3D Magnetic Scalar Potential Finite Element Formulation for Conducting Shells Coupled with an External Circuit

Transcription

1 3D Magnetic Scalar Potential Finite Element Formulation for Conducting Shells Coupled with an External Circuit Christophe Guérin CEDRAT, Meylan, France Gérard Meunier Grenoble Electrical Engineering Laboratory - UMR CNRS Grenoble-INP / UJF / CNRS ENSE3, Grenoble, France

2 Context of the work Radiated and conducted electromagnetic fields in large systems (EMC) Presence of electric circuits (cable) and large conductive surfaces Examples Lightning on composite or aluminum aircraft Magnetic scattering in car Which is the method to used? FEM PEEC BEM or Volume Integral Methods 08/07/20 2

3 Efficiency of PEEC method (using FMM) Car Headlight LED PCB modeling in InCa3D, general view of the geometry and mesh details (courtesy of Valeo) elements Total computational time : 85s PC Intel Core 2 - 2GB memory (The memory requirement does not exceed 240 MB) T-S. Nguyen, J-M. Guichon, O. Chadebec, G. Meunier, B. Vincent, An Independent Loops Search Algorithm 08/07/20 for Solving Inductive PEEC Large Problems, Compumag 20,Sidney, Poster PA9-2 3

4 PEEC Which method for shell coupled with circuit? Well adapted for modeling complex electrical circuits Successfully used for study lightning on aircraft (and power electronic applications) Full Matrix but interactions can be compressed (ex: FMM) No adapted to model magnetic and conductive shell with δ << e FEM General method, needs to mesh all domain Efficient for magnetic and conductive shell (δ << e -> δ >> e) with scalar magnetic potential (small computation time and memory requirement) Not well adapted for multiscale geometries of conductors with large dimension ratio Integral method Promising with matrix compression 08/07/20 4

5 Some previous works with FEM or BEM t-φ or t-t 0 -φ volumic formulations φ or t 0 -φ surface impedance formulations φ or t 0 -φ formulations for conducting shells δ<< L (δ independent) Simply connected conductors and not coupled with circuits Preston 982 (t-φ) Luong 997 (t-t 0 -φ) Krähenbühl 988 (BEM) Rodger 99 Guérin 993 Krähenbühl 990 (BEM) Mayergoyz 995, Guérin 995 Multiply connected conductors or coupled with circuits Meunier 2003 Biro 2004 Meunier 2008 This work 08/07/20 5

6 Our work : modeling shell with circuit coupling by using the FEM Four major steps:. FEM formulation in presence of shells σ µ 2. Add voltage-current relations for current in the shell between two terminals or around holes 3. Add voltage-current relations for coils in presence of shell 4. Add the equations for the electric circuit 08/07/20 6

7 AKNOWLEGMENTS This work was done for a contract with the PSA Peugeot Citroën group 08/07/20 7

8 Outline Formulation Magnetic scalar finite element formulation with shells Source field computations with terminals and holes Voltage-current relations in presence of shell Final system to solve Examples Conclusions and future works 08/07/20 8

9 Typical problem and notations e Γ Thin region Γ 2 Ω a Ω c µ 0 φ σ µ φ 2 08/07/20 9

10 Surface impedance boundary conditions for a shell σ and µ are assumed to be linear and isotropic Use of analytical solution of the conducting plate submitted to uniform sinusoidal time varying fields on both sides e e 2 h = h 0 cos (ωt+φ ) h 2 = h 02 cos (ωt+φ 2 ) z n 2 n 08/07/20 0

11 Surface impedance boundary conditions for a shell Surface impedance boundary conditions (IBC) for a shell: they couple tangential e and e 2 to tangential h and h 2 e e s 2s = = ( Z h Z h ) s 2 2s n ( Z22 h2s Z2 hs ) n2 e s e 2s h s n 2 h 2s n Impedances for each surface Impedances: link between the two surfaces I.D. Mayergoyz and G. Bedrosian, On calculation of 3-D eddy currents in conducting and magnetic shells, IEEE Trans. Magn., vol.3, no.3, 995. C. Guérin, G. Tanneau, G. Meunier A shell element for computing 3D eddy currents - application to transformers, IEEE Trans. Magn., vol. 3, no.3, /07/20

