Rotational symmetry applied to boundary element computation for nuclear fusion plasma

Transcription

1 Bouda Elemets ad Othe Mesh Reductio Methods XXXII 33 Rotatioal smmet applied to bouda elemet computatio fo uclea fusio plasma M. Itagaki, T. Ishimau & K. Wataabe 2 Facult of Egieeig, Hokkaido Uivesit, Japa 2 Natioal Istitute fo Fusio Sciece, Japa Abstact A efficiet bouda elemet techique has bee poposed to deal with a poblem whe geomet ad bouda coditios have a otatioal smmet. I this techique, o bouda elemets eed to be defied o the smmet suface. This idea has bee applied to a poblem of magetic field i uclea fusio plasma. A useful liea tasfomatio fo the otatioal smmet i the tooidal diectio has bee deived i tems of the -, - ad -compoets of vecto potetial i a 3-D Catesia coodiate sstem. Applig this liea tasfomatio to the discetied set of bouda itegal equatios, oe ca dasticall educe the sie of mati, the umbe of ukows ad the computig cost. Also, accuate solutios ca be epected. Results of test calculatios demostate the validit of the peset fomulatio. Kewods: bouda elemet method, otatioal smmet, vecto potetial, liea tasfomatio, uclea fusio plasma, Cauch coditio suface method, magetic seso. Itoductio The peset wok is a pat of the authos eseach pla to develop a ivese aaltic techique to idetif the bouda shape of uclea fusio plasma fom sigals of magetic sesos located outside the plasma. Fo this pupose, the Cauch coditio suface (CCS) method [] has alead bee established fo a tokamak-tpe fusio device. The geomet of tokamak plasma is aismmetic so that the aalsis ca be made i a 2-dimesioal (2-D), - sstem. O the othe had, 3-D aalses ae equied fo o-aismmetic plasma, e.g., i a helical tpe device such as the Lage Helical Device (LHD). The applicatio of doi:0.2495/be0002

2 34 Bouda Elemets ad Othe Mesh Reductio Methods XXXII the CCS method to a 3-D space aalsis, howeve, becomes quite challegig because the 3-D sstem has a much lage umbe of ukows tha a 2-D sstem. The authos have oticed that the LHD has a /5 otatioal smmet. I Sectio 2 of the peset pape, a efficiet bouda elemet techique is give fo solvig a simple potetial poblem havig otatioal smmet. Whe thee is a -fold otatioal smmet, the sstem mati becomes a cclic oe. Because of this, the umbe of ukows is educed to /. I this case, o bouda elemets eed to be defied o the smmet suface. I Sectio 3, this idea is eteded to uclea fusio plasma that has a otatioal smmet i the tooidal diectio. A useful liea tasfomatio is deived i tems of the -, - ad -compoets of vecto potetial i a Catesia coodiate sstem. Applig this liea tasfomatio to the discetied set of bouda itegal equatios, oe ca educe the umbe of ukows ad the computig cost damaticall. Also, accuate solutios ca be epected. Numeical eamples ae show i Sectio 4 to demostate the validit of the peset fomulatio to deal with the otatioal smmet. 2 Rotatioal smmet i a simple potetial poblem Fist of all, oe hee cosides a simple potetial poblem. Coespodig to the 2-D Laplace equatio u u the bouda itegal equatio is give b 0, * * u u i cu i i ui u d, (2) which discetied fom ca be witte i a mati equatio fom [2] Hu Gq. (3) Hee, the quatities u ad q deote vectos of the Diichlet coditio u ad the Neuma coditio q u/, espectivel o the bouda. Whe the bouda is divided ito segmets, (, 2,, ), eq (3) is give b H, H, 2 H, u G, G, 2 G, q H2, H2,2 H2, u2 G2, G2,2 G2, q2 H H H u G G G q,,2,,,2, usig submatices H i, ad G i,. If thee is a -fold otatioal smmet i the geomet as well as the bouda coditios ude cosideatio, oe fids (4)

