Design Charts for Sheet Piles-Downstream Blanket
Systems
Imad Habeeb Obead
University of Babylon - College of Engineering
Abstract
The work reported in this research presents numerical investigations on the effect of Sheet Pilesdownstream blanket of any length and location under a concrete dam of negligible thickness and
limited length, on the uplift heads under dam, seepage discharges, and exit gradients through the
complex foundation with a finite depth of soil layer. The problem is solved numerically with aid of the
finite element method.
The results are presented in a form of design charts, with dimensionless parameters; maximum
L

), relative length of downstream blanket( b ), relative depth of upstream and
H
Lf
ie
d
downstream cutoff walls( 2 ), relative exit gradient(
), and relative seepage
d1
H / Lf
q
quantities(
). These design charts can be used to estimate maximum uplift head around upstream
KH
uplift head (
and downstream Sheet Piles respectively, seepage discharge, and exit gradient under concrete dam. The
applicability and validity of the charts were examined and compared with relevant existing
approximations from literature and showed good performance.
‫الخالصة‬
‫م ارسدة عمميدة حدو تدثثرر الددمراا الحوحيدة ووسدمؤم مدلجر الدرحدما تحدا السدم ال دو رحت‬
‫يقدم العمد الموثد فد هداا البحد‬
‫الترب دة غرددر المتممثحددة والمتدم سددة عحددق مددي ضددغو االصددعمم تحددا السد عدم وعحددق ميددما الميددم‬
‫(بغددا ال ددر عددا سددم السددم ج د‬
‫ تد تطدوحر مدوا عدمم يح د دمهر التسدر جد الطبقدة المحصدور واسدتجمما طرحقدة الع مصدر‬.‫المتسدربة جد طبقدة اسسدمس‬
‫م ح يددما‬
‫تمثرد ال تددمؤم المستحصدحة بشد‬
‫إا تد‬
‫الشد‬
‫مثحد‬
‫مد ع صددر جطد‬
‫مي شح ة االصعمم الع مدق حدو الددمراا الحوحيدة فد مقدم ومدلجر السدم و ميدة الميدم‬
‫تد فحدو واجتبدمر صد حية الم ح يدما التصدميمية مد بعدا الححدو‬
‫بملتعممد‬
‫المعمملددة الحم مددة والد‬
‫المحدمم لحد‬
‫ يم ا استجما ها الم ح يما لحسم‬.‫تصميمية‬
‫المتسربة أسف السم وتمر المجر ف مدلجر الم شدث الدردمرولي‬
. ‫المتوفر ااا الع مة وأعطا تمؤم درم‬
1- Introduction
Study of seepage of water through soil is important for the following engineering
problems(Manna et al.,2003).
(a) Determination of rate of settlement of a saturated compressible soil layers.
(b) Calculation of seepage through the body of dams and stability of the slopes.
(c) Calculation of uplift pressures under hydraulic structures and their safety against
piping.
(d) Groundwater flow toward wells and drainage of soils.
The concrete dam is subjected mostly to uplift and piping effects. One method
to maintain the safety of the concrete structure against these effects is to extend the
length of creep. This safety is established by constructing sheet piles or extending the
floor length (using downstream blanket) or Sheet Piles-blanket system. The last
system was considered the most practical one(Shehata,2006). The amount of seepage
through anisotropic foundation soil under the dam floor equipped with these systems
always has been more important in design considerations of the Geotechnical
engineers. Moreover, when hydraulic gradients in the soil foundation are significant,
knowledge of their magnitude is essential to guard against heave or piping
phenomena. In this study, an numerical solution for the seepage problem under
concrete dam with two end Sheet Piles furnished with downstream blanket and
founded on complex formations was carried out. The numerical solution was
presented to investigate the effect of drawdown state, and proper length of a
downstream blanket. The effects of the downstream blanket on seepage under the dam
were obtained. Also, the effect of the anisotropic soil foundation on seepage flow
under the dam was indicated. Finally a set of design charts have been presented and
their use was simple and practical.
2- Statement of the problem
Seepage forces change stress state, deformation and change permeability of
soil elements which would affect the amount of the seepage rate and stability of
structure. In routine analyses, steady-state conditions commonly are considered
ignoring the time required to reach the steady-state condition, but actually, moving
water trough soil voids is a time consuming procedure, increase of the water table
height in a dam reservoir is gradual and time dependent, so transient analysis is
essential to yield reliable results. In addition, in uncouple systems there is only one
degree of freedom in the governing equation which is the hydraulic potential, so in
order to determine deformations of the structure due to seepage forces it is essential to
consider element equilibrium equations besides the fluid continuity equation, in this
case a set of partial differential equations including the continuity and equilibrium
equations must be solved at the same time(Ahad et al.,2007) . For the present study,
the geometry of the problem model and material properties are illustrated in figure(1).
Up Stream water level
H
Down Stream blanket
Concrete Dam
Lb
d1
Anisotropic
layer
d2
T
Lf
Impervious
Figure(1) : Geometry and material conditions of the problem
It is expected that correlations of the blanket length (Lb) with parameters
implying anisotropic soil thickness (T), hydraulic conductivity of soil in horizontal
direction (Kx), hydraulic conductivity of soil in vertical direction (Ky), upstream
sheetpile depth (d1), downstream sheetpile depth (d2), concrete dam length (Lf), the
head upstream the concrete dam (H0), the head loss downstream the concrete dam
(Hds), the discharge of seepage flow (Q) and exit gradient downstream the blanket (Ie)
would be concluded.
The governing equation for transient confined flow for two dimensional case
can be expressed by the following linear partial differential equation(Rao,2004).

