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Kalman Filter Finite Element Method
Applied to Dynamic Motion of Ground
Yusuke KATO , Mutsuto KAWAHARAand Naoto KOIZUMI
Department of Civil Engineering, Chuo University,Kasuga 1-13-27,
Bunkyo-ku, Tokyo 112-8551, JAPAN
E-mail : [email protected]
Sato Kogyo co.,Ltd4-12-20 Nihonbashi-honcho
Chuo-ku,Tokyo,JAPAN
Summary
This paper presents an estimation of dynamic elastic behavior of
the ground body usingthe Kalman filter finite element method. In
this research, as the state equation, thebalance of stress
equation, strain - displacement equation, the stress strain
equation areapplied. For the temporal discretaization, the Newmark
method is used and for thespatial discretization the Galerkin
method is applied. The Kalman filter finite elementmethod is the
combination of the Kalman filter and the finite element method.
This iscapable of estimation not only in time but also in space
directions, which was confirmedby the application to the
Futatsuishi quarry site.
KEY WORDS: Finite Element Method; Kalman Filter Finite Element
Method; Balanceof strain equation; Strain-displacement equation;
Stress-strain equation;Futatsuishi quarry site
1 INTRODUCTION
Generally, observation data obtained at natural practical site
is distorted by noise, andfurther, a lot of state variables are not
able to observe in direct way. The Kalman filter isthe method of
estimating unknown variable using the observation data distorted by
noise[1],[2]. The Kalman filter can be applied to the state space
model, which consists of bothsystem and observation equations. It
is necessary to find out suitable coefficients so as to
minimize square of the error.This coefficient is called the
Kalman gain. Improved valueis obtained by multiplying estimated
value by the Kalman gain. However, the Kalmanfilter is not able to
estimate the state values in space direction.Therefore by
combiningthe Kalman filter and the finite element method, the
Kalman filter finite element methodis can be obtained to estimatie
not only in time but also in space directions. Kawaharaand this
group have presented some papers concerned with the applications
([3]-[8]). TheNewmark method was applied to the temporary
discretization and the Galerkin methodwas applied to the spatial
discretization. The observation at the Futatsuishi quarry siteis
used in this research, which is located in Mt.Minowa in Miyagi
prefecture, Japan. Theblast examination was carried out on
September, 12th through 15th, 2005. The observation
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acceleration is measured by the accelerometer, which was set at
two points. Observationdata at one point is used as an observation
data, and at another point is used as anreference. The estimation
value is estimated by the Kalman filter finite element methodusing
observation data of acceleration. The estimation value at the
estimation point iscompared with data actually observed at another
observation point. Effectiveness of theKalman filter finite element
method is verified.
2 THE KALMAN FILTER
2.1 State Space Model
The Kalman Filter is based on a set of two systems. The system
equation can be expressedstate of the phenomena. The observation
equation is dependent on the observation data.System equation is as
follows;
xk+1 = Fkxk + Gkwk (1)
and observation equation is
yk = Hkxk + vk (2)
where xk is state vector at time k, Fk is state transition
matrix which represents the finiteelement equation, Gk is driving
matrix and wk is system noise, and yk is observation vectorat time
k, Hk is observation matrix and vk is an observation
noise,respectively.System noise wk is assumed:
E{wk} = 0 (3)
cov{wk, wj} = E{wk, wTj }
= Qkkj (4)
and observation noise vk is
E{vk} = 0 (5)
cov{vk, vj} = E{vk, vTj }
= Rkkj (6)
with
E{wk, vj} = 0 (7)
where E{ } means expectation operation, kj is the Kroneckers
delta function, in which
kj =
1 k = j0 otherwise
The optimal estimate xk is the average of xk giving the
observation data Yk,2
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xk = E{xk|Yk} (8)
The covariance Pk is written as follows ;
Pk = cov{xk|Yk}
= E{(xk xk)(xk xk)T} (9)
where Pk is called estimated error covariance. The estimate
x
k is an average of xk givingthe observation data Yk1,
xk = E{xk|Yk1} (10)
The covariance k is written as follows;
k = cov{xk|Yk1}
= E{(xk xk)(xk x
k)T} (11)
where k is called as predicted error covariance.
