Application of the finite-element method within a two-parameter regularised inversion algorithm for electrical capacitance tomography

Transcription

1 Appliation of the finite-element method within a two-parameter regularised inversion algorithm for eletrial apaitane tomograph D Hinestroza, J C Gamio, M Ono Departamento de Matemátias, Universidad del Valle, Cali, Colomia. dohin@mafalda.univalle.edu.o Instituto Meiano del Petróleo, Ee Central L Cárdenas Nte #5, Col San Bartolo Atepehuaan, Méio, D.F. gamio@imp.m. Introdution In this paper, we present a theoretial stud of the use of finite-element tehniques in the numerial solution of the inverse prolem of alulating the permittivit oeffiient in eletrial apaitane tomograph ECT [], partiularl applied to the minimization of a speial funtional that depends on two regularization parameters. The theoretial analsis is developed and the orresponding numerial suroutines in MATLAB are given. In ECT, we have a sensor fig. onsisting of an eletriall non-onduting pipe with a irular arra of ontiguous retangular sensing eletrodes, separated small gaps, attahed to its outer surfae. The ring of eletrodes is loated etween two grounded lindrial end-guard eletrodes that eliminate an end-effets, allowing the use of -dimensional -D modelling []. The whole asseml is in turn surrounded a grounded sreen to avoid eternal interferene. The aim of ECT is to reonstrut, using a suitale algorithm, a ross-setional image of the unnown eletrial permittivit distriution inside the non-onduting pipe at the zone of the eletrode ring, from the nowledge of the mutual apaitane that eists etween all possile sensing-eletrode pairs, whih must e previousl measured using a suitale instrument. The eletrial permittivit distriution otained will reflet the phase distriution of a miture ontained in the sensor. To measure the mutual apaitane values i,, for a -eletrode sensor lie that of figure, first a nown eitation voltage is applied to eletrode while eeping all the others at zero potential and the harge on eletrodes to is measured. These measurements divided the eitation voltage diretl represent, to,. Net, the eitation voltage is applied to eletrode while eeping all the others at zero and the harge on eletrodes to is measured, representing, to,. This proedure is repeated, appling voltage to eletrode n and measuring the harge on eletrodes n + to, until, as a final step, voltage is applied to eletrode and the harge of eletrode is measured. In this wa, 66 independent mutual apaitane values are otained.

2 a Fig. ECT sensor: a omplete asseml and ross-setion view The rest of this paper is organised as follows: Setion presents the mathematial model of the prolem, Setion desries the two-parameter regularised inversion method, Setion 4 deals with the appliation of the finite element method in this ontet. And Setion 5 shows several numerial eamples of interest.. Mathematial Model The tomograph sensor of figure an e modelled figure as the irular region with radius R orresponding to the imaging area, surrounded two annular regions: with inner and outer radiuses R and R, respetivel orresponding to the insulating pipe, and with inner and outer radiuses R and R, respetivel orresponding to the area etween the pipe and the eternal sreen.

3 R R R Figure Sensor model Let us then onsider the ounded domain figure with { z, : z < R }, { z, : R < z < R } and { z, : R < z < R }. We shall neglet the width of the inter-eletrode gaps. Then, we an onsider that eletrode i is desried the ar S i { z, : z R, π i- /N argz π i /N }. The following partial differential equation desries the sensor under the oundar onditions i ε, u, for i,..., N u i, in in, S in, R, R, i, S i where ε, is the relative eletri permittivit or simpl permitivit, for short. Q The mutual apaitanes are given the formula, i or Vi i u, i K ε ds S n i where K is a onstant, and u is the solution of the equation -. The ar S orresponds to the urve surrounding eletrode. The inverse prolem is: given ½NN- values, i, i,,,, N, i <, of the mutual apaitanes etween the eletrodes S i and S, determine approimatel the value of ε,. N is the numer of eletrodes in our partiular ase. In order to find ε, we will onsider the spaes X H, E { ε H : ε ε ma}, Y H, N i Z R N N, and the funtion

