Reconstruction of the eddy current profile on the vacuum vessel in a nuclear fusion device using only external magnetic sensor signals

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1 Boundary Elements and Other Mesh Reducton Methods XXXVII 149 Reconstructon of the eddy current profle on the vacuum vessel n a nuclear fuson devce usng only external magnetc sensor sgnals M. Itaga 1, A. Sanpe, S. Masamune & K. Watanabe 3 1 Faculty of Engneerng, Hoado Unversty, Japan Kyoto Insttute of Technology, Japan 3 Natonal Insttute for Fuson Scence, Japan Abstract In a reversed feld pnch devce, the ordnary Cauchy-condton surface (CCS) method cannot reconstruct the magnetc flux profle accurately due to the strong eddy current flow on the shell (vacuum vessel). Boundary ntegrals of the eddy current densty along the shell are added to the orgnal CCS method formulaton. The eddy current profle s unnown n advance but dentfed usng only the sgnals of magnetc sensors located outsde the plasma. As the sensors are closely adjacent to the shell, the near sngular boundary ntegrals along the shell should be accurately evaluated. Ths sngularty can be damped out wth the algorthm based on the dstance functon proposed by Ma and Kamya. The modfed truncated sngular value decomposton technque of Hansen et al. s an effectve way to solve an ll-condtoned matrx equaton when a large number of nodal ponts exst on the shell. The capablty of the new method s demonstrated for a test problem modellng the RELAX devce. Keywords: Cauchy-condton surface, eddy current, magnetc sensor sgnal, sngular ntegral, modfed truncated sngular value decomposton technque. 1 Introducton In a nuclear fuson devce, the magnetc feld or flux profle outsde the plasma and hence the plasma boundary shape are hghly mportant for operatng control and dagnoss. Such nformaton should be deduced from sgnals of magnetc sensors located outsde the plasma, snce the drect measurement of physcal quanttes nsde the plasma s usually dffcult. The Cauchy condton surface ISSN X (on-lne) WIT Transactons on Modellng and Smulaton, Vol 57, 14 WIT Press do:1.495/be37131

2 15 Boundary Elements and Other Mesh Reducton Methods XXXVII (CCS) method [1, ] s one such dea for reconstructng the flux dstrbutons around the plasma boundary. The method has already been establshed for operatng control of JT-6U, a toama-type devce. In the method, however, the eddy current generated on the vacuum vessel was neglected. In a reverse feld pnch (RFP) devce such as the RELAX [3] at Kyoto Insttute of Technology, a strong eddy current s generated on the shell (vacuum vessel) wall, whch s closely related to the local magneto-hydrodynamc (MHD) equlbrum. In the present paper the authors propose an advanced CCS method where the eddy current term s gven by a boundary ntegral along the shell n the polodal drecton. The eddy current profle s not gven n advance but completely unnown before one starts the analyss. One solves the Cauchy condtons and the eddy current profle smultaneously. The authors ntroduce two deas to overcome the numercal dffcultes encountered n ths nverse problem. One s an accurate boundary ntegral scheme to damp out the near sngularty occurrng at the sensor poston very close to the shell. The other s the modfed truncated sngular value decomposton (MTSVD) technque to solve an ll-condtoned matrx equaton when a large number of nodal ponts exst on the shell. Problem specfcatons One here consders a problem to model a lmter confguraton of the RELAX devce [3], as an example of a reversed feld pnch devce. The shell (vacuum vessel) s regarded as axsymmetrc n the torodal drecton and ts cross secton s a crcle wth radus.5m, whch s centred at (r, z) = (.51m,.51m) as shown n Fgure 1. One assumes that a lmter havng a length of 1cm s located at the poston (r, z) = (.75m,.51m) on the nner wall of the shell. The reference dstrbutons of magnetc flux nsde the shell and the eddy current on the shell were analysed beforehand usng the RELAX-Ft code [4]..8 z (m).6.4.5m Plasma boundary (.51,.51).15m Shell (.75,.51) CCS. Magnetc sensors.1m Lmter r (m) Fgure 1: Image of the RELAX lmter confguraton. WIT Transactons on Modellng and Smulaton, Vol 57, 14 WIT Press ISSN X (on-lne)

