Reconstruction of capacitance tomography images of simulated two-phase flow regimes

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Applied Numerical Mathematics 46 (2003) 197–208www.elsevier.com/locate/apnum

Reconstruction of capacitance tomography imagesof simulated two-phase flow regimes

Susana Gomeza,∗, Michiyo Onob, Carlos Gamiob, Andres Fraguelac

a IIMAS–UNAM, Mexicob IMP, Mexico

c BUAP, Puebla, Mexico

Abstract

Reconstruction of electrical capacitance tomography (ECT) images is performed using simulateddistributions. An inverse problem has to be solved to find the permittivity coefficient, using measuremthe capacitances. The least squares optimal solution is sought using a Gauss–Newton method, with adescent condition and a backtracking for the steplength. The Tikhonov regularisation method is used, to comeasurement error propagation due to the ill-posednes of the inverse problem. It is shown that the reconis very sensitive to the Tikhonov regularisation parameter and theL-curve method to find its value is used. Whthe optimal regularisation parameter is used, convergence is attained to points where no further precisipermittivity parameter is possible. Simulation examples using typical two-phase flow regimes are presenthe approximated images as well as the range of values for the regularization parameter for different regshown. 2003 Published by Elsevier B.V. on behalf of IMACS.

1. Introduction

Electrical capacitance tomography (ECT) is an emerging technique aimed at the non-invasivevisualisation of industrial processes like mixing, separation and two-phase flow [14,15]. Onapplication to flow imaging will be considered here. The basic principle of this method is to psensor containing an array of between 8 to 16 contiguous sensing electrodes around the pipe cothe process fluids, at the cross-section to be investigated (see Figs. 1 and 2). The pipe walbe electrically non-conducting in the zone of the electrodes. The electrodes are typicallylong. The sensor also has a couple of cylindrical end guards adjacent to the electrodes and

* Corresponding author.E-mail address: [email protected] (S. Gomez).

0168-9274/03/$30.00 2003 Published by Elsevier B.V. on behalf of IMACS.doi:10.1016/S0168-9274(02)00253-2

198 S. Gomez et al. / Applied Numerical Mathematics 46 (2003) 197–208

ectrice mutual

ents the. Theruction

le

eresidered

that thea two-s the

(when

d,

Fig. 1. Schematic diagram of a 12-electrode ECT sensor (the outer screen is not shown for clarity).

cylindrical metallic screen cover (not shown in Fig. 1), all of which are always kept at an elpotential of zero volts. The sensing electrodes are connected to an apparatus that allows all thcapacitances between the different electrode pairs to be measured. From this set of measuremelectrical permittivity distribution inside the sensor is obtained using a suitable inversion algorithmpermittivity distribution lects the spatial arrangement of the two phases in the flow. Image reconstcan thus be regarded as aninverse permittivity problem.

In order to measure the mutual capacitanceCij between electrodesi andj , the potential of electrodei(source) is set to a known valueV and the chargeQ induced on electrodej (detector) is measured, whikeeping the potential of all the other electrodes at zero volts.Ci,j is then given by

Ci,j = Qj

Vi

. (1)

BecauseCi,j = Cj,i holds, there are onlyn = N(N −1)/2 independent mutual capacitance values, whN is the number of electrodes. In the remainder of this paper, a 12-electrode sensor will be conand thusn = 66.

The use of the cylindrical guards at the ends of the sensing electrodes (and assumingphase distribution changes slowly in the axial direction) allows the sensor to be represented bydimensional model [13], like the one shown in Fig. 2. The application of Maxwell’s equations yieldgoverning equation for the sensor:

∇ · [ε(x, y)∇φ(x, y)] = 0, (2)

subject to the following Dirichlet boundary conditions imposed by the measurement procedureelectrodei is the source):

ϕi ={

V (x, y) ⊆ Γ,

0 (x, y) ⊆ all Γj(j �= i) and(x, y) ⊆ Γs,(3)

where ε(x, y) and φ(x, y) are the permittivity and potential distributions in the sensor, whileΓk

(k = 1,2,3, . . . ,12) andΓs are the spatial locations of the electrodes and the screen, respectively.The forward problem is to determine the mutual capacitancesCi,j given a known permittivity distri-

bution ε(x, y). To do this, Eq. (2) subject to (3) can be solved forφ using the finite element metho

S. Gomez et al. / Applied Numerical Mathematics 46 (2003) 197–208 199

valuating

n thetherithms

osedsolution

ularisein the

Fig. 2. Cross-section of a 12-electrode ECT sensor.

and then Gauss’s law can be used to calculate the charges induced on the detection electrodes, enumerically the line integral:

Qj =∫

(x,y)⊆Γj

ε(x, y)∇φ(x, y) · −−−→dΓj . (4)

Finally, the mutual capacitances can be found applying Eq. (1).