4 λ ε f o u ε f u ε 4 where u ε u ε 5 N u ε and M N,, M N, λ ε M 6 K N, N M, N, N N, N N, N N, N. Inversion Method The proposed inversion method is ased on the minimization of the following funtional that the depends on two parameters α and β. f u ε + α ε + β u ε β M ε M α ε o λos Z X Y 7 os Here, λ os, where os is the vetor of ½NN- oserved mutual apaitanes for a K given permittivit distriution in the imaging area. A main prolem related with this funtional has to do with finding its derivative. Appling the Gateau derivative, given

5 δm M ε + tδε M ε t ε lim M ' ε δε t 8 then, after doing all the tehnial alulations involved we get δ M ε δ λ, λ ε λ os + β δ u, u ε + α δε, ε 9 where δ λ λ' ε δε and N i i δ u, u ε u δ u d d. i Using the sensitivit equations [], we have to solve the differential equation with i i ε δ u + δε u i δ u on i i Solving the previous prolem forδ u, requires the nowledge of u andδε. B solving i i the equation ε, u, we otain u. To find δε we shall introdue the i onept of the adoin equations, hoosing a funtion Ψ, that satisfies the equation Ψ ε i + Ψ ε i εψ i β u i ε with the oundar onditions Ψ i on. From equations to 4, it follows that δm δm λ ' ε δε, λ ε λ os λ ' ε δε, λ ε λ os N i Ψ N i i Ψ i u i Ψ + u i i i u δε d d δε d d + α δε, ε + α δε, ε 4 Having otained the differential δm M 'ε δε, we now need to find the minimum of M, whih must satisf M 'ε. In pratie, the proess of minimisation an e arried out

6 using the Gauss-Newton Method. In order to do this, the prolem must first e disretised. For this, we shall use the finite-element method. 4. Appliation of the Finite-Element Method In order to appl the finite element method to approimate the solution of the equation ε, u 5 we need to proet the wea form of this differential equation onto a finite-dimensional spae. To find the wea form, we onsider an aritrar funtion v in some adequate spae, suh as H, and given that and using Green's formula, we have that v ε u v εu + εu v 6 u v ε u d d v εu d d + ε u v d d ε v ds 7 n Maing u u i i and sine ε, u, we have that i i u ε u. v d d ε v ds for i,..., N. 8 n If we define a L-dimensional spae V H L, the wea form of the differential i equation requires that u and v e in the spae V L instead of onsidering it in H. Sine V is finite-dimensional, there eists a finite asis { with. Maing L v,,,..., L, in equation 8 we have } L V L i i u ε u. d d ε 9 n B onsidering the epansion of the unnown solution in the form u i L i, u, We denote the vetor of the omponents of the funtion u i in the asis { as l l l } L

7 i u i u M i ul Sustituting in 9 we otain the linear sstem of equations u L i i ε l ul ε l n,,... L. We an write these sstem equations in the form i i Au, for i,, N with A a l where al ε l, with l,,, L, and i is the olumn u i vetor with omponents ε,,, L. n We shall onsider U as the matri whose olumns are the vetors u i, and B as that whose olumns are the vetors i, i.e., [ N N U u,..., u ] and B [,..., ] 4 Then, we an form the larger sstem of equations 4. Seletion of the asis { } L AU B 5 Let s onsider a triangular mesh on the region, made su-dividing it using a set of T non-overlapping triangles { P } T, suh that the verte of an triangle does not lie on the edge of another triangle. We shall onsider as the asis for V L the set of hat funtions whih are linear on eah triangle and tae the value at all nodes, eept for where. This onstrution implies that L i i u ul l l u 6 i

