A COMPARISON OF TWO THREE-DIMENSIONAL SHELL-ELEMENT TRANSIENT ELECTROMAGNETICS CODES*

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A COMPARISON OF TWO THREE-DIMENSIONAL SHELL-ELEMENT

TRANSIENT ELECTROMAGNETICS CODES*

J. J. Yugo

Oak Ridge National Laboratory Post Office Box 2009

Oak Ridge, TN 37831-8071 (615) 576-5499

ABSTRACT

Electromagnetic forces due to eddy currents strongly influence the design of components for the next generation of fusion devices. An effort has been made to benchmark two computer programs used to generate transient electromagnetic loads: SPARK and EddyCuFF. Two simple transient field problems were analyzed, both of which had been previously analyzed by the SPARK code with results recorded in the literature. A third problem that uses an ITER inboard blanket benchmark model was analyzed as well. This problem was driven with a self-consistent, distributed multifilament plasma model generated by an axisymmetric physics code. The benchmark problems showed good agreement between the two shell-element codes. Variations in calculated eddy currents of 1-3% have been found for similar, finely meshed models. A difference of 8% was found in induced current and 20% in force for a coarse mesh and complex, multifilament field driver. Because comparisons were made to results obtained from literature, model preparation and code execution times were not evaluated.

I. INTRODUCTION

Electromagnetic forces due to eddy currents strongly influence the design of components for the next generation of fusion devices. During the Conceptual Design Activities phase of the International Thermonuclear Experimental Reactor (ITER) project, benchmark analyses were performed to compare the axisymmetric, three-dimensional (3-D) shell, and fully 3-D electromagnetics codes in use by the four parties.1

Results indicated that differences in problem formulation can have a significant effect on the accuracy of current and force distributions. In order to minimize differences due to solution methods or inputs, we compared two 3-D shell-element codes, SPARK2 and EddyCuFF3. Two simple transient field problems

were analyzed, both of which had been previously analyzed by the SPARK code with results recorded in the user manual or in literature. We also analyzed a third problem that was designed to verify the results obtained with the ITER benchmark model, but with a self-consistent, distributed plasma multifilament model generated by the Tokamak Simulation Code (TSC).4 In the TSC code the plasma is

represented by about two thousand filamentary currents which were condensed, for use in the eddy current codes, into 253 filaments representing the plasma and 40 filaments representing the poloidal field coils, divertor, internal control coils, passive stability loops, and active control coils. The time varying currents in these 293 filaments drive the solution for .currents in the finite element models of the vacuum vessel and

blanket.

* Research sponsored by the Office of Fusion Energy, U.S. Department of Energy, under contract DE-AC05-84OR21400 with Martin Marietta Energy Systems, Inc.

D. E. Williamson

Martin Marietta Energy Systems, Inc. Post Office Box 2009

Oak Ridge, TN 37831-8055

(615)574-0295 CCNF-9?n*n-7_{« W 0 7 — 5}

010686

II. ANALYSIS CODES

The SPARK computer program uses a network mesh method to analyze the 3-D conducting structure. The surface is modeled with triangular or quadrilateral elements, the sides of which represent the branches of the electrical network. Resistive and inductive properties of the network are calculated, including mutual inductances between each pair of elements. The differential circuit equations are solved for the value of mesh loop currents. Interaction of the loop currents with self- and background magnetic fields produces nodal force results for use with MSC/NASTRAN.5 Graphical

pre-and post-processing of results is also performed by SPARK. EddyCuFF employs an integral technique, first solving the circuit equations for the normal component of the current vector potential, then using energy equations to find the linear current density throughout the surface. The conducting structure is represented by linear triangular elements, and both element and nodal force results can be input tc MSC/NASTRAN. The authors have used a commercial pre-and post-processor, PATRAN,6 to interface with EddyCuFF

However an integrated processor, EDDYTRAN,7 is alsc

available. Table I summarizes the major features of the twc analysis codes.