2. Inverse problem

The inverse problem involves finding the permittivity distribution inside the pipe based oknowledge of all the 66 mutual capacitancesCi,j (note that in the pipe itself and in the area betweenpipe and the screen the permittivity distribution is known). This can be done using iterative algobased on optimisation theory, as discussed in this section.

Given some measurements of the capacitance with measurement errorsCmeas= C + δ, we want tofind the permittivityε, solving a least-squares data fitting minimisation. This is a non-linear ill-pinverse problem. Due to the ill-posedness, the measurement errors can be propagated, and theobtained can be far from the real solution [2]. To avoid this error propagation, it is necessary to regthe optimisation problem using, for example, Tikhonov regularisation [11], which adds a termobjective function to control the size of the solution. This regularisation depends on a parameterα.

The regularised optimisation problem we have to solve is the following: findε that solves

Min f (ε) = 1

2

∥∥C(ε) − Cmeas

∥∥22 + α‖Lε‖, (5)

whereC(ε) is the computed capacitance for a given value ofε,L is a regularisation matrix andα is theregularisation parameter.

The solution of this optimisation problem is calculated iteratively by

εk+1 = εk + �ε, (6)


where the descent direction�ε is found solving the Tikhonov Gauss–Newton (or Levenberg–Marquardt)linear system of equations

er to

, that is

sthethe

finds apose of

o-eology

[J T

(εk

)J(εk

) + αLTL]�ε = −J T

(εk

)[C(ε) − Cmeas

] − αLTLε, (7)

whereJ is the Jacobian off with respect toε.If the initial approximationε0 is sufficiently close to the solution, the method converges. In ord

guarantee global convergence from any initial approximation, it is necessary to find a step lengthλ, thatguarantees sufficient descent of the objective function and that satisfies the Wolfe conditions [10]f (εk+1) has to satisfy the following inequality

f(εk+1

)� f

(εk

) − 10−3λJ T(εk

)[C

(εk

) − Cmeas]T

�ε. (8)

The iteration will then be

εk+1 = εk + λ�ε. (9)

To findλ, we backtrack using the bisection method, described in [7].The minimisation process will stop when the 2-norm of the gradient‖∇f ‖ < tol1 (optimal solution),

when no further precision can be achieved‖εk+1 − εk‖ < tol2, or when the Jacobian at some iterationεk

becomes singular (convergence to a non-stationary singular point).

3. Determination of the regularisation parameter α

The solution of the inverse problem (5) is very sensitive to the regularisation parameterα, and thus theproblem of finding its value accurately is extremely important as will be shown numerically later.

TheL-curve algorithm [8,9] was designed to compute the regularisation parameterα in the Tikhonovmethod. The main idea is to construct a parametric curve of the norm of the regularised solution‖εk‖versus the norm of the residual‖C(εk) − Cmeas‖. This curve is formed by discrete points and whenproblem is ill-posed has an “L” shape, which gives the name to the algorithm [7]. The corner ofL-curve balances the minimisation of the residual norm and the norm of the solution, that is,balance between the approximation to the solution and the error propagation, which is the purregularisation.

To construct theL-curve, we start from a largeα0, and minimise problem (1) to obtainε∗(α0). Wethen plot‖C(ε∗(α0))−Cdata‖ versus‖ε∗(α0)‖ to obtain a point in the curve. We then decreaseα makingα1 = α0/10 and minimise again to obtainε∗(α1), and continue in this fashion until anL-curve has beenformed. The optimalα, denoted by ofα∗, will be at the corner, and the solution is the permittivityε∗(α∗).

It will be shown in the numerical results, that the solution is very sensitive to the value ofα, and goodimages are only obtained forε∗(α∗).

