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Bielefeld University
D2 PHYSICS
Hybrid FEM-BEM
approach for open
boundary magnetostatic
problems Two- and three-dimensional
formulations
A. Weddemann, D. Kappe, A. Hütten
Presented at the 2011 COMSOL Conference
Boston, MA
Motivation 2
Have ever experienced this?
More than you want to know …
10/13/2011 A. Weddemann
… and the area of
interest is somewhere
there …
You have some electromagnetic problem …
This may be fine as long as we are
only interested in the properties of
the electromagnetic field. But
usually these equations form only a
small part of a much larger system.
Is it possible to eliminate the
degrees of freedom in the auxiliary
domain?
Equilibrium state of a
ferromagnetic ring
… or you do not know how
to choose the exterior
domain in order to keep the
additional error small?
Open boundary approaches 3 Governing equations
10/13/2011 A. Weddemann
0B
Maxwell’s equations of magnetostatics:
0H φH
0( )µB M H
φ M
magVr
nr
φˆ ˆn n M
φ φ φ1 2
φ1 M
magn Vrφ1 0
magVr
\
magVr
φˆ 1 ˆn n M
φ φ
mag2 1( ) ( ) ( , )
ˆVG dr r r r r
n
n = 2
Greens function G depends
on the system dimension n:
φ2 0
π
1( , ) ln(| |)2
G r r r r
π
1 1( , )4 | |
G r rr r
n = 3 Let’s try the decomposition with
Ωπ φ14 1( ) 1r
nr
nr
Vmag
∂Vmag
n
Open boundary approaches 4 Implementation into COMSOL Multiphysics
10/13/2011 A. Weddemann
φ1 M
magn Vrφ1 0
magVr
\
magVr
φˆ 1 ˆn n M
φ φ
mag2 1( ) ( ) ( , )
ˆVG dr r r r r
n
φ2 0
Ωπ φ14 1( ) 1r
nr
nr
magVr
magVr
% Integration coupling variables
expr{1} = -phi1/(4*pi)* ...
(sign(x)*abs(nx)*(dest(x)-x) ...
+sign(y)*abs(ny)*(dest(y)-y) ...
+sign(z)*abs(nz)*(dest(z)-z))*( ...
(dest(x)-x)^2+ ...
(dest(y)-y)^2+ ...
(dest(z)-z)^2)^(-3/2);
fem.weak{1} = test(phi1x)*(phi1x - Mx) ...
test(phi1y)*(phi1y – My) ...
test(phi1z)*(phi1z – Mz);
No longer need for
auxiliary domain
Solution is exact ϕ1-mode
ϕ2-mode hybrid
FEM-BEM approach
magVr
Benchmark model 5 Outer regions
10/13/2011 A. Weddemann
- locally vanishing magnetic moment
- in-plane confinement for high aspect ratio
Domain Dirichlet data Solution
Arbitrary exterior
domain with angular
surface …
… that does not affect
the solution of the
magnetic stray field …
… due to an appropriate
choice of Dirichlet
boundary conditions.
Benchmark model 6 Performance analysis
10/13/2011 A. Weddemann
Error analysis
( ) | | 1mag
1
( )Δ h h H VH φ φ φ
#Elem.
97
324
1251
9098
18119
#Bnd. Elem.
80
152
320
1240
3495
ΔH1(ϕh)
0.0757
0.0470
0.0324
0.0230
0.0036
tsol [s]
1.466
1.981
3.113
17.191
34.725
|
mag
-1/22| dh
Vφ φ r
> 40000
0.0079
3.2
pure FEM
Sparsity plots
43725²-matrix
9966²-matrix
Presolving step
Influence of boundary topology 7 Homogeneously magnetized cube
10/13/2011 A. Weddemann
Ωπφ φ φ
mag
142 1 1( ) ( ) ( , ) ( ) 1
ˆVG dr r r r r r
n
Ωπ
π
4 inner point( )
2 along smooth surfacer
2π
½π
Such a definition
does not make
sense for L²-
functions
unphysical
features
...
% Solid angle, Boundaries
fem.appl{3}.mode = FlPDEWBoundary;
... % use values along edges as
% boundary values
% Solid angle, Edges
fem.appl{4}.mode = FlPDEWEdge;
... % use values at corners as
% boundary values
High aspect geometries 8 Weak equations for finite element discretization
10/13/2011 A. Weddemann
φ φ1 1( ) ( , )x yr
Γ Γ
φφ φ
π1
1 1 2 2 3/2ˆ 4 ( / 4)z
xy z
a dx dyGd
ar
n r
Γ Γ
φ φπ|| ||
1 1 3
ˆˆ 4 | |
xyz
xy
aGd d
n rr r
n r
ˆ ˆ( ) ( )xy
x x y yr x y
High aspect ratio geometries:
Integral decomposition
FEM-BEM: 0.17 s
FEM: 1.72 s
ax = 1.2, ay = 0.8, az = 0.1
Increased performance due to reduced dimensionality,
thickness only enters as numerical parameter.
www.spinelectronics.de Bielefeld University
D2 PHYSICS
Conclusion & Outlook
Hybrid FEM-BEM approaches can be implemented into COMSOL Multiphysics
So far, for full three-dimensional problems, no increase of performance can be reported due to decreased sparsity of the matrix
Increased performance for geometries of high aspect ratios
Further optimization of solver
parameters and settings
Implementation of hybrid FEM-BEM
methods in ferromagnetic systems
10/13/2011
9
A. Weddemann
Conclusion Outlook