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7/30/2019 Talk Js Trafo

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Eddy Current Simulation in

Power Transformers by Multigrid Methods

Peter HambergerVA TECH EBG

Transformatoren GmbH

Joachim SchoberlSFB Scientific Computing

Johannes Kepler University Linz

Joachim Schoberl Page 1


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Eddy Current Simulation in

Power Transformers by Multigrid Methods

Peter HambergerVA TECH EBG

Transformatoren GmbH

Joachim SchoberlSFB Scientific Computing

Johannes Kepler University Linz



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The equations

Time-harmonic, low frequency Maxwell equations:

curl E = iH

curl H = ji + E

Ferromagnetic casing is modelled as a non-linear surface impedance boundary condition:

H n = jS(E)



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The equations

Time-harmonic, low frequency Maxwell equations:

curl E = iH

curl H = ji + E

Ferromagnetic casing is modelled as a non-linear surface impedance boundary condition:

H n = jS(E)

Magnetic vector potential A such that

E = iA H = 1 curl A



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Weak formulation

Weak formulation on H(curl):

1 curl A curl v dx +

i Av dx +

jS(A)vds =

jiv dx

Setting > 0 everywhere gives an implicit gauging.



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Weak formulation

Weak formulation on H(curl):

1 curl A curl v dx +

i Av dx +

jS(A)vds =

jiv dx

Setting > 0 everywhere gives an implicit gauging.

Core and shields have high permeability (r = 35000)

By now, there are no conducting materials. Conductivity of laminated parts is neglected. We set verysmall (0.01S).

Nonlinear surface impedance b.c. at casing, and internal magnetic parts (pressing plates).

Current density in coils is known.



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H(curl) finite elements

First order Nedelec elements:

Vh = {v H(curl) : v|T = aT + bT x}

first order approximation for A-field and B-field



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First order Nedelec elements of second type:

Vh = {v H(curl) : v|T P1}

second order for A-field, first order for B-field



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First order Nedelec elements of second type:


second order for A-field, first order for B-field

Second order Nedelec elements of second type:


third order for A-field, second order for B-field



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Vertex-Edge-Face-Cell shape-functions

V-E-F-C shape functions for a scalar field: linear vertex shape functions

edge shape functions bi-orthogonal to V-E dofs, some extension to F-C

face shape functions bi-orthogonal to V-E-F dofs, some extension to C

cell shape functions bi-orthogonal to all dofs

WE

WF



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Vertex-Edge-Face-Cell shape-functions

V-E-F-C shape functions for a scalar field: linear vertex shape functions

edge shape functions bi-orthogonal to V-E dofs, some extension to F-C

face shape functions bi-orthogonal to V-E-F dofs, some extension to C

cell shape functions bi-orthogonal to all dofs

WE

WF

E-F-C shape functions for the H(curl) field:

bi-orthogonal to dofs associated with lower-dimensional geometric entities

Splitting shall be compatible with differential operator :

WE VE and WF VF

V

VE

F



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Finite element problem

Plug fe space into weak formulation. Find Ah Vh s.t. for all vh Vh:

1 curl Ah curl vh dx +

iAh vh dx +

jS(Ah)vh ds =

jivh dx

System of equations:

K(A) = j

nonlinear due to non-linear surface impedeancy b.c.

Solved by outer Newton iteration.

In every Newton step one has to solve a complex-symmetric linear equation.

Solved with multigrid preconditioned QMR iteration.



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Hiptmair / Arnold-Falk-Winther smoothers for H(curl) problems

Block Gauss-Seidel smoother with blocks Vi such that

j : Wj Vi,

where the scalar space splits as Wh =

Wj.

Gradient ofvertex shape function:Hiptmair blocks

1

1

1

11

Arnold-Falk-Winther:Use large blocks:



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Hiptmair / Arnold-Falk-Winther smoothers for H(curl) problems

Block Gauss-Seidel smoother with blocks Vi such that

j : Wj Vi,

where the scalar space splits as Wh =

Wj.

