UNIVERSITÀ DEGLI STUDI DI CAGLIARI

Transcription

1 UNIVERSITÀ DEGLI STUDI DI CAGLIARI Dottoato di iceca in INGEGNERIA ELETTRONICA ED INFORMATICA XV Ciclo Implementation in the ANSYS finite element code of the electic vecto potential T-Ω,Ω fomulation and its validation with the magnetic vecto potential A-V,A fomulation. Relatoi Pof. Ing. Alessanda Fanni Tesi di dottoato di Dott. Ing. Pieto Testoni

2 Implementation in the ANSYS finite element code of the electic vecto potential T-Ω,Ω fomulation and its validation with the magnetic vecto potential A-V,A fomulation. Pieto Testoni Gennaio 2003

3 To my mothe, my siste and my bothes

4 Acknowledgements Acknowledgements I wish to expess my deepest gatitude to Pofesso Piegiogio Sonato fo poposing and supevising this wok and fo his appeciated suggestions. I would also wamly thank D. John Swanson fo his vey helpful coespondence about the ANSYS pogammable capabilities. Last, but not least, I would like to thank my family fo always suppoting me.

5 Contents Contents Intoduction. Fomulations fo the eddy cuent poblem 3. Poblem definition 3.2 The T-Ω, Ω fomulation 6.3 The A,V-A fomulation.4 The A,V-Ψ fomulation 5.5 The A,V-A-Ψ fomulation The T-Ω, Ω finite element fomulation Finite element epesentation of the scala Ω and vecto T potentials The Galekin s method Discetization of the T-Ω, Ω fomulation equations Assembling element equations Tansient and hamonic analysis Bounday conditions Deived fields fom the degee of feedom solution Implementation in ANSYS of the T-Ω, Ω fomulation The Uec outines The Uel outines Validation of the implementation of the T-Ω,Ω fomulation Conducting cube in homogeneous magnetic field 43

6 Contents 4.2 Hollow sphee in unifom field The FELIX bick Conclusions 60 Refeences 6

7 Intoduction Intoduction Accoding to Faaday s law a time vaying magnetic field poduces an induced electic field which causes an electic cuent to flow in a conducting mateial. If the conducting mateial is not a wie filament, but athe a massive conducting body this cuent is distibuted inside the conducto and is efeed to as eddy cuent. Eddy cuents occu in a wide ange of electomagnetic appaatus and devices (synchonous machines, tansfomes, fusion machines) and have seveal applications (magnetic levitation, non destuctive testing, biomedicine, induction heating ). The effect of the eddy cuents can be beneficial o dangeous. Fo example, they ae necessay fo the opeation of the electic machines, such as tansfomes, but eddy cuents also cause hamful effects in them: the ciculating cuents dissipate enegy though ohmic losses and they geneate a eaction magnetic field in the opposite diection of the inducting field leading to a decease in the machine efficiency. Also, eddy cuents geneate a nonunifom cuent distibution in the conductos coss section causing additional losses. On the othe hand, eddy cuents can have beneficial effects: they can be used in metallugy fo induction heating by using the heat geneation fo melting metal objects o to contol the motion of molten metals. They have application in the magnetically levitated tains and can be used on non destuctive testing to get infomation about the mateials homogeneity and dimension they flow though. A good knowledge of space and time distibution of both electic cuent and magnetic flux is necessay in ode to obtain efficient and economical designs of all the appaatus whee eddy cuents ae involved. The inteaction between electical and magnetic field is descibed by the Maxwell s equations. Basically, thee ae two ways to solve this system of equations: analytical and numeical methods. Unfotunately, analytical solutions ae possible only fo poblems involving simple two-dimensional geometies, linea media and steady state poblems. Thee dimensional, nonlinea and tansient poblems ae vey difficult to be solved analytically and algoithms exist only fo specific poblems [0]. The limitations of analytical methods can be ovecome by using numeical techniques, which have been developed in the last yeas with the digital computing advent. Numeical methods can handle any geometical configuation, time-depending poblems and both linea and non-linea cases. The numeical appoach consists of dividing the field

8 Intoduction egion in discete elements and e-fomulating the poblem in each element. Thee ae seveal numeical methods: the finite diffeence method, the bounday element method, the moment method, and so on. The most poweful is the finite element method because of its majo flexibility. Basically, the solution of the electomagnetic poblem can be found e-fomulating the Maxwell s equations in tems of potentials. Potentials ae auxiliay functions that expess vaious physical popeties in ways leading a moe wokable solution to a poblem involving patial diffeential equations. The potentials can be chosen in a wide vaiety of ways, and this choice will affect the computational pocedue of the field poblem leading advantages and disadvantages. Numeical analysts, mathematicians and enginees ae involved in the study of vaious fomulations in ode to constuct efficient and economical numeical algoithms since the system of equation to be solved is usually vey lage. Thee dimensional eddy cuent poblems can be mathematically fomulated in vaious ways. The vaiable to be solved may be a vecto potential, a scala potential o a combination of those. Seveal finite element commecial codes ae available on the maket. The fomulations they implement ae not eve geneally applicable to all kind of poblems and not eve ae the most convenient in tems of availability and powe of computing esouces. One of them is ANSYS, which is maybe the most poweful and wide used finite element commecial package. Fo the eddy cuent study, ANSYS implements the magnetic vecto potential fomulation which uses in the non-conducting egions thee degee of feedom, the magnetic vecto potential components, and adds an exta degee of feedom, the time-integated electic voltage, in the conducting egions. This is the most widespead fomulation in all the electomagnetic finite element commecial packages. This wok deals with the implementation of a new fomulation (the T-Ω,Ω) in ANSYS by using its customization capabilities fo ceating new element types and adding them in the ANSYS libay. The goal of this wok has been to implement in ANSYS a simple and economical method fo calculating 3-D eddy cuents educing the numbe of degees of feedom, fom thee to one, in the non-conducting egions. On the othe hand, the implementation of the new fomulation in ANSYS allows to take advantage of the excellent and seveal capabilities of the commecial code itself: mesh geneation, post-pocessing, gaphics window, optimization, coupling and so on. 2

9 Cap. Fomulation fo the eddy cuent poblem. Fomulation fo the eddy cuent poblem. Poblem definition The inteaction between magnetic fields and electical phenomena is descibed by the following subset of Maxwell s equation: H = J [..] B E = t [..2] B =0 [..3] In eq. [..] the displacement cuent tem is neglected; in fact, if the dimension of the eddy cuent egions ae small compaed with the wavelength of the pescibed fields, the D / t tem can be neglected because it is sufficiently small. The fist equation is the Ampee s law, whee H is the magnetic intensity of the field poduced by a cuent density J, the second is the Faaday s law, whee E is the electic field poduced by a changing magnetic flux density B and the thid is the Gauss s law fo magnetism, which eflects the fact that thee ae no point souce of magnetic flux. The field vectos ae not independent since they ae futhe elated by the mateial constitutive elationship: B = µ H [..4] J = σ E [..5] whee µ and σ ae the mateial pemeability and conductivity. They may be field dependent and may vay in space, hysteesis and anisotopy ae hee neglected. Fo the solution of the poblem we conside also the continuity condition: which states the cuent density solenoidality. J = 0 [..6] A typical eddy cuent poblem consists (Fig...) of an eddy cuent egion Ω with non zeo conductivity σ and magnetic pemeability µ bounded by a suface S 2 and a suounding egion Ω 2 fee of eddy cuents, which may contain souce cuents J s, bounded by a suface S o which may be extended to infinity. The whole poblem domain is done by the sum of Ω and Ω 2 and will be denoted by Ω 0. The S o suface can be divided into two pats in accodance of the two types of bounday condition of pactical impotance: on S B the nomal component of the flux density is pescibed, on S H the tangential component of the 3

10 Cap. Fomulation fo the eddy cuent poblem magnetic field intensity is pescibed. S B S H Ω 2 µ 2 J s Ω σ µ j s S 2 S o Fig... Electomagnetic field egions The Ω 0 bounday conditions ae elated to the field values of the components nomal o tangential to the boundaies: B nˆ = 0 on S B [..7] H nˆ = 0 on S H [..8] Of couse also the inteface conditions between the conductive and non conductive media must be satisfied: B nˆ ˆ + B2 n2 H n + H nˆ = ˆ 2 2 = 0 0 on S 2 [..9] on S 2 [..0] J nˆ = 0 on S 2 [..] whee nˆ is the oute nomal on the coesponding suface. These inteface conditions can be easily found expessing eq. [..], [..3] and [..6]: H = J [..] in integal fom espectively: B =0 [..3] J = 0 [..6] H dl = I [..2] C S B ds = 0 [..3] 4