12 Surface impedance boundary conditions for a shell Surface impedances (linear σ and µ) : Z = Z22 = σ a th(ae) Z2 = Z2 = σ a sh(ae) with a = + δ j δ = 2 (skin depth) ωµσ 08/07/20 2

13 Finite element formulation for shells The div b = 0 equation is solved in regions Ω a (air) Application of Galerkine s method Ω a wi div b dω = 0 e Γ Thin region Ω a Ω c σ µ 0 w i µ w i : nodal test functions associated to the magnetic scalar potential φ Γ 2 Ω grad wi b dω + wi b n2 dγ + wi b n dγ = a Γ 2 Γ 0 (with Green s formula) Boundary terms which will be coupled by the impedance boundary conditions for shells 08/07/20 3

14 Finite element formulation for shells Transformation of boundary terms Γ wi b n dγ wi b n2 Γ 2 dγ Maxwell-Faraday s equation b n b n 2 = curle n jω = curle n jω 2 e e s 2s = = ( Z h Z h ) Surface IBC s 2 2s n ( Z22 h2s Z2 hs ) n2 e s h s e 2s h 2s jω Γ grad s w i ( Z h Z h ) dγ grad w ( Z h Z h ) s 2 2s jω Γ 2 s i 22 2s 2 s dγ 08/07/20 4

15 Finite element formulation for air and shell Scalar reduced potential formulation. As we have conducting shells, there is a jump of the magnetic potential Φ. Φ =Φ i 2 Ω System of equations non meshed coil e Ω o φ3 φ φh φ6 φ2 φ4 φb φ2h φ5 φ2b gradwi µ o h dω + swi s 2 2s s i 22 2s 2 s = jω grad jω a Γ ( Z h Z h ) dγ + grad w ( Z h Z h ) dγ 0 Γ 2 h = T gradw φ o j. j j φ3h φ3b Pre-calculation for the source field T o curlt Φ =Φ 2 allows to ensure that the surface current density is tangential to the edge of the shell T o o = jin Ω o xn = 0 on Ω o 08/07/20 5

16 Outline Formulation Magnetic scalar finite element formulation with shells Source field computations with terminals and holes Voltage-current relations in presence of shell Final system to solve Examples Conclusions and future works 08/07/20 6

17 to-φ formulation : our typical case Holes Non meshed conductor Electric terminals 08/07/20 7

18 Shell t 0 -φ formulation with circuits i c i b i c2 i b4 i c5 i c3 i c4 i b2 i b3 Magnetic field h in air must be reduce by t bi of coils and by t 0i of paths of currents h = i bi t bi + icit0i gradφ coils current paths of Where t 0i and t bi are source fields produce by currents of A 08/07/20 8

19 Source field computation for a current carrying shell i c i c2 i c3 i c5 i c4 n h : number of holes n h paths of current around each hole n t : number of terminals n t - paths of current between two terminals One additional unknown i ci for each path of current n h + n t - additional unknowns i ci 08/07/20 9

20 Source field computation for a current carrying shell For each path of current :. Determination of source current j 0 (electric conduction problem) 2. Determination of source field t 0 produced by j 0 08/07/20 20

21 Source field computation for a current carrying shell j 0i is computed assuming a current of A, solving a electric conduction problem, using scalar electric potential j 0i = σ grad (v 0i ), div j 0i = 0 solved with FE j 0i is assumed constant through thickness For j 0i current around a hole: with a line cut with doubled nodes v = V For j 0i current between two terminals: st terminal: v = 0V 2 nd terminal: v = V Shell v=0v v=v Shell v v -v 2 =V v 2 Line cut: 08/07/20 2

