Maximum entropy based finite element analysis of porous media

doi:10.1007/s11709-018-0470-x

Frontiers of Structural and Civil Engineering

2019, Vol. 13

Issue (2): 364-379 https://doi.org/10.1007/s11709-018-0470-x

本期目录

Maximum entropy based finite element analysis of porous media

Emad NOROUZI, Hesam MOSLEMZADEH, Soheil MOHAMMADI(

)

High Performance Computing Laboratory, School of Civil Engineering, Faculty of Engineering, University of Tehran, Tehran, Iran

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Abstract：

The maximum entropy theory has been used in a wide variety of physical, mathematical and engineering applications in the past few years. However, its application in numerical methods, especially in developing new shape functions, has attracted much interest in recent years. These shape functions possess the potential for performing better than the conventional basis functions in problems with randomly generated coarse meshes. In this paper, the maximum entropy theory is adopted to spatially discretize the deformation variable of the governing coupled equations of porous media. This is in line with the well-known fact that higher-order shape functions can provide more stable solutions in porous problems. Some of the benchmark problems in deformable porous media are solved with the developed approach and the results are compared with available references.

Key words： maximum entropy FEM fully coupled multi-phase system porous media

收稿日期: 2017-07-30 出版日期: 2019-03-12

Corresponding Author(s): Soheil MOHAMMADI

引用本文:

. [J]. Frontiers of Structural and Civil Engineering, 2019, 13(2): 364-379.
Emad NOROUZI, Hesam MOSLEMZADEH, Soheil MOHAMMADI. Maximum entropy based finite element analysis of porous media. Front. Struct. Civ. Eng., 2019, 13(2): 364-379.

链接本文:

https://academic.hep.com.cn/fsce/CN/10.1007/s11709-018-0470-x
https://academic.hep.com.cn/fsce/CN/Y2019/V13/I2/364

Fig.1

Fig.2

base	position	x direction	y direction	water pressure
rigid base	top	free	rigid	constant $P w$ =0.0 Pa
	bottom	rigid	free	impermeable
	left	rigid	free	impermeable
	right	rigid	free	impermeable
smooth base	top	free	rigid	constant $P w$ =0.0 Pa
	bottom	rigid	free	impermeable
	left	free	free	impermeable
	right	free	free	impermeable

Tab.1

item	symbol	value
porosity	n	0.3
biot coefficient	Α	1
elastic module of solid (N/m²)	E	1e4
Poissoin’s ratio	Ν	0.0
solid density (kg/m³)	$ρ s$	2700
water density (kg/m³)	$ρ w$	1000
intrinsic permeability	k	1e?6
dynamic viscosity of water (Pa·s)	$μ w$	1.1e?3

Tab.2

Fig.3

Fig.4

Fig.5

position	x direction	y direction	water pressure	Heat condition
top	free	rigid	constant $P w = 0.0$ Pa	constant $T = 50$ K
bottom	rigid	free	impermeable	impermeable
left	rigid	free	impermeable	impermeable
right	rigid	free	impermeable	impermeable

Tab.3

item	symbol	value
porosity	n	0.3
biot coefficient	Α	1
elastic module of solid (N/m²)	E	6e6
Poissoin’s ratio	$ν$	0.4
solid density (kg/m³)	$ρ s$	2000
water density (kg/m³)	$ρ w$	1000
intrinsic permeability	k	4e?9
dynamic viscosity of water (Pa·s)	$μ w$	1e?3
Effective special heat capacity (kcal/m·K·s)	$(ρ c) eff$	40
Effective thermal conductivity (kcal/m·K·s)	$λ eff$	0.2

Tab.4

Fig.6

Fig.7

Fig.8

Fig.9

item	symbol	value
porosity	n	0.2975
biot coefficient	α	1
elastic module of solid (N/m²)	E	1.3e6
Poissoin’s Ratio	$ν$	0.4
density of soil (kg/m³)	$ρ s$	2000
density of water (kg/m³)	$ρ w$	1000
density of gas (kg/m³)	$ρ g$	1.22
bulk module of solid (N/m²)	$K s$	1e12
bulk module of water (N/m²)	$K w$	2e9
bulk module of gas (N/m²)	$K g$	0.1e6
intrinsic permeability (m²)	k	4.5e?13
dynamic viscosity of water (Pa·s)	$μ w$	1e?3
dynamic viscosity of gas (Pa·s)	$μ g$	1.8e?5