Gradient ofvertex shape function:Hiptmair blocks

1

1

1

11

Arnold-Falk-Winther:Use large blocks:

Hiptmair: one iteration is cheaper, less memory

Arnold-Falk-Winther: less iterations, simpler implementation



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Low-order high-order two-grid method

Assemble high order matrix K, e.g., Nedelec 2b elementsAssemble low order matrix K1 with Nedelec 1 elements

A good preconditionier for K is obtained by the two-grid iteration:

Do a block Gauss-Seidel iteration w.r.t. edge-face-(cell) blocks

Do a coarse grid correction with the low order matrix



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Low-order high-order two-grid method

Assemble high order matrix K, e.g., Nedelec 2b elementsAssemble low order matrix K1 with Nedelec 1 elements

A good preconditionier for K is obtained by the two-grid iteration:

Do a block Gauss-Seidel iteration w.r.t. edge-face-(cell) blocks

Do a coarse grid correction with the low order matrix

Requires factorization of low-order matrix, which is about 9 times smaller.

fast convergence, if shape functions fulfill

WE VE and WF VF



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Anisotropic meshing of thin layers

Coil and thin shield:

Thin prisms in shield, pyramid transition elements to connect to tet-mesh:



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Anisotropic meshing of thin layers

Coil and thin shield:

Thin prisms in shield, pyramid transition elements to connect to tet-mesh:

Problem: Are there higher order H(curl)-elements for pyramids available ?

Lowest order, type one by Hiptmair, Lowest order, type two by adding gradients.



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Semi-coarsening plus line-smoothing for anisotropic mesh refinement

Standard iterative methods (including standard multigrid) break down for highly anisotropic meshes.

Way out: Semi-coarsening, line-smoothing, or a combination of both (Pflaum, B orm+Hiptmair,Apel+Schoberl)



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Local mesh refinement based on a posteriori error estimators

Generate initial mesh

c

Compute FE Solution

c

Compute Error Estimator Refine Mesh

cdd

dddd

dd

dd

dd

Accuracy

reached?

no

T

'



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Local mesh refinement based on a posteriori error estimators

Generate initial mesh

c

Compute FE Solution

c

Compute Error Estimator Refine Mesh

cdd

dddd

dd

dd

dd

Accuracy

reached?

no

T

'

Zienkiewics-Zhu error estimator:Compute smoothed

A = S(A) and

B = S(B) (sub-domain by sub-domain). Then, the element error

contributions are

T =

T

1|B B|2 + |A A|2 dx.



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Software

When using standard software, you do not have the possibility to use your favorite algorithms. You need

source code access, or, at least an application programming interface (API).



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Software



In Linz, we develop FE software, namely

NETGEN: An automatic tetrahedral mesh generator

Constructive Solid Geometry (CSG) modeling Delaunay and advancing frond mesh generation algorithms

Free for universities, licensed by 300 groups.



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Software







NGSolve: A finite element module

Mechanical and magnetic field problems Adaptivity and geometric multigrid solvers Intensively object oriented C++ (Compile time polymorphism by templates)



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Software







NGSolve: A finite element module

Mechanical and magnetic field problems Adaptivity and geometric multigrid solvers Intensively object oriented C++ (Compile time polymorphism by templates)

Pebbles: Algebraic Multigrid Solver [S. Reitzinger]

only needs finite element matrix (sometimes also mesh topology) for scalar and vector valued problems, edge elements, massively parallel



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Coarse mesh of transformer generated by Netgen:

22k elements, 26k complex dofs



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Anisotropic elements for thin shields

Height of shields 2m, thickness of shields 1.6cm, distance to wall 0.4cm.

Prism elements in and behind shields, pyramid transition elements



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Simulation data

Second order type 2 - Nedelec elements

4 Levels of adaptive refinement, 500k complex dofs

2 Newton iterations per level

about 20 QMR-Multigrid iterations per Newton iteration

Simulation time on Pentium III, 1GHz: 25 min, memory: 1 GB



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Magnetic flux in shields



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Eddy losses in casing



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Magnetic flux density



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Loss density in pressing plates



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Eddy current density



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Ongoing work: Homogenization of laminated core

Framework: 2-Level FEM by Morgan-Babuska, Schwab-Matache

Isolation

Fe



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Isolation

Fe Oscillating vector potential:

A(x)

A (x)

A (x)1

0



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Isolation

Fe Oscillating vector potential:

A(x)

A (x)

A (x)1

0

Expansion: A(x,y,z) = A0(x,y,z) + (x/p)A1(x,y,z)

A0, A1 . . .standard FEM functions

(s) . . .fine scale oscillation



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Work in progress / future

Higher order elements, mixed order

Homogenization of laminated material

Approximation with higher order harmonics

Automatic topology/shape optimization of shields



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Work in progress / future

Higher order elements, mixed order

Homogenization of laminated material

Approximation with higher order harmonics

Automatic topology/shape optimization of shields

Thank you


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