BOUNDARY-ONLY INTEGRAL EQUATION APPROACH BASED ON POLYNOMIAL EXPANSION OF PLASMA CURRENT PROFILE TO SOLVE THE GRAD-SHAFRANOV EQUATION

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1 Ttle Bounday-only ntegal equaton appoach based on po Gad Shafanov equaton Authos)Itagak, Masafu; Kasawada, Jun-ch; Okawa, Shu CtatonNuclea Fuson, 443): Issue Date 4-3 Doc UR Rghts 4 IOP Publshng td. Ths s an autho-ceated Nuclea Fuson. The publshe s not esponsble fo veson deved fo t. The Veson of Recod s av Type atcle autho veson) Fle Infoaton Nucl. Fuson, 44, ).pdf Instuctons fo use Hokkado Unvesty Collecton of Scholaly and Aca

2 BOUNDARY-ONY INTEGRA EQUATION APPROACH BASED ON POYNOMIA EXPANSION OF PASMA CURRENT PROFIE TO SOVE THE GRAD-SHAFRANOV EQUATION Masafu ITAGAKI, Jun-ch KAMISAWADA, Shun-ch OIKAWA Gaduate School of Engneeng, Hokkado Unvesty, Kta 3, Nsh 8, Kta-ku, Sappoo 6-868, JAPAN Tel , Fax E-al: Ths pape contans 7 pages of text, table and 9 fgues.

3 ABSTRACT A new type of bounday eleent ethod has been appled to solve the Gad-Shafanov equaton and to gve a dstbuton of agnetc flux functon n a Tokaak nuclea fuson devce. The quantty µ j elated to the ϕ plasa cuent pofle s expanded nto two-densonal polynoal. Usng the patcula soluton of the Gad-Shafanov equaton wth ths nhoogeneous polynoal souce and applyng Geen s second dentty, the doan ntegal elated to the plasa cuent s tansfoed nto an equvalent bounday ntegal. Doan dscetaton s not equed n ths foulaton, thus pesevng all the advantages of the bounday eleent ethod. Nuecal coputatons of all bounday ntegals ae only equed n the ntal stage of the egenvalue teaton, so that the nube of egenvalue teatons does not hape the total coputng te. Test calculatons deonstated that the pesent ethod povdes stable and accuate solutons. Key wods: tokaak, Gad-Shafanov equaton, bounday eleent ethod, polynoal expanson, patcula soluton, bounday-only ntegal,

4 . INTRODUCTION The MHD equlbu n axsyetc plasa lke a tokaak s descbed by the Gad-Shafanov equaton [-4] n tes of the agnetc flux functon ψ. Analytc and nuecal solutons fo ths equaton ae potant n explong the plasa confguatons. The wdely used nuecal technques to solve ths equaton s based on a doan type solutons [5, 6], such as fnte eleents and fnte dffeences, howeve, soe eseaches ecently attepted to apply the bounday eleent ethod BEM) [7] to the analyss [8-]. As the nae ples, n the BEM the govenng dffeental equaton s tansfoed nto a bounday ntegal equaton that s appled ove the bounday. The bounday s dvded nto sall bounday segents bounday eleents) fo the nuecal ntegaton, and then one solves a syste of lnea algebac equatons. The ost potant featue of the ethod s that t eques dscetaton of the suface only athe than of the volue. Ths advantage s patculaly potant n a sees of fequent analyses; geoety data geneaton and odfcatons ae easly pefoed. That s, the ethod s well suted fo on-lne plasa equlbu analyss that eques effcent data pepaaton and coputaton followng the change n plasa shape dung the opeaton of an actual fuson devce. A dffculty ases when one attepts to tansfo the Gad-Shafanov equaton nto the bounday ntegal equaton. The nhoogeneous te µ j ϕ elated to the plasa cuent ϕ j stll eans n the ntegal equaton as a doan ntegal. If nothng s done fo the doan ntegal, one cannot take advantage of the BEM that only the bounday dscetaton s equed. As the plasa cuent s ultpled by the fundaental soluton Geen s functon fo an nfnte syste) that has a sngulaty, to pefo nuecally the doan ntegal s qute toublesoe, dffcult to obtan accuate esults, and futheoe, causes an ense consupton of coputng te. Ths hapes the accoplshent of eal te coputng n futue fuson eacto opeatons. As fa as the authos ae awae, howeve, n ost of the attepts to apply the BEM to solve the Gad-Shafanov equaton, they dscete the doan ntegal as t s wthout changng the doan ntegal to any bounday one [], o to apply to the equaton fo a vacuu egon,.e. the equaton whch has ognally no nhoogeneous plasa cuent te [8-]. 3