2.2 Formulation
The Bayes rule is shown as follows;
P(xk|Yk) =P(yk|xk)P(xk|Yk1)
P(yk|Yk1). (12)
Optimal estimated value xk, Kalman-gain Kk, estimated error
covariance Pk and predictederror covariance k+1 are derived from
the assumptions and the algorithm can be writtenas;
{xk} = {x
k} + [Kk]({yk [Hk]{x
k})
[0] = [v0], {x1} = {x0}
[Kk] = [k][Hk]T([Rk] + [Hk][k][Hk]
T)1
[Pk] = ([I] [Kk][Hk])[k]
[k+1] = [Fk][Pk][Fk]T + [Gk][Qk][Gk]
T
(13)
where Qk is system error covariance and Rk is observation error
covariance,respectively.
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3 STATE EQUATION
Here and in next three sections, indecial notation and the
summation convention withrepeated indices are used. The equilibrium
of stress equation is expressed as,
ij,j bi + ui = 0, (14)
where ij , bi, , ui denote total stress, body force, density of
the ground, acceleration,respectively. The strain - displacement
equation can be described in the following form,
ij =1
2(ui,j + uj,i), (15)
where ij and ui are strain and displacement, respectively. The
stress - strain equation is
ij = Dijklkl, (16)
where Dijkl expresses coefficient of elastic stress - strain
relation and can be written as,
Dijkl = ijkl + (ikjl + iljk), (17)
in which ij is Kroneckers delta, and Lames constant and are
=E
(1 2)(1 + ), (18)
=E
2(1 + ), (19)
where E is the Youngs modulus and is Poisson ratio,
respectively. The boundary S canbe divided into SU and ST. On
theses boundaries, the following condition are specified
ui = ui on SU, (20)
ti = ijnj = ti on ST, (21)
where ui and ti mean the known values on the boundary and ni is
the external unit vectorto the boundary.
4 FINITE ELEMENT EQUATION
Applying the finite element method, the discretized equation
with the linear triangleelement is obtained as follows;
Mikuk + Kikuk = i, (22)
where uk denotes the displacement at node in k
direction.Considering the effect of damping, eq.(22) can be
expressed as,
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Mikuk + Cikuk + Kikuk = i, (23)
Mik = V
(NN)dV, (24)
Cik = 0M + 1Kik, (25)
Kik =
V
(N,jDijklN,l)dV, (26)
i =
V
(Nbi)dV
S
(N ti)dS, (27)
in which Ni is the interpolation function of the finite element
method. For the damping,eq.(25) is assumed, where 0,1 are damping
coefficient.
5 NEWMARK METHOD
In this paper, Newmark method is applied to the finite element
equation. In Newmark method, velocity and displacement at time
(n+1) time are expressed as follows,
u(n+1)i = u
(n)i + u
(n)i t +
1
2u(n)i t
2 + t2(u(n+1)i u
(n)i ), (28)
u(n+1)i = u
(n)i + u
(n)i t + t(u
(n+1)i u
(n)i ), (29)
where u(n+1)i , u(n+1)i are displacement and velocity at (n+1)
time pint, substituting theseinto the finite element equation, the
following equation can be derived.
(u(n+1)k ) = E
1ikAik(u
(n)k ) E
1ikBik(u
(n)k ) E
1ikKik(u
(n)k ) + E
1iki, (30)
where Eik, Aik, Bik can be written as,
Eik = M +t
2
Cik+t2
4
Kik, (31)
Aik = t
2Cik+
t2
4Kik, (32)
Bik = Cik+ tKik (33)
in which Mik, Cik, Kik are expressed in eqs.(24)-(26).
Acceleration at (n+1) time is
substituted into eqs.(28) and (29) to calculate u(n+1)k andu
(n+1)k .
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6 THE KALMAN FILTER FINITE ELEMENT METHOD
6.1 State Transition Matrix Fik
Applying the finite element method to the Kalman filter, the
finite element equation isused for the state transition matrix.
From eq.(30), the state transition matrix is given as
follows ;
{uk}n+1 = [Fik] {uk}
n + [fik]
ukuk
n+ [i] (34)
Fik = [Eik]1
t
2Cik+
t2
4Kik
(35)
6.2 Algorithm
The algorithm of the Kalman filter finite element method is
written as follows. In thisalgorithm, [Fn] is used in place of Fik
in eq.(34).
1). [0] = [v0], {u1} = {u0}
2). Calculate un and un by eq.(28),eq.(29)
3). [Kn] = [n][Hn]T([Rn] + [Hn][k][Hn]
T)1
4). [Pn] = ([I] [Kn][Hn])[n]
5). [n+1] = [Fn][Pn][Fn]T + [Gn][Qn][Gn]
T
6). {un} = [Fn1]{un1} + [fn]{un} + [gn]{un}
7). {un} = {u
n} + [Kn]({yn [Hn]{u
n})
where n expresses the time cycle, un , nu and un represent
acceleration, velocity anddisplacement at n time, respectively.
