orthotropic seepage analysis using hybrid finite element...

Columbia International Publishing Journal of Advanced Mechanical Engineering (2015) Vol. 2 No. 1 pp. 1-13 doi:10.7726/jame.2015.1001

Research Article

______________________________________________________________________________________________________________________________ *Corresponding e-mail: [email protected]; [email protected] 1 Department of Engineering Mechanics, Henan University of Technology, Zhengzhou, 450001, China 2 Research School of Engineering, Australian National University, Canberra, ACT 2601, Australia

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Orthotropic Seepage Analysis using Hybrid Finite Element Method

Ya-Ting Gao1, Hui Wang1*, and Qing-Hua Qin2*

Received 22 August 2014; Published online 17 January 2015 © The author(s) 2015. Published with open access at www.uscip.us

Abstract A solution procedure using the hybrid finite element method is presented for two-dimensional steady-state linear seepage analysis of orthotropic dams. In the present algorithm, fundamental solutions for orthotropic seepage problems are employed to construct the element interior hydraulic head field and the conventional shape functions are used for the element frame hydraulic head field, then these two independent fields are connected by a hybrid functional. According to the physical definition of fundamental solutions, the elemental interior field can naturally satisfy the orthotropic seepage governing equation and as a result, all two-dimensional element domain integrals in the present algorithm can reduce to one-dimensional element boundary integrals. Finally, the present hybrid finite element method with orthotropic fundamental solution kernels is verified by way of analytical solutions and subsequently, numerical experiment is carried out for a practical seepage problem of an orthotropic dam. Keywords: Seepage; Orthotropic; Hybrid finite element method; Fundamental solution

1. Introduction Seepage exists widely in engineering problems associated with water. In particular, the drawdown seepage of a dam will directly influence the security of the dam, which depends on the condition of water level change. Therefore, the research on dam seepage problems is of particularly interest to geotechnical engineers in the design process of dams. Due to complicated boundary conditions and geometrical domain in most seepage problems, analytical solutions are difficult to be obtained. Numerical methods including the finite element method (FEM) (Neuman 1973; Bathe and Khoshgoftaar 1979; Qin 2000; Dhanasekar et al. 2006; Luo et al. 2008) and the boundary element method (BEM) (Brebbia and Chang 1979; Chen et al.

Y.T. Gao, H. Wang, and Q.H. Qin / Journal of Advanced Mechanical Engineering (2015) Vol. 2 No. 1 pp. 1-13

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1994; Tsay et al. 1997) are thus widely employed to analyze seepage problems in dams. For examples, Bath and Khoshgoftaar developed an effective finite element solution procedure for nonlinear free surface seepage problems without mesh iteration (Bathe and Khoshgoftaar 1979). Based on the finite element method, Luo et al. developed a three dimensional computational model of transient seepage for deep foundation pit dewatering in the Yangtze River Delta (Luo et al. 2008). Brebbia and Chang proposed a boundary element formulation for seepage problem in anisotropic soils (Brebbia and Chang 1979). Chen et al. constructed a boundary element formulation for seepage problems by introducing a dual integral equation with hypersingular integrals (Chen et al. 1994). In addition to the FEM and BEM mentioned above, meshless/meshfree methods including the method of fundamental solutions (Young et al. 2006; Chaiyo et al. 2011), the natural element method (Jie et al. 2013) and the hybrid boundary node method (Tan et al. 2010) were also developed for the numerical solution of seepage problems. In this study, another numerical method different to the FEM and BEM mentioned above, called as the fundamental solution-based hybrid finite element method, was extended to two-dimensional seepage problems in orthotropic media. The hybrid finite element formulation with fundamental solutions was firstly proposed by Wang and Qin in 2009 (Wang and Qin 2009) and then was applied to analyze elastic problems (Wang and Qin 2010; Wang and Qin 2011; Wang and Qin 2012), heat transfer problems (Wang and Qin 2011; Wang et al. 2012; Wang et al. 2013) and bioheat problems (Wang and Qin 2010; Wang and Qin 2012) with general or special elements to achieve the purpose of high accuracy and mesh reduction. In the present work, fundamental solutions of the orthotropic seepage governing equation under consideration are employed to construct the element interior hydraulic head field and the conventional shape functions are used for the element frame hydraulic head field. These two independent fields are connected by a hybrid functional. Since the fundamental solutions are used as the interpolation function of the elemental hydraulic head of interior field, it can naturally satisfy the orthotropic seepage governing equation and thus, all element domain integrals in the hybrid functional can be converted into element boundary integrals, which have lower dimension than domain integrals. Most importantly, all integrals are nonsingular, which are easily evaluated using the standard numerical Gaussian integration. In comparison with conventional FEM and the BEM, the present method inherits their advantages, such as lower-dimensional boundary integrals and domain discretization, which is important for solving problems involving multimaterials. Further, the method constructs easily n-side polygonal elements, which are different from the traditional triangular and quadrilateral elements in the FEM.