5 The an pupose of the pesent wok s to popose an elegant way of tansfong the above doan ntegal nto an equvalent bounday one. Ths dea s based on the polynoal expanson of the µ j ϕ te n the Gad-Shafanov equaton. The tck to tansfo the doan ntegal nto a bounday one s to apply Geen s second dentty fo the doan ntegal, as found n Secton 3. Matheatcal quantty whch plays an potant ole n ths tansfoaton s a patcula soluton of agnetc flux functon whch satsfes the Gad-Shafanov equaton wth the above polynoal souce te. The detaled fo of ths patcula soluton s deved n Secton 5. The poble to solve the Gad-Shafanov equaton as a fxed bounday value poble s at the sae te an egenvalue poble, snce the nhoogeneous plasa cuent te s a functon of the unknown agnetc flux functon. Ths type of egenvalue seach eques qute a nube of teatons, howeve, all nuecal values of bounday ntegals equed to asseble the atx equaton, whch s the dsceted fo of the bounday ntegal equaton, ae calculated at the ntal stage of teaton. Only the coeffcents of the polynoal expanson ae updated though the teaton. Anothe atx to be used fo ths expanson coeffcent detenaton s also nvaant though the teaton. Theefoe the nube of teatons hadly affects the total coputng te. As wll be descbed n Secton 6, one eques anothe type of doan ntegal to evaluate the above egenvalue. Howeve, t wll be shown that ths doan ntegal can be also tansfoed nto a bounday one. The ethod poposed n the pesent pape does not eque any coputaton of doan ntegal. Thee s a possblty that the BEM shows anothe et when appled to an nvese poble to epoduce the pofles of agnetc flux and/o plasa cuent fo data fxed along the plasa bounday, as the eason fo ths wll be suggested n Secton 8. The a n the pesent wok s, howeve, to popose a new foulaton based on a bounday-only ntegal equaton as the soluton to the Gad-Shafanov equaton, and to deonstate ts valdty. Because of ths, nuecal exaples n the pesent pape ae lted to fowad pobles,.e., pobles to seek the agnetc flux dstbuton wthn a plasa doan unde the assupton of fxed bounday shape and wth an appopately paaeted plasa cuent pofle as a functon of agnetc flux. 4

6 . BOUNDARY INTEGRA EQUATION Fo an axsyetc, ) syste the Gad-Shafanov equaton s gven by ψ dp d I ψ µ µ = dψ dψ j ϕ, ) whee the agnetc flux functon ψ s defned as ψ = A wth the toodal coponent of vecto potental ϕ A ϕ, j ϕ denotes the toodal coponent of the plasa cuent, P the plasa pessue and I the polodal cuent functon. The a n ths secton s to deve the bounday ntegal equaton whch coesponds to Eq.). One hee ntoduces the fundaental soluton ψ whch satsfes a subsday equaton ψ = δ, ) whee δ s Dac s delta functon wth the spke at the pont, say, the coodnates b) a,. Physcally Eq.) descbes the agnetc flux functon fo an abtay feld pont, ) caused by a unt toodal cuent located at the pont a, b). The detaled fo of the fundaental soluton wth whee K k) and k) ψ s gven by [8-] a k ψ = K k ) E k ) 3) π k k a =, 4) 4 a) b) E ae the coplete ellptc ntegals of the fst and the second knd, espectvely. The second-ode patal dffeental pat Equaton 5) ncludes the aplace opeato ψ can be educed to = ψ =, ψ. 5) and ths s qute convenent to apply Geen s second dentty whch wll be shown as Eq.7). Usng Eq.5), one 5

7 fnds the elatonshp Ω ψ ψ ψ ψ ψ ψ ψ ψ )d Ω= d Ω Ω. 6) Applyng Geen s second dentty φ ψ ψ φ φ ψ ) dω = ψ φ dγ Ω Γ n n 7) to the RHS of Eq.6), one obtans ψ ψ ψ ψ ψ ψ ψ ψ The RHS of Eq.6) = dγ = dγ Γ n n Γ n n, whee n s the outwadly dected noal decton on the bounday Γ. That s, nstead of Geen s second dentty, one heeafte can use the followng ecpocal elatonshp fo the Gad-Shafanov opeato: Multplyng Eq.) and Eq.) by doan Ω, one has ψ ψ n n Γ ψ ψ ψ ψ ) dω= ψ ψ dγ 8) Ω ψ / and ψ /, espectvely, subtactng, and ntegatng ove the ψ ψ ψ ψ ψ ψ dω= µ j dω d Ω ) j ) Ω Ω Ω, and next applyng Eq.8) to the HS, one obtans ψ ψ n ψ ψ n ψ dγ = µ j ) j dω Γ Ω Ω ψ d dω. 9) It should be noted that d Ω and d Γ n Eq.9) ean dω = π dd and d Γ = π d d, espectvely,.e., Eq.9) should be consdeed n a thee-densonal space. Equaton 9) can then be ewtten as ψ ψ ψ ψ ψ π jj ) dd πψ. d d = π n n µ Γ Ω Note that one can eove π fo both sdes of ths equaton. Now one denotes 6

8 d Ω' = dd and d Γ ' = d d n the - plane, and heeafte one also edefnes as d Ω = dd and d Γ = d d, ottng π fo splcty. Theefoe the axsyetc veson of the ntegal equaton can be wtten n the fo: ψ ψ ψ ψ ψ ψ = dγ µ j dω Γ n n Ω ) j. ) The eason why d Ω on the second te of the RHS has not been changed to d Ω s that Eq.8) o Geen s second dentty) need be agan appled to the ognal Ω Γ syste to tansfo ths doan ntegal nto a bounday one. Equaton ) s vald fo any pont n the doan Ω, howeve, one needs to odfy Eq.) when the pont s located on the bounday Γ. Moe geneal fo of the bounday ntegal equaton s gven by ψ ψ ψ ψ ψ c ψ = dγ µ j ) dω j, ) Γ n n Ω whee the constant c depends on the local bounday geoety unde consdeaton: c =. fo an ntenal pont, whle c =/ on a sooth bounday. The devaton of ths geneal fo s shown n Appendx A. 3. BOUNDARY INTEGRA EXPRESSION OF POYNOMIA SOURCE TERM 3. Polynoal expanson The spatal dstbuton n the RHS of Eq.) s expessed usng a polynoal wth espect to ξ and η : jj α, ξ η, ) ), The absolute values of the densonless coodnates ξ = η / /, = ) 7

