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A Domain Decomposition Analysis of
a Nonlinear Magnetostatic Problem with
100 Million Degrees of Freedom
H.KANAYAMA *, M.Ogino*, S.Sugimoto** and
J.Zhao*
*
**
Kyushu University
The University of Tokyo
1
Contents

Introduction





Backgrounds
Objectives
DDM Applications to Magnetic Field
Problems
Numerical Examples
Conclusions
2
Backgrounds

A large-scale complicated model
A transformer (Moriyasu, S., 2000)
Model offer by Japan AE Power Systems Corporation
and Fuji Electric Advanced Technology Co., Ltd.3
Objectives

Development of ADVENTURE_Magnetic
for analysis of large-scale magnetic field
problems


Parallel computing
Analysis of models with about 100 million
degrees of freedom (DOF)
4
Contents


Introduction
DDM Application to Magnetic Field
Problems





Magnetostaic Problems
DDM
HDDM
Numerical Examples
Conclusions
5
Non-Linear Magnetostatic Analysis

Formulation


Solution for non-linear equations


Newton method
On the interface



A method
Conjugate Gradient (CG) method
A simplified block diagonal scaling
In each subdomain


The mixed formulation with the Lagrange multiplier p
Skyline method with partial pivoting
6
Formulation
rot  rot A  J

div A  0
A n  0
 rot A n  0

 A  n  0
GE
in W,
in W,
on GE ,
on GN ,
on GN ,
W1：air or vacuum
W2：magnetic
material
GN
A
: Magnetic vector potential [Wb/m]
J
: Electric current density [A/m 2 ],
div J  0
in W, J  n  0
n

W
on GN
: Unit normal vector
: Magnetic reluctivit y [m/H]
constant

ν |rot A|
in W1
in W 2
7
Weak formulation
A weak formulation is constructed by the
introduction of the Lagrange multiplier p:
Find  A, p  V  Q such that,for any A* , p* V  Q



 
 

*
*
*


rot
A
,
rot
A

grad
p
,
A

J
,
A
,


*

A
,
grad
p
 0,

( . , . ): the real valued L2-inner product.




V  v  L W ;

2
3
Q  q  H 1 W ;


on G 
rot v  L W ,
q0
3
2
vn  0
on GE
E
8

Finite element approximation
Ah:Nedelec elements of simplex type
ph：Conventional piecewise linear tetrahedral elements
D.O.F.
9
Finite element approximation
 
 rot A , rot A    grad p , A    J , A  ,


 A , grad p   0,

Find  Ah , ph  Vh  Qh such that,for any Ah* , ph* Vh  Qh
*
h
h
h
h
*
h
*
h
*
h
Vh , Qh: Finite element spaces corresponding to V and Q,
Ah : Finite element approximation of A by the Nedelec
elements of simplex type,
ph : Finite element approximation of p by the
conventional piecewise linear tetrahedral elements.
10
Finite element approximation


Elimination of the Lagrange multiplier ph
Correction of electric current density
~
J h  J h  grad I h ,

 

~ *
 h rot Ah ,rot A  J h , Ah .
*
h
11
Newton iteration

Adoption of the Newton iteration to solve
the nonlinear equation
n








n
n 1
*
n

1
n
*
 h rot Ah , rot Ah      Ah rot Ah , rot Ah 

   A 







n





~ *     n 
n
*
 J h , Ah    Ah rot Ah , rot Ah .

   A 








Solver for linear simultaneous equations
12
DDM
(Domain Decomposition Method)
K u  f 
Domain decomposition
 K II1

 0



 0
 1 1T
 RB K IB

0

0




 
K II N 
  RB N  K IB N T
K IB1 RB1T
 1
 f I1 
  uI  





 
















K IB N  RB N T    N    f I N  
u I   N
N

i  i  i T 
i  i  

R
K
R
R

B
BB B   u B 
 B f B 
i 1

 i 1

I: corresponding to inner DOF
B: corresponding to interface DOF
13
DDM
(Domain Decomposition Method)

On the interface


 N i  i 
i T
i  †
i  i T 
R
K

K
K
K
 B BB
IB
II
IB RB  u B
 i 1

 
N

 
i T
  RBi  f Bi   K IB
K IIi 
i 1
†
f Ii 
SuB  g

The interface problem

In each subdomain

(i ) i T
K II(i )uI(i )  f I(i )  K IB
RB uB

The subdomain problem
14
IDDM
(Iterative Domain Decomposition Method)
for n  0,1,.....;
0
B
Choose u ;
In each subdomain
In each subdomain
ComputewIi n by
Comput eu Ii 0 by
i  i 0
K II u I
i 
i 
i T
 f I  K IB RB u
0
B
;(a)
i T i 0
i  i T 0
r i 0  K IB
u I  K BB
RB u B  f Bi 
N
r   RB r
0
i  i 0
(b)
i  i T n
q i n  K IBi T wIi n  K BB
RB w ;
N
q   RBi q i n ;
n
;
i 1
 n  r n  r n wn  q n ;
i 1
T
w r ;
0
K IIi wIi n   K IBi  RBi T wn ;
0
T
u Bn 1  u Bn   n wn ;
r n 1  r n   n q n ;
If r n 1   r 0 , break ;
 n  r n 1  r n 1 r n  r n ;
T
T
wn 1  r n 1   n wn ;
end;
15
The modification for the subdomain problem

