High-Frequency Modeling of Granular Soft Magnetic Materials With Local Model Order Reduction

A full-wave electromagnetic field equation in a cross section of a ferrite core, modeled as an array of grains, is solved with the finite element (FE) method utilizing local model order reduction (LMOR) techniques. With the traditional FE method and a high number of grains, the resulting system becomes resource-intensive. The reduced approach greatly decreases the amount of degrees of freedom (DOFs) and the computational time for solving the resulting system.


I. INTRODUCTION
L OCAL model order reduction (LMOR) methods are methods which can be applied to eliminate excess amounts of degrees of freedom (DOFs) inside subregions of a domain [1], [2].They can be applied to a wide range of problems such as eliminating the internal DOFs of windings in magnetoquasistatic problems [3].The need for special techniques like LMOR typically arises when the problem has multiscale features which, in traditional finite element (FE) approaches, will result in an extremely high number of DOFs.In this article, we will use a randomized basis generation technique, which differs somewhat from the highly popular proper orthogonal decomposition (POD) technique which is used, e.g., in [4] and [5].
Manganese-zinc ferrite cores are widely used in high-frequency power electronic applications because they offer low losses and high permeability at high frequency [6].This kind of material can be modeled as a structure consisting of a large number of equivalent small-sized grains separated by a thin boundary layer [7].Modeling ferrite cores while considering such grain-scale structure is heavily expensive when standard FE solvers are used.Thus, fast and accurate solvers are needed to tackle such problems.
A popular way to tackle the multiscale problem is homogenization [8].Homogenization techniques assume that the field quantities inside a subdomain containing a microscale feature can be assumed homogeneous over the subdomain.It is however not always evident a priori when such an assumption is justifiable.Therefore, homogenization is not applicable to all problems of this nature.
In this article, a ferrite core is modeled while considering the grain-scale microstructure by taking a single grain which is modeled and meshed.Next, a reduced order model of the grain is computed.The resulting grain model is then replicated and the grain domains are connected accordingly using Lagrange multipliers to form the grain structure.In this approach, the amount of DOFs is significantly reduced.Hence, it can be used to fill the gap between homogenization methods and traditional FE analysis when homogenization may not be fully justified.The novelty of this article is to demonstrate the combination of the natural domain decomposition technique utilizing Lagrange multipliers, emerging from the ferrite's grain structure, in combination with LMOR techniques in order to speed up the simulation of granular magnetic materials.Furthermore, we demonstrate that the axisymmetric nature of the problem can be approximated as an affine dependency and hence the proposed ROM can be precomputed efficiently in the offline stage and then be used in the online stage with the optimal computational complexity [9].

II. COMPUTATIONAL MODELS
The problem under study is an impedance measurement of a disk-shaped ferrite sample (see Fig. 1) with h = 7.5 mm.The number of individual grains in the array (see Fig. 2, left) is denoted with G.The radius R varies based on the dimensions δ g , δ b (see.Fig. 3).The sample is placed between two conducting plates.A sinusoidal voltage is applied to the plates and the device determines the impedance of the sample based on the voltage, current, and the phase difference between the two.
Frequency domain analysis and cylindrical coordinates r, ϕ and z are used.Considering the grain-scale microstructure of the core (see Fig. 2, right), the rectangular cross section in the r z plane can be modeled as an array of square grains shown in Fig. 2, left.A 2-D axisymmetric FE model is constructed for solving the electromagnetic full-wave equation in the rectangular cross section of the core where Fig. 2. Left: a grain array a with boundary conditions for H ϕ .Because the same net current passes through each layer of grains in the z-direction, only one layer of grains is considered in the simulation.Right: a scanning electron microscope image of the ferrite material from [10].
The axisymmetric model is well justified due to the currents flowing through the sample in the r z plane, from the top electrode to the bottom.The magnetic field is considered to have only a circumferential component: H = H ϕ (r, z)e ϕ , where e ϕ is the unit vector in the ϕ-direction.Furthermore, based on Ampère's law, we can conclude that the H ϕ at r = R does not depend on z for the whole outer edge of the sample.Therefore, we can consider modeling only a single z-directional layer of the grain array and we can scale the results we obtain in the layer for the whole sample thickness h.The governing equations for the magnetic field strength H in a 2-D cross section of a disk core are The impedance is calculated as follows: where V is the sample disk volume, and ||•|| denotes the L 2norm.
In the classical FE model, ( 1) is discretized with the Galerkin FEM with nodal elements for H ϕ in the whole grain array.The weak form of the problem is in the whole grain array a (see Fig. 2, left), where ( a ) denotes the set of test functions in a .