9 do not exceed. when,, ae taken as the outeost plasa adus n -decton, the half length of -dectonal plasa wdth and the -coodnate of the plasa cente, espectvely. Table Pascal s tangle fo polynoal functons In oe detal, one assues hee that the quantty µ jϕ s expanded up to a cetan level of the coplete polynoal that s specfed accodng to Pascal s tangle shown n Table. Fo exaple, level eans that µ jϕ s appoxated as: µ ϕ j α αξ α η α ξ αξη α η. 3. Patcula soluton and the applcaton of Eq.8) Usng the above sple polynoal expanson, one ewtes Eq.) n the fo: ψ α, ξη =, 3) Then the second te on the RHS of the bounday ntegal equaton ) s ewtten as Q ψ ψ = j dω = dω j ) α, ξ η. 4) Ω Ω, One hee assues the exstence of a patcula soluton, ) ϕ whch satsfes, ) ϕ ξη =. 5) The detaled atheatcal fo of, ) ϕ wll be dscussed n Secton 5. Applyng Eq.5) and utlng the ecpocal elatonshp gven by Eq.8), the quantty, Q, ) ψ ) d ξη = Ω Ω can be aanged as follows: Q ψ = dω, ), ) ϕ Ω ϕ ψ ϕ ϕ ψ = Ω Γ Ω, ), ), ) ψ d d. n n Γ 6) 8

10 Now endng Eq.), Eq.6) can be educed to, ), ), ), ) ϕ ψ ϕ ϕ ψ Q = d Ω dγ. 7) n n Ω Γ Recallng d Ω = dω, dγ = dγ and ntoducng the sngulaty constant c, one obtans, ), ), ), ) ψ ϕ ϕ ψ Q = cϕ dγ. 8) n n Γ That s, the doan ntegal elated to the polynoal souce s tansfoed nto an equvalent bounday ntegal:, ), ), ), ) ψ ϕ ϕ ψ Q = α, Q = c α, ϕ α, dγ. 9),,, n n Γ Consequently the bounday ntegal equaton coespondng to the Gad-Shafanov equaton wth the polynoal souce te can be gven n the fo:, ), ) ψ ψ ψ ψ Γ, ) ψ ϕ ϕ ψ c Γ = ψ d α, cϕ dγ n n Γ. ), n n It should be stessed that Eq.) s expessed only by cuvlnea ntegals. The devaton pocess shown hee s equvalent to what s called the Dual Recpocty Method DRM) n the bounday eleent eseach feld [3], although the ethod was ognally appled to solve the Posson-type equaton. The nae of dual ecpocty coes fo the fact that the ecpocty theoe Eq.8) o Geen s second dentty) s appled to both sdes of the equaton to take all tes to the bounday. That s, once the theoe s appled to the HS of the Gad-Shafanov equaton as descbed n secton, next t s appled to the RHS wth the help of patcula solutons. 3.3 Soe ntepetatons of Eq.) When all the expanson coeffcents α, take eo values, Eq.) s conveted to the bounday ntegal equaton fo the agnetc flux functon ψˆ n a vacuu egon 9

11 ψ ψˆ ψˆ ψ c ˆ ψ dγ =, ) Γ n n whch coespondng to the hoogeneous patal dffeental equaton: ψ ˆ =. ) It s nteestng to pont out that Eq.) can also be deved n a sple way statng wth Eqs.) and ). Couplng the equaton fo a plasa egon ψ = j j = α,, ξ η wth the equaton whch the patcula solutons satsfy, ) α ϕ, =,, α, ξ η, 3) one knows the elatonshp:, ) ψ α, ϕ =. 4), Note that Eqs. ) and 4) have the sae fo. Then, substtutng, ) ψ ˆ = ψ α ϕ nto Eq.), one can easly each Eq.). Ths devaton pocess s dentcal to the patcula soluton technque fo the Posson equaton, whch s descbed n the lteatue [3]. Anothe nteestng ntepetaton of Eq.) s as follows. If the pocess to deve Eq.) s agan appled to the equaton fo a patcula soluton ϕ,,, = j µ j j, nstead of the ognal Gad-Shafanov equaton, one also eaches the followng ntegal equaton ψ j j ψ ψ cj = dγ µ j dω ) j n n. 5) Γ Ω Both Eq.) and Eq.5) nclude the sae te ψ / ) µ j ) Ω j dω, then one easly fnd

