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Journal of Applied Mathematics and Physics, 2013, 1, 11-14
Published Online November 2013 (http://www.scirp.org/journal/jamp)
http://dx.doi.org/10.4236/jamp.2013.16003
Open Access JAMP
Apply of Explicit Finite Element in Seismic Ground
Motion Computation*
Yuzhu Bai, Xiwei Xu
Active Tectonics and Volcano Lab, Key Lab of China Earthquake Administration, Beijing, China
Email: yuzhubai2008@126.com
Received August 2013
ABSTRACT
In this paper, we will use the explicit finite element to compute ground motion due to Tangshan earthquake. The expli-
cit finite-element method uses one integration point and an hourglass control scheme. We implement the coarse-grain
method in a structured finite-element mesh straightforwardly. At the same time, we also apply the coarse-grain method
to a widely used, slightly unstructured finite-element mesh, where unstructured finite elements are only used in the ver-
tical velocity transition zones. By the finite-element methods, we can compute the ground velocity with some distance
to the seismogenic fault in Tangshan earthquake. Through the computation, we can find the main character of ground
motion for the strike slip earthquake events and the high frequency vibration motion of ground motion.
Keywords: Finite-Element Method; Fault; Ground Motion; Tangshan Earthquake
1. Introduction
In earthquake research, we often mainly apply the finite-
difference and finite-element method to calculate the
ground motion due to the movement of the seismic faults.
The finite-element method, one of the most popular me-
thods in the engineering science, is increasingly used in
earthquake computation (see [1-3]) and earthquake dy-
namic rupture propagatios (see [4,5]). With the help of
irregular elements of different size, geometry, and order
of approximations, the finite-element is efficient to mod-
el complicated geometrical boundary conditions at present.
The explicit finite-element method with second-order
elements and one-point integration is widely used. This
method combines the flexibility of the finite-element
method and the efficiency of finite-difference method.
As its efficiency and versatility, this method has been
widely implemented and applied to transient analyses in
engineering and seismology. Here, we refer to this par-
ticular implementation as the explicit finite-element me-
thod.
The following algorithm for the explicit finite-element
method is easy. Usually, it uses for-node quadrilateral
elements in two dimensions and eight-node hexahedral
elements in three dimensions and applies one integration
point and an hourglass control scheme. Because of the
computational costs and the numerical noise resulting
from the ad hoc mass lumping necessary to generate a
diagonal mass matrix (see [6]), higher-order elements are
rarely used in researching transient problems. But one-
point integration provides tremendous computational
benefits in dynamic simulations. Comparing with the full
integration, one-point integration has a 3/4 reduction in
computational time in two dimensions and 7/8 reduction
in three dimensions. At the same time, one-point integra-
tion also provides the efficient way to simulate nonlinear
material response (see [7]).
In fact, the one-point integration in the elements has its
own drawback which is a mesh instability known as
hourglassing. But the hourglass modes can be eliminated
by well-developed hourglass control schemes (see [8]).
Kosloff and Frazier [9] had showed that a one-point in-
tegration implementation coupled with a stiffness hour-
glass control scheme can produce a more accurate flex-
ural response than fully integrated elements. So in this
article, we will apply the finite-element with one-point
integration developed by Ma [10] to compute the ground
velocity of some locations varying with time in the Tang-
shan earthquake.
2. Explicit Finite-Element Algorithm and
Tectonic Background
2.1. Explicit Finite-Element Algorithm
Here we will use the explicit finite element developed by
Ma [10], so we will introduce his method in a simple
*Subsidized by “Discern and evaluation of earthquake risk zone
(2012BAK15B01)”.