4. Numerical results

In this work we use the software packageEIT-2D for two-dimensional electrical impedance tomgraphy (EIT) image reconstruction, developed as part of theEIDORS project by researchers at thUniversity of Kuopio, Finland, and the University of Manchester Institute of Science and Techn


(UMIST), UK [12]. The package consists of a series of routines written in MATLAB.EIT-2D supportstwo EIT variants: electrical resistance tomography (ERT) and electrical capacitance tomography (ECT).

he direct

atter

mplescanrved for

wardacitancesnd the

In this paper only the latter is considered.The package uses the quadtree mesh generator [5], and the finite element method to solve t

problem.We have modified EIT-2D software, to find the step lengthλ required to force sufficient descent

each iteration of the minimisation, and to use theL-curve to find the Tikhonov regularisation parameα, to solve typical examples of two-phase flow regimes.

If the modification to find the step length is not included, the iteration (6) diverges in all the exatested. Also, if the optimal regularisation parameterα∗ is not found accurately, the images obtainedbe completely wrong as will be shown for Examples 2 and 6, although this phenomena was obseall the examples.

All the examples are synthetic, and thus for a known permittivity distribution, we solve the forproblem (2)–(3), and calculate the capacitances. Then the exercise consists of, given these capwith a random normal distribution error of 2% added (to simulate the measurement errors), fipermittivity.

Fig. 3. One object at the center. Calculated image:α = 10−6, Max= 2.1509, Min= 0.9180.


methode

ttion

Fig. 4. Two objects at the center. Calculated image:α = 10−7, Max= 1.6088, Min= 0.9223.

We now describe the examples and report the number of iterations taken by the optimisationfor each example. In the figures we report the exact solution, theL-curve obtained where we show thoptimalα∗ found, and the images obtained for this optimalα∗.

Although the algorithm stopped for all examples, when no better precision forε was possible, thais when‖εk+1 − εk‖2 < 10−6 when usingα∗, that did not happen in all cases when the optimisa


Fig. 5. Two objects, at the center and at the boundary, equal permittivity. Calculated image:α = 10−7, Max = 2.3474,Min = 0.8837.

Fig. 6. Two objects, at the center and at the boundary, different permittivity. Calculated image:α = 10−7, Max = 2.8073,Min = 0.8273.


Fig. 7. Six objects, two large at the boundary. Calculated image:α = 10−6, Max= 2.1793, Min= 0.6918.

Fig. 8. Stratified flow. Calculated image:α = 10−2, Max= 2.2093, Min= 0.9216.


ow,p

Fig. 9. Annular flow. Calculated imageα = 10−1, Max= 2.1135, Min= 0.8777.

was carried out with differentα to generate theL-curve. For the examples concerning stratified flannular flow and full flow, it was not possible to use smaller values ofα, as the optimisation would stoat non-stationary singular points, that is at points when the Jacobian became singular.


ittivityactual

ft, thety

29le

y

is

s

Fig. 10. Full flow. Calculated imageα = 100, Max= 2.0032, Min= 1.9958.

The solution images have in their legend, the maximum and minimum value of the permobtained (at some points of the image), to show not only the shape of the image, but also theapproximated value of the permittivity obtained. In Figs. 3–10, the real image is shown on the leL-curve is shown in the middle, indicating the optimalα∗, while the right image shows the permittivicalculated withα∗.

Example 1 consists of an object with permittivityε = 2 placed at the centre using an area oftriangles. In the rest of the grid the permittivity isε = 1. The minimisation process in this exampstopped after 4 iterations forα = 10−6. See Fig. 3.

Example 2 consists of two objects in the center, both withε = 2. In the rest of the grid the permittivitis ε = 1. The minimisation process in this example stopped after 4 iterations forα = 10−7. See Fig. 4.The calculated images for differentα are also shown in Fig. 5.

Example 3 consists of two objects, one at the center and one near the boundary,

(a) both withε = 2. In the rest of the grid the permittivity isε = 1. The minimisation process in thexample stopped after 6 iterations forα = 10−7. See Fig. 5.

(b) One object with permittivityε = 2 and one withε = 1.5. In the rest of the grid the permittivity iε = 1. The minimisation process in this example stopped after 6 iterations forα = 10−7. See Fig. 6.


Example 4 consists of six objects, four at the center and two big at the boundary with permittivityε = 2. In the rest of the grid the permittivity isε = 1. The minimisation process stopped after 7 iterations

s for

e

s

al valuetionsn of the

erwisehe most

ically is

es, is

for α = 10−6. See Fig. 7.Example 5 consists of stratified flow (approximately 30%) with permittivityε = 2. In the rest of the

grid the permittivity isε = 1. The minimisation process in this example stopped after 5 iterationα = 10−2. See Fig. 8.

