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Procedures to interface plasma disruption simulations and finite element

electromagnetic analyses

R. Roccella1, M. Roccella

2, G. Sannazzaro

1, M. Sugihara

1

1 ITER Organization, Route de Vinon sur Verdon, 13115 St. Paul lez Durance, France ;

2L.T.

Calcoli, Merate, Italy

E-mail contact of main author: riccardo.roccella@iter.org

Abstract. The interface procedures presented in this paper are based on the Secondary Excitations (SE) method that has been developed to solve the issues related to the transferring information between axis-symmetric

Magneto-Hydro-Dynamic (MHD) outputs and Finite Element (FE) codes. The SE is a thin toroid made of an

array of axisymmetric current filaments included in the FE model that must reproduce the EM transient for the

FE analysis. The SE method can be used in two different approaches to an EM analyses: a) for a Global Analysis

(GA) in which all the field sources are inside the SE while the sample to be analysed and all the conducting

structures relevant for the analysis are outside, and for a Zooming Analysis (ZA) in which the sample is inside the

SE toroid while the plasma and most of other conducting structures are outside. In this paper the SE method and

the EMAG code, developed to ease its implementation and validation are presented.

1. Introduction

Very high mechanical loads act on the ITER vacuum vessel and in-vessel components during

EM transients (mainly plasma disruptions) because of the interaction between the currents

induced in the conducting structures by those EM transients and the external high magnetic

fields, typical of the tokamak operations. For the design of many ITER components the EM

loads are the most challenging and thus great care must be taken in their assessment. While

the plasma disruptions are mainly simulated (based on extrapolations of measurements done

in existing tokamaks) by means of 2D axis-symmetric MHD codes the load assessment on the

structures is usually carried out with 3D FE analyses. The accuracy of the method used to

interface the MHD and FE codes has been proven critical for the reliability of the EM analysis

results.

The interface procedures presented in this paper are based on the Secondary Excitations (SE)

method that has been developed to solve the issues related to the transferring of information

between the axis-symmetric MHD and the FE codes.

The SE method has also shown to be suitable to perform detailed analyses (zooming) of small

components inside complex structures like ITER vessel and cryostat. The way to perform

the EM zooming is described in the second part of this paper. This method allows the EM

analysis of small components, while avoiding the huge and time consuming work required by

the contextual modelling of their environment.

2. The Secondary Excitations (SE) method

In the MHD codes the plasma is usually reproduced by means of a cloud of current filaments,

the Primary Excitations (PE), whose currents and positions can change during the time. The

output of a MHD disruption simulation is provided with a very tight time stepping (typically

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in the order of a tenth of millisecond). Such very fine time stepping, while producing a huge

CPU time consumption, is usually of no importance in the FE analyses, where the EM

transients involved and the eddy current time constants are of the order of milliseconds. In

the output of DINA (the MHD code used in ITER to simulate plasma disruptions) the position

and number of filaments varies during the simulation. This fact makes their direct use difficult

in any finite element EM analysis. A moving or very detailed meshing of the plasma region

would be required. For these reasons the use of interfaces to adapt the MHD outputs to the

Finite Element Model (FEM) fixed calculation grids is needed. To perform reliable analyses

the interface procedure must ensure a correct reproduction of the EM transient (in terms of

magnetic field and magnetic field time derivative) taking into account the effects of the

environment in which the component will operate. The implementation of the interface is

normally left to the analyst expertise. It is difficult to check the method reliability and the

related uncertainty can be a major cause of discrepancies in the results of the EM analyses.

The aim of the SE method described in this work is twofold: i) to provide a reliable and easily

checkable procedure to interface the MHD outputs to the FE EM codes and ii) to introduce a

method that can be standardized avoiding dispute in case of discrepancies.

The SE method can be used for two different approaches to the solution of an EM analysis:

a. For analyses where a full tokamak sector is modelled (Global Analyses) and the plasma region is completely included in the SE;

b. For analyses of single components (like a port, or the divertor) where only a local model of the component and the closely surrounding structure is performed. In these

analyses the SE enclose the local model while the plasma region and most part of the

other conducting structures are external to the SE (Zooming Analyses).