Y. Z. BAI, X. W. XU
Open Access JAMP
12
way. The details of method introduction can be found in
his article. For an eight-node hexahedral isoparametric
element, the trilinear shape function is given in the ref-
erence plane by
( )( )( )
[1 1 1 ]8
I III
N
ξξ ηη ζ ζ
=+++
(1)
In (1),
( )
,,
ξηζ
is the coordinate of an arbitrary point
within the element in the reference plane and
( )
,,
III
ξηζ
is the coordinate of node I. The range of the
upper-case subscripts is {1 - 8}. The transformation be-
tween the physical plane and the reference plane is fol-
lowing:
;;
II II II
x xN y yN z zN= = =
(2)
In (2),
( )
,,xyz
is the coordinate of an arbitrary point
within the element in the physical plane and
( )
,,
III
xyz
is the coordinate of node. Summation over repeated in-
dices is applied. In the element, the velocity field can be
expressed by the same shape function as:
i il I
v vN=
(3)
where
il
v
is the velocity of node. The range of lower
case subscripts is {1, 2, 3}. In the one-point integration
finite-element scheme, the velocities are located at the
element nodes and the stresses are all defined at the cen-
ter of the element.
If defined the matrix as
,0
|
iI I i
BN
ξηζ
= = =
=
(4)
where the comma denotes differentiation, then the veloc-
ity gradient at the element center is
,i j il jI
v vB=
(5)
By the Equation (4), it can be found that the B matrix
has the antisymmetry properties. The detailed derivation
of the B matrix is shown in appendix A of [10]. Through
the velocity gradient, we can calculate the element stress
rate at the element center by following
( )
, ,,ij ij l l i j j i
v vv
σ λδ µ
⋅
= ++
(6)
where λ and μ are the Lame’s constants of the element
and δij is the Kronecker delta. The nodal force rate
caused by the stress rate within the element is given by
stress
il jI ij
f VB
σ
⋅⋅
=
(7)
where V is the element volume. The applying of one-
point integration can result in certain deformation modes
remain stressless. These zero-stress modes are hourglass
modes. The amplitudes of the hourglass modes in the
element are given by
i il I
qv
αα
ϕ
⋅
=
(8)
where Greek subscripts have a range of {1, 2, 3, 4} and
the hourglass base vector
α
ϕ
is defined as:
{ }
{ }
{ }
{ }
1
1
1
1
11111111
1 1 11 111-1
11111111
11 111 11 1
ϕ
ϕ
ϕ
ϕ
= −−−−
= −− −
= −−−−
=− − −−
In the viscous hourglass control scheme, the hourglass
forces can be approximated by
23vhg
il p i I
f VV q
αα
χρ ϕ
⋅
=
(9)
where ρ and VP are the density and the P-wave velocity
of the element, and χ is a tunable parameter that is usual-
ly set in the range 0.05 - 0.15 (see [7]).
If a stiffness hourglass control scheme is used, the
stiffness hourglass force rates can be approximated by:
( )
13
2
16
ehg
f Vq
i
iI I
κλµ ϕ
αα
⋅
= +
(10)
where the tunable parameter κ is usually 0.3. The total
elastic nodal force rate caused by both the stress and
hourglass modes is given by
ehg
elastic stress
f ff
jI jI jI
= +
(11)
2.2. The Tectonic Background of Tangshan
Earthquake
On 28 July 1976, a destructive earthquake struck the city
of Tangshan, in mainland China, 160 km east of Beijing.
The focal depth of the MS = 7.8 Tangshan earthquake
was 10 km [Bulletin of the International Seismological
Center (ISC)]. A prominent surface rupture crossed the
city of Tangshan, it completely destroyed the city and
heavily damaged the surrounding areas. The largest af-
tershock (Ms = 7.1 Luanxian earthquake) occurred ap-
proximately 45 km northeast of the mainshock on the
same day. Another large aftershock (MS = 6.9 Ningho
earthquake) happened on 15 Novermber 1976 and was
located southwest of Tangshan near Ningho. The 1976
Tangshan, China, earthquake of MS 7.8 killed 242,000
persons, seriously injured 164,000 persons, and caused
direct property losses totaling 8 billion Yuan Ren Min
Bi.
Figure 1 shows a map view of the Tangshan fault and
the fault near to Tangshang. Tangshan lies near the
northern boundary of the China Basin, a seismically ac-
tive region with at least nine large, destructive earth-
quakes since 1600 A D. The tectonics is characterized by
active subsidence in right step-overs between predomi-
natly north-northeast trending, right-lateral strike-slip
faults. After the Tangshan earthquake, the mechanisms of
the Tangshan events had been studied by many scholars.
