Parameter Identification of Physical

Kawahara Lab. Vol.3 (Nov.9 2002)
Chuo Univ.
Parameter Identification of Physical-Properties
around Tunnel Face
Atsushi HIKAWA*
*Deptartment of Civil Engineering, Chuo University
Kasuga 1-13-27, Bunkyo-ku, Tokyo 112–8551, JAPAN
E-mail: [email protected]
Abstract
This paper presents a parameter identification method for the physical-properties of natural
ground around a tunnel face. As a numerical model, the 3-D linear elastic analysis is used
to identify the Young’s modulus. The parameter identification is utilized to minimize
the differance between the computed and measured displacements. In order to solve the
minimization problem, the Conjugate Gradient ( CG ) method is applied. As the algorithm
for the CG method, the Fletcher-Reeves method is employed. The advantages of this
method are a high convergence rate and a simple algorithm. In this research, the finite
elament method is employed. The Galerkin method is applied to the discretization in
space. As the numerical example, the natural ground has four layers. As the result, the
Young’s modulus is identified, respectively.
Keywords: Parameter identification; 3-D linear elastic analysis;
Finite Element method; Galerkin method;
Conjugate Gradient method; Fletcher-Reeves method
1
Introduction
There are many civil engineering structures in the world. In this paper, its attention was paid to the
tunnel of underground structures in it. In recent years, the digging technology of tunnel is progressing
every day. In the main, the NATM ( New Austrian Tunneling Method ) is used in many site. The
NATM is the safe and rational construction method. Especially, it becomes safer by applying the
tunnel support like the lock bolt and the shotcrete at an early stage. But, it is always accompanied
by danger. In fact, many crash accidents have occured in the site. The large deformation occurs to
natural ground when a tunnel face is approached the soft layer. In civil engineering, to secure people
is very important at the site. Therefore, the means which can be predicted and prevented the crash
accidents is important. In this research, the backward analysis using 3-D model is performed. These
days, computer is advanced. Therefore, the backward analysis is used as an effective means. And
the actual behavior of a tunnel is the three dimensional action. For such occasions, the backward
analysis using the 3-D tunnel model attracts notice. As a result of the analysis, the 3-D model is
expected to verify more exact by the properties at the cross section of a tunnel face. At first, the
displacements of the tunnel surface are computed by the forward analysis. Then, the tunnel support
is not considered. The displacements are made into the observation data. And the backward analysis
1
is performed, following the algorithm of CG method. The problem is to minimize difference between
the computed and measured displacements. In this paper, the identified parameter is the Young’s
modulus. The Young’s modulus shows the intensity of the ground. And a tunnel of natural ground
is most influenced in the Young’s modulus. The analysis domain assumes four layer. So the Young’
modulus is identified, respectively. As a result of this development, a computational system to predict
the physical-properties around a tunnel face has been obtained.
2
Basic Equation
Balance of stress equation is expressed as ;
σij,j − ρbi = 0,
(1)
where σij is total stress tensor, bi is acceleration that produced body force, ρ is the density of solid.
Strain - displacement equation can be described in the following form ;
1
εij = − (ui,j + uj,i ),
2
(2)
where εij is strain tensor, ui is displacement of solid.
Stress - strain equation is ;
e
εkl ,
σij = Dijkl
(3)
e
is elastic stress - strain tensor and can be written as the following equation ;
where Dijkl
e
= λδij δkl + µ(δik δjl + δil δjk ),
Dijkl
(4)
in which δij is Kronecker’s delta, Lame’s constant λ, µ can be written as follows ;
λ=
νE
,
(1 − 2ν)(1 + ν)
(5)
E
,
2(1 + ν)
(6)
µ=
where E is Young’s modulus and ν is Poisson ratio.
3
Boundary condition
Basic equations are solved on following boundary conditions. The boundary S can be divided into SU
and ST . These boundaries satisfy following conditions ;
ui = ûi
on SU ,
(7)
ti = σij nj = t̂i
on ST ,
(8)
where the ûi , t̂i and ni mean the known values on the boundaries, the displacement, the surface force
and the external unit vector to the boundary.
