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Magnetic Field Conforming Multiscale Formulations for Locally-Confined Nonlinear Eddy Current Problems Using the FE-HMM Method
Innocent Niyonzima, Gérard Meunier, Antoine Marteau, Ruth V. Sabariego, Olivier Chadebec, Nicolas Galopin, Christophe Geuzaine
TL;DR
The paper addresses nonlinear magnetic composites with significant confined eddy currents and associated homogenization and convergence challenges. It proposes an h-conforming HMM using magnetoquasistatic and magnetostatic mesoscale problems with relaxed Newton–Raphson schemes. The method accurately captures key quantities in 2D and 3D tests, while macroscale convergence and periodic-cell assumptions remain important boundaries.
Problem
Homogenizing nonlinear magnetic composites with significant confined eddy currents is difficult because scale separation can fail and conventional approaches may not represent the problem adequately.
Method
The paper proposes an h-conforming HMM with magnetoquasistatic upscaling of BM, magnetostatic upscaling of macroscale incremental reluctivity, and relaxed Newton–Raphson schemes at both scales.
Results
The method accurately captures eddy-current losses, magnetic power, and induced voltage; eddy-current-loss relative errors remain below 0.2% for linear and 5% for nonlinear 2D cases.
Takeaways & Limitations
The formulation advances modeling of 3D composites with nonlinear magnetic laws and significant confined eddy currents within periodic cells.
Takeaways & Limitations
Macroscale convergence is difficult and computational cost rises when additional nonlinear iterations require repeated mesoscale solves; the present work assumes periodic cell geometry.
Abstract
from arXiv · showhide
Magnetic composites used for the conversion of electrical energy often incorporate ferromagnetic inclusions insulated from each other to mitigate eddy current losses. Numerical models for these composites must be robust enough to address potential convergence issues arising from the presence of nonlinear magnetic inclusions and presence of significant confined eddy currents in the cell. This paper introduces an $\mathbf{h}$-conforming multiscale formulation for magnetic composites in a periodic framework. The proposed method uses the Heterogeneous Multiscale Method (HMM) with two mesoscale problems: a magnetoquasistatic problem for upscaling the homogenized magnetic flux density $\mathbf{B}_M$ and a magnetostatic problem for upscaling the macroscale incremental reluctivity $(\partial \mathbf{B}_M/\partial \mathbf{H}_M)$. Additionally, the method uses relaxed Newton--Raphson schemes at both macro and mesoscale levels to mitigate the well-known NR convergence issue linked to nonlinear BH constitutive laws in $\mathbf{h}$-conforming formulations. The accuracy and performance of the formulation are evaluated using 2D and 3D idealized periodic soft magnetic composites with linear and nonlinear BH curves. Furthermore, the paper demonstrates that the magnetoquasistatic mesoscale problem can be replaced by a magnet a
1. Introduction.
The paper addresses homogenization challenges caused by nonlinear materials, complex 3D fields, limited scale separation, and confined eddy currents. It proposes an h-conforming HMM formulation for 3D multiscale electromagnetic problems.
- Research challenges: Homogenizing Maxwell’s equations is challenged by nonlinear materials, complex 3D field distributions, multiscale fields, limited scale separation, confined eddy currents, and stochastic inclusions.The paper addresses the first four challenges using idealized periodic soft magnetic composites for low-frequency problems.
- Research challenges: Classical asymptotic homogenization can handle some fully conducting multiscale problems but fails for locally confined eddy currents because the homogenized conductivity vanishes.The locally confined setting involves a disconnected conducting domain and can invalidate standard homogenization approaches.
- Prior approaches: Existing mean-field methods cannot correctly handle 3D nonlinear problems with strongly confined eddy currents, while classical approaches are not suited to arbitrary periodic or nonperiodic cells.These limitations motivate a formulation adapted to magnetic composites with confined currents.
- Contribution: The paper proposes an h-conforming multiscale formulation using the Heterogeneous Multiscale Method for 3D multiscale electromagnetic problems.The formulation targets nonlinear magnetic materials and significant eddy currents that can introduce a phase shift between magnetic flux density and magnetic field.
- Paper scope: The paper develops the formulation after presenting a classical h−ϕ finescale reference problem and carrying out dimensional and asymptotic analyses of Maxwell’s equations.Later sections propose homogenized formulations and their spatial discretization.
