fft-based solution of 3d problems in type-ii ... - bgu

FFT-based Solution of 3D Problems inType-II Superconductivity

L. Prigozhin and V. Sokolovsky

Ben-Gurion University of the Negev, Israel

Vestgarden et al. 2012,2013: an FFT-based method for thinfilm magnetization problems;

This work: a new FFT-based method for 3D bulkmagnetization problems.

ICSM 2018 FFT method for 3D problems

Nonlinear 3D eddy current problem: formulation

Let Ω be the SC domain, Γ its boundary, Ωout = R3 \ Ω,h - the magnetic field, j - the current density, e - the electric field.

For simplicity, here we assume:1. the SC domain Ω is countourwise simply connected;

2. SC obeys e = ρ(j )j relation with ρ = e0jc

(|j |jc

)n−1in Ω;

3. the external magnetic field he(t) is uniform.

Our formulation is written for the magnetic field but differs fromthe popular h-formulation.Let h be known at time t and j = ∇× h = 0 in Ωout.Knowing how to find ∂th for a given h we can integrate h in time.An outlook of the numerical method:Since j = ∇× h (Ampere’s law) we can find e = ρ(j )j in Ω.Although e in Ωout remains unknown, it should be such that

∂th = −µ−10 ∇× e (the Faraday law)

yields∂t j = ∇× ∂th = 0 in Ωout.

This condition is used to find the time derivative ∂th iteratively.




Our formulation is written for the magnetic field but differs fromthe popular h-formulation.Let h be known at time t and j = ∇× h = 0 in Ωout.Knowing how to find ∂th for a given h we can integrate h in time.

An outlook of the numerical method:Since j = ∇× h (Ampere’s law) we can find e = ρ(j )j in Ω.Although e in Ωout remains unknown, it should be such that







Our formulation is written for the magnetic field but differs fromthe popular h-formulation.Let h be known at time t and j = ∇× h = 0 in Ωout.Knowing how to find ∂th for a given h we can integrate h in time.An outlook of the numerical method:Since j = ∇× h (Ampere’s law) we can find e = ρ(j )j in Ω.Although e in Ωout remains unknown, it should be such that





Iterative algorithm

Let h at time t be known; then j = ∇× h. We set

e0 =

ρ(j )j in Ω,ρoutj in Ωout,

with a sufficiently high fictitious resistivity ρout and set, as aninitial approximation,

∂th0 = −µ−10 ∇× e0.

A very high ρout can suppress j in Ωout but makes the evolutionaryproblem for h stiff, which inhibits the integration.Instead, we used a lower value of ρout and enforced the condition

∂t j |Ωout = ∇× ∂th|Ωout = 0

by iterative updating ∂th|Ωout .The role of ρout - suppressing the stray current in a thin outsideboundary layer around Ω.


Iterative algorithm: the details

The Biot-Savart law can be written as h = he(t) + Φ[j ] with

Φ[j ] = ∇×∫R3

G (r − r ′)j (r ′, t) dr ′,

where G (r) = (4π|r |)−1 is the Green function. Clearly,

∇× h = ∇×Φ[j ] = j .

To find ∂th, on the i-th iteration we set

∂t j i−1in =

∇× ∂thi−1 in Ω,0 in Ωout

and use the time derivative of the Biot-Savart law to compute

∂thi |Ωout = dhe/dt + Φ[∂t j i−1in ]|Ωout .

If these iterations converge, which is usually fast, we obtain∇× ∂th|Ωout = ∂t jin|Ωout = 0 as desired.


Fourier transform

The operator

Φ[j ] = ∇×∫R3

G (r − r ′)j (r ′, t) dr ′

can be expressed by means of convolutions in R3 and computed inthe Fourier space. Since

F (G ) :=

∫R3

G (r)e ik·rdr =1

|k2|

we obtain

Φ[j ] =

jz ∗ ∂yG − jy ∗ ∂zGjx ∗ ∂zG − jz ∗ ∂xGjy ∗ ∂xG − jx ∗ ∂yG

= F−1

i

|k |2

kyF (jz)− kzF (jy )kzF (jx)− kxF (jz)kxF (jy )− kyF (jx)

We use this representation of Φ for our iterations

∂thi |Ωout = dhe/dt + Φ[∂t j i−1in ]|Ωout .