h

h
h
(1)
(K x )  (K y )  S
x
x y
y
t
where: h is the piezometric head(m), S is the storage coefficient, and t is the time. The
solution of Eq.(1) represents time and space distribution of piezometric head in
homogeneous, anisotropic medium with confined transient flow.
3- Finite element formulation
Two dimensional transient seepage beneath a concrete dam was solved in the present
study by the finite element method. Using Galerkin's method, the flow domain were
subdivided in finite number of elements of a domain e and the boundary e. The
polynomial approximation of the solution is of the form:
n
e
e
(3)
h e ( x, y, t )  i 1 hi (t )N i ( x, y)
in which h e i are the piezometric head values at the nodes and N e i are the shape
function over the finite element with n number of nodes. The derived form of
equation(1) can be expressed as:
 w
h w
h
h 
(4)
e  x ( K x x )  y ( K y y )  wS t dxdy  e wqn ds  0


in which qn is the discharge occurred in the direction cosine between the normal and
(x) direction, s is the boundary portion over which integration takes place, and w is the
weight function, which can be expressed as:
e
(5)
w  Ni ( x, y)
The coefficient of hydraulic conductivity in both directions i.e., Kx and Ky are
assumed to be constant within one finite element. Substitution equation(3) and
equation(5) in equation(4) yields:
n
n
n
N j
N j
 N i

N i
h
(
K
h
)

(
K
h
)

SN
(
N j ) dxdy   N i qn ds  0
j
y j
i 
e  x x 
x
y
y
j 1
j 1
j 1 t


e
(6)
The matrix form for equation(6) can be expressed as(Obaed,2002):
e
e
e
e  h 
e
(7)
[ D ]{h }  [ M ]
  {F }
 t 
in which,
N i N j
N i N j
D e ij   [ K x
 Ky
]dxdy
(8)

x

x

y

y
e

M e ij   SN i N j dxdy
(9)
F e i   N i qn ds
(10)
e

e
At first stage of the present study, development of the flow net by subdivide
the flow domain in to finite elements by using linear three nodes triangular element
(Fig. 2).
Y
h3(x3,y3) 
3
L1
L2
90°
1 h1(x1,y1)
L3
2
h2(x2,y2)

