Various Two Dimensional Multiscale Finite Element Formulations for the Eddy Current Problem in Iron Laminates

Transcription

1 Various Two Dimensional Multiscale Finite Element Formulations for the Eddy Current Problem in Iron Laminates Karl Hollaus and Joachim Schöberl Institute for Analysis and Scientific Computing Vienna University of Technology, Austria April 13, / 63

2 Motivation Aim: Simulation of eddy current losses in large transformers Large transformer Laminated core FE-Model with 1 laminates Detail, lower right corner 2 / 63

3 Outline 1 Introduction to the Multiscale Finite Element Method MSFEM 2 Eddy Current Problem in 2D (Reference Solution 3 MSFEM with A in 2D 4 Higher Order Multiscale Approach with A in 2D 5 MSFEM with Single Component Current Vector Potential T 6 MSFEM for Mixed Formulation with A and J in 2D 7 MSFEM with A in 3D 8 Benchmark for Iron Laminates 3 / 63

4 Introduction to the Multiscale Finite Element Method MSFEM Introduction to the Multiscale Finite Element Method MSFEM: Large Scale Microshape functions φ i for one periode d + d Multiscale finite element approach: n m u h (x = u ij ϕ i (xφ j (x = i j n i m u ij ψ ij (x j Standard polynomials ϕ i Special functions, micro-shape functions φ j u ij are the coefficients of the approximated solution u h New basis functions ψ ij 4 / 63

5 Eddy Current Problem in 2D (Reference Solution Eddy Current Problem in 2D (Reference Solution: Large scale dimensions: W... width L... length Small scale dimensions: d + d... iron + air k f =... fill factor d d+d Conductivity: σ... iron... air σ Permeability: µ... iron µ... air Draft of the problem in 2D Quantities: B... Magnetic flux density J... Current density 5 / 63

6 Eddy Current Problem in 2D (Reference Solution Boundary Value Problem: curl 1 µ curl A + jωσa = J in Ω = Ω m Ω, A... Magnetic vector potential Weak Form: A n = α on Γ Find A h V α := {A h V h : A h n = α h on Γ}, such that 1 µ curl A h curl v h dω + jω σa h v h dω = J v h dω Ω for all v h V := {v h V h : v h n = on Γ}, where V h H(curl, Ω. Ω Ω The solution serves as a reference solution for the multiscale finite element methods. 6 / 63

7 MSFEM with A in 2D Multiscale Approach: Eddy currents in laminates with edge effect, detail Micro-shape function for one periode d + d à = A + φ 1 ( A 1 + (φ 1 w 1 Average value A Scalar quantities A 1 and w 1 Micro-shape function φ 1 7 / 63

8 MSFEM with A in 2D Boundary Conditions: Ã = A + φ 1 ( A 1 + (φ 1 w 1 Outer boundary Γ: A n = α Interface Γ m : Natural boundary conditions 8 / 63

9 MSFEM with A in 2D Weak Form of the MSFEM: Inserting the multiscale approach into the weak form yields Find (A h, A 1h, w 1h V B := {(A h, A 1h, w 1h : A h U h, A 1h V h, w 1h W h and A h n = α h on Γ}, such that 1 µ curl à h curl ṽ h dω + jω σã h ṽ h dω = J ṽ h dω Ω for all (v h, v 1h, q 1h V. Finite element subspaces: A h, v h U h H(curl, Ω A 1h, v 1h V h L 2 (Ω m w 1h, q 1h W h H 1 (Ω m and φ 1 H per (Ω m Ω Ω 9 / 63

10 MSFEM with A in 2D Finite Element System: Ω 1 µ curl à h curl ṽ h dω + jω σã h ṽ h dω = J ṽ h dω Ω Ω Ω ( 1 µ curl ( +jω σ Ω Ω J ( A h + φ 1 A 1h ( ( curl v h + φ 1 v 1h ( A 1h ( v h + φ 1 v 1h A h + φ 1 ( + (φ 1 w 1h + (φ 1 q 1h + (φ 1 w 1h + (φ 1 q 1h ( v h + φ 1 ( v 1h + (φ 1 q 1h dω dω dω = 1 / 63

