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Topology OptimizationRadhouane KHLISSA, Frdric GILLON,
Noureddine TAKORABET
L2EP, GREEN
Emails: [email protected],
[email protected],
[email protected]
Introduction
The art of structure is where to put holes Robert Le RicoletA
part of the cogging torque in the permanent magnet synchronous
machines is due to the voltage ripple caused by the magnetic flux
wave shape. The shape of the air gap directly affects the quality
of
the torque generated by the motor.
Topology optimization is a technique used to find the optimal
topology of a structure submitted to certain constraints, without
taking any restriction. It is widely used in mechanical structures
design. This
technique is a promising approach to solve the problem of
cogging torque due to the air gap shape.
In this paper, topology optimization of the magnet with the
homogenization method is presented and explained. The
implementation and test are being realized. A FEM model is used for
analysis. The
model used is of a machine designed and conceived by GREEN in
Nancy. Homogenizat ion method is implemented with some constraints
imposed buy the FE software limits . Optimization is realized
under Matlab, with SQP algorithm.
Magnetic flux saturation
Of the iron
Dependence to the air gap shape
Magnetic flux density wave form
in the air gap
Homogenization method
Implementation
Algorithm
Penalization
Conclusion
Magnetic flux density harmonics
Topology optimization consistsof optimizing material layout
within a given designspace, andfor a given
set of constraints suchthat the resulting structure meetsa
prescribed set of performance targets.
Homogenization is one of the most used methods of topology
optimization: it consists in dividing the
design domain in segments, for each segment a material density i
is attributed . Optimization process
finds the optimal values of densities minimizing the objective
function and respecting the constraints.
To obtain a discrete material, a penalizationprocess is
applied.
Domain: continue
Material : discrete
Domain: continue
Material : continue
Domain: discrete
Material : continue
Domain: discrete
Material : discrete
RELAXATION DISCRETIZATION PENALISATION
Optimization problem: Minimization of flux density harmonics in
order to minimize torque ripple
due to air gap shape.
- Design domain: Magnet domain.
- Design variable: Magnet density- Constraints: Maximum flux
density
Used tools :
FEM: Vector Fields PC Opera13
Optimization: Matlab
L2EP Matlab-Opera coupling plate-form
Optimization process Begin
FE Analysis
FE Analysis
Sensitivity Analysis
Optimization solver
SQP
Sensitivity Analysis
Optimization solverSQP
Converge ?
Converge ?
yes
yes
NO
NO
Initial densities vector in input
Densities variables change
Optimal densities vector input,
Introduction of penalization
parameter p
Densities variables change
End
HOMOGENIZATION
PENALIZATION
Topology optimization is a new tool for fully automatic
design.
This approach permits to explore new topologies .
Optimization execution time rises when resolution is higher.
Quality of results is be better with higher number of elements
.
Results are to be given after the completion of works.
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Homogenizat ion process gives continuous values of magnet
density, to converge
to a discrete range of density values, a penalization is
applied.
Density values are taken to the power of a penalization factor,
this will force these
values to converge to 0 or 1.
Values given by the homogenization are taken as initial
point.
Extra optimization iterations are needed.
cogem
tknj
n
k n
bnrt TTeBkNi
BhT
)(.
The Torque is given by:
Harmonics of Magnetic flux density affects directly torque
quality: to reduce cogging torque a suitable
winding and a sinusoidal magnetic flux density wave form are
needed.
Magnet is no longer one piece, it
is discretized, Resolut ion used
depends on software limits.
Optimization process decides
between magnet or air to be put
in each element.
Relaxation and discretization
[Il-Han Park, Jin-KyuByun, Joonsun S. Kang, and Wansoo S. Nah:
Topology Optimization Method for Superconducting System With
Critical Current Condition]
[T.Touya: " Mthodes d'optimisation pour l'espace et
l'environnement " , rapport de thse, Universit de Toulouse 2008.
]
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