Example 6 consists of an annular flow type I with permittivityε = 2. In the rest of the grid thpermittivity is ε = 1. The minimisation process in this example stopped after 4 iterations forα = 10−1.See Fig. 9. The images obtained for differentα are also shown in Fig. 9.

Example 7 consists of a full flow with permittivityε = 2. In the rest of the grid the permittivity iε = 1. The minimisation process in this example stopped after 10 iterations forα = 10−1. See Fig. 10.

5. Conclusions

These results show that it is possible to get good images and a good approximation to the reof the permittivity using the EIDORS program, if the optimisation is performed with the condion the steplength to assure global convergence and using regularisation to avoid propagatiomeasurements error.

If Tikhonov regularisation is used, the regularization parameter has to be found accurately, oththe solution can be completely wrong. From all the tests we have made (we only reported here trepresentative), it can be concluded that:

(a) If the flux has several objects the optimal regularisation parameter has to be takenα∗ = 10−7 or 10−8.(b) For a stratified flow and for an annular flow,α∗ = 10−1 or 10−2.(c) For a full flowα∗ = 101.

In the case of a real application, where real time software is needed, the use of theL-curve to find theregularization parameter could be expensive, and an algorithm to find the corner point automatneeded [1]. An alternative regularisation method as multiscale [6] can also be used.

A new mathematical model for the Electrical Capacitance Tomography of two-phase flow regimbeing developed by our group [3,4].

Acknowledgements

We thank Nelson del Castillo for his help in the ellaboration of this article.

References

[1] L. Castellanos, S. Gomez, V. Guerra, The triangle method for finding the corner of theL-curve, Appl.Numer. Math. 43(2003) 359–373.

[2] H. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems, Kluwer, Dordrecht, 1996.


[3] A. Fraguela, J. Oliveros, S. Gomez, A new approach to the solution of the inverse problem in capacitance tomographyimaging of two phase flow regimes, Inverse Problems, submitted.

ation toPhysics,

ethods

inted by

–74..1035–

ORS

rement

ipeline

Part 1:

[4] A. Fraguela, C. Gamio, D. Hinestroza, The inverse problem of electrical capacitance tomography and its applicgas–oil 2-phase flow imaging, in: Proceedings of the 4th WSEAS Conference on Mathematics and Computers inMexico, 2002.

[5] P.L. George, Automatic Mesh Generation, Application to Finite Element Methods, Wiley, New York, 1991.[6] S. Gomez, A. Perez, F. Dilla, R.M. Alvarez, On the automatic calibration of a confined aquifer, in: Computational M

for Water Resources, Vol. IX, Kluwer Academic, Dordrecht, 2002.[7] P.C. Hansen, Rank-Deficient and Discrete Ill-posed Problems, SIAM, Philadelphia, PA, 1998.[8] C.L. Lawson, R.J. Hanson, Solving Least Squares Problems, Prentice-Hall, Englewood Cliffs, NJ, 1974, Repr

SIAM, Philadelphia, PA, 1995.[9] K. Miller, Least squares methods for ill-posed problems with a prescribed bound, SIAM J. Math. Anal. 1 (1970) 52

[10] J. Nocedal, S.J. Wright, Numerical Optimisation, Springer Series in Operations Research, Springer, Berlin, 1999[11] A.N. Tikhonov, Solution of incorrectly formulated problems and the regularisation method, Soviet Math. Dokl. 4,

1038; English translation: Dokl. Akad. Nauk. USSR 151 (1963) 501–504.[12] M. Vauhkonen, W.R.B. Lionheart, L.M. Heikkinen, P.J. Vauhkonen, J.P. Kaipio, A MATLAB package for the EID

project to reconstruct two-dimensional EIT images, Phys. Measurement 22 (2001) 107–111.[13] W.Q. Yang, M.S. Beck, M. Byars, Electrical capacitance tomography: from design to applications, Measu

Control 28 (1995) 261–266.[14] W.Q. Yang, A.S. Sttot, C.G. Xie, M.S. Beck, Development of capacitance tomographic imaging systems for oil p

measurements, Rev. Sci. Instrum. 66 (1995) 4326–4332.[15] C.G. Xie, A. Plaskowski, M.S. Beck, 8-electrode capacitance system for two-component flow identification

Tomographic flow imaging, IEE Proc. A 136 (1989) 173–183.

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Reconstruction of capacitance tomography images of simulated two-phase flow regimes