In step 0
Compute uIi 0 by
 K II(i )
 ( i )T
 K II p
K II(i )p  u I(i ) 0   f I(i )   K IB(i ) RB(i )T u B0 


(i )   (i ) 0   
K I p I p  u I p   0  
0

(a`)
i T i 0
i  i T 0
r i 0  KIB
uI  KBB
RB uB  f Bi 

In step n
ComputewIi n by
 K II(i )
 ( i )T
 K II p
K II(i p)  wI(i ) n 
 K IB(i ) RB(i )T wn 

(i )   (i ) n   
K I p I p  wI p 
0


(b`)
i T i n
i  i T n
qi n  KIB
wI  KBB
RB w
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HDDM
(Hierarchical Domain Decomposition Method)

Introduction of HDDM (Hierarchical
Domain Decomposition Method) for
computing in parallel environments



Single processor mode (S-mode)
Parallel processor mode (P-mode)
Hierarchical processor mode (H-mode)
17
Contents



Introduction
DDM Applications to Magnetic Field Problems
Numerical Examples





TEAM Workshop Problem 20
Linear Magnetostatic Analysis
Nonlinear Analysis of the model with 100 million DOF
Checking for the accuracy
Conclusions
18
TEAM Workshop Problem 20
Yoke
Center pole
Coil
SS400
polyimide electric wire
|J| = 1,000 [A]
19
TEAM Workshop Problem 20

Elect riccurrentdensity: J  1,000A 
Magnet icreluctivity
: air, coil
yoke,centerpole
Initialvaluesfor thefirst linear iteration


:  1 4  107 m/H
: 0  100m/H
: Ah0  0Wb/m
20
TEAM Workshop Problem 20
Elements
DOF
Subdomains
Model 1
471,541
559,848
8×300
Model 2
952,845
1,125,501
8×600
Model 3
1,769,871
2,083,209
8×1,100
Model 4
9,326,492
10,945,318
56×830
Model 5
38,232,019 44,676,346 56×3,400
Model 6
86,570,893 100,818,053 56×7,730
21
Linear Magnetostatic Analysis
(Computational conditions)
Solver: The Interface problem:
CG method
Judge of convergence  1.0  105
A simplified block diagonal
scaling preconditioner
The subdomain problem:
The proposed method: Skyline method with partial pivoting
The previous method: ICCG method
HDDM: P-mode
PC cluster:
Intel Core2Duo E6600
Memory 8GB
The number of PCs: 4
22
Linear Magnetostatic Analysis
(CPU time)
CPU time by the
previous method
[s]
CPU time by the
proposed method
[s]
Ratio
(%)
Model 1
Model 2
81.6
230.7
53.6
153.3
33.9
33.6
Model 3
497.3
332.1
33.2
Ratio = ( | The previous method – The proposed method |/|
The previous method | ) ×100
23
Linear Magnetostatic Analysis
(Averaged memory)
Ave. memory
by the previous
method [MB]
Ave. memory
by the proposed
method [MB]
Times
Model 1
Model 2
25.3
56.3
53.2
107.4
2.10
1.91
Model 3
104.5
199.3
1.91
Times = ( |The proposed method |/| The previous method | )
24
Linear Magnetostatic Analysis
(Iteration figure of the interface problem)
25
Nonlinear analysis of the model with
100 million DOF
Solver: Non-linear
:Newton method
Judge of convergence  1.0  10-5
The interface problem:
CG method
Judge of convergence  1.0  10 4
A simplified block diagonal
scaling preconditioner
The subdomain problem: Skyline method with partial pivoting
HDDM: P-mode
PC cluster:
Intel Core2Duo E6600
Memory 8GB
The number of PCs: 28
26
Residual norms
Nonlinear analysis of the model with
100 million DOF
Step 0
Step 1
Step 2
Model 4
Step 0
Step 1
Step 2
Step 0
Step 1
Step 2
Model 5
Model 6
Iteration counts on the interface
27
Nonlinear analysis of the model with
100 million DOF
Iteration counts
(Newton method)
CPU time
[s]
Memory per
CPU
[MB]
Model 4
Model 5
2
2
4,495
15,777
197
812
Model 6
2
46,361
1,840
28
Checking for the accuracy
Measured Bz
[T]
I
Relative error[%]
Model 1
Model 2
0.571
0.609
I
20.6
15.4
Model 3
Model 4
Model 5
Model 6
0.628
0.669
0.679
0.681
12.8
7.1
5.7
5.4
0.72
II
Computed
Bz [T]
0.71
II
19.5
14.2
11.6
8.5
4.4
4.1
29
Checking for the accuracy
(VS.Ⅰ)
Model 1
The relative error
Model 2
Model 3
Model 4
Model 5
Model 6
9.8×1.0-4
1.9×1.0-3
The average length of edge [m]
3.9×1.0-3
30
Checking for the accuracy
(VS.Ⅱ)
The relative error
Model 1
Model 2
Model 3
Model 4
Model 5
Model 6
9.8×1.0-4
1.9×1.0-3
3.9×1.0-3
The average length of edge [m]
31
Conclusions



Improvement of ADVENTURE_Magnetic
Demonstration of the possibility of largescale analysis in magnetic field problems
with over 100 million DOF
Future work


Application of strong preconditioners
Coupled analysis of magnetic field and other
phenomena (ex. solid, fluid …etc.)
32