Tab.5

Fig.10

position	x direction	y direction	water pressure
top	free	rigid	impermeable
bottom	rigid	free	constant $P w$ =0.0 Pa
left	rigid	free	impermeable
right	rigid	free	impermeable

Tab.6

Fig.11

Fig.12

Fig.13

Fig.14

Fig.15

position	x direction	y direction	water pressure	air pressure
top	free	rigid	impermeable	constant $P air$ =0.0 Pa
bottom	rigid	free	constant $P w$ =0.0 Pa	constant $P air$ =0.0 Pa
left	rigid	free	impermeable	impermeable
right	rigid	free	impermeable	impermeable

Tab.7

Fig.16

Fig.17

Fig.18

Fig.19

Fig.20

Fig.21

1	JTouma, M Vauclin. Experimental and numerical analysis of two-phase infiltration in a partially saturated soil. Transport in Porous Media, 1986, 1(1): 27–55 https://doi.org/10.1007/BF01036524
2	C RFaust, J H Guswa, J W Mercer. Simulation of three-dimensional flow of immiscible fluids within and below the unsaturated zone. Water Resources Research, 1989, 25(12): 2449–2464 https://doi.org/10.1029/WR025i012p02449
3	BAtaie-Ashtiani, D Raeesi-Ardekani. Comparison of numerical formulations for two-phase flow in porous media. Geotechnical and Geological Engineering, 2010, 28(4): 373–389 https://doi.org/10.1007/s10706-009-9298-4
4	L JDurlofsky. A triangle based mixed finite element—finite volume technique for modeling two phase flow through porous media. Journal of Computational Physics, 1993, 105(2): 252–266 https://doi.org/10.1006/jcph.1993.1072
5	P AForsyth, Y Wu, KPruess. Robust numerical methods for saturated-unsaturated flow with dry initial conditions in heterogeneous media. Advances in Water Resources, 1995, 18(1): 25–38 https://doi.org/10.1016/0309-1708(95)00020-J
6	PJenny, S H Lee, H A Tchelepi. Adaptive multiscale finite-volume method for multiphase flow and transport in porous media. Multiscale Modeling & Simulation, 2005, 3(1): 50–64 https://doi.org/10.1137/030600795
7	WKlieber, B Rivière. Adaptive simulations of two-phase flow by discontinuous Galerkin methods. Computer Methods in Applied Mechanics and Engineering, 2006, 196(1‒3): 404–419 https://doi.org/10.1016/j.cma.2006.05.007
8	YEpshteyn, B Rivière. Fully implicit discontinuous finite element methods for two-phase flow. Applied Numerical Mathematics, 2007, 57(4): 383–401 https://doi.org/10.1016/j.apnum.2006.04.004
9	.Li OX, Zienkiewicz. Multiphase flow in deforming porous media and finite element solutions. Computers & Structures, 1992, 45(2): 211–227 https://doi.org/10.1016/0045-7949(92)90405-O
10	N ARahman, R W Lewis. Finite element modelling of multiphase immiscible flow in deforming porous media for subsurface systems. Computers and Geotechnics, 1999, 24(1): 41–63 https://doi.org/10.1016/S0266-352X(98)00029-9
11	LLaloui, G Klubertanz, LVulliet. Solid–liquid–air coupling in multiphase porous media. International Journal for Numerical and Analytical Methods in Geomechanics, 2003, 27(3): 183–206 https://doi.org/10.1002/nag.269
12	GOettl, R Stark, GHofstetter. Numerical simulation of geotechnical problems based on a multi-phase finite element approach. Computers and Geotechnics, 2004, 31(8): 643–664 https://doi.org/10.1016/j.compgeo.2004.10.002
13	RStelzer, G Hofstetter. Adaptive finite element analysis of multi-phase problems in geotechnics. Computers and Geotechnics, 2005, 32(6): 458–481 https://doi.org/10.1016/j.compgeo.2005.06.003
14	CCallari, A Abati. Finite element methods for unsaturated porous solids and their application to dam engineering problems. Computers & Structures, 2009, 87(7‒8): 485–501 https://doi.org/10.1016/j.compstruc.2008.12.012
