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Development and Applications of Computational Electromagnetics
Development and Applications of Computational Electromagnetics
Numerical electromagnetic field analysis is now essential in electrical and electronic engineering. Advances in computing have enabled increasingly detailed simulations, but larger and more accurate models still require faster algorithms. My research explores the mathematical structure of electromagnetic governing equations to develop efficient numerical methods and apply them to the optimal design of electromagnetic devices.
Research Themes
Research Themes
Fast eddy current analysis of litz wire using integral equations
Topology and parameter optimization of electromagnetic devices
Mathematical foundations of Cauer ladder network model reduction
Efficient preconditioners for large linear systems arising from Maxwell’s equations
Fast Eddy Current Analysis of Litz Wire Using Integral Equations
Fast Eddy Current Analysis of Litz Wire Using Integral Equations
Litz wire consists of many thin, twisted strands and is used to reduce power loss at high frequencies. We develop integral equation methods that represent each strand as a one-dimensional curve in space and model proximity-effect eddy currents within the strand through magnetization. This makes it possible to analyze three-dimensional twisting patterns without finely meshing each strand’s cross-section.
Accounting for Circulating Currents and External Magnetic Fields
Accounting for Circulating Currents and External Magnetic Fields
Our supervised student’s research has been accepted by IEEE Transactions on Magnetics and published in Early Access. Differences in strand position and magnetic exposure can drive circulating currents between strands connected at their ends, producing additional losses. The new method combines an impedance matrix describing electromagnetic coupling between strands with terminal circuit equations to account for circulating currents and the proximity effect together. It handles both terminal-voltage excitation and currents induced by a uniform external magnetic field.
The method was compared with the finite element method using a four-conductor model and then applied to a 49-strand litz wire. Under terminal-voltage excitation, a single twisting stage mainly alleviates current imbalance among strands. Under a transverse external magnetic field, two-stage twisting, which exchanges the positions of the bundles, effectively suppresses circulating currents between bundles. The twisting pattern needed to reduce loss therefore depends on the excitation.
At 100 kHz under a uniform external magnetic flux density of 1 mT (peak), with zero applied voltage and zero net terminal current, the calculated losses were 2.186 mW without twisting, 1.891 mW with one-stage twisting, and 0.085 mW with two-stage twisting. The figures below show the current distribution at 100 kHz and the frequency dependence of the loss.
Related Publication
Naoki Hada, Shingo Hiruma, Shoma Ida, Nobutaka Onishi, and Masanao Yamashita, “Eddy Current Analysis of Litz Wire Considering Circulating Currents Using Integral Equations,” IEEE Transactions on Magnetics, Early Access, 2026.
Read on IEEE Xplore | DOI: 10.1109/TMAG.2026.3735702
How Twisting Changes the Current Distribution
Current density magnitude in 49 strands at 100 kHz under a uniform external magnetic flux density of 1 mT (peak), with zero applied voltage and zero net terminal current. Two-stage twisting (top left) greatly suppresses circulating currents compared with one-stage twisting (top right) and no twisting (bottom). Proximity-effect currents within individual strands remain.
Source: N. Hada et al., “Eddy Current Analysis of Litz Wire Considering Circulating Currents Using Integral Equations,” IEEE Transactions on Magnetics, Early Access, 2026, Fig. 5. doi:10.1109/TMAG.2026.3735702. © 2026 IEEE.
Frequency Dependence of Loss in an External Magnetic Field
Uniform external magnetic flux density: 1 mT (peak); applied voltage and net terminal current: zero. Red dashed: no twisting; green dash-dotted: one-stage twisting; blue solid: two-stage twisting. Loss reduction with two-stage twisting is pronounced around 100 kHz, and its effectiveness varies with frequency.
Source: N. Hada et al., “Eddy Current Analysis of Litz Wire Considering Circulating Currents Using Integral Equations,” IEEE Transactions on Magnetics, Early Access, 2026, Fig. 4. doi:10.1109/TMAG.2026.3735702. © 2026 IEEE.
Topology Optimization of Electromagnetic Devices Using Evolutionary and Gradient-Based Methods
Topology Optimization of Electromagnetic Devices Using Evolutionary and Gradient-Based Methods
Designing Magnet Shape and Rotor Structure Together
Designing Magnet Shape and Rotor Structure Together
The torque and smooth operation of permanent magnet (PM) motors depend strongly on magnet geometry and the air cavities, or flux barriers, inside the iron core. Finding a good combination by hand is difficult, motivating automated design methods for motors used in electric vehicles, industrial robots, and other equipment.
Parameter optimization (PO) adjusts quantities such as magnet width, angle, and position while retaining a prescribed, manufacturable shape. Topology optimization (TO) changes the distribution of material and air, allowing complex flux barriers to emerge. Our hybrid approach combines these complementary capabilities: PO represents the magnets, while TO describes the surrounding flux barriers.
A normalized Gaussian network (NGnet) represents the material distribution as a continuous function. Its parameters and the magnet geometry parameters are optimized together using a genetic algorithm (GA). Automatic remeshing adapts the finite element mesh to each candidate geometry, supporting accurate electromagnetic analysis with smooth material boundaries.
Applied to interior permanent magnet (IPM) motor rotors, simultaneous optimization of magnet curvature and flux-barrier geometry produced higher average torque and lower torque ripple than topology optimization alone in the investigated examples. The resulting designs make effective use of reluctance torque. Studies of practical U-shaped and V-shaped magnet configurations also explored the balance between torque performance and manufacturability.
