Global sensitivity analysis of certain and uncertain factors for a circular tunnel under seismic action
Nazim Abdul NARIMAN1(), Raja Rizwan HUSSAIN2, Ilham Ibrahim MOHAMMAD1, Peyman KARAMPOUR3
1. Department of Civil Engineering, Tishk International University Sulaimani, Sulaimaniya 46001, Iraq 2. Civil Engineering Department, College of Engineering, King Saud University, Riyadh 11421, Saudi Arabia 3. Department of Mechanical Engineering, Faculty of Engineering, Arak University, Arak 38156-875, Iran
There are many certain and uncertain design factors which have unrevealed rational effects on the generation of tensile damage and the stability of the circular tunnels during seismic actions. In this research paper, we have dedicated three certain and four uncertain design factors to quantify their rational effects using numerical simulations and the Sobol’s sensitivity indices. Main effects and interaction effects between the design factors have been determined supporting on variance-based global sensitivity analysis. The results detected that the concrete modulus of elasticity for the tunnel lining has the greatest effect on the tensile damage generation in the tunnel lining during the seismic action. In the other direction, the interactions between the concrete density and both of concrete modulus of elasticity and tunnel diameter have appreciable effects on the tensile damage. Furthermore, the tunnel diameter has the deciding effect on the stability of the tunnel structure. While the interaction between the tunnel diameter and concrete density has appreciable effect on the stability process. It is worthy to mention that Sobol’s sensitivity indices manifested strong efficiency in detecting the roles of each design factor in cooperation with the numerical simulations explaining the responses of the circular tunnel during seismic actions.
. [J]. Frontiers of Structural and Civil Engineering, 2019, 13(6): 1289-1300.
Nazim Abdul NARIMAN, Raja Rizwan HUSSAIN, Ilham Ibrahim MOHAMMAD, Peyman KARAMPOUR. Global sensitivity analysis of certain and uncertain factors for a circular tunnel under seismic action. Front. Struct. Civ. Eng., 2019, 13(6): 1289-1300.
tunnel lining (concrete) modulus of elasticity (GPa)
X3
17
31
tunnel lining concrete density (kg/m3)
X4
2300
2500
Tab.2
uncertain factors
symbol
minimum value
maximum value
peak ground acceleration (m/s2)
X5
0.981
4.905
soil density (kg/m3)
X6
1800
2200
soil modulus of elasticity (GPa)
X7
0.2
0.25
Tab.3
sensitivity indices
maximum principal stress
maximum overall displacement
first effect X1
3.37
0.41
first effect X2
2.94
88.71?
first effect X3
83.76?
0.68
first effect X4
0.92
7.81
sum of first effects
90.99?
97.61?
interaction between X1 and X2
0.12
0.01
interaction between X1 and X3
0.40
0.00
interaction between X1 and X4
0.54
0.00
interaction between X2 and X3
0.68
0.00
interaction between X2 and X4
1.15
1.26
interaction between X3 and X4
2.05
0.00
total effect of X1
4.43
0.42
total effect of X2
4.89
89.98?
total effect of X3
86.89?
0.68
total effect of X4
4.66
9.07
sum of total effects
100.87??
100.15??
Tab.4
Fig.3
Fig.4
sensitivity indices
maximum principal stress
maximum overall displacement
first effect X5
40.81?
99.81?
first effect X6
30.3
0.07
first effect X7
26.62?
0.09
sum of first effects
97.73?
99.97?
interaction between X5 and X6
0.89
0.01
interaction between X5 and X7
0.55
0.01
interaction between X6 and X7
0.56
0.00
total effect of X5
42.25?
99.83?
total effect of X6
31.75?
0.09
total effect of X7
27.73?
0.10
sum of total effects
101.73??
100.02??
