Summary ELECTROMAGNETIC FIELDS AT THE WORKPLACES. System layout: exposure to magnetic field only. Quasi-static dosimetric analysis: system layout

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1 Internatonal Workshop on LCTROMGNTIC FILDS T TH WORKPLCS 5-7 September 5 Warszawa POLND 3d approah to numeral dosmetr n quas-stat ondtons: problems and eample of solutons Dr. Nola Zoppett - IFC-CNR, Florene, Ital Summar pproah to resoluton of quas-stat dosmetr problems n 3D. Standard approah. Overvew on man features and rtal aspets. Detals not eplaned n ltterature. Bod model Feld dstrbuton n the bod model volume and/or boundar ondtons on bod surfae Quas-stat dosmetr analss: sstem laout Numeral methods appled n the segmented bod model regon Quas-stat felds an be omputed separatel from the problem s soluton nsde the bod model volume. Soure/s model Post proessng Results Sstem laout: eposure to magnet feld onl Φ Φ ( ) l n m n ω m m Contents of presentaton:. Bass of the method. Dsretzaton 3. ample of mplementaton: overvew and rtal aspets Current paths J J J z Post proessng Φ Φ + ω l m Φ Φ 3 + ω l m 3 Φ Φ 5 + ω l m 5 J

2 : bas assumptons Quas stat ondtons:. l>>l where L s the problem ma. trasversal dmenson, usuall m are taken for L (alwas true f are true the net ondtons). The nternal magnet feld has not to be nfluened b ndued urrents 3. For ever tssue (or at leas for the man ones) must be vald that s>>we (good ondutors) The equaton an be wrtten down n a general form that goes beond the quas-stat assumpton: ω B ω + ω Φ Φ ω ( + ω ) H ( + ωε ) ( H ) ( + ωε) ) equaton n the general form [( + ωε )( φ ω) ] Φ ω >> ωε n quas stat ondtons [( + ωε )( φ ω) ] The permettvt dstrbuton plas no further role n the present analss ( ( φ ω )) B nternal B nternal

3 : boundar ondtons on bod surfae If an eternal eletr feld s onsdered, on bod surfae nˆ must be mposed the ondton: ωρ (Charge ontnut) s Where r s s a surfae harge denst that an be alulated solvng a stat potental problem (Laplae problem) wth the bod surfae replaed b a perfet ondutor (a surfae harge denst an be assumed onl f s>>we). If eternal magnet feld onl s onsdered, no supplementar boundar ondtons are needed on bod surfae. For good ondutors the harge ontnut ondtons beome the urrent ontnut espresson that s equvalent to the epresson tself. φ ω J ( ( )) ( ) ( ) n quas stat ondtons: urrent denst epresson One salar potental has been alulated: J ( φ ω ) Dsretzaton of the salar equaton ntegral and dfferental form n quas-stat ondtons The dsretzed form of the salar relaton gven b the method an be obtaned startng from both the dfferental and the ntegral form. dfferental equaton : ( ( φ ω )) ntegral equaton : S [( )( φ ω )] d S Closed surfae 3

4 Unknown salar potental are appled on ell s edges We want to wrte down the epresson of the salar potental F n n n terms of:. Unknown potental F on neghbourng edges n (..).. Vetor potental (magnet feld ). 3. Cell s ondutvtes. : D dsretzaton ver blak edged square represents a ell wth ts own ondutvt The dsretzed relaton we are searhng for an be obtaned from the ntegral form. The flu of the quantt φ ω ( )( ) : D dsretzaton Can be epressed as the sum of the streams though the four edges of the red ontour. Sne ever red edge overlap two ells the mean ondutvt s onsdered. The generated vetor potental s appled n ponts m (..). Onl the omponent perpendular to the red edge s onsdered. S [( )( φ ω )] ds Closed ontour S φ ds ω m Φn Φ l n S + : D dsretzaton Φn Φ l ds n l + m m m Consderng: +, + + +, m, + +, and: ( m ) ˆ l ( φ nˆ ) ω l ( nˆ ) m where: nˆ Φ n3 Φ n + l ω ( m ) ˆ m nˆ ˆ ˆ Φ n Φ l +, +, + + +,, + ( + ) m m nˆ nˆ n ˆ m ( m ) ˆ 3 ( m ) ˆ ˆ m m : D dsretzaton The last epresson s the dsretzed form, that an be rewrtten n the followng was: Φ n m m n Usuall + Φ n [ Φ ( ) ω l ] { m n m } + Φ n3 + m n l ( + ) m m n ω so that the new dsretzed equaton s salar.