3 Boundary Elements and Other Mesh Reducton Methods XXXVII 151 The sgnals of magnetc sensors were also nown before the present nverse analyses. The sensor locatons are also llustrated n Fgure 1. One assumes 4 sensor postons around two crcles wth a common centre at (r, z) = (.51m,.51m) nsde the shell: ponts are around the crcle wth a radus of.4m at even ntervals, whle the other are at a dstance of.48m from the centre. Each poston s 8mm and mm away from the shell, respectvely. One here hypothetcally assumes that both a torodal flux loop and a tangental probe are located at each of the 4 postons. That s, a total of 8 magnetc sensors are assumed. The tangental probe detects the magnetc feld component that s tangental to the shell surface n the polodal drecton. 3 Method In ths secton, one descrbes how the effect of the eddy current on the shell s ncorporated nto the orgnal CCS method. The eddy current dstrbuton on the shell, n the same way as all other physcal quanttes, s assumed to be axsymmetrc n the torodal drecton. 3.1 The set of boundary ntegral equatons The Cauchy-condton surface (CCS), where both the Drchlet and the Neumann condtons (.e. the magnetc flux functon and ts normal dervatve / n ) are unnown, s hypothetcally placed n a doman that can be supposed to be nsde the plasma. In the analyss, no plasma current s assumed outsde ths CCS, where n realty plasma current does exst. Instead, the CCS plays the same role as the plasma current n causng the feld outsde the plasma. For an axsymmetrc (r, z) system, the dfferental form of Ampere s law j B can be reduced to a partal dfferental equaton 1 rr r z r r jpl jcol jeddy ( ) (1) n terms of magnetc flux functon.here, j Pl, j Col, and j Eddy denote the torodal components of the plasma current, the external col current and the eddy current respectvely. The quantty s the permeablty of a vacuum. In the followng formulaton one uses the magnetc flux [Wb] nstead of the magnetc flux functon [Wb/rad] because the physcal quantty drectly measured s the magnetc flux. Also, nstead of the magnetc feld sgnal B, a quantty B B s defned. To evaluate and / n at sx ponts along the CCS ( CCS ), three types of boundary ntegral equatons (BIEs) can be gven usng the sensor sgnals and the external polodal col current data, as shown n the followng. WIT Transactons on Modellng and Smulaton, Vol 57, 14 WIT Press ISSN X (on-lne)

4 15 Boundary Elements and Other Mesh Reducton Methods XXXVII () For the magnetc flux sgnal at ponts : ( ) ( ) ( ). (a) W d js s s d s r n r n r r r r CCS Shell () For the magnetc feld sgnal B at ponts : B B B W d j ( ) B ( ) d ( ), (b) B S s s s r n r n r r r r CCS Shell usng the quantty B B, where B n / r, B / n r wth n beng the assgned vector normal to the drecton of the magnetc probe located at the pont. () For ponts on the CCS: 1 ( ) ( ) ( ). (c) C W d js s s d s r n r n r r r r CCS Shell In eqns (a) (c), W, B W and C W are the contrbutons of the external col currents to the pont. In each equaton, denotes the fundamental soluton whch satsfes a subsdary equaton 1 rr r z r r, (3) where Drac s delta functon means ( r a) ( z b) wth the spe at the pont, where s defned as havng the coordnates (a, b). The detaled form of s gven by [1, ] ar 1 K E (4) wth 4ar ra zb, (5) WIT Transactons on Modellng and Smulaton, Vol 57, 14 WIT Press ISSN X (on-lne)

