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Sliding Interfaces for Eddy Current Simulations Raffael Casagrande Supervisor: Prof. Dr. Ralf Hiptmair Seminar of Applied Mathematics ETH Zrich April 17th, 2013 Raffael Casagrande (ETH Zrich) Sliding Interfaces for Eddy Current April

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SLIDE 1

Sliding Interfaces for Eddy Current Simulations

Raffael Casagrande Supervisor: Prof. Dr. Ralf Hiptmair

Seminar of Applied Mathematics ETH Zürich

April 17th, 2013

Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 1 / 25

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SLIDE 2

Outline

1

Introduction Motivation

2

Deriving the eddy current model Maxwell’s Equations in a moving frame The eddy current model in a moving frame

3

Discontinuous Galerkin Formulation DG Theory Aspects of the implementation

4

Results and Conclusion

Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 2 / 25

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SLIDE 3

Motivation

Generator circuit breakers

◮ translational motion

Electric engines

◮ rotation Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 3 / 25

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SLIDE 4

Motivation

Generator circuit breakers

◮ translational motion

Electric engines

◮ rotation Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 3 / 25

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SLIDE 5

Outline

1

Introduction Motivation

2

Deriving the eddy current model Maxwell’s Equations in a moving frame The eddy current model in a moving frame

3

Discontinuous Galerkin Formulation DG Theory Aspects of the implementation

4

Results and Conclusion

Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 4 / 25

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SLIDE 6

Outline

1

Introduction Motivation

2

Deriving the eddy current model Maxwell’s Equations in a moving frame The eddy current model in a moving frame

3

Discontinuous Galerkin Formulation DG Theory Aspects of the implementation

4

Results and Conclusion

Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 5 / 25

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SLIDE 7

Maxwell’s Equations

div B = 0 curl E + ∂B ∂t = 0 div E = ρ ε0 curl B − 1 c2 ∂E ∂t = µ0(jf + ji). ⇓ jf = σE, c → ∞    Quasistatic model for slowly varying Electric fields (High conductivities) Eddy Current Model div B = 0 curl E + ∂B ∂t = 0 curl B = µ0

  • σE + ji

div E = ρ ε0

Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 6 / 25

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SLIDE 8

Maxwell’s Equations

div B = 0 curl E + ∂B ∂t = 0 div E = ρ ε0 curl B − 1 c2 ∂E ∂t = µ0(jf + ji). ⇓ jf = σE, c → ∞    Quasistatic model for slowly varying Electric fields (High conductivities) Eddy Current Model div B = 0 curl E + ∂B ∂t = 0 curl B = µ0

  • σE + ji

div E = ρ ε0

Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 6 / 25

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SLIDE 9

Maxwells equations are invariant under Lorentz transformation if E and B transform as ˜ E = γ(E + V × B) − (γ − 1)(E · ˆ V)ˆ V ˜ B = γ

  • B − V × E

c2

  • − (γ − 1)(B · ˆ

V)ˆ V γ := 1

  • 1 − v2/c2

ˆ V = ˆ

V/|ˆ V|

⇓ c → ∞ ˜ E = E + V × B ˜ B = B It can be shown that the eddy current model is also invariant under Rotation!!!

Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 7 / 25

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SLIDE 10

Maxwells equations are invariant under Lorentz transformation if E and B transform as ˜ E = γ(E + V × B) − (γ − 1)(E · ˆ V)ˆ V ˜ B = γ

  • B − V × E

c2

  • − (γ − 1)(B · ˆ

V)ˆ V γ := 1

  • 1 − v2/c2

ˆ V = ˆ

V/|ˆ V|

⇓ c → ∞ ˜ E = E + V × B ˜ B = B It can be shown that the eddy current model is also invariant under Rotation!!!

Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 7 / 25

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SLIDE 11

Two eddy current formulations

Temporal gauged Potential formulation : curl 1 µ curl A + σ∂A ∂t = ji A(t = 0) = 0 curl A × n = 0

  • n ∂Ω

H-formulation : curl 1 σ curl H + µ∂H ∂t = curl 1 σji H(t = 0) = 0 H = 0

  • n ∂Ω

Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 8 / 25

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SLIDE 12

Two eddy current formulations

Temporal gauged Potential formulation (Rest frame): curl 1 µ curl A + σ∂A ∂t = ji + σV × curl A A(t = 0) = 0 curl A × n = 0

  • n ∂Ω

H-formulation (Rest frame): curl 1 σ curl H + µ∂H ∂t = curl 1 σji + curl (µV × H) H(t = 0) = 0 H = 0