12 ψ ψ ψ ψ ψ ϕ ϕ ψ ψ ϕ n n n n Γ Γ. 6) c dγ= c dγ Substtutng ϕ = α, ϕ,, ) nto Eq.6), t wll be copletely dentcal to Eq.). 4. DISCRETIZATION - BOUNDARY EEMENT METHOD The followng nuecal schees ae bascally the sae as the ones n the wdely used bounday eleent ethod [7]. 4. Dscetaton usng constant bounday eleent Fo a dgtal coputaton, one sply dscetes Eq.) usng constant bounday eleents,.e., the bounday Γ s dvded nto N staght lne segents. The values of ψ and ψ / n ae assued to be constant on each eleent and equal to the value at the d-node of the eleent. The dsceted fo can be wtten as c ψ = N j= ψ n j Γj ψ dγ ψ dγ n N, ), ), ) ψ j j ψ α, cj dγ. =,,, n) j n n = j Γ, N j= ψ j Γj 7) Equaton 7) s splfed as usng the followng notatons: q n j= H n jψ j G, j q j = Q =,,, ), 8), n j= = ψ / n), ˆ ), ˆ j H, j = H, j c j H, j = H, j c j) j = ˆ ψ ψ H, j = dγ, Γ, j = n Γj Γj dγ. It should be noted that Eq.8) epesents a set of n sultaneous algebac equaton fo n unknowns, and can be wtten n the atx fo:

13 H ψ Gq = Q 9) 4. Two stages of BEM coputaton as Reodeng Eq.9) n such a way that all the unknowns ae on the left-hand sde, Eq.9) can be expessed Ax = f, 3) whee x s the vecto of the unknowns ψ and q, f s the contbutons of bounday condtons added to Q. As the absolute value of the scala flux functon ψ s abtay, one can sply pose ψ = along the plasa bounday. In ths case the fst stage of BEM coputaton s to solve nstead of Eq.3). Once all the values of ψ and q = ψ / n Gq Q = 3) ) on the ente bounday have becoe known by solvng Eq.3) o Eq.3), the values of agnetc flux at any ntenal pont can be calculated usng the dsceted fo of Eq.) wth c =. : n, j j j= j= n ψ = G q Hˆ, jψ j Q, 3) In suay, the BEM calculaton can be dvded nto two stages: the fst calculaton to seek all bounday values of ψ and ψ / n and the second calculaton fo ntenal ponts, as llustated n Fg.. Fg. Two stages of bounday eleent calculaton 5. PARTICUAR SOUTION FOR A MONOMIA SOURCE TERM The key to success to eale the bounday-only fo gven by Eq.) s to fnd an actual patcula soluton ϕ, ) whch satsfes the Gad-Shafanov equaton wth a onoal souce:, ), ) ϕ ϕ = = =., ) ξ η 33) The elatonshp

14 ) ) ) = η ξ η ξ η ξ 34) s easly found. Substtutng nto Eq.34), one obtans the ecuence foula: ) ) ) ) ) = η ξ η ξ η ξ Next, applyng ths elatonshp tself to the second te of the HS, one fnds ) 3) 4) ) ) ) ) 4) 3) ) ) ) ) ) = η ξ ξ η ξ η ξ Ths pocess can be epeated successvely, and t should be notced that the last te of the RHS vanshes befoe too long f s an even nube. Even f s an odd nube, the absolute values of the last te deceases apdly. In ths way, one fnally obtans the followng patcula soluton as an nfnte sees: = = = ), ) ) ) ). ) ) k k s s s s s ξ η η ξ ϕ. 35) Because of the geoety of usual tokaak fuson devces, one fnds. < = ξ η n Eq.35), and ths fact helps the stable convegence of the sees. The devatves of ), ϕ n the - and -dectons ae gven espectvely by = = = ), ) ) ) ) ) ) ) k k s s s s s k ξ η η ξ ϕ 36a) and = = = ), ) ) ) ) ) ) ) k k s s s s s k ξ η η ξ ϕ. 36b) 3

15 6. EIGENVAUE ITERATION The RHS of the Gad-Shafanov equaton s often appoxated n a sple fo, e.g. [8], µ jφ = α{ β p R β p )}exp γ X ) ), 37) whee β s the polodal beta, R a chaactestc adus of the achne, and p α, γ ae adjusted paaetes. The noaled flux functon X s defned by X = ψ ψ ) / ψ ψ ), whee ψ a s the value of ψ at the agnetc axs whle b a b ψ b s the one on the plasa bounday. As suggested fo the ognal fo of the Gad-Shafanov equaton, Eq.), µ j ϕ s a functon of the unknown agnetc functon ψ. Because of ths, one needs to solve the equaton teatvely as an egenvalue poble descbed below. 6. Powe teatve schee to fnd egenvalue One hee ewtes the Gad-Shafanov equaton usng the egenvalue n) λ, as n) n) n) n) ψ = λ f, ψ ) S, n ) 38) whee f, ψ n) ) n) n) µ jj / λ, 39) n) and the values of f, ψ ) can be calculated usng a sple coelaton foula such as Eq.37). The teaton to seek the egenvalue s pefoed n such a way that the elatonshp 4) n ) n n) ) n) λ f, ψ ) dω= λ f, ψ ) dω Ω Ω s peseved though the teaton. That s, the egenvalue s updated as follows: = λ n) n) Ω λ. 4) n) Ω f, ψ f, ψ n) ) dω ) dω A unfo souce s assued as the ntal estate of n) ) S,.e., const. S =, then, solvng Eq.38) usng the BEM, one obtans the dstbuton of agnetc flux functon ) ) ψ and then f, ψ ). Saplng the ) values of f, ψ ) fo ponts n the plasa doan, one detenes the expanson coeffcents α, n 4