5 Boundary Elements and Other Mesh Reducton Methods XXXVII 153 where K ( ) and E( ) are the complete ellptc ntegrals of the frst and the second nd, respectvely. The second term on the RHS n each of eqns (a), (b) and (c) descrbes the effect of the eddy current on the shell. In the quantty ( rs r ), r s means an arbtrary pont on the shell. The quantty js( r s) denotes the lnear densty [MA/m] dstrbuton of the eddy current on the shell, whch s ntegrated n the polodal drecton along the shell ( ). 3. Accurate computaton of boundary ntegrals along the shell Shell The boundary ntegrals along the shell (the second term on the RHS n each of eqns (a), (b) and (c)) should be performed very carefully. Snce the dstance ( ra) ( z b) between the sensor poston (a, b) and an ntegraton pont (, rz ) on the shell s very short, the followng sngulartes [5] a log, (6) a 1 ar ( a) 1 log 4 (7) and b az ( b) 1, (8) arse n the ntegraton ernels when. In a boundary element ( Shell, j ) along the shell, one here uses the notatons G( ) and ( ) for the Jacoban of the coordnate transformaton and one of the nterpolaton functons, respectvely. Suppose that B ( ) s the fundamental soluton or ts dervatve, whle FS ( ) denotes the correspondng asymptotc functon,.e. eqns (6), (7) or (8). The general form of the boundary ntegral over can be rearranged as Shell, j ( ) G( ) B ( ) d ( ) G( ) B ( ) GFS( ) d G FS( ) d, (9) where G and are the values of G( ) and ( ) at the poston on the boundary element that s the nearest to the locaton of the magnetc sensor under consderaton. The asymptotc functon s subtracted from the orgnal ntegrand n the frst ntegral on the RHS of eqn (9), and ths subtracton s compensated by WIT Transactons on Modellng and Smulaton, Vol 57, 14 WIT Press ISSN X (on-lne)

6 154 Boundary Elements and Other Mesh Reducton Methods XXXVII the analytcal ntegral of the second ntegral on the RHS. The total ntegrand of the frst ntegral has no sngularty and can therefore be evaluated wth the ordnary Gaussan quadrature wth 16 ntegraton ponts for each boundary element. Boundary element Sensor poston d Fgure : The mnmum dstance boundary element. from the sensor poston (a, b) to a Ma and Kamya [6] proposed the use of an approxmated dstance functon for the boundary element adjacent to the sensor poston (a, b) as d( ) G ( ) ( d / G ), (1) where d s the mnmum dstance from the pont (a, b) to the boundary element as shown n Fgure, whch corresponds to the local coordnate. Equaton (1) agrees wth n eqns (6), (7) and (8) when and d. The quanttes n eqns (7) and (8) can be rewrtten n the followng forms: ar ( a) 1 a 1( ) G ( ) ( d / G) ( ) ( d / G) (11) and az ( b) 1 a 1( ). (1) G ( ) ( d / G) ( ) ( d / G) The terms havng sngulartes can be ntegrated analytcally, e.g. 1 1 ( ) d d d log (1 ) log (1 ). ( ) ( d / G) G G (13) WIT Transactons on Modellng and Smulaton, Vol 57, 14 WIT Press ISSN X (on-lne)