2
4
Finite Element equation
Applying the Galerkin method, the discretization in space can be performed with the linear tetrahedron
elements. Then the finite element equation is obtained as follows ;
Kαiβk uβk = Γ̂αi ,
Kαiβk =
Γ̂αi =
V
V
(9)
e
(Nα,j Dijkl
Nβ,l )dV,
(Nα ρbˆi )dV −
(10)
ST
(Nα tˆi )ds,
(11)
where N is the shape function.
5
Parameter Identification
Parameter identification is the minimization problem to minimize performance function J. This performance function J is the square residual between the computed and measured displacements, which
is given as follows ;
J=
V
1
(u − u∗ )T (u − u∗ )dV.
2
(12)
Algorithm of the Conjugate Gradient method can be written as in the following manner. The conjugate
gradient (CG) method is employed for minimization algorithm. Using this method, Young’s modulus
E can be solved, which is expressed as follows ;
{P }T = {E}T = {E1 , E2 , E3 , · · ·, Em },
(13)
where m is the number of Young’s modulus to solved.
As a next step, the initial gradient of performance function for parameter E0 is given as following
equation ;
(0)
{d}
∂J
=−
∂P
(0)
∂u(P )
=−
∂P
T
(u(P ) − u(P )∗ ).
(14)
To solve Eqs.(14), boundary condition can be given as follows ;
∂u(P )
=0
∂P
on
SU ,
(15)
)
where [ ∂u(P
∂P ] is called as sensitivity matrix.
At next step, by using Taylor series expansion, the scalar function J(P + αd) with respect to the step
size α can be developed as follows ;
∂J T
{d}
J({P } + α {d}) = J + α
∂P
∂u(P )
∂u(P )
1
∗ T
(u(P ) + α
{d} − u ) (u(P ) + α
{d} − u(P )∗ )
=
2
∂P
∂P
3
(16)
(17)
So as to minimize the this function, Eqs.(17) should be partially differentiated with respect to step
size α. And setting this differentiated equation equal to zero, as the result, step size α can be equal
represented as following equation ;
α=−
{d}T
{d}T
∂u(P ) T
∂P
∂J
∂P
∂u(P )
∂P
{d}
.
(18)
The parameter E is renewed using d and α, which are obtained from Eqs.(14) and Eqs.(18) respectively.
The new parameter E is expressed as follows ;
P (i+1) = P (i) + αd(i) .
(19)
After second iteration, the gradient of performance function can be solved using Flecher-Reeves
method.
This gradient is expressed as follows ;
(i+1)
{d}
∂J
=−
∂P
(i+1)
+ β{d}(i) ,
(20)
where β is as follows ;
β=
∂J
∂P
(i+1) ,
∂J
∂P
(i) ,
∂J
∂P
∂J
∂P
(i+1) (i) .
(21)
Next step is to check for convergence. If |d(i+1) | is less than ε, then calculation can be stopped.
Otherwise calculation continues from determination of α.
The algorithm of conjugate gradient method is summarized as follows ;
[1] Assume initial material parameter P (0) , set allowable constant εj and i = 0.
[2] Compute initial displacement u(P )(0) and initial performance function J (0) .
[3] Compute initial sensitive matrix
[4]
[5]
[6]
[7]
∂u(P ) (0)
∂P
under the constraint of
Compute initial gradient of performance function {d}(0) .
Determine α to minimize J(P (i) + αd(i) ).
Compute parameter P (i+1) = P (i) + αd(i) .
Compute displacement u(P )(i+1) and performance function J (i+1) .
∂u(P )
∂P
= 0 on SU .
(i+1)
)
.
[8] Compute sensitive matrix ∂u(P
∂P
[9] Compute β to renew gradient of performance function.
[10] Compute gradient of performance function {d}(i+1) .
[11] If J (i+1) < εj then stop.
[12] Else, set i = i + 1 and go to [5].