15	V PNguyen, H Lian, TRabczuk, SBordas. Modelling hydraulic fractures in porous media using flow cohesive interface elements. Engineering Geology, 2017, 225: 68–82 https://doi.org/10.1016/j.enggeo.2017.04.010
16	SSamimi, A Pak. Three-dimensional simulation of fully coupled hydro-mechanical behavior of saturated porous media using Element Free Galerkin (EFG) method. Computers and Geotechnics, 2012, 46: 75–83 https://doi.org/10.1016/j.compgeo.2012.06.004
17	MGoudarzi, S Mohammadi. Weak discontinuity in porous media: an enriched EFG method for fully coupled layered porous media. International Journal for Numerical and Analytical Methods in Geomechanics, 2014, 38(17): 1792–1822 https://doi.org/10.1002/nag.2281
18	MGoudarzi, S Mohammadi. Analysis of cohesive cracking in saturated porous media using an extrinsically enriched EFG method. Computers and Geotechnics, 2015, 63: 183–198 https://doi.org/10.1016/j.compgeo.2014.09.007
19	SSamimi, A Pak. A three-dimensional mesh-free model for analyzing multi-phase flow in deforming porous media. Meccanica, 2016, 51(3): 517–536 https://doi.org/10.1007/s11012-015-0231-z
20	TMohammadnejad, A Khoei. Hydro-mechanical modeling of cohesive crack propagation in multiphase porous media using the extended finite element method. International Journal for Numerical and Analytical Methods in Geomechanics, 2013, 37(10): 1247–1279 https://doi.org/10.1002/nag.2079
21	MGoodarzi, S Mohammadi, AJafari. Numerical analysis of rock fracturing by gas pressure using the extended finite element method. Petroleum Science, 2015, 12(2): 304–315 https://doi.org/10.1007/s12182-015-0017-x
22	TMohammadnejad, A R Khoei. An extended finite element method for hydraulic fracture propagation in deformable porous media with the cohesive crack model. Finite Elements in Analysis and Design, 2013, 73: 77–95 https://doi.org/10.1016/j.finel.2013.05.005
23	XZhuang, Q Wang, HZhu. 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
24	HZhu, Q Wang, XZhuang. 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
25	HBayesteh, S Mohammadi. Micro-based enriched multiscale homogenization method for analysis of heterogeneous materials. International Journal of Solids and Structures, 2017, 125: 22–42 https://doi.org/10.1016/j.ijsolstr.2017.07.018
26	PFatemi Dehaghani, SHatefi Ardakani, HBayesteh, SMohammadi. 3D hierarchical multiscale analysis of heterogeneous SMA based materials. International Journal of Solids and Structures, 2017, 118–119: 24–40 https://doi.org/10.1016/j.ijsolstr.2017.04.025
27	A IBeltzer. Entropy characterization of finite elements. International Journal of Solids and Structures, 1996, 33(24): 3549–3560 https://doi.org/10.1016/0020-7683(95)00193-X
28	C EShannon. Communication theory of secrecy systems. Bell Labs Technical Journal, 1949, 28(4): 656–715 https://doi.org/10.1002/j.1538-7305.1949.tb00928.x
29	NSukumar. Construction of polygonal interpolants: a maximum entropy approach. International Journal for Numerical Methods in Engineering, 2004, 61(12): 2159–2181 https://doi.org/10.1002/nme.1193
30	MArroyo, M Ortiz. Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods. International Journal for Numerical Methods in Engineering, 2006, 65(13): 2167–2202 https://doi.org/10.1002/nme.1534
31	DMillán, N Sukumar, MArroyo. Cell-based maximum-entropy approximants. Computer Methods in Applied Mechanics and Engineering, 2015, 284: 712–731 https://doi.org/10.1016/j.cma.2014.10.012
32	AOrtiz, M Puso, NSukumar. Maximum-entropy meshfree method for compressible and near-incompressible elasticity. Computer Methods in Applied Mechanics and Engineering, 2010, 199(25): 1859–1871 https://doi.org/10.1016/j.cma.2010.02.013