16 Eq.), that wll be used n the next ) ψ coputaton. The detaled pocedue to detene the coeffcents s descbed n the next secton. Thanks to ths polynoal expanson, the doan ntegals n Eq.4) can be also pefoed usng only bounday ntegals, as descbed n Secton 6.3. Once ) λ has been calculated n ths way, one agan coputes ψ ) usng the BEM schee. The above pocess s epeated untl a gven convegence cteon, e.g., λ λ λ n) n) n) 5 ε = < 4) n) s satsfed. The above pocess s vey sla to the fsson souce teatve schee to fnd the ctcal egenvalue n nuclea fsson eacto analyss [4]. 6. Detenaton of the polynoal expanson coeffcents In evey egenvalue teaton, the polynoal expanson coeffcents α, n Eq.) can be detened as follows. Fst, one defnes a ectangula doan that encloses the plasa egon Ω unde consdeaton. Next, one geneates any saplng ponts unfoly wthn the ectangula doan. The ponts outsde the doan Ω ae autoatcally excluded wth the ad of the esdue theoe [5]. That s, whethe a pont w = esdes nsde o outsde Ω can be detened by the esult of the followng, ) coplex ntegal: dw = π fo w Ω w w Γ = fo w Ω. 43) Fo all saplng ponts nsde Ω, one calculates the values of µ j accodng to a coelaton foula, ϕ fo exaple, Eq.37). Based on the esultant dstbuton of µ j ϕ, the coeffcents, α n Eq.) ae detened usng the sngula value decoposton SVD) technque [6]. 6.3 Anothe bounday ntegal fo egenvalue calculaton It s nteestng to pont out that the doan ntegal defned by 5

17 f, ψ) dω= α ξη dω Ω Ω,, can be also tansfoed nto a bounday one. If one fnds a patcula soluton, ) φ whch satsfes φ, ) = ξ η, 44) one can apply Gauss theoe to the doan ntegal, and then one obtans the elatonshp:, ) φ α ξη dω = α φ dω= α dγ. 45) n, ),,, Ω Ω Γ,,, The patcula soluton s gven by φ, ) = s s s s csξ η d η sξ s= s=. 46) In Eq.46), [ ] denotes the ntege pat of the aguent. The coeffcents c s and d s ae evaluated usng the ecuence elatonshp: c s s 4) s 3) = c ) s s, s) s ) c = 47a) ) ) and d s s 4) s 3) = d ) s s, s) s ) d =, 47b) ) ) whee and denote the absolute lengths defned n Secton NUMERICA EXAMPES The followng pobles wee solved usng the pesent bounday-only type BEM so as to conf ts valdty. 7. Rectangula Plasa Suppose hypothetcal ectangula plasa as shown n Fg.. The bounday condton ψ = s posed along each sde of the ectangle. In ths case, as shown n Appendx B, the analytc soluton exsts fo the equaton wth a onoal souce te 6

18 ψ ψ ψ =, ). Fg. Rectangula plasa One hee assues the ses: a =.5[], b=.5[], R=.[]. Each sde of the ectangle was equally dvded n such a way that each constant bounday eleent has a length of.5[]; thus a total of 8 eleents wee eployed. Copasons wee ade between the analytc and the bounday eleent solutons fo all cobnatons of nteges and n the ange 8. As an exaple, Fg.3 shows the contou ap of the BEM soluton of ψ fo a onoal souce 3.Relatve devaton fo the analytc soluton n ths case s llustated n Fg.4. The devaton, defned by BEM-Analytc)/Analytc), s less than. ~.% n the geate pat of the doan. Devaton lage than % s found nea the edges and cones, howeve, the absolute values of ψ ae exteely sall n these places. Alost the sae level of accuacy was also deonstated fo othe cobnatons of and. Fg.3 Bounday eleent soluton of ψ pofle fo 3 Fg.4 Relatve devaton between the BEM and the analytc soluton 7. Tokaak Geoety As a oe ealstc test poble, one hee consdes a poble to odel a tokaak-type devce. By the coutesy of Japan Atoc Enegy Reseach Insttute JAERI), efeence data of plasa bounday, dstbutons of plasa cuent and agnetc flux functon wee fstly povded, whch wee calculated usng a elable equlbu code, SEENE [5]. Ths equlbu coputaton was ade based on the cuent pofle paaetaton that has the fo µ j = β β. Hee,.6 j c{ p p R )} X ) X = ψ ψ M ) / ψ S ψ M ) n whch ψ M and ψ S ae the values of ψ on the agnetc axs and on the bounday, and β p =.6) and R = 3.3 ) denote the polodal beta and the chaactestc ajo adus, espectvely. Ths poble was agan analyed usng the BEM as a fxed bounday poble. Only the bounday shape 7