Tab.5
Fig.5
Fig.6
Fig.7
Fig.8
1
T Akhlaghi, A Nikkar. Effect of vertically propagating shear waves on seismic behavior of circular tunnels. Hindawi the Scientific World Journal, 2014, 2014, 1–10
2
Y M A Hashash, J J Hook, B Schmidt, J I-Chiang Yao. Seismic design and analysis of underground structure. Tunnelling and Underground Space Technology, 2001, 16(4): 247–293 https://doi.org/10.1016/S0886-7798(01)00051-7
3
N A Nariman, R R Hussain, M A Msekh, P Karampour. Prediction meta-models for the responses of a circular tunnel during earthquakes. Underground Space, 2019, 4(1): 31–47 https://doi.org/10.1016/j.undsp.2018.06.003
4
N A Nariman, A Ramazan, I I Mohammad. Application of coupled XFEM-BCQO in the structural optimization of a circular tunnel lining subjected to a ground motion. Frontiers of Structural and Civil Engineering, 2019 (in press)
5
G Mollon, D Dias, A H Soubra. Probabilistic analysis and design of circular tunnels against face stability. International Journal of Geomechanics, 2009, 9(6): 237–249 https://doi.org/10.1061/(ASCE)1532-3641(2009)9:6(237)
6
Z Kalab, P Stemon. Influence of seismic events on shallow geotechnical structures. Acta Montanistica Slovaca, 2017, 22(4): 412–421
7
H Ghasemi, H S Park, T Rabczuk. A level-set based IGA formulation for topology optimization of flexoelectric materials. Computer Methods in Applied Mechanics and Engineering, 2017, 313: 239–258 https://doi.org/10.1016/j.cma.2016.09.029
8
H Ghasemi, R Brighenti, X Zhuang, J Muthu, T Rabczuk. Optimum fiber content and distribution in fiber-reinforced solids using a reliability and NURBS based sequential optimization approach. Structural and Multidisciplinary Optimization, 2015, 51(1): 99–112 https://doi.org/10.1007/s00158-014-1114-y
9
C Zhang, S S Nanthakumar, T Lahmer, T Rabczuk. Multiple cracks identification for piezoelectric structures. International Journal of Fracture, 2017, 206(2): 151–169 https://doi.org/10.1007/s10704-017-0206-2
10
S Nanthakumar, X Zhuang, H Park, T Rabczuk. Topology optimization of flexoelectric structures. Journal of the Mechanics and Physics of Solids, 2017, 105: 217–234 https://doi.org/10.1016/j.jmps.2017.05.010
11
S Nanthakumar, T Lahmer, X Zhuang, H S Park, T Rabczuk. Topology optimization of piezoelectric nanostructures. Journal of the Mechanics and Physics of Solids, 2016, 94: 316–335 https://doi.org/10.1016/j.jmps.2016.03.027
12
S Nanthakumar, T Lahmer, X Zhuang, G Zi, T Rabczuk. Detection of material interfaces using a regularized level set method in piezoelectric structures. Inverse Problems in Science and Engineering, 2016, 24(1): 153–176 https://doi.org/10.1080/17415977.2015.1017485
13
S Nanthakumar, N Valizadeh, H Park, T Rabczuk. Surface effects on shape and topology optimization of nanostructures. Computational Mechanics, 2015, 56(1): 97–112 https://doi.org/10.1007/s00466-015-1159-9
14
S S Nanthakumar, T Lahmer, T Rabczuk. Detection of flaws in piezoelectric structures using extended FEM. International Journal for Numerical Methods in Engineering, 2013, 96(6): 373–389 https://doi.org/10.1002/nme.4565
15
N Vu-Bac, T Lahmer, X Zhuang, T Nguyen-Thoi, T Rabczuk. A software framework for probabilistic sensitivity analysis for computationally expensive models. Advances in Engineering Software, 2016, 100: 19–31 https://doi.org/10.1016/j.advengsoft.2016.06.005
16
N Vu-Bac, M Silani, T Lahmer, X Zhuang, T Rabczuk. A unified framework for stochastic predictions of mechanical properties of polymeric nanocomposites. Computational Materials Science, 2015, 96: 520–535 https://doi.org/10.1016/j.commatsci.2014.04.066
17
N Vu-Bac, R Rafiee, X Zhuang, T Lahmer, T Rabczuk. Uncertainty quantification for multiscale modeling of polymer nanocomposites with correlated parameters. Composites. Part B, Engineering, 2015, 68: 446–464 https://doi.org/10.1016/j.compositesb.2014.09.008
18