X
Figure(2):Three-noded triangular element
The integral of above equations are performed using the natural coordinates
L1, L2, and L3 which can be expressed briefly as(Smith, and Griffiths,2004):
1
D e ij 
[ K x e i e j  K y e i e j ]
(11)
4 e
In which e is the area of finite element,  and  are the local coordinates with corner
nodes values =1, =1. The relationship between local and global coordinates can
be expressed as:
 e i  yi  yk 
(12)
 (i  j  k )
 e i  xk  x j 
 S
M e ij  e
(13)
12
F e ij  0 for all nodes in the flow domain with following conditions:
1- No flow boundary.
2- Entire nodes within flow domain.
3- Specified head boundaries.
Q e ij  0
The assembled form of equation(7) becomes:
 h 
[ D]{h}  [ M ]   {F }
 t 
in which,
m
D   Dij
e
(14)
(15)
e 1
Where m is the total number of elements.
m
M   M ij
e
(16)
e 1
m
F   Fi
e
(17)
e 1
4- Boundary and initial conditions
Figure(3) shows a schematic diagram of a concrete dam which founded on anisotropic
layer. The various boundary and initial conditions of the dam section are as follows.
N.W.L(H0)
H1= falling water level at t=t0+t
Y
H2=falling water level at the end of drawdown process
Lr
Ll
A
Hds
H
G
B
Z
C C1
Lb
I
J
K
F F1
d1
d2
tp
Ts D
D1
E
T
E1
Lf
L
X
O
Lt
Figure(3): Proposed case study showing the boundary and initial conditions
h
on OL
0  x  Lt
( x,0, t )  0
y
Ll  L f  x  Lt
on HK
h( x,Ts  Z , t )  H ds
h
( Ll  L f , y, t )  0
on HF
Ts  y  Ts  Z
x
h
Ll  x  Ll  L f
on CF
( x, Ts , t )  0
y
tp
h t p
h
on C1 D1 and CD
( x, , t )  ( x, , t )  0
0  x  d1
y
2
y
2
h
( Ll , y, t )  0
on BC
Ts  y  Ts  Z
x
For the boundary AB, if the upstream water level is constant at level H0, then
on AB
h(Ts  Z , t )  H 0
0  x  Ll
For the boundaries AO and LK , they are considered constant head in the present study
and will have the form:
on AO
h(0, y, t )  H 0
0  y  Ts  Z
h
( Lt , y, t )  H ds
on LK
0  y  Ts  Z
x
For the further investigation, if they are considered as impervious boundaries, they
will expressed as:
h(0, y, t )  0
on AO
0  y  Ts  Z
h
( Lt , y, t )  0
0  y  Ts  Z
on LK
x
5- Integration with implicit time evaluating schemes
The derivative {
h
} in equation(14) can be approximated by aid of Taylor series as:
t
h h t  t  h t

(18)
t
t
in which t is the discrete time step. In order to obtain stable solution for equation
(14), an implicit numerical technique is used; which is the backward finite difference
method. The solution of equation (14) after t time step becomes(Alsenousi, and
Mohamed,2008):
1
1
1

 

{h}t  t  [ D]  [ M ] {F }  [ M ]{h}t 
(19)
t
t

 