  • n ∂Ω

Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 8 / 25

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SLIDE 13

Two eddy current formulations

Temporal gauged Potential formulation (Moving frame) : ˜ curl 1 µ ˜ curl˜ A + σ∂ ˜ A ∂t = ˜ ji ˜ A(t = 0) = 0 ˜ curl˜ A × n = 0

  • n ∂Ω

H-formulation (Moving frame): ˜ curl 1 σ ˜ curl ˜ H + µ∂ ˜ H ∂t = ˜ curl 1 σ ˜ ji ˜ H(t = 0) = 0 ˜ H = 0

  • n ∂Ω

Note: If ji is smooth enough, 1

µ curl A = H

⇒ Do the same simulation and compare the two models (Primal & Dual formulation).

Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 8 / 25

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SLIDE 14

Transformation laws

The coordinates of the moving frame (˜ x) are related to the rest frame (x) by x = T(t)˜ x + r(t). T: Rotation matrix.

Transformation laws

T˜ E = E + V × B T˜ B = B T˜ ji = ji T ˜ H = H T˜ jf = jf T˜ V = −V T˜ A = A − T t TT grad (V · A) ⇒ Use transformation laws to derive transmission conditions at sliding interface.

Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 9 / 25

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SLIDE 15

Transformation laws

The coordinates of the moving frame (˜ x) are related to the rest frame (x) by x = T(t)˜ x + r(t). T: Rotation matrix.

Transformation laws

T˜ E = E + V × B T˜ B = B T˜ ji = ji T ˜ H = H T˜ jf = jf T˜ V = −V T˜ A = A − T t TT grad (V · A) ⇒ Use transformation laws to derive transmission conditions at sliding interface.

Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 9 / 25

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SLIDE 16

Transformation laws

The coordinates of the moving frame (˜ x) are related to the rest frame (x) by x = T(t)˜ x + r(t). T: Rotation matrix.

Transformation laws

T˜ E = E + V × B T˜ B = B T˜ ji = ji T ˜ H = H T˜ jf = jf T˜ V = −V T˜ A = A − T t TT grad (V · A) ⇒ Use transformation laws to derive transmission conditions at sliding interface.

Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 9 / 25

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SLIDE 17

Outline

1

Introduction Motivation

2

Deriving the eddy current model Maxwell’s Equations in a moving frame The eddy current model in a moving frame

3

Discontinuous Galerkin Formulation DG Theory Aspects of the implementation

4

Results and Conclusion

Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 10 / 25

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SLIDE 18

DG Formulation of the Eddy Current Model

σ∂A ∂t + curl 1 µ curl A = ji curl A × n = 0

  • n ∂Ω

DG Variational formulation

Find A(i)

h ∈ Vh, i = 1, . . . , N such that for all A′ h ∈ Vh, we have

  • σA(i+1)

h

− A(i)

h

δt , A′

h

  • + aSWIP

h

(A(i+1)

h

, A′

h) =

  • ji,(i+1), A′

h

  • Where Vh :=
  • Pk

3(Th)

3, Pk

d(Th) :=

  • v ∈ L2(Ω)
  • ∀T ∈ Th, v|t ∈ Pk

d(T)

  • .

Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 11 / 25

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SLIDE 19

Symmetric-Weighted-Interior-Penalty Bilinear form aSWIP

h

aSWIP

h

(Ah, A′

h) =

  • Ω

1 µ curlh Ah · curlh A′

h

−

  • F∈Fi

h

  • F

1 µ curlh Ah

  • ω

·

  • A′

h

  • T

−

  • F∈Fi

h

  • F

1 µ curlh A′

h

  • ω

· [Ah]T +

  • F∈Fi

h

ηγµ,F hF

  • F

[Ah]T ·

  • A′

h

  • T

{Ah}ω = ω1Ah,1 + ω2Ah,2, [Ah]T = nF × (Ah,1 − Ah,2) (1) ω1 = µ1 µ1 + µ2 , ω2 = µ2 µ1 + µ2 , γµ,F = 2 µ1 + µ2 (2)

Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 12 / 25

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SLIDE 20

Convergence

Under regularity conditions on the mesh sequence (matching) and assuming the exact solution A is smooth enough we can prove

  • √σ(A(N) − A(N)

h

)