T Rabczuk, J Akkermann, J Eibl. A numerical model for reinforced concrete structures. International Journal of Solids and Structures, 2005, 42(5–6): 1327–1354 https://doi.org/10.1016/j.ijsolstr.2004.07.019
T Q Thai, T Rabczuk, Y Bazilevs, G Meschke. A higher-order stress-based gradient-enhanced damage model based on isogeometric analysis. Computer Methods in Applied Mechanics and Engineering, 2016, 304: 584–604 https://doi.org/10.1016/j.cma.2016.02.031
21
N A Fleck, J W Hutchinson. A phenomenological theory for strain gradient effects in plasticity. Journal of the Mechanics and Physics of Solids, 1993, 41(12): 1825–1857 https://doi.org/10.1016/0022-5096(93)90072-N
22
T Rabczuk, J Eibl. Simulation of high velocity concrete fragmentation using SPH/MLSPH. International Journal for Numerical Methods in Engineering, 2003, 56(10): 1421–1444 https://doi.org/10.1002/nme.617
23
T Rabczuk, J Eibl, L Stempniewski. Numerical analysis of high speed concrete fragmentation using a meshfree Lagrangian method. Engineering Fracture Mechanics, 2004, 71(4–6): 547–556 https://doi.org/10.1016/S0013-7944(03)00032-8
24
T Rabczuk, S P Xiao, M Sauer. Coupling of meshfree methods with nite elements: Basic concepts and test results. Communications in Numerical Methods in Engineering, 2006, 22(10): 1031–1065 https://doi.org/10.1002/cnm.871
25
T Rabczuk, J Eibl. Modelling dynamic failure of concrete with meshfree methods. International Journal of Impact Engineering, 2006, 32(11): 1878–1897 https://doi.org/10.1016/j.ijimpeng.2005.02.008
C Miehe, M Hofacker, F Welschinger. A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits. Computer Methods in Applied Mechanics and Engineering, 2010, 199(45–48): 2765–2778 https://doi.org/10.1016/j.cma.2010.04.011
28
F Amiri, D Millan, M Arroyo, M Silani, T Rabczuk. Fourth order phase-field model for local max-ent approximants applied to crack propagation. Computer Methods in Applied Mechanics and Engineering, 2016, 312(C): 254–275 https://doi.org/10.1016/j.cma.2016.02.011
29
P Areias, T Rabczuk, M Msekh. Phase-field analysis of finite-strain plates and shells including element subdivision. Computer Methods in Applied Mechanics and Engineering, 2016, 312(C): 322–350 https://doi.org/10.1016/j.cma.2016.01.020
30
M A Msekh, M Silani, M Jamshidian, P Areias, X Zhuang, G Zi, P He, T Rabczuk. Predictions of J integral and tensile strength of clay/epoxy nanocomposites material using phase-field model. Composites. Part B, Engineering, 2016, 93: 97–114 https://doi.org/10.1016/j.compositesb.2016.02.022
31
K Hamdia, M A Msekh, M Silani, N Vu-Bac, X Zhuang, T Nguyen-Thoi, T Rabczuk. Uncertainty quantication of the fracture properties of polymeric nanocomposites based on phase-field modeling. Composite Structures, 2015, 133: 1177–1190 https://doi.org/10.1016/j.compstruct.2015.08.051
32
M A Msekh, M Sargado, M Jamshidian, P Areias, T Rabczuk. ABAQUS implementation of phase-field model for brittle fracture. Computational Materials Science, 2015, 96: 472–484 https://doi.org/10.1016/j.commatsci.2014.05.071
33
F Amiri, D Millán, Y Shen, T Rabczuk, M Arroyo. Phase-field modeling of fracture in linear thin shells. Theoretical and Applied Fracture Mechanics, 2014, 69: 102–109 https://doi.org/10.1016/j.tafmec.2013.12.002
34
K M Hamdia, X Zhuang, P He, T Rabczuk. Fracture toughness of polymeric particle nanocomposites: Evaluation of Models performance using Bayesian method. Composites Science and Technology, 2016, 126: 122–129 https://doi.org/10.1016/j.compscitech.2016.02.012
35
T Rabczuk, T Belytschko, S P Xiao. 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
36
T Rabczuk, T Belytschko. Adaptivity for structured meshfree particle methods in 2D and 3D. International Journal for Numerical Methods in Engineering, 2005, 63(11): 1559–1582 https://doi.org/10.1002/nme.1326
37
