The Applcato of hybrd / methods to solve Electrcal Impedace Tomography s forward problem for the huma head S.R. Arrdge, R.H. ayford, L. Horesh ad J. Skora March, Abstract The forward problem Electrcal Impedace Tomography (EIT) requres a accurate estmato of ts soluto. Although aalytcal solutos est for a lmted umber of cases, there s o sutable aalytcal soluto for the geometry of the huma head. A commoly adopted method s the use of Fte Elemet Methods (). However, the huma head cossts of layers, whch clude the scalp ad CSF, whch are appromately mm thck. Meshg these layers s techcally dffcult ad troduces sgfcat errors to the soluto. A terestg alteratve s the use of hybrd methods combg both ad. The oudary Elemet Method () s a alteratve to. Ths method essetally uses Gree s theorem to map boudary dstrbutos to boudary dstrbutos, usg the eplct form of the Gree s fuctos. Although has the dsadvatage of dese matrces, ad assumes the rego s homogeous, t would be of great advatage for the th layer of the huma such as the scalp ad CSF, whch are homogeous. The larger regos are better suted to, whch are composed of o-uform coductvty dstrbuto. A logcal approach for ths forward problem s to combe the ad to produce hybrd code, whch assumes some rego, have cotact values. We preset results for D spheres ad dscuss the requred modfcatos of order to mprove relatvely hgh error at the, we also show for spheres that wth certa dscretzato soluto error for a mm th layer could be reduced. Itroducto There s a lot of dscusso about the advatages ad dsadvatages of the E method whe compared to the FE oe. Clearly there are certa applcatos where oe techque s more sutable tha the other. ut for electrcal mpedace or optcal tomography problems, combg both techques the same computer program, would be the most effcet way of modellg th layers lke scull or CSF a huma head. I order to take advatage of ad, ther couplg has bee vestgated etesvely several egeerg felds, such as geomechacs [, 3], sold mechacs [], fracture mechacs [] ad electromagetcs [,,,, ] There are several dfferet method of couplg ad [9,, ]. - couplg Ths problem s closely related to the mult rego problem of the E method such as preseted Fg.. The mult rego aalyss has to fulfl cotuty ad equlbrum codtos alog the le betwee Ω ad Ω regos. Ths results the
Ω Ω Ω Ω approach we have assume that o the we have addtoal ukow flu epressed by Neuma boudary codtos. Normally to solve FE system the boudary codtos have to be mposed allow us to solve the system of equatos. So ow the FE system of equatos ts matr form could be epressed as follows A (FE) Φ (FE) (FE) Φ(FE) = + F Ω FE () Fg. : The mult rego boudary elemet aalyss (left) ad hybrd dscretzato (rght) followg two relatoshps Φ () = Φ () () Φ () = Φ() Let the sub rego Ω be dscretzed by the Fte Elemets ad the Ω by the oudary Elemets. Alog the commo has tobesatsfedeq.(). Cotuty of the state fucto Φ ca be mataed by usg the same order of bass fuctos both FE ad E formulatos. Thus, f a three ode soparametrc quadratc boudary elemet s used a equvalet fte elemet such as for eample eght ode quadrlateral quadratc elemet or s odes soparametrc tragle has to be used for the fte elemet appromato. The essetalty of the problem les the fact that the terpolato for the dervatves of the potetal for the les oe order lower tha the order of the potetal tself, whereas for the formulato developed here, the terpolato fuctos has the same order ot oly for the potetal but also for ts dervatves. Such uequal terpolato of the ormal dervatves o the mplats a error to the resultg system of equatos. ecause alog the the cotuty ad equlbrum codtos have to be fulflled