Uncertainty assessment in hydro-mechanical-coupled analysis of saturated porous medium applying fuzzy finite element method

doi:10.1007/s11709-019-0601-z

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 Author(s): 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.

URL:

https://academic.hep.com.cn/fsce/EN/10.1007/s11709-019-0601-z
https://academic.hep.com.cn/fsce/EN/Y2020/V14/I2/387

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
$ρ s (kg / m 3)$		2000	2000
$ρ f (kg / m 3)$		1000	1000
$k f (P a)$		2.1E9	2.1E9
$k s (P a)$		1.0E20	1.0E20
n		0.3	0.3
$k γ (m 3 s / kg)$	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)	$k γ (m 3 s / kg)$
coarse	10–10³	10⁻⁶–10⁻⁴
fine gravel, coarse, and medium sand	10⁻²–10	10⁻⁹–10⁻⁶
fine sand, loose silt	10⁻⁴–10⁻²	10⁻¹¹–10⁻⁹

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
$ρ s (kg / m 3)$		2000	2000	2000
$ρ f (kg / m 3)$		1000	1000	1000
$k f (P a)$		2.1E9	2.1E9	2.1E9
$k s (P a)$		1.0E20	1.0E20	1.0E20
n		0.25	0.3	0.35
$k γ (m 3 s / kg)$	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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