11 Cap. Fomulation fo the eddy cuent poblem S J ds = 0 [..4] The fist one states that the integal of H along a closed path C equals the cuent I enclosed in it, the othe equations state that the flux of B and J ove any closed suface S is zeo. We conside the inteface between two mateials and 2 with diffeent popeties and conside the ectangula closed path in Fig...2. By applying eq. [..2], as the height of the ectangle goes to zeo thee will be no contibution fom the vetical sides and eq. [..0] must be veified if the suface cuent density is neglected. 2 ˆn 2 ˆn 2 Fig...2 Magnetic field at a bounday Fig...3 Magnetic field and cuent density flux at a bounday Also, if we conside the disk in Fig...3, by applying eq. [..3] and [..4] as its height goes to zeo no flux leaves the volume though the cylinde side and eq. [..9] and [..] must be veified. Paticula cae must be taken in poblems involving multiply connected egions. A egion Λ is defined as simple connected if any line l belonging to Λ can be educed in a point by a continuous defomation of l itself. In othe wods, if any closed line l in Λ has an open suface which belongs to Λ and has l as contou. If it is not, the egion Λ is called as multiply connected. The cylinde in Fig...4 epesents a simple connected egion, while the tous in Fig...5 is a multiply connected egion. Λ Λ Fig...4 Simply connected egion Fig...5 Multiply connected egion 5

12 Cap. Fomulation fo the eddy cuent poblem.2 The T-Ω, Ω fomulation Maxwell s equations ae fist ode coupled diffeential equations which can be vey difficult to solve in bounday values poblems. The way fo educing mathematical complexity is theefoe to fomulate poblems in tems of potentials. Potentials ae auxiliay functions, which ae used fequently fo calculating electomagnetic fields. They, in fact, pemit the constuction of efficient and economical numeical algoithms. One of the widespead fomulations is the A,V A fomulation, which uses the magnetic vecto potential A both in Ω and Ω 2 as well the electic scala potential in Ω. It is defined expessing the flux density in tems of an auxiliay vecto A : B = A [.2.] The A,V A fomulation togethe with othe useful fomulations will be descibed in detail in the next paagaphs. This wok deals with the T-Ω, Ω fomulation, which has the advantage to pemit a eduction of computing cost by deceasing the degees of feedom fom thee to one in all the nonconducting egion. Stating fom [..6 ] J = 0 [..6] we can expess the cuent density in tems of an auxiliay vecto T, which is called the electic vecto potential: J = T. [.2.2] by using the well known vecto identity T = 0, whee T is any sufficiently diffeentiable vectoial function. We note that fom eq. [..] H = J as well, so T and H diffe by the gadient of a scala and have the same units: H = T Ω in Ω [.2.3] whee Ω is a magnetic scala potential. Using [..2 ], [..4] and [..5]: B E = [..2] t B = µ H [..4] 6

13 Cap. Fomulation fo the eddy cuent poblem we obtain: and fom [..3]: J = σ E [..5] T + µ σ t ( T Ω) = 0 in Ω [.2.4] we obtain: B =0 [..3] ( Ω) = 0 µ T in Ω [.2.5] It should be noted that by taking the divegence of both sides of [.2.4] the solenoidality of the magnetic flux density [.2.5] is satisfied and would be at this point supefluous. In cuent-fee egions the magnetic field can be found fom the scala potential: whee Ω esults fom [.2.5]: H = Ω in Ω 2 [.2.6] µ Ω = 0 in Ω 2 [.2.7] The potential T is not yet fully defined because a vecto field can be uniquely defined only defining its divegence and its cul, and by detemining suitable bounday conditions on the inteface between conducting and non conducting egions. The divegence of T is not yet defined and consequently T and Ω emain ambiguous. Defining the divegence of T in addition to its cul is efeed to as a choice of gauge. Any of this value may be chosen without affecting the physical poblem. On the othe hand this choice will affect the computation ease. The best known gauge condition used in electomagnetics is the Coulomb gauge: This condition pemits to append [2] the left-side of [.2.4] by a tem T σ : T = 0 [.2.8] T + µ σ t ( T Ω) = 0 in Ω [.2.4] Taking the divegence of [.2.9]: T T + µ σ σ t ( T Ω) = 0 in Ω [.2.9] 7

14 Cap. Fomulation fo the eddy cuent poblem 2 T T + ( µ ( T Ω ) = 0 in Ω [.2.0] σ σ t The fist tem of [.2.0]is zeo fo any vecto, the thid is zeo as well consideing [..3], so the scala T σ satisfies Laplace s equation 2 B =0 [..3] T = 0 σ in Ω [.2.] If we set the bounday condition T = 0 on the inteface between the conducting and σ non-conducting media, then fom [.2.] we conclude that T = 0 in Ω. This satisfies σ the gauge condition in the conducto. The bounday condition of the electic vecto potential [7] is detemined by [..] J nˆ = 0 [..] and can be expessed as: nˆ T = 0 [.2.2] By taking the divegence of both sides of [.2.9] the solenoidality of the magnetic flux density is no moe satisfied and it must be enfoced. The inteface conditions [..9], [..0] B nˆ ˆ + B2 n2 H n + H nˆ = 0 ˆ 2 2 = 0 [..9] [..0] must be satisfied in the inteface suface between the conducting and non conducting media. Substituting equations [..4 ], [.2.3 ], [.2.6 ] B = µ H [..4] H = T Ω [.2.3] in [..9] and [..0] gives: H = Ω [.2.6] ( T ) ˆ ( ) ˆ Ω n + µ Ω n2 = 0 ( Ω ) n + ( Ω ) nˆ 0 µ on S 2 [.2.3] ˆ 2 = Also the bounday conditions [..7] and [..8] T on S 2 [.2.4] B nˆ = 0 on S B [..7] 8

15 Cap. Fomulation fo the eddy cuent poblem H nˆ = 0 on S H [..8] must be satisfied, by using [.2.3 ], [.2.6 ] H = T Ω in Ω [.2.3] the fist one can be witten as: H = Ω in Ω 2 [.2.6] ( Ω) nˆ = 0 µ T on S B [.2.5] ( Ω ) nˆ = 0 µ on S B [.2.6] fo conducting and non-conducting egions espectively, the second one is: ( Ω) nˆ = 0 T on S H [.2.7] ( ) nˆ = 0 Ω on S H [.2.8] fo conducting and non-conducting egions espectively. In the discetization by the finite element method [.2.3] is satisfied implicitly, the continuity of the scala potential togethe with [.2.2] ensues that [.2.4]is satisfied too. The diffeential equations [.2.5], [.2.7] and [.2.9], ( Ω) = 0 µ T in Ω 2 [.2.5] µ Ω = 0 in Ω 2 [.2.7] T T + µ σ σ t ( T Ω) = 0 in Ω [.2.9] the inteface conditions[.2.3] and [.2.4], ( T ) ˆ ( ) ˆ Ω n + µ Ω n2 = 0 ( Ω ) n + ( Ω ) nˆ 0 µ on S 2 [.2.3] ˆ 2 = T on S 2 [.2.4] 9

16 Cap. Fomulation fo the eddy cuent poblem the bounday conditions [.2.5], [.2.6], [.2.7] and [.2.8] ( Ω) nˆ = 0 µ T on S B and σ 0 [.2.5] ( Ω ) nˆ = 0 µ on S B and σ=0 [.2.6] ( Ω) nˆ = 0 T on S H and σ 0 [.2.7] ( ) nˆ = 0 Ω on S H and σ=0 [.2.8] ae the T-Ω, Ω fomulation equations. 0

17 Cap. Fomulation fo the eddy cuent poblem.3 The A,V-A fomulation The A,V A fomulation uses as degees of feedom the magnetic vecto potential A both in Ω and Ω 2 as well the electic scala V potential in Ω. Stating fom [..3] B =0 [..3] we can expess the flux density in tems of an auxiliay vecto A : B = A [.3.] by using the well known vecto identity A = 0. With this substitution equation [..2] B E = [..2] t becomes: A E+ =0 [.3.2] t and using the second vecto useful identity V = 0, whee V is any sufficiently diffeentiable scala function, we can wite: A E= V [.3.3] t The vaiables A and V ae usually efeed to as the magnetic vecto potential and the electic scala potential. Substituting [.3.] and [.3.3] in [..], [..4] and [..5] H = J [..] B = µ H [..4] J = σ E [..5] esults: A ν A + σ + V = 0 in Ω [.3.4] t ν A = in Ω2 [.3.5] The equation [.3.4] automatically satisfies the divegence-fee popety of the cuent density in Ω and would be supefluous to specify it. The bounday conditions [..7], [..8], [..9], [..0] J s