Using this method the analyst replaces the axisymmetric filamentary conductors of the MHD

output (PE) with an array of fixed axisymmetric conductors (SE), in green fig.1 left and in

purple in fig. 1 right, that, will surround the plasma region for GA (at left), or part of the

tokamak structure for ZA(at right). The currents evaluated by means of the SE method and

imposed in the array of fixed conductors will reproduce (outside in case of GA and inside in

case of ZA) the same field and the same field time derivative of the original PE.

3. The EMAG code

The SE method has been checked in several works ([3],[4]). At the end a code, EMAG, has

been developed in the C++ programming language to ease the application of the SE procedure

and to assist the analyst in the preparation of the input both for Global and Zooming FE

Analyses. This code is an integral axis-symmetric code based upon well checked semi

analytical routines for the calculus of fields, forces and mutual inductances between axis-

symmetric circuits of any cross section [1]. The EMAG code provides tools to:

• pre-process the MHD disruption simulation outputs of several codes (Max FEA, DINA[5] and others) to optimise the selection of the time steps to be solved in the EM

FE analysis (transient sampling);

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• validate the time step selection by comparing the plasma current and position before and after the transient sampling;

• evaluate, by flux conservation, the currents induced (at each selected time step) by the PE (plasma current filaments and, if the case, other axisymmetric conductors) on each

secondary excitation

FIG. 1. Up: FE models for which the SE method has been used. On the up left the SE (light green

elements) surround the plasma region in a GA for the assessment of EM loads on blanket shield modules

(not shown in the picture); on the up right the SE (purple elements) surround the divertor, lower port

and part of the VV in a ZA for the assessment of divertor EM loads. Down: Corresponding 2D models

used in EMAG to evaluate the currents in the SE. In the lower figures the PE, the plasma filaments

clouds and, for the ZA, the vessel part external to the zooming region, are shown in blue; in green are

the SE.

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• check and compare the field (and its time derivative) generated by the primary and secondary excitations

• write in text files, for each selected time step, the currents to be imposed in each of the SE (the format can be selected to be read bythe most used FE codes for EM anlysis).

For the EM analyses of halo currents DINA provides the position vs. time of ten poloidal

sectors, in which the area wetted by the halo currents is subdivided, and the halo currents

flowing in these sectors is calculated. In another DINA outputs the poloidal flux map vs. time

in the plasma region is given.

As the poloidal current is a flux function, EMAG interpolates the halo current distribution vs.

time on the plasma facing surfaces of the FEM, and write down the results for the 3-D

analyses.

4. Evaluation of SE

The main scope of the EMAG code is anyway the implementation of the SE method which is

based on the evaluation of currents induced in an array of axisymmetric filaments by another

array of filaments in which the current are imposed. To this end a routine that evaluates the

mutual inductance between two axisymmetric filaments has been generalized for

axisymmetric filaments of finite cross section of any shape, following the method described in

[1]. The linear system of differential equations to be solved is given below in matrix form:

(1) M·İ+R·I=N·İpla

where M is square matrix of mutual inductances between the filaments for which the current

must be evaluated, R the diagonal matrix of the resistances, I is the vector of unknown

currents, Ipla is the vector of currents of the field sources and N is the rectangular matrix of

mutual inductance between the source filaments and the filaments of unknown currents. In

case of pure inductive limit (resistance of filaments is negligible) the equation 1 reduces to an

ordinary system of linear equations whose solution is straightforward given by:

(2) I(t)- I(t0)=M-1·N·(Ipla(t)- Ipla(t0))

where M-1

is the inverse matrix of M, evaluated numerically using the LU decomposition

method described in [2]. This solution is used in the global analyses where the unknown

currents are only the current of the secondary excitations that, to compensate exactly the field

variations produced by the field sources must have zero resistance.

In the zooming analyses, the unknown currents in the axisymmetric conductors (like

the vessel) totally or partially external to the zoomed region must be previously evaluated.