22 Source field computation for a current carrying shell t 0i j t 0i2 n For a current carrying shell t 0i is computed with FEM such that: curl t 0i = j 0i in Ω 0 (t 0 domain) t 0i x n = 0 on Ω 0 Transmission condition: k 0i = n t 0i2 n t 0i where k 0i = e j 0i Fonctionnal minimized (linear system solved with FEM): Terms coupling both sides of the shell with the transmission condition F [ ] 2 dω + ( n t + n t + k ) 2 ( t ) ( curl t ) + ( div t ) 0i = dγ 2 Ω o 0i 0i 2 Γ 0i 2 0i2 0i 2 08/07/20 22

23 Counting holes of a conductor Algorithm which determines the number of holes of a shell: determination of the first Betti number b For volume conductors Algorithm which determines the number of holes = number of needed surface cuts Algorithm which automatically create cuts For conducting shells Adaptation of the Algorithm which determines the number of holes = number of needed line cuts 08/07/20 23

24 Counting holes Euler s number χ of the boundary of a volume (surface characteristic): F: number of faces E: number of edges on the boundary of the volume P: number of points χ' = F E + P Algorithm which determines the number of holes of a volume: b 0 : number of connected components = b : number of holes b 2 : number of cavities in the volume χ : Euler s number of the boundary of the volume b 0 2 χ = b + b '/ 2 = + b '/ 2 b 2 χ For counting holes for a shell we adapt volume algorithm, considering its volume through thickness 08/07/20 24

25 Automatic cut detection Manual cuts : very difficult for the user Automatic cuts algorithm for volumes : by modeling the inflation of a virtual balloon Automatic cuts algorithm for shell : not operational (in progress) Example for a car : 44 holes! 08/07/20 25

26 Automatic cuts detection for J 0 calculation Exemple with a volume (NDT application) Holes Eddy currents Anh Tuan Phung, Patrice Labie, Olivier Chadebec, Yann Le Floch, Gerard Meunier, "On the Use of Automatic Cuts Algorithm for To-T-Φ Formulation in Nondestructive Testing by Eddy Current", Intelligent Computer Techniques in Applied 08/07/20 Electromagnetics, Springer, Volume 9/2008 July 2008 pp

27 Outline Formulation Magnetic scalar finite element formulation with shells Source field computations with terminals and holes Voltage-current relations in presence of shell Final system to solve Examples Conclusions and future works 08/07/20 27

28 Voltage-current relation for shells The voltage-current relation for a bulk solid conductor u 2 u Shell Ω c Ω a u 3 u 4 =0 u 5 =0 u i = j0i edω + Ω c Ω t 0i db dt dω u i = 0 for a hole u i = j0i edω + jω t0i b dω + jω Ω c Ω c Ω a t 0i b dω G.Meunier, Y. Le Floch, C. Guérin, "A non linear circuit t-t0-φ formulation for solid conductor", IEEE Trans. Magn., vol. 39, n 3, 2003 O. Biro, K. Preis "Voltage-driven coils in finite-element formulations using a current vector and a magnetic scalar potential", IEEE Trans. Magn., vol. 40, No 2, /07/20 28

29 Voltage-current relation for shells Transformations of the voltage-current relation for a shell (/2) u i = j0i edω + jω t0i bdω + jω Ω c Ω c Ω a t 0i bdω curl t 0i = j 0i Ω c curl t 0 edω i Ω c t 0i curl e Maxwell-Faraday s equation curl e = db/dt dω n 2 Γ Ω a Ω c div ( t 0 e) dω i n Γ 2 Thin region Ω c Γ ( e n ) t Γ ( e n ) t dγ 0i d 2 2 0i2 Γ 2 08/07/20 29

30 Voltage-current relation for shells Transformations of the voltage-current relation for a shell (2/2) u i ( e n ) t dγ ( e n ) t dγ + jω t bdω = 0i 2 2 0i2 Γ Γ 2 Ω a 0i Surface IBCs couple e s and h s on both sides of the shell e e s 2s = = ( Z h Z h ) s 2 2s n ( Z22h2s Z2hs ) n2 e s e 2s h s h 2s u i Ω Γ [ t ( Z h Z h ) + t ( Z h Z h )] = jω t bdω + dγ a 0i 0is s 2 2s 0i2s 22 2s 2 s 08/07/20 30