By treating magnet shape and iron-core topology together, this research expands the design space available for efficient, high-performance motors.
Interactive Topology Optimization Example
Interactive Topology Optimization Example
Explore the motor topology optimization example (Japanese interface). The same interactive example is available on the Japanese research page.
Mathematical Foundations of Cauer Ladder Network Model Reduction
Mathematical Foundations of Cauer Ladder Network Model Reduction
From Electromagnetic Field Models to Efficient Equivalent Circuits
From Electromagnetic Field Models to Efficient Equivalent Circuits
Eddy currents in motors, transformers, and inductors cause energy loss and heating. Finite element analysis can resolve these effects, but repeatedly solving models with hundreds of thousands or millions of degrees of freedom can be expensive. Cauer ladder network (CLN) model reduction replaces such field models with compact resistor–inductor equivalent circuits that reproduce their impedance behavior to the required accuracy.
Our 2020 study connected the symmetric positive-definite structure of the finite element systems to Krylov subspace methods, particularly the Lanczos algorithm. This relationship provides a systematic way to derive the resistances and inductances of a Cauer ladder directly from the electromagnetic equations. The continued-fraction representation of impedance explains how the resulting circuit retains a physically meaningful structure.
Increasing the number of ladder stages improves accuracy, but an effective reduction method must also determine when enough stages have been retained. Our 2022 study developed a guaranteed error estimator based on electromagnetic energy and the mathematical properties of the ladder representation. An upper bound on the approximation error can be evaluated from the circuit’s terminal-stage elements.
The estimator was tested using analytical and finite element models of copper foil. These studies verified that it bounds the actual error in the tested cases, providing a basis for selecting the number of stages needed to meet a prescribed tolerance.
Efficient Preconditioners for Large Linear Systems Arising from Maxwell’s Equations
Efficient Preconditioners for Large Linear Systems Arising from Maxwell’s Equations
Finite element simulations of electromagnetic devices repeatedly solve large systems of linear equations. Long conductors and fine meshes that resolve the skin effect can slow iterative convergence. We develop preconditioners that exploit the physical and mathematical structure of these systems to reduce solution time.
Extending EMD to a Wider Range of Electromagnetic Problems
Extending EMD to a Wider Range of Electromagnetic Problems
Our new paper was published in IEEE Transactions on Magnetics as an Early Access article on 18 August 2026. The electro-magnetic decoupling (EMD) preconditioner separates the magnetic vector-potential block from the scalar-potential block associated with current continuity. It applies an inexpensive incomplete Cholesky (IC) preconditioner to the larger magnetic block and a stronger method, such as algebraic multigrid (AMG), to the smaller scalar block.
The new work extends the approach from eddy current analysis coupled to external circuits to nonlinear transient analysis, full Maxwell formulations, and stabilized Darwin formulations. A one-dimensional Laplacian model and three-dimensional singular-value computations help explain deteriorating convergence in long conductors. A lower bound on the smallest nonzero singular value provides a theoretical rationale for strengthening the scalar-block preconditioner.
For a nonlinear reactor over 20 time steps, the combined linear solution time for the time-step and Newton solves decreased from 4157.5 s to 1819.1 s. For a high-frequency inductor at 1 MHz, the time decreased from 364.4 s to 196.5 s for full Maxwell (EMD-V, approximately 1.85× faster) and from 557.2 s to 247.8 s for stabilized Darwin (EMD-W, approximately 2.25× faster). These values are linear solver times for the models and computing environment used in the paper; they exclude preconditioner setup, matrix construction, and related preparation.
A stronger preconditioner reduces the number of iterations but costs more per iteration. In these tests, a V-cycle gave the shorter solution time for full Maxwell, while a W-cycle was faster for stabilized Darwin. Efficient design therefore requires balancing convergence improvement against the computational cost of applying the preconditioner.
Related Publication
Shingo Hiruma, Takeshi Mifune, and Tetsuji Matsuo, “A Unified Electro-Magnetic Decoupling Preconditioner for Maxwell-Type Finite-Element Systems,” IEEE Transactions on Magnetics, Early Access, 2026.
Read on IEEE Xplore | DOI: 10.1109/TMAG.2026.3724614
High-Frequency Inductor Model
An inductor containing a metal composite magnetic material. Preconditioner performance was evaluated at 1 MHz and a terminal voltage of 1 V using both full Maxwell and stabilized Darwin formulations.
Source: S. Hiruma, T. Mifune, and T. Matsuo, “A Unified Electro-Magnetic Decoupling Preconditioner for Maxwell-Type Finite-Element Systems,” IEEE Transactions on Magnetics, Early Access, 2026, Fig. 4. doi:10.1109/TMAG.2026.3724614. © 2026 IEEE.
Balancing Preconditioner Strength and Solution Time
Breakdown of linear solver time. FM: full Maxwell; SD: stabilized Darwin. EMD-V and EMD-W apply AMG V- and W-cycles to the scalar block. EMD-V was fastest for FM, while EMD-W was fastest for SD, illustrating the balance between fewer iterations and the cost of preconditioning. Setup and matrix construction are excluded.
Source: S. Hiruma, T. Mifune, and T. Matsuo, “A Unified Electro-Magnetic Decoupling Preconditioner for Maxwell-Type Finite-Element Systems,” IEEE Transactions on Magnetics, Early Access, 2026, Fig. 5. doi:10.1109/TMAG.2026.3724614. © 2026 IEEE.
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