11 MSFEM with A in 2D Finite Element System: 1 Ω µ curl +jω jω Ω Ω ( A h + φ 1 ( A 1h ( ( σ A h + φ 1 + (φ A 1 w 1h 1h Ω ( ( curl v h + φ 1 dω v 1h ( ( v h + φ 1 + (φ v 1 q 1h 1h ( T ( ( curl Ah ν νφ1x curl vh dω + A 1 νφ 1x νφ 2 1x v 1 dω= T (A h x σ σφ 1x σφ 1 (v h x (A h y σ σφ 1 σφ 1 (v h y A 1h σφ 1 σφ 2 1 σφ 2 1 v 1h w 1h σφ 1x σφ 2 1x σφ 1x φ 1 q 1h dω w 1hx σφ 1 σφ 1x φ 1 σφ 2 1 q 1hx w 1hy σφ 1 σφ 2 1 σφ 2 1 q 1hy with ν = µ / 63

12 MSFEM with A in 2D Averaging Averaging of coefficients λ, λ φ 1, λφ 1, etc. over the periode p = d + d (φ 1x := φ 1 : λ = 1 p λφ 1x = 1 p λφ 1 = 1 p λφ 2 1x = 1 p λφ 1x φ 1 = 1 p λφ 2 1 = 1 p p p p p p p λ(xdx = λ Fed + λ d p λ(xφ 1x (xdx = 2 λ Fe λ p λ(xφ 1 (xdx = λ(xφ 2 1x(xdx = 4 p (λ Fe d + λ d λ(xφ 1x (xφ 1 (xdx = λ(xφ 2 1(xdx = λ Fed + λ d 3p 12 / 63

13 MSFEM with A in 2D Finite Element System after Averaging: ( T ( ( curl Ah ν νφ1x curl vh Ω A 1h νφ 1x νφ 2 dω, + v 1x 1h T (A h x σ σφ 1x σφ 1 (v h x (A h y σ σφ 1 σφ 1 jω A 1h σφ 1 σφ 2 1 σφ 2 (v h y 1 Ω w 1h σφ w 1hx 1x σφ 2 v 1h 1x σφ 1x φ 1 q 1h dω σφ 1 σφ 1x φ 1 σφ 2 1 q 1hx w 1hy σφ 1 σφ 2 1 σφ 2 q 1hy 1 with ν = µ 1. Now, the coefficients are constant in the laminated medium. Consequently, a very coarse finite element mesh suffices to approximate the solution accurately! 13 / 63

14 MSFEM with A in 2D Finite Element Models: FE model for the reference solution (RS. FE model for the MSFEM. Discretisation in y-direction is the same for both FE models. 14 / 63

15 MSFEM with A in 2D Relative Errors in Percentage: Potentials: u MS u RS L2 u RS L2 1% Fluxes: curl (u MS curl (u RS L2 curl (u RS L2 1% Losses: P MS P RS P RS 1%... evaluated in iron only! RS... reference solution MS... multiscale solution 15 / 63

16 MSFEM with A in 2D Simulations: Exact vs. Averaging à = A + φ 1 ( A 1 + (φ 1 w 1 Order of the standard basis: A... 1, A , w / 63

17 Higher Order Multiscale Approach with A in 2D Higher Order Multiscale Approach: First order: Ã = A +φ 1 ( A 1 + (φ 1 w 1 Higher order: Ã = A + φ 1 ( A 1 + φ 3 ( A 3 + φ 5 ( A 5 + (φ 1 w 1 + (φ 3 w 3 + (φ 5 w 5 Higher order micro-shape functions: Integrated Lengendre polynomials 17 / 63

18 Higher Order Multiscale Approach with A in 2D Higher Order Multiscale Approach: Ã = A + φ 1 ( A 1 + (φ 1 w 1 + φ 3 ( A 3 + (φ 3 w 3 ( + φ 5 + (φ A 5 w 5 5 Finite element subspaces: A h U h H(curl, Ω A 1h, A 3h, A 5h V h L 2 (Ω m w 1h, w 3h, w 5h W h H 1 (Ω m and φ 1, φ 3, φ 5 H per (Ω m 18 / 63

