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Frontiers of Structural and Civil Engineering

Front. Struct. Civ. Eng.    2020, Vol. 14 Issue (2) : 387-410     https://doi.org/10.1007/s11709-019-0601-z
RESEARCH ARTICLE
Uncertainty assessment in hydro-mechanical-coupled analysis of saturated porous medium applying fuzzy finite element method
Farhoud KALATEH(), Farideh HOSSEINEJAD
Faculty of Civil Engineering, University of Tabriz, Tabriz 51666-16471, Iran
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Abstract

The purpose of the present study was to develop a fuzzy finite element method, for uncertainty quantification of saturated soil properties on dynamic response of porous media, and also to discrete the coupled dynamic equations known as u-p hydro-mechanical equations. Input parameters included fuzzy numbers of Poisson’s ratio, Young’s modulus, and permeability coefficient as uncertain material of soil properties. Triangular membership functions were applied to obtain the intervals of input parameters in five membership grades, followed up by a minute examination of the effects of input parameters uncertainty on dynamic behavior of porous media. Calculations were for the optimized combinations of upper and lower bounds of input parameters to reveal soil response including displacement and pore water pressure via fuzzy numbers. Fuzzy analysis procedure was verified, and several numerical examples were analyzed by the developed method, including a dynamic analysis of elastic soil column and elastic foundation under ramp loading. Results indicated that the range of calculated displacements and pore pressure were dependent upon the number of fuzzy parameters and uncertainty of parameters within equations. Moreover, it was revealed that for the input variations looser sands were more sensitive than dense ones.

Keywords fuzzy finite element method      saturated soil      hydro-mechanical coupled equations      coupled analysis      uncertainty analysis     
Corresponding Authors: Farhoud KALATEH   
Just Accepted Date: 17 January 2020   Online First Date: 23 April 2020    Issue Date: 08 May 2020
 Cite this article:   
Farhoud KALATEH,Farideh HOSSEINEJAD. Uncertainty assessment in hydro-mechanical-coupled analysis of saturated porous medium applying fuzzy finite element method[J]. Front. Struct. Civ. Eng., 2020, 14(2): 387-410.
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http://journal.hep.com.cn/fsce/EN/10.1007/s11709-019-0601-z
http://journal.hep.com.cn/fsce/EN/Y2020/V14/I2/387
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Farhoud KALATEH
Farideh HOSSEINEJAD
Fig.1  Definition of triangular fuzzy number [a,b,c].
Fig.2  Structure chart for incremental form of Biot analysis with fuzzy finite element method.
Fig.3  Communications between (a) deterministic and (b) fuzzy solvers.
Fig.4  Soil column subjected to a surface step loading; the geometry and boundary condition [47].
Fig.5  Membership function of fuzzy input parameters in soil column. (a) Poisson’s ratio; (b) Young’s modulus (E); (c) permeability coefficient (k).
material properties region 1 region 2
E (Pa) l 1.0E7 5.0E7
m 3.0E7 6.0E7
h 5.0E7 8.0E7
υ l 0.2 0.3
m 0.3 0.35
h 0.4 0.4
ρs(kg/ m 3) 2000 2000
ρf(kg/ m 3) 1000 1000
kf (Pa) 2.1E9 2.1E9
ks (Pa) 1.0E20 1.0E20
n 0.3 0.3
k γ( m3s/kg) l 1.02E–9 1.02E–10
m 1.02E–8 1.02E–9
h 1.02E–7 1.02E–8
Tab.1  Material properties of soil column
soil Poisson’s ratio, υ Young’s modulus, E (MPa)
gravel?????loose 0.2–0.35 30–80
?????????medium dense 80–100
?????????dense 0.3–0.4 100–200
sand?????loose 0.2–0.35 10–30
??????medium dense 30–50
?????????dense 0.3–0.4 50–80
fine sand????loose 0.25 8–12
??????medium dense 12–20
??????dense 20–30
Tab.2  Elastic constants for various soils (after AASHTO, 1995)
soil permeability coefficient (mm/s) k γ( m3s/kg)
coarse 10–103 106–104
fine gravel, coarse, and medium sand 102–10 109–106
fine sand, loose silt 104–102 1011–109
Tab.3  Typical values of permeability coefficient for various soils
Fig.6  Time history of pressure at (a) node 46, (b) node 36, (c) node 16, (d) node 6, and time history of displacement for (e) node 11, (f) node1 of soil column in deterministic solution.