19 aong the SEENE coputng esults was tansfeed to the BEM coputaton as nput data. The bounday condton ψ = was posed at each nodal pont along the bounday. The sae cuent pofle paaetaton shown above was agan assued, and the coplete polynoal of the 8-th ode was adopted to appoxate the µ j ϕ dstbuton. That s, the level defned n Secton 3. s 8, and hence the polynoal conssts of a total of 45 tes. To detene the polynoal expanson coeffcents, a total of 63 saplng ponts wee autoatcally geneated wthn the doan, followng the pocedue descbed n Secton 6.. The plasa bounday s appoxated by a polygon that has 8 sdes,.e., a total of 8 constant eleents wee eployed. A total of 8 teatons wee equed n the BEM analyss when the egenvalue devaton defned by Eq.4) was educed to less than 5. The CPU te consued fo ths coputaton was ~6.9s wth Alpha CPU-64A 6MH), whle the coputng te devoted fo the SEENE calculaton was ~.8s wth the sae CPU. The pesent BEM coputaton s not always supeo to the SEENE coputaton fo the vewpont of the coputng te. The ajo pat of the BEM coputng s devoted to the bounday ntegatons and ths nuecal ntegaton has not yet fully opted. Snce such bounday ntegatons can be ade ndependently fo each bounday node pont, the CPU te could be dastcally educed f one adopted a paallel coputng n futue. Consdeng also the pogess n copute pocessng capablty, the authos ae not pessstc about the poble of coputng te. In the BEM coputaton, only about 5.5% of the total coputng te was consued afte the fst stage of egenvalue teaton, whch was anly devoted fo bounday ntegal coputatons, and ths fact shows that the nube of teatons hadly affects the total coputng te. The pofle of agnetc flux functon thus obtaned fo the BEM calculaton s copaed wth the SEENE calculaton esults, as shown n Fg.5. The plasa cuent pofles ae also copaed n Fg.6. In each Fgue, the sold lnes show the BEM solutons, whle the dashed lnes denote the esults obtaned usng the SEENE code. The BEM esults show good ageeent wth the efeence data, and ths deonstates the valdty of the pesent bounday-only ntegal foulaton, especally of the polynoal expanson 8

20 appoxaton of µ j ϕ. The dstbutons of the agnetc flux functons and the plasa cuent along the lne = ae shown espectvely n Fg.7 and Fg.8. The efeenced SEENE esults ae plotted by dots, whle the sold lnes denote the BEM solutons based on the level-8 polynoal and the dashed lnes ae the BEM ones obtaned usng the level- to -7 polynoals. Fg.5 Contous of agnetc flux functon Fg.6 Contous of plasa cuent densty Fg.7 Results of agnetc flux functon along the lne = Fg.8 Results of plasa cuent densty along the lne = 8 CONCUSION AND FURTHER REMARKS A new type of bounday eleent ethod pesented n ths wok does not eque any coputaton of doan ntegal. The fnal fo of the bounday ntegal equaton has no doan ntegal, as shown n Eq.). In addton, even the pocess of the egenvalue coputaton s also based on a bounday-only ntegal, as descbed n Secton 6.3. When atteptng to pefo an egenvalue teaton, one needs to dstbute any saplng data ponts n the plasa doan fo detenng the coeffcents of polynoal expanson of µ j ϕ, howeve, the coodnates of these saplng ponts ae autoatcally geneated n the copute poga. Thus the data saplng neve contadcts the advantage of the pesent ethod that t eques dscetaton of the bounday only. The poga use has only to pepae bounday eleent data that specfy the shape of the last closed agnetc suface. The bounday values of ψ n and the polynoal expanson coeffcents ae updated n each stage of the egenvalue teaton, howeve, the coponents n the syste atx elated to the bounday ntegal equaton and also the ones n the atx fo polynoal coeffcent detenaton ae nvaant though the teaton. Thus the bounday values and the expanson coeffcents ae obtaned only fo sple 9

21 ultplcatons of atx and souce vecto, wth the esult that the nube of teatons hadly affects the total coputng te. Test calculatons ndcate that the pesent bounday-only ntegal equaton appoach povdes stable and accuate nuecal solutons. The authos used constant bounday eleents fo the dscetaton n the pesent wok, howeve, a new veson of FORTRAN code based on sopaaetc quadatc bounday eleents s now unde developent to odel the plasa bounday cuvatue oe accuately wth a salle nube of bounday nodes. Nuecal exaples n the pesent pape ae lted to fxed bounday pobles; howeve, the applcaton of the pesent ethod can be extended to solve a fee bounday poble by addng an teatve seach functon to the copute code. The addton of polodal col cuent tes nto the equaton s also useful to expand the applcaton. Takng advantage of the pesent ethod that eques only the bounday dscetaton, t s nteestng to consde a poble of ovng bounday,.e., the plasa bounday that shape s changng te-dependently. How the bounday-only ntegal equaton ) can be appled to an nvese poble to econstuct the plasa cuent densty pofle? The authos futue plan s as follows. Kuhaa s Cauchy condton suface CCS) ethod [] s fo the detenaton of the shape of plasa bounday, howeve, t can also estate values of ψ / n as well as the agnetc flux functon ψ on the plasa bounday. Ths eans, once the bounday shape s fxed by the ethod, Eq.) has no unknowns any oe except fo the polynoal expanson coeffcents α,. The coeffcents and then the pofle of µ jϕ can be easly estated, although one needs to add soe a po nfoaton to successfully obtan a unque soluton. The followngs ae canddates of a po nfoaton and physcal constants we can take nto account. ) The total plasa cuent s known. ) Zeo-cuent along the plasa bounday. 3) Constants deved fo the equlbu J B= p. Fo ths pupose, we can adopt the sple scala elatonshp poposed by K. Kuhaa [8] to connect the cuent densty wth the agnetc flux. Ths

