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Fast Angular Sweep Monostatic Radar Cross Section Computation via Model Order Reduction

Noelia Naranjo, Valentin de la Rubia

arXiv:2610.10917v1physics.comp-pheess.SP

TL;DR

Computing monostatic RCS over dense angular sweeps is costly with full-wave methods. The paper combines RBM and DEIM in a non-intrusive reduced model, reporting a 4000-point F-16 sweep in 27 minutes compared with 50 MoM angles in 30 minutes.

  • Problem

    Full-wave MoM computation is costly for large systems and repeated incidence angles, while dense angular RCS evaluation is needed.

    Source: §I · p. 1, §II · p. 2

  • Method

    The non-intrusive approach combines RBM and DEIM with a black-box integral equation solver, using selected MoM solutions and reduced-space far-field evaluation.

    Source: §I · p. 1, §V · p. 4, §VII · p. 7

  • Results

    For the F-16 model, the ROM with DEIM evaluated 4000 angular points in 27 min, while MoM evaluated 50 incidence angles in 30 min.

    Source: D. F-16 Aircraft Model Problem · p. 7

  • Takeaways & Limitations

    The F-16 sweep demonstrates finer angular resolution than 50 full-order simulations, while the ROM reproduces the main features of the angular response.

    Source: D. F-16 Aircraft Model Problem · p. 7

    Localized mismatches can occur in the turbine-inlet comparison, possibly because numerical errors in MoM snapshot currents are inherited by the ROM.

    Source: C. Turbine Inlet Model Problem · p. 6

Abstract

from arXiv · show

The radar cross section (RCS) is a fundamental parameter in radar engineering, as it determines the detectability of a given target by a radar system. Increasing interest in very low observability (VLO) targets has arisen recently. As a result, especial effort is put in developing numerical tools to accurately predict the electromagnetic scattering from perfect electric conductor objects. Integral equation methods are commonly used in this task. However, full-wave computation of the monostatic RCS at a single angle can be rather time-consuming, nevermind computing the RCS in a specific angular interval with a fine sampling. In fact, this is what is needed in industrial applications. A reduced-order model for fast full-wave monostatic RCS evaluation is proposed to easily achieve fine details in the desired angular domain. An open source integral equation solver, taken into account as a black box, is used to get the electromagnetic scattering in non-penetrable objects at some specific incident angles. Several radar targets will show the possibilities and capabilities of these model order reduction approaches.

I. INTRODUCTION

Radar cross section (RCS) characterizes a target’s detectability, but computing it with full-wave methods is costly, especially across many incidence angles. The paper introduces reduced-order modeling to accelerate these angular evaluations.

I. INTRODUCTION

  • RCS measures scattered power in a direction and is used to characterize target detectability and properties.The introduction identifies applications including stealth design and evaluating communication and defense systems.Source: §I · p. 1
  • Method of Moments (MoM) accurately solves scattering integral equations but becomes computationally demanding for complex geometries and many frequencies or incidence angles.The paper frames its task as obtaining scattered fields from PEC targets across multiple incident angles.Source: §I · p. 1
  • The paper uses the Reduced Basis Method (RBM) to approximate induced currents from selected MoM solutions and applies DEIM to interpolate parameter-dependent terms efficiently.The introduction also outlines examples including a turbine inlet and an F-16 aircraft model.Source: §I · pp. 1–2

II. PROBLEM STATEMENT

The problem statement formulates PEC scattering through the EFIE and its MoM discretization. The resulting dense system is expensive to assemble and solve, motivating reduced-order techniques.

II. PROBLEM STATEMENT

  • For PEC targets, the Stratton–Chu representation leads to an EFIE boundary integral equation, expressed in weak form over an admissible function space.The space is a trace space of H(curl, Ω).Source: §II · p. 2
  • MoM discretizes the surface current using RWG basis functions on adjacent mesh triangles and represents the unknown current through basis coefficients.Testing the weak formulation against the same basis functions produces a discrete system.Source: §II · p. 2
  • The discrete MoM system uses the current solution vector I(φ), impedance matrix Z, and excitation vector V(φ).The excitation depends on incidence angle φ.Source: §II · p. 2
  • The MoM matrix is dense and large, making assembly and solution computationally expensive; once the current is computed, the scattered field and RCS can be obtained.This cost motivates reduced-order methods such as RBM and DEIM.Source: §II · p. 2

III. REDUCED BASIS METHOD

Because the current varies relatively smoothly with incidence angle, the Reduced Basis Method approximates the full solution family using a low-dimensional space of selected MoM snapshots. A greedy residual-based procedure selects snapshots across the angular band.

