estimation of tidal current using kalman filter finite element method with aic ryosuke suga chuo...
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Estimation of Tidal Current using Kalman Filter
Finite Element Method with AIC
Ryosuke SUGAChuo university
DEPARTMENT OF CIVIL ENGINEERING
FACULTY OF SCIENCE AND ENGINEERING
Second MIT Conference on Computational Fluid and Solid Mechanics
IntroductionIntroduction
In a seaside area, various structures are built in Japan.
To grasp the state of the sea around a seaside areaJAPAN
The mechanical and the manual error are included in the observation data. (observation noise)
A numerical model cannot express the physical phenomena completely. (system noise)
It is difficult to set many observation points economically.
Important
Kalman filter finite element method
Second MIT Conference on Computational Fluid and Solid Mechanics
Kalman FilterKalman Filter
To be presented by Kalman and Bucy in 1960’s
the filtering algorithm based on stochastic process including noises
Since the noise is taken into consideration,Kalman filter can remove the noise.
It can estimate the state estimative valueonly in time series.
aerospace science, control engineering and civil engineering
Kalman filter
Observation data
Estimation value
(+noise)
Second MIT Conference on Computational Fluid and Solid Mechanics
Kalman Filter Finite Element MethodKalman Filter Finite Element Method
The conventional Kalman filter can not estimate the state values in space model.
Kalman filterKalman filter + finite element method + finite element method
It is possible to estimate the state estimative values not only in time series but also in space model.
This method is possible even if it is a large-scale domain.
Second MIT Conference on Computational Fluid and Solid Mechanics
PurposePurpose
To present Kalman filter finite element method
To estimate of tidal current using Kalman filter finite element method
Second MIT Conference on Computational Fluid and Solid Mechanics
kkkkk wGxFx 1
kkkk vxHy
<system equation>
<observation equation>
The state-space model of the Kalman filter
kx FG kw
ky H kv
: State vector at time k : State transition matrix
: Driving matrix : System noise
: Observation vector : Observation matrix : Observation noise
Kalman FilterKalman Filter
Second MIT Conference on Computational Fluid and Solid Mechanics
Applying the finite element equationApplying the finite element equation
Second MIT Conference on Computational Fluid and Solid Mechanics
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AlgorithmAlgorithm
0, ii gu
0, iihu
<momentum equation>
<continuity equation>
Shallow water equationShallow water equation
h
g
hu : Water velocity : Gravitational acceleration
: Water elevation : Water depth
Second MIT Conference on Computational Fluid and Solid Mechanics
Basic EquationBasic Equation
Finite Element MethodFinite Element Method
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tvSuShtMM
The spatial discretization
The temporal discretization
Galerkin methodGalerkin method
BTD methodBTD method
Finite element equation
Second MIT Conference on Computational Fluid and Solid Mechanics
Kalman Filter + Finite Element MethodKalman Filter + Finite Element Method
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Finite element equation
F kx
State transition matrix State vector
Finite element matrix
Second MIT Conference on Computational Fluid and Solid Mechanics
}ˆ{}ˆ{],[][ 0100 xxV 1.1)]][][[]([]][[][ T
kT
kk HHRHKT
kkkk HKIHKIP ])][[]]([])[][[]([][
TTkk GQGFPF ]][][[]][][[][ 1
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Second MIT Conference on Computational Fluid and Solid Mechanics
Tkk KRK ]][][[
AlgorithmAlgorithm
Second MIT Conference on Computational Fluid and Solid Mechanics
Akaike Information CriterionAkaike Information Criterion(AIC)(AIC)
AIC is the criterion of the selected one from the models which applied maximum log-likelihood method.
Maximum log-likelihood method is the way to choose the value of mother group that has the possibility to produce observed sample larger than something else.
Numerical ExampleNumerical ExampleEstimation of tidal current using KF-FEM
Onjuku coast
The water pollution moves ahead with the inflow of pollution material.
Second MIT Conference on Computational Fluid and Solid Mechanics
Numerical ModelNumerical Model
Onjuku coast
Onjuku Coast
Iwawada Port
No.5
No.4No.3
No.2
No.1
Onjuku Port
Second MIT Conference on Computational Fluid and Solid Mechanics
Node : 407Element : 728
(m)
Finite Element Mesh and Water DepthFinite Element Mesh and Water Depth
Second MIT Conference on Computational Fluid and Solid Mechanics
Observation Data at NO.1Observation Data at NO.1
Onjuku Coast
Iwawada Port
No.5
No.4No.3
No.2
No.1
Onjuku Port
Observation Estimation
Onjuku Coast
Iwawada Port
No.5
No.4No.3
No.2
No.1
Onjuku Port
Observation Estimation
ResultResult
Second MIT Conference on Computational Fluid and Solid Mechanics
ResultResult
ConclusionConclusion
The tidal current was estimated using KF-FEM
The tidal current at the Onjuku coast have been analyzed.
KF-FEM is able to estimate a large-scale domain.
KF-FEM has been presented.
Second MIT Conference on Computational Fluid and Solid Mechanics
Second MIT Conference on Computational Fluid and Solid Mechanics
Second MIT Conference on Computational Fluid and Solid Mechanics
}ˆ{}ˆ{],[][ 0100 xxV 1.1)]][][[]([]][[][ T
kT
kk HHRHK
]])[][[]([][ kkk HKIP TT
kk GQGFPF ]][][[]][][[][ 1
if ][][ 1kk PtrPtr go to 6then
else go to 2
2.
3.
4.
5.
Off-lineOff-line
}ˆ]{[}{ 1*
nn xFx
})]{{[}]({[}{}ˆ{ **nnnn xHyKxx
6.
7.On-lineOn-line
AlgorithmAlgorithm
ResultResult
Onjuku Coast
Iwawada Port
No.5
No.4No.3
No.2
No.1
Onjuku Port
Observation Estimation