19 Higher Order Multiscale Approach with A in 2D Simulations: Ã = A + φ 1 ( A 1 + (φ 1 w 1 + φ 3 ( A 3 + (φ 3 w 3 + φ 5 ( A 5 + (φ 5 w 5 Order of standard basis: A... 1, A i... 1, w i / 63

20 Higher Order Multiscale Approach with A in 2D Higher Order Multiscale Approach: Ã = A + φ 1 ( A 1 + φ 3 ( A 3 + φ 5 ( A 5 + (φ 1 w 1 + (φ 3 w 3 + (φ 5 w 5 Numerical Data: Table: Number of degrees of freedom. Total No. H(curl, Ω L 2 (Ω m H 1 (Ω m RS 24, 745 a 24, MSFEM 1, 675 b c 181 c a For 6 th order H(curl - elements. b For the 5 th higher order MSA, static condensation eliminates all degrees of freedom of L 2 (Ω and possible higher order ones of H(curl, Ω and H 1 (Ω m respectively. c Three times for the 5 th higher order MSA. The improvement in computational costs is clearly visible, but not spectacular. 2 / 63

21 Higher Order Multiscale Approach with A in 2D Simulations: Nonlinear Large Problem with 1 Laminates FE model for RS. FE model for MSFEM. 21 / 63

22 Higher Order Multiscale Approach with A in 2D Simulations: Nonlinear Large Problem with 1 Laminates Magnetization curve. Eddy current losses. 22 / 63

23 Higher Order Multiscale Approach with A in 2D Simulations: Nonlinear Large Problem with 1 Laminates Numerical Data: Ã = A + φ 1 ( A 12 + φ 3 ( A 32 + (φ 1 w 1 + (φ 3 w 3 Table: Number of degrees of freedom for the most accurate approach. Total No. H(curl, Ω L 2 (Ω m H 1 (Ω m RS 164, , MSFEM 1, 738 a b 241 b a For the 3 rd higher order MSA. b Two times for the 3 rd higher order MSA. + The improvement is impressive! The computational requirements for MSFEM are almost the same as in the small example with only 1 laminates. 23 / 63

24 MSFEM with Single Component Current Vector Potential T Single Component Current Vector Potential: T = Te z T : Ω R 2 C, (x, y T (x, y ( T curl T : Ω R 2 C 2, (x, y y T (x, y x Boundary value problem: Weak form: curl( 1 σ curl T + jωµt = in Ω = Ω m Ω, T = T on Γ Find T h V Th := {T h V h : T h = T h on Γ}, such that 1 σ curl T h curl t h dω + jω µt h t h dω = Ω for all t h V := {t h V h : t h = on Γ}, where V h H 1 (Ω. The solution serves as a reference solution for the multiscale finite element methods. 24 / 63 Ω

25 MSFEM with Single Component Current Vector Potential T Multiscale Approach: T (x, y = T (x, y + φ 2 (xt 2 (x, y + φ 4 (xt 4 (x, y leading to the multiscale current density ( ( Ty φ J = curl Te z = + 2 T 2y + T x φ 2x T 2 φ 2 T 2x ( φ 4 T 4y. φ 4x T 4 φ 4 T 4x Micro-shape functions φ 2 and φ / 63

26 MSFEM with Single Component Current Vector Potential T Multiscale Approach: T (x, y = T (x, y + φ 2 (xt 2 (x, y + φ 4 (xt 4 (x, y leading to the multiscale current density ( ( Ty φ J = curl Te z = + 2 T 2y + T x φ 2x T 2 φ 2 T 2x Weak Form: ( φ 4 T 4y. φ 4x T 4 φ 4 T 4x Find (T h, T 2h, T 4h V h,t := {(T h, T 2h, T 4h : T h, T 2h and T 4h U h, T h = T on Γ and T 2h = and T 4h = on Γ m,1 Γ m }, such that 1 σ curl T h curl t h dω + jω µ T h t h dω = Ω for all (t h, t 2h, t 4h V h,, where U h is a subspace of H 1 (Ω m and φ 2 and φ 4 H 1 per (Ω m. Ω 26 / 63

27 MSFEM with Single Component Current Vector Potential T Boundary Conditions: T (x, y = T (x, y + φ 2 (xt 2 (x, y + φ 4 (xt 4 (x, y Outer boundary Γ: T = T Interface Γ m,1 : T 2 = and T 4 = required for edge effect! 27 / 63