2. Mathematical Model 2.1 Basic formulation of orthotropic seepage problems

Considering a bounded domain in a two-dimensional space 2 which refers to the rectangular

coordinate 1 2,X X , the generalized steady-state water flow in homogeneous orthotropic solids is

expressed as (Mao 2003)

2 2

2

1 2 1 22 2

1 2

0 ( , )H H

k k X XX X

(1)


3

with the following boundary conditions

on

on

H

q

H H

q q

(2)

where H is the hydraulic head in the domain , and H and q are the specified hydraulic head and

seepage flow, respectively. 0ik denote orthotropic hydraulic conductivities of materials. Besides, in

Eq. (2), H q .

The normal flow flux in Eq. (2) is defined by

1 1 2 2 1 1 2 2

1 2

H Hq q n q n k n k n

X X

(3)

or in matrix form

11 1

1 2 1 2

2 2

2

0

0

H

Xq kq n n n n

q k H

X

(4)

where in are the unit outward normal to the boundary .

Fig. 1 shows a classic problem of two-dimensional seepage flow of a dam, in which the material is defined to be orthotropic along the two coordinate axes, and the bottom surface of the dam is assumed to be impermeable.

Fig. 1. Classical seepage problems of an orthotropic dam


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2.2 Fundamental solution for orthotropic seepage problems The fundamental solution for a two-dimensional orthotropic seepage problem is obtained by considering

2 * 2 *

1 22 2

1 2

( , ) 0H H

k k P QX X

(5)

where 1 2,P X X and 1 2,s sQ X X are the field point and source point, respectively.

By means of variables transformation, the induced seepage hydraulic head field *H at the field point P can be given by (Brebbia et al, 1984)

* 2 2

2 1 1 2

1 2

1, ln

2H P Q k r k r

k k (6)

in which ( 1,2)s

i i ir X X i .

Making use of Eq (6), the derivatives of the hydraulic head fundamental solution are written as

*

2 1

2 2

1 2 1 1 21 2

*

1 2

2 2

2 2 1 1 21 2

1

2

1

2

k rH

X k r k rk k

k rH

X k r k rk k

(7)

3. Hybrid Finite Element Formulation In the present hybrid finite element formulation, two independent element fields are

independently defined and are linked by a hybrid functional, from which the stiffness equation in

terms of unknown nodal potential and the relationship of these two fields are established.

3.1 Hydraulic head within the element

In the interior of an element, say element e , the field of hydraulic head can be approximated by

a linear combination of fundamental solutions of the governing equation (5), that is,

*

1

,sN

i i e

i

H P c H P Q

N c (8)

where sN is the number of source points locating outside the element domain, ic represents

unknown coefficient, and


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* * *

1 2

T

1 2

, , ,s

s

N

e N

H P Q H P Q H P Q

c c c

N

c

(9)

Differentiating Eq. (8) yields the following normal seepage flow

eq P Q c (10)

where

11

1 2

2

2

0

0

Xkn n

k

X

N

QN

(11)

with

** *

1 2,, ,

( 1,2)sN

i i i i

H P QH P Q H P Qi

X X X X

N (12)

3.2 Hydraulic head along the element boundary

The frame flow field H is independently defined in terms of element nodal hydraulic head ed by

eH P N d (13)

where { }N is an interpolation function vector consisting of the conventional shape functions used in

the FEM or BEM. For example, let’s consider a hybrid element with 5 edges and each edge has 3 nodes.

For the edge defined by nodes 3, 4, 5, the hydraulic head distribution on the quadratic edge can be

expressed in terms of the interpolation vector { }N and the nodal hydraulic head vector { }ed :

1 2 3

T

1 2 3 4 5 6 7 8 9 10

0 0 0 0 0 0 0

e

N N N

d d d d d d d d d d

N

d (14)

where iN ( 1,2,3)i stands for conventional quadratic shape functions in terms of local natural

coordinate

2

1 2 3

(1 ) (1 ), 1 ,

2 2N N N

(15)


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3.3 Hybrid functional

For a given element e, which occupying a sub-domain e with boundary e , the hybrid functional is

defined as(Qin 2005; Qin and Wang 2008)

2 2

1 2

1 2

1d d d

2 e qe ee

H Hk k qH q H H

X X

(16)

Applying the Gauss theorem, we have

2 2

1 22 2

1 2

1 1d d d d

2 2e qe e ee

H Hk k H qH qH qH

X X

(17)

Due to the physical definition of orthotropic fundamental solutions, the internal hydraulic head analytically satisfies the governing equation (1), thus the domain integral in Eq. (17) is equal to zero. As a result, one has

1

d d d2 e qe e

e qH qH qH

(18)

in which only element boundary integrals are involved. The substitution of the interior field (8) and the frame field (13) into the functional (18) yields

T T T1

2e e e e e e e e e c H c d g c G d (19)

Subsequently, the minimization of the functional e with respect to ec and ed respectively

gives

T

T

T

ee e e e

e

ee e e

e

H c G d 0c

G c g 0d

(20)

from which the optional relationship between ec and ed

1

e e e e

c = H G d (21)

and the element stiffness equation in terms of nodal hydraulic head

e e eK d = g (22)

are obtained. In Eq. (22), T 1

e e e e

K = G H G .


