50 3D Structural Optimization in Electromagnetics
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50 3D Structural Optimization in Electromagnetics
Thirteenth International Conference on Domain Decomposition Methods
Editors: N. Debit, M.Garbey, R. Hoppe, J. P磂riaux, D. Keyes, Y. Kuznetsov
c
2001 DDM.org
50 3D Structural Optimization in Electromagnetics
R.H.W. Hoppe
1
, S. Petrova
2
, V. Schulz
3
Introduction
We consider the optimal design and layout of high power electronic devices that are based on
the pulse width modulation technique such as DC-AC converter modules used in applications
as electric drives for high power electromotors. The design objective is to minimize power
losses caused by eddy currents that build up in the device due to fast switching times and steep
current ramps (cf., e.g., [BFS99, BHM01, DGH98]).
In mathematical terms this leads to a topology optimization problem with the electric conduc-
tivity of the material as the design parameter and the electric and the magnetic eld as the state
variables that are supposed to satisfy the quasistationary limit of Maxwells equations. With
the optimal design of mechanical structures described by continuum mechanical models being
by now a well established discipline (cf., e.g., [BEN95] and the references therein), not much
work has been done with regard to the optimization of systems whose operational behavior is
governed by Maxwells equations. Moreover, the use of modern discretization and numerical
solution techniques such as multigrid and domain decomposition methods for optimization
problems with PDE constraints is still in its infancy (cf., e.g., [HEI00, HPS01, MAS00]).
In this paper, we focus on an approach relying on a primal-dual Newton interior-point method
for the discretized optimization problem where the discretization of the eddy currents equa-
tions is taken care of by curl-conforming edge elements. Domain decomposition methods on
nonmatching grids can be used for the numerical solution of the discretized eld equations
which is an integral part of the optimization routine featuring logarithmic barrier functions
and simultaneous sequential quadratic programming.
The topology optimization problem
We consider a DC-AC converter module consisting of specic semiconductor devices such
as IGBTs (Insulated Gate Bipolar Transistors) and GTOs (Gate Turn-Off Thyristors) that are
interconnected and linked to the high power source as well as the load by copper made bus
bars (cf. Figure 1).
Each bus bar contains a certain number of ports where currents are either supplied to or taken
off the bar. The IGBTs and GTOs serve as valves for the currents which can be in the range
of several kA. During operation of the module, electromagnetic elds
and
are generated
that can be described by the eddy currents equations
％
％Θ
curl
div
curl
!
(1)
1
Institute of Mathematics , University of Augsburg, D-86159 Augsburg, Germany
2
Institute of Mathematics , University of Augsburg, D-86159 Augsburg, Germany ; on leave from the Bulgarian
Academy of Sciences, Soa, Bulgaria
3
Weierstrass Institute Berlin, D-10117 Berlin, Germany
478
HOPPE, PETROVA, SCHULZ
PSfrag replacements
Figure 1: DC-AC converter module
ⅰ
ぃ
(2)
where
and
stand for the magnetic induction and the current density,
denotes the mag-
netic permeability, and
is the electric conductivity.
Considering a module
Е┄
with
bars
"!#"
, each bar containing
ports
$
%
&(')0
, and introducing a scalar electric potential
1
and a magnetic vector
potential
2
according to
3
grad
143
2
curl
2
we are led to the following coupled system of PDEs
div
56
grad
187
in
(3)
9BA
grad
1
C
3ED
%
5
7
on
$
%
else
(4)
2
curl
8F
curl
2
C
3@
grad
1
in
in
GIHQPR
(5)
with appropriate initial and boundary conditions.
Note that in (4) we refer to
D
%
as the uxes associated with the ports
$
S%
satisfying
T
U
T
8V
%W
D
S%
.
The total inductivity caused by the eddy currents can be described by the functional
X
56
Y1
Y2`7ba
dce8f
g
%
f
h
g
i
p
q
rts
X
S%ug
h
i
5
7
s
vxw
y
Y
v
(6)
3D STRUCTURAL OPTIMIZATION IN ELECTROMAGNETICS
479
Here,
X
S%ug
h
i
5
7
are the generalized transient inductivity coefcients
X
S%ug
h
i
5
7Ra
F
q
V
S%
5
ⅰ
78A
5
7
h
i
5
ⅰ
7
w
where
%
denotes the current density generated by
D
S%
at the port
$
S%
of the bus bar
and
5
A
7
is the solution operator associated with (5).
The design objective is to distribute the material in terms of the electric conductivity
as the
design parameter in such a way that the total inductivity is minimized
うエ
g
g
X
5
1
Y2I7
(7)
subject to the equality constraints
1
and
2
satisfy the state equations (3),(4),(5)
(8)
q
w
(9)
and the inequality constraints
!