for the FE where Φ (FE) ad Φ(FE) are colum matrces cotag the odal values for the potetal (electrc potetal or photo desty) ad ts ormal dervatves (electrc curret or curret photo). The correspodg boudary tegral equato for the sub doma s gve by A (E) (E) Φ(E) Φ = + q o E (3) where Φ (E) ad Φ(E) are the odal potetals ad thers ormal dervatve vectors respectvely.. Goverg equatos The boudary value problem for the FE sub rego s defed by the secod order dfferetal equato [7] D(r) Φ(r)+k Φ(r) = () cojucto wth the boudary codtos Φ = Φ o Φ = ψ o () where D(r) s the coductvty (EIT) or dffuso coeffcet (OT), k teds to zero case of Laplace s equato (EIT). The equvalet varatoal problem for the boudary value problem defed above s gve by δf(φ) = ()
cojucto wth the boudary codtos Eq.(), form where F (Φ) = [ D Φ + k Φ ] dω+ Φψd (7) Ω (FE) To dscretze the fuctoal Eq.(7), the FE sub rego s dvded to elemets ME wth M odes ad boudary (see Fg. ) s broke to MS segmets wth M umber of odes. Usually, M s much larger tha M. Wth each area elemet, for eample s ode tragle, the feld s epressed as Φ(, y) = N e (, y)φ e = {N e } T {Φ e } = = = {Φ e } T {N e } () ad o each le segmet o the the feld s epressed as Φ (, y) = 3 N (, y)φ = {N } T {Φ } = = = {Φ } T {N } (9) Assumg that the boudary s a smooth cotour, the ormal dervatve, whch s ψ, s well defed at each ode ad therefore ca also be epressed as ψ (, y) = 3 N (, y)ψ = {N } T {ψ } = = = {ψ } T {N } () Ths s a weak pot of ths approach because the Φ ad ts ormal dervatve ψ are appromated by the same shape fuctos. Results of such approach wll be demostrated later. Substtutg Eq.( ) to Eq.(7), we obta F = M e= MS E{Φ e } T [A e ]{Φ e } + {Φ } T [ ]{ψ } () = where case of D space matr [A e ] wll take the ad [A e ] = + [ ]= [ { N }{ } e N e T D + Ω e { }{ } ] N e N e T ddy y y Ω e k {N e }{N e } T ddy () {N }{N } T d (3) Provded that the elemet legth of the s small, the Jacoba of trasformato to local coordate system may be assumed costat ad take out of the tegral sg Eq.(3) wthout causg sgfcat errors. Therefore, by substtutg the eplct epressos for the shape fuctos, t s easy to perform the dcated tegratos aalytcally. So the etres of matr [ ] case of the quadrlateral three odes soparametrc elemets of local coordate system are defed by [ ] = = N N NN NN 3 N N N N N N 3 J(ξ) = N3 N N3 N N3 N 3 J(ξ) () Tha, performg the assembly Eq.() ca be wrtte as F = {Φ}T [ A (FE)] {Φ} + {Φ} T [ (FE)] {ψ} () where A (FE) s a M M square matr, (FE) s a M M rectagular matr, Φ s a colum vector represetg the odal values of electrc potetal or photo testy ad ψ s a colum vector represetg the odal values of Φ o the M odes of the. Dfferetatg F wth respect to each 3
odal value of Φ ad equatg the resultg epresso to zero yelds a system of lear equatos A (FE) Φ (FE) (FE) Φ(FE) + = () As a result we wll get the matr wth the followg structure (see Fg. ). l u Fg. : The matr structure (left) ad a eemplary mage for the dscretzato show Fg. 3 Matr form of the E sub rego ca be wrtte as A (E) Φ (E) (E) Φ(E) = q (7) Now the system of boudary equatos could be corporated to the system of fte elemets. A (FE) (FE) A (FE) (FE) A (E) (E) A (E) (E) Φ (FE) Φ (FE) Φ (E) Φ (E) = = q q Note that the superscrpts (E) ad (FE) dcate to whch subrego partcular matr s prescrbed. The matr s usymmetrcal wth much bgger badwdth wth two addtoal group of o zero elemets caused by the betwee FE ad E sub domas. Rearragemet of resultg matr s very smlar to that requred mult rego problems. The smplest approach to solvg Eq.(7) ad Eq.() s to solve them smultaeously, that s, to solve a (M +3M +M ) (M +3M +M ) matr system, where M s the umber of ukows o the boudary Fg.. As prevous secto, the matrces wll eed some rearragemet to accommodate the codtos Eq.