MSFEM With MOR and DEIM to Solve Nonlinear Eddy Current Problems in Laminated Iron Cores

The computational costs to determine the eddy currents (ECs) in nonlinear laminated iron cores could be reduced essentially using multiscale finite element methods (MSFEMs). However, they are still too high for routine tasks in the design of electrical devices. Therefore, this article investigates the feasibility of the MSFEM with model order reduction (MOR) using the discrete empirical interpolation method (DEIM) to facilitate a convenient solution to the nonlinear problem. We propose structure preserving MOR to avoid the known large errors due to applying the DEIM to update the nonlinear right-hand side of the fixed point method. Numerical simulations show very satisfactory results.


I. INTRODUCTION
C OMPUTATIONAL costs to compute the eddy currents (ECs) in laminated iron cores can be very high [1].Our aim is a fast and accurate computation of the ECs in laminated nonlinear iron cores.It turned out that the reduction of the computational costs with the aid of the multiscale finite element method (MSFEM) is not sufficient.
The complexity of a nonlinear EC problem (ECP) could be significantly reduced by model order reduction (MOR) in [2].Proper orthogonal decomposition (POD) is very successful to construct reduced-order models for linear problems.However, to assemble the nonlinear term of the finite element (FE) system remains expensive.Therefore, the discrete empirical interpolation method (DEIM) presented in [3] is investigated to substantially improve the efficiency of MOR in the context of MSFEM.MOR with DEIM was successfully applied to a nonlinear static magnetic field problem in [4] and to an ECP in [5].
The algorithm in [3] was used to select the degrees of freedom (DoFs) of the MSFEM for the DEIM.Results of the numerical example in Section III demonstrate that MSFEM and MOR with DEIM are capable of providing very accurate results.On the other hand, DEIM is also very sensitive.Very large errors can occur if the dimension of the snapshot matrix for the right-hand side (RHS) is too small.A gappy-POD approach was proposed in [5] to eliminate the large errors.Our idea is to apply structure preserving MOR [6], in short, structural MOR (SMOR), which exploits the MSFEM approach and has low additional costs.
The simulation results show that MOR and DEIM significantly reduce the memory requirements and computation times of MSFEM, and SMOR is able to significantly minimize large errors.

II. NUMERICAL METHODS
An ECP, such as shown in Fig. 1, is to be solved in the time domain based on the magnetic vector potential (MVP) A. The initial boundary value problem reads where ν is the magnetic reluctivity, σ the electric conductivity, J 0 a given current density, the domain of the problem consisting of a conducting domain (iron) c and a non-conducting domain 0 with the boundary ∂ = N ∪ D .

A. Multiscale Finite Element Method
The first order MSFEM approach marked by the tilde for the MVP A is used, where A 0 ∈ H (curl, ), A 1 ∈ H (curl, m ) and w 1 ∈ H 1 ( m ) are suitably approximated by FE spaces, see [7] and [8], respectively.The micro-shape function φ 1 is a periodic, piecewise linear and continuous function [1], i.e., φ 1 ∈ H per ( m ).The laminated domain m consists of iron sheets and insulation layers in between.Essential boundary conditions are prescribed by means of A 0 exclusively, and only natural boundary conditions are provided for A 1 and w 1 .To obtain the weak form of MSFEM, the approach ( 2) is inserted into the associated weak form of (1), see e.g., [1].

B. Fixed Point Method
The weak form of the MSFEM of ( 1) with ( 2) yields the non-linear ordinary differential equation system with a nonlinear stiffness matrix A(ν), a linear mass matrix M, both of dimension n × n, and the filamentary current i in Fig. 1.Applying the fixed point method (FPM) [9] to (3) moves the nonlinear part to the RHS.The Euler method was used as a time stepping method (TSM).Thus, (3) becomes with the constant fixed point reluctivity ν FP , the time step t, the superscript k for the time instant t k = k t, which yields u (k) = u(t k ) and the fixed point iteration l.Thus, the nonlinear algebraic equation system (4) has to be solved iteratively at each time instant t k .

C. Model Order Reduction
To reduce the effort to solve (4), MOR and DEIM are used.A snapshot matrix S = u (1) , u (2) , . . ., u (m) (5) with m solutions of (4) and simultaneously associated snapshots of the RHS are computed.The dimension of S and F equals n × m.Simply selecting S for the projection of (4) leads to the reduced order model where K ∈ R m×m with m ≪ n.The reduced system (8) maintains the nonlinearity of the original large system (3).
A POD based on the singular value decomposition (SVD) is too expensive for large problems and therefore avoided.The application of the Gram-Schmidt orthogonalization method to (5) of the numerical problem in Section III did not help at all.If the dimension of the matrix S would be too small, an adaptive MOR could be considered.This has not been applied in the present work.
The most expensive part to solve ( 8) is to update the nonlinear RHS.To cope with this burden DEIM is investigated for MSFEM.