7 Boundary Elements and Other Mesh Reducton Methods XXXVII 155 Fgure 3 compares the ntegrands n terms of / b wth an nterpolaton functon (1 )(1 ) before and after dampng out the sngularty for a case where 1 boundary elements are used along the shell. The strong sngularty s effcently damped out. As the modfed ntegrand n Fgure 3(b) stll has a small edge at.5, the frst ntegral on the RHS of eqn (9) s dvded nto two ntegral ntervals, [ 1, ] and [, 1] as long as 1. The ntegrands.1 The modfed ntegrand where the sngularty s damped out. The modfed ntegrand where the sngularty s damped out -.1 The orgnal ntegrand The orgnal -.4 ntegrand -.3 ( eddy current -.6 ( eddy current nodes are used.) nodes are used.) Dmensonless local coordnate ( Dmensonless local coordnate ( (a)., d 8mm (b).5, d mm The ntegrands.6.4 Fgure 3: Integrands before and after dampng out the sngularty. 3.3 Dscretzaton Usng the quadratc boundary elements, eqns (a), (b) and (c) are dscretzed, coupled and can be expressed n a matrx form Dp g, (14) where D s an m n matrx. The soluton vector p contans N S nodes of lnear D eddy current densty on the shell as well as the set of N C flux values and ther N C normal dervatves on the CCS. Then, the total number of unnowns s n N C N S, whle the number of BIEs s gven by m N NB NC wth N and N B beng the numbers of flux loops and feld sensors, respectvely. The quanttes W B and B W are stored n the vector g n eqn (14). The matrx equaton (14) s solved usng the sngular value decomposton T (SVD) technque [7]. The matrx D s decomposed as D UΛV, where U and T V are orthogonal matrces and Λ s a dagonal matrx wth postve sngular values or zero components. In the so-called truncated SVD (TSVD) technque [7], the regularzed soluton s gven by WIT Transactons on Modellng and Smulaton, Vol 57, 14 WIT Press ISSN X (on-lne)

8 156 Boundary Elements and Other Mesh Reducton Methods XXXVII p VΛ U g. (15) 1 T Here Λ means that the sngular values smaller than n Λ are omtted so that the condton number s not larger than a certan value. Once all the values of the Cauchy condtons on the CCS and the lnear eddy current denstes on the shell nodes are nown, the magnetc flux for arbtrary ponts outsde the CCS can be calculated usng eqn (a). 3.4 Solvng an ll-condtoned matrx equaton the MTSVD method Fgure 4 shows the behavour of the sngular values whch appeared n the SVD process for varous numbers of assumed eddy current nodes, N. The vertcal axs S represents the sngular values whose maxmum value s normalzed to unty. The 3 smallest sngular value becomes less than 1 when N s greater than 4. S 1 Normalzed sngular value current nodes current nodes Gap of sngular values Cond. no. = current nodes th sngular values Fgure 4: The sngular values wth the number of eddy current nodes. 3 A gap s commonly observed n the vcnty of 1 of the normalzed sngular values. When the number of current nodes N S exceeds 4, the condton number (the rato of the largest to the smallest sngular values; the recprocal of the 4 normalzed sngular value) jumps up to over1. If one truncates the sngular values smaller than the gap threshold observed n Fgure 4, the mproved condton numbers are around 6 1 n all cases. Even wth ths TSVD technque, however, a numercal oscllaton of p s observed when the smallest sngular value s smaller than the gap threshold. One dea to suppress such an oscllaton s to ntroduce a constrant, mn Lp, n WIT Transactons on Modellng and Smulaton, Vol 57, 14 WIT Press ISSN X (on-lne)

9 Boundary Elements and Other Mesh Reducton Methods XXXVII 157 addton to mn Dp g, where L means a dfferental operator. Based on ths dea, Hansen et al. [8] derved the modfed TSVD (MTSVD) soluton pl, p V( LV) Lp p Vz, (16) where V [ V,, V ] and z 1 n s the soluton of ( LV ) z Lp. (17) One uses the followng ( n6) n matrx for a total of n unnowns: L (N) L (D) L L (E). (18) In eqn (18), L (N) (D) (E), L and L are dscrete approxmatons to the second dervatve operator appled to the Neumann condtons on the CCS, the Drchlet condtons on the CCS and the current densty soluton on the shell surface, respectvely, each of whch has the trdagonal form L (E) (19) 1 1 to obtan a smooth soluton. Fgure 5 llustrates the effectveness of the MTSVD method when appled to a test case where 6 eddy current nodes are assumed. The sold grey curve that oscllates at hgh frequency s the orgnal soluton ( p ) that s produced usng the ordnary TSVD technque. The number of peas n ths curve, 6, agrees wth the number of assumed eddy current nodes. The dotted curve depcts the correcton vector that s gven by Vz n eqn (16). Subtractng the dotted curve from the grey curve,.e. followng eqn (16), one obtans the sold blac curve ( p L, ) where the numercal oscllaton has been drastcally damped out. Although there are slght rpples n the resultant curve, t s worth mentonng that the number of rpples agrees not wth the number of current nodes any more but wth the number of sensor locatons (= 4), whch are the sngular ponts,, ndcated n eqns (a) and (b). WIT Transactons on Modellng and Smulaton, Vol 57, 14 WIT Press ISSN X (on-lne)