() ad stadard boudary codtos. 3 Numercal eamples Let us cosder several eamples D space, startg from the smplest oe ad tha move to the stadard EIT or optcal tomography bechmark. 3. echmark problem (?) As a bechmark problem let us cosder the steady state for a dffuso equato a square rego (see Fg. 3) Φ() μ a D Φ() = Φ() k Φ() = () where k s so called a wave umber ad for the EIT k become zero. The followg boudary codtos are mposed: Drchlet boudary codtos Φ() = = Φ() =a = (9) ad Neuma boudary codtos Φ() = Φ() = y = Φ() = y = () =a Ths problem, due to the geometry ad boudary codtos s oe dmesoal fact, but t would be solved D space ad results wll compare wth the aalytcal soluto whch s Φ() = ( Φ(a) D D e k(a ) J(a) k e k(a ) + + Φ() D e k + J() ) k e k () where for Optcal Tomography, usually dffuso coeffcet D =.3cm ad μ a =.cm so k =
μa /D.7cm,ad J() = Φ() = kd (cosh(kδ)φ() Φ(a)) sh(kδ) J(a) = kdsh(kδ)φ() + () + cosh(kδ)kd (cosh(kδ)φ() Φ(a)) sh(kδ) I order to acheve the followg results wth the ad of, rego has to be dscretzed by elemets wth 33 odes (badwdth equal 3) ad wth the ad of, rego has to be dscretszed by elemets ad odes. Results of calculatos are preseted Fg. 3. We ca observe how the error crease o the le. A A couplg Aalytc soluto 3 7 9 couplg 3 Aalytc soluto 3 7 9 relatve error [%] 3 3 3 7 9 3 7 9 Fg. : Dscretzato ad soluto alog the le A A for (left) ad alog the le (rght) for dese mesh relatve error [%] 9 couplg Aalytc soluto 3 7 3 7. 9 relatve error [%] 3 7. 9 relatve error [%] 7 3 3. 3 7 9 9. Fg. 3: Dscretzato ad soluto for the coarse mesh (left) ad dese (rght) for the hybrd soluto 3. Two cocetrc squares Now let us cosder slghtly more advaced eample whe the FE rego s mmersed the E rego as t s show Fg.. Ifluece of the gap dmeso o the error soluto s show Fg.??. What s terestg ad a very promsg feature of the hybrd approach, s the fact that the error of the soluto for such cofgurato as Fg. s ot sestve o dscretzato. Cocluso???? Fg. : Dscretzato ad soluto alog the le for dese mesh ad the small gap betwee ad sub regos Refereces [] M. H. Alabad. The oudary Elemet Method; Volume ; Applcatos Solds ad Structures. Joh Wley & Sos, LTD,. [] G. eer. Programmg the oudary Elemet Method. A Itroducto for Egeers. Joh Wley & Sos,. [3] G. eer ad J.O. Watso. Itroducto to Fte ad oudary Elemet Methods for Egeers. Joh Wley & Sos, 99. [] C.P. radley, G.M. Harrs, ad A.J. Pulla. The computatoal performace of a hgh order coupled fem/bem procedure elektropotetal
problems. IEEE Trasactos o omedcal Egeerg, ():3, November. [] M.V.K. Char ad S.J. Salo. Numercal Methods Electromagetsm. Academc Press,. [] I. Guve ad E. Madec. Traset heat coductg aalyss a pecewse homogeeous doma by a coupled boudary ad fte elemet method. It. Jour. for Numercal Meth. Egeerg, :3 3, 3. [7] Jamg J. The Fte Elemet Method Electromagetcs. Joh Wley & Sos, 993. [] S. Kurz ad S. Russeschuck. Accurate calculato of magetc felds the ed regos of supercoductg accelerator magets usg the bem fem couplg method, 999. Proceedgs of the 999 Partcle Accelerator Coferece, New York. [9] O.K. Paagoul ad P.D. Paagotopulos. The fem ad bem for fractal boudares ad s. applcatos to ulateral problems. Computers ad Structures, ( ):39 339, 997. [] M. Trlep, L. Skerget,. Kreča, ad. Hrberk. Hybrd fte elemet boudary elemet method for olear electromagetc problems. IEEE Trasactos o Magetcs, 3(3):3 33, May 99. [] O.C. Zekewcz, D.W. Kelly, ad P. ettess. The couplg of the fte elemets method ad boudary soluto procedures. It. Jour. for Numercal Meth. Egeerg, pages 3 37, 977.