Fig.7  Time history of pore pressure at (a) node 36, (b) node 46, (c) node 6, (d) node 16, and time history of displacement at (e) node 1, (f) node 11, (g) node 36, (h) node 46 of soil column by three fuzzy input parameters in FFEM model for five membership grades.
Fig.8  Fuzzy number of vertical displacement at (a) node 1, (b) node 6, (c) node 11, (d) node 36, and (e) node 46 of soil column within 200 s after loading. (Left side: results in five modes of analysis: E, ν, k fuzzy; E, ν fuzzy; E fuzzy; ν fuzzy; k fuzzy. Right side: results in three modes of analysis: E, ν fuzzy; E fuzzy; ν fuzzy.
Fig.9  Results in five modes of analysis with E, v, k fuzzy; E, v fuzzy; E fuzzy; v fuzzy; k fuzzy and fuzzy number of pressure at (a) node 6, (b) node 11, (c) node 36, and (e) node 46 of soil column within 200 s after loading
Fig.10  (a) Elastic foundation subject to a surface step loading; (b) geometry and boundary condition.
Fig.11  (a) Young’s modulus (E); (b) Poisson’s ratio; (c) permeability coefficient (k).
material properties region 1 region 2 region3
E (Pa) l 10.0E6 20.0E6 60.0E6
m 20.0E6 40.0E6 100.0E6
h 30.0E6 60.0E6 160.0E6
υ 1 0.15 0.15 0.15
m 0.2 0.2 0.2
h 0.25 0.25 0.25
ρs(kg/ m 3) 2000 2000 2000
ρf(kg/ m 3) 1000 1000 1000
kf (Pa) 2.1E9 2.1E9 2.1E9
ks (Pa) 1.0E20 1.0E20 1.0E20
n 0.25 0.3 0.35
k γ( m3s/ kg) 1 0.50E–8 2.50E–8 1.00E–7
m 1.00E–8 5.00E–8 2.00E–7
h 2.00E–8 10.00E–8 4.00E–7
Tab.4  Elastic foundation material properties
Fig.12  Pore pressure history of FFEM model in different membership grades (left) and membership grade of 0.9 (right) at different nodes of elastic foundation within three fuzzy input parameters: (a) node 35; (b) node 67, (c) node 99; (d) node 131.
Fig.13  Displacement history of FFEM model in different membership grades (left) and membership grade of 0.9 (right), at different nodes of elastic foundation within three fuzzy input parameters: (a) node 35; (b) node 67; (c) node 131.
Fig.14  Vertical displacement (a) node 35, (b) node 67, (c) node 99, (d) node 131, and pressure (e) node 35, (f) node 67, (g) node 99, (h) node 131 fuzzy number in five modes of analysis including: E, ν, k fuzzy; E, ν fuzzy; E fuzzy; ν fuzzy; k fuzzy for different nodes of elastic foundation within 100 s after loading.
Fig.15  Variation ratio of (a) horizontal displacements, (b) vertical displacements, (c) pore water pressure in membership grade of 0.9 for foundation.
Fig.16  Upper and lower bounds of (a) pressure, (b) vertical displacements, and (c) horizontal displacements of elastic foundation in membership grade of 0.9 by assuming E, ν, k as a fuzzy inputs within one second after loading (upper bound in left side and lower bound in right side).
Fig.17  Upper and lower bounds of (a) pressure, (b) vertical displacements, and (c) horizontal displacements of elastic foundation in membership grade of 0.9 by assuming E, ν, k as a fuzzy inputs within 100 s after loading (upper bound in left side and lower bound in right side).