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Assembling the element stiffness equation (22) over all elements gives the final global stiffness equation from which the nodal hydraulic head can be obtained, and then the unknown coefficients

ec are determined using Eq. (21).

4. Numerical Experiments 4.1 Verification of the proposed model

As the first example, an orthotropic medium with domian 0,10 0,11 is considered to verify

the present algorithm. Under the assumption of orthotropic hydraulic conductivities 1 1m/sk and

2 5m/sk , the exact solution of the problem is given by

2 2

1 2 1 2 1 2, 10 2 3H X X X X X X (23)

which is also used to apply the mixed boundary conditions on the boundary of the rectangular plate. Fig. 2 shows the geometry of the domain and the specified boundary conditions for this example.

Fig. 2. Rectangular domain and boundary conditions

In the computation, 36 hybrid elements with 4 quadratic edges are used to model the computing domain, as shown in Fig. 3. Numerical solutions from the present method are compared with exact results and listed in Table 1, from which we can see that the present method has a very high accuracy. Besides, the convergence of the present algorithm is investigated and the results are plotted in Fig. 4,

in which the average relative error ( Arerr ) is defined as


8

2 2

1

( ) ( ) / ( )N N

numerical exact i exact i

i

Arerr H H H H (24)

where N is the number of sampling points. It's obvious that the error decreases with an increase in the number of elements. Finally, the entire distribution of the hydraulic head in the rectangular domain is plotted in Fig. 5 for reference.

Fig. 3. Illustration of elements division

Table 1 Hydraulic head along 2 0X

Coordinate Hydraulic head Difference

1X 2X Numerical solutions Exact solutions

0.0000 0.0000 0.0000 0.0000 0.00E+00 0.8333 0.0000 -6.9428 -6.9444 -1.64E-03 1.6667 0.0000 -27.7826 -27.7778 4.82E-03 2.5000 0.0000 -62.4978 -62.5000 -2.20E-03 3.3333 0.0000 -111.1167 -111.1111 5.59E-03 4.1667 0.0000 -173.6087 -173.6111 -2.40E-03 5.0000 0.0000 -250.0062 -250.0000 6.20E-03 5.8333 0.0000 -340.2750 -340.2778 -2.74E-03 6.6667 0.0000 -444.4513 -444.4444 6.88E-03 7.5000 0.0000 -562.4968 -562.5000 -3.20E-03 8.3333 0.0000 -694.4517 -694.4444 7.31E-03 9.1667 0.0000 -840.2749 -840.2777 -2.76E-03

10.0000 0.0000 -1000.0000 -1000.0000 0.00E+00


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Fig. 4. Convergence demonstration of the proposed approach

Fig. 5. The hydraulic head obtained from the proposed approach

4.2 Seepage in a trapezoidal dam In this example, a typical two-dimensional model of water flow through a dam shown in Fig. 6 is considered. The hydraulic conductivities of two coordinate directions are

5 5

1 24.0 10 m/s , 1.0 10 m/sk k . The hydraulic head is 15m at the upstream and 6m at the

downstream. The other edges of the dam are assumed to be impermeable.


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Fig. 6. Geometrical configuration of the dam

In the computation, 25 quadratic elements are employed to discretize the dam domain, as shown in Fig. 7. Fig. 8 displays the contour map of the hydraulic head. It is found that there is a large hydraulic head gradient in the interaction region of the right slope and the water level, while in the region close to the top surface of the dam, the hydraulic head gradient becomes small.

Fig. 7. Mesh division for the dam


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Fig. 8. Contour map of the hydraulic head in the dam

5. Conclusions A novel hybrid finite element model using fundamental solution kernels is developed for solving orthotropic plane seepage problem in this study. In the algorithm, the use of orthotropic fundamental solutions makes all integrals to be calculated along the element boundary only, and also it is easy to generate arbitrary-shaped hybrid elements with multiple edges and nodes. The obtained numerical results are compared with analytical solutions to verify the correctness and convergence performance of the present algorithm. The seepage in a practical dam is conducted to represent the change of hydraulic head gradient in the dam. Numerical experiments show that the proposed method is effective for two-dimensional orthotropic steady-state linear seepage analysis. A further study on nonlinear seepage with free surface using the presented algorithm is under way.

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