6
Numerical Model
The numerical model is shown in Figure 1 (a), (b). These figures show the scale and boundary condition
of numerical model. The boundary condition are displacement and surface force. The displacement is
free to the longitudinal and lateral direction. The surface force condition is the same pressure around
4
natural ground. The same pressure are given on x, y and z directions. The condition of tunnel support
is not considered. After all, the tunnel surface is imposed as the traction free condition because arch
action happen on the tunnel surface. In this research, it is assumed that the natural ground has four
layers. It is for bringing the natural ground close to the more nearly actual ground. The section of a
tunnel face is located at 10.0(m) from entrance.
20.0kN/m2
20.0kN/m2
10m
20.0kN/m 2
20.0kN/m2
2.0m
LAYER 1
LAYER 2
LAYER 3
LAYER 4
z
10m
y
z
20m
x
(b) Z-X section
y
10m
x
(a) Y-Z section
Figure 1
3-D Numerical model
7
7.1
Numerical Example
Numerical Example 1
As numerical example 1, the behavior of tunnel are shown in natural ground has four layers. The
difference between the two kind natural ground is researched about displacement and stress. The one
kind natural ground has a soft layer. The Young’s modulus is set as 2500(kN/m2 ) in Layer3. The
other kind that has a hard layer. The Young’s modulus is set as 500(kN/m2 ) in Layer3. After all,
there is a soft layer ahead of a tunnel face in example 1-(2). The physical-properties of natural ground
are shown in Table 1 and 2. The ground pressure is 10.0(kN/m2 ). The finite element mesh ( 17833
nodes, 94481 elements ) was employed.
10
Table 1 Parameter of example 1-(1)
8
6
Z
E
ν
4
LAYER 1
2000
0.30
LAYER 2
1500
0.30
LAYER 3
2500
0.30
LAYER 4
1800
0.30
2
Table 2 Parameter of example 1-(2)
0
10
5
5
0 0
Y
20
X
Z
Y
10
15
E
ν
X
Figure 2
Finite element mesh
5
LAYER 1
2000
0.30
LAYER 2
1500
0.30
LAYER 3
500
0.30
LAYER 4
1800
0.30
7.2
Numerical Example 2
From numerical example 1, it is found that tunnel surface is much danger when a tunnel face is confronted with soft layer. Then, it is necessary that the Young’s modulus of each layers are predicted
and the strength of natural ground is checked. In numerical example 2, the backward analysis predicting the Young’s modulus of each layers is analyzed. The problem is parameter identification of
the Young’s modulus. The displacements on tunnel surface are computed using parameter of Table 2.
The displacements are made into the observation data. As the backward analysis, the initial value of
the Young’s modulus are set as Einit = 100(kN/m2 ) in each layer, respectively.
8
8.1
Numerical Result
Result 1
Results of the forward analysis using 3-D model are shown in Figure 3, 4, 5, 6. In Figure 3, 4, the
deformation of the tunnel is represented. Figure 5, 6 show the distribution of stress. In example 1-(1),
the large deformation doesn’t occur because there is a hard layer ( E = 2500kN/m2 ) ahead a tunnel
face. But, in example 1-(2), the large deformation hapens because a soft layer ( E = 500kN/m2 )
exists ahead a tunnel face. In particular, it is shown that a tunnel face is swollen behind. In the actual
site, it is very danger and the crash accident happens. The stress results also show the same thing.
In example 1-(2), the stresses of tunnel surface is very large. In short, it is necessary to estimate the
Young’s modulus of natural ground by employng the backward analysis. Parameter identification is
very important.
8.2
Result 2
Results of the backward analysis using 3-D model are shown in Figure 7. Figure 7-(a) shows history
of performance function. Figure 7-(b) shows history of the identified Young’s modulus, respectivery.
The performance function is reduced and converged equal to 0.0. The Young’s modulus is converged
to the target value, respectivery. Table 3 shows terminal values, respectively. After all, the Young’s
modulus of each layers could be estimated by the displacements of tunnel surface. This result indicates
that the strength ahead a tunnel face is predicted and the safe excavating means can be projected in
the next gradual when a tunnel face is approached with soft layer. The safety of people are obtained
in the site.