18 Cap. Fomulation fo the eddy cuent poblem B nˆ = 0 on S B [..7] H nˆ = 0 on S H [..8] B ˆ ˆ n + B2 n2 = 0 on S 2 [..9] H n + H nˆ 0 on S 2 [..0] ˆ 2 2 = ae now fomulated in tems of the magnetic vecto potential as: nˆ A = 0 on S B [.3.6] ν A nˆ = 0 on S H [.3.7] A + nˆ A 0 on S 2 [.3.8] nˆ 2 2 = ν A n + ν A nˆ 0 on S 2 [.3.9] ˆ 2 2 = Having A itself continuous between the two egions ensues the automatic satisfaction of the inteface conditions [.3.8] and [.3.9]. To fully define the potential A we have to define the divegence of A in addition to its cul. Also in the A,V A fomulation [3] the Coulomb gauge is used : A = 0. This condition pemits to append the left-side of [.3.4] and [.3.4] A ν A + σ + V = 0 t in Ω [.3.4] ν A = J s in Ω2 [.3.5] by a tem ν A : A ν A ν A + σ + V = 0 t in Ω [.3.0] ν A ν A = in Ω 2 [.3.] The divegence-fee popety of the cuent density is not automatically satisfied, and must be enfoced. Substituting [.3.3] in [..5] and taking the divegence we obtain: A E= V t J s [.3.3] J = σ E [..5] 2

19 Cap. Fomulation fo the eddy cuent poblem A σ σ V = 0 in Ω [.3.2] t Moeove, also the fact that the nomal component of the cuent density along the inteface suface is zeo must be enfoced, substituting [.3.3] and [..5]in[..] J nˆ = 0 [..] gives: A n σ σ V = 0 t in Ω [.3.3] Taking the divegence of [.3.0] and [.3.], consideing [.3.2] and the fact that J s is divegence-fee we have: 2 ( A ) = 0 ν in Ω [.3.4] To ensue the Coulomb gauge satisfaction in the whole egion Ω o pope bounday conditions must be enfoced on ν A along the bounday S o. O. Bio and K. Peis have shown in [3] that the satisfaction of the Coulomb gauge can be ensued by imposing: and: n ( A) = 0 ν on S H [.3.5] ν A = 0 on S B [.3.6] The uniqueness of the solution depends also by the imposition of suitable bounday conditions on A itself along S o. The bounday condition [.3.6] nˆ A = 0 can be eplaced by the: n ˆ A = 0 on S B [.3.6] on S B [.3.7] The bounday condition[.3.7] on S H is then eplaced by: ν A nˆ = 0 on S H [.3.7] nˆ A = 0 on S H [.3.8] 3

20 Cap. Fomulation fo the eddy cuent poblem On summay, the diffeential equations [.3.0], [.3.] and [.3.2], A ν A ν A + σ + V = 0 in Ω [.3.0] t ν A ν A = J s in Ω 2 [.3.] A σ σ V = 0 in Ω [.3.2] t the bounday conditions [.3.5], [.3.6], [.3.7] and [.3.8] n ν A = 0 on S B [.3.6] n ˆ A = 0 on S B [.3.7] ( A) = 0 ν on S H [.3.5] nˆ A = 0 on S H [.3.8] the inteface condition [.3.8], [.3.9] and [.3.3] nˆ A ˆ + n2 A2 = 0 on S 2 [.3.8] ν A ˆ ˆ n + ν A2 n2 = 0 on S 2 [.3.9] A n σ σ V = 0 in S 2 [.3.3] t ae the A,V-A fomulation equations. 4

21 Cap. Fomulation fo the eddy cuent poblem.4 The A,V-Ψ fomulation The A,V-Ψ fomulation pemits to educe the computational cost of the poblem solution by using a scala magnetic potential in the fee of eddy cuents egion Ω 2. In this egion, in fact, it is possible to split the magnetic field intensity vecto H into two pats: H = H s + H m [.4.] whee H s is the field geneated by the souces cuent in Ω 2 and satisfies [..] H = J s [..] thus H is iotational m H =0 [.4.2] and can be expessed as the gadient of the educed magnetic scala potential φ : m H = φ [.4.3] m and so [.4.] becomes: H H = s φ [.4.4] The field H s can be computed in any point by using the Biot-Savat law: J s H s = dvols vols 3 4π [.4.5] P Q dvol s J s Whee (Fig..4.): position vecto fom the cuent souce point Q to the node point P J cuent souce density vecto in d ( ) s Fig..4. Biot Savat law application vol s vol s volume whee the cuent souce is defined. 5

22 Cap. Fomulation fo the eddy cuent poblem Substituting [.4.5] in [..4] B = µ H [..4] and taking the divegence gives: µ φ = µh s [.4.6] Eq. [.4.6] is a genealized fom of the Poisson equation. In magnetic mateials the two pats of the field H s and H m tend to be of simila magnitude but opposite in diection, so that cancellation occus in computing the field intensity H, giving a loss in accuacy, especially when the magnetic pemeability µ is lage [4]. This poblem can be ovecome if the feomagnetic egion is fee of souce cuents. In this case, in fact, the total field can be epesented by a scala potential called as the total magnetic scala potential Ψ since [..]: H = [..] can be witten as: J s and thus: Substituting [.4.8] in [..4] and taking the divegence gives: H =0 [.4.7] H = Ψ [.4.8] B = µ H [..4] µ Ψ = 0 in Ω 2 [.4.9] Eq. [.4.9] is a genealized fom of the Laplace equation. In the egion Ω which is descibed by the magnetic vecto potential [.3.4] is still valid A ν A + σ + V = 0 in Ω [.3.4] t Again, the eq. [.3.4] automatically satisfies the divegence-fee popety of the cuent density in Ω and would be supefluous to specify it. The bounday conditions [..7], [..8] can now be specified in tem of the scala potential: B nˆ = 0 on S B [..7] H nˆ = 0 on S H [..8] ( Ψ ) nˆ = 0 µ on S B [.4.0] 6

23 Cap. Fomulation fo the eddy cuent poblem which can also be expessed as: ( ) nˆ = 0 The inteface conditions [..9] and[..0] B nˆ ˆ + B2 n2 H n + H nˆ Ψ on S H [.4.] Ψ = 0 on S H [.4.2] = 0 ˆ 2 2 = 0 on S 2 [..9] on S 2 [..0] ae now fomulated in tems of the magnetic vecto and the magnetic scala potentials as: nˆ ˆ A n2 µ 2 ( Ψ ) = 0 on S 2 [.4.3] ν A n Ψ nˆ 0 on S 2 [.4.4] ˆ 2 = As fo the A,V-A fomulation the uniqueness of the solution depends by the imposition of suitable bounday conditions on A itself along the bounday and by a choice of gauge. The same bounday conditions that have been found fo the A,V-A fomulation can be hee applied, howeve it has to be taken into account that A is defined only in egion Ω and its bounday S 2 is the inteface suface between the conducting and non conducting media. It can be noted that fom [.4.4] it is possible to define the tangential component of H and [.4.3] can be teated as a bounday condition fo the magnetic scala potential in Ω 2. This means that the inteface suface must be teated as S H in the A,V-A fomulation whee [..8] H nˆ = 0 on S H [..8] must be applied. Theefoe, to ensue the uniqueness of A, the nomal component of A itself must be pescibed on the inteface suface and the divegence of A must be enfoced to be zeo in Ω (Coulomb gauge). By ewiting [.3.0] and [.3.2] A ν A ν A + σ + V = 0 t in Ω [.3.0] A σ σ V = 0 t in Ω [.3.2] Taking the divegence of [.3.0], consideing [.3.2] we have: 2 ν A = in Ω [.3.4] ( ) 0 Taking the nomal component of [.3.0] along the inteface suface yields to: 7