33	AOrtiz, M Puso, NSukumar. Maximum-entropy meshfree method for incompressible media problems. Finite Elements in Analysis and Design, 2011, 47(6): 572–585 https://doi.org/10.1016/j.finel.2010.12.009
34	GQuaranta, S K Kunnath, N Sukumar. Maximum-entropy meshfree method for nonlinear static analysis of planar reinforced concrete structures. Engineering Structures, 2012, 42: 179–189 https://doi.org/10.1016/j.engstruct.2012.04.020
35	ZUllah, W Coombs, CAugarde. An adaptive finite element/meshless coupled method based on local maximum entropy shape functions for linear and nonlinear problems. Computer Methods in Applied Mechanics and Engineering, 2013, 267: 111–132 https://doi.org/10.1016/j.cma.2013.07.018
36	FAmiri, C Anitescu, MArroyo, S P ABordas, TRabczuk. 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
37	FAmiri, D Millán, YShen, TRabczuk, MArroyo. 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
38	FAmiri, D Millán, MArroyo, MSilani, TRabczuk. Fourth order phase-field model for local max-ent approximants applied to crack propagation. Computer Methods in Applied Mechanics and Engineering, 2016, 312: 254–275 https://doi.org/10.1016/j.cma.2016.02.011
39	CWu, D Young, HHong. Adaptive meshless local maximum-entropy finite element method for convection-diffusion problems. Computational Mechanics, 2014, 53(1): 189–200 https://doi.org/10.1007/s00466-013-0901-4
40	OKardani, M Nazem, MKardani, SSloan. On the application of the maximum entropy meshfree method for elastoplastic geotechnical analysis. Computers and Geotechnics, 2017, 84: 68–77 https://doi.org/10.1016/j.compgeo.2016.11.015
41	MNazem, M Kardani, BBienen, MCassidy. A stable maximum-entropy meshless method for analysis of porous media. Computers and Geotechnics, 2016, 80: 248–260 https://doi.org/10.1016/j.compgeo.2016.08.021
42	PNavas, S López-Querol, R CYu, BLi. Meshfree Methods Applied to Consolidation Problems in Saturated Soils. In: Weinberg K, Pandolfi A, eds. Innovative Numerical Approaches for Multi-Field and Multi-Scale Problems. Springer, 2016, 241–264
43	PNavas, S López-Querol, R CYu, BLi. B-bar based algorithm applied to meshfree numerical schemes to solve unconfined seepage problems through porous media. International Journal for Numerical and Analytical Methods in Geomechanics, 2016, 40(6): 962–984 https://doi.org/10.1002/nag.2472
44	PNavas, R C Yu, S López-Querol, BLi. Dynamic consolidation problems in saturated soils solved through u–w formulation in a LME meshfree framework. Computers and Geotechnics, 2016, 79: 55–72 https://doi.org/10.1016/j.compgeo.2016.05.021
45	NZakrzewski, M Nazem, S WSloan, MCassidy et al. On application of the maximum entropy meshless method for large deformation analysis of geotechnical problems. In: Gu Y, Guan H, Sauret E, Saha S, Zhan H, Persky R, eds. Applied Mechanics and Materials. 2016. Trans Tech Publ, 331–335
46	R WLewis, B A Schrefler. The finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media. John Wiley& Sons, 1998
47	E TJaynes. On the rationale of maximum-entropy methods. Proceedings of the IEEE, 1982, 70(9): 939–952 https://doi.org/10.1109/PROC.1982.12425
48	S FGull, J Skilling. Maximum entropy method in image processing. In: IEE Proceedings F- Communications, Radar and Signal Processing. IET, 1984
49	AGolan, G G Judge, D Miller. Maximum Entropy Econometrics. John Wiley & Sons, 1996
50	. JKarmeshu. Entropy Measures, Maximum Entropy Principle and Emerging Applications. Springer Science & Business Media, 2003
51	E TJaynes. Information theory and statistical mechanics. Physical Review, 1957, 106(4): 620–630 https://doi.org/10.1103/PhysRev.106.620