Using the above algorithm, calculation consideringan observation
value can be performed.
7 NUMERICAL STUDY 1
Verification of the Kalman filter finite element method is
carried out. The numericalresults by the Kalman filter finite
element method are compared with the observationdata, which were
obtained by the forward analysis.The computational model and
observation points are shown in Fig.1, which tunnel ismodeled.
Total number of nodes and elements are 2577 and 12008,respectively.
Boundary
condition of upper surface is assumed to be the ground, and the
boundary condition onother surfaces is assumed as slip conditions.
Time increment t is 0.001(s).Dampingcoefficient 0, 1 is set as
0.0,0.005,respectively. The elastic modulus, Poisson ratio
anddensity of ground are set as 6.0104[kN/m2], 0.3 and
6.0[g/cm3],respectively. Observationerror covariance R is 1.0 103
and system error covariance Q is 1.0 103.As an external force,
uniformly distributed load of 1000[KN/m2] is added to tunnel
face,which is shown in Fig.2. This external force is an image of
blasting excavation. Estimationdata is computed at the point shown
in Fig.3. Acceleration and velocity, displacementare estimated at
estimation point using observation data. As the observation data,
thecomputed acceleration obtained by the finite element method at
points Nos.1 and 2 adding
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the white noise is used. Fig.4 shows the observation data of the
acceleration in x-directionat No.1. At a glance, it is impossible
to distinguish the average value in Fig.4.
8 RESULT 1
From Figs.5-7 show the comparison of acceleration,velocity and
displacement betweencomputed value using the finite element method
and estimation value using the Kalmanfilter finite element method
at estimation point. Figs.5-7 show the acceleration, velocityand
displacement in x-direction, respectively. Looking at those
figures, the noise as shownin Fig.4 has been clearly filtered by
the present method.
9 NUMERICAL STUDY 2
In this research, the Futatsuishi quarry site is used as a model
in numerical study 2. TheFutatsuishi quarry site is located in
Mt.Minowa in Miyagi prefecture, Japan. The sitephotograph is shown
in Fig.16. This area is about 400 200[m]. The blast was done
fourtimes at the blasting point. Observation velocity and
acceleration are measured by theaccelerometer and
velocity-meter.The computational model is shown in Fig.9. Total
number of nodes and elements are 5056and 24961,respectively.
Boundary condition of the bottom surface is assumed slip
condi-tions, which means both displacements in x and y directions
are free. Time increment tis 0.001(s).Damping coefficient 0 and 1
are set as 0.143 and 0.00173,respectively. Thereare two elastic
moduli zones, denoted by CL and CM. The elastic moduli of CM and
CLzone , Poisson ratio, density of ground is set as 3.0106[kN/m2]
and 8.0105[kN/m2], 0.3,2.0103[Kg/m3],respectively. Observation
error covariance R is 1.0103 and system er-ror covariance Q is
1.0103,respectively. As an external force, uniformly distributed
loadof 3.2 106[N/m2] is assumed as shown in Fig.10. For the
temporal direction, the impul-sive external force as shown in
Fig.11 is applied. This external force is an image of blasting
excavation. Observation data were obtained at the blast on
September 15, 2005. In thisresearch, observation data of
acceleration is used. Observation and estimation points areshown in
Fig.12,respectively. Figs.13-15 show the observation data at
observation point,which are acceleration of x-direction,y-direction
and z-direction,respectively.
10 RESULT 2
Figs.16-18 show the comparison of acceleration between
estimation value using Kalmanfilter finite element method and
observation value at the estimation point. Estimationcan explain
the inclination of the filtered data. In this estimation analysis,
it is importantto estimate the external force. The peak value is
assumed and some iteration has been
carried out to determine the peak value.
11 CONCLUSION
In this research, the Kalman filter finite element method
applied to the dynamic elasticbehavior of the ground is presented.
The balance of stress equation,strain-displacementequation and the
stress strain equation are used as state equation. As a result,
observationand estimation are good in agreement, and noise included
in the observation could beremoved at the estimation
point.Therefore, the effectiveness of the Kalman filter
finiteelement method has been shown. Estimation problem of
acceleration at the Futatsuishi
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quarry site is carried out. Comparing with the estimated and
observed acceleration, itis clearly shown that the noise of the
observation has been removed. It is necessaryto estimate the
external force in advance before the estimation analysis. The
externalforce determination should be introduced. The present
technique can be extended to thedetermination of the elastic
modulus of the ground.