24 Cap. Fomulation fo the eddy cuent poblem A n ν A ν A + n σ + V = 0 on S 2 [.4.5] n t the fist tem is zeo by using [.3.7] ν A nˆ = 0 on S H [.3.7] and because it can be eaanged [3] as n ν A = on S 2 [.4.6] the thid tems is zeo fom [.3.3] A n σ σ V = 0 t and so[.4.5]can be ewitten as: ( n ν A) in Ω [.3.3] ν A = 0 on S 2 [.4.7] n Eq. [.3.4] with [.4.7] ensues that ν A = cost in Ω, since the nomal component of A is supposed to be zeo on S 2, this constant can only be zeo. This ensue the Coulomb gauge satisfaction. On summay, the diffeential equations [.3.0], [.4.9] and [.3.2], A ν A ν A + σ + V = 0 in Ω [.3.0] t µ Ψ = 0 in Ω 2 [.4.9] A σ σ V = 0 in Ω [.3.2] t the bounday conditions [.4.0], [.4.2], ( Ψ ) nˆ = 0 µ on S B [.4.0] Ψ = 0 on S H [.4.2] 8

25 Cap. Fomulation fo the eddy cuent poblem the inteface conditions [.3.3], [.3.8], [.3.7] and [.3.8] nˆ ˆ A n2 µ 2 = ν A ˆ ˆ n Ψ n2 = 0 A n σ σ V = 0 t ( Ψ ) 0 in S 2 [.4.3] in S 2 [.4.4] in S 2 [.3.3] nˆ A = 0 on S 2 [.3.8] ae the A,V-Ψ fomulation equations. 9

26 Cap. Fomulation fo the eddy cuent poblem.5 The A,V-A-Ψ fomulation The magnetic scala potential can not be used in the non conducting egion if the conducting egion is multiply connected. This poblem can be ovecome if in the non conducting holes the magnetic vecto potential fomulation is used. In Fig..5. the conducting egion Ω is multiply connected and egion Ω 3, whee the magnetic vecto potential fomulation hold, is selected to have in conjunction with Ω a simple connected egion. S o S B S 2 Ω 2 µ 2 J s S 2 S H Ω σ µ j s S 23 S 3 S 23 Ω 3 A Ω σ µ j s S 23 S 3 Fig..5. Electomagnetic field egions with multiply connected conducto So, the magnetic scala fomulation can be used in the non conducting egion Ω 2, which now suounds a simple connected egion done by the sum of Ω and Ω 3. The above consideations yield in a mixed fomulation combining the A,V-A and the A,V-Ψ fomulations and the equation to be solved came fom them. This equations ae hee summaized. 20

27 Cap. Fomulation fo the eddy cuent poblem The diffeential equations [.3.0], [.3.], [.4.9] and [.3.2], A ν A ν A + σ + V = 0 t in Ω [.3.0] A σ σ V = 0 t in Ω [.3.2] µ Ψ = 0 in Ω 2 [.4.9] ν A ν A = 0 in Ω 3 [.3.] the bounday conditions [.3.5], [.3.6], [.3.7], [.3.8], [.4.0] and [.4.2], n ν A = 0 on S B3 [.3.6] n ˆ A = 0 on S B3 [.3.7] ( A) = 0 ν on S H3 [.3.5] nˆ A = 0 on S H3 [.3.8] ( Ψ ) nˆ = 0 µ on S B2 [.4.0] Ψ = 0 on S H2 [.4.2] the inteface conditions [.3.3], [.3.8], [.3.7] and [.3.8] 2

28 Cap. Fomulation fo the eddy cuent poblem nˆ A ˆ + n2 A2 = 0 on S 2 [.3.8] ν A ˆ ˆ n + ν A2 n2 = 0 on S 2 [.3.9] nˆ ˆ 3 A n2 µ 2 ( Ψ ) = 0 on S 2 and S 23 [.4.3] ν A ˆ ˆ n3 Ψ n2 = 0 on S 2 and S 23 [.4.4] A n σ σ V = 0 on S 2 and S 3 [.3.3] t nˆ A = 0 on S 2 and S 23 [.3.8] ae the A,V-A-Ψ fomulation equations. 22

29 Cap.2 The T-Ω, Ω finite element fomulation 2. The T-Ω, Ω finite element fomulation The finite element method is a numeical appoach by which geneal diffeential equations can be solved in an appoximate manne. It s a chaacteistic featue of the finite element method to divide the field egion whee the diffeential hold into a numbe of finite elements which have the same dimension (one, two o thee-dimensional) of the poblem to be solved (Fig. 2..). Each element has a cetain numbe of nodes which ae often located at its boundaies. The collection of all elements and nodes is called as the finite element mesh. nodes Poblem egion elements Fig. 2.. Poblem egion discetization Instead of seeking the appoximations that hold diectly the whole egion, the appoximation is caied out ove each element. The unknown field quantity is epesented within each element as a combination of intepolatoy functions known as shape functions. Such functions ae chosen in a elative simple fom by using polynomial expansions. In fact, polynomial functions give an excellent desciption of the field unde investigation, simultaneously pemitting the pocess of integation and diffeentiation. A elationship involving the unknown field quantity at the nodal points is then obtained fo the poblem fomulation fo a typical element. The numbe of nodes fo each element define the vaiation law of the shape functions: linea, quadatic, cubic. The next step is the assembly of the element equations to obtain the equations fo the oveall system. The imposition of the bounday conditions leads to the final system of equations to be solved. The solution can be achieved by minimizing a functional with espect to each of the nodal potentials o applying diectly the Galekin [3] pocedue. Once the poblem of finding the potential in each 23

30 Cap.2 The T-Ω, Ω finite element fomulation element node is solved it is possible to compute othe desied quantities and epesent them in tabula o gaphical fom. 2. Finite element epesentation of the scala Ω and vecto T potentials The finite element implementation of the T-Ω, Ω fomulation has been caied out by using fou-noded, fist-ode, tetahedal elements. A tetahedal element in the global x, y, z system is shown in Fig The numbes on the element indicate the local numeation of the nodes. Inside each element the magnetic scala potential Ω and the electic vecto potential T can be expessed by a linea combination of the shape functions associated with the nodes. 3 y 4 2 z x Fig. 2.. Finite tetahedal element Within an element the scala potential Ω is appoximated as: Ω = m i= Ωi Ni [2..] whee Ni is the nodal shape function coesponding to node i. The index m is the numbe of the element nodes and m = 4 fo tetahedal elements. The coefficient Ωi is the degee of feedom and it is the value of the magnetic scala potential Ω on the node i. The electic scala potential T is teated as thee scala components, T x, T y, T z in the Catesian co-odinate system. Each node then has thee degees of feedom instead of one. In each element the electic vecto potential T can be appoximated as: 24

31 Cap.2 The T-Ω, Ω finite element fomulation m m T = Ti Ni = i= i= ( Txi x + Tyi y + Tzi z ) Ni [2..2] whee the coefficient T i is the value of the electic vecto potential T on the node i, Txi, Tyi, Tzi ae the components of T i. Fo a tetahedal element the shapes functions ae defined as: ai + bi x + ci y + di z Ni = 6 vol whee all the paametes depend on the element node co-odinates. [2..3] The element volume multiplied by six [5] is given by the deteminant of the coefficient matix: 6 vol = x x2 x3 x4 while the a, b, c, d constants can be obtained calculating the cofactos of the coefficient matix: y y2 y3 y4 z z2 z3 z4 a = x2 x3 x4 y2 y3 y4 z2 z3 z4 b = y2 y3 y4 z2 z3 z4 c = x2 x3 x4 z2 z3 z4 d = x2 x3 x4 y2 y3 y4 the othe coefficients ae deived by cycling intechange of the subscipts. 2.2 The Galekin s method The finite element method uses vaiational fomulations o weighted esidual methods to solve bounday-value poblems. In the vaiational method the patial diffeential equations of the field poblem is fomulated in tems of an equivalent enegy elated expession called a functional, which in some applications may epesent the stoed enegy o the dissipated powe in the system. This functional has the popety of being stationay with the coect set of function epesenting the equied solution of the poblem. The solution of the field poblem is then obtained by minimizing the functional with espect to a set of tial solutions. 25