Equation (19) can be solved at each time step by using the matrices solution.
6- Concrete dam operation conditions
There are two cases considered in the present study for the upstream water condition:
1- Normal operation water level (H0) at the upstream side(steady state), the initial
condition corresponding such case expressed as:
(20)
h( x, y,0)  H ds
2- Transient upstream water level: as shown in Fig.(3), the initial and final water
levels at the upstream face are H0 and H2 respectively, and H1 is a water level in
between. In order to analyze the seepage flow at water level H1. It is required to apply
the same initial condition as in case(1) above with boundary condition on AB as
follows:
(21)
h( x,Ts  Z , t )  H 0
when steady state is exist, the values of piezometric head obtained are considered as
initial condition and the following boundary condition will be applied on AB :
(22)
h( x,Ts  Z , t )  H 2
The time required to fill or empty the reservoir can be defined as TFE where
the specific water level occurs during this time is H1, and TSP is the time at which H1
occurs. Assuming constant rate to fill or empty the reservoir, then for the case of
filling the reservoir:
[ H  H 0 ]  TSP
H1  2
 H0
(23)
TFE
And for the case of emptying the reservoir,
[ H  H 2 ][TFE  TSP ]
H1  0
 H2
(24)
TFE
Figure (4) shows the finite element mesh used in the present study which was
modified many times in order to calibrate the model. The total number of elements
and their geometry are not constant, they depends on the geometry of the study area,
dam floor length, sheetpile length, a convergence study was performed in table (1) to
obtain adequate finite element mesh.
Table(1): Results of convergence study
Seepage discharge Q (m3/sec.)
No. of
No. of
elements
nodes
Mesh of equal
spacing
Mesh of distorted
spacing
168
318
402
394
732
915
0.00197
0.00201
0.00199
0.0033
0.00261
0.00255
Concrete Dam
4m
9m
15 m
9m
15 m
Figure(4): Discretization of finite element mesh composed of 3-node elements
7- Calculation of exit gradient and factor of safety against piping
The exit gradient can be obtained as(Harr,1962):
h  H ds
ie  i
(25)
yi
in which hi are the values of piezometric head at nodes vertically below exit nodes,
and yi are the vertical distance as shown in figure(5). The critical exit gradient icr is
given by expression:
G 1
icr  s
(26)
1 e
in which Gs is the relative density of the soil particles, and e is the void ratio, then the
factor of safety against piping is expressed as:
i
FS  cr
(27)
ie
The value of factor of safety between (3) to(4) is enough for the safety of the
structure(Al-Musawi,2002).
Hds Downstream blanket
y i
hi
Downstream
cutoff wall
Figure(5): Sketch for calculation of exit gradient
A computer program was developed by the author using FORTRAN 90 language, the
compiler used is the FORTRAN Power Station-Microsoft Developer Studio-94.
8- Characteristics of seepage design charts
The maximum uplift head for both U/S & D/S cutoff walls has been determined, the
exit gradient as well as seepage discharge can be determined for both operation
conditions as mentioned above.
1- Steady state : the parameters of these problems are expressed in dimensionless
form with following ranges.
d
a) The relative depth of D/S to U/S cutoff walls 2 =[0.5,1,2]
d1
b) The soil anisotropy is modeled using different coefficients of hydraulic
conductivity in horizontal and vertical directions, and expressed by anisotropy degree
Ky
ranges [0.2,0.6,1].
R
Kx
L
c)The relative length of downstream blanket b from[0 to 1].
Lf
d
Figure(7- a, b, and c) indicates that for a given ( 2 ), R, the maximum uplift
d1
head on U/S cutoff , or D/S cutoff walls decreases with increasing of D/S blanket
length, which mean that more safety for Sheet Piles sections in case of using
downstream of concrete dam. In addition to increase of anisotropy degree causes
strongly decreasing of maximum uplift head on U/S & D/S cutoff walls.
L
For most values of ( b ), maximum uplift head on U/S cutoff wall decreases slightly
Lf
L
with increasing ( b ), while the rate of decreasing of maximum uplift head on D/S
Lf
cutoff wall is significant.
The trend of the curves in case of isotropic soil is slightly changed than the
L
anisotropic soil (R=0.2) which means that the increasing of ( b ) is more effective in
Lf
case of anisotropy.
0.7
U/S Cutoff
D/S Cutoff
0.6
Max. Uplift Head
R=0.2
0.5
0.4
0.3
0.2
0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Relative length of D/S Blanket
