The work, carried out by David Jin Tao Leong and Doreen Ying Ying Sim of the University of Nottingham Malaysia, tackles a stubborn weakness in the growing field of simulation-driven antenna optimization. Physics-aware active learning, in which analytical antenna knowledge guides a surrogate-assisted search, has already proven it can cut the cost of full-wave electromagnetic analysis. But the sampling rules and probe locations in such systems are usually tuned to one specific design case. Constants that work beautifully for an FR4 patch at 2.4 gigahertz may fail completely on a low-loss Rogers laminate at 5.8 gigahertz, quietly reintroducing the manual tuning the optimizer was supposed to eliminate.

The researchers’ answer is an integrated workflow that combines four ideas: online calibration, matching-first recovery, physics-informed probes, and a lightweight surrogate model. Instead of prescribing where the optimizer should look for a matched design, the workflow estimates sampling bands from its own observations after every batch of simulations. When too few designs achieve a good impedance match, the system switches into a recovery mode that focuses the search on the best near-miss geometry until matches appear. A radial-basis-function surrogate, seeded with closed-form patch equations and deterministic physical transforms such as effective permittivity and electrical length, then ranks candidate designs so that each expensive simulation is spent where it is most informative.

The headline experiment was deliberately austere. Four optimizers, the integrated workflow, random search, a budget-limited genetic algorithm, and an unstructured surrogate search, were each given exactly 37 attempted simulations per run using the open-source openEMS solver. They competed on four substrate and frequency cases, spanning FR4 and Rogers laminates at 2.4 and 5.8 gigahertz, with ten replicate random seeds per case. The integrated workflow won decisively: it produced the lowest mean and median final objective in every single case and returned a properly matched design in all forty runs. The baselines, while competitive on individual seeds, showed far heavier upper tails and between one and eight unmatched endpoints depending on the case and method.

The statistical treatment was unusually careful for an engineering study. Using exact Mann-Whitney rank tests with Holm adjustment within each case, the integrated workflow was significantly lower than the three baselines in nine of twelve directional comparisons. Six contrasts remained significant in a stricter two-sided unpaired analysis, and seven in a paired-by-seed sensitivity analysis. The authors are candid about the limits: comparisons on the easier fourth case against the genetic and unstructured surrogate baselines were not statistically resolved, suggesting the method shines brightest when an unfamiliar substrate or band makes impedance matching genuinely hard.

The second major contribution is a multi-fidelity screening protocol that exploits the huge cost gap between coarse and fine simulation meshes. The workflow runs its entire 37-call search on a coarse mesh, then promotes the twelve best candidates to full resolution for confirmation. Measured in equivalent high-fidelity simulations, this cut the mean cost by 19 to 46 percent across cases, an overall reduction of 36.9 percent, while still returning a matched full-resolution design in 39 of 40 runs. The researchers are equally clear about the trade-off: endpoint objective quality was not uniformly preserved, with one case showing a modestly worse average result than a full-resolution search. Cost savings and final quality, they show, must be reported together rather than cherry-picked.

To check that the optimized designs were not artifacts of one particular numerical setup, the team subjected the four selected finalists to twelve stability checks involving a finer mesh, a 50 percent larger air domain, and perfectly matched layer boundaries. All twelve checks preserved the impedance match, with maximum changes of just 0.180 decibels in broadside directivity and 0.121 decibels in realized gain. Nominal far-field calculations gave directivities of 6.51 to 7.00 dBi and realized gains of 3.59 to 5.77 dBi. Notch depth and threshold-defined bandwidth proved more sensitive, varying by up to 3.7 decibels and 27.8 percent respectively, a useful reminder that some antenna metrics are inherently fragile under small numerical perturbations.

Two exploratory extensions probe how far the approach can stretch. A ten-variable E-shaped patch case, adding slot geometry and two-dimensional probe placement to the search, matched all ten integrated-workflow runs and beat random search with statistical robustness, though comparisons with the adaptive baselines were inconclusive. A five-seed pilot that folded realized gain into the acquisition loop alongside impedance produced a directionally favorable but non-significant hypervolume trend, establishing feasibility rather than a demonstrated multi-objective win. The authors frame both studies as bounded extensibility tests, not claims of high-dimensional or multi-objective superiority.