32 Cap.2 The T-Ω, Ω finite element fomulation The weighted esidual method is a moe geneal and a moe univesally applicable method than the vaiational appoach because thee ae a wide numbe of cases in which the vaiational expession does not exist and the weighted esidual method can be applied. The Galekin s method is a special case of the weighted esidual methods; in such kind of methods a weighted eo on the solution domain has to be minimized. Let us assume that the govening patial diffeential equation of the electomagnetic system we ae dealing with is of the fom: Lu = f [2.2.] whee L is a diffeential opeato, f is a known function (the excitation) and u is the unknown function. When L is chosen, it specifies the actual fom of the diffeential equation given by [2.2.]. In the weighted esidual methods the unknown function u is appoximated in tems of a linea combination of n functions called as basis o tial functions. u u app = n i= Φ i qi [2.2.2] Hee q qn ae unknown paametes and Φ Φn ae the tial functions, each defined in the domain of L. The equation [2.2.] may be taken as an element-wise appoximation, in this case q qn would be the values of u at the nodal points and the tial functios would become the shape functions: this is the application of the weighted esidual method to the finite element method. Substituting [2.2.2] in [2.2.] we obtain a esidual: since app u u R = Lu app f [2.2.3]. To detemine the coefficients q qn so that app u is a appoximation of u the esidual is foced to be zeo, in an aveage sense, by setting weighted integals of the esidual equal to zeo, i.e: w m, R = 0 [2.2.4] whee w (m = n) is a set of weighting functions and the notation m w m, R indicates an inne poduct defined as w m Rdv = 0 in the domain solution. Assuming L is linea and substituting [2.2.2] and [2.2.3] in [2.2.4] we obtain: n w, qilφ i f = 0 o m i= n i= qi w m, n i= LΦ i = w, f [2.2.5] m The set of equations given by [2.2.5] can be cast into a matix fom as: 26

33 Cap.2 The T-Ω, Ω finite element fomulation whee: [ K ] = w LΦ dv w2lφdv.. w LΦ dv n [K] [q] = [C] [2.2.6] w LΦ dv w2lφ 2dv.. w LΦ dv n w LΦ ndv w2lφ ndv.. w LΦ dv n n [ C] = w fdv w2 fdv.. w fdv n and [q] is the column vecto of the coefficients q qn. The equation [2.2.6] consists of n linea equations fom which the vecto [q] can be detemined, once [q] has been obtained eq. [2.2.2] povides the equied appoximate solution. A special case of the above descibed weighted esidual method is the Galekin s method in which the weighting functions ae chosen to be identical to the basis functions. 27

34 Cap.2 The T-Ω, Ω finite element fomulation 2.3 Discetization of the T-Ω, Ω fomulation equations The Galekin s fom of equations [.2.9], [.2.5] and [.2.7] T T + µ ( T Ω) = 0 [.2.9] σ σ t µ T Ω = [.2.5] ( ) 0 µ Ω = 0 [.2.7] is: vol N i T T + µ ( T Ω) dvol = 0 σ σ t [2.3.] vol N i µ ( T Ω) dvol = 0 [2.3.2] t vol N i µ ( Ω) dvol = 0 [2.3.3] t By using the divegence of coss poduct ule: P F = P F P F [2.3.4] ( ) ( ) ( ) whee P and F ae vecto functions of position, integating ove the volume and applying the divegence theoem to the left-hand side and eaanging, gives: P F dvol = P Fdvol P F nˆ vol ( ) ( ) ( ) ds vol whee S is a closed suface with outwad nomal nˆ. Making the substitution [2.3.] can be witten as: P Q dvol vol ( ) = ( P) ( Q) dvol ( P Q) nˆ ds vol S S [2.3.5] F = Q eq. [2.3.6] The fist tem on the left side of eq. [2.3.] can be ewitten fom [2.3.6] by substituting P with the shape functions N i and Q with T : σ vol N i T dvol = ( N i ) T dvol N i T nˆ2ds [2.3.7] σ σ σ vol The second tem on the left side of equation can be ewitten by using the Geen Gauss theoem: S2 28

35 Cap.2 The T-Ω, Ω finite element fomulation vol u Qdvol = vol ( u) Qdvol + uq nˆ ds S [2.3.8] whee u and Q ae sufficiently diffeentiable function of position. Substituting Q with N i and u with T gives: σ vol T N idvol = T N idvol + T N i σ σ σ and eaanging: vol S nds ˆ [2.3.9] vol N i T dvol = N i T dvol + N i nˆ Tds σ σ σ vol Eq. [2.3.]can now be ewitten as: vol S2 ( N ) T + ( N ) T + N µ ( T Ω) N i i σ nˆ T ds + σ S2 N i i σ T nˆ σ 2 i ds S2 t dvol [2.3.0] [2.3.] Equations [2.3.2] and [2.3.3] can be ewitten by using the vecto integation by pats ule: vol N vol i N µ ( T Ω) dvol + ( Ω) ˆ = 0 µ T n [2.3.2] t t µ ( Ω) dvol + ( Ω) ˆ = 0 n [2.3.3] t t i µ by imposing the inteface conditions on the nomal components of B and J and on the tangential H and the condition nˆ T = 0 on the inteface suface between the conducting and non-conducting media all the suface integals disappea and Eq. [2.3.], [2.3.2] and [2.3.3] can be ewitten as: du K u + D = 0 [2.3.4] dt if the tems multiplying the potentials ae collected in a matix K (the stiffness matix) and the tems multiplying the time deivatives of the potentials in a matix D (the damping matix); u is the vecto of the nodal values potentials. The name of these matix has its oigin in the stuctual and mechanical field, whee the finite element method was applied fistly. The method was not applied in electomagnetism until 968, in fact such kind of 29

36 Cap.2 The T-Ω, Ω finite element fomulation electomagnetic poblems equied not yet existing poweful calculus systems due to the fact that also the ai suounding the conductos has to be modelled. The [2.3.4] matix equation descibe the behaviou of each element independently of the othes. 2.4 Assembling element equations A equation system like [2.3.4] valid fo the whole egion whee the poblem is defined can be found by assembling togethe all the element equations and enfocing the conditions that adjacent elements ae connected ensuing the nodal potential continuity. To do that, we can ewite [2.3.4] fo the whole egion whee the poblem is defined: du g K gu g + D g = 0 [2.4.] dt whee the subscipt g indicates global paametes. The assembling matix pocess is best illustated by consideing a simple example (Fig. 2.4.) of a thee tiangula finite elements in a two dimensional egion Fig Assembly of thee element The numbeing of node, 2, 3, 4, 5 is called global numbeing. It is also possible to define, fo each element, a local element numbeing (Fig ) in counteclockwise sequence stating fom any node. 3 2 Fig Local numbeing of the element 30

37 Cap.2 The T-Ω, Ω finite element fomulation Fo example, fo element 2 in Fig. 2.4., the global numbeing, 3, 4 coespond to the local numbeing, 2, 3 of the element in Fig So, a helpful table can be built: Local Numbeing 2 3 Global Numbeing of element 4 2 Global Numbeing of element Global Numbeing of element Table 2.4. Local and global numbeing coespondence In this case, the dimension of the global vecto of the nodal values potentials u g is 5 and the global stiffness and damping matices have dimension 5x5 since five nodes ae involved. The global stiffness matix K can be witten as: g K + K K3 K2 K2 + K3 0 K 3 K 33 0 K 32 0 = K g K K 22 K K 23 K3 K [2.4.2] K 2 + K 3 K 23 K 32 + K 3 K 22 + K 33 + K 33 K K 2 K 23 K 22 The global matix coefficients can be found by using the fact that the potential distibution must be continuous acoss the element boundaies. Fo example, elements and 2 have node in common; hence: K g = K + K 2 Nodes and 4 belong simultaneously to elements and 2; hence: K g = K + K Since thee is no coupling between nodes 2 and 3: K g 23 = 0 The subscipt in the fist side of each matix tem indicates the nodal global numeation. In the second side, the supescipt indicates the involved element, while the subscipt indicates the local numeation. The global damping matix can be assembled in the same way. It can be shown that: 3

38 Cap.2 The T-Ω, Ω finite element fomulation - the global stiffness and damping matices ae symmetic - the global stiffness and damping matices ae spase and banded. 2.5 Tansient and hamonic analysis A tansient dynamic analysis is used to detemine the dynamic esponse of the system unde the action of any geneal time-dependent loads. It is possible to use this type of analysis to detemine the time-vaying quantities such as magnetic field intensity, the magnetic flux density and the cuent density in the poblem definition domain as they espond to any combination of static, tansient, and hamonic loads. The basic equation solved by a tansient dynamic analysis is eq. [2.4.] du g K gu g + D g = 0 [2.4.] dt At any given time, t, this equation can be thought of as a "static" equilibium equation. The ANSYS pogam uses a time integation method to solve this equation at discete timepoints. The time incement between successive time-points is called the integation time step. To specify loads and bounday conditions, it is possible to divide the load-vesus-time cuve into suitable load steps. Each "cone" on the load-time cuve is one load step, as shown in Fig Fig Load-time cuve By specifying moe than one load step, it is possible to step o amp the load itself : 32