22 condton eques teaton to copute altenatvely the agnetc flux pofle and the cuent densty dstbuton.) 4) Assue that the cuent densty f possble) o othe physcal quanttes closely elated to the cuent densty can be easued at a cetan nube of ponts n the plasa doan. The authos pesonal opnon s that ths condton s vey essental to ensue the unqueness of the cuent densty soluton.) All of these condtons can be descbed usng the polynoal fo, and then they should be ncopoated nto the algebac equatons gven fo Eq.) to detene the coeffcents α,. The sngula value decoposton SVD) technque [6] s well suted to solve the esultant atx equaton, and n ths case the Tkhonov egulaaton [9] can be also eployed to stable the nuecal ll-posedness. The detaled ethodology based on the above pocedue s now unde developent. Appendx A: DERIVATION OF THE SINGUAR POINT PARAMETER c Equaton ) s vald fo any pont n the doan Ω, but one ust odfy the equaton fo a pont on the bounday. One dvdes the bounday Γ nto two pats, Γ = Γ Γ,.e., whee ψ ψ ψ ψ ψ ψ n n n dγ = l dγ l dγ ε ε, A) Γ Γ Γ Γ s an agnay fan-shaped bounday wth an angle of π θ) as llustated n Fg.A. The bounday pont ab, ) s assued to be at the cente of the ccle and aftewad the adus ε s educed to eo. Fg.A Bounday pont augented by a sall seccle The coodnates, ) of an abtay pont on Γ can now be gven by = a εcos θ, = b εsnθ, then the coplete ellptc ntegals of the fst and the second knds ae educed to

23 ) K k π cos a aε θ ε = dθ ln = ln sn k θ k ε and ) k E k π = sn θdθ, espectvely when ε. The fundaental soluton gven by Eq.3) can then be educed to 4a 4aεcosθ ε 4 4a 4aεcosθ ε ψ = ln. π ε Consdeng that the noal devatve of the fundaental soluton s wtten n the fo and usng dγ = εdθ, one obtans ψ ψ ψ = cosθ sn θ, n n ψ ψ θ = = a εcosθ π ψ ψ l d Γ ε Γ πθ l ψ cosθ snθ εdθ ψ. ε A) Futhe, one knows ψ ψ ψ ψ l d Γ = VP.. d Γ, ε Γ n Γ n whee the ght-hand-sde s the Cauchy pncpal ntegal. A3) The sae pocedue fo the te ψ ψ d Γ Γ of Eq.), howeve, does not ntoduce any new te n n Eq.). Substtutng Eqs.A) and A3) nto Eq.A), one obtans Γ ψ ψ ψ ψ θ d Γ VP.. dγ ψ n Γ n π, then Eq.) s changed to Eq.) wth c = θ π. In Eq.), the sybol V.P. s otted. The value of c = s taken on a sooth bounday θ = π ), and c = fo an ntenal pont. Appendx B: ANAYTIC SOUTION FOR RECTANGUAR PASMA

24 The followng analytc soluton has been ognally deved by the fst autho fo the pesent eseach. Suppose hypothetcal ectangula plasa as shown n Fg.. The bounday condton ψ = s posed along each sde of the doan. In ths case the analytc soluton of the patal dffeental equaton wth a onoal, ψ ψ ψ =,, ) B) can be gven n the fo ψ = R b) Σ n= b ) n g n s) sn nπ t, B) usng densonless vaables s / R, t = b) /b). The expanson coeffcents n Eq.B), ) b n, s calculated as )! ) b = n nπ Σ j= j)! n π ) The functon g n s) n Eq.B) s gven by j n j { ) ) j xx ) X ~ X ) ~ x X g s) = s[ Ce I x) Ce K x)] x x g s n np } ). B3) B4) In Eq.B4), one denotes the quanttes a a x = Bns, X = Bn ), X = Bn ) wth R R R B n = nπ ), b ~ x ~ x whle the functons I x) = e x I ) and K x) = e xk ) ae defned usng the fst ode x x odfed Bessel functons of the fst and the second knds, espectvely. The patcula soluton n Eq.B4) s coputed as follows: o g np k k s) = x Σ Π j ) j) x :even ntege) B5a) B n k = j = 3

25 g np s) = B B n n x Π k = Σ k = k Π j= j ) j) x k k ) k) x p x) I x)). :odd ntege) B5b) One uses the fst ode odfed Stuve functon [7], ), hee n Eq.B5b). The unknown coeffcents C and C n Eq.B4) can be detened n such a way that the bounday condton ψ = s satsfed at = R ± a. x ACKNOWEDGEMENTS The authos wsh to expess the gattude to D. H. Nnoya and D. K. Kuhaa of Japan Atoc Enegy Reseach Insttute JAERI), Pofesso T. Hona and D. H. Igaash of Hokkado Unvesty fo the valuable and helpful coents on ths wok. Futhe, D. Kuhaa kndly povded the wth the efeence tokaak plasa data used n the nuecal deonstaton n Secton 7.. The authos have to enton hee that the latte pat of the descpton n Secton 3.3 was kndly suggested by one of the efeees n hs evew lette to the authos. Specal thanks ae also due to D. C.A. Bebba of Wessex Insttute of Technology, U.K., who guded the fst autho to the bounday eleent eseach feld, fo hs contnuous encouageent though ths wok. REFERENCES ) SHAFRANOV, V.D., Sov. Phys. JETP 37 96) 775. ) Mukhovatov, V.S., Shafanov, V.D., Nucl. Fuson 97) 65. 3) Shafanov, V.D., Plasa Phys. 3 97) 757 4) WESSON, J, Tokaaks Second edton), The Oxfod Engneeng Sees 48, Claendon Pess, Oxfod 997). 5) AZUMI, M., KURITA, G., MATSUURA, T. et al., A Flud Model Nuecal Code Syste fo Tokaak 4

