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Technical Communication

Treatment of transitional element with the Monte Carlo method for

FEM-based seepage analysis

Yu-xin Jie a, Xu-dong Fu a, Gang Deng b,

a State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084, Chinab State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin, China Institute of Water Resources and Hydropower Research, Beijing 100048, China

a r t i c l e i n f o

Article history:

Received 2 January 2013

Received in revised form 15 February 2013

Accepted 21 February 2013

Available online 30 March 2013

Keywords:

Monte Carlo method

Numerical integration

Seepage

Finite element method (FEM)

Free surface

a b s t r a c t

In this paper, The Monte Carlo method is incorporated into the finite element method (FEM) to conduct

seepage analysis with a free surface. For the transitional element cut by the free surface, it is used to cal-

culate the composite permeability coefficient, as well as to perform the integration directly using Monte

Carlo integration. This new algorithm requires less iteration procedures for convergence. The conver-

gence of the method is also proved for cases where there is a significant difference between the perme-

ability coefficient above and below the free surface.

2013 Elsevier Ltd. All rights reserved.

1. Introduction

One of the major difficulties in FEM-based seepage analysis of

levees and dams is that the free surface is not known a priori. In

general, two approaches have been developed to deal with such

problems. One assumes a free surface initially and adjusts it itera-

tively, resulting in a moving-mesh procedure that continues until

the boundary conditions are sufficiently satisfied on the free sur-

face[1]. The other is a fixed mesh method in which the finite ele-

ment meshes are fixed during the iterations. For example, Desai

presented residual flow schemes[2,3], while Bathe and Khoshgof-

taar[4]proposed a model considering meshes on the free surface

to be composed of composite materials. Saturated and unsaturated

theory can also be used to solve this problem with a fixed mesh, in

which the zero pressure contour serves as the free surface [5,6].

There are many other techniques that are also used to solve theseproblems, such as the variational inequality method, the numerical

manifold method, the boundary-fitted coordinate transformation

method, the finite-volume method, and the meshless method[7

11].

The fixed mesh finite element method is more convenient when

the seepage problem is coupled with stressstrain analysis, e.g., in

consolidation analysis with a free surface. In the method proposed

by Bathe and Khoshgoftaar[4], the composite permeability coeffi-

cient of the transitional element cut by the free surface is deter-

mined by the ratio between the parts above and below the free

surface. For solving the problem with saturated and unsaturated

theory, there is also a marked difference in the permeability be-

tween the upper and lower parts of the transitional element. In

fact, Bathe and Khoshgoftaars method can be considered as a spe-

cial case of saturated and unsaturated theory. The method for esti-

mating the composite permeability correctly is crucial in the

numerical analysis. If the permeability coefficient is not suitably

calculated, iterations may fail leading to a wrong result being

obtained.

In this paper, we deal with the problem through integrating the

Monte Carlo method with finite element analysis. This proposed

method can be used to calculate the composite permeability coef-

ficient of the transitional element cut by the free surface, or to do

the integration directly by Monte Carlo integration. A program

written by the authors was used to analyze several seepage prob-lems with a free surface, and the computational examples show

that this method improves the convergence rate greatly and sim-

plifies the program.

2. The Monte Carlo method

The Monte Carlo method approximates solutions to mathemat-

ical or physical problems by using statistical sampling theory. It is

based on the law of large numbers in probability theory [12,13],

and is promising in many aspects and simple to implement.

0266-352X/$ - see front matter 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compgeo.2013.02.005

Corresponding author. Tel.: +86 10 68785126.

E-mail address:dgang@iwhr.com(G. Deng).

Computers and Geotechnics 52 (2013) 16

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The Monte Carlo method provides a simple approach to con-

ducting numerical integration called Monte Carlo integration. For

the integration

I

Z ba

fxdx; 1

the value ofIcan be considered to be the product of b a and theaverage value offx over a; b. The average value offx can be ob-tained by generating a random sample x1, . . .,xn that is uniformly

distributed over a; b. One then obtains:

I bafx1 fxn

n : 2

Ifn !1, Iwill converge to Iwith probability 1. The error in theintegration is proportional to the standard error of Iand is inver-

sely proportional toffiffiffi

np

, so we can achieve arbitrary accuracy with

sufficiently many sample points.