39 Cap.2 The T-Ω, Ω finite element fomulation - If a load is stepped, then its full value is applied instantaneously - If a load is amped, then its value inceases gadually with the full value occuing at the end of the load step In the case of hamonic analysis the potentials can be consideed vaying sinusoidally with time and can be epesented by phasos. The vecto u g can be divided into a eal and an imaginay pat: u g = u + ju [2.5.] gre gim and eq. [2.4.] can be witten as: Ku jω Du = 0 [2.5.2] g + g whee ω is the fequency of the potential field vaiation. Substituting [2.5.] in [2.5.2] we can see that a hamonic analysis povides two sets of solution: the eal and imaginay pat of a complex solution. The measuable quantity ( x, y, z, t) can be descibed as the eal pat of u g the complex function: u g {( u ju ) e j ω + t } ( x, y, z, t) = RE [2.5.3] gre gim 2.6 Bounday conditions Electomagnetic fields ae aely tuly bounded, but in most cases they exist in an infinitely extending space. Howeve, in the solution of electomagnetic poblems bounday conditions can be imposed by using symmety conditions and inteface popeties between diffeent mateials. Fo example, an ion body in ai does not bound the magnetic field. But since the pemeability of the ion is much highe than the one of the ai, the flux lines in ai ae pependicula to the inteface suface between ion and ai. So, a bounday condition can be imposed in this inteface suface if only the field distibution in ai is of inteest. In othe cases, symmety lines o planes pemit to impose bounday conditions to have the magnetic field pependicula o paallel to them. Fo example in a fusion machine (Fig. 2.6.) the magnetic field is geneated by a cental solenoid and by some coils which suound the vacuum chambe. Since all the geomety is tooidal, an axis-symmetic model can be built. In the whole egion of the poblem the fields 33

40 Cap.2 The T-Ω, Ω finite element fomulation distibution can be descibed by the magnetic vecto potential A. Fig Magnetic field distibution in a fusion machine Since the cuent density posses only the component pependicula to the plane of the model, the magnetic vecto potential has only the component diected in the same diection of the cuent density. The magnetic vecto potential fomulation educes in this case to a Poisson equation: whee Az and ν A = [2.6.] z J z J z ae scala and ae the component of A and J in the tooidal diection. The vetical line bisects the coils and the cental solenoid, with exactly simila but oppositely diected cuent on its left and ight sides. It must theefoe be the sepaatix cuve which sepaates the flux lines belonging to the ight side fom the flux line belonging to the left side, so the magnetic field must be paallel to this line. A paallel magnetic field bounday condition must be enfoced in it, enfocing the fact that this line must be a flux line along which the vecto potential zeo, so that: Az has a fixed value. Fo convenience, this value may be taken to 34

41 Cap.2 The T-Ω, Ω finite element fomulation A = 0 [2.6.2] z Bounday conditions in which the potential is pescibed ae often efeed to as Diichlet conditions [6]. The bottom edge of the model is also a symmety line: the cuents in the coils ae diected in the same sense above and below this line, so the flux lines must be pependicula to this lines. A pependicula magnetic field bounday condition must be enfoced in it by imposing: A z = 0 [2.6.3] n whee n is the nomal to the suface. Bounday conditions in which the nomal deivative of the potential is pescibed ae often efeed to as Neumann conditions [6]. In the finite element solution and in this paticula case, this type of bounday condition is called as natual because it does not need to be specified by the use. Consideing Fig , it is possible to summaize how to apply Diichlet and Neumann bounday conditions in thee-dimensional poblem egions involving the magnetic vecto o scala potential fomulations. Y X Z Fig Bounday conditions on a thee dimensional egion Diichlet bounday condition. 35

42 Cap.2 The T-Ω, Ω finite element fomulation This is a pependicula magnetic field condition. It consists in imposing the following condition: - in the case of the magnetic scala potential fomulation: Ω = cost on the plane whee the magnetic field has to be pependicula - in the case of the vecto scala potential fomulation: A x = 0 o A y = 0 o A z = 0 if the plane whee the condition has to be imposed is pependicula to the X, Y o Z axis. Neumann bounday condition. This is a paallel magnetic field condition. It consists in imposing the following condition: - in the case of the magnetic scala potential fomulation this is a natual bounday condition because it will be imposed by the finite element solution pocess itself. - in the case of the vecto scala potential fomulation: A x = 0 and A y = 0 if Z = cost is the plane whee the magnetic field has to be pependicula. The conditions on the othes planes can be easily found by otation of the X, Y and Z indexes. 2.6 Deived fields fom the degee of feedom solution The deived electomagnetic field esults ae the magnetic field intensity H, the magnetic flux density B and the cuent density J. The magnetic field intensity is defined as eq. [.2.3] and [.2.6] H = T Ω [.2.3] espectively fo conducting and non conducting egions. The gadient of the magnetic scala potential function as: H Ω H = Ω [.2.6] = m i= Ω is computed using the element shape Ni Ωi [2.5.] 36

43 Cap.2 The T-Ω, Ω finite element fomulation the pat which depends fom the vecto electic potential is computed as: H T = m n Ni( x j, y j, z j ) T i m j= i= [2.5.2] whee m and n equals the numbe of element nodes. In the non conducting egions we have H = H Ω, while in the conducting egions H = H T + H Ω. The magnetic flux density can be easily computed fom the equation B = µ H. The cuent density is defined in eq. [.2.2] J = T [.2.2] as the cul of the electic vecto potential and can be computed using the element shape functions as: J J J x z y m Ni = T i= y m Ni = T i= z m Ni = T i= x zi xi yi Ni T z yi Ni T x Ni T y zi xi [2.5.3] 37

44 Cap. 3 Implementation in ANSYS of the T-Ω, Ω fomulation 3 Implementation in ANSYS of the T-Ω, Ω fomulation The ANSYS code allows to wite outines and suboutines in FORTRAN 77 o in C and eithe to compile and link them into the pogam, ceating a custom vesion of the ANSYS pogam. This capability is called as Use Pogammable Featues (UPFs). UPFs allow to ceate new elements, add them to the ANSYS element libay and use them as egula elements. The customisation of the new fomulation has been done by modifying the uel0, uel02, uec0 and uec02 suboutines which eside in the subdiectoy /ansys56/customize/use. The uec outines allow to descibe the element chaacteistics such us: 2-D o 3-D geomety, numbes of nodes, degee of feedom set, and so on. The outine elccmt, which eside in the subdiectoy /ansys56/customize/include, descibes the input fo these outines in detail. The uel outines pemit to calculate the element stiffness and damping matices and the element load vecto. The element pintout also is geneated, and the vaiables to be saved ae calculated and stoed in the esults file. The ANSYS distibution medium has also include-decks which can be included in the suboutines. This include-decks also called commons contain impotant amounts of data, such us: solution options, output contol infomations, element chaacteistics and so on. The include-decks pemit the communication between the option defined by the use duing the model ceation and analysis solution and the costumized outines. The modified outines and also the anscust.bat, makefile, ansysex.def, ansysb.dll and mnflib.dll files (which ae on the distibution medium) have to be moved in a woking diectoy whee the FORTRAN files will be compiled and a custom ANSYS executable vesion will be ceated by unning the anscust.bat file. In this way, two elements have been ceated: the USER0 fo nonconducting egions and the USER02 fo conducting egions. To un the ANSYS executable vesion the following sting: ansys56cust.exe -custom <path of the woking di>\ansys.exe -p ansysuh has to be typed in the commands pompt of the opeative system. It is also possible to un the executable ANSYS vesion in inteactive mode selecting the ansys.exe file fom the Execute a costumized ANSYS executable command of the un window of ANSYS. The ANSYS output of the use-linked vesion will include the following note: This ANSYS vesion was linked by licence. 38

