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Magnetic Field Synthesis in an Open-Boundary Region
Exploiting Thévenin Equivalent Conditions
Paolo DI BARBA, PhD Dept of Industrial and Information Engineering University of Pavia, Italy paolo.dibarba@unipv.it
Outline of the presentation
Optimal shape design with Thévenin like conditions Conditions of equivalence Finite element (FE) equations of the analysis problem Optimal design procedure The H-shaped electromagnet as a benchmark Controlling the FE mesh
Case study: maglev device Field analysis Field synthesis Optimisation problem Optimisation results Conclusion
Outline of the presentation (II)
INTRODUCTION When solving analysis problems in electricity and
magnetism by means of finite-element (FE), often a small part of the field domain includes the region of main interest.
Nonetheless, the analysis of the whole domain has to be performed, even in subdomains of little or no interest. In the case of repeated field analyses, like e.g. in design problems, the computational burden grows up.
In (Santini and Silvester, 1996) a principle of field diakoptics was presented: the region which is not of interest is replaced by means of the generalized Thévenin theorem.
The systematic use of Thévenin-like conditions in field analysis is now proposed as a useful method to solve optimal shape design problems.
OPTIMAL SHAPE DESIGN
WITH THEVENIN-LIKE CONDITIONS
Given a field region W3 , let the controlled region Wc , and the region Wd in which the design variables x are defined, belong to a subdomain W2 included in W3 . Moreover, let the complementary subdomain W1 = W3 \ W2 be filled in by linear materials.
Then, an optimal shape design problem defined in W3 can be solved acting only on the finite-element grid discretizing W2 , after replacing W1 with the corresponding Thévenin n-pole N1 .
Field model
G1
G2
G12
W1
W2
Wc
Wd
Thévenin model FE model
W2
1
2
n
N1
linear n-pole
CONDITIONS OF EQUIVALENCE
When J12 is the same in both domains W1 and We , the potential at the common boundary G12 is the same in
W1UW2 and in WeUW2
with u magnetic potential. In particular, an equivalent domain We replacing W1 can be identified which has the same line current J12 as W1
After a cut along the common boundary G12 , the magnetic effect of W1 on W2 (and vice versa) can be restored by means of a line current source J12 acting along G12 i.e.
21222
1
2 W inJJu 11211
1
1 W inJJuand
2122222 W inJCJuK1121111 W inJCJuKand
FINITE-ELEMENT EQUATIONS OF THE ANALYSIS PROBLEM
1
11
1t1e CKCK
11
1t1ee JKCKJ
e222
t
2e22 JCJuCKCK
eq.s for Thévenin-like conditions
equations for domain W2
nT continuity equations along G12
2pn governing equations
Stiffness matrix and current density vector for equivalent domain We
Augmented system
modified stiffness matrix
modified current density vector
FE model
No matrix should be explicitly inverted
0
J
J
J
u
u
0IC
IK0
C0K
e
2
12
e
2
nt2
ne
22
T
T
2pn
Ke is a dense matrix
2pT nn
THEVENIN EQUIVALENCE FOR THE ANALYSIS PROBLEM
1
11
1t1e CKCK
11
1t1ee JKCKJ
Stiffness matrix and current density vector for the equivalent domain We replacing domain W1
It is not necessary to invert matrix K1 explicitly. In fact, it turns out to be
011
1
112
J
uJK 01112
J
t
ee uCKJ
W1 disconnected
W1 inner source free and unit outer source
1,012111121
JJ
JCuK 1,0111
1
11121
JJ
tt uUCCKC
The only matrix to be explicitly inverted is of size (nT,nT) but nT << np2
1,011
121 JJ
t uUC
NON-LINEAR CASE
If W2 incorporates a non-linear magnetic material, stiffness matrix K2 depends on the unknown potential u2 . Formally, the solving system modifies as follows:
Given an initial estimate of u2 , an iterative procedure can be started to solve the system of non-linear equations (e.g. Newton-Raphson algorithm).
e
t
e JCJuCKCuK 2222222
OPTIMAL DESIGN PROCEDURE
Derivative-free optimisation algorithm
K2 , C2 , and J2 are updated according to the variations determined in W2 by the optimisation algorithm
Minimum of the objective function
Core equation for field analysis
Arrays Ke and Je are computed only at the start
Magnetic field in the reduced domain W2
e
t
e JCJuCKCK 222222
OPTIMAL DESIGN PROCEDURE (II)
Tentative cost of a derivative-free opt procedure
2pifcv3 3
nnnn,nkc on the whole field domain W3
on the field subdomain W2 2pifcv2 2
nnnn,nkc
1n
n
c
c2
p
p
3
2
3
2
possible cost reduction
variables
constraints
objectives
convergence it.s
mesh nodes
THE H-SHAPED ELECTROMAGNET
AS A BENCHMARK
The device
The objective function, to be minimised
Design variables: rectangular coordinates of a set of moving nodes discretizing the pole profile
Prescribed induction field at the air gap
0,BB 0
dc201 ,x,,xBsupB,xBsupf
cc
WWWW
H-SHAPED ELECTROMAGNET - RESULTS
Magnetic potential along G12
Improved pole shape and flux lines
History of the W2 mesh
History of the improvement
FE MESH CONTROL
The number of nodes along the Thévenin boundary should be kept constant, and their position should be prescribed for the equivalence to be exact.
Using a commercial code of FEA the constraint of prescribed node distribution along the Thévenin boundary should be relaxed; moreover, the construction of appropriate connectivity matrices, adjusting the nodes of subdomain W2 is necessary.
When W1 and W2 are connected, let the common boundary G12 exhibit nT nodes. In turn, when W1 and W2 are not connected, let nT1 (nT2) be the number of nodes along the corresponding boundary of W1 (W2).