Normally for these conductors the resistance cannot be neglected. In EMAG the equation 1 is

solved using the eigenvalues method. The eigenvalues λi of the matrix M-1·R (the λi represent

the time constants of the conductor array), and the associated eigenvectors Λi (i=1…n), where

n is the number of unknown currents, are evaluated numerically by the inverse iteration

method [2]. The equation (1) can be put in diagonal form by the matrix ΛΛΛΛ ≡{Λik} where

Λik is the component k of eigenvector ΛΛΛΛi.. Using the relationship

tΛ ·Λ =U

where tΛ is the transpose of Λ and U the identity matrix, the eq. 1 becomes:

(3) Y =- Λ·(M-1·R)

tΛ·Y+Φ

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Where Y = Λ·I is the transform of the unknown current vector I and Φ= Λ·(M-1·N)· tΛ· Λ·Ipla

is the transform of the known source term vector. Since the transform matrix Λ·(M-1·R)·

tΛ in

eq. 3 is diagonal and the eq. 3 can be written down component by component as follows:

(4) Ẏi=- λiYi+ φi(t)

where Yi is the component i of vector Y and φi is the component i of the source term Φ. Once

equation 4 has been integrated EMAG evaluates the unknown currents in the real circuits by

the inverse transform of the vector Y: I = tΛ ·Y. In the ZA these currents will be added to the

plasma and used as PE to evaluate the currents in the SE.

5. Global and zooming analyses

GA or ZA start with the preparation of the FEM for the EM analyses where the geometry of

the SE is first defined (see fig. 1 top). The geometrical data of the SE are saved in a format

suitable to be imported in EMAG. In parallel, the disruption simulation to be analysed is

selected and the corresponding output file is imported in EMAG for pre-processing. The

EMAG approach can be summarized in two main steps: the first step consists in the selection

of the time steps to be analysed in the FE analysis (transient sampling); the second consists in

the evaluation of the currents vs. time induced by the PE on the SE that will be used as field

sources in the FE analysis (SE evaluation for GA and for ZA).

5.1 Transient sampling

Generally the time steps number can be reduced by a factor between 5 and 20 with respect to

the time steps number of the MHD output without losing any relevant information for the FE

EM analysis. Finer time steps must be used only where abrupt changes in plasma current

and/or position occur. To aid the analyst finding the intervals where a more fine time stepping

is needed, the EMAG code provides, after reading the MHD output, the plot (called IRZ plot)

of total plasma current (Ipla) and plasma current centroid radial (Rpla) and vertical (Zpla)

position:

Where the index i refers to the generic current filament number and npla to the total number

of filaments reproducing the plasma configuration in the MHD output (at the generic time

step).

A dedicated step control window (fig. 3) helps the selection of the time steps while, after the

selection, the IRZ plot (fig. 2) shows the comparison of original (MHD output) plasma current

and position with the ones obtained after the time step selection. In case the superposition of

the two sets of curves shows evident mismatches (fig. 2 left, red circles) the selection can be

refined through the step control window until reaching enough accuracy (fig. 2 right).

Figure 2 (left side) shows that constant time steps of 5 ms (implying the solution of ~180 steps

for a slow + fast downward VDE IV) don’t allow a very accurate reproduction of the transient

while using the variable time stepping of fig. 2 (right side) much better reproduction is

obtained with lower number of steps (86).

pla

npla

ii

pla

pla

npla

ii

pla

npla

iplaI

ZI

ZI

RI

RII

∑∑∑ ===

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5.2 SE evaluation for GA

To evaluate the currents to be imposed in the SE of a GA, see Fig. 1 bottom left, EMAG

solves at each selected time step the linear system equation (2). Once the SE are evaluated the

equivalence between the PE and the SE must be checked in terms of field (B) and field time

derivative (Bdot) produced at any spacial location outside the SE during the EM transient.

This check can be performed in EMAG where a specific routine compares the fields produced

by primary and SE by means of the Biot-Savart law. A specific window assists the analyst in

the selection of the field sources (plasma, poloidals or SE) to be taken into account in the

comparison. The final check is performed comparing the field produced by the PE evaluated

in EMAG using the original DINA with the field produced in the ANSYS FE analysis by the

SE, in absence of eddy currents. In fig. 4 is shown an example of this check. In this example

the two sets of curves match almost perfectly (the maximum error is lower than 5*10-4

). A

larger error can occur mainly depending on precision of the 3D model.