28 MSFEM with Single Component Current Vector Potential T Simulations: T (x, y = T (x, y + φ 2 (xt 2 (x, y Order of standard basis: T... 1 T T φ 4 (xt 4 (x, y 28 / 63

29 MSFEM with Single Component Current Vector Potential T Simulations: T (x, y = T (x, y + φ 2 (xt 2 (x, y Order of standard basis: T... 1 T T φ 4 (xt 4 (x, y 29 / 63

30 MSFEM with Single Component Current Vector Potential T Simulations: T (x, y = T (x, y + φ 2 (xt 2 (x, y Order of standard basis: T... 1 T T φ 4 (xt 4 (x, y 3 / 63

31 MSFEM with Single Component Current Vector Potential T Simulations: Edge Effect Edge effect considered. Edge effect neglected. 31 / 63

32 MSFEM with Single Component Current Vector Potential T Simulations: Edge Effect T (x, y = T (x, y + φ 2 (xt 2 (x, y Edge Effect neglected! + φ 4 (xt 4 (x, y 32 / 63

33 MSFEM with Single Component Current Vector Potential T Simulations: Edge Effect T (x, y = T (x, y + φ 2 (xt 2 (x, y Edge Effect considered! + φ 4 (xt 4 (x, y 33 / 63

34 MSFEM for Mixed Formulation with A and J in 2D Weak Form of the Mixed Formulation: Remember the standard weak form: Find A h V α := {A h V h : A h n = α h on Γ}, such that 1 µ curl A h curl v h dω + jω σa h v h dω = J v h dω Ω for all v h V, where V h H(curl, Ω. Ω Ω Introducing the current density jωσa h = J h as unknown leads to 1 µ curl A h curl v h dω with Ω Ω J h v h dω = A h g h dω + j 1 Ω ω Ω σ J h g h dω =. Ω J v h dω 34 / 63

35 MSFEM for Mixed Formulation with A and J in 2D Weak Form of the Mixed Formulation: Find (A h, J h V α := {(A h, J h : A h U h, J h M h and A h n = α h on Γ B }, such that 1 Ω µ curl A h curl v h dω J h v h dω = J v h dω Ω Ω A h g h dω + j σj h g h dω = Ω ω Ω for all (v h, g h V. A h, v h U h H(curl, Ω J h, g h M h H(div, Ω m 35 / 63

36 MSFEM for Mixed Formulation with A and J in 2D Weak Form of the Mixed Multiscale Formulation: Approaches à = A + φ 1 ( A 1 + (φ 1 w 1 J = J + curl(φ 2 ψ 2 e z yield the weak form of the mixed multiscale formulation: Find (A h, A 1h, w 1h, J h, ψ 2h V αh := {(A h, A 1h, w 1h, J h, ψ 2h : A h U h, A 1h V h, w 1h and ψ 2h W h, J h M h and A h n = α h on Γ}, such that 1 Ω µ curl à h curl ṽ h dω J h ṽ h dω = J ṽ h dω Ω Ω Ã h g h dω + j 1 Ω ω Ω σ J h g h dω = p div J h div g h dω = Ω for a sufficiently large p R and for all (v h, v 1h, q 1h, g h, β 2h V. 36 / 63

37 MSFEM for Mixed Formulation with A and J in 2D Weak Form of the Mixed Multiscale Formulation: Finite element subspaces: A h, v h U h H(curl, Ω J h, g h M h H(div, Ω A 1h, v 1h V h L 2 (Ω m w 1h, q 1h, ψ 2h, β 2h W h H 1 (Ω m and φ 1, φ 2 H per (Ω m 37 / 63

38 MSFEM for Mixed Formulation with A and J in 2D Simulations: Mixed MSFEM Mixed MSFEM with A and J. MSFEM with A. 38 / 63

39 MSFEM with A in 3D Model Problem: σ = S/m µ r = 5, α = Vs/m n = 1 No. Laminates d = d + d =.25mm ff =.9 (Fill factor Problem with dimensions in mm. 39 / 63

40 MSFEM with A in 3D Magnetic Field: Challenges: 1. Large main magnetic field 2. Significant magnetic stray field 3. Pronounced boundary layers Magnetic flux density B. 4 / 63