(a) Maximum uplift head on U/S & D/S cutoff walls with various relative length of
D/S blanket for (R=0.2)
0.5
U/S Cutoff
D/S Cutoff
Max. Uplift Head
0.4
R=0.6
0.3
0.2
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Relative Length of D/S Blanket
(b) Maximum uplift head on U/S & D/S cutoff walls with various relative length
of D/S blanket for (R=0.6)
0.4
U/S Cutoff
D/S Cutoff
R=1
Max. Uplift head
0.3
0.2
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Relative Length of D/S Blanket
(c) Maximum uplift head on U/S & D/S cutoff walls with various relative length
of D/S blanket for (R=1)
Figure(6): Effect of various length of D/S blanket on max. uplift head with
different anisotropy degree for both U/S & D/S cutoff walls
Lb
) has significant effect on
Lf
the relative exit gradient(Ier1), in which decreases with roughly constant rate for
L
isotropic soil, and for R=0.2 due to increase of ( b ), this mean more safety against
Lf
piping for isotropic and anisotropic soils.
Figure (7) shows that the relative length of D/S blanket (
Relative Exir Gradient, Ier1
2.4
R=0.2
2.2
R=0.6
2.0
R=1
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Relative Length of D/S Blanket
Figure(7): Effect of various length of D/S blanket on relative exit gradient with
different anisotropy degree for both U/S & D/S cutoff walls
Figure (8-a, and b)shows the effect of change in isotropy degree on the relative
seepage discharge under concrete dam, it is seen that the value of relative discharge
was decreasing steeply until R= 0.6 then the rate of change become slightly, the
L
relative discharge decreases significant with increasing of blanket length ( b ), which
Lf
mean the safety against piping phenomena can be improved by increasing the length
of D/S blanket.
13
R=0.2
Relative Seepage Discharge, Qr
12
11
10
9
8
7
6
5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Relative Length of Downstream Blanket
(a) Relative discharge with various relative length of D/S blanket for (R=0.2)
2.4
R=0.6
2.3
R=1
2.1
Relative Seepage Discharge, Qr
2.0
1.8
1.7
1.5
1.4
1.2
1.1
0.9
0.8
0.6
0.5
0.3
0.2
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Relative Length od D/S Blanket
(b) Relative discharge with various relative length of D/S blanket for (R=0.6 , 1)
Figure(8): Effect of D/S length on seepage discharge for different anisotropy
degrees
2- Transient state : the dimensionless parameters of such state for both constant and
variable upstream head are expressed as:
TK
a) time factor t r 
, in which ( K  K y K x ) is the equivalent hydraulic
H
conductivity. For a time period (T=100days), anisotropy degree R=0.2, and H=25m,
the time factor tr=1. By the same way, the time factor corresponding to time periods
T=500,100, 1500 days are tr =5,10,15 respectively.
ie
b) relative exit gradient I er 2 
.
(H / d2 )
L
c) D/S blanket length to D/S cutoff wall depth ratio( b ) ranges [0 to 6].
d2
L
d) relative length of D/S blanket( b ) ranges[0 to 1].
Lf
H
e) reservoir–level fluctuation ratio (
) ranges[1/3 , 1/2, 2/3, 1].
Lf
The reservoir-level fluctuations investigated by consider the water table in reservoir
1
drawdown gradually from[H to H ] during a time period of 1500 days.
3
The maximum uplift heads vs. relative length of D/S blanket for constant
upstream head are illustrated in figure(9), the maximum uplift heads acting on the U/S
cutoff wall are slightly decrease as the D/S blanket length increased, results show that
for beginning of transient process large area of pore water pressure in dam foundation
still has not promptly drained out, it is induces the uplift head to slightly decrease at
U/S cutoff which reaches (0.947- 0.921) after 1500 days. On the other hand, the max.
uplift head acting along D/S cutoff significantly decrease as length of D/S blanket
increase.
1.0
U/S Cutoff
D/S Cutoff
0.9
R=0.2
tr=1
0.8
Max. Uplift Head
tr=5
0.7
0.6
0.5
tr=10
0.4
tr=15
0.3
0.2
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Relative Length of D/S Blanket
Figure(9): Effect of D/S blanket length on max. uplift head for constant
upstream head conditions
It can be seen from figure(10) the influence of D/S blanket length to D/S cutoff depth
ratio on relative exit gradient. The value of relative exit gradient increases as D/S
blanket length increase , lengthening the D/S blanket to certain extent, namely
Lb
 0.75 in figure(10) results maximum increase in relative exit gradient values,