What makes the study notable is its intellectual honesty. The authors state explicitly that all evidence is numerical and openEMS-based, that no commercial-solver replication, fabricated prototype, or anechoic-chamber measurement is claimed, and that the advantage belongs to the integrated workflow as a whole rather than to any single component. They also disclose that one legacy implementation detail involving solver-inert feature columns was corrected before the primary comparisons, and that the second through fourth cases should be read as cross-case transfer evaluations rather than a formal held-out validation set.

The implications reach beyond patch antennas. As wireless systems multiply across the Internet of Things, 5G and 6G bands, and satellite constellations, the demand for rapidly optimized radiators on wildly different substrates keeps growing. A workflow that transfers across materials and frequencies without case-specific retuning, and that documents its own statistical evidence with per-seed tables and exact rank tests, offers a template for how simulation-hungry engineering design might adopt machine learning responsibly. The authors point toward tolerance and yield studies, richer multi-objective acquisition, and tests on arrays and multilayer structures as the next frontier. For now, the message is clear: with the right physics in the loop, even 37 simulations can be enough to tame an antenna.

Subject of Research: Self-calibrating multi-fidelity active learning for low-budget full-wave optimization of microstrip patch antennas across substrates and frequency bands

Article Title: Self-calibrating multi-fidelity active learning for microstrip patch optimization across substrates and bands

Article References: Leong, D. J. T., & Sim, D. Y. Y. (2026). Self-calibrating multi-fidelity active learning for microstrip patch optimization across substrates and bands. Frontiers in Antennas and Propagation, 4, Article 1935948. https://doi.org/10.3389/fanpr.2026.1935948

Image Credits: AI Generated

DOI: 10.3389/fanpr.2026.1935948

Keywords: active learning, antenna design optimization, microstrip patch antenna, full-wave electromagnetic simulation, multi-fidelity simulation, surrogate modeling, openEMS, impedance matching, Bayesian-inspired optimization, radial basis function, computational electromagnetics, self-calibrating priors

The research team, led by Adnan M. Taha and including Mahdi Ghafourivayghan, Konstantin Burlakov, Sergey Shabunin, and Mohammad Alibakhshikenari, set out to address a persistent bottleneck in computational electromagnetics. Commercial full-wave solvers such as ANSYS HFSS, CST, and FEKO discretize an entire three-dimensional volume into a finite-element mesh, and their cost balloons when reflected waves must be tracked near a boundary. When the antenna operates in its near or intermediate zone, the amplitude and phase of the field cannot be inferred from a simple aperture distribution; every field component must be computed numerically, which dramatically inflates processing time and memory demands. For applications like radar sensors, subsurface radiolocation, and level gauges, where the far-zone condition is simply never met, this is more than an inconvenience.

The core of the new method is the Green’s tensor function, a mathematical object that encodes how an infinitesimal electric or magnetic current source radiates through a specified environment. In unbounded homogeneous space, the Green function takes a comparatively simple form. In a layered medium, however, it must absorb the thickness, number, and electromagnetic parameters of every layer, including permittivity, permeability, and conductivity. The authors build these properties into characteristic functions g(z, z’) and f(z, z’), which solve Sturm-Liouville-type differential equations and automatically satisfy all boundary conditions at each interface. Crucially, the modal conductances are recalculated recursively through the stack using an equivalent transmission-line model, in which spectral components of the field are mapped onto voltages and currents in equivalent circuits. Once this analytical groundwork is laid, the field can be evaluated only where it is actually needed, rather than everywhere in space.

To model a real aperture antenna, the team begins with the Huygens element, a pair of orthogonally crossed electric and magnetic dipoles that together reproduce the radiation of an elementary patch of wavefront. The electric field is obtained by integrating the external electric and magnetic currents over the source region, weighted by the appropriate Green’s tensor components. Because a Huygens element radiates a curved, non-uniform wavefront rather than a plane wave, its reflection from a nearby interface behaves nothing like the textbook plane-wave diffraction problem, and the Green’s-function framework captures this distinction exactly. The formulation also yields closed-form far-zone expressions via the saddle-point method, providing a convenient analytical check: the radial field component vanishes, and the radiation maximum points toward the interface, exactly as physics demands.