3 Bouda Elemets ad Othe Mesh Reductio Methods XXXII 35 u u2 u ad q q2 q. Also, the mati o each side i eq (4) becomes a cclic oe. That is, eq (4) ca be ewitte as H, H,2 H, u G, G,2 G, q H, H, H, u G, G, G, q. (5) H,2 H,3 H, u G,2 G,3 G, q Equatio (5) ca the be simplified as H, H, 2 H, { u} G, G, 2 G, { q}. (6) Now the mati sie ad the umbe of ukows ae educed to / 2 ad /, espectivel. It should be oted that i this case oe eed ot to defie a bouda elemets o the smmet suface, so that the peiodic bouda coditios ae atuall satisfied. Suppose ow J mesh poits, (, ), (, 2,, J), have bee give fo the bouda to geeate the submatices H, ad G,. Othe mesh poit coodiates (, ), ( k 2, 3,, ;, 2,, J) to compute [ H,2, H,3,, H, ] ad [ G,2, G,3,, G, ] ae calculated usig the well-kow liea tasfomatio ( k ) 0 cos( k ) si( k ) 0 ( k ) 0 si( k ) cos( k ) (7) 0 with the agle 2 / ad the otatio ais ( 0, 0), as show i fig.. 2 : u, q 3 : u, q ( u, q ) ( u2, q2) 3 3 H,3, G,3 H, G,2,2 (, ) k : u, q ( u, q ) k k 2 H, G, k, k (, ) H, G ( 0, 0) H, G,,,, ( k ) : u, q : u,q ( u,q) Figue : Bouda segmetatio with -fold otatioal smmet.

4 36 Bouda Elemets ad Othe Mesh Reductio Methods XXXII 3 Rotatioal smmet of 3-D vecto potetial The fudametal idea descibed i Sectio 2 is ow applied to a poblem of magetic field i uclea fusio plasma, which is descibed usig vecto potetial i a 3-D space. This sectio descibes how the otatioal smmet is itoduced to educe the umbe of ukows i the 3-D CCS method fomulatio that is give b a set of bouda itegal equatios fo poits alog the CCS ad fo the magetic seso positios. The bouda itegal equatios fomulated i this wok ae all descibed i a Catesia coodiate sstem. The easo fo this is give i the Appedi. A outlie of 3-D CCS method is foud i the liteatue [3]. A, A A, A ( k ) O ( k ) A, A A, A Figue 2: Rotatioal tasfomatio of vecto potetials. 3. Liea tasfomatio of vecto potetial ( k) Now oe descibes the vecto potetial ( A, A, A ) i the k-th segmet i tems of the vecto potetial ( A, A, A ) i the fist segmet. The elatioship betwee ( A, A ) ad ( A, A ) i the diffeet coodiate sstems ae ( k ) descibed usig the tooidal agle, as ( k) k k ( k) A cos si A ( k) k k ( k) A si cos A Cosideig the otatioal smmet, i.e., A A ad A ( k ) ( k ) k k A cos si A ( k ) k k A si cos A. (8) A ( k ), oe fids. (9)

5 Bouda Elemets ad Othe Mesh Reductio Methods XXXII 37 With the elatioship betwee ( A, A ) ad ( A, A ), eq (9) becomes ( k ) k k A cos si cos si A ( k ) k k. (0) A si cos si cos A This fomula ca be tasfomed ito ( k ) k k A cos si A ( k ) k k () A si cos A usig the additioal theoem of tigoometic fuctios with a otatio agle k k. Icludig the -compoet, oe fiall obtais ( k) A cos si 0A ( k) A si cos 0 A. (2) ( k ) A 0 0 A ( k) This is the liea tasfomatio fom ( A, A, A ) to ( A, A, A ) i a Catesia coodiate sstem. 3.2 Applicatio to bouda elemet equatios i 3-D CCS method The set of bouda itegal equatios fo the 3-D CCS method ca also be tasfomed ito a mati equatio Hu Gq, (3) whee ( ) ( ) ( ) T u A, A, A,, A, A, A (4) ad ( ) ( ) ( ) A A A A A A q,,,,,,. (5) ( k) Hee, ( A, A, A ) meas the set of all vecto potetials withi the k-th segmet. The LHS of eq (3), fo eample, ca be witte as Hu H H2 H3 H,32 H,3 H,3 A H2 H22 H23 H2,32 H2,3 H 2,3 A H3 H32 H33 H3,32 H3,3 H 3,3 A, (6) Hl, Hl,2 Hl,3 Hl,32 Hl,3 H l,3 ( ) A ( ) HL, HL, 2 HL,3 HL,32 HL,3 H L,3 A ( ) HL, HL,2 HL,3 HL,32 HL,3 L,3 A H T