In this case, a line G12 exhibiting nT nodes can still be defined as a hinge layer between W1 and W2 (node mismatch) .
FE MESH CONTROL (II)
00
0
0 2
12
2
122
1
222
ee
t
t
e
t
J
J
J
u
u
BCB
BK
BCK
np2 eq.s in domain W2
nT1 eq.s for Thévenin conditions
nT continuity eq.s along G12
The system of equations governing the FE model must be recast in matrix form as follows:
K2 stiffness matrix (np2,np2) of domain W2 C2 connectivity matrix (np2,nT2) relating the nT2 boundary nodes of W2 to the np2 nodes of W2 B2 matching matrix (nT,nT2) relating the nT2 boundary nodes of W2 to the nT nodes of G12
u2 node potential vector of domain W2 J2 nodal current density vector of domain W2 Ke stiffness matrix (nT1,nT1) of the equivalent domain replacing W1 B1 matching matrix (nT,nT1) relating the nT1 boundary nodes of W1 to the nT nodes of G12
ue node potential vector at the boundary of W1 when W1 and W2 are connected Je nodal current density vector of the equivalent domain replacing W1 J12 nodal current density vector at G12 when W1 and W2 are connected
FE MESH CONTROL (III)
e
tt
e
t JBCJuCBKBCK 222222222
modified core equation for field analysis in W2
augmented stiffness matrix
loss of sparsity
condition number higher than that of K2
If nT1 = nT and , then matrix B1 is nothing but the identity of order nT1 . As a consequence, the governing equation of vector potential u2 in domain W2 can be deduced in a compact form:
TT nn 2
FE MESH CONTROL (IV)
HTSC
PM PM FC FC FC
Thevenin boundary
-0.03 -0.02 -0.01 0.01 0.02 0.03 0
x1 [m]
x2 [
m]
0.25
0.245
0.24
0.235
0.23
0.225
0.22
CASE STUDY: A MAGLEV DEVICE
Device geometry and materials (NdFeB magnet PM, ferromagnetic field corrector FC, high-temperature supercon HTSC)
Field diakoptics: incorporate the excitation system of the maglev device (i.e. PMs and FCs) in a subregion bounded by a closed line, along which Thévenin–like conditions in terms of potential are applied.
The HTS sample (48 mm times 6 mm) is supposed to be in the zero-field state; accordingly, it is modelled as a perfectly diamagnetic material.
2 mm wide air gap
FIELD ANALYSIS
Complementary domain W1 : FE mesh (5,617 nodes, 129 Thévenin nodes)
Reduced domain W2 :
FE mesh (2,014 nodes, 93 Thévenin nodes) and flux lines
FIELD ANALYSIS (II)
21 x,xu
120Je121
ee1
ee uuJKJKu12
In principle, the analysis of the magnetic field in W2 is based on the Poisson’s equation in terms of vector potential u, subject to non-homogeneous Dirichlet’s condition
In the FE scheme, the nodal approximation to function is given by vector ue , which has two contributions:
no-load potential ( W1 and W2 not connected )
on-load potential ( W1 and W2 connected )
along G12
as well as symmetry condition u = 0 along x1 = 0
FIELD ANALYSIS (III)
In practice, the analysis of the magnetic field in W2 is based on the direct solution of the FE equation in terms of nodal vector potential u2 , subject to homogeneous Dirichlet’s condition u2 = 0 at x1 = 0 (symmetry axis).
e
tt
e
t JBCJuCBKBCK 222222222
FIELD ANALYSIS (IV)
As a result of the interaction between HTS and magnets, a repulsion effect is originated. Elementary levitation force density:
dSuDJdSJBdF 2212
specific current carried by the sheet modelling the magnet
derivative component D2 acts on the elementary shape function interpolating potential u2
FIELD SYNTHESIS
Given a sample of HTS, which is assumed to be in the zero-field state, find the magnetic field distribution such that the levitation force density, i.e. the force density F2 acting on the HTS in the x2 direction, is maximised.
In practice, the field synthesis problem is converted into a problem of optimal shape design, in which the dimensions of permanent magnets and field correctors are selected as design variables (six-dimensional design vector ).
OPTIMISATION PROBLEM
,22 xFf
Design vector
Objective function
1 and 5 half-width and height of the central field corrector 2 and 6 width and height of the magnets 3 and 4 width and height of the lateral field corrector
subject to suitable bounds
6
levitation force density acting on the HTSC, to be maximised wrt
An evolution strategy of lowest order, which has proven to be cost-effective and global-optimum oriented, was used as the optimisation algorithm: a (1+1) scheme, in which the fittest individual between parent and offspring survives to the next generation.
OPTIMISATION RESULTS
In a typical optimisation run, the evolution of the mesh discretizing domain W2 makes the number of nodes along the Thévenin boundary G12 oscillate.
History of the number of Thévenin nodes during the maximisation of the objective f (levitation force density acting on HTSC).
Initial geometry with flux lines F2 = 1.833 Nmm-1
Final geometry with flux lines F2 = 3.446 Nmm-1
OPTIMISATION RESULTS (II) Discrete-valued design variables (step equal to 1 mm). Search tolerance at convergence:
The maximisation of levitation force density implies to increase the permanent magnet size and modify the field corrector shape.
410/
CONCLUSION
Thévenin-like conditions are a useful tool to reduce the field domain when field analysis must be repeated, as in the case of synthesis problems.
The example examined refers to magnetic field synthesis in an open-boundary domain.
The reduced field domain can be both linear or non-linear, while the complementary domain must be linear.
A still critical aspect is the construction of connectivity matrices adjusting the nodes of subdomain to the Thévenin boundary.
Suitable meshing techniques, in fact, would help improve the solution of the synthesis problem.
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