The extension from the elementary Huygens source to a practical pyramidal horn is where the study claims its principal advance over the authors’ earlier work. The horn flares in both principal planes, combining the properties of E-plane and H-plane sectoral horns, and its aperture field carries a quadratic phase variation that reflects the true curvature of the wavefront. The researchers derive a rigorous aperture integration in which the path-length difference across the aperture is expanded binomially, and the resulting integrals are evaluated in terms of cosine and sine Fresnel integrals. This captures interference structure near the reflector that a simple Huygens source fundamentally cannot represent, making the tool genuinely design-ready rather than a purely theoretical exercise.

Numerical stability is handled through a documented three-region spectral truncation strategy. The integration domain in wave-number space is split into an evanescent region of exponentially decaying near-field contributions, a narrow singularity gap around the branch point where split-path integration is applied, and a propagating region of oscillatory far-field radiation. Convergence analysis shows the truncation error decays exponentially with the upper limit, achieving one percent accuracy well within the adopted defaults, while the singularity treatment remains stable for gap sizes down to ten to the minus twenty-fifth. A parametric robustness study across horn heights from half a wavelength to two wavelengths, frequencies from 0.5 to 3 gigahertz, and substrate permittivities from 1 to 10 confirms that the chosen settings sit on a stable plateau and need no geometry-specific retuning.

The validation against ANSYS HFSS is the study’s most persuasive evidence. On identical observation grids, the proposed method required 30.9 seconds of wall-clock time against 88 seconds for HFSS, a roughly threefold speed-up, and used 200 megabytes of memory against 749 megabytes, a nearly fourfold reduction. The speed advantage stems from the region-of-interest strategy: the spectral integral is evaluated only at the user-specified observation points, whereas the finite-element solver must discretize the entire computational volume. Error budgets quantify the agreement precisely. The L2-norm relative error for the electric field magnitude is 2.78 percent, with maximum deviation of 2.29 percent, while the phase component shows an L2 error of 4.50 percent, a figure the authors attribute to the notorious sensitivity of phase near field nulls.

Even more telling is how the error scales with antenna height above the interface. At a separation of half a wavelength, deep in the reactive near field, the L2 error against HFSS is about 2.6 to 2.9 percent. At three-quarters of a wavelength it drops to roughly 0.65 percent, and at one and a half wavelengths it falls below 0.01 percent as the observation domain moves into the Fresnel region. Notably, the error shows only weak dependence on frequency and substrate permittivity, with higher permittivity actually marginally reducing the error at close separations because stronger field confinement diminishes the relative weight of the evanescent spectral tail. For radar-sensor and level-gauge designers, these numbers suggest the method can be trusted across a decade of bandwidth without recalibration.

The comparison also exposed an interesting divergence between the two approaches at a perfect electric conductor boundary, where the Green’s-function method delivered cleaner wave patterns than the finite-element reference. The authors point out that HFSS results depend on mesh symmetry and on the correct selection of the bounding box around the analyzed object; an asymmetric mesh around a structure symmetric with respect to the reference plane can introduce spurious field asymmetry. The analytical method, by contrast, is electrodynamically exact and automatically enforces all boundary conditions, eliminating such numerical artifacts. In benchmark runs over a dielectric boundary, the conventional solver took roughly twice as long at a comparable spatial step.

The authors are careful to delineate the method’s limits. Its computational advantage is specific to planar, stratified reflecting surfaces; for finite-sized, curved, or laterally inhomogeneous reflectors, edge diffraction and geometric scattering fall outside the stratified-media kernel, and full-wave solvers remain the appropriate tools. Substrates containing vias, patches, or etched patterns would require augmenting the transmission-line recursion with periodic or moment-method treatments. Within its domain, however, the framework offers a unified and efficient foundation that extends naturally to printed and slot antennas, transmitarrays, and reflectarrays, whose multiple functional layers are handled natively by stratified-media Green’s functions. As near-field applications multiply, from high-resolution electromagnetic imaging and secure short-range communication to over-the-air testing of 5G and 6G arrays, tools that trade brute-force meshing for analytical rigor are likely to become indispensable to the antenna community.