(10)
where
#")$%'&
and
$!
refers to the conductivity of copper.
Note that (10) represents relaxed constraints on the design parameter, since allowing only
!
or
t$(
would lead to an ill-posed optimization problem. In practice, we
scale the conductivity by means of
)
567
5
3#
min
10
max
3
min
7
2"
0
&
(11)
with an appropriately chosen
354
.
The primal-dual Newton interior-point method
The discretization of the state equations (3),(4),(5) is performed as follows: For the interior-
exterior domain problem (5) we use a domain decomposition approach on nonmatching grids
featuring individual edge element discretizations of the interior and exterior domain prob-
lems with respect to simplicial triangulations
6
78@9
A
and
6
7(BC9
A
whereas the discretization in
time is done by the backward Euler scheme. Moreover, the elliptic boundary value problem
(3),(4) is discretized by means of nonconforming Crouzeix-Raviart elements. The electric
conductivity
serving as the design parameter is discretized by elementwise constants, i.e.,
D
A
5
7
9
A
S
7
E
9
A
7
p
F3
A
a
card
6
78@9
A
. Comprising the discrete state variables
D
1
A
and
D
2
A
to a vector
D
G
A
5
D
1
A
D
2
A
7
p
, the discretized state equations can be stated in compact form
H
A
5
D
A
7
D
G
A
D
I
A
(12)
480
HOPPE, PETROVA, SCHULZ
If we further denote by
X
A
5
D
A
D
1
A
D
2
A
7
the discretized objective functional, the topology
optimization problem in the discrete regime reads as follows:
う
E
g
E
g
E
X
A
5
D
A
D
1
A
D
2
A
7
(13)
subject to the constraints
D
G
A
5
D
1
A
D
2
A
7
p
satises
5
7
(14)
A
5
D
A
7
a
E
f
s
イ
s
7
9
A
(15)
min
D
A
D
A
max
D
A
(16)
where
ě
6
7(8
9
A
u
3
A
, and
D
A
a
5
S
S
7
p
.
Among the most efcient numerical solution techniques for constrained optimization prob-
lems like (13)-(16) are primal-dual Newton interior-point methods (cf., e.g., [ETT96, FOG98,
GOW98]). The idea is to take care of the inequality constraints (16) by parametrized logarith-
mic barrier functions
A
5
D
A
D
G
A
7
a
X
A
5
D
A
D
1
A
D
2
A
7
3
log
5
D
A
3А$%
D
A
7
log
5S!
D
A
3
D
A
7
and to couple the equality constraints (14),(15) by Lagrangian multipliers. This gives rise to
the saddle point problem
う
E
g
E
"!$#
%
E
g
&
E
'
7
9
A
5
D
A
D
G
A
D
(
A
)
A
7
(17)
for the Lagrangian
'
7
9
A
5
D
A
D
G
A
D
(
A
)
A
7
a
)
A
5
D
A
D
G
A
7
D
(
p
A
5
H
A
5
D
A
7
D
G
A
3
D
0
A
7
)
A
5
A
5
D
A
7&3
7R
For the solution of the above primal-dual interior-point approach we use simultaneous se-
quential quadratic programming in the sense that Newtons method is applied to the Karush-
Kuhn-Tucker conditions associated with (13). Denoting the Newton increments by
1
D
2
A
a
5
31
D
G
A
41
D
(
A
41
D
A
51
)
A
7
p
, this gives rise to a linear system
6
A
1
D
2
A
D
7
A
(18)
which is solved iteratively by right transforming iterations
1
D
2
98U
A
1
D
2
A
6)@
A
5
BA
7
9
A
7
F
5
D
7
A
3
6
A
1
D
2
A
7
(19)
based on a regular splitting
6
A
6
@
A
CA
7
9
A
3
DA
7
v
9
A
involving an appropriately chosen right
transform
6
@
A
.
The
new
iterate
D
2
(new)
A
a
5
D
G
(new)
A
D
(
(new)
A
D
(new)
A
)
(new)
A
7
p
is then obtained by a line search
D
2
7
new
9
A
g
D
2
7
old
9
A
g
FE
5
G1
D
2
A
7
H
HI
(20)
where the steplengths are tested by means of a hierarchy of merit functions. We refer to
[HPS00, HPS01] for details.
3D STRUCTURAL OPTIMIZATION IN ELECTROMAGNETICS
481
Domain decomposition on nonmatching grids
The simultaneous sequential quadratic programming approach being integral part of the primal-
dual Newton interior-point method, described in the previous section, requires an iterative
solver of the discretized state equations. In this section, we briey sketch a domain decom-
position technique on nonmatching grids for the implicitly in time discretized equation (5)
with respect to a nonoverlapping geometrically conforming decomposition
with
skeleton