V P Nguyen, T Rabczuk, S Bordas, M Duflot. 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
38
X Zhuang, Y Cai, C Augarde. A meshless sub-region radial point interpolation method for accurate calculation of crack tip elds. Theoretical and Applied Fracture Mechanics, 2014, 69: 118–125 https://doi.org/10.1016/j.tafmec.2013.12.003
39
X Zhuang, H Zhu, C Augarde. An improved meshless Shepard and least square method possessing the delta property and requiring no singular weight function. Computational Mechanics, 2014, 53(2): 343–357 https://doi.org/10.1007/s00466-013-0912-1
40
X Zhuang, C Augarde, K Mathisen. Fracture modelling using meshless methods and level sets in 3D: Framework and modelling. International Journal for Numerical Methods in Engineering, 2012, 92(11): 969–998 https://doi.org/10.1002/nme.4365
41
L Chen, T Rabczuk, S Bordas, GR Liu, KY Zeng, P Kerfriden. Extended finite element method with edge-based strain smoothing (Esm-XFEM) for linear elastic crack growth. Computer Methods in Applied Mechanics and Engineering, 2012, 209–212(4): 250–265
N Vu-Bac, H Nguyen-Xuan, L Chen, C K Lee, G Zi, X Zhuang, G R Liu, T Rabczuk. A phantom-node method with edge-based strain smoothing for linear elastic fracture mechanics. Journal of Applied Mathematics, 2013, 2013, 978026
45
S P A Bordas, S Natarajan, P Kerfriden, C E Augarde, D R Mahapatra, T Rabczuk, S D Pont. On the performance of strain smoothing for quadratic and enriched finite element approximations (XFEM/GFEM/PUFEM). International Journal for Numerical Methods in Engineering, 2011, 86(4–5): 637–666 https://doi.org/10.1002/nme.3156
46
S P A Bordas, T Rabczuk, N X Hung, V P Nguyen, S Natarajan, T Bog, D M Quan, N V Hiep. Strain Smoothing in FEM and XFEM. Computers & Structures, 2010, 88(23–24): 1419–1443 https://doi.org/10.1016/j.compstruc.2008.07.006
47
T Rabczuk, G Zi, A Gerstenberger, W A Wall. A new crack tip element for the phantom node method with arbitrary cohesive cracks. International Journal for Numerical Methods in Engineering, 2008, 75(5): 577–599 https://doi.org/10.1002/nme.2273
48
T Chau-Dinh, G Zi, P S Lee, T Rabczuk, J H Song. Phantom-node method for shell models with arbitrary cracks. Computers & Structures, 2012, 92–93: 242–256 https://doi.org/10.1016/j.compstruc.2011.10.021
49
J H Song, P M A Areias, T Belytschko. A method for dynamic crack and shear band propagation with phantom nodes. International Journal for Numerical Methods in Engineering, 2006, 67(6): 868–893 https://doi.org/10.1002/nme.1652
50
P M A Areias, J H Song, T Belytschko. Analysis of fracture in thin shells by overlapping paired elements. Computer Methods in Applied Mechanics and Engineering, 2006, 195(41–43): 5343–5360 https://doi.org/10.1016/j.cma.2005.10.024
51
T Rabczuk, P M A Areias. A meshfree thin shell for arbitrary evolving cracks based on an external enrichment. CMES-Computer Modeling in Engineering and Sciences, 2006, 16(2): 115–130
52
G Zi, T Rabczuk, W A Wall. Extended meshfree methods without branch enrichment for cohesive cracks. Computational Mechanics, 2007, 40(2): 367–382 https://doi.org/10.1007/s00466-006-0115-0
53
T Rabczuk, S Bordas, G Zi. A three-dimensional meshfree method for continuous multiple-crack initiation, propagation and junction in statics and dynamics. Computational Mechanics, 2007, 40(3): 473–495 https://doi.org/10.1007/s00466-006-0122-1
54
T Rabczuk, G Zi. A meshfree method based on the local partition of unity for cohesive cracks. Computational Mechanics, 2007, 39(6): 743–760 https://doi.org/10.1007/s00466-006-0067-4
55
T Rabczuk, P M A Areias, T Belytschko. A meshfree thin shell method for nonlinear dynamic fracture. International Journal for Numerical Methods in Engineering, 2007, 72(5): 524–548 https://doi.org/10.1002/nme.2013
56