10 158 Boundary Elements and Other Mesh Reducton Methods XXXVII Lnear densty of eddy current (A/m) 5 5 The resultant curve (p -V z ) The orgnal curve (p ) Correcton vector (V z ) Polodal angle (rad) Fgure 5: Subtracton of correcton vector from the orgnal soluton. 4 Numercal demonstraton for the RELAX devce In the present wor the CCS approxmates a crcle havng a radus of.15m and centre (r, z) = (.51m,.51m), as also shown n Fgure 1. The crcle s dvded nto 3 contnuous quadratc boundary elements, so that the total number of nodes s 6 (the number of unnowns on the CCS becomes 1). 4.1 Reconstructed profles of eddy current densty and magnetc flux Fgures 6(a) (d) show the varaton n the eddy current densty on the shell surface for the cases assumng, 3, 4 and 6 eddy current nodes, respectvely. In each fgure the vertcal axs denotes the current densty, whle the abscssa means the polodal angle that vares n the clocwse drecton whose startng pont ( ) on the shell s at the top (r, z) = (.51m,.76m). The dashed and the sold curves n Fgure 6 denote the reference and the reconstructed varaton n the eddy current densty, respectvely. Note here that the MTSVD technque s appled n cases where one truncates the sngular values smaller than the gap threshold shown n Fgure 4,.e. the cases where the number of current nodes s greater than 4. That s, Fgure 6(d) s the result when usng the MTSVD technque. Fgures 7(a) and 7(b) show the reconstructed flux profles for the 4 and 6 current node cases. In each fgure the dashed contours show the reference soluton, whle the sold contours ndcate the reconstructed flux. In an ordnary CCS method analyss, the reconstructed flux soluton s naccurate n the doman where the plasma current exsts. However, n the present RELAX analyses, accurate reconstructons can be observed even deep nsde the plasma regon. Ths s because the eddy current effect s domnant over the plasma current effect for the formaton of the flux dstrbuton n the RELAX devce. WIT Transactons on Modellng and Smulaton, Vol 57, 14 WIT Press ISSN X (on-lne)

11 Boundary Elements and Other Mesh Reducton Methods XXXVII 159 Lnear densty of eddy current (A/m) 5 5 Reference soluton Reconstructed Lnear densty of eddy current (A/m) 5 5 Reference soluton Reconstructed 4 6 Polodal angle (rad) (a) current nodes 4 6 Polodal angle (rad) (b) 3 current nodes Lnear densty of eddy current (A/m) 5 5 Reference soluton Reconstructed Lnear densty of eddy current (A/m) 5 5 Reference soluton Reconstructed 4 6 Polodal angle (rad) (c) 4 current nodes 4 6 Polodal angle (rad) (d) 6 current nodes Fgure 6: Reconstructon of the eddy current profle..8 Reconstructed flux Shell.8 Reconstructed flux Shell.6.6 z (m) z (m).4. Current node Reference flux Sensors r (m) r (m) (a) 4 current nodes (b) 6 current nodes Fgure 7: Reconstructon of the magnetc flux profle..4. Current node Reference flux Sensors WIT Transactons on Modellng and Smulaton, Vol 57, 14 WIT Press ISSN X (on-lne)