31 Voltage-current relation for coils in the presence of shells Voltage-current relation in presence of a shell Shell Ω c Ω a u i u i db = Ri ibi + t dω Ω bi a dt = Ri ibi + jω tbi bdω + jω Ω a Ω c t bi bdω i b i b2 i b3 Coils u By using Maxwell Faraday equation and surface IBCs we obtain i = Ri ibi + jω tbi bdω + Ω a Γ t bis ( Z h Z h + Z h Z h ) dγ s 2 2s 22 2s 2 s O. Biro, K. Preis "Voltage-driven coils in finite-element formulations using a current vector and a magnetic scalar potential", IEEE Trans. Magn., vol. 40, No 2, 2004 G. Meunier, C. Guérin and Y. Le Floch, Circuit-coupled t0-φ formulation with surface impedance condition, IEEE Trans. Magn., vol. 44, no.6, /07/20 3

32 Remark on the calculation current-voltage relation (/2) Consider the volume integration of source field Ω a t 0i bdω = Ω a t 0i µ o gradφ)i dω This calculation needs fine mesh around conductor, since the source field t 0 is varying very quickly j ( t 0 j j Supposing no volume magnetic material, and an integration over all the domain, t 0 is the Biot-and-Savart field and : 0 0i 0 j 0 t0i. t0 j dωa = Mij = dωld 4π r Ω Ωi Ωj µ a µ j. j Ω k Classical PEEC term Where M ij is the mutual inductance in air 08/07/20 32

33 Remark on the calculation current-voltage relation (2/2) Compute by PEEC (analytical or numerical) technique has significant advantages Good precision Not necessary to refine the mesh around conductors This technique can be adapted in presence of magnetic material -> apply on a common mode filter Not yet implemented for this work Than Son Tran, Gérard Meunier, Patrice Labie, "A Efficient FEM-PEEC Coupled Method", IEEE Transaction on Magnetics, Vol. 46, April 200, Number 4, pp /07/20 33

34 Outline Formulation Magnetic scalar finite element formulation with shells Source field computations with terminals and holes Voltage-current relations in presence of shell Final system to solve Examples Conclusions and future works 08/07/20 34

35 Adding electric circuit equations Unknowns for circuit: electric potential integrated in time ψ = t 0 v(t)dt 2 i c N : nodes-branches topological matrix (n- nodes, b branches) i b5 3 i c2 i c3 i b4 4 N = Voltages expression function of ψ: { u} = N t { ψ} d dt Kirchhoff s first law (conservation of currents) N { I} = 0 Exterior electric circuit equations can be added 08/07/20 35

36 Ω Final formulation gradwi µ hdω + swi s 2 2s s i 22 2s 2 s = jω grad jω a System of equations: Γ Finite element equations for air and shell regions ( Z h Z h ) dγ + grad w ( Z h Z h ) dγ 0 Γ 2 Paths of current in shell Ω a t 0i µ hdω + jω Γ t [ t ( Z h Z h ) + t ( Z h Z h )] dγ + N { ψ} = 0 0si s 2 2s 02si 22 2s 2 s i Ri i jω bi + Ω a t bi µ hdω + jω Γ [ t ( Z h Z h ) + t ( Z h Z h )] dγ + N { ψ} = 0 bsi s 2 2s b2si 22 2s 2 s t i N { I} = 0 Kirchhoff s first law Current in coils 08/07/20 36

37 Final formulation The magnetic field h is expressed function of the unknowns i bi, i ci and φ: h = i bi t bi + coils paths current i ci of t 0 i grad φ in Ω h s = i bi t bis + Coils Paths current i ci of t 0 is grad s φ on Γ and Γ 2 Symmetric system A A A A A A A A A A A A A φ ib ic ψ = B B B Available in 08/07/20 37

38 Outline Formulation Examples Conclusions and future works 08/07/20 38

39 Numerical examples Rectangular loop 8,E-05 7,E-05 Coil One hole Loop shell J A/mm² 6,E-05 5,E-05 4,E-05 3,E-05 J "shell formulation" J t-t0-phi formulation x (mm) j on path at the top face of the loop (0 Hz) With shell formulation W With t-t 0 -φ formulation W Difference.38 % Joule losses in the loop at 0 Hz 08/07/20 39