The Monte Carlo integration needs a lot of sampling data, which

lowers its computational efficiency in most cases in contrast to the

Gaussian integration. With the development of computer hard-

ware, restrictions on the running speed have become less and less

important. The Monte Carlo method has become more and moreattractive due to its simplicity and adaptability, and the fact that

the probability and rate of convergence are independent of the

dimension of the problem. It has been used to calculate the area

or volume of a geometric object as well as the location of the cen-

troid of a region. For example, we found that the Du et al.s method

for determining Centroidal Voronoi tessellations (CVTs) can be

considered as a probabilistic version of Lloyds method[14,15],

the essence of which is actually computing the centroids of Voro-

noi regions by the Monte Carlo integration.

In the field of geomechanics, the Monte Carlo method has

mainly been used to solve problems involving spatial and temporal

variability of rock and soil parameters [1623], but has not yet

been directly used in seepage analysis with a free surface.

3. Saturated steady seepage

3.1. Governing equations

The differential equation governing steady-state saturated

water flow is

@

@x kx

@h

@x

@

@y ky

@h

@y

@

@z kz

@h

@z

0; 3

wherekx, ky, kzare the permeability coefficients in the x , y , and z

directions, h =z+p/cw is the water head, p is the pore water pres-

sure, and cw is the unit weight of water. Note that here we assume

the coordinate system (x,y,z) corresponds to the principal direc-

tions of the hydraulic conductivity tensor.The finite element equation is

Kfhg fQg; 4

whereK is the element characteristic matrix,fhg is the vector ofnodal heads, and fQg is the element flux vector.

3.2. Treatment of transitional elements

In the adjusting permeability method proposed by Bathe and

Khoshgoftaar [4], the permeability coefficient of the transitional

element is adjusted to approximate the real seepage field. For the

portion of the element below the free surface, the soil is supposed

to be saturated with a permeability coefficient ofk, while for the

portion of the element above the free surface the permeabilitycoefficient is gk, generally with g = 1/1000. That is,

g 1; pP 0;

g 1=1000; p< 0; 5

wherep is the pore water pressure.

However, there are often difficulties with the convergence of

this method, as the abrupt jump in the permeability coefficient

leads to oscillation of the solution during iterations. There are

two general approaches to solving this problem. One is to replace

the step function for g with a continuous function (see Fig. 1), to

control the change in the permeability coefficient during the itera-

tions. The other is to use more Gaussian integration points to cap-

ture the change in permeability in the transitional element cut by

the free surface. For the transitional elements inFig. 2, if all of the

integration points are above the free surface (see Fig. 2a), the ele-

ment is considered to be an unsaturated material and the influence

of the lower part of the element is totally omitted. Conversely, if all

of the integration points are below the free surface (see Fig. 2b), the

element is considered to be a saturated material and the influence

of the upper part of the element is omitted. In general, this omis-

sion may be remedied to some extend by adding more Gaussian

points. However, the effect of adding Gaussian points is limited.

Since the number of Gaussian points is unlikely to be extremely

large (e.g., up to 10,000 points), thus the situations indicated inFig. 2will always occur.

An possible alternative method is to determine the composite

permeability coefficient of the transitional element according to

the volume ratio between the two parts above and below the free

surface, whereas it is troublesome to calculate the volume of the

irregular geometric parts as well as the pattern with which the

transitional element is cut by the free surface [24,25]. Therefore,

the Monte Carlo method seems promising in these cases.

The Monte Carlo method is feasible for the following two

approaches:

(1) The Monte Carlo method is used to perform numerical inte-

gration on the transitional elements, while Gaussian integra-

tion is used for the other elements.(2) Gaussian integration is used on all the elements. The perme-

ability coefficient of the elements below and above the free

surface are specified to be k and gk, respectively, while for

the transitional elements, the composite permeability coeffi-

cient is determined by the Monte Carlo method. For exam-

ple, as shown inFig. 3, we generaten random points in the

element. Assume that the number of points below the free

surface is n1, and that the number above the free surface is

n2 (n= n1+ n2). Then the composite permeability coefficient

of the transitional element is thought to be

kn1nk1

n2nk2; 6

wherek1= k andk2= gk.Obviously, n1=nandn2=nare approximations of the volume ra-

tiosV1/VandV2/V, respectively, whereV1 is the volume below the

free surface, V2 is the volume above the free surface, and V= V1+ V2is the total volume of the transitional element. Thus kin Eq.(6)is

actually determined by the volume ratio between the parts above

and below the free surface.

The advantage of the Monte Carlo method is that it avoids the

problems of calculating the volume of the transitional element as

well as how the element is cut by the free surface. The location

and the related permeability coefficient of the sample points can

be simply determined from the pore water pressure p. For exam-

ple, if p> 0, the point is below the free surface, and ifp< 0, the

point is above the free surface. Additionally, in comparison with

Gaussian points, the number of points can be very large by theMonte Carlo method, e.g., 10,000, even 100,000 or 1,000,000 for

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an element, thus the situations indicated in Fig. 2 can be largelyavoided.