6 38 Bouda Elemets ad Othe Mesh Reductio Methods XXXII whee L deotes the umbe of lies that depeds o the umbe of odal poits o the CCS ad the umbe of magetic sesos as well. Assumig a -fold otatioal smmet, i.e., applig eq (2) to the vecto o the RHS i eq (6), eq (6) is descibed ol usig A, A ad A. Fo istace, the l th lie of Hu ca be witte i the fom ( Hu) l H cos H si, l,3k2 l,3k k A Hl,3k2 si Hl,3k cos, A k A Hl,3k k. (7) The Gq o the RHS of eq (3) ca also be ewitte i the same wa. Both the umbes of colums ad ukows ca be educed to / if ol givig the ( ) otatioal agle k. Whe a sigula poit i is located o the CCS, the bouda itegal equatio fo each of A, A ad A holds idepedetl, ad all equatios commol use the same fudametal solutio. Because of this, i eq (6) the potio elated to the CCS epesets a 3 3 squae mati equatio: Hu CCS H 0 0 H, ,3 0 H H A 0 0 H H3,3 A A. (8) ( ) H , 0 H3 2,3 2 0 A ( ) 3,2 0 H H 3,3 A 0 0 3, ,3 ( ) H H A The, itoducig the otatioal smmet, eq (8) ca be simplified as CCS Hu23 H,3k2cos H,3k2si 0 k k A. (9) H2,3k si H2,3k cos 0 A k k A 0 0 H3,3k k

7 Bouda Elemets ad Othe Mesh Reductio Methods XXXII 39 Fiall, the equatios fo poits o the CCS as well as the oes fo the magetic seso positios ae solved simultaeousl. Oce all the values o the CCS ae kow, oe ca calculate the magetic field fo abita poits. 4 Numeical eample Oe hee cosides a poblem to model the o-aismmetic plasma i the LHD, a helical-tpe device. The thee compoets of magetic field i the LHD wee ecostucted usig the 3-D CCS method. As the magetic field pofile is ot aismmetic, this poblem is challegig ad equies a lage umbe of ukows. The esults wee compaed with the efeece solutio obtaied usig the HINT code [4]. Figue 3 shows a eample of the efeece solutio obtaied usig the HINT code. This figue gives the cotous of the -compoet of the magetic field B o the - plae at the tooidal agle of 8 deg. B (T) (m) Plasma egio (m) Figue 3: Refeece pofile of magetic field B at 8 deg. Fo ivese aalses, oe hee assumes that 20 magetic flu loop sesos ad 45 magetic field sesos ae aaged outside the plasma. Each of field sesos is hpotheticall assumed to detect all of the thee compoets of magetic field. I this case the umbe of seso sigals is A tube-shaped CCS was placed withi a domai that ca be supposed to be iside the actual plasma.

8 40 Bouda Elemets ad Othe Mesh Reductio Methods XXXII Fist, a CCS that coves 360-deg. was divided ito 60 bouda elemets (Case A). As the umbe of ukows, 8640, is lage tha the umbe of equatios, 5693, as show i Table, it was impossible to solve this poblem. Net, cosideig a /5 otatioal smmet, ol 72-deg. potio of the CCS tube was modelled ad this potio was divided ito 32 bouda elemets (Case B). Figue 4 shows the ecostucted cotous of the -compoet of the magetic field B o the - plae at the tooidal agle of 8 deg., which coespod to the efeece cotous i fig. 3. As the magetic fields computed usig the CCS method has o phsical meaig iside the plasma bouda [3], the ae ot daw iside the plasma i fig. 4. This ecostucted field pofile agees well with the efeece oe i fig. 3. Table : Calculatio coditio fo plasma i LHD. Case A B C Rotatioal smmet 360-deg. /5 smmet 360-deg. No. of seso sigals No. of bouda elemets No. of equatios No. of ukows Solutio accuac Not solved Acceptable Poo B (T) (m) Plasma egio (m) Figue 4: Pofile of ecostucted magetic field B at 8 deg. (Case B).