S Bordas, T Rabczuk, G Zi. Three-dimensional crack initiation, propagation, branching and junction in non-linear materials by an extended meshfree method without asymptotic enrichment. Engineering Fracture Mechanics, 2008, 75(5): 943–960 https://doi.org/10.1016/j.engfracmech.2007.05.010
57
T Rabczuk, G Zi, S Bordas, H Nguyen-Xuan. A geometrically non-linear three dimensional cohesive crack method for reinforced concrete structures. Engineering Fracture Mechanics, 2008, 75(16): 4740–4758 https://doi.org/10.1016/j.engfracmech.2008.06.019
58
T Rabczuk, R Gracie, J H Song, T Belytschko. Immersed particle method for fluid-structure interaction. International Journal for Numerical Methods in Engineering, 2010, 81(1): 48–71
59
T Rabczuk, S Bordas, G Zi. On three-dimensional modelling of crack growth using partition of unity methods. Computers & Structures, 2010, 88(23–24): 1391–1411 https://doi.org/10.1016/j.compstruc.2008.08.010
60
F Amiri, C Anitescu, M Arroyo, S Bordas, T Rabczuk. XLME interpolants, a seamless bridge between XFEM and enriched meshless methods. Computational Mechanics, 2014, 53(1): 45–57 https://doi.org/10.1007/s00466-013-0891-2
61
H Talebi, C Samaniego, E Samaniego, T Rabczuk. On the numerical stability and mass-lumping schemes for explicit enriched meshfree methods. International Journal for Numerical Methods in Engineering, 2012, 89(8): 1009–1027 https://doi.org/10.1002/nme.3275
62
N Nguyen-Thanh, K Zhou, X Zhuang, P Areias, H Nguyen-Xuan, Y Bazilevs, T Rabczuk. Isogeometric analysis of large-deformation thin shells using RHTsplines for multiple-patch coupling. Computer Methods in Applied Mechanics and Engineering, 2017, 316: 1157–1178 https://doi.org/10.1016/j.cma.2016.12.002
63
N Nguyen-Thanh, N Valizadeh, M N Nguyen, H Nguyen-Xuan, X Zhuang, P Areias, G Zi, Y Bazilevs, L De Lorenzis, T Rabczuk. An extended isogeometric thin shell analysis based on Kirchho-Love theory. Computer Methods in Applied Mechanics and Engineering, 2015, 284: 265–291 https://doi.org/10.1016/j.cma.2014.08.025
64
Y Jia, C Anitescu, S Ghorashi, T Rabczuk. Extended isogeometric analysis for material interface problems. IMA Journal of Applied Mathematics, 2015, 80(3): 608–633 https://doi.org/10.1093/imamat/hxu004
65
S Ghorashi, N Valizadeh, S Mohammadi, T Rabczuk. T-spline based XIGA for fracture nalysis of orthotropic media. Computers & Structures, 2015, 147: 138–146 https://doi.org/10.1016/j.compstruc.2014.09.017
66
T Rabczuk, T Belytschko. Cracking particles: A simplified meshfree method for arbitrary evolving cracks. International Journal for Numerical Methods in Engineering, 2004, 61(13): 2316–2343 https://doi.org/10.1002/nme.1151
67
T Rabczuk, 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 Engineering, 2006, 30(11): 1159–1172 https://doi.org/10.1002/nag.522
68
T Rabczuk, T Belytschko. Application of particle methods to static fracture of reinforced concrete structures. International Journal of Fracture, 2006, 137(1–4): 19–49 https://doi.org/10.1007/s10704-005-3075-z
69
T Rabczuk, 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
70
T Rabczuk, P M A Areias, T Belytschko. A simplied meshfree method for shear bands with cohesive surfaces. International Journal for Numerical Methods in Engineering, 2007, 69(5): 993–1021 https://doi.org/10.1002/nme.1797
71
T Rabczuk, E Samaniego. Discontinuous modelling of shear bands using adaptive meshfree methods. Computer Methods in Applied Mechanics and Engineering, 2008, 197(6–8): 641–658 https://doi.org/10.1016/j.cma.2007.08.027
72
T Rabczuk, J H Song, T Belytschko. Simulations of instability in dynamic fracture by the cracking particles method. Engineering Fracture Mechanics, 2009, 76(6): 730–741 https://doi.org/10.1016/j.engfracmech.2008.06.002
73
T Rabczuk, G Zi, S Bordas, H Nguyen-Xuan. A simple and robust three-dimensional cracking-particle method without enrichment. Computer Methods in Applied Mechanics and Engineering, 2010, 199(37–40): 2437–2455 https://doi.org/10.1016/j.cma.2010.03.031