D. Discrete Empirical Interpolation Method
The matrix F is used to enable a fast update of f .For feasibility of the DEIM the approximation should hold for any RHS f as good as possible.Since ( 9) is strongly overdetermined, the algorithm in [3] was chosen in order to still determine a unique vector c.Simply speaking, the algorithm recursively searches for m dominant entries in the column vectors of F resulting in a regular matrix G of dimension m × m.Thus, only the m entries of f corresponding to the previous search really have to be determined and the nonlinear ν( A) of the associated FEs updated.Then, c can be uniquely calculated with G.The remaining entries of f are obtained inexpensively by means of ( 9) with the known c.For a more detailed explanation we refer to [3].

E. Structural MOR
The idea is to enlarge the space spanned by a given snapshot matrix F without computing more snapshots f i , which would be expensive.Therefore SMOR has been applied such that an RHS vector is decomposed according to the unknown variables A 0 , A 1 and w 1 in the MSFEM approach (2) and consequently F can be written as where T means transposed.Using SMOR leads to the approximation with the unknown vectors c A 0 , c A 1 and c w 1 , respectively, which are uniquely determined with the decompositions (10) and (11) and using (12) analog to c in (9) with the algorithm in [3].Projection of (12) based on the decomposition of SMOR results in where

III. NUMERICAL SIMULATIONS AND RESULTS
The numerical example with details is presented in Fig. 1.The used magnetization curve can be found in [1].A conductivity of σ = 2•10 6 S/m was selected.The nonlinear iterations l were stopped as soon as the criterion Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.was met.This criterion was also used with the DEIM.To evaluate the feasibility of MSFEM and MOR with DEIM, the time behavior of the EC losses was studied.

A. Losses and Fields
The first 10 solutions from the TSM with t = 0.5 ms were selected as snapshots S as well as the corresponding RHS vectors for F.
EC losses obtained by the standard FEM (SFEM) served as a reference solution (RS).A comparison with those obtained by MSFEM and MOR with DEIM are shown in Fig. 2. The agreement of the losses is very satisfactory.The influence of the magnetization curve is clearly visible.There is also a satisfactory agreement of the magnetic flux density B obtained by MSFEM and MOR with DEIM compared with MSFEM only as shown in Fig. 3.The same holds for the current density J, see Fig. 4.

B. Computational Costs
The required number of DoFs are summarized in Table I.It shows clearly the reduction of the size of the system of equations to be solved.The computation time of MSFEM and MOR with DEIM is less than a quarter of MSFEM only as shown in Table II.The parameter ε in (15) had to be set to 10 −4 for î = 5 A to obtained a sufficiently accurate solution.The FPM required a moderate number of iterations as can be seen in Fig. 5 to solve the nonlinear problem (4).

C. Visualization of DEIM DoFs
To fix the locus of a DoF in space the center of gravity , where r = (x, y, z) T (17) of the respective basis function v was computed.The loci of the DEIM DoFs are shown in Fig. 7.The single DoF 2 in air belongs to A 0 , all other DoFs to A 1 .DoFs of w 1 seem not to be relevant for the used algorithm and given problem.The distribution of the DEIM DoFs to the components of (2) remains the same for all excitations î as shown in Table III.The affected DEIM DoFs of A 1 vary with the excitation î.The    DEIM patches in terms of the required number of FEs and DoFs in order to update the material for the DEIM DoFs can be found in Table III, compare also with Fig. 6.The number of FEs for the DEIM-DoFs is different depending on whether the FEs are on the boundary of m or not.

D. Snapshots and SMOR
The choice of snapshots is examined in Fig. 8.The poor results show how critical the choice of m snapshots is in F. The pronounced large errors occur with only six snapshots at î = 10 A. The reason is a too small dimension of F. The large errors are significantly minimized by SMOR with six snapshots, as shown in Fig. 9.However, SMOR also exhibits a slightly larger error for a higher number of snapshots in F.

IV. CONCLUSION
The large reduction in computational cost and the very good results demonstrate the usefulness of MSFEM and MOR with DEIM.The sensitivity of DEIM with respect to the number of snapshots in the RHS matrix F can be significantly minimized with SMOR at negligible additional cost.However, the field distributions obtained with SMOR show a larger error than those obtained with MOR shown in Figs. 3 and 4.

Fig. 2 .
Fig. 2. Losses obtained by SFEM and MSFEM using MOR with DEIM for different excitations î in A.

Fig. 6 .
Fig. 6.Sketch of an FE DEIM patch with the FEs FE 1 and FE 2 .The DEIM DoFs belong to an A 0 -and an A 1 -component of the MSFEM approach (2).

Fig. 7 .
Fig. 7. Loci of the 10 DEIM DoFs, front view (top, note that some of the DoFs are hidden by others), top view (bottom).See legend in Fig. 6.

TABLE I NO
. OF DEGREES OF FREEDOM

TABLE II COMPUTATION
TIME IN s

TABLE III NO
. DEIM DOFS AND SIZE OF DEIM DOF PATCHES