TABLE I

Features of the SPARK and EddyCuFF Codes Feature/Code Formulation Solution Variable Governing Equations Time Difference Method Element Type

Pre- and Post-Processing Structural and Thermal Analysis Computer Systems SPARK Network Mesh Loop Current Biot-Savart and Kirchhoff Variable-Step Adams Method Quadrilateral or Triangular Integrated in Code Interface to NASTRAN Cray (CTSS, UNICOS), VAX (VMS) EddyCuFF Integral Equation Current Vector Potential Biot-Savart and Energy Expansion by Eigenvector Triangular Interface to PATRAN Interface to NASTRAN Cray (UNICOS). VAX (VMS), Workstation (UNIX)

III. BENCHMARK MODELS

A simple electromagnetics problem, for which a numerical solution using SPARK is well documented, is that of a flat plate in a uniform, decaying field.8 As shown in Fig. 1, the

copper plate is 0.120 X 0.020 X 0.001 m thick, and is subjected to a magnetic field change of 0.24 T in 0.003 s. We compared the magnitude of the induced eddy currents calculated by SPARK and EddyCuFF. A second problem that has been addressed with the SPARK code is that of a circular torus subjected to a step function toroidal field.9 An

EddyCuFF model of arbitrary dimensions (Fig. 2) was prepared and analyzed to determine the vertical force on the top or bottom half of a torus due to poloidal eddy currents. This result was also verified analytically. Calculations using a simple model of the iTER vacuum vessel and inboard blanket/shield modules made up the third analysis. Tables II and III summarize the main geometry and field driver parameters used in the analysis. The ITER vacuum vessel is toroidally continuous while the inboard blanket/shield is segmented toroidally into 96 electrically isolated sections. The proper use of boundary conditions and symmetry specifications allow a significant reduction in the size of this problem. As shown in Fig. 3, the model represents a 1/192 segment of the toroidally continuous vessel and 1/2 of one blanket/shield assembly. Reflection and 96-fold toroidal rotation symmetries are used to represent the complete vacuum vessel with 96 isolated blanket/shield modules. The SPARK model consists of 418 nodes and 282 quadrilateral meshes. The EddyCuFF model is composed of 760 nodes and 770 triangular elements. A multifilament model of plasma motion and current decay was developed, using data generated by TSC, and applied to both the SPARK and EddyCuFF simulations. Results for comparison include net circulating current and forces in the vessel and blanket box.

Table II

Vacuum Vessel and Inboard Blanket Model Parameters

COPPER PLATE

0.24 T FIELD 3 MS LINEAR DECAY

1 2 c m

-0.1 cm

Fig. 1. Model of Flat Copper Plate

I • 1E6 AMP STEP CHANGE Component Vacuum Vessel Inboard Blanket: Front Sides Rear Top/Bottom Material Stainless Steel Stainless Steel Stainless Steel Stainless Steel Stainless Steel Thickness (cm) 4.0 1.2 2.0 22.8 2.0 Resistivity (|i£2-cm) 100 100 100 100 100 Table III

Plasma Model Parameters Plasma Current Decay Rate, MA/ms Simulation Duration, s

Major Radius, m Minor Radius, m Plasma Current. MA Magnetic Field on Axis, T Poloidal Beta

Plasma Internal Inductance (I/2)

0.6 0.63 6.0 2.15 22 4.85 0.47 0.31 2.3

Fig. 3. ITER Blanket and Vacuum Vessel Model Geometry

IV. RESULTS • IV.A. Flat Plate

EddyCuFF results for streamlines of current induced in the flat plate are illustrated graphically in Fig. 4. The variation of total current with time is shown in Fig. 5. A comparison with the SPARK results indicates good agreement, with only a 3% difference in total current at 0.003 s. The SPARK model employs a coarse mesh of 15 elements, while the EddyCuFF moael uses 24 elements.

Fig. 4. Eddy Currants in a Flat Plate Calculated by EddyCuFF.

vector forces at 0.60 s on the SPARK vacuum vessel model shown in Fig. 9 have a poioidal variation which differs slightly from the EddyCuFF calculation. Differences between the codes in applying the exciting field to the coarse vacuum vessel mesh may also contribute to the variation. The eddy current patterns in the ITER inboard blanket module and the resultant forces are shown schematically in Fig. 10. Blanket forces are in much better agreement, as shown in Fig. 11. SPARK and EddyCuFF results for vertical force are identical, indicating good agreement on the magnitude of induced current in each box. The circulating current is the result of the interaction of the varying vertical field with the blanket module. The vertical force results from the interaction of circulating currents with the static toroidal magnetic field. Saddle currents are the result of the interaction of the varying radial field with the blanket module. The radial force is due to saddle currents in the front and back walls of the box interacting with the static toroidal magnetic field. SPARK and EddyCuFF results for radial force vary only slightly.

0.3 0.2 o.i - 0.0 2 -0.1 -0.2 -0.3

j

N

\

i

1

1

Fig. 5. SPARK and EddyCuFF Currents vs Time in a Flat

Plate.