1 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
2 K M Hamdia, M Silani, X Zhuang, P He, T Rabczuk. Stochastic analysis of the fracture toughness of polymeric nanoparticle composites using polynomial chaos expansions. International Journal of Fracture, 2017, 206(2): 215–227
https://doi.org/10.1007/s10704-017-0210-6
3 K M Hamdia, H Ghasemi, X Zhuang, N Alajlan, T Rabczuk. Sensitivity and uncertainty analysis for flexoelectric nanostructures. Computer Methods in Applied Mechanics and Engineering, 2018, 337: 95–109
https://doi.org/10.1016/j.cma.2018.03.016
4 S S Rao, L Berke. Analysis of uncertain structural systems using interval analysis. AIAA Journal, 1997, 35(4): 727–735
https://doi.org/10.2514/2.164
5 S S Rao, L Chen, E Mulkay. Unifed finite element method for engineering systems with hybrid uncertainties. AIAA Journal, 1998, 36(7): 1291–1299
https://doi.org/10.2514/2.513
6 R L Muhanna, R L Mullen, M V R Rao. Nonlinear interval finite elements for beams. Vulnerability, Uncertainty, and Risk ASCE, 2014: 2227–2236
7 A Sofi, G Muscolino. Static analysis of Euler–Bernoulli beams with interval Young’s modulus. Computers & Structures, 2015, 156: 72–82
https://doi.org/10.1016/j.compstruc.2015.04.002
8 Y Cheng, L L Zhang, J H Li, L M Zhang, J H Wang, D Y Wang. Consolidation in spatially random unsaturated soils based on coupled flow-deformation simulation. International Journal for Numerical and Analytical Methods in Geomechanics, 2017, 41(5): 682–706
9 M Papadopoulos, S Francois, G Degrande, G Lombaert. Analysis of stochastic dynamic soil-structure interaction problems by means of coupled finite lements perfectly matched layers. In: VII European Congress on Computational Methods in Applied Sciences and Engineering. Crete Island: ECCOMAS Congress, 2016, 5–10
10 M Effati Daryani, H Bahadori, K Effati Daryani. Soil probabilistic slope stability analysis using stochastic finite difference method. Modern Applied Science, 2017, 11(4): 23–29
https://doi.org/10.5539/mas.v11n4p23
11 S H Jiang, D Q Li, L M Zhang, C B Zhou. Slope reliability analysis considering spatially variable shear strength parameters using a non-intrusive stochastic finite element method. Engineering Geology, 2014, 168: 120–128
https://doi.org/10.1016/j.enggeo.2013.11.006
12 K Fujita, K Kojima, I Takewaki. Prediction of worst combination of variable soil properties in seismic pile response. Soil Dynamics and Earthquake Engineering, 2015, 77: 369–372
https://doi.org/10.1016/j.soildyn.2015.06.009
13 D Behera, S Chakraverty, H Z Huang. Non-probabilistic uncertain static responses of imprecisely defined structures with fuzzy parameters. Journal of Intelligent & Fuzzy Systems, 2016, 30(6): 3177–3189
https://doi.org/10.3233/IFS-152061
14 Z Luo, S Atamturktur, C H Juang, H Huang, P S Lin. Probability of serviceability failure in a braced excavation in a spatially random field: Fuzzy finite element approach. Computers and Geotechnics, 2011, 38(8): 1031–1040
https://doi.org/10.1016/j.compgeo.2011.07.009
15 Z Qiu, P C Muller, A Frommer. An approximate method for the standard interval eigenvalue problem of real non-symmetric interval matrices. Communications in Numerical Methods in Engineering Banner, 2001, 17(4): 239–251
16 S Valliappan, T D Pham. Fuzzy finite element analysis of a foundation on an elastic soil medium. International Journal for Numerical and Analytical Methods in Geomechanics, 1993, 17(11): 771–789
https://doi.org/10.1002/nag.1610171103
17 L A Zadeh. Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems, 1978, 1(1): 3–28
https://doi.org/10.1016/0165-0114(78)90029-5
18 S Valliappan, T D Pham. Elasto-plastic finite element analysis with fuzzy parameters. International Journal for Numerical Methods in Engineering, 1995, 38(4): 531–548
https://doi.org/10.1002/nme.1620380403
19 A Cherki, G Plessis, B Lallemand, T Tison, P Level. Fuzzy behavior of mechanical systems with uncertain boundary conditions. Computer Methods in Applied Mechanics and Engineering, 2000, 189(3): 863–873
https://doi.org/10.1016/S0045-7825(99)00401-6
20 B Möller, W Graf, M Beer. Fuzzy structural analysis using α-level optimization. Computational Mechanics, 2000, 26(6): 547–565
https://doi.org/10.1007/s004660000204
21 M Hanss. Applied Fuzzy Arithmetic: An Introduction with Engineering Applications. Berlin: Springer-Verlag, 2005