All spatial derivatives can also be computed in the Fourier space.ICSM 2018 FFT method for 3D problems

Implementation (in Matlab)

To make this algorithm practical, we

define a uniform Nx × Ny × Nz grid in a rectangular domaincontaining Ω and some empty space around it;

compute the operator Φ and all spatial derivatives in theFourier space (with Gaussian smoothing for the derivatives);

replace Fourier transform by its discrete counterpart on thisgrid and use the standard FFT software;

employ a standard ODE solver for integration in time (themethod of lines).

Implementation of this method is not complicated: our Matlabprogram is short (≈ 120 command lines + comments).


FFT vs FEM; benchmark problem, other examples

Finite element methods for 3D magnetization problems:Pecher et al. 2003; Kashima and Elliot 2007,2008; Zhang andCoombs 2011; Grilli et al. 2013; Stenvall et al. 2014; Pardo andKapolka 2017, 2018; Kapolka et al. 2017; Olm et al. 2017, ...

Benchmark problem 5 (H.T.S Modeling Workgroup).Our solution (LP and Sokolovsky, SuST, 2018) is similar to thoseobtained using two different FEM methods by Kapolka, Pardo,Zermeno, Grilli, Morandi and Ribani (2016). The FFT method issimpler; our preliminary comparison suggests the efficiency may becomparable.

Two new examples: magnetic shield and magnetic lens. Scaling:

(x ′, y ′, z ′) =(x , y , z)

l, t ′ =

t

t0, e′ =

ee0, j ′ =

jjc, h′ =

hjcl,

where l is the characteristic size and t0 = µ0jcl2/e0; n = 30.


Example 1: Magnetic shield

Hollow sc cylinder closed from one end: Gozzelino et al. (2016).

1 he first grows along the cylinder axis, then perpendicular to it;

2 he first grows perpendicularly to the axis, then along the axis.

Gray: SC cylinder, black: shielded

area. Scaled dimensionless units.

Left: case 1, right: case 2.

Dashed line - |he|, solid line -

average |h| in the shielded zone.


Example 1: Magnetic shield, case 1

Domain: −1.6 ≤ x,y,zR

≤ 1.6Mesh:1283: CPUtime 2h;1923: CPUtime 12h

|j|jc

(left) and|h|jc R

(right).

Top: hejc R

= (0, 0, 0.09);

Bottom:hejc R

= (0.09, 0, 0.09).


Example 2: Magnetic lens

SC cylinder with a coaxial biconical hole and a narrow slit toprevent shielding circumference current: ZY Zhang et al. (2012).

Gray: superconductor, black: lens

central area. Dimensionless units.

Dashed line - |he|, blue - < |h| >in the central area, red - measured.


Example 2: Magnetic lens

|j |jc

(left) and |h|jcR

(right) forhe,zjcR

= 0.1;

Mesh: 128× 128× 256 in the domain |x|,|y |R ≤ 1.8, |z|

R ≤ 3.6CPU time 6h (for 0 ≤ t ≤ 0.2).


Conclusion

A new FFT-based method for 3D magnetization problems hasbeen derived, tested, and applied to two realistic problems;

The main advantages are:

simplicity: much easier to implement than the FE methods,although further comparison is desirable;generality: applicable also to multiply connected SC domains,the power law can be replaced by another e(j ,h) relation;

The method can be efficient; its (further) parallelization is oneof the improvements we are going to explore.

Extending the FFT method to 3D modeling of SC/FM hybridsystems is a direction of our future research.

Thank you!1. LP and V. Sokolovsky, SuST 31 (2018) 055018;2. LP and V. Sokolovsky, ArXiv 1803.01346


fft-based solution of 3d problems in type-ii ... - bgu

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