12 16 Boundary Elements and Other Mesh Reducton Methods XXXVII 4. Influence of the sensor sgnal nose The effect of measurement errors on the reconstructon was also studed. Nose was numercally generated and added to all feld and flux loop sgnals. A nose-added sgnal s gven by b j bj(1 G), where b j s the orgnal sgnal, G denotes a unt Gaussan random number, and s the standard devaton of the Gaussan nose. Wthout sensor sgnal nose, the reconstructed current densty profle shown n Fgure 6(a) under the adopton of current nodes seems to be n good agreement wth the reference profle. However, t s premature to mae a concluson that the best choce s ths number of current nodes. Fgure 8 shows the relatve errors of reconstructed flux and eddy current densty as functons of the number of current nodes under the assumpton of 3% nose. Unfortunately the solutons are senstve to the sgnal nose f the number of current nodes s less than 4. In Fgure 8, the dashed curve ndcates the varaton n the condton number multpled by 3%, whch s the theoretcal maxmum of the error caused by the 3% nose. (The condton numbers wth over 4 current nodes mean the results after truncatng the small sngular values.) It should be notced that, n spte of the comparatvely small condton numbers, the observed errors wth fewer than 4 nodes are much larger than those wth over 4 nodes. Accordngly, ths phenomenon cannot be explaned by the magntude of the condton number. Rather, t s suggested that ths s caused by the lac of nformaton as a constrant n the nverse analyss. One should adopt a number of current nodes whch s large enough to ensure that all sngular values larger than the gap threshold, whch have meanngful physcal nformaton, are taen nto account n the analyss for obtanng a robust soluton (%) x Cond. No. Gap threshold of sngular values 1 3 Relatve error (%) Max. flux error MTSVD method s appled Max. current error 1 Ave. current error 1 1 Ave. flux error No. of eddy current nodes 1 Fgure 8: Influence of the sensor sgnal nose when 3% σ nose s mposed. WIT Transactons on Modellng and Smulaton, Vol 57, 14 WIT Press ISSN X (on-lne)

13 Boundary Elements and Other Mesh Reducton Methods XXXVII Concluson (1) As the sensors are closely adjacent to the shell, the near sngular boundary ntegrals along the shell should be accurately evaluated. Ths sngularty s damped out effectvely wth the algorthm based on the approxmated dstance functon. () If the smallest sngular value s smaller than the gap threshold, a numercal oscllaton of the eddy current profle s observed. Ths oscllaton s elmnated effectvely by applyng the MTSVD technque. (3) To obtan a soluton stable aganst sensor sgnal nose, one should adopt enough number of eddy current nodes to ensure that all sngular values larger than the gap threshold are taen nto account. The present technques are applcable to the problem of eddy current flow n a conductor located close to a magnetc sensor n many other devces. Acnowledgements Ths research was performed wth the support and under the auspces of the NIFS Collaboraton Research Program (NIFS1KLPP4 & NIFS1KNWP), also supported by the Mnstry of Educaton, Culture, Sports, Scence and Technology, Grant-n-Ad for Scentfc Research (C), , 1. References [1] Kurhara, K., Fuson Eng. Des ,. [] Itaga, M., Yamaguch, S. & Fuunaga, T., Nucl. Fuson 45, , 5. [3] Masamune, S., Sanpe, A., Iezono, R., Onch, T., O, K, Yamashta, T., Shmazu, H., Hmura, H. & Paccagnella, R., J. Fuson Energ. 8, 187, 9. [4] Sanpe, A., O, K., Iezoe, R., Onch, T., Murata, K., Shmazu, H., Yamashta, T., Fujta, S., Hmura, H., Masamune, S. & Anderson, J. K., J. Phys. Soc. Jpn. 78[1], 1351, 9. [5] Itaga, M. & Shmoda, H., Eng. Anal. Boundary Elements 33, 845, 9. [6] Ma, H. & Kamya, N., Eng. Anal. Boundary Elements 5, 833, 1. [7] Hansen, P.C., Ran-Defcent and Dscrete Ill-Posed Problems Numercal Aspects of Lnear Inverson (Phladelpha, SIAM, 1998). [8] Hansen, P.C., Se, T. & Shbahash, H., SIAM J. Sc. Stat. Comput., 13, 114, 199. WIT Transactons on Modellng and Smulaton, Vol 57, 14 WIT Press ISSN X (on-lne)