8 This means that if we appl the finite element method, we otain the value of the funtion u i at the mesh nodes. Note that the asis funtions vanish in all triangles that do not have the node. Then, the entries a l of matri A an e alulated onl in the triangles that ontain the node l. This means that a l is zero eept when l and Therefore, A is ver sparse. 4. Calulation of the matries A and B are nodes of the same triangle. Realling that the matri A is given A a l where al ε l, and epanding the permittivit in the form where then we have ε, ε χ, 7 NL P if, P χ P, 8 if, P a l ε l d d ε l d d. 9 T P We an oserve from equation 9, that to find the elements of the matri A we need to alulate the integral l d d l,,, L,,, T P Let s onsider the triangle P with verties V V with oordinates, V, V,, V,, V,. Over the triangle P we have the asis funtions,,,, suh that,, a + +,

9 For the verte V, we have the linear sstem a a a equivalent to a 4 Similarl we ontinue with the other verties, otaining the general sstem a a a 5 If we denote the P the matri, the olumns of the inverse, P -, will give the oeffiients of the asi funtions. The gradients of the asis funtions l, are given,,,, 6 These gradients appear in the seond and third rows of P -. If we define the matri S 7 then P S G 8

10 In order to otain the inner produts we multipl G its transpose. Then we otain l l G G T 9 Clearl this matri is smmetri. Also we have det l l P d d P G G T 4 where. det P P area Sine to eah triangle P orresponds the same permittivit ε, multipling 4 the diagonal matri D with all the elements in its diagonal equal to ε, we otain part of the matri A, orresponding to this triangle P. det l l D P A 4 Clearl A has the entries p a, where.,,, p To find the vetor i in equation, with omponents n u i ε, we onsider the indies of vetor u i orresponding to the points on the oundar, taing into aount that the and that over the omponents of u i are equal to zero. Then, the alulation of the integral is done onsidering onl the verties of the ith-eletrode. B denoting I the indies orresponding to the triangle whih verties are on the, we now the omponent i n U for ever I n. If we represent I I *U,I AI n u i i ε 4 Appling Choles deomposition of R R I I A T, and epresing ˆ I U U we redue the sstem to the matri equation

11 R T RUˆ B 4 If we mae Z RUˆ we have to solve the sstem R T Z B whih solution is given T Z R B. The vetor Uˆ is given Uˆ T R R B 44 Calulation of the ase funtions l over the triangle P Numerial Approimation for the Calulation of the Gradient of funtional M α ε. β 4. Calulation of the adoin prolem The adoin prolem is given u with the oundar i ψ in. 46 i We ontinue in similar wa as we alulate u. i For this ase we epand ψ in the form Using the Green formula we otain i i ε, ψ β * 45 M i i ψ ψ, 47 l l l i i i i ψ. ε lψ l εψ + εψ ε 48 n Replaing the differential equation for i ψ we have M M i l i ψ β * Ul l + ψ l εl ε 49 l l n We an oserve that we an otain a sstem equation of the form A Ψ + CU F 5

12 where the matri A is the same matri define efore. The matri C has entries β i ψ and the vetor F has omponents ε. We an determine this vetor in the same n wa as we did with the vetor B. Otaining the sstem AΨ F CU 5 B using the Choles deomposition if the matri A we have that Ψ R R T F CU. 5 l B oserving that l l and following the alulations in we have M P area P l + δ l 5 P the δ l tae the value if l and zero if l orresponding to the indies of the verties of the triangle P. Sine area P det B we have det l 4

13 , P χ h, 59 Otherwise then using 5 we find that M ε λ λ ε λ ε os N Ψ i P i u i + αε 6 It is not diffiult to prove that λ i i ε u P u dd 6 Using the epansion of i u in the ases we have that λ ε 5 Numerial Eamples i u P i u dd M M i U l l m U m P lm 6 In this wor we have uilt new programs to our two-parameter minimization prolem, and we use the G-Newton method for the minimization proedure. We adapt the software paage EIT-D for two-dimensional eletrial apaitane tomograph EIT image reonstrution, developed as part of the EIDORS proet researhers at the Universit of Kuopio, Finland, and the Universit of Manhester Institute of Siene and Tehnolog UMIST, UK []. All the eamples are simulated, and thus for a nown permittivit distriution, we solve the forward prolem - 4, and alulate the apaitanes. Then, given these apaitanes with an added % random normal-distriution error to simulate the measurement errors, we find the permittivit. We desrie some eamples and report the image-reonstrution error in L for a partiular seletion of values for the parameters α and β. We selet, for eah eample, the est values for these parameters, the ones that result in the smallest reonstrution error. In the figures, we have the eat solution, and we show the optimal α and β seleted from the tale of errors as well as the reonstruted images otained using these optimal parameters. Eample onsists of an oet with permittivit ε plaed at the enter using an area of 9 triangles. In the rest of the grid the permittivit is ε. We tae for this eample, α and β -8. It is ver interesting to see that the onl non-zero parameter atuall has a regularizing effet with respet to the potential and not the permittivit. See Figure and tale of errors.

