SLIDE 1 Electrical Impedance Tomography, inverse prob- lems, material characterization & structural health monitoring
Aku Sepp¨ anen
Department of Applied Physics University of Eastern Finland Kuopio, Finland Finnish Inverse Problems Summer School 2019
UEF // University of Eastern Finland
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Inverse problems
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Linnanm¨ aki
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Contents
Electrical Impedance Tomography(EIT) Ill-posedness of the EIT inverse problem EIT-imaging of concrete EIT-based sensing skin
SLIDE 8 Electrical Impedance Tomography (EIT)
In EIT electric currents I are applied to electrodes on the surface
- f the object and the resulting potentials V are measured using
the same electrodes. The conductivity distribution σ = σ(x) is reconstructed based on the potential measurements.
SLIDE 9 The forward model in EIT
∇ · (σ∇u) = 0, x ∈ Ω u + zℓσ∂u ∂ν = U(ℓ), x ∈ eℓ, ℓ = 1, 2, . . . , L
∂ν dS = I (ℓ), ℓ = 1, 2, . . . , L σ∂u ∂ν = 0, x ∈ ∂Ω\ ∪L
ℓ=1 eℓ
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Finite element approximation of the EIT forward model
FE-approximation of the complete electrode model ⇒ V = U(σ) where σ ∈ RN is a finite dimensional approximation of the conductivity.
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Solution of the forward problem
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Solution of the forward problem
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Solution of the forward problem
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Solution of the forward problem
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Solution of the forward problem
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Solution of the forward problem
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Solution of the forward problem
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Two different targets & electrode potentials
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Inverse problem of EIT
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Two different targets & electrode potentials
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Two different targets & electrode potentials
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Two different targets & electrode potentials
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Two different targets & electrode potentials
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Two different targets & electrode potentials
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Two different targets & electrode potentials
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Two different targets & electrode potentials
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Two different targets & electrode potentials
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Inverse problem of EIT
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Modeling errors in EIT
Example 1: Modeling error due to unknown contact impedances z (true z = 1, assumed z = 0.01).
Left: True conductivity distribution. Middle: EIT reconstruction based on correct model (z=1). Right: EIT reconstruction based on incorrect model (z=0.01).
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Modeling errors in EIT
Example 2: Modeling error due to inaccuracy in the injected current I (error level in I: 0.5%).
Left: True conductivity distribution. Middle: EIT reconstruction based on correct model. Right: EIT reconstruction based on incorrect model.
SLIDE 31 Modeling errors in EIT
Example 3: Modeling error due to unknown boundary shape.
Left: Photograph of a target. Middle: EIT reconstruction based on correct
- geometry. Right: EIT reconstruction based on circular model geometry.
◮ Nissinen et al 2010
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Inverse problem of EIT
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Solution of the inverse problem of EIT
SLIDE 34 Solution of the inverse problem of EIT
Reconstructing the conductivity σ based on noisy observations Vobs is an ill-posed inverse problem. The solution of the inverse problem is typically written in the form σMAP = arg min
σ>0{Ln(Vobs − U(σ))2 + A(σ)}
The functional A(σ) models the prior information on the conductivity distribution σ. Non-linear, constrained optimization problem
SLIDE 35 Iteration step 1
Left: estimated conductivity distribution. Right: Measured vs. computed potentials.
SLIDE 36 Iteration step 2
Left: estimated conductivity distribution. Right: Measured vs. computed potentials.
SLIDE 37 Iteration step 3
Left: estimated conductivity distribution. Right: Measured vs. computed potentials.
SLIDE 38 Iteration step 4
Left: estimated conductivity distribution. Right: Measured vs. computed potentials.
SLIDE 39 Iteration step 5
Left: estimated conductivity distribution. Right: Measured vs. computed potentials.
SLIDE 40 Final estimate
Figure: Left: Photo of the true target; Right: estimated conductivity distribution.
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The resolution of EIT is usually not very high...
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”Blobology”
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”Blobology”
However, if feasible prior information on the resistivity is available, the resolution can be improved...
SLIDE 44 Concrete
The most extensively used construction material in the world About 7.5 cubic kilometers of concrete made each year In the United States, more than 55,000 miles of highways paved with concrete $ 35-billion industry. In the United States, 2 million workers Evaluation, repair and restoration: 35 %
- f the total volume work in building
industry
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Concrete, need for evaluation/testing/monitoring
On-site testing & evaluation
Crack detection Prediction of rebar corrosion risk, etc...
Material characterization in lab scale
Evaluation of transport properties – esp. the ability of concrete to impede the ingress of water
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EIT imaging of concrete
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EIT imaging of 3D moisture flow in concrete
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ECT imaging of 3D moisture flow in concrete
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ECT imaging of 3D moisture flow in concrete
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ECT imaging of 3D moisture flow in concrete
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Results with X-ray CT...
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Imaging of cracks
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EIT-based sensing skin for damage detection
Electrically conductive material (e.g. copper tape, CNT film, copper/silver paint) is applied on the surface of concrete The cracking of concrete breaks also the sensing skin Detecting of cracks in the surface material with EIT
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EIT-based sensing skin for damage detection
We choose the painted sensing skin (Easy to apply & applicable to a large scale). 2D EIT imaging problem
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Case 1: Sensing skin on plexi-glass
Sensing skin painted on plexi-glass 16 electrodes for EIT Synthetic cracks made by scratching the paint
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Case 1
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Case 1
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Case 1
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Case 1
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Case 1
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Case 1
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Case 1
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Case 1
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Case 1
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No blobology!
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Case 1: difference vs absolute reconstructions
SLIDE 70 How?
We fit homogeneous conductivity distribution σref to reference EIT data Vref Denote the discrepancy between Vref and the modeled data by ǫ ǫ = Vref − U(σref) This error is mostly due to inhomogeneity
An approximative modeling error correction; observation model V = U(σ) + ǫ + n
SLIDE 71 How?
MAP estimate σMAP = arg min
0<σ<σref{1
2Ln(V − U(σ) − ǫ)2 + A(σ)} where A(σ) is a potential function related to a total variation prior A(σ) = α
∇σdr A(σ) promotes sparsity of ∇σ.
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Case 2: Notched concrete beam in 4-point bending
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Case 2: Notched concrete beam in 4-point bending
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Case 2: Notched concrete beam in 4-point bending
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Case 2: Notched concrete beam in 4-point bending
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Case 2: Notched concrete beam in 4-point bending
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Case 2: Photo vs. EIT reconstruction
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Case 2: Photo vs. EIT reconstruction
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Case 2: Photo vs. EIT reconstruction
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Case 2: Photo vs. EIT reconstruction
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Case 2: Photo vs. EIT reconstruction
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Case 2: reconstructions, denser FE mesh
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Temperature sensing experiment
Sensing skin was exposed to temperature changes by contact with a heat source. Temperature of the heat source could be controlled within 2◦C, when in contact with the temperature sensor. Reconstructed conductivities were converted to temperature maps based on an experimentally determined T vs. σ curve.
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Local temperature change 77◦C
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Local temperature changes 37◦C and 77◦C
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H2020 project, Science for clean energy
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H2020 project, Science for clean energy
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Thank you!