41 MSFEM with A in 3D Multiscale Approach for 3D 2D: à = A + φ 1 ( A 1 + (φ 1 w 3D: (straightforward extension à = A + φ 1 A 12 + (φ 1 w A 13 Finite element subspaces: A h U h H(curl, Ω A 12h, A 13h V h L 2 (Ω m w h W h H 1 (Ω m and φ 1 H per (Ω m 41 / 63

42 MSFEM with A in 3D Multiscale Approach for 3D 3D: (straightforward extension à = A + φ 1 A 12 + (φ 1 w A 13 Table: Comparison of Eddy Current Losses RS MSA Losses in W Rel. Error in % / 63

43 MSFEM with A in 3D Study of Different Multiscale Approaches: MSA1 Ã = A + φ 1 + (φ 1 w, w H 1 (Ω m A 12 A 13 MSA2 Ã = A + φ 1 A 12 A 13 MSA3 Ã = A + φ 1 + w A 12 A 13 MSA4 Ã = A + φ 1 + w Finite element subspaces: A 12 A 13 φ 1x φ 1x, w L 2 (Ω m, w H 1 (Ω m A h U h H(curl, Ω, A 12h, A 13h V h L 2 (Ω m and φ 1 H per (Ω m 43 / 63

44 MSFEM with A in 3D Study of Different Multiscale Approaches: MSA1 Ã = A + φ 1 + (φ 1 w, w H 1 (Ω m A 12 A 13 Table: Comparison of Eddy Current Losses. RS MSA1 MSA2 MSA3 MSA4 Losses in W Rel. Error in % / 63

45 MSFEM with A in 3D Study of Different Multiscale Approaches: MSA2 Ã = A + φ 1 A 12 A 13 Table: Comparison of Eddy Current Losses. RS MSA1 MSA2 MSA3 MSA4 Losses in W Rel. Error in % / 63

46 MSFEM with A in 3D Study of Different Multiscale Approaches: MSA3 Ã = A + φ 1 + w A 12 A 13 φ 1x Table: Comparison of Eddy Current Losses., w L 2 (Ω m RS MSA1 MSA2 MSA3 MSA4 Losses in W Rel. Error in % / 63

47 MSFEM with A in 3D Study of Different Multiscale Approaches: MSA4 Ã = A + φ 1 + w A 12 A 13 φ 1x Table: Comparison of Eddy Current Losses., w H 1 (Ω m RS MSA1 MSA2 MSA3 MSA4 Losses in W Rel. Error in % / 63

48 MSFEM with A in 3D Study of Different Multiscale Approaches: MSA1 MSA2 MSA3 MSA4 Finite element subspaces: curl à = curl A + φ 1x curl à = curl A + φ 1x curl à = curl A + φ 1x curl à = curl A + φ 1x A 13 A 12 A 13 A 12 A 13 A 12 A 13 A 12 + curl( (φ 1 w, w H 1 (Ω m + curl wφ 1x, w L 2 (Ω m + φ 1x w z, w H 1 (Ω m w y A h U h H(curl, Ω, A 12h, A 13h V h L 2 (Ω m and φ 1 H per (Ω m 48 / 63

49 MSFEM with A in 3D Higher Order Multiscale Approach in 3D: Only odd terms are considered... φ 1x à = A + φ 1 A 12 + φ 3 A 32 + w 1 + w 3 A 13 A 33 Finite element subspaces: A h U h H(curl, Ω A 12h, A 13h, A 32h, A 33h V h L 2 (Ω m w 1h, w 3h W h H 1 (Ω m and φ 1, φ 3 H per (Ω m φ 3x 49 / 63

50 MSFEM with A in 3D Anisotropic Material: Unit vectors: e... perpendicular to the lamination (e = e x e... any other vector (e e = Conductivity: σ σ = σ, σ =, σ = k f σ Fe σ Reluctivity: ν = ν ν, ν ν = k f µ Fe + 1 k f µ, ν = 1 µ 1 k f µ Fe 5 / 63

51 MSFEM with A in 3D Higher Order Multiscale Approach in 3D: Ã = A + φ 1 A 12 + φ 3 A 32 + w 1 A 13 A 33 φ 1x + w 3 φ 3x Comparison of the eddy current losses in the frequency domain 51 / 63