d2
beyond this range the relative exit gradient will decrease rapidly . The reduction in
exit gradient will occurs as time period increased.
0.18
Unsteady State
tr=1
0.16
Steady State
tr=5
R=0.2
Relative Exit gradient, Ier2
0.14
0.12
tr=10
0.10
0.08
0.06
tr=15
0.04
0.02
0.00
0
1
2
3
4
5
6
Ratio of D/S Blanket Length to D/S Cutoff Depth
Figure(10): Effect of D/S blanket length to D/S cutoff depth ratio on relative exit
gradient for constant upstream head conditions
H
)is clearly indicated in figures(11,
Lf
and 12). Maximum uplift head decreases as relative length of D/S blanket increase,
The effect of reservoir level fluctuation ratio(
the variation of relative exit gradient follows a general trend in figure(10) which
H
means that the increasing of (
)is more effective in case of reducing exit gradient.
Lf
1.0
Unsteady State
0.9
Steady State
R=0.2
0.8
Max. uplift Head
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Relative Length of D/S Blanket
Figure(11): Effect of D/S blanket length on max. uplift head for variable
upstream head conditions
0.20
Unsteady State
0.18
Relative Exit Gradirnt, Ier2
Stadeay State
0.16
R=0.2
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
0
1
2
3
4
5
6
Ratio of D/S Blanket Length to D/S Cutoff Depth
Figure(12): Effect of D/S blanket length to D/S cutoff depth ratio on relative exit
gradient for variable upstream head conditions
9- Validity of present model
The reliability of the present work was verified by comparing the results with analytic
solution(Salem and Ghazaw,2001),figure(13) presents the comparisons for uplift head
on D/S cutoff wall.
1.0
0.9
0.8
Relative Uplift Head
0.7
0.6
0.5
0.4
0.3
Present Work
0.2
Analytic solution
0.1
0.0
0.0
0.3
0.5
0.8
1.0
1.3
1.5
1.8
2.0
Relative depth of cutoff walls
Figure(13): Comparison the present work and past works results
10- Conclusions
1- The maximum uplift head acting on U/S and D/S sheetpile walls decreases with
increasing of D/S blanket length which mean that more safety for cutoff wall
sections in case of using blanket downstream the concrete dam.
2- It is very necessary to conclude that the design concept which deal with anisotropic
soil as isotropic gives unsafe sections of Sheet Piles which may be less than 30%
of safe section.
L
3- The exit gradient, after a distance ( b  0.8 )downstream from the concrete dam
Lf
d
toe, decreases with decreasing( 2 ), and the reduction increase as the depth of U/S
d1
sheetpile(d1) increases. The resultant is an increase in the safety factor against
piping phenomenon.
4- The errors associated with the present method of solving seepage problem with the
consideration of the soil anisotropy, and nonlinearity of unsteady seepage are
determined by the form of the boundary conditions, degree and form of nonlinearity
of the process, its duration and fineness of the time interval.
11- References
Ahad, O., Toufigh, M. M., Ali, N.,2007." An investigation on the effect of the
coupled and uncoupled formulation on transient seepage by the finite element
method" , American journal of applied sciences Vol.4, No.12, pp.950-956.
Al-Musawi, W. H.H., 2002." Optimum design of control devices for safe seepage
under hydraulic structures", M.Sc. thesis, Department of civil engineering,
Babylon university.
Alsenousi, K.F., and Mohamed, H.G., 2008."Effects of inclined cutoffs and soil
foundation characteristics on seepage beneath hydraulic structures", Twelfth
international water technology conference, IWTC12, Alexandria, Egypt.
Harr, M. E., 1962. "Groundwater and Seepage", McGraw-Hill, New York.
Manna, M. C., Bhattacharya, A.K., Choudhury, S., and Maji, S.C., 2003.
"Groundwater flow beneath a sheetpile analyzed using six-noded triangular
finite elements" IE(I) Journal, Vol. 84, August.
Obaed,I.H.,2002." Stability of zoned earth dam coupled with unsteady drawdown"
M.Sc. thesis, Department of civil engineering, Babylon university.
Rao, S.S., 2004." The finite element method in engineering", 4th Edition, Elsevier
science & technology books.
Salem, A.A.S., Ghazaw, Y.,2001." Stability of two consecutive floors with
intermediate filters", Journal of hydraulic research, Vol.39, No.5, pp. 549-555.
Shehata, A.K.,2006."Design of downstream blanket for overflow spillway founded on
complex foundations" Journal of applied sciences research, Vol.2, No.12,
pp.1217-1227.
Smith I. M., and Griffiths D.V.,2004."Programming the finite element",4th Edition,
John Wiley & Sons, New York.