A fast and accurate boundary element method framework for benchmarking the Radar Cross Section (RCS) computations for various Perfect Electric Conductor (PEC) targets, including sharp-edge models with electrically large sizes, is proposed. The approach employs a high-order Chebyshev-based Boundary Integral Equation (CBIE) formulation of the Magnetic Field Integral Equation (MFIE). Acceleration is achieved using both the H-matrix framework and distributed parallelism for filling inadmissible blocks, substantially improving computational efficiency in both time and memory usage. We refer to the resulting solver as H-CBIE and demonstrate its O(hp) error convergence, scalability, and accuracy through extensive numerical results, h being the mesh element size and p the order of discretization. Localized hp-refinement is incorporated for sharp-edge problems to maintain accuracy near edges and junctions. We include a comprehensive complexity analysis of the high-order accelerated framework, detailing the error-dependent wall time scaling for each stage of the solution process. The resulting scaling dependencies show that the total solution wall time grows quadratically in N with the target accuracy when high-order discretizations are used, whereas it rises exponentially for traditional low-order methods, underscoring the substantial efficiency gains of the proposed approach and its suitability for benchmarking.

This paper presents a