74
Y Cai, X Zhuang, H Zhu. A generalized and ecient method for nite cover generation in the numerical manifold method. International Journal of Computational Methods, 2013, 10(5): 1350028 https://doi.org/10.1142/S021987621350028X
75
G Liu, X Zhuang, Z Cui. 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
76
B H Nguyen, X Zhuang, P Wriggers, T Rabczuk, M E Mear, H D Tran. Isogeometric symmetric Galerkin boundary element method for three-dimensional elasticity problems. Computer Methods in Applied Mechanics and Engineering, 2017, 323: 132–150 https://doi.org/10.1016/j.cma.2017.05.011
77
B H Nguyen, H D Tran, C Anitescu, X Zhuang, T Rabczuk. An isogeometric symmetric Galerkin boundary element method for two-dimensional crack problems. Computer Methods in Applied Mechanics and Engineering, 2016, 306: 252–275 https://doi.org/10.1016/j.cma.2016.04.002
78
H Zhu, W Wu, J Chen, G Ma, X Liu, X Zhuang. Integration of three dimensional discontinuous deformation analysis (DDA) with binocular photogrammetry for stability analysis of tunnels in blocky rock mass. Tunnelling and Underground Space Technology, 2016, 51: 30–40 https://doi.org/10.1016/j.tust.2015.10.012
79
W Wu, H Zhu, X Zhuang, G Ma, Y Cai. A multi-shell cover algorithm for contact detection in the three dimensional discontinuous deformation analysis. Theoretical and Applied Fracture Mechanics, 2014, 72: 136–149 https://doi.org/10.1016/j.tafmec.2014.03.004
80
Y Cai, H Zhu, X Zhuang. A continuous/discontinuous deformation analysis (CDDA) method based on deformable blocks for fracture modelling. Frontiers of Structural and Civil Engineering, 2013, 7(4): 369–378 https://doi.org/10.1007/s11709-013-0222-x
81
H Nguyen-Xuan, G R Liu, S Bordas, S Natarajan, T Rabczuk. An adaptive singular ES-FEM for mechanics problems with singular eld of arbitrary order. Computer Methods in Applied Mechanics and Engineering, 2013, 253: 252–273 https://doi.org/10.1016/j.cma.2012.07.017
82
P Areias, T Rabczuk. Steiner-point free edge cutting of tetrahedral meshes with applications in fracture. Finite Elements in Analysis and Design, 2017, 132: 27–41 https://doi.org/10.1016/j.finel.2017.05.001
83
P Areias, J Reinoso, P Camanho, T Rabczuk. A constitutive-based element-by-element crack propagation algorithm with local mesh refinement. Computational Mechanics, 2015, 56(2): 291–315 https://doi.org/10.1007/s00466-015-1172-z
84
P M A Areias, T Rabczuk, P P Camanho. Finite strain fracture of 2D problems with injected anisotropic softening elements. Theoretical and Applied Fracture Mechanics, 2014, 72: 50–63 https://doi.org/10.1016/j.tafmec.2014.06.006
85
P Areias, T Rabczuk, D Dias-da-Costa. Element-wise fracture algorithm based on rotation of edges. Engineering Fracture Mechanics, 2013, 110: 113–137 https://doi.org/10.1016/j.engfracmech.2013.06.006
86
P Areias, T Rabczuk, P P Camanho. Initially rigid cohesive laws and fracture based on edge rotations. Computational Mechanics, 2013, 52(4): 931–947 https://doi.org/10.1007/s00466-013-0855-6
87
P Areias, T Rabczuk. Finite strain fracture of plates and shells with congurational forces and edge rotation. International Journal for Numerical Methods in Engineering, 2013, 94(12): 1099–1122 https://doi.org/10.1002/nme.4477
88
M Silani, H Talebi, A S Hamouda, T Rabczuk. Nonlocal damage modeling in clay/epoxy nanocomposites using a multiscale approach. Journal of Computational Science, 2016, 15: 18–23 https://doi.org/10.1016/j.jocs.2015.11.007
89
H Talebi, M Silani, T Rabczuk. Concurrent multiscale modelling of three dimensional crack and dislocation propagation. Advances in Engineering Software, 2015, 80: 82–92 https://doi.org/10.1016/j.advengsoft.2014.09.016
90
M Silani, H Talebi, S Ziaei-Rad, A M S Hamouda, G Zi, T Rabczuk. A three dimensional extended Arlequin method for dynamic fracture. Computational Materials Science, 2015, 96: 425–431 https://doi.org/10.1016/j.commatsci.2014.07.039