Subject of Research: Electromagnetic near-field analysis of pyramidal horn antennas above layered reflecting media using dyadic Green's functions

Article Title: Analysis of pyramidal horn antennas near a reflecting surface using Green’s functions

Article References: Taha, A. M., Ghafourivayghan, M., Burlakov, K., Shabunin, S., & Alibakhshikenari, M. (2026). Analysis of pyramidal horn antennas near a reflecting surface using Green’s functions. Results in Engineering, 32, Article 113033. https://doi.org/10.1016/j.rineng.2026.113033

Image Credits: AI Generated

DOI: 10.1016/j.rineng.2026.113033

Keywords: pyramidal horn antenna, Green's functions, near-field, reflecting surface, stratified media, ANSYS HFSS, electromagnetic modeling, radar sensors, spectral integration, computational electromagnetics, 5G, aperture antennas

The study, published in the International Journal of Aeronautical and Space Sciences by Seung-Hoon Kang of Sejong University and Younggeun Park and SangJoon Shin of Seoul National University, tackles a problem that has grown steadily worse as stealth analysis has become more demanding. To certify how well a design hides from radar, analysts must compute the monostatic radar cross-section, or RCS, for a sweeping range of illumination angles. Each angle corresponds to a different incident radar wave, and each incident wave requires solving a large system of linear equations derived from a finite-element model of the aircraft or missile. In the language of numerical linear algebra, the analyst faces a system with many right-hand sides, one for every direction the radar might come from.

The finite-element method discretizes the volume of space around and inside the target into millions of small elements, turning Maxwell’s equations of electromagnetism into a gigantic sparse linear system. Solving that system once is already a serious undertaking. Solving it hundreds of times, once per angular sample, multiplies the cost into something that can dominate an entire design cycle. The researchers describe this as the central obstacle in angular monostatic RCS analysis for large-scale finite-element models, and it is precisely the obstacle their new methods are built to remove.

Their starting point is an established technique called the finite element tearing and interconnecting method, or FETI, originally devised in the early 1990s by Charbel Farhat and François-Xavier Roux as a way of tearing a large structural problem into smaller subdomains and then interconnecting the pieces through interface conditions. The dual-primal refinement, FETI-DP, introduced in 2001, added primal constraints at selected interface points that dramatically improved convergence, and the electromagnetic adaptation, FETI-DPEM, extended the approach to wave problems governed by Maxwell’s equations. In FETI-DPEM, the computational domain is partitioned into subdomains, each subdomain problem is solved independently, and an interface problem enforces continuity of the electromagnetic fields across the artificial boundaries between subdomains. The interface problem is solved iteratively with a Krylov-subspace method, a family of algorithms that build up an approximate solution from a sequence of increasingly refined search directions.

The key insight behind the new work is that the many right-hand sides arising from angular sweeps are not independent of one another. When radar illuminates a target from ten degrees and then from eleven degrees, the two incident waves produce scattered-field solutions that are highly correlated. Information generated while solving the first system, in the form of Krylov subspace vectors, is therefore valuable for solving the second. Conventional practice throws that information away after each solve. The researchers’ first proposed variant, which they call recycling FETI-DPEM, or rFETI-DPEM, instead retains and reuses Krylov subspace information from previously solved angles. As each new right-hand side arrives, the method augments the stored subspace with the new search directions using a modified Gram–Schmidt orthogonalization procedure, keeping the recycled vectors orthonormal and computationally cheap to maintain. The result is a sequential solver in which each successive angular solve starts much closer to its answer than a cold start would.