Y. Z. BAI, X. W. XU
Open Access JAMP
13
These studies showed that the mainshock consisted of
two subevents, separated by about 10 sec, with right-
lateral strike-slip motion on two near-vertical north-
northeast trending faults. The first subevent initiated near
the junction of the two segments and the junction to the
north.
3. Earthquake Model and Focus Parameters
In the Figure 1, we will computation → compute the
zone which covers 140 km along strike direction, 210 km
vertical to strike direction and 40 km in depth direction.
In the finite element model, the calculating parameters
are listed in Table 1. In our explicit finite-element me-
thod, we set the spatial and time step is 500 m and 0.05 s
respectively. The duration of computation is 40 seconds.
Figure 1. The fault location of Tangshan earthquake.
Table 1. The calculating parameters.
Type of Parameters value
Earthquake magnitude 7.8
Focus location 39.6˚
118.2˚
Fault size 48 km(L)
20 km(W)
Focus depth 15 km
Strike direction N40˚E
Average stress drop 4.5 MPa
Model size 210 km × 140 km × 140 km
Space step 1 km
Time step 0.1s
Dynamic friction coefficient 0.6
Static friction coefficient 0.8
Initial shear stress 28.5 MPa
Initial normal stress −35 MPa
Viscous parameters 0.08
Stiffness parameters 0.3
4. The Computation Result
Applying explicit finite-element method, we compute the
ground motion of Tangshan earthquake. Here, we give
the velocity of ground motion at the 4 km to the seismic
fault from the east side of central of Tangshan fault. The
Figures 2 and 3 are the ground motion component in
vertical and parallel to the fault strike direction. Figure 4
is the ground motion component in vertical ground sur-
face direction. Figures 2-4 give the velocity varying with
time in 40 seconds.
Figure 2 is the velocity of ground motion component
vertical to the strike direction. From the Figure 2, we can
find that the strike slip motion of fault can also bring the
ground motion vertical to the fault strike direction. The
maximum of ground motion vertical to strike direction is
0
10
20
30
40
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
Figure 2. The ground velocity component vertical to fault
strike direction.
0.4
0
10
20
30
40
0.2
-0.4
-0.2
0.6
0.8
1.0
0
Figure 3. The ground velocity component parallel to fault
strike direction.
-0.6
0
10
20
30
40
-0.4
0.4
-0.2
-0.8
-1.0
0
0.2
Figure 4. The ground velocity component vertical to ground
surface.
Y. Z. BAI, X. W. XU
Open Access JAMP
14
0.65 m/s. Figure 3 is the velocity of ground motion
component parallel to the strike direction. Because the
Tangshan earthquake is a special strike event, the ground
motion parallel to strike is the mainly character of ground
motion. The maximum of ground motion parallel to
strike direction is 0.9 m/s and is larger than that of com-
ponent vertical to strike direction and ground surface.
Figure 4 is the velocity of ground motion component
vertical to the ground surface. The maximum of ground
motion vertical the ground surface is 0.85 m/s and larger
than that of component vertical to strike direction. Com-
paring the maximum of our computational result with
other scholar’s computation, we find our computational
result is very near to theirs, which shows our achieve-
ment is reasonable.
Comparing the above three computational result fig-
ures, when the earthquake happens due to the strike slip
movement of strike fault, the main motion of ground
surface is parallel to the fault strike direction and vertical
to the ground surface. And these two kind motions have
the high frequency vibration at the same time, which can
be seen from the Figures 3 and 4. Since the computa-
tional spot is 4 km to the Tangshan fault, the three
ground motion component almost begin at the 1 - 2
second. Because in earthquake, the S wave is mainly
factor to cause damage, the corresponding time of max-
imum of ground motion vertical to ground surface and
parallel to strike direction almost equals to the time of
S-wave travel to the computational spot.
5. Acknowledgements
This work is supported by the China Earthquake Admin-
istration through “Earthquake Backbone Technology
Professionals Foundations” and “Discern and evaluation
of earthquake risk zone (2012BAK15B01)”. Thank Shuo
Ma for his providing the computation code to calculate.
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