IV.B. Circular Torus

The vertical force on a top/bottom half of a thin-shell torus due to poloidal currents is given in Ref. 9 as:

Fz = (B0R0)2 In [(Ro + a)/(R0 - a)] / (2^)

where Bo is toroidal field on axis, Ro is major radius, and a is the minor radius. With the values of Ro = 6 m, a = 4.8 m, and

Bo = 0.3 T, the ratio of force calculated by EddyCuFF to the

analytic value was 0.9636. This compares to a ratio of 0.9832 given in Ref. 2 for a SPARK analysis with a similar mesh. IV.C. ITER Vessel and Inboard Blanket Model

The analyses show that induced current in the vessel peaks at about 54% of the initial plasma current for a plasma current decay rate of 0.6 MA/ms. The values of induced vacuum vessel current given by the two codes differ by about 8% (Fig. 6). Net radial and vertical forces differ by about 20% (Fig. 7), due in part to the higher induced current calculated by EddyCuFF and also to a slightly different poloidal distribution of current and resultant force. Eddy currents on the SPARK vacuum vessel model at 0.544 s are shown in Fig. 8. The

570 580 590 600 610 62X1 630

0.0

Fig. 6. Comparison of Induced Vacuum Vessel Currents

-40.0

570 580 S90 600 610 620 630 Time(ms)

V. CONCLUSIONS

The benchmark problems analyzed have shown good agreement between the two shell-element codes. Variations in calculated eddy currents of 1-3% have been found for similar, finely meshed models. A difference of 8% was found n induced current and 20% in force for a coarse mesh and complex, multifilament field driver. Because some of the comparisons were made to results obtained from literature, •nodel preparation and code execution times were not evaluated.

" • 0 1 / 9 1 1 1 0 2 : 1 3 P S I C r o r c NG= ' N P = 0 T= 0 5 4 4 e + 0 0 I TEH VV.INID I I M T : : C « M 101i::TIC C«»« 0701: :f ATHAH 10-14-11 01:01 AH

C o n t o u r ( 1 0 t o t o l ) i n c l e m e n t - 0 4 J J e + 0 5 A n p t r e

Fig. 8. SPARK Vacuum Vessel Model with Toroidal Eddy Currents at 0.544 s.

ITER Case 1016: Vacuum Vessel (1/192) ik = 191 time = 0.600 s VAHYINO FIELD STATIC FIELD VARYIttO FIILD \

CURRENT FORCE CURRENT FORCE

Fig. 10. Illustration of ITER Blanket Eddy Current Patterns, Background Magnetic Field and Resultant Forces.

-0.5

570 580 590 600 610 620 630 TfcTie(ms)

Fig. 11. Blanket Forces Calculated by SPARK and EddyCuFF.

.2

4 1 -L

-•• a t - s c a n 3 2fl624137e+05

O 3O <£>

Fig. 9. Force Vectors on the SPARK Vacuum Vessel at 0.6 s.

REFERENCES

1. L BOTTURA and S. CHIOCCHIO, "Eddy Currents Benchmark Analysis in ITER," presented at COMPUMAG-91 (proceedings to be published in IEEE

Trans. Magn.)

2. D.W. WEISSENBURGER, U.R. CHRISTENSEN, "A Network Mesh Method to Calculate Eddy Currents on Conducting Surfaces", IEEE Trans. Magn., MAG-18. 422-425(1982).

3. A. KAMEARI, "Transient Eddy Current Analysis on Thin Conductors with Arbitrary Connections and Shapes", J Comput. Phys., 42, 124-140 (1981). 4. R. O. SAYER, "ITER Disruption Modeling with

TSC," ITER-IL-PF-4-0-91.IAEA, Vienna (1991). 5. MSC/NASTRAN Handbook for Linear Analysis.

MacNeal-Schwendler Corp., Los Angeles, CA (1985).

5. PATRAN Plus User Manual. PDA Engineering, Costa Mesa. CA(1987).

7. A. KAMEARI. "EDDYTRAN Program System for Eddy Current, Electromagnetic Force, and Structural Analysis," Proceedings of the 1 1!" Symposium on

Fusion Engineering, 1985, IEEE. New York (1985). 3. J.M. BIALEK and D.W. WEISSENBURGER, "The

Coupling of Mechanical Dynamics and Induced Currents in a Cantilever Beam." PPPL-2187, Princeton Plasma Physics Laboratory (1985).

9. D.W. WEISSENBURGER. "Self-Fields. Self-Forces, and Other Features of SPARK 1.2." Proceedings of the IEEE Thirteenth Symposium on Fusion Engineering, Knoxville, 1989, IEEE, New York. Vol 1. p 124 (1990).

DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsi-bility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Refer-ence herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recom-mendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.