The second variant goes further by attacking the multiple right-hand sides simultaneously. Block FETI-DPEM, or bFETI-DPEM, gathers the incident-wave vectors into a block and first applies a rank-revealing QR decomposition to that block. Rank-revealing QR is a classical matrix factorization technique that identifies which columns of a matrix carry genuinely independent information and which are nearly redundant. In the angular RCS setting, this means the method detects the linear dependency among the incident waves and collapses the block of right-hand sides to a much smaller set of independent representatives. The reduced system is then solved with a block iterative solver, a block version of the BiCGSTAB algorithm that advances all right-hand sides together, sharing search-direction information across the entire set. Because the block dimension has already been pruned, the block solver operates on a far leaner problem than the original sweep would imply.

The numerical experiments reported in the paper show that both variants deliver on their promise. In test cases ranging from canonical scattering objects to large-scale missile geometries, rFETI-DPEM and bFETI-DPEM reproduced the RCS results of the original FETI-DPEM method to the expected accuracy, confirming that the acceleration does not come at the price of fidelity. The speed-ups, however, differed markedly between the two. The block approach proved the more robust of the pair, and in the largest missile analyses it achieved a speed-up factor of up to 19.6 times compared with the baseline method. The researchers attribute this superiority to the block strategy’s ability to exploit the correlation structure among all the right-hand sides at once, rather than only sequentially, and to the dimensionality reduction provided by the rank-revealing QR step, which strips out redundant work before the expensive iterative phase begins.

The significance of such an improvement extends well beyond a single benchmark. Angular RCS analysis sits at the heart of stealth certification, and the cost of that analysis constrains how finely engineers can sample angles, how detailed their finite-element models can be, and how many design iterations they can afford. A nearly twenty-fold acceleration in the large-scale regime effectively multiplies the design exploration budget. It allows analysts to sweep angles more densely, catching narrow specular lobes and grazing-angle signatures that a coarse angular sampling might miss, and it makes it feasible to evaluate radar signatures early in the design process rather than late, when geometric changes are expensive. The work also connects to a broader current in computational electromagnetics, in which domain decomposition, Krylov recycling, and block iterative techniques are being combined to push the frontier of solvable problem sizes.

The methodological lineage of the paper is deep. The recycling idea traces back to foundational work by Parks and colleagues on reusing Krylov subspaces for sequences of linear systems, and recycling BiCGSTAB has found applications from parametric model-order reduction to computational fluid dynamics. Block Krylov methods for multiple right-hand sides have been developed since the early 2000s, including block variants of BiCGSTAB used in lattice quantum chromodynamics. What the Korean team contributes is the careful integration of these ingredients into the electromagnetic FETI-DP framework, together with the observation that angular radar sweeps provide exactly the kind of correlated right-hand-side structure that makes recycling and blocking pay off. The rank-revealing QR reduction, drawn from classical numerical linear algebra, acts as the bridge that makes the block method efficient rather than merely elegant.

For the aerospace community, the message is that the computational wall standing in front of large-scale stealth analysis has just become considerably lower. The authors position the framework as a highly efficient and robust solution for radar stealth analyses of complex aerospace structures, and the reported speed-ups suggest that full-aircraft angular signature studies that once strained computational budgets could become routine. As stealth requirements spread from dedicated attack aircraft to missiles, drones, radomes, and coated components, tools that turn weeks of computation into days, or days into hours, will shape what engineers can design and verify. This study demonstrates that sometimes the most powerful upgrade to a stealth platform is not a new coating or a new shape, but a smarter way of solving the equations that describe it.

Subject of Research: Accelerated domain decomposition methods for angular radar cross-section stealth analysis of aerospace structures

Article Title: Improved Dual-Primal Domain Decomposition Method for the Angular Radar Stealth Analysis

Article References: Kang, S.-H., Park, Y., & Shin, S. (2026). Improved Dual-Primal Domain Decomposition Method for the Angular Radar Stealth Analysis. International Journal of Aeronautical and Space Sciences. https://doi.org/10.1007/s42405-026-01236-1

Image Credits: AI Generated

DOI: 10.1007/s42405-026-01236-1

Keywords: radar cross-section, stealth technology, domain decomposition, FETI-DP, finite element method, Krylov subspace recycling, block iterative solvers, rank-revealing QR, computational electromagnetics, multiple right-hand sides, aerospace engineering, parallel computation