22 Y Liu, Z D Duan. Fuzzy finite element model updating of bridges by considering the uncertainty of the measured modal parameters. Science China. Technological Sciences, 2012, 55(11): 3109–3117
https://doi.org/10.1007/s11431-012-5009-0
23 D Behera, S Chakraverty. Fuzzy finite element analysis of imprecisely definedstructures with fuzzy nodal force. Engineering Applications of Artificial Intelligence, 2013, 26(10): 2458–2466
https://doi.org/10.1016/j.engappai.2013.07.021
24 L Yang, G Li. Fuzzy stochastic variable and variational principle. Applied Mathematics and Mechanics, 1999, 20(7): 795–800
https://doi.org/10.1007/BF02454902
25 H Huang, H Li. Perturbation finite element method of structural analysis under fuzzy environments. Engineering Applications of Artificial Intelligence, 2005, 18(1): 83–91
https://doi.org/10.1016/j.engappai.2004.08.033
26 S Abbasbandy, A Jafarian, R Ezzati. Conjugate gradient method for fuzzy symmetric positive definite system of linear equations. Applied Mathematics and Computation, 2005, 171(2): 1184–1191
https://doi.org/10.1016/j.amc.2005.01.110
27 I Skalna, M V Rama Rao, A Pownuk. Systems of fuzzy equations in structural mechanics. Journal of Computational and Applied Mathematics, 2008, 218(1): 149–156
https://doi.org/10.1016/j.cam.2007.04.039
28 N Mikaeilvand, T Allahviranloo. Solutions of the fully fuzzy linear system. In: The 39th Annual Iranian Mathematics Conference. Kerman: Shahid Bahonar University of Kerman, 2009
29 N Mikaeilvand, T Allahviranloo. Non zero solutions of the fully fuzzy linear systems. Applied and Computational Mathematics, 2011, 10(2): 271–282
30 W Verhaeghe, M D Munck, W Desmet, D Vandepitte, D Moens. A fuzzy finite element analysis technique for structural static analysis based on interval fields. In: The 4th International Workshop on Reliable Engeering Compuataions, 2010, 117–128
31 A Kumar, A Bansal. A method for solving fully fuzzy linear system with trapezoidal fuzzy numbers. Iranian Journal of Optimization, 2010, 2: 359–374
32 P Senthilkumar, G Rajendran. New approach to solve symmetric fully fuzzy linear systems. Sadhana, 2011, 36(6): 933–940
https://doi.org/10.1007/s12046-011-0059-8
33 L Farkas, D Moens, D Vandepitte, W Desmet. Fuzzy finite element analysis based on reanalysis technique. Structural Safety, 2010, 32(6): 442–448
https://doi.org/10.1016/j.strusafe.2010.04.004
34 A S Balu, B N Rao. High dimensional model representation based formulations for fuzzy finite element analysis of structures. Finite Elements in Analysis and Design, 2012, 50: 217–230
https://doi.org/10.1016/j.finel.2011.09.012
35 N Babbar, A Kumar, A Bansal. Solving fully fuzzy linear system with arbitrary triangular fuzzy numbers (m,α,β). Soft Computing, 2013, 17(4): 691–702
https://doi.org/10.1007/s00500-012-0941-2
36 M A Biot. Theory of propagation of elastic waves in a fluid-saturated porous solid. Journal of the Acoustical Society of America, 1956, 28(2): 168–178
https://doi.org/10.1121/1.1908239
37 O C Zienkiewicz, T Shiomi. Dynamic behavior of saturated porous media; the generalized Biot formulation and its numerical solution. International Journal for Numerical and Analytical Methods in Geomechanics, 1984, 8(1): 71–96
https://doi.org/10.1002/nag.1610080106
38 H Ghasemi, H S Park, T Rabczuk. A multi-material level set-based topology optimization of flexoelectric composites. Computer Methods in Applied Mechanics and Engineering, 2018, 332: 47–62
https://doi.org/10.1016/j.cma.2017.12.005
39 H Badnava, M A Msekh, E Etemadi, T Rabczuk. An h-adaptive thermo-mechanical phase field model for fracture. Finite Elements in Analysis and Design, 2018, 138: 31–47
https://doi.org/10.1016/j.finel.2017.09.003
40 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
41 X Zhuang, R Huang, C Liang, T Rabczuk. A coupled thermo-hydro-mechanical model of jointed hard rock for compressed air energy storage. Mathematical Problems in Engineering, 2014, 2014: 179169
https://doi.org/10.1155/2014/179169
42 A R Khoei, S M Azami, S M Haeri. Implementation of plasticity based models in dynamic analysis of earth and rockfill dams: A comparison of Pastor-Zienkiewicz and cap models. Computers and Geotechnics, 2004, 31(5): 384–410
https://doi.org/10.1016/j.compgeo.2004.04.003