5 : D dsretzaton The last epressons an be obtaned also applng the fnte dfferene epressons to the epanded dfferental form, that s: Φ Φ Φ + + ( ( φ ω )) Φ ω That aproah leads to the same dsretzed epressons but entals more algebr operaton. : 3D dsretzaton The shfted voel, that s the losed surfae of the ntegral, espresson s entered n the voel s verte (nde ) for wh we are wrtng the dsretzed epresson. The 3D dsretzaton s a smple etenson of the D ase The means of the onutvtes are done on ells nstead than two ells as n the D ase. For ever entral verte (nde ), s neghbourng verte have to be onsdered Φ [ Φ ( ) ω l ] { m m } m J, Post-proessng ( ˆ + ˆ ), m where: Φ Φ l +,, m + ω m Φ Φ l, +, + ω m 3 ample of mplementaton: overvew and rtal aspets Dfferent post-proessng approahes are applable. 5

6 Bod models: dmensons, fles organzaton and memor oupaton Four model are avalable wth 58 reognzed tssues. One bte s requred for ever ell of the model. Cell s poston s dstlled from the bte poston. The struture of the fle s desrbed n fle s headng. Cell sze n mm ternal, medum, nternal le varable Model NUMX NUMY NUMZ n_ells Model memor oupaton [Mb] Head mm , Man 3mm ,3 Man mm , Man mm ,8 ample: a ube wth edges of ell, ells of mm 3 Bod models: wrtng and readng les If eternal le varable s Z, medum le varable s Y and nternal le varable s X, then the fle s wrtten and read wt a le lke for(nt z ; z<numz; z++) { for(nt ; <NUMY; ++) { for(nt ; <NUMX; ++) { } } } In that ase the fle s organzed n aal setons Soures models: wred models and numeral ntegraton The soure potental s related to the soure urrents b: s µ J s That has the soluton (onsderng the ondutor thkness neglgble): ( Q) π s µ I dc Q P Condutor The latter epresson an be ntegrated numeral. If a ondutor s represented b a segmented lne, the analtal soluton ests. urrent I flowng along the porton of the z as, between z a and z b generates a vetor potental n Q gven b: ( z b) ( z b) ( z a) ( z a) µ I s (,, z) ln + + ˆ z π + + f a and b + f a and b + µ I s (,, z) ln( ) + ˆ µ I z s (,, z) ln( + z z) ˆ z π π The epresson ror arbtrar orented elements an be derved usng rotaton and translaton Soures model: multphase ase Sne the dsretzed equaton has real oeffents, the etaton due to the n-phase and quadrature omponents of the soure an be omputed separatel. p + q Φ Φ p + Φ q The resultng solutons an then be ombned at the post-proessng stage to eld the omple ndued feld and denst urrent. J + Φ p ω + Φ q ω [ ] [( ) ( )] p q That aproah s usuall faster than mplementng the method for omple numbers (unless s avalable a fast omple arthmet tool). p q

7 m n + Salar potental boundar ondtons on the borders of the onsdered volume n + n3 + n ω m n l ( + ) m m Salar potental boundar ondtons on the borders of the onsdered volume Omogeneous ube wth s,5 S/m eposed to a unform magnet feld of mt. Dfferent shell s thkness were appled: Salar potental values n ondutve regons (borders nluded) are not nfluened b values n not ondutve regons. The bod model an be norporated n a shell of vauum wth thkness of l. The shell s added for omputatonal reason: we don t wont to go outsde the dsretzed domne border durng teraton. Salar potental on the border s fed (usuall to ) and teraton stops one ell before. l 3 l l m + Implementaton of the 3D method :memor patterns + ω l m5 5 + m m ( + + ) m m For ever node are needed: 7 offents for the salar potental values at the nodes (an be redued to thanks to the struture of the nodes dsposton) known term salar potental value Dfferent approahes an be used to memorze and reall those oeffents durng the problem resoluton. ompromze between speed and memor oupaton has usuall to be founded. m5 m5 m m Implementaton of the 3D method :memor patterns ample of memor pattern on a 3 bt platform usng double preson: amount of memor for ever grd node. double for ever oeffent (8 oeff. 5 f the smmetr of the matr s taken nto aount and n partular that s s s m for the neghbourng ell). double for the unknown salar potental value n the ell 8 btes for ever ell Other patterns an be used and the number of btes needed for ever ell an be dereased to 5 Model n n nz n_ells Model memor oupaton [Mb] Demanded re[mb] Head mm , Man 3mm ,3 Man mm , Man mm ,8 78 7

8 Computaton tme Omogeneous ondutve ube eposed to unform magnet feld of mt (Hz) n n nz kells teratons tme [s] 7 3 3, , ,8 5, , , , ,78 VHP head ( mm) model eposed to a unform magnet feld of mt ( Hz) more than hours. 8