45 Cap. 3 Implementation in ANSYS of the T-Ω, Ω fomulation 3. The Uec outines The Uec outines pemit to define the element chaacteistics by flagging the positions of an aay which is called ielc and which is an agument of the outine itself. The flags desciption can be found in the elccmt outine. The meaning of the most impotant Uec suboutines flags fo the USER0 and USER02 ae epoted below: c GEOMETRY ielc(kdim)=3 ielc(ishap)=jtet ielc(idegen)= define a thee-dimensional element define a fou-noded element no degeneations is allowed c ELEMENT USAGE ielc(kelsto)= oientation of the quantities global coodinates ielc(matrqd)=200 a not well defined mateial popety must be input fo this element c ELEMENT NODES ielc(nmndmx)=4 maximum numbe of nodes pe element ielc(nmndmn)=4 minimum numbe of nodes pe element ielc(nmndac)=4 active numbe of nodes pe elements (nodes that have dofs) ielc(nmdfpn)= maximum numbe of degees of feedom pe node fo this element ielc(matrxs)=0 matices possible fo this element type ielc(kdofs)= MAG degee of feedom set fo this element c ELEMENT LOADS & SURFACES ielc(nmptsf)=3 load vaying linealy ove tiangula suface c POSDATA FILE ielc(nmndno)=4 ielc(kconit)= ielc(kmagc)= numbe of nodes that have nodal output stoed on postdata file contouable nodal items ae pesent element suppoting eddy cuents ielc(nmsmis)=5 maximum numbe of items on the ecod fo save vaiables 39

46 Cap. 3 Implementation in ANSYS of the T-Ω, Ω fomulation ielc(nmnmis)=20 ielc(nmnmup)=24 ielc(ncptm)=299 ielc(ncptm2)=354 ielc(jstpr)= c NUMBER OF SAVED VARIABLES ielc(nmmsvr)=8 numbe of saved vaiables fo linea magnetic effects 3.2 The Uel outines The Uel outines have the function to compute element matices, load vectos and esults and to maintain the element solution data. The Uel outines input aguments ae: elem: element label (numbe) ielc: aay of element type chaacteistics elmdat: aay of element data eomask: bit patten fo element output nodes : aay of element node numbes locsv: location of the saved vaiables keleq: matix and load vecto fom equests kelfil: keys indicating incoming matices and load vectos n: matix and load vecto size xyz: nodal coodinates u: element nodal solution values The Uel outines output aguments ae: kelout: keys indicating ceated matices and load vectos zs: stiffness matix damp: damping matix zsc: applied load vecto 40

47 Cap. 3 Implementation in ANSYS of the T-Ω, Ω fomulation zscn: applied load imaginay vecto elvol: element volume elmass: element mass cente: centoid location elene: element enegies edindx: element esult data file indexes lcest: position on esult file The Uel outines call othe suboutines, the most impotant ae: PROPEV suboutine pemit to get the element mateial popeties, thei input aguments ae: iel: the element numbe mt: the mateial numbe lp: keys fo which specific value is equested tem: tempeatue at which to evaluate mateial n: numbe of mateial popeties to be evaluate the output agument is: pop: values of mateial popeties SVGIDX suboutine pemit to get the index fo saved vaiables, its input agument is: locsv: pointe to location of index its output agument is: svindx: the index of the saved vaiables SVRGET suboutine pemit to fetch the saved vaiables fo an element, thei input agument ae: svindx: the index of the saved vaiables nset: the set numbe in the index nsv: numbe of dp wods expected in the set its output agument is: sv: data in the set 4

48 Cap. 3 Implementation in ANSYS of the T-Ω, Ω fomulation ELDWRT suboutine pemit to output the element data to the esult file, thei input aguments ae: ielem: the element numbe edtype: the element type lcest: pointe to esults file position edindx: the index to esults file data nval: total numbe of values the output agument is: value: the output value SVRPUT suboutine pemit to wite an element s saved vaiable set, thei input aguments ae: svindx: the index of the saved vaiables nset: the set numbe in the index leng: numbe of dp wods in the set sv: data in the set its output agument is: svindx: the update index SVPIDX suboutine pemit to wite out the saved vaiables index vecto, thei input aguments ae: locsv: pointe to stat of the saved vaiable fo element svindx: the index of the saved vaiables fo this element its output agument is: locsv: pointe to stat of the saved vaiable fo the next element 42

49 Cap. 4 Validation of the implementation of the T-Ω,Ω fomulation 4. Validation of the implementation of the T-Ω,Ω fomulation To validate the implementation in ANSYS of the new T-Ω,Ω fomulation, thee benchmak poblems have been consideed. Results obtained ae always compaed with the esults fom the commecial fomulations in ANSYS. The fist poblem is the analysis of a conducting cube in homogeneus magnetic field the second benchmak is the hollow sphee in unifom magnetic field, the last is the FELIX bick expeiment. The fist one has been poposed by O. Biò and K. Peis in a published pape on eddy cuent numeical computation [3], the last two benchmak ae fom some Intenational Wokshops fo the compaison of eddy cuent codes. In paticula, the hollow sphee analysis is the poblem numbe 6, while the FELIX bick expeiment is the poblem numbe 4. The idea to develop cetain benchmak poblems to validate 3-D eddy cuent compute codes was bon in 985 in the Office of Fusion Enegy, U.S. Depatment of fusion. The goal of the benchmaks is: to show the effectiveness of numeical techniques and associated compute codes in solving electomagnetic field poblems, and to gain confidence in thei pedictions. [5] Results obtained with the new T-Ω,Ω fomulation ae in vey good ageement with those fom othe codes and with analytical solutions and expeimental solution whee them ae available. 4. Conducting cube in homogeneous magnetic field A conducting cube is immesed in a unifom magnetic field of T with hamonic time vaiation of 50 Hz. Symmety conditions pemit to discetize only /8 of the entie cube. The model used fo the analysis is shown in Fig. 4.. This poblem is studied by O. Biò and K. Peis in [3]. Each side of the conducting egion is cm long, while the boundaies ae 5 cm fathest fom the axes oigin. The model includes 5872 elements and 202 nodes. The cube conductivity is σ= 5, S/m. 43

50 Cap. 4 Validation of the implementation of the T-Ω,Ω fomulation Fig. 4.. Conducting Cube Model The magnetic field and the eddy cuents distibutions in the conducting cube (eal and imaginay pats) obtained, by using the same mesh, with the implemented T-Ω,Ω fomulation and with the ANSYS commecial A-V,A fomulation ae shown in the following pictues. On the left we have esults fom the T-Ω,Ω fomulation, on the ight fom the A- V,A fomulation. Fig B Re pat T-Ω,Ω fomulation Fig B Re pat A-V,V fomulation 44

51 Cap. 4 Validation of the implementation of the T-Ω,Ω fomulation Fig. 4.,4 B Imag pat T-Ω,Ω fomulation Fig B Imag pat A-V,V fomulation Fig J Re pat T-Ω,Ω fomulation Fig J Re pat A-V,V fomulation Fig J Imag pat T-Ω,Ω fomulation Fig J Imag pat A-V,V fomulation As can be noted esults obtained with the two diffeent fomulations ae in faily 45

52 Cap. 4 Validation of the implementation of the T-Ω,Ω fomulation coespondence. The eal and imaginay pats of the X component of the cuent density along the conducting cube Z axis is epoted in Fig ,0E+00 0, 0,4 0,7,0-2,0E+07 Jx [A/m 2 ] -4,0E+07-6,0E+07-8,0E+07 J eal J imag -,0E+08 -,2E+08 X [cm] Fig Cuent density along the x axis cube The cuent densities distibutions depicted in the plot ae in excellent ageement with those (Fig. 4..) epoted by O. Biò in [3]. Fig. 4.. Cuent density along the x axis cube as epoted in [3] The compute stoage and the CPU time (s) as computed by ANSYS in obtaining the 46

53 Cap. 4 Validation of the implementation of the T-Ω,Ω fomulation solution fo both the T-Ω,Ω fomulation and the A-V,A fomulation ae epoted in Table 4..: Solution file size (MB) CPU time pocessing CPU time solution CPU time esults CPU time total T-Ω,Ω fomulation A-V,A fomulation Table. 4.. Runtime statistics and compute stoage In Table 4.. the total CPU time and also the CPU time fo the element pocessing phase, the CPU time fo solving the equation system and the CPU time fo calculating the deived quantities have been consideed. Moeove, also the compute stoage size in MB of the files which ANSYS uses to obtain the solution and to wite esults has been epoted in table 4... In Table 4..2 the degees of feedom numbe in ai and in the conductive mateial is shown fo both the fomulations: DOFs in ai DOFs in conducto Total numbe of DOFs T-Ω,Ω fomulation A-V,A fomulation Table Degees of feedom The A-V,A fomulation equies an highe CPU time because of the element nodes in ai equie thee degees of feedom instead of one of the T-Ω,Ω fomulation. 47