2. The finescale formulations.
The finescale formulation defines the magnetoquasistatic Maxwell problem on conducting and non-conducting subdomains, with current-driven sources and assumptions ensuring a unique solution. Its discrete h-conforming approximation combines finite-element representations for conducting, topological, and stranded-inductor contributions.
- Strong and weak formulations: The h-conforming strong form combines Faraday’s law, Ampère’s law, and magnetic-flux conservation through ∂tbε + curl eε = 0, curl hε = jε, and div bε = 0.The resulting weak form seeks hε in a time-dependent function space and tests it against functions in V0(Ω).
- Problem definition: The finescale fields are magnetic field hε, electric field eε, magnetic flux density bε, and current density jε on a bounded 2D or 3D domain.The domain is decomposed into conducting and non-conducting regions, with source current density defined in inductors.
- Assumptions: Existence and uniqueness rely on bounded positive conductivity in conducting regions, zero conductivity in non-conducting regions, strong monotonicity, Lipschitz continuity, and regular divergence-free sources.The source regularity assumption is suitable for sine sources but not for PWM sources.
- Source implementation: The source-current solenoidal condition can be enforced by expressing js as the curl of a source magnetic field, with the static field obtained from a projection problem.The projection may use a Coulomb or tree-cotree gauge.
- Boundary and source conditions: The model is restricted to current-driven problems, imposes zero tangential magnetic field on the boundary, and represents imposed currents through circulation functionals around conductors.The current functionals specify net currents in massive and stranded inductors.
- Finite-element discretization: The discrete approximation combines first-order Nédélec elements in conductors, nodal P1 elements for a gradient contribution, global cut basis functions, and stranded-inductor terms.These components represent conducting fields, scalar-gradient contributions, domain topology, and stranded-inductor magnetic fields.
3. Dimensional and asymptotic analysis.
The analysis identifies scale separation and locally confined eddy currents as central homogenization challenges, then distinguishes problem categories by their macroscale and mesoscale PDEs. It shows that homogenized magnetic-flux-density errors depend more directly on the eddy-current-to-magnetic-power ratio than on skin-depth scaling alone.
- Scale separation: Classical scale separation requires the macroscale length, skin depth, and wavelength to be much larger than the mesoscale cell size.For eddy-current problems, the skin depth may be comparable to the cell size, invalidating this assumption.
- Problem classification: Category 1 includes fully conducting or multiply connected domains with macroscale eddy currents, while category 2 has disconnected conducting regions and a magnetostatic macroscale problem.Category 1 can be homogenized through material properties or electromagnetic fields; category 2 requires field-based upscaling.
- Problem classification: Locally confined eddy currents are especially difficult because homogenization produces a magnetostatic macroscale problem coupled to magnetoquasistatic mesoscale problems.The paper therefore recommends upscaling electromagnetic fields rather than only material properties for this category.
- Scope boundaries: When the macroscale skin depth is comparable to or smaller than the cell size, homogenization of problems with global currents makes little sense.A high conductivity contrast between matrix and inclusions can also make category 1 resemble locally confined-current problems.
- Upscaling implications: For a small power ratio Peddy_m/Pmag_m = 10^-3, magnetoquasistatic and magnetostatic calculations produce homogenized magnetic flux densities in good agreement.With Peddy_m/Pmag_m ≈ 1, the two homogenized flux densities do not coincide, despite the same skin-depth ratio δc/lc.
- Upscaling implications: The homogenized magnetic field H_M can likely be obtained by volume-averaging h_m when the cell magnetization induced by ∂_tB_M is negligible relative to the macroscale magnetic contribution.The supporting condition is expressed as <h_c>_Ωm · ∂_tB_M ≪ H_M · ∂_tB_M.
4. The multiscale formulations.
The formulation replaces the finescale problem with a macroscale problem coupled to periodic mesoscale problems, using HMM to upscale magnetic quantities and incremental reluctivity. It addresses locally confined eddy currents through a macroscopically non-conducting homogenized domain and uses relaxed Newton–Raphson iterations for nonlinear convergence.
- HMM formulation: HMM replaces the finescale problem with a coarse macroscale problem and many fine-mesh mesoscale problems located at macro-element Gauß points or barycenters.Macroscale quantities are downscaled to mesoscale source terms, while homogenized quantities are upscaled from mesoscale solutions.
- Domain treatment: The homogenized domain is macroscopically non-conducting, while eddy currents remain restricted to the massive inductors.This construction is used for locally confined eddy currents and separates homogenized material regions from remaining conducting regions.