FIG. 3. EMAG Step Control window: the original DINA time steps for a slow + fast downward

VDE IV and the selected time steps are shown in the top and bottom half respectively

FIG. 2. Transient sampling for EM analysis: on the left a fix time step of 5 ms; on the right a more

accurate sampling with variable time stepping

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5.3 SE evaluation for ZA

In EM analyses, as the field is transmitted through both structures and vacuum, all the space

including the component to be analysed and the field sources, up to boundary surfaces where

known boundary conditions can be applied, must be included in the FE model. Furthermore

the EM transient produced on a generic structure by a field source can be strongly influenced

by the other conductors in the space. Due to this fact the EM analysis of any component, even

if small and of very simple geometry, imply a huge work to model the surrounding

environment. The zooming approach can be used to facilitate the modelling work. In this case

the SE set encloses the component to be analysed together with the main surrounding

structures (see right side of fig. 1). Outside the SE enclosure no structures need to be modelled

and the space from the SE set up to the boundary surfaces is filled with a free and very coarse

mesh. The evaluation of the SE for a ZA is performed by EMAG in two steps: the first step

consists in the assessment of the eddy currents induced by the original MHD plasma current in

the main toroidal continuous structures (VV, divertor rail, triangular support) by solving the

linear differential equation system (3); in the second step the currents induced in the SE by the

plasma and by the eddy currents in the conducting structures outside sample region are

evaluated. These currents are added to the plasma source and used as a new PE set to evaluate

by means of equation (1) the current in the SE. These currents reverted in sign, will reproduce

the EM transient inside the sample region and can be used for the FE EM analysis of the

components included in the zooming region. In fig. 1 (bottom right) an axisymmetric view

produced by EMAG in the preparation of the SE for a divertor ZA is shown: the SE (in green)

surround the divertor, the lower port and part of the VV while the PE (in dark blue) are both

the plasma current filaments and the VV eddy currents external to the SE enclosure.

In fig. 5 is reported the field check performed in EMAG by comparing the field B vs. time

produced by PE (including plasma current and the VV eddy currents external to the zoomed

region) with the field produced by SE in five points in the divertor region. The field

reproduction appears very accurate with perfect superposition of the two sets of curves.

FIG. 4. Check of the magnetic field reproduction in three points close to the SE set during a fast

downward VDE II (between DINA output and SE used in an ANSYS FE EM analysis);

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6. Conclusions

The SE interface procedures have proven to be very reliable; the error introduced by the

interface is negligible when applying the method both to global and to zooming analyses. For

all codesfor which the excitations are made of external circuits independent on the FE model,

the same SE set and the same input can be used in all models to analyze the same EM

transient. The transient sampling can be easily accomplished and checked by IO using the

EMAG code, thus allowing a reliable standardization of the outputs of the MHD codes to be

used in all EM analyses performed in IO or in the domestic agency.

7. ITER disclaimer

The views and opinions expressed herein do not necessarily reflect those of the ITER

Organization.

8. References

[1] Frederick W. Grover, “Inductance calculations, working formulas and tables”, Dover publications (1962)

[2] W. H. Press., et al., “Numerical Recipes in C, The Art of Scientific Computing”, CAMBRIDGE UNIVERSITY PRESS (1992)

[3] R. Roccella et al., “Assessment of EM loads on the EU HCPB TBM during plasma disruption and normal operating scenario including the ferromagnetic effect”

Proceedings of the 8th

International Symposium of Fusion Nuclear Technology, ISFNT-

8 SI

[4] M. Roccella et al, Detailed Electromagnetic numerical evaluation of eddy currents induced by toroidal and poloidal magnetic field variation and Halo currents, Fusion

Engineering and Design, Volume 83, Issues 10-12, December 2008, Pages 1625-1630

[5] Khayrutdinov, R.R., Lukash, V.E., J. Comp. Physics 109 (1993) 193

FIG. 5. Check of the magnetic field reproduction in EMAG for a ZA: the field produced by PE and

SE is compared in five points in the divertor region