43 J Grabe, T Hamann, A Chmelnizkij. Numerical simulation of wave propagation in fully saturated soil modeled as a two-phase medium. In: Proceedings of the 9th International Conference on Structural Dynamics. EURODYN, 2014, 631–637
44 J Ye, D Jeng, R Wang, C Zhu. Validation of a 2-D semi-coupled numerical model for fluid-structure-seabed interaction. Journal of Fluids and Structures, 2013, 42: 333–357
https://doi.org/10.1016/j.jfluidstructs.2013.04.008
45 M Y Fattah, S F Abbas, H H Karim. A model for coupled dynamic elasto-plastic analysis of soils. Journal of GeoEngineering, 2012, 7(3): 89–96
46 A Rahmani, O Ghasemi Fare, A Pak. Investigation of the influence of permeability coefficient on thenumerical modeling of the liquefaction phenomenon. Scientia Iranica, 2012, 19(2): 179–187
47 A R Khoei, E Haghighat. Extended finite element modeling of deformable porous media with arbitrary interfaces. Applied Mathematical Modelling, 2011, 35(11): 5426–5441
https://doi.org/10.1016/j.apm.2011.04.037
48 R L Muhanna, R L Mullen. Formulation of fuzzy finite-element methods for solid mechanics problems. Computer-Aided Civil and Infrastructure Engineering, 1999, 14(2): 107–117
https://doi.org/10.1111/0885-9507.00134
49 M Hanss, K Willner. A fuzzy arithmetical approach to the solution of finite element problems with uncertain parameters. Mechanics Research Communications, 2000, 27(3): 257–272
https://doi.org/10.1016/S0093-6413(00)00091-4
50 A Bárdossy, A Bronstert, B Merz. l-, 2- and 3-dimensional modeling of water movement in the unsaturated soil matrix using a fuzzy approach. Advances in Water Resources, 1995, 18(4): 237–251
https://doi.org/10.1016/0309-1708(95)00009-8
51 A Arman, N Samtani, R Castelli, G Munfakh. Geotechnical and Foundation Engineering Module1-Subsurface Investigations. Report No. FHWA-HI-97–021. 1997
52 M Das Braja. Advanced Soil Mechanics. 3rd ed. London: Taylor & Francis, 2008
53 A R Khoei, S A Gharehbaghi, A R Tabarraie, A Riahi. Error estimation, adaptivity and data transfer in enriched plasticity continua to analysis of shear band localization. Applied Mathematical Modelling, 2007, 31(6): 983–1000
https://doi.org/10.1016/j.apm.2006.03.021
54 S Zhou, X Zhuang, H Zhu, T Rabczuk. Phase field modelling of crack propagation, branching and coalescence in rocks. Theoretical and Applied Fracture Mechanics, 2018, 96: 174–192
https://doi.org/10.1016/j.tafmec.2018.04.011
55 S Zhou, X Zhuang, T Rabczuk. A phase-field modeling approach of fracture propagation in poroelastic media. Engineering Geology, 2018, 240: 189–203
https://doi.org/10.1016/j.enggeo.2018.04.008
56 S Zhou, T Rabczuk, X Zhuang. Phase field modeling of quasi-static and dynamic crack propagation: COMSOL implementation and case studies. Advances in Engineering Software, 2018, 122: 31–49
https://doi.org/10.1016/j.advengsoft.2018.03.012
57 S W Zhou, C C Xia. Propagation and coalescence of quasi-static cracks in Brazilian disks: An insight from a phase field model. Acta Geotechnica, 2018, 14(4): 1–20
58 Z Fu, W Chen, P Wen, C Zhang. Singular boundary method for wave propagation analysis in periodic structures. Journal of Sound and Vibration, 2018, 425: 170–188
https://doi.org/10.1016/j.jsv.2018.04.005
59 Z Fu, Q Xi, W Chen, A H D Cheng. A boundary-type meshless solver for transient heat conduction analysis of slender functionally graded materials with exponential variations. Computers & Mathematics with Applications (Oxford, England), 2018, 76(4): 760–773
https://doi.org/10.1016/j.camwa.2018.05.017
60 M G Katona, O C Zienkiewicz. A unified set of single step algorithms. Part 3: the beta-m method, a generalisation of the newmark scheme. International Journal for Numerical Methods in Engineering, 1985, 21(7): 1345–1359
https://doi.org/10.1002/nme.1620210713
61 M Huang, O C Zienkiewicz. New unconditionally stable staggered solution procedures for coupled soil-pore fluid dynamic problems. International Journal for Numerical Methods in Engineering, 1998, 43(6): 1029–1052
https://doi.org/10.1002/(SICI)1097-0207(19981130)43:6<1029::AID-NME459>3.0.CO;2-H
62 A H C Chan. A unified finite element solution to static and dynamic problems of geomechanics. Dissertation for the Doctoral Degree. Swansea: University of Wales, 1988
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