III. LOCAL MODEL ORDER REDUCTION
In order to create the ROM for a grain, we consider only a single-grain domain shown in Fig. 3.We use POD with an oversampling domain [2].The ROM is parametrized with respect to the angular frequency ω.Because of the axisymmetry, each grain is located in different radial positions, and hence the effect of r is visible in the equations.The r dependency of the model is nonaffine and it therefore needs to be approximated to obtain an affine decomposition, e.g., by using polynomials [9, p. 194].Here, a Taylor polynomial is used.The Taylor approximation for (1/r ) in the centerpoint r i of the grain i is given by the following equation: It turns out that in this case the zeroth-order term is sufficient to produce well matching results.The zeroth-order approximation means that the r -coordinate is actually considered as the constant (1/r i ) in each grain.Using this approach, the approximated problem in a single grain i is expressed as follows: where = b ∪ g .

A. POD With an Oversampling Domain
In the reference grain cell, we use an oversampling domain ⋆ (see Fig. 3) to get a better reduction rate of the DOFs in the domain.The technique on obtaining the local reduced basis (RB) consists of setting randomized Dirichlet boundary conditions on the left and right sides of the grain.If the Dirichlet data are set directly to l and r , the resulting RB tends to have at least as many basis functions as there are DOFs on the boundary l ∪ r .The purpose of ⋆ is to let the randomized Dirichlet data to develop and smoothen out before entering the actual domain and hence the captured snapshots correspond to more naturally occurring excitations on the grain boundaries.This results in a more condensed RB.The reduction rate depends on the dimensions of the oversampling domain, which are chosen heuristically.A thicker oversampling layer results in less basis functions but increases the error.Typically, the oversampling domain in FE-context is chosen to be in the order of a few layers of elements.The oversampling domain is here considered to be vacuum.
The matrix form of (6 from which (1/r i ) can be neglected, hence, it will not affect the obtained RB.Note that the matrix S ⋆ depends on the angular frequency ω.The equation can be split as follows: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Next h ⋆ i is solved in a single grain with different frequencies ω and random boundary values on ⋆ l and ⋆ r as follows: From h ⋆ i , we extract only the values of DOFs inside b ∪ g and add them to the snapshot matrix X.The RB U is then obtained by orthogonalizing X using a modified Gram-Schmidt algorithm introduced in [11].Next, we check if the basis is good enough by computing the approximation error for a precomputed set of solutions H QA , i.e., The set H QA is obtained by randomly choosing a set of samples of ω, h ⋆ l , h ⋆ r and computing the corresponding solutions in b ∪ g .Since the quality assurance samples are not the same as the snapshot samples, they offer an objective measure for the level of error of the ROM in each iteration.If condition (10) is false, we compute a new snapshot, add it to X, recompute U, and recheck.When condition ( 10) is met, we consider the RB ready.
This approach differs from the usual way the POD basis is computed, which uses the singular value decomposition (SVD).After sampling the parameters and computing the snapshots, SVD is applied and only the significant singular vectors are chosen into the RB.In SVD-based approaches, one needs to have a set of snapshots computed completely before generating the RB.If new snapshots are added, one needs to recompute the basis for the whole snapshot set.This is computationally expensive especially when there are multiple parameters to be sampled.
In the approach presented in this article, we extend the basis gradually until it is good enough.Therefore, we do not need any a priori information on the location or amount of the snapshot parameters.The randomized approach is best suited when the ROM has many parameters or the associated high-fidelity FE problem is very time-consuming to solve.

B. Assembling the Reduced System
To utilize the affine decomposition in the grain array level, we introduce matrices K and T as follows: where φ k ∈ ( ) and φ l ∈ ( ) are FE trial and test functions, respectively.Next, we introduce the reduced matrices The matrices K and T are reused for all different frequencies.The resulting frequency dependent reduced system matrix in grain i can now be computed as follows: Next, a block-diagonal matrix S = diag ( S1 , . . ., SG ) is defined.The matrices R r and R l pick the entries of the DOF vector of the internal regions of b ∪ g corresponding to r and l , respectively.We denote the matrix which maps the values of the DOFs in r to l as D lr .This matrix is used to ensure the continuity of the field between grains.The meshes of l and r are made conforming.The total reduced system is then where consists of the couplings between the grains and equations to set the boundary conditions A high-fidelity model and the ROM were implemented in Python utilizing GMSH's Python API [12].To solve the resulting equation systems, both models are using SuperLU through the SciPy library.