- Homogenized constitutive law: The macroscale constitutive law uses the original magnetic law outside the homogenized domain and a volume-averaged mesoscale flux density at each Gauß point inside it.The mesoscale cell problem uses the macroscale magnetic field as input and supplies the homogenized magnetic law.
- Coupling and discretization: The macroscale problem is weakly coupled to independently solvable mesoscale problems after linearization, enabling parallel mesoscale resolution at the cost of iterative solves.The weak form is posed for the macroscale field and periodic mesoscale correction fields.
- Nonlinear solution: Backward Euler time discretization and possibly under-relaxed quasi-Newton–Raphson iterations solve the resulting nonlinear macro- and mesoscale systems.Relaxation factors below one are used to prevent divergence or oscillatory behavior during iterations, particularly with magnetic saturation.
- Upscaling: The method computes homogenized magnetic flux density by averaging the magnetoquasistatic mesoscale flux density and incremental reluctivity through finite-difference magnetostatic problems.Three mesoscale magnetostatic problems are solved to obtain the directional derivatives of the homogenized constitutive law.
5. Results.
The proposed HMM formulation is validated on 2D and 3D periodic soft magnetic composites, including eddy currents and nonlinear magnetic laws, with accuracy and parallel-performance evaluations. HMM reproduces key homogenized quantities accurately, while macroscale convergence and speed remain practical constraints.
- Validation setup: Validation covers 2D composites with conducting, magnetic disc inclusions and 3D composites with conducting, magnetic ball inclusions.The tests use idealized periodic soft magnetic composites with linear and nonlinear magnetic laws.
- Validation setup: Reference results use fine meshes, while homogenized results combine macroscale meshes with mesoscale cell-problem meshes.For the 2D case, the reference mesh has 184685 elements, the macroscale mesh has 64 macro-elements, and the cell mesh has 7099 elements.
- Accuracy: HMM accurately captures eddy-current losses, magnetic power, induced voltage, and homogenized magnetic quantities across the tested cases.The evaluation compares time-dependent global quantities and their relative errors against reference solutions.
- Accuracy: Relative eddy-current-loss errors remain below 0.2 % for the linear case and below 5 % for the nonlinear case.The paper reports this result for the evaluated 2D and 3D configurations.
- Accuracy: Magnetostatic and magnetodynamic homogenized magnetic induction values are similar when cell eddy-current losses are small, but discrepancies emerge when those losses are non-negligible.This comparison motivates retaining the magnetoquasistatic mesoscale treatment when confined eddy currents matter.
- Accuracy: In the 3D linear case, Joule losses and voltage agree closely with reference results, with errors smaller than 0.035 % and 0.2 %, respectively.The 3D tests also show agreement for homogenized powers, voltages, and currents.
- Performance: Macroscale convergence is substantially slower than expected for a single-scale nonlinear h-conforming problem, increasing computational cost because mesoscale problems are solved at each iteration.Alternative stopping indicators based on field increments and homogenized quantities were used alongside residual criteria.
- Performance: Strong scaling shows that HMM cost decreases as the number of CPUs increases, although these small-period test cases are not faster than the reference approach.The reported HMM complexity is independent of the number of material periods, whereas the reference problem becomes intractable for real-world heterogeneous materials.
6. Conclusions and perspectives.
The formulation combines two mesoscale problems to upscale magnetic flux density and incremental reluctivity, with a magnetostatic replacement possible when cell eddy currents are negligible. It achieves accurate global quantities relative to brute force, while convergence and scaling remain parameter- and processor-dependent.
- Conclusions: Two mesoscale problems upscale the homogenized magnetic flux density B_M and macroscale incremental reluctivity.A magnetoquasistatic problem provides B_M, while a magnetostatic problem provides ∂B_M/∂H_M.
- Conclusions: The magnetoquasistatic mesoscale problem can be replaced by a magnetostatic problem when cell eddy currents are negligible relative to magnetic power.
- Conclusions: Multiscale solutions produced eddy current losses, magnetic power, and currents/voltages consistent with brute force results.
- Perspectives: Convergence depends on parameters including macroscale mesh resolution, and weak scaling is not fully guaranteed across CPU counts.Computational cost increased noticeably from 2 to 16 CPUs and stabilized from 16 to 32 CPUs.
- Perspectives: Future work includes stronger convergence studies, sensitivity-based reluctivity evaluation, model-order reduction, and extension beyond periodic cell geometries.