52	DGawin, P Baggio, B ASchrefler. Coupled heat, water and gas flow in deformable porous media. International Journal for Numerical Methods in Fluids, 1995, 20(8–9): 969–987 doi:10.1002/fld.1650200817
53	AKhoei, T Mohammadnejad. Numerical modeling of multiphase fluid flow in deforming porous media: a comparison between two-and three-phase models for seismic analysis of earth and rockfill dams. Computers and Geotechnics, 2011, 38(2): 142–166 https://doi.org/10.1016/j.compgeo.2010.10.010
54	B ASchrefler, RScotta. A fully coupled dynamic model for two-phase fluid flow in deformable porous media. Computer Methods in Applied Mechanics and Engineering, 2001, 190(24–25): 3223–3246 https://doi.org/10.1016/S0045-7825(00)00390-X
55	R HBrooks, A T Corey, 0. Hydraulic properties of porous media and their relation to drainage design. Transactions of the ASAE. American Society of Agricultural Engineers, 1964, 7(1): 26–28 https://doi.org/10.13031/2013.40684
56	J RBooker, J Small. Finite layer analysis of consolidation. I. International Journal for Numerical and Analytical Methods in Geomechanics, 1982, 6(2): 151–171 https://doi.org/10.1002/nag.1610060204
57	JBooker, J Small. A method of computing the consolidation behaviour of layered soils using direct numerical inversion of Laplace transforms. International Journal for Numerical and Analytical Methods in Geomechanics, 1987, 11(4): 363–380 https://doi.org/10.1002/nag.1610110405
58	RGibson, R Schiffman, SPu. Plane strain and axially symmetric consolidation of a clay layer on a smooth impervious base. Quarterly Journal of Mechanics and Applied Mathematics, 1970, 23(4): 505–520 https://doi.org/10.1093/qjmam/23.4.505
59	BAboustit, S Advani, JLee. Variational principles and finite element simulations for thermo-elastic consolidation. International Journal for Numerical and Analytical Methods in Geomechanics, 1985, 9(1): 49–69 https://doi.org/10.1002/nag.1610090105
60	A CLiakopoulos. Transient Flow Through Unsaturated Porous Media. Dissertation for PhD degree. University of California, Berkeley. 1964
61	T NNarasimhan, PWitherspoon. Numerical model for saturated-unsaturated flow in deformable porous media: 3. Applications. Water Resources Research, 1978, 14(6): 1017–1034 https://doi.org/10.1029/WR014i006p01017
62	BSchrefler, L Simoni. A unified approach to the analysis of saturated-unsaturated elastoplastic porous media. Numerical Methods in Geomechanics, 1988, 1: 205–212
63	OZienkiewicz, Y M Xie, B A Schrefler, A Ledesma, NBicanic. Static and dynamic behaviour of soils: a rational approach to quantitative solutions. II. Semi-saturated problems. In: Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences. The Royal Society, 1990
64	BSchrefler, Zhan X. A fully coupled model for water flow and airflow in deformable porous media. Water Resources Research, 1993, 29(1): 155–167 https://doi.org/10.1029/92WR01737
65	DGawin, B A Schrefler, M Galindo. Thermo-hydro-mechanical analysis of partially saturated porous materials. Engineering Computations, 1996, 13(7): 113–143 https://doi.org/10.1108/02644409610151584
66	.Wang WX, B Schrefler. Fully coupled thermo-hydro-mechanical analysis by an algebraic multigrid method. Engineering Computations, 2003, 20(2): 211–229 https://doi.org/10.1108/02644400310465326
67	WEhlers, T Graf, MAmmann. Deformation and localization analysis of partially saturated soil. Computer Methods in Applied Mechanics and Engineering, 2004, 193(27): 2885–2910 https://doi.org/10.1016/j.cma.2003.09.026

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