  • L2(Ω) +
  • Cstabδt

N

  • i=1
  • A(i) − A(i)

h

  • 2

SWIP

1/2 ≤ Ct

1/2

F

  • C1hk + C2δt
  • where C1 = maxt∈[0,tF] |A(t)|Hk+1(Ω) and C2 = maxt∈[0,tF]
  • ∂2A(t)

∂t2

  • L2(Ω) The

constants C1,C2 and C are independent of h and δt. |A|SWIP :=  

  • 1

õ curlh A

  • 2

L2(Ω)

+

  • F∈Fh

γmu,F hF [A]T2

L2(F)

 

1/2 Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 13 / 25

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SLIDE 21

Convergence

Under regularity conditions on the mesh sequence (matching) and assuming the exact solution A is smooth enough we can prove

  • √σ(A(N) − A(N)

h

)

  • L2(Ω) +
  • Cstabδt

N

  • i=1
  • A(i) − A(i)

h

  • 2

SWIP

1/2 ≤ Ct

1/2

F

  • C1hk + C2δt
  • where C1 = maxt∈[0,tF] |A(t)|Hk+1(Ω) and C2 = maxt∈[0,tF]
  • ∂2A(t)

∂t2

  • L2(Ω) The

constants C1,C2 and C are independent of h and δt. |A|SWIP :=  

  • 1

õ curlh A

  • 2

L2(Ω)

+

  • F∈Fh

γmu,F hF [A]T2

L2(F)

 

1/2 Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 13 / 25

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SLIDE 22

Aspects of the implementation

In comparison to FEM, DG has much more degrees of freedom

◮ Use DG only along the non-matching interfaces.

Incorporate the transformation formulas for the moving frame into the DG fluxes.

◮ No convective terms appear

2D spatial discretization:

◮ 1st order Edge functions of the first kind for vectorial problem. ◮ 1st order Lagrange elements for scalar problem.

Use NGSolve and Netgen.

Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 14 / 25

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SLIDE 23

Outline

1

Introduction Motivation

2

Deriving the eddy current model Maxwell’s Equations in a moving frame The eddy current model in a moving frame

3

Discontinuous Galerkin Formulation DG Theory Aspects of the implementation

4

Results and Conclusion

Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 15 / 25

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SLIDE 24

Convergence analysis with analytical solution

Construct an analytical radial solution, Hz = Hz(|x|) to 2D scalar H-formulation (TE). Let Ω1 rotate at ω = 20rad/s. Measure rate of convergence in L2 and SWIP-norm.

(a) The subdomains Ω1 and Ω2 (b) The solution at time

t = 0.05 sec(δt = 0.0005, h = 0.0361371)

Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 16 / 25

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SLIDE 25

(a) δt convergence, h = 0.0180874 (b) h convergence, δt = 2.5e − 4

Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 17 / 25

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SLIDE 26

Comparison with F. Rapetti et al.

F . Rapetti used Mortar Method to deal with non-conforming mesh. 2D, scalar H-formulation (Transverse magnetic). Simulation time: 0.2s. ji = 0

Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 18 / 25

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SLIDE 27

(a) DG, background shows |curl2D Hz| (b) Mortar method Figure: Visualization of curl2D Hz for ω = 630rad/s

Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 19 / 25

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SLIDE 28

(a) DG, background shows |curl2D Hz| (b) Mortar method Figure: Visualization of curl2D Hz for ω = 6300rad/s

Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 20 / 25

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SLIDE 29

Complex rotational setting

Circle rotating in square (as before). Compare H-formulation with temporal gauged A-formulation by measuring

  • 1

µ curl2D A − Hz

  • L2(Ω).

◮ A: 2D vector ◮ H: 1D scalar

Excitation by impressed current ji = (4y, −2x). ω = 4π, tend = 1. T˜ A = A − T t

0 TT grad (V · A)

Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 21 / 25

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SLIDE 30

(Movies)

Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 22 / 25

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SLIDE 31

Convergence

Figure: δt convergence at tend = 1 (one full rotation), h = 0.0254402.

Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 23 / 25

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SLIDE 32

Conclusion

DG approach is a viable alternative to Mortar methods for simulating sliding interfaces. The H-formulation is equivalent to the temporal gauged potential formulation if the correct transformation rules are used. O(h) and O(t) convergence was proven for a system at rest. Outlook

◮ Coulomb gauged potential formulation: No time integration is

needed

◮ Extension to 3D Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 24 / 25

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SLIDE 33

Questions ?

Raffael Casagrande (ETH Zürich) Sliding Interfaces for Eddy Current April 17th, 2013 25 / 25