26 Fuson Reseach n Coputng Methods n Appled Scence and Engneeng, p.335, Noth-Holland, Asteda / New Yok / Oxfod 98). 6) McCAIN, F.W., BROWN, B.B., GAQ, A Copute Poga to Fnd and Analye Axsyetc MHD Plasa Equlba, GA-A 449, Geneal Atoc Copany 977). 7) BREBBIA, C.A., The Bounday Eleent Method fo Engnees, Pentech Pess, ondon 978). 8) BRAAMS, B.J., Intepetaton of tokaak agnetc dagnostcs, epot IPP 5/, Max Plank Insttute fu Plasa Physcs 985). 9) HAKKARAINEN, S.P., FREIDBERG, J.P., Reconstucton Of Vacuu Flux Sufaces Fo Dagnostc Measueents In A Tokaak, epot PFC/RR-87-, MIT Plasa Fuson Cente 987). ) KURIHARA, K., A New Shape Repoducton Method Based on the Cauchy-Condton Suface fo Real-Te Tokaak Reacto Contol, Fuson Eng. Des., 5-5 ) 49. ) KURIHARA, K., Tokaak Plasa Shape Identfcaton on the Bass of Bounday Integal Equatons, Nuclea Fuson, 33[3] 993) 399. ) TAKEDA, T., TOKUDA, S., Coputaton of MHD equlbu of tokaak plasa, Jounal of Coputatonal Physcs, 93[] 99) -7. 3) PARTRIDGE, P.W., BREBBIA, C.A., WROBE,.C., The Dual Recpocty Bounday Eleent Method, Coputatonal Mechancs Publcatons, Southapton / Boston, Co-publshed wth Elseve Appled Scence, ondon / New Yok 99). 4) ITAGAKI, M., Bounday Eleent Methods Appled to Two-Densonal Neuton Dffuson Pobles, J. Nucl. Sc. Tech., [6] 985) ) GIPSON, G.S., Use of the esdue theoe n locatng ponts wthn an abtay ultply-connected egon, Advances n Engneeng Softwae, 8[] 986) 73. 6) PRESS, W.H., FANNERY, B.P., TEUKOSKY, S.A., VETTERING, W.T., Nuecal Recpes - The At of Scentfc Coputng, Cabdge Unvesty Pess, Cabdge 986). 7) ABRAMOWITZ, M., STEGUN, I.A.: Handbook of Matheatcal Functons, Dove Publcatons, New Yok 965). 5

27 8) KURIHARA, K., Cuent Pofle Repoducton Study on the Bass of a New Expanson Method wth the Egenfunctons Defned n the Tokaak Plasa Inteo, Fuson Technology, ) ) HANSEN, P.C., Rank-Defcent and Dscete Ill-Posed Pobles Nuecal Aspects of nea Inveson, SIAM, Phladelpha 998). 6

28 st of Tables Table Pascal s tangle fo polynoal functons st of Fgues Fg. Two stages of bounday eleent calculaton Fg. Rectangula plasa Fg.3 Bounday eleent soluton of ψ pofle fo 3 Fg.4 Relatve devaton between the BEM and the analytc soluton Fg.5 Contous of agnetc flux functon Fg.6 Contous of plasa cuent densty Fg.7 Results of agnetc flux functon along the lne = Fg.8 Results of plasa cuent densty along the lne = Fg.A Bounday pont augented by a sall seccle

29 Table Pascal s tangle fo polynoal functons evel ξ η ξ ξη η 3 ξ 3 ξ η ξη η 3 4 ξ 4 ξ 3η ξ η ξη 3 η 4

30 Path of ntegaton Bounday node Intenal pont Fg. Two stages of bounday eleent calculaton 3

31 Z θ R a a b b Fg. Rectangula plasa 4

32 .5 ) ) Fg.3 Bounday eleent soluton of ψ pofle fo 3 5

33 .5.%.% ).% ) Fg.4 Relatve devaton between the BEM and the analytc soluton 6

34 .5.5 ) ) Fg.5 Contous of agnetc flux functon 7

35 .5 ) ) Fg.6 Contous of plasa cuent densty 8

36 evel 3 Magnetc flux Wb).6.4. evel evel evel evel 4~ ) Fg.7 Results of agnetc flux functon along the lne = 9

37 .8 evel 7 evel 3 evel 4 Cuent densty MA/ ).6.4. evel evel 5 evel 6 evel evel ) Fg.8 Results of plasa cuent densty along the lne =

38 Γ Γ Γ Γ ab, ) ε θ, ) θ Γ Fg.A Bounday pont augented by a sall seccle