52 MSFEM with A in 3D Higher Order Multiscale Approach in 3D: Ã = A + φ 1 A 12 + φ 3 A 32 + w 1 A 13 A 33 φ 1x + w 3 φ 3x Eddy current losses with respect to time. 52 / 63

53 MSFEM with A in 3D Finite Element Models: FE model for the reference solution (RS. FE model for MSFEM. 53 / 63

54 MSFEM with A in 3D Higher Order Multiscale Approach in 3D: Ã = A + φ 1 A 12 + φ 3 A 32 + w 1 A 13 A 33 Numerical Data: φ 1x Table: Number of degrees of freedom. + w 3 φ 3x Total No. H(curl, Ω L 2 (Ω m H 1 (Ω m RS 4 41, , MSFEM 161, , 774 1, 62 a 11, 711 b a Four times. b Two times. 54 / 63

55 Benchmark for Iron Laminates Benchmark for Iron Laminates: preliminary design! Geometry of the Benchmark. Dimensions of the Benchmark. 55 / 63

56 Benchmark for Iron Laminates Benchmark for Iron Laminates: preliminary design! Coils: No. turns per coil N = 1 Wire d = 3.mm Laminates: Magnetization curve of non-oriented iron. No. n = 3 Thickness d =.3mm Fill factor ff = 96% σ = S/m µ is isotropic 56 / 63

57 Benchmark for Iron Laminates Benchmark for Iron Laminates: preliminary design! - Biot-Savart field - Newton method - Network coupling: voltag excitation - Hexahedral edge finite elements of 2 nd order unknowns 57 / 63

58 Acknowledgments: Markus Schöbinger, PhD Student Haik Davtjan, Master Student 58 / 63

59 Thank you for your attention! 59 / 63

60 Various Two Dimensional Multiscale Finite Element Formulations for the Eddy Current Problem in Iron Laminates Workshop MSHOM216 Advances in Multi-Scale Methods and Homogenization for Laminates and Windings in magnetic Felds Technische Universita t Wien, Vienna, Austria, on Sept , 216 Riesenrad, Giant Ferris Wheel Wiener Staatsoper, Vienna State Opera Stefansdom, St. Stephen s Cathedral in Vienna Schloß Scho nbrunn, Scho nbrunn Palace 6 / 63

61 Various Two Dimensional Multiscale Finite Element Formulations for the Eddy Current Problem in Iron Laminates Workshop MSHOM216 Advances in Multi-Scale Methods and Homogenization for Laminates and Windings in magnetic Felds Technische Universita t Wien, Vienna, Austria, on Sept , 216 Map of Location New Building for Electrical Engineering 61 / 63

62 Workshop MSHOM216 Advances in Multi-Scale Methods and Homogenization for Laminates and Windings in magnetic Felds Technische Universität Wien, Vienna, Austria, on Sept , 216 Aim: The aim of the workshop is to bring together experts in the field of mathematical and numerical modeling of static or quasi-static magnetic fields in laminates and windings, to discuss recent developments and industrial applications, to identify challenging mathematical and engineering problems and new research directions. Keynote Speakers: Prof. Oszkár Bíró Prof. Patrick Dular Prof. Kay Hameyer Prof. Joachim Schöberl Workshop Organizer: Dr. Karl Hollaus Workshop Venue: Gußhausstraße 29, 14 Vienna 62 / 63

63 Workshop MSHOM216 Advances in Multi-Scale Methods and Homogenization for Laminates and Windings in magnetic Felds Technische Universität Wien, Vienna, Austria, on Sept , 216 Topics: Contributions dealing with one of the following topics are welcome: 1 Numerical Techniques: Formulations 2 Geometry: Step-Lap-Joints, Ventilation Ducts, etc. 3 Material Modeling: Anisotropy, Hysteresis, etc. 4 Coupled Multi-physics Problems 5 Benchmark Problems Call for presentations: 1 Abstract: about 2 words,... Important Dates: 1 Deadline for the abstract: June 5, Notification of acceptance: June 3, Registration deadline: Sept. 9, 216 Workshop homepage 63 / 63