III. REDUCED BASIS METHOD

  • RBM uses a limited set of current snapshots I(φ_i) to approximate solutions over the angular band in a lower-dimensional space.The paper motivates this reduction by the relatively smooth variation of current with angle.Source: §III · p. 2
  • Galerkin projection forms a reduced system with dimension r much smaller than n, which can be solved less expensively than the full MoM system.The reduced solution coefficients can be used to compute the current at an angle of interest.Source: §III · pp. 2–3
  • The reduced coefficients are chosen to minimize the residual norm, providing the criterion used to assess the RBM approximation.The residual evaluation uses inner products computed once and stored in memory.Source: §III · p. 2
  • The greedy algorithm adds a MoM snapshot at the angle with the largest residual, repeating basis updates until the residual tolerance is met or the procedure stops.The residual norm can be evaluated using precomputed inner products, making checks across angles computationally simple.Source: §III · pp. 2–3

IV. DISCRETE EMPIRICAL INTERPOLATION METHOD

RBM still requires repeated evaluation of the full-dimensional excitation vector, limiting efficiency for many-angle sweeps. DEIM approximates that vector from a low-dimensional basis and computes only selected components at each angle.

IV. DISCRETE EMPIRICAL INTERPOLATION METHOD

  • RBM’s reduced system still contains an angle-dependent excitation vector whose full-dimensional evaluation is repeated during multi-angle RCS analysis.This repeated evaluation can limit overall computational efficiency.Source: §IV · p. 3
  • DEIM approximates angle-dependent excitation vectors in a low-dimensional subspace, with basis dimension m much smaller than the MoM dimension n.The approach combines projection and interpolation.Source: §IV · p. 3
  • The DEIM basis is built from excitation snapshots at sampled angles, with angular sampling chosen to balance capturing excitation variation against offline cost.The paper adopts an angular sampling criterion for selecting the snapshot count.Source: §IV · p. 3
  • A truncated POD/SVD of the excitation snapshots supplies dominant modes, and DEIM selects interpolation indices from that projection basis.The modes are linear combinations of snapshots, with coefficients determined by singular values and right singular vectors.Source: §IV · p. 3
  • For each incidence angle, DEIM computes only m selected components of the excitation vector instead of all n components.The angle-independent reduced projection factor can be precomputed.Source: §IV · pp. 3–4

V. DEIM APPROXIMATION OF THE FAR-FIELD OPERATOR

The method extends DEIM to the far-field operator so scattered fields can be evaluated from reduced current coefficients without rebuilding the full MoM current at each angle. It reuses excitation-snapshot information to avoid a separate far-field SVD and supports efficient monostatic RCS evaluation.

V. DEIM APPROXIMATION OF THE FAR-FIELD OPERATOR

  • A direct far-field evaluation would require assembling the MoM-dimension current for each angle, so the operator is approximated with DEIM instead.The far-field quantity is then evaluated from the reduced coefficients.Source: §V · p. 4
  • The far-field DEIM applies to the transpose of the row-vector operator and reuses the interpolation matrix previously obtained for the excitation vector.The paper attributes reuse of the same interpolation indices to reciprocity.Source: §V · p. 4
  • Far-field modes reuse expansion coefficients from the excitation-snapshot SVD, while the angle-independent product is precomputed and only m operator components are evaluated per angle.This avoids an additional SVD of far-field snapshots.Source: §V · p. 4
  • The proposed far-field DEIM evaluates the scattered field directly from reduced current coefficients, avoiding MoM-dimension current reconstruction at each angle.The authors state that this avoids unnecessary full-dimension computation while preserving RCS accuracy.Source: §V · p. 4

VI. NUMERICAL RESULTS

The evaluation tests the methodology on PEC targets using an open-source solver as a black box. Each test compares a 4,000-angle ROM sweep with MoM reference evaluations at 50 angles and reports timings for two ROM strategies.

VI. NUMERICAL RESULTS

  • Puma-EM supplies the MoM elements as a black box, with users specifying simulation parameters and Gmsh meshing geometries for EFIE or CFIE solutions.The procedure is described as fully automatic and non-intrusive.Source: §VI · p. 4
  • Computations ran on a desktop with an 11th Generation Intel Core i7-11700 processor at 2.50 GHz and 16 GB memory.Source: §VI · p. 4
  • Each test uses the ROM for 4,000 angular samples and MoM at 50 reference angles, timing both current-reconstruction and direct far-field DEIM strategies.The direct strategy evaluates RCS from the reduced current solution without reconstructing the full-order current first.Source: §VI · p. 5

A. Cavity Problem

For the cavity test, the reduced model closely matches the MoM RCS across the angular range while evaluating many more samples. The greedy procedure reduces the system substantially and the far-field DEIM strategy completes the sweep faster than the naive ROM strategy.