9 Bouda Elemets ad Othe Mesh Reductio Methods XXXII 4 Whe oe took ol 40 bouda elemets o the CCS fo 360-deg. (Case C), the umbe of ukows is of couse smalle tha that of equatios. Ufotuatel, howeve the accuac of the ecostucted solutio i this case was ve poo, as show i fig. 5. That is, without cosideig the otatioal smmet, it is difficult to ealie a good accuac i the 3-D aalsis of oaismmetic plasma, which equies a lage umbe of ukows. B (T) (m) Plasma egio (m) Figue 5: Pofile of ecostucted magetic field B at 8 deg. (Case C). 5 Coclusio A efficiet bouda elemet techique has bee applied to poblems of magetic field i 3-D uclea fusio plasma that has a otatioal smmet i the tooidal diectio. I this techique, o bouda elemets eed to be defied o the smmet suface. The liea tasfomatio i tems of the -, - ad - compoets of 3-D vecto potetial plas a impotat ole to educe the mati sie ad the umbe of ukows i the discetied set of bouda itegal equatios. This eables oe to ealie ot ol the eductio of the computig cost but the impovemet of umeical solutio accuac. Results of test calculatios fo uclea fusio plasmas i a helical-tpe device demostate the validit of the peset fomulatio. It has bee foud that oe should icopoate the otatioal smmet ito the aalsis of oaismmetic 3-D plasma i ode to obtai a acceptable accuac of the solutio.

10 42 Bouda Elemets ad Othe Mesh Reductio Methods XXXII Ackowledgemets This eseach was suppoted b the Miist of Educatio, Cultue, Spots, Sciece ad Techolog, Gat-i-Aid fo Scietific Reseach (C), , This wok was also pefomed with the suppot ad ude the auspices of the NIFS Collaboatio Reseach Pogam (NIFS08KLHH308). Appedi: Wh the bouda itegal equatios i this wok ae descibed i a Catesia coodiate sstem? The vecto Laplacia i a Catesia coodiate sstem has a simple elatioship ( A ) k Ak ( k,, ). (A) That is, the vecto Laplacia ca be give b a set of the scala Laplacia of each Catesia scala compoet. I a clidical o a spheical coodiate sstem, o the othe had, the epessio of the vecto Laplacia is ot so staightfowad. I a clidical sstem, fo eample, the compoets of the vecto Laplacia ae witte i complicated foms, as 2 A A ( A ) A, (A2a) 2 A ( ) A A A (A2b) ad ( A ) A. (A2c) If oe uses this coodiate sstem, the bouda itegal equatios coespodig to - ad - compoets will iclude domai itegal tems. To ealie a bouda-ol itegal fomulatio, it is bette ot to use a clidical o a spheical sstem. Because of this, the authos adopt a 3-D Catesia coodiate sstem fo the aalsis to obtai the 3-D distibutio of vecto potetial. Howeve, it is eas to tasfom the esult, oce calculated i a Catesia coodiate sstem, ito oe i aothe coodiate sstem. Refeeces [] Kuihaa, K, Fusio Eg. Des., 5-52, pp , [2] Bebbia, C.A., The Bouda Elemet Method fo Egiees, Petech Pess, Lodo, 978 [3] Itagaki, M., Maeda, T., Wakasa, A., Wataabe, K., Poc. BEM/MRM XXXI, Southampto, pp , [4] Haafui, K., Haashi T., Sato T., J. Comput. Phs., 8, pp.69-92, 989.