Magnetic-Structural Parallel Coupled Analysis ofSimplified MRI model with Mesh Smoothing
Shin-ichiro Sugimoto 1, Hideki Fujii 1, Shunji Kataoka 1 and Shinobu Yoshimura 1
1

1

The University of Tokyo, Japan

Introduction

There are various general-purpose computational mechanics
systems for simple phenomenon analysis such as fluid analysis,
structural analysis, heat conduction analysis, magnetic fieldanalysis, and so on. However, because most actual phenomena
are mutual related compound phenomena with more than twokinds of phenomena, the coupled analysis that contains themutual related compound phenomena is required for obtaining
characteristics of the actual phenomena in detail. Therefore, weare developing general-purpose coupled analysis systems which
integrate more than two kinds of solvers. In this study, we
consider to develop a general purpose Magnetic-Structural
coupled analysis system that is particularly useful to a medical
Magnetic Resonance Imaging (MRI) equipment, where thevibrations of the yoke, due to the electromagnetic force
generated by the gradient coil.
Moreover, considering recent numerical analysis, acomputational object like the MRI is a highly complicatedmechanical object. In addition, subdivision of the mesh is
performed for the improvement of accuracy. Therefore, largescale computations are increasingly important. To reply thisrequirement, we will use solvers that adopt Hierarchical Domain
Decomposition Method (HDDM) [1][2] that achievements tosolve problems with over 1 billion Degrees of Freedom (DOF).
As a solver for magnetic analysis, ADVENTURE_Magnetic
module is used [3][4]. And as a solver for structural analysis,
ADVENTURE_Solid module is used [2][5]. These modules arefinite element analysis solvers designed in the ADVENTURE
Project [6] to perform the magnetic analysis and the structural
analysis.
In the previous study, we have considered the magneticstructural coupled analysis without the mesh smoothing [7].
Therefore, we were able to handle only very small vibration. In
this paper, in order to handle a larger vibration, the meshsmoothing is introduced into the magnetic-structural coupledanalysis system. As the mesh smoothing method, we consider
the Jacobian-based stiffening [8][9].
2Magnetic-Structural
Mesh Smoothing

CoupledAnalysisf) If difference of the displacement between new one andprevious one is enough small, this procedure is finished.
Otherwise, this procedure returns to c).
Because only the displacement of the structures is considered inthis procedure, elements in the air domain that is on boundariesof the structures and the air domain are crushed or reversed
when the structural displacement is large. Therefore, we wereable to handle only very small vibration.

Air domainStructures(a) Field for the magnetic field analysis.

(b) Field for the structural analysis.

withIn the previous study, the magnetic-structural tow-way
coupled static analysis was performed in the following
procedure [7]:a) Electromagnetic force that works on the structures isobtained by the magnetic field analysis of the structures
and the air domain (Fig. 1 (a)).
b) Structural displacement is obtained by the structural
analysis of the structures (Fig. 1 (b)).
c) The mesh for the magnetic field analysis is modified
with the displacement of the structures.
d) The magnetic field analysis is performed again.
e) The structural analysis is performed again.

(c) Field for the mesh smoothing.

Fig. 1. Fields for analyses.
In order to handle a large vibration, the mesh smoothing isintroduced. In this paper, we consider the Jacobian-based
stiffening [8][9] as the mesh smoothing method. In this method,
we consider the linear elasticity analysis of the air domain (Fig.
1 (c)). The structural displacements on the boundaries of thestructures and the air domain are given as the Dirichlet boundary
conditions. Furthermore, to make this method more robust,
selective mesh deformation based on the element sizes isimplemented. Generally, the element sizes around the structuresthat may move greatly are small. On the other hand, theybecome large as they become far from the structures. If small
elements around the structures are deformed, because they arecrushed or mesh quality turns worse, the coupled analysis fails
easily. Therefore, the Young’s modulus is made large as
element size becomes small. Hereby, even if small elements
move greatly, it is hard to deform them. In this paper, we
assume the reciprocal number of the volume of the element theYoung’s modulus.
And now, new procedure is as follows:a) Electromagnetic force that works on the structures isobtained by the magnetic field analysis of the structures
and the air domain (Fig. 1 (a)).
b) Structural displacement is obtained by the structural
analysis of the structures (Fig. 1 (b)).
c) Mesh smoothing of the air domain (Fig. 1 (c)).
d) The mesh for the magnetic field analysis is modified
with the displacements of the structures and the air
domain.
e) The magnetic field analysis is performed again.
f) The structural analysis is performed again.
g) If difference of the displacement between new one andprevious one is enough small, this procedure is finished.
Otherwise, this procedure returns to c).
3Numerical Result with Simplified Model

To verify our method basically, a simplified model (Fig. 2(a)) is considered. This model consists of a plate and apermanent magnet. The plate is transformed by electromagnetic
force of the permanent magnet.
Fig. 3 shows the transformation of the plate. By introductionof the mesh smoothing, even if the displacement is larger thanelement sizes, we were able to obtain actual solutions without
failure of analysis.
4[7]S. Sugimoto, V. Magron and S. Yoshimura, Parallel Vibration
Analysis of Magnetic-Structural Coupled Phenomena of MRI
Model, 9th Wrold Congress on Computational Mechanics and 4th
Asian Pacific Congress on Computational Mechanics(WCCM/APCOM2010), CD-ROM, 2010.
[8] T. Tezduyar, M. Behr, S. Mittal, and A. Johnson, Computation ofUnsteady Incompressible Flows with the Finite Element
Methods—Space-Time Formulations, Iterative Strategies andMassively Parallel Implementations, New Methods in Transient
Analysis, Vol. 246, pp. 7–24, 1992.
[9] K. Stein, T. Tezduyar, and R. Benney, Mesh Moving Techniques
for Fluid-Structure Interactions with Large Displacements, Journalof Applied Mechanics, Vol. 70, No. 1, pp. 58-63, 2003.
[10] K. Miyata, K. Ohashi, A. Mureaoka and N. Takahashi, 3-D
magnetic field analysis of permanent-magnet type of MRI taking
account of minor loop, IEEE Transactions on Magnetics, VOL.42,No.4, pp.1451-1454, 2006.

AirPlate

Permanentmagnet(a) Simplified model.

Conclusion

In this paper, we have introduced the mesh smoothing to themagnetic-structural coupled analysis. In the future work, we will
adopt this system to the MRI model [10].

(b) Mesh for the magnetic field analysis.

Acknowledgements
The support of “Research and Development of
Innovative
Simulation
Software”
project
isgratefully acknowledged.
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Fig. 2. Simplified model.

Fig. 3. Transformation of the plate (x100).