Table 3 Terminal value
E
9
LAYER 1
2000.001
LAYER 2
1500.002
LAYER 3
500.002
LAYER 4
1799.992
Conclusion
In this analysis, a behavior of tunnel in the natural ground with four layers were researched using 3-D
model at first. The two cases where Layer3 is hard, and when soft, were considered. As a results,
the tunnel surface was deformed and had a large stress when a layer was soft ground ahead a tunnel
face. Especially, a tunnel face was stick out to the entrace. The big accident will be caused if this
phenomenon actually occurs. So the backward analysis which predicts the intensity of the layers
which is the main purpose of this research was performed in numerical example 2. In consequence,
to forecast the Young’s modulus of each layers from the observed values was accomplished. But since
the computed displacements were used as the observed data, the actual data want to be employed.
6
As a future work, an excavating analysis is first taken in the forward analysis and it is brought to a
more nearly actual action. About the backward analysis, it considers identifying the position of a soft
layer rather than identifying the strength of layers. It will be a very effective means in the actual site
if the backward analysis is successful.
References
[1] Konishi, S. and Tamura, T., Analysis of Threedimensional Effect of Tunnel Face Stability on
Sandy Ground with a Clay Layer by the Three-dimensional Rigid Plastic Finite Elament Method,
Proceedings of the International Symposium of Modern Tunneling Science and Technology(ISKyoto), Vol.1, pp121-126(2001).
[2] Hisatake, M. and Murakami, T., Non-destructive evaluation of tunnel lining stresses, Proc. 4th
Int. Symp. on Numerical Models in Geomech., NUMOG IV, pp685-695(1992).
[3] Nakata, M., Sano, N. and Sato, J., Investigation for The Behavior of The Ultra Large Tunnels
with Three Demensional FEM, Proc. 9th Symp. on Rock Engineering, (1994).
[4] Sato Kogyo Co., Ltd., A Guide of NATM Constraction, pp1-29, pp102-107, (1984).
[5] Terzaghi, K., Peck, R.B., Soil Mechanics in Engineering Practice, John Wiley and Sons, Inc.,
pp72-73(1967).
[6] Horikawa, S. and Kawahara, M., Shape Optimization Problem of Hole Structure in the Ground,
(1999).
[7] Akai, K. and Tamura, T., Numerical analysis of multidimensional consolidation accompanied
elasto-plastic constitutive equation, Proc. of JSCE, pp95-104(1978).
[8] Asaoka, A., Observational procedure of settlement prediction, Soils and Foundation, Vol.18, No.4,
pp87-101(1978).
[9] R.Fletcher and C.M.Reeves., Function Minimization by Conjugate Gradients, Computer J.,
pp149-154(1964).
[10] Nojima, K. and Kawahara, M., Mesh Generation of Three-dimensional Underground Tunnels
Based on the Three-Dimensional Delaunay Tetrahedration, Vol.5, Journal of Applied Mechanics
JSCE, pp253-262(2002).
