1. 1 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

2. 2 Contents Introduction Backgrounds Objectives DDM Applications to Magnetic Field Problems Numerical Examples Conclusions

3. 3 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.

4. 4 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)

5. 5 Contents Introduction DDM Application to Magnetic Field Problems Magnetostaic Problems DDM HDDM Numerical Examples Conclusions

6. 6 Non-Linear Magnetostatic Analysis Formulation A method Solution for non-linear equations Newton method On the interface 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

7. 7 Formulation  1 ： air or vacuum  2 ： magnetic material EE NN 

8. 8 Weak formulation A weak formulation is constructed by the introduction of the Lagrange multiplier p: (.,. ): the real valued L 2 -inner product.

9. 9 Finite element approximation A h :Nedelec elements of simplex type p h ： Conventional piecewise linear tetrahedral elements D.O.F.

10. 10 Finite element approximation V h, Q h : Finite element spaces corresponding to V and Q, A h : Finite element approximation of A by the Nedelec elements of simplex type, p h : Finite element approximation of p by the conventional piecewise linear tetrahedral elements.

11. 11 Finite element approximation Elimination of the Lagrange multiplier p h Correction of electric current density

12. 12 Newton iteration Adoption of the Newton iteration to solve the nonlinear equation Solver for linear simultaneous equations

13. 13 DDM (Domain Decomposition Method) I: corresponding to inner DOF B: corresponding to interface DOF Domain decomposition

14. 14 DDM (Domain Decomposition Method) On the interface In each subdomain The interface problem The subdomain problem

15. 15 IDDM (Iterative Domain Decomposition Method) (a) (b)

16. 16 The modification for the subdomain problem In step 0 In step n (a ` ) (b`)

17. 17 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)

18. 18 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

19. 19 TEAM Workshop Problem 20 Yoke SS400 Center pole Coil polyimide electric wire |J| = 1,000 [A]

20. 20 TEAM Workshop Problem 20

21. 21 TEAM Workshop Problem 20 ElementsDOFSubdomains Model 1471,541559,8488×300 Model 2952,8451,125,5018×600 Model 31,769,8712,083,2098×1,100 Model 49,326,49210,945,31856×830 Model 538,232,01944,676,34656×3,400 Model 686,570,893100,818,05356×7,730

22. 22 Linear Magnetostatic Analysis (Computational conditions) The number of PCs: 4

23. 23 Linear Magnetostatic Analysis (CPU time) CPU time by the previous method [s] CPU time by the proposed method [s] Ratio (%) Model 181.653.633.9 Model 2230.7153.333.6 Model 3497.3332.133.2 Ratio = ( | The previous method – The proposed method |/| The previous method | ) ×100

24. 24 Linear Magnetostatic Analysis (Averaged memory) Ave. memory by the previous method [MB] Ave. memory by the proposed method [MB] Times Model 125.353.22.10 Model 256.3107.41.91 Model 3104.5199.31.91 Times = ( |The proposed method |/| The previous method | )

25. 25 Linear Magnetostatic Analysis (Iteration figure of the interface problem)

26. 26 Nonlinear analysis of the model with 100 million DOF The number of PCs: 28

27. 27 Nonlinear analysis of the model with 100 million DOF Residual norms Iteration counts on the interface Step 0 Step 1 Step 2 Step 1 Step 2 Step 0 Model 4 Model 5 Model 6

28. 28 Nonlinear analysis of the model with 100 million DOF Iteration counts (Newton method) CPU time [s] Memory per CPU [MB] Model 424,495197 Model 5215,777812 Model 6246,3611,840

29. 29 Checking for the accuracy Measured B z [T] Computed B z [T] Relative error[%] IIII Model 1 0.720.71 0.57120.619.5 Model 20.60915.414.2 Model 30.62812.811.6 Model 40.6697.18.5 Model 50.6795.74.4 Model 60.6815.44.1

30. 30 Checking for the accuracy (VS. Ⅰ ) The average length of edge [m] The relative error 9.8×1.0 -4 1.9×1.0 -3 3.9×1.0 -3 Model 1 Model 2 Model 3 Model 4 Model 5 Model 6

31. 31 Checking for the accuracy (VS. Ⅱ ) The average length of edge [m] The relative error 9.8×1.0 -4 1.9×1.0 -3 3.9×1.0 -3 Model 1 Model 2 Model 3 Model 4 Model 5 Model 6

32. 32 Conclusions Improvement of ADVENTURE_Magnetic Demonstration of the possibility of large- scale 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.)