40 Numerical examples With shell formulation With t-t 0 -φ formulation Ratio min 58 s 9 h 4 min 53 s 582 Solving CPU times at 0 Hz With PC 64 bits, 2 Intel Xeon 5507 (quad core), 2.26 GHz, 48 Gb RAM. 08/07/20 40

41 Numerical examples Coils with currents with a 90 phase difference d -shape rectangular loop First line cut Loop shell Second line cut Shell cross section Two holes: one in the center and one in the middle of the "tube : 2 line cuts Arrows of current density 08/07/20 4

42 Numerical examples Coils with currents with a 90 phase difference d -shape rectangular loop First line cut Loop shell Second line cut Shell cross section Two holes: one in the center and one in the middle of the "tube : 2 line cuts Arrows of current density at 2 times (0 and π/2) 08/07/20 42

43 Simple automotive numerical example Geometry and electric circuit Current source : I = 5 2 /2 (53.3 ) Round wire: non meshed coil m Example created for PSA Peugeot Citroën group Sheet iron: steel or aluminium e = 2 mm 08/07/20 43

44 Simple automotive numerical example Frequencies and δ/e ratios with steel Frequency 0 Hz 00 Hz 000 Hz 0 khz 00 khz δ/e δ > e δ << e 08/07/20 44

45 Simple automotive numerical example Current density j at 00 khz with steel Surface region with Shell formulation Volume region with surface t 0 -φ impedance formulation 08/07/20 45

46 Simple automotive numerical example Current density k/e obtained with FEM ( shell formulation ) and with PEEC method with aluminium 0 Hz 00 khz FEM PEEC 08/07/20 46

47 Simple automotive numerical example Need to take into account the permeability in case of magnetic material in the sheet 6,E-06,E-0 L (H) 5,E-06 4,E-06 3,E-06 2,E-06 L (Mur=500) L (Mur=) R (Ohm),E-02 R (Mur=500) R (Mur=),E-06 0,E+00,E+0,E+02,E+03,E+04,E+05 f (Hz),E-03,E+0,E+02,E+03,E+04 f (Hz),E+05 R, L and lzl of the wire in series with the sheet, for : µ r =500 µ r =,E+00,E-0 f(hz),0e+0,0e+02,0e+03,0e+04,0e+05 µr = µr = 500 (same conductivity) Z,E-02,E-03 08/07/20 47

48 Realistic automotive numerical example Application example case: EMC analysis of a car body (courtesy of PSA) Result: Magnetic scattering Current density repartition Impedance of car body 0 Hz à 00 khz 08/07/20 48

49 Conclusion The FE coupled circuit shell formulation has been developed and validated with results obtained by other methods: volume finite element formulations PEEC method With the shell formulation the EMC simulation of a car body has been performed (first results) Due to the use of the magnetic scalar potential, the method needs the determination of cuts (in presence of holes) 08/07/20 49

50 Future works Development of Integral Methods for modeling shell and coupling it with PEEC method and FMM. First works: A integral formulation which allows to take into account non magnetic shell element (δ independent) T. Le-Duc, G. Meunier, O. Chadebec, J-M. Guichon, A New Integral Formulation for Eddy Current Computation in Thin Conductive Shells, Compumag 20 Sidney, Poster PC-20 Coupling PEEC with calculation of eddy current in a conductive and magnetic shell by integral method (δ>>e) T. Le-Duc, O. Chadebec, J-M. Guichon, G. Meunier, Coupling Between PEEC Method and an integrodifferential approach for solving electromagnetic problem, CEM 20, Wroclaw, Poland 24E3 7E3 0E3 03E3 96E3 88E3 8E3 74E3 67E3 60E3 46E3 39E3 32E3 25E3 8E3 E3 4E3 (Voltage applied) PEEC-Integral Method 600 elements FEM elements Loss : W 08/07/20 Loss : W 50

51 Additional slides 08/07/20 5