A. Cavity Problem

  • 338,035 MoM unknowns are reduced to a 39-dimensional ROM, with 47 excitation components evaluated by DEIM for the 23-GHz cavity.The target electrical size is 15.45λ.Source: A. Cavity Problem · p. 5
  • 265 min completes the 4,000-angle sweep with far-field DEIM, versus 276 min for naive ROM reconstruction and 335 min for 50 MoM angles.The greedy residual decreases over the angular interval and terminates after 39 basis vectors.Source: A. Cavity Problem · p. 5
  • The 4,000-sample ROM RCS agrees excellently with the MoM reference across the complete angular range, whose reference uses 50 incidence angles.The MoM values are plotted as dots and the ROM samples as continuous curves.Source: A. Cavity Problem · p. 5

B. NASA Almond Problem

The NASA almond example uses a 35-GHz angular sweep and reduces a large MoM problem to a compact ROM. The proposed far-field DEIM strategy computes 4,000 samples faster than the 50-angle MoM reference, while the reduced and full-order RCS are compared.

B. NASA Almond Problem

  • At 35 GHz (29.57λ), the target is evaluated with θ = 90° and φ from −45° to 45°; the 222,624-unknown MoM problem is reduced to a 33-dimensional ROM using 45 DEIM components.MoM evaluates 50 angular samples in 20 min, while the ROM computes 4,000 samples in 15 min using the DEIM far-field approximation.Source: B. NASA Almond Problem · p. 6
  • 222,624 MoM unknowns are reduced to a 33-dimensional ROM, and excitation DEIM uses 45 interpolation components.Source: B. NASA Almond Problem · p. 6
  • 15 min computes 4,000 almond samples with far-field DEIM, compared with 20 min for 50 MoM angles; the reduced and full-order RCS are also compared.The supplied text does not state the outcome of that RCS comparison.Source: B. NASA Almond Problem · p. 6

C. Turbine Inlet Model Problem

The turbine inlet example tests the reduced model on a 5.5 GHz target over a 180° angular interval. It evaluates 4000 angles faster than the full-order model evaluates 50, while retaining the overall RCS behavior despite localized mismatches.

C. Turbine Inlet Model Problem

  • 106 min for 50 angular samples with MoM versus 74 min for 4000 angles with the ROM, using a 35-dimensional reduced model and 41 DEIM components.The full-order problem has 89942 unknowns; the analysis is at 5.5 GHz over φ ∈ [0°, 180°].Source: C. Turbine Inlet Model Problem · p. 6
  • The residual evolves during greedy construction as the reduced basis is enriched.The residual-per-iteration plot is identified in Fig. 8.Source: C. Turbine Inlet Model Problem · p. 6, Figure 8 · p. 7
  • The ROM reproduces the full-order RCS behavior across the angular range, but localized mismatches may reflect errors in the MoM current snapshots.The paper says such snapshot inaccuracies can be inherited by the reduced model, which still reproduces the main characteristics and overall angular evolution.Source: C. Turbine Inlet Model Problem · p. 6, Figure 9 · p. 7

D. F-16 Aircraft Model Problem

For the F-16 model, the ROM computes a 4000-point angular sweep in less time than MoM needs for 50 angles. Its RCS response agrees well with the MoM reference samples and reproduces the main angular features.

D. F-16 Aircraft Model Problem

  • 27 min for a 4000-point ROM sweep versus 30 min for MoM at 50 incidence angles.The full-order problem has 125721 unknowns; the reduced system has dimension 41, with 45 DEIM components retained.Source: D. F-16 Aircraft Model Problem · p. 7
  • The F-16 case is analyzed at 500 MHz, with electrical size 19.65λ and θ = 90° over φ ∈ [0°, 90°].The ROM uses DEIM applied to the far-field operator for the reported angular sweep.Source: D. F-16 Aircraft Model Problem · p. 7
  • The monostatic RCS comparison shows good agreement between MoM reference samples and the ROM's continuous response, which reproduces the main angular features.Fig. 11 tracks residual evolution through the greedy iterations.Source: D. F-16 Aircraft Model Problem · p. 7

VII. CONCLUSIONS

The paper presents a non-intrusive RBM–DEIM strategy for fast full-wave monostatic RCS sweeps. It builds a compact current basis from selected MoM solutions and extends DEIM to far-field evaluation so the online calculation remains reduced.

VII. CONCLUSIONS

  • The non-intrusive RBM–DEIM framework uses the full-order MoM solver at selected incidence angles and a low-dimensional model for the final angular response.The solver is treated as a black box.Source: §VII · p. 7
  • RBM builds a compact current basis from representative full-order solutions selected by a greedy algorithm guided by a residual-error estimator.The estimator identifies angles where additional information is required.Source: §VII · pp. 7–8
  • DEIM is extended to far-field evaluation, allowing the scattered field and monostatic RCS to be computed from reduced current coefficients without reconstructing the full current vector.This keeps the online stage within the reduced space.Source: §VII · p. 8
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