IV. RESULTS AND DISCUSSION
The ROM was verified against traditional FE simulations.The grain dimensions and material parameters are shown in Table I.The properties of the ROM obtained for the grain are presented in Table II.The performance was compared in four cases with 100, 200, 1000, and 2200 grains.All computations were run in a Lenovo ThinkPad T14 with an i7-1165G7 CPU and 32 GB of RAM.The simulated impedance can be seen in Fig. 4 left.Fig. 4 right shows the error in a single grain in the grain array with respect to its r -position.As expected the highest error, 2.54%, is near the symmetry axis.Fig. 5 shows field plots of the magnitudes of H ϕ and its curl ∇ × H = J+(∂D/∂t).At 1 MHz, there is a resonance effect happening, which can be seen in the distribution of the magnetic field.
To  III.The single grain problem was reduced approximately by a factor of five, but the full grain array problem only by approximately the factor of three.This is because the full system includes also the Lagrange multipliers needed to couple the grains together.The frequency range of the simulations was 1 kHz-10 MHz from which we sampled 84 frequencies.A ROM valid for a larger frequency range requires more snapshots and therefore produces a larger RB for the ROM.Also, utilizing the affine decomposition for additional parametric dependencies, e.g., σ g , σ b , ε g , ε b , µ g , µ b will increase the dimension of the RB further.
The computational times in Tables II and III exclude the meshing times which grow larger for grain arrays with a high number of grains.In the ROM, it is only required to mesh one grain, including the oversampling domain.
Table III shows the computed relative root-mean square error (RRMSE) of the impedance given by the full FE and reduced FE models for the four different cases.The accuracy of the obtained ROM is high, but reducing the number of basis functions further from 14, in order to, e.g., trade accuracy for speed, causes a complete breakdown of the model.

V. CONCLUSION
A randomized LMOR technique was applied to model a ferrite disk using an axisymmetric 2-D model consisting of simplified ferrite grains.The LMOR technique reduces the number of DOFs and speeds up the computation of the impedance of the sample for a frequency sweep.The technique seems promising whenever there is a naturally emerging way to split the problem domain into repeating subdomains.It may be useful to speed up parameter optimization or the forward problem of an inverse problem where repeated solving of the system with varying parameters is required.

Fig. 3 .
Fig. 3. Single grain cell.The grains are connected to each other through l and r .The oversampling domain ⋆ is used only for the creation of the ROM.
where λ 0 and λ R are the Lagrange multipliers used to set the boundary conditions on the left boundary of the first grain, and the right boundary of the last grain, respectively.The blocks consisting of zeros are left empty for clarity.The vectorF = • • • F T 0 F T R T consists ofthe aforementioned boundary values.Matrices K, T, and B can all be precomputed in the offline phase, and they are reused for different frequencies in the impedance computation.
analyze the reduction of computational resources, let us first analyze the number of DOFs in the ROM and the highfidelity model.Let M denote the number of DOFs in a single grain, M b denote the number of DOFs on l ∪ r , and N denote the number of DOFs in the ROM of the grain.The full FE-model contains (M − M b )G + M b DOFs.The ROM contains N G +(G −1)M b +2M b DOFs.The numbers of DOFs for the three considered cases are presented in Table

Fig. 4 .
Fig. 4. Left: impedance computed with a full FE model and the LMOR model.Right: the RRMSE of |H ϕ | with respect to the position of the grain when G = 1000.

Fig. 5 .
Fig. 5. Field plots of |H ϕ | and ||∇ × H|| in the ferrite disk using the reduced and full models in the G = 1000 case.

TABLE I DIMENSIONS
AND MATERIAL PARAMETERS OF THE GRAINSTABLE II PROPERTIES OF THE REDUCED ORDER MODEL FOR THE GRAIN

TABLE III NUMBERS
OF DOFS BETWEEN THE HIGH-FIDELITY MODEL AND THE ROM WITH DIFFERENT NUMBER OF GRAINS IN THE ARRAY