12 acknowledgment
Authors would like to express our gratitude to Tohoku Regional
Agricultural Administra-tion Office,government of Japan and Sato
Kogyo co.,Ltd for the collection of observationdata at the
Futatsuishi quarry site.
References
[1] Grewal M, Andrews A. Kalman Filtering Theory and
Practice.
[2] Kalman RE. A New Approach to Linear Filtering and Prediction
Problems.
T rans.ASME , J.BasicEng.82D(1) 1960; 34-45
[3] Fujimoto M, Kawahara M. Tidal Flow Analysis Using KF-FEM
with Domain Decom-position Method. International Series
Mathematical Sciences and Applications Computaional Methods for
Control Applications 2001; Vol.16,199-218
[4] Yonekawa K, Kawahara M. Application of Kalman Filter Finite
Element Methodand AIC. International Journal of Computational Fluid
Dynamicsr. 2003; Vol.17.Number 4,307-317
[5] Hayakawa Y, Kawahara M. Tidal Flow Analysis Using Kalman
Filter. Third Asian Pacif ic Conf erence on Computational M
echanics, Seoul, Korea 1996.
[6] Funakoshi Y, Kawahara M. Estimation of Incident Flow of
Tidal Current Using Ex-tended Kalman Filter with Finite Element
Method. 4th International conference onHydro science and
Engineering.
[7] Suga R, Kawahara M. Estimation of Tidal Current Using Kalman
Filter Finite ElementMethod with AIC. Second Mit Conference on
Computational Fluid and Solid Mechanic 2003.
[8] Wakita H, Kawahara M. Estimation of the River Flow Using
Kalman Fil-ter Finite Element Method. T he Sixth International
Conference on Hydro science and Engineering 2004.
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Fig.1, Finite Element Mesh
Fig.2, External Force
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Fig.3, Estimation Point
Fig.4, x-acceleration10
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-2
0
2
4
6
8
10
12
0 0.05 0.1 0.15 0.2
Acceleration[m/sec
2]
Time[s]
estimationovservation
Fig.5, x-acceleration
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0.01
0.012
0 0.05 0.1 0.15 0.2
Velocity[m/sec]
Time[s]
estimationovservation
Fig.6, x-velocity
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-2e-05
0
2e-05
4e-05
6e-05
8e-05
0.0001
0.00012
0 0.05 0.1 0.15 0.2
Displacement[m]
Time[s]
estimationovservation
Fig.7, x-displacement
Fig.8, Futatsuishi Site
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Nodes:5056 Element:24961Fig.9, Finite Element Mesh
Fig.10, External Force
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-1e+07
-5e+06
0
5e+06
1e+07
1.5e+07
2e+07
2.5e+07
3e+07
3.5e+07
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2
Externalforce[KN/m2]
Time[s]
External force
Fig.11, Time History of External Force
Fig.12, Observation and Estimation Points
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-1.5
-1
-0.5
0
0.5
1
1.5
0 0.05 0.1 0.15 0.2
Acceleration[m/sec2]
Time[s]
Observation data
Fig.13, x-acceleration at observation point
-1
-0.5
0
0.5
1
1.5
2
2.5
3
0 0.05 0.1 0.15 0.2
Acceleration[m/sec2]
Time[s]
Observation data
Fig.14, y-acceleration at observation point
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-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
0 0.05 0.1 0.15 0.2
Acceleration[m/sec2]
Time[s]
Observation data
Fig.15, z-acceleration at observation point
-8
-6
-4
-2
0
2
4
6
8
0 0.05 0.1 0.15 0.2
Acceleration[m/sec2]
Time[s]
KF-FEMObservation data
Fig.16, x-acceleration at estimation point
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-8
-6
-4
-2
0
2
4
6
8
10
0 0.05 0.1 0.15 0.2
Acceleration[m/sec2]
Time[s]
KF-FEMObservation data
Fig.17, y-acceleration at estimation point
-12
-10
-8
-6
-4
-2
0
2
4
6
8
10
0 0.05 0.1 0.15 0.2
Acceleration[m/sec2]
Time[s]
KF-FEMObservation data
Fig.18, z-acceleration at estimation point
17