54 Cap. 4 Validation of the implementation of the T-Ω,Ω fomulation 4.2 Hollow sphee in unifom field A hollow conducting sphee is immesed in a unifom field of T oscillating at 50 Hz. Symmety conditions pemit to discetize only one abitay cicumfeential slice of 20 degees of the entie sphee. The model used fo the analysis is shown in Fig and Fig Fig Hollow Sphee Model Fig Hollow Sphee Model detail The sphee has an inne adius of 0.05 m and an oute adius of m. A ectangula exteio bounday is located at a distance of 0.6 m whee the extenal field load is applied. The conducting sphee has a conductivity of S/m. The model includes 2506 elements and 792 nodes. This is a typical case of multiply connected egions; the egion inside the conducto is teated by intoducing an atificial egion of low conductivity (one thousand smalle than the sphee egion) endeing the poblem simply connected. This is the benchmak poblem 6 defined in the Intenational Wokshops fo Eddy Cuent Code Compaison and has been poposed by C. R. I. Emson [5]. The poblem consists in detemining the magnetic field and eddy cuent distibution, in detemining the peak flux density at the cente of the sphee and the aveage powe loss within the sphee. Results can be compaed with esults fom othe codes and also with the analytical solution. ANSYS solves this poblem by using the mixed fomulation. The conducting egion is modelled with the vecto potential fomulation. The egion of ai inside and outside of the sphee is modelled with the Reduced Scala Potential. The two egions ae intefaced with 00 inteface elements at the sphee-ai inteface bounday. The esults of the magnetic field and the eddy cuents distibutions in the conducting cube (eal and imaginay pats) obtained with the implemented T-Ω,Ω fomulation and with the 48

55 Cap. 4 Validation of the implementation of the T-Ω,Ω fomulation ANSYS commecial mixed fomulation ae showed in the following pictues. On the left we have esults fom the T-Ω,Ω fomulation, on the ight fom the mixed fomulation. Fig B Re pat T-Ω,Ω fomulation Fig B Re pat mixed fomulation Fig B Imag pat T-Ω,Ω fomulation Fig B Imag pat mixed fomulation Fig J Re pat T-Ω,Ω fomulation Fig J Re pat mixed fomulation 49

56 Cap. 4 Validation of the implementation of the T-Ω,Ω fomulation Fig J Imag pat T-Ω,Ω fomulation Fig J Imag pat mixed fomulation Fig B Real pat detail T-Ω,Ω fomulation Fig B Real pat detail mixed fomulation Fig B Imag pat detail T-Ω,Ω fomulation Fig B Imag pat detail mixed fomulation As can be noted esults obtained with the two diffeent fomulations ae in faily coespondence. In table 4.2. the peak flux density at the cente of the sphee and the aveage powe loss within the sphee as computed fom the two fomulations ae compaed with those computed analytically. 50

57 Cap. 4 Validation of the implementation of the T-Ω,Ω fomulation T-Ω,Ω fomulation Mixed fomulation Analytical solution B y (0,0), T 5.43E E E-02 Powe loss, W Table 4.2. Results compaison The compute stoage and the CPU time (s) as computed by ANSYS in obtaining the solution fo both the T-Ω,Ω fomulation and the A-V,A fomulation ae epoted in Table 4.2.2: Solution file size (MB) CPU time pocessing CPU time solution CPU time esults CPU time total T-Ω,Ω fomulation A-V,A fomulation Not epoted In table the degees of feedom numbe in ai and in the conductive mateial is shown fo both the fomulations: Table Runtime statistics and compute stoage DOFs in ai DOFs in conducto DOFs in inteface elements Total numbe of DOFs T-Ω,Ω fomulation A-V,A fomulation Table Degees of feedom The CPU time in obtaining the solution fo the T-Ω,Ω fomulation is quite the same than the one fo the A-V,A fomulation even if the numbe of degees of feedom is bigge, this is because in the mixed fomulation thee ae the inteface elements that incease the total element numbe and also because the use element suboutine fo the T-Ω,Ω fomulation is moe pasimonious. 5

58 Cap. 4 Validation of the implementation of the T-Ω,Ω fomulation 4.3 The FELIX bick A ectangula aluminium bick with a ectangula hole is immesed in a unifom field which decays exponentially with time. Symmety conditions pemit to build the model of only one eighth of the bick. The model used fo the analysis is shown in figue 4.3. and Fig Whole FELIX Bick Model Fig FELIX Bick Model The bick is made of aluminium alloy 600, of esistivity Ohm m. Dimensions in the X, Y and Z diections ae espectively m, 0.06 m and m. A ectangula hole m x m penetates the bick though the cente of the lage faces. The model includes 6668 elements and 364 nodes. The egion inside the conducto is teated by intoducing an atificial egion of low conductivity to ende the poblem simply connected. This is the benchmak poblem 4 defined in the Intenational Wokshops fo Eddy Cuent Code Compaison and has been poposed by A. Kameay [5]. The poblem has been solved by diffeent compute codes. The poblem consists in detemining the time evaluations of the magnetic, the aveage powe loss field and eddy cuent at diffeent locations. The applied magnetic field is diected in the Z diection and it vaies as: Bz=Bo fo t<0 Bz= Bo e -t/τ fo t>0 whee Bo = 0. T and τ = 0.09 s. The tansient analysis poblem solution has been obtained consideing a 20 ms time ange and by dividing it in 20 load steps, so using 20 iteations. The time evolution of the total powe loss in the bick as calculated by using the T-Ω,Ω 52

59 Cap. 4 Validation of the implementation of the T-Ω,Ω fomulation fomulation and the one fom the othe codes [5] ae shown in Fig and Fig Loss Powe [W] Time [ms] Fig Total powe loss time evolution Fig Total powe loss time evolution as epoted in [5] 53

60 Cap. 4 Validation of the implementation of the T-Ω,Ω fomulation The total magnetic flux density at t = 4, 8, 2 and 20 ms as calculated by using the T-Ω,Ω fomulation and the one fom the othe codes [5] ae shown in Fig and Fig ms 8 ms 2 ms 6 ms 20 ms 0,0 0,09 0,08 0,07 Bz [T] 0,06 0,05 0,04 0,03 0,02 0,0 0,00 0 0,05 0, 0,5 0,2 0,25 Z [m] Fig Magnetic field on Z axis Fig Magnetic field on Z axis as epoted in [5] 54

61 Cap. 4 Validation of the implementation of the T-Ω,Ω fomulation In table 4.3. the peak of the induced flux density at the cente of the hole and at diffeent locations, the aveage powe loss within the bick as computed fom the two fomulations ae computed and compaed with the mean values fom the othe codes. T-Ω,Ω fomulation Mean B z (Z=0.0), T 3.95E-02 (2) 3.88 E-02 B z (Z=0.027), T 3.72E-02 () 3.63 E-02 B z (Z=0.0254), T 2.97E-02 () 2.96 E-02 Powe loss, W 2.4 (0) 0.6 Table Results compaison The time evolution of the magnetic field and the eddy cuents distibutions in the conducting bick obtained with the implemented T-Ω,Ω fomulation and with the A-V,A ANSYS commecial fomulation ae shown in the following pictues. On the left we have esults fom the T-Ω,Ω fomulation, on the ight fom the commecial fomulation. Fig B 4 ms T-Ω,Ω fomulation Fig B 4 ms A-V, A fomulation Iteation numbe 55

62 Cap. 4 Validation of the implementation of the T-Ω,Ω fomulation Fig B 8 ms T-Ω,Ω fomulation Fig B 8 ms A-V, V fomulation Fig B 0 ms T-Ω,Ω fomulation Fig B 0 ms A-V, V fomulation Fig B 2 ms T-Ω,Ω fomulation Fig B 2 ms A-V, V fomulation 56

63 Cap. 4 Validation of the implementation of the T-Ω,Ω fomulation Fig B 6 ms T-Ω,Ω fomulation Fig B 6 ms A-V, V fomulation Fig J 4 ms T-Ω,Ω fomulation Fig J 4 ms A-V, A fomulation Fig J 8 ms T-Ω,Ω fomulation Fig J 8 ms A-V, V fomulation 57

64 Cap. 4 Validation of the implementation of the T-Ω,Ω fomulation Fig J 0 ms T-Ω,Ω fomulation Fig J 0 ms A-V, V fomulation Fig J2 ms T-Ω,Ω fomulation Fig J 2 ms A-V, V fomulation Fig J 6 ms T-Ω,Ω fomulation Fig J 6 ms A-V, V fomulation 58