7
Z
Z
X
Y
Y
X
10
8
6
Z
4
2
20
0
15
10
10
5
5
0 0
X
Y
0
(a) Deformation of tunel
5
10
(b) Precise deformation in the plain
Figure 3
Result of numerical example 1-(1)
Z
Z
X
Y
Y
X
10
8
6
Z
4
2
20
0
15
10
8
10
6
4
5
2
0
X
Y
5
(a) Deformation of tunel
10
(b) Precise deformation in the plain
Figure 4
Result of numerical example 1-(2)
10
10
8
8
6
6
10
8
6
4
2
2
Z
Z
Z
4
4
2
20
0
15
10
10
5
5
0
0
X
Y
Z
Y
X
(a) stress-x
20
0
STX
182.732
170.792
158.851
146.911
134.97
123.03
111.09
99.1493
87.2089
75.2685
63.3281
51.3877
39.4473
27.5069
15.5665
15
10
10
5
5
0
0
Y
X
Z
Y
X
STY
185.062
173.732
162.402
151.072
139.742
128.412
117.082
105.751
94.4213
83.0912
71.7611
60.431
49.1009
37.7707
26.4406
20
0
15
10
10
5
5
8
X
Z
Y
(b) stress-y
0
0
Y
X
(c) stress-z
STZ
203.302
190.243
177.185
164.127
151.069
138.011
124.953
111.895
98.8367
85.7786
72.7205
59.6624
46.6043
33.5462
20.4881
10
10
8
8
6
6
10
8
6
Z
Z
4
2
2
Z
4
4
2
20
0
15
10
10
5
0
0
X
Y
Z
X
Y
20
0
STXY
25.6243
21.2159
16.8075
12.3991
7.9907
3.5823
-0.826093
-5.23449
-9.64289
-14.0513
-18.4597
-22.8681
-27.2765
-31.6849
-36.0933
5
15
10
10
5
5
0
0
X
Y
Z
X
Y
(d) shearing stress-xy
20
0
STYZ
68.2332
62.7049
57.1766
51.6483
46.1201
40.5918
35.0635
29.5353
24.007
18.4787
12.9505
7.42218
1.89391
-3.63436
-9.16263
15
STZX
53.7084
47.3819
41.0554
34.7289
28.4025
22.076
15.7495
9.42298
3.09649
-3.23
-9.55649
-15.883
-22.2095
-28.536
-34.8624
10
10
5
5
0
0
X
Y
Z
X
Y
(e) shearing stress-yz
(f ) shearing stress-zx
Figure 5
Result of numerical example 1-(1)
10
10
10
8
8
8
6
6
6
Z
Z
Z
4
4
2
2
20
0
10
STX
181.836
169.876
157.917
145.957
133.997
122.038
110.078
98.1182
86.1585
74.1987
62.239
50.2793
38.3196
26.3599
14.4002
5
5
0
0
X
Y
Z
X
Y
2
20
0
15
10
4
15
STY
184.302
173.018
161.734
150.451
139.167
127.883
116.599
105.315
94.0309
82.747
71.463
60.1791
48.8952
37.6113
26.3273
5
5
0
0
X
Y
Z
X
Y
(a) stress-x
10
8
8
6
6
2
2
20
15
10
5
5
0
0
X
Y
Z
X
Y
X
Y
8
6
4
2
20
15
10
10
STYZ
73.9881
68.0615
62.1348
56.2082
50.2816
44.3549
38.4283
32.5016
26.575
20.6483
14.7217
8.79503
2.86839
-3.05826
-8.9849
5
5
0
0
X
Y
Z
X
Y
(d) shearing stress-xy
X
Z
10
0
STXY
26.8676
22.7306
18.5935
14.4565
10.3194
6.18239
2.04533
-2.09172
-6.22877
-10.3658
-14.5029
-18.6399
-22.777
-26.914
-31.0511
0
0
Y
Z
4
(e) shearing stress-yz
20
0
15
10
10
STZX
54.3088
48.6521
42.9953
37.3385
31.6818
26.025
20.3683
14.7115
9.05473
3.39796
-2.2588
-7.91557
-13.5723
-19.2291
-24.8859
5
5
0
0
X
Y
Z
X
Y
(f ) shearing stress-zx
Figure 6
Result of numerical example 1-(2)
0.12
2500
J
0.1
0.08
YOUNGS MODULUS
PERFORMANCE FUNCTION
2000
0.06
1500
LAYER 1
LAYER 2
LAYER 3
LAYER 4
1000
0.04
500
0.02
0
0
0
5
10
15
20
ITERATION
25
STZ
186.316
174.27
162.223
150.176
138.13
126.083
114.037
101.99
89.9436
77.897
65.8504
53.8039
41.7573
29.7107
17.6641
10
5
5
(c) stress-z
Z
Z
4
10
15
10
(b) stress-y
10
0
20
0
10
10
30
35
40
0
(a) History of performance function
5
10
15
25
30
35
(b) History of Young’s modulus
Figure 7
Result of parameter identification
9
20
ITERATION
40