91
M Silani, S Ziaei-Rad, H Talebi, T Rabczuk. A semi-concurrent multiscale approach for modeling damage in nanocomposites. Theoretical and Applied Fracture Mechanics, 2014, 74: 30–38 https://doi.org/10.1016/j.tafmec.2014.06.009
92
H Talebi, M Silani, S Bordas, P Kerfriden, T Rabczuk. A computational library for multiscale modelling of material failure. Computational Mechanics, 2014, 53(5): 1047–1071 https://doi.org/10.1007/s00466-013-0948-2
93
H Talebi, M Silani, S P A Bordas, P Kerfriden, T Rabczuk. Molecular dynamics/XFEM coupling by a three-dimensional extended bridging domain with applications to dynamic brittle fracture. International Journal for Multiscale Computational Engineering, 2013, 11(6): 527–541 https://doi.org/10.1615/IntJMultCompEng.2013005838
94
S W Yang, P R Budarapu, D R Mahapatra, S P A Bordas, G Zi, T Rabczuk. A meshless adaptive multiscale method for fracture. Computational Materials Science, 2015, 96: 382–395 https://doi.org/10.1016/j.commatsci.2014.08.054
95
P Budarapu, R Gracie, S Bordas, T Rabczuk. An adaptive multiscale method for quasi-static crack growth. Computational Mechanics, 2014, 53(6): 1129–1148 https://doi.org/10.1007/s00466-013-0952-6
96
P R Budarapu, R Gracie, S W Yang, X Zhuang, T Rabczuk. Ecient coarse graining in multiscale modeling of fracture. Theoretical and Applied Fracture Mechanics, 2014, 69: 126–143 https://doi.org/10.1016/j.tafmec.2013.12.004
97
X Zhuang, Q Wang, H Zhu. Multiscale modelling of hydro-mechanical couplings in quasi-brittle materials. International Journal of Fracture, 2017, 204(1): 1–27 https://doi.org/10.1007/s10704-016-0139-1
98
H Zhu, Q Wang, X Zhuang. A nonlinear semi-concurrent multiscale method for fractures. International Journal of Impact Engineering, 2016, 87: 65–82 https://doi.org/10.1016/j.ijimpeng.2015.06.022
99
X Zhuang, Q Wang, H Zhu. A 3D computational homogenization model for porous material and parameters identification. Computational Materials Science, 2015, 96: 536–548 https://doi.org/10.1016/j.commatsci.2014.04.059
100
V Kouznetsova, M G D Geers, W A M Brekelmans. Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme. International Journal for Numerical Methods in Engineering, 2002, 54(8): 1235–1260 https://doi.org/10.1002/nme.541
J Amani, E Oterkus, P Areias, G Zi, T Nguyen-Thoi, T Rabczuk. A non-ordinary state-based peridynamics formulation for thermoplastic fracture. International Journal of Impact Engineering, 2016, 87: 83–94 https://doi.org/10.1016/j.ijimpeng.2015.06.019
103
H Ren, X Zhuang, T Rabczuk. A new Peridynamic formulation with shear deformation for elastic solid. Journal of Micromechanics and Molecular Physics, 2016, 1(2): 1650009 https://doi.org/10.1142/S2424913016500090
104
H Ren, X Zhuang, Y Cai, T Rabczuk. Dual-horizon peridynamics. International Journal for Numerical Methods in Engineering, 2016, 108(12): 1451–1476 https://doi.org/10.1002/nme.5257
105
H Ren, X Zhuang, T Rabczuk. Dual-horizon peridynamics: A stable solution to varying horizons. Computer Methods in Applied Mechanics and Engineering, 2017, 318: 762–782 https://doi.org/10.1016/j.cma.2016.12.031
106
G Glen, K Isaacs. Estimating sobol sensitivity indices using correlations. Journal of Environmental Modelling and Software, 2012, 37: 157–166 https://doi.org/10.1016/j.envsoft.2012.03.014
107
J Nossent, P Elsen, W Bauwens. Sobol sensitivity analysis of a complex environmental model. Journal of Environmental Modelling and Software, 2011, 26(12): 1515–1525 https://doi.org/10.1016/j.envsoft.2011.08.010
108
X Y Zhang, M N Trame, L J Lesko, S Schmidt. Sobol sensitivity analysis: A tool to guide the development and evaluation of systems pharmacology models. CPT: Pharmacometrics & Systems Pharmacology, 2015, 4(2): 69–79 https://doi.org/10.1002/psp4.6
