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