25C07
Shape Reconstruction Method Based on Magnetic Measurement and Boundary
Element Method for the Peripheral Region in Non-Axisymmetric Fusion Plasma
磁気計測と境界要素法を用いた非軸対称核融合プラズマの周辺磁場領域形状の同定
Masafumi Itagaki1), Gaku Okubo1), Masayuki Akazawa1), Yutaka Matsumoto1),
Kiyomasa Watanabe2), Ryosuke Seki2) and Yasuhiro Suzuki2)
1)
板垣正文 ，大久保岳1)，赤澤眞之1)，松本裕1)，渡邊清政2)，關良輔2)，鈴木康浩2)
1)

2)

Hokkaido University, Kita 13, Nishi 8, Kita-ku, Sapporo 060-8628, Japan
National Institute for Fusion Science, 322-6 Oroshi-cho, Toki-city, GIFU 509-5292, Japan1)
北海道大学, 〒060-8628 札幌市北区北13条西8丁目2)
核融合科学研究所, 〒509-5292 岐阜県土岐市下石町322-6

The Cauchy condition surface (CCS) method, originally developed for axisymmetric tokamak plasma, hasbeen expanded to reconstruct the 3-D magnetic field profile outside the non-axisymmetric plasma in theLarge Helical Device. The boundary integral equations (BIEs) in terms of 3-D vector potential for fieldsensors, flux loops and points along the CCS are solved simultaneously. With the use of ‘twisted CCS’, thereconstructed magnetic field shows acceptable accuracy. The magnetic field line tracing using thereconstructed field indicates the outer surface of the stochastic region precisely, and the last closed
magnetic surface agrees fairly well with the reference one.

1. Introduction
Usually the plasma boundary shape in a nuclear
fusion device is indirectly estimated with the aid ofcomputing from signals of magnetic sensors locatedoutside the plasma. Kurihara [1] proposed the Cauchy
condition surface (CCS) method [1,2]. The method
focuses on tokamaks, i.e., axisymmetric plasmas, sothat the analyses can be made in a 2-dimensional(2-D), r-z system. On the other hand, 3-D analyses are
required for non-axisymmetric plasma. In a helicaltype device such as the Large Helical Device (LHD),
it is important to consider the following characteristicsof the plasma current:(i) The plasma current itself is much weaker than
the toroidal current in a tokamak device.
(ii) The dominant plasma current is the so-called
Pfirsch-Schülter current, the average of which
over a magnetic surface is zero. However, thiscurrent still has a 3-D profile.
The present work is an extension of the CCS method
to non-axisymmetric, 3-D fusion plasma. This 3-D
analysis consumes a huge number of unknowns, andthen the problem becomes very ill-conditioned.
2. Three-Dimensional CCS Method
The Cauchy condition surface (CCS), where boththe Dirichlet and the Neumann conditions areunknown, is hypothetically placed in a domain thatcan be supposed to be inside the plasma. In theanalysis, no plasma current is assumed outside thisCCS, where in reality plasma current does exist.
A 3-D Cartesian coordinate system is adopted torealize a boundary-only integral formulation. The firststep of the analysis is to solve the following boundary

integral equations (BIEs) and obtain the values of thevector potential and its derivative on the CCS in sucha way that they will be consistent with the sensor
signals [3].
(i) For magnetic field sensor locations i :B j  W j B  *  A    Lkj i*  k  Ak  Lkj i  d .
nk CCS  n  (1)