14 Eample onsists in stratified flow approimatel % with permittivit ε. In the rest of the grid the permittivit is ε. For this ase α β -8. See Figure 4 and tale of errors. Eample onsists in annular flow with permittivit ε plaed at the enter using an area of 9 triangles. In the rest of the grid the permittivit is ε. For this ase, α and -8. One again, in this eample the onl non-zero parameter has a regularizing effet with respet to the potential and there is no regularization assoiated with the permittivit. See Figure 5 and tale of errors. Eample 4 onsists of a full flow with permittivit ε. In the rest of the grid the -8 permittivit is ε. For this ase α. β. See figure 4 and tale errors 4. Real image Error image Reonstruted image α, β -8 Figure : Eample One oet at the enter

15 β α Tale : L -errors for eah α and β in eample Real image Error image Reonstruted image α β 8

16 Figure 4: Eample Stratied flow β α Tale : L -errors for eah α and β in eample Real image Error image Reonstruted image -8 α β

17 Figure 5: Eample Annular flow β α Tale : L -errors for eah α and β in eample Real image Error image Reonstruted image -8 α. β Figure 6: Eample 4 homogeneous flow

18 β α Conlusions and Future Wor We thin that the idea of finite element method using two regularization parameters for the minimization prolem using the onugate gradient method and parallel omputing tehniques to improve the aura and proessing speed of the algorithms. BIBLIOGRAPHY. Alessandrini G., An Identifiation Prolem for an Ellipti Equation in two Variales. Ann. Mat. Pure Appl. Vol. 45,986.. Chavent G., Kunish K., Regularization in State Spae. Mathematial Modeling and Numerial Analsis. Vol. 7, No. 5, 99.. Engl H., Hane M. and Neuauer A., Regularization Inverse Prolems. 996, Dordreht:Kluwer. 4. Gamio J.C., A High-Sensitivit Fleile-Eitation Eletrial Capaitane Tomograph Sstem, PhD Thesis, UMIST, Hinestroza D., Murio D. A., Zhan, S., Regularization Tehniques for Nonlinear Prolems. Journal of Comput. Math. Appli., Hinestroza D., Gamio C. Regularization Tehniques Aplied to Eletrial Capaitane Tomograph. Parametri Optimization and Related Topis VII. Marh. 7. Hinestroza D., Gamio C., On the Numerial Solution of Ill-Posed Prolems Using Regularization Tehniques with Two Parameters Applied to Eletrial Capaitane Tomograph. To e send it for puliation. 8. Neuauer A., Tihonov Regularization for Non-Linear Prolems: Optimal Convergene Rates and Finite-Dimensional Approimation. Inverse Prolems V

19 9. Spin, D.M. and Noras, J.M., Reent Developments in the Solution of the Forward Prolem in Capaitane Tomograph and Impliations for Iterative Reonstrution, Nondestr. Test. Eval., 998, Vol. 4, pp Yang W.Q., Be M.S., and Bars M., Eletrial apaitane tomograph: From design to appliations, Measurement + Control, 8, 995, pp Vauhonen M., Lionheart W.R.B., Heiinen L.M., Vauhonen P.J. and Kaipio J.P.,, A MATLAB paage for the EIDORS proet to reonstrut twodimensional EIT images, Phsiologial Measurement,, pp. 7-