89)90015-2
57
J CSimo, T J R Hughes. Computational Inelasticity. New York: Springer, 1998
58
D GFredlund, H Rahardjo. Soil Mechanics for Unsaturated Soils. New York: John Wiley & Sons , 1993
59
XSong, R I Borja. Mathematical framework for unsaturated flow in the finite deformation range. International Journal for Numerical Methods in Engineering, 2014, 97(9): 658–682 https://doi.org/10.1002/nme.4605
R IBorja, J A White. Continuum deformation and stability analyses of a steep hillside slope under rainfall infiltration. Acta Geotechnica, 2010, 5(1): 1–14 https://doi.org/10.1007/s11440-009-0108-1
62
CJacquard. Experimental study in laboratory of a capillary barrier. Dissertation for the Doctoral Degree. Paris: Ecole Mines Paris, 1988 (in French)
63
MBourgeois. The concept of capillary barrier: study by numerical model. Dissertation for the Doctoral Degree. Paris: Ecole Mines Paris, 1986 (in French)
64
PWei, W Xiao. Area calculation of three dimensional polygon. Chinese Mathematics Bulletin, 1984, 2: 18–21 (in Chinese)
65
DWang, P Xie, HLu. Meshfree consolidation analysis of saturated porous media with stabilized conforming nodal integration formulation. Interaction and Multiscale Mechanics, 2013, 6(2): 107–125 https://doi.org/10.12989/imm.2013.6.2.107
66
S WChi, T Siriaksorn, S PLin. Von Neumann stability analysis of the u-p reproducing kernel formulation for saturated porous media. Computational Mechanics, 2017, 59(2): 335–357 https://doi.org/10.1007/s00466-016-1349-0
67
SKawamura, S Miura, TIshikawa, SYokohama. Rainfall-induced failure of unsaturated volcanic slope subjected to freeze-thaw action and its mechanism. JSCE Journal of Geotechnical and Geoenvironmental Engineering, 2010, 66(3): 577–594 https://doi.org/10.2208/jscejc.66.577
68
W CLi, H J Li, F C Dai, L M Lee. Discrete element modeling of a rainfall-induced flowslide. Engineering Geology, 2012, 149–150: 22–34 https://doi.org/10.1016/j.enggeo.2012.08.006