( j  r ,  , z ; k  x, y , z )The operator

corresponds to a term in B   AjLk

with A expressed in Cartesian coordinates, W j B  isthe contribution of external coil currents, and i* thefundamental solution of the Laplace equation.
(ii) For flux loops:
For example, the BIE for a loop in the toroidal
direction is written as (Tor )  W Tor  Ak    n     d   A    n d d  (2)k  CCS20 2

kik0ki ( k  x, y )with x   R sin  and  y  R cos  .
(iii) For points i on the CCS ( CCS ): A
 * 
1Ak , i    i* k  Ak i d . (k  x, y, z ) (3)2nn CCS The above three types of BIEs are discretized,
coupled and expressed in a matrix equation form.
Considering the 10-fold rotational symmetry of theLHD in the toroidal direction, the number of
unknowns is reduced by a factor of 10.
The matrix equation to be solved has the form(4)Dp  g ,where the solution vector p contains the vector
potentials and their normal derivatives on the CCS.
This equation is solved using the truncated singularvalue decomposition (TSVD) technique. The matrix
D is decomposed as D  UΛV T , where U and V T
are orthogonal matrices and Λ is a diagonal matrix
with positive singular value or zero components.
The regularized solution is given by(5)p  VΛk 1UT g .
Here Λ k means that the singular values smaller thank in Λ are omitted so that the condition number is
not larger than 105 .
Once all the components in p are known, themagnetic fields for arbitrary points can be calculatedusing Eq.(1) again.

Figure 1 shows the Poincaré plots of the field line on
the r-z plane at   18 (the horizontal elongatedcross section). The white and the black round symbols
are the results following the reference field and thereconstructed field, respectively. The white closed line
indicates the LCMS given from the reference field.
The reconstructed outer surface of the stochastic
region shows a good agreement with the referenceone.

3. Use of Twisted CCSAn axisymmetric CCS having a circular cross sectionis the simplest model [3]. However, the better idea is a‘twisted CCS’ whose elliptic cross section rotates withthe variation in vacuum vessel geometry in thetoroidal direction. Independent of the toroidal angle,
this CCS can keep a certain distance from its surface
to the plasma boundary, so that a reduction in thenumerical error can be expected. The CCS is divided
into 48 boundary elements, each of which has 9 nodal
points.

In Fig. 2, the dashed closed line shows the LCMS
for the vacuum field, i.e.,   0% . This is shifted
outward when the beta takes the nonzero value,  2.7% , i.e., the reference LCMS in this case is
the solid closed line. The round symbols show theresults of the trace originating at the same starting
point as the reference LCMS for   2.7% , but
based on the reconstructed field obtained using the3-D CCS method. They do not form a sharp closed
surface, however, the round symbols are distributed
almost along the reference LCMS for   2.7% .

4. Numerical ExamplesOne here considers the plasma with a volumeaveraged beta being   2.7% in the LHD. The
reference field and the sensor signals (126 flux loopsand 440 field sensors) for this condition had been
calculated beforehand using the HINT2 code [4].
In the greater part of the region outside the LCMS
(last closed magnetic surface), the absolute errors ofBr , B and Bz reconstructed using the 3-D CCS
method were less than 0.01T.

z (m)

0.5

LCMS(Vacuum)Starting
point

0-0.5

ReconstructedLCMS
33.5r (m)44.5

Fig.2. Field line tracing (2): LCMS

5. Conclusion
A prototype of 3-D CCS method code has beendeveloped, in which the formulation is based on the3-D distribution of vector potential. A 3-D test
calculation has been made for non-axisymmetricplasma in the LHD. The magnetic field outside theplasma can be reconstructed with a fairly acceptableaccuracy if a large number of magnetic sensors can belocated outside the plasma. The magnetic field line
tracing indicates the plasma boundary precisely, andthe LCMS agrees fairly well with the reference one. It
should be stressed that they were reconstructed using
only sensor signals.
References[1] K. Kurihara: Fusion Eng. Des. 51-52 (2000) 1049.
[2] M. Itagaki et al.: Nucl. Fusion 45 (2005) 153.
[3] M. Itagaki et al.: Plasma Phys. Control. Fusion 53(2011) 105007.
[4] Y. Suzuki et al.: Nucl. Fusion 46 (2006) L19.

Fig.1. Field line tracing (1): Peripheral region

Magnetic field line tracing was then carried out.