Upload
shashank11
View
227
Download
0
Embed Size (px)
Citation preview
7/30/2019 Talk Js Trafo
1/38
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
7/30/2019 Talk Js Trafo
2/38
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
7/30/2019 Talk Js Trafo
3/38
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)
Joachim Schoberl Page 2
7/30/2019 Talk Js Trafo
4/38
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
Joachim Schoberl Page 2
7/30/2019 Talk Js Trafo
5/38
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.
Joachim Schoberl Page 3
7/30/2019 Talk Js Trafo
6/38
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.
Joachim Schoberl Page 3
7/30/2019 Talk Js Trafo
7/38
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
Joachim Schoberl Page 4
7/30/2019 Talk Js Trafo
8/38
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
First order Nedelec elements of second type:
Vh = {v H(curl) : v|T P1}
second order for A-field, first order for B-field
Joachim Schoberl Page 4
7/30/2019 Talk Js Trafo
9/38
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
First order Nedelec elements of second type:
Vh = {v H(curl) : v|T P1}
second order for A-field, first order for B-field
Second order Nedelec elements of second type:
Vh = {v H(curl) : v|T P2}
third order for A-field, second order for B-field
Joachim Schoberl Page 4
7/30/2019 Talk Js Trafo
10/38
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
Joachim Schoberl Page 5
7/30/2019 Talk Js Trafo
11/38
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
Joachim Schoberl Page 5
7/30/2019 Talk Js Trafo
12/38
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.
Joachim Schoberl Page 6
7/30/2019 Talk Js Trafo
13/38
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:
Joachim Schoberl Page 7
7/30/2019 Talk Js Trafo
14/38
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
Joachim Schoberl Page 7
7/30/2019 Talk Js Trafo
15/38
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
Joachim Schoberl Page 8
7/30/2019 Talk Js Trafo
16/38
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
Joachim Schoberl Page 8
7/30/2019 Talk Js Trafo
17/38
Anisotropic meshing of thin layers
Coil and thin shield:
Thin prisms in shield, pyramid transition elements to connect to tet-mesh:
Joachim Schoberl Page 9
7/30/2019 Talk Js Trafo
18/38
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.
Joachim Schoberl Page 9
7/30/2019 Talk Js Trafo
19/38
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)
Joachim Schoberl Page 10
7/30/2019 Talk Js Trafo
20/38
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
'
Joachim Schoberl Page 11
7/30/2019 Talk Js Trafo
21/38
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.
Joachim Schoberl Page 11
7/30/2019 Talk Js Trafo
22/38
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).
Joachim Schoberl Page 12
7/30/2019 Talk Js Trafo
23/38
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).
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.
Joachim Schoberl Page 12
7/30/2019 Talk Js Trafo
24/38
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).
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.
NGSolve: A finite element module
Mechanical and magnetic field problems Adaptivity and geometric multigrid solvers Intensively object oriented C++ (Compile time polymorphism by templates)
Joachim Schoberl Page 12
7/30/2019 Talk Js Trafo
25/38
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).
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.
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
Joachim Schoberl Page 12
7/30/2019 Talk Js Trafo
26/38
Coarse mesh of transformer generated by Netgen:
22k elements, 26k complex dofs
Joachim Schoberl Page 13
7/30/2019 Talk Js Trafo
27/38
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
Joachim Schoberl Page 14
7/30/2019 Talk Js Trafo
28/38
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
Joachim Schoberl Page 15
7/30/2019 Talk Js Trafo
29/38
Magnetic flux in shields
Joachim Schoberl Page 16
7/30/2019 Talk Js Trafo
30/38
Eddy losses in casing
Joachim Schoberl Page 17
7/30/2019 Talk Js Trafo
31/38
Magnetic flux density
Joachim Schoberl Page 18
7/30/2019 Talk Js Trafo
32/38
Loss density in pressing plates
Joachim Schoberl Page 19
7/30/2019 Talk Js Trafo
33/38
Eddy current density
Joachim Schoberl Page 20
7/30/2019 Talk Js Trafo
34/38
Ongoing work: Homogenization of laminated core
Framework: 2-Level FEM by Morgan-Babuska, Schwab-Matache
Isolation
Fe
Joachim Schoberl Page 21
7/30/2019 Talk Js Trafo
35/38
Ongoing work: Homogenization of laminated core
Framework: 2-Level FEM by Morgan-Babuska, Schwab-Matache
Isolation
Fe Oscillating vector potential:
A(x)
A (x)
A (x)1
0
Joachim Schoberl Page 21
7/30/2019 Talk Js Trafo
36/38
Ongoing work: Homogenization of laminated core
Framework: 2-Level FEM by Morgan-Babuska, Schwab-Matache
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
Joachim Schoberl Page 21
7/30/2019 Talk Js Trafo
37/38
Work in progress / future
Higher order elements, mixed order
Homogenization of laminated material
Approximation with higher order harmonics
Automatic topology/shape optimization of shields
Joachim Schoberl Page 22
7/30/2019 Talk Js Trafo
38/38
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
Joachim Schoberl Page 22