3.3. Examples

Fig. 4shows a finite element model for a rectangular dam. The

model consists of 4-node linear elements. The width and the height

are 4 m and 6 m, respectively. The upstream (left) water level is

6 m, and the downstream(right) water level is 1 m. The permeabil-

ity coefficient is 1 103 cm/s, while g in Eq.(5)is assigned to be1/1000. The error tolerance is 0.1%.

The following three schemes are used and compared in the

analysis:

Scheme 1: Gaussian integration is used on all the elements, andthe number of Gaussian integration points in each element is

2 2.Scheme 2: Monte Carlo integration is employed on the transi-

tional elements, while Gaussian integration is still used on

other elements.

Scheme 3: Gaussian integration is used on all the elements as in

Scheme 1. For transitional element, the composite permeability

coefficient is obtained by the Monte Carlo method according to

Eq.(6).

The computational procedure is as follows:

1. Assume an initial release point on the downstream boundary.

2. Compute the water head fhg1according to Eq.(4),in which thepermeability coefficient of each element is set to be k.

3. Determine the relationship between each element and the free

surface forfhg1: an element is below the free surface if all thenodal water pressures are bigger than 0, and is above the free

surface if all the nodal water pressures are smaller than 0. An

element is a transitional element cut by the free surface if some

of the nodal water pressures are bigger than 0 and the others

are smaller than 0.

4. Compute the water head (fhg2) again, where the permeabilitycoefficient of elements below the free surface and those above

the free surface are set to be k and gk, respectively, and the

transitional elements are dealt with according to the three

schemes explained above.5. If the difference between fhg2 and fhg1 is bigger than the error

tolerance, setfhg1 fhg1 fhg2=2 to determine the perme-ability coefficient of each element.

6. Repeat steps 25 until the error tolerance is met.

7. Check the boundary conditions on the downstream boundary. If

the boundary conditions are not fulfilled, modify the release

point.

8. Repeat steps 27 until the boundary conditions are fulfilled.

In our analysis, the number of random points for the Monte Car-

lo method was set to be 10,000.

We found that convergence is usually achieved in 250 itera-

tions, and generally less than 100 iterations, by Schemes 2 and 3,

and that the free surface obtained is quite smooth (see Fig. 5a).On the other hand, convergence failed even after 1000 iterations

k

k

p

k

k

k

p

k

(a) The step function (b) The continuous function

Fig. 1. Permeability coefficient vs. pore water pressure.

(a) Integration points above the

free surface

(b) Integration points below the

free surface

Free surface

Free surface

Fig. 2. Free surface and the integration points.

V1

V2 Free surface

Fig. 3. Transitional element cut by the free surface.

Fig. 4. Finite element mesh for a rectangular dam (in m).

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by Scheme 1. Fig. 5b also shows the free surface obtained by

Scheme 1. The curve is obviously rough, although the location of

the free surface and the release point are not incorrect. However,

the non-convergent results are theoretically unreliable and should

be rejected.

Fig. 6is a finite element model for a trapezoidal dam. The up-

stream(left) water level is 18 m, and the downstream (right) water

level is 0 m. Again, convergence is achieved in less than 100 itera-

tions by Schemes 2 and 3, and fails by Scheme 1. The free surfaceobtained by Schemes 2 and 3 is again quite smooth (seeFig. 7).

Further calculations show that for the two examples inFigs. 4

and 6, convergence is achieved within 200 iterations by Schemes

2 and 3 for g= 104, while convergence is achieved within 200 iter-ations by Scheme 3 and within 500 iterations by Scheme 2 for

g= 105. Even in the extreme case of g= 1010, convergence canstill be achieved within 200 iterations by Scheme 3 although con-

vergence difficulties emerges for Scheme 2.

The Monte Carlo method is also feasible for analyzing seepage

problems in three dimensions. Fig. 8presents a 3-D finite element

model for a rectangular dam, where the width and height in thexz-

plane are 4 m and 6 m, as in Fig. 4, and the thickness in theydirec-

tion is 2 m. The upstream (left) water level is 6 m, and the down-

stream (right) water level is 1 m, again as in Fig. 4. Convergencefails after 1000 iterations by Scheme 1, while Scheme 3 continues

to show good convergence characteristics. The free surface ob-

tained by Scheme 3 is shown inFig. 9.

4. Unsaturated unsteady seepage

For unsaturated seepage, the zero pressure contour serves as

the free surface. The permeability coefficient is a function of pore

water pressure (seeFig. 10). In fact, the curve ofk vs. p in Fig. 1

can be considered to be a special case for unsaturated soil.

The differential equation governing unsaturated unsteady

water flow is

@

@x kx

@h

@x

@

@y ky

@h

@y

@

@z kz

@h

@z

@nSr

@t ; 7

where nis the porosity of the soil, and Sris the degree of saturation.As does in Eq. (3), here we also assume the coordinate system

(a) Schemes 2 and 3 (b) Scheme 1

Fig. 5. Calculated free surface (in m).

Fig. 6. Finite element mesh for a trapezoidal dam (in m).

Fig. 7. Calculated free surface by Schemes 2 and 3 (in m).

Fig. 8. 3-D finite element mesh for a rectangular dam.

Fig. 9. Calculated 3-D free surface by Scheme 3.

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(x,y,z) corresponds to the principal directions of the hydraulic con-

ductivity tensor.In numerical analysis of unsaturated soil, the permeability coef-

ficient is also adjusted according to the calculated pore water pres-

sure, not only for the transitional element but also for an element

with negative pore water pressure. Consequently, a convergence

problem occurs when there is a big change in the permeability

coefficient during iterations. Again, the Monte Carlo method can

be helpful in these cases, and we can use it to compute the com-

posite permeability coefficient or to perform integration on the

element directly.

Fig. 11shows the finite element mesh for the sand box model

test done by Akai et al. [26]. The model was 3.15 m long, 0.23 m

wide and 0.33 m high. Porosity of the uniform sand was 0.44,

and the saturated permeability coefficient was 3.3 103 m/s.The variation in degree of saturation Srand the relative permeabil-ity coefficientkrvs. pore water pressure p are shown inTable 1.

The initial water level on the left side was 0.1 m, and it reached

0.3 m instantaneously during the test. The water level on the right

side was kept at 0.1 m throughout. The computed results are pre-

sented inFig. 12, in which the curves are the free surfaces calcu-

lated at t= 30 s, 60 s, 120 s, 240 s, 600 s and 4800s after the

water level reaches 0.3 m. Comparing the test results indicates that

the Monte Carlo method performs well for analyzing unsaturated

and unsteady seepage. Not surprisingly, a similar application is

to conduct consolidation analysis of unsaturated soil. In fact, the

numerical simulation of water-filling in an earth-rock dam re-

ported in Ref.[27]has actually employed the Monte Carlo method

to deal with the transitional element.

5. Conclusions

The Monte Carlo method is used in this paper to improve the

convergence of seepage analysis. It is effective for regions where

permeability varies greatly on some transitional elements that

are cut by the free surface.

(1) The Monte Carlo method is a simple but effective method. It

can be used to calculate the area or volume of an irregularregion, and be used as an alternative to Gaussian integration.

The Monte Carlo method can sample a huge number of

points, which makes it able to capture the information

involved in an element in detail. The property of what the

Monte Carlo method has facilities its application to regions

where exists great gradient in permeability. The examina-

tion has proved its applicability to seepage problems with

free surfaces and those unsaturated and unsteady flows

where permeability varies greatly on some elements.

(2) A sufficient number of random points is necessary, prefera-

bly at least 10,000. Though a large number of points require

more time at each step, the decreased number of iterations

actually saves time. It is thus more efficient as a whole in

many cases.

(3) The Monte Carlo method may also be useful for other appli-

cations such as consolidation analysis and stressstrain

analysis when the material properties vary greatly on some

elements. Further development and application of this

method looks promising.

(4) Though there are only linear elements employed in this

study, the proposed method may still be suitable for high-

order elements. The theorem of Monte Carlo integration

and the Monte Carlo based volume calculation is indepen-

dent of the element type. By using the corresponding inter-

polation method to determine the water head (or pore water

pressure) in the element, the Monte Carlo method can be

directly employed in the numerical analysis with high-order

elements.

Acknowledgments

The supports of Natural Science Foundation of China

(51039003), National Basic Research Program of China (973 Pro-

gram 2013CB036402) and the State Key Laboratory of Hydro-

science and Engineering (2013-KY-4) are gratefully acknowledged.

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Table 1

Degree of saturation Srand relative permeability coefficient krof the sand.

Pore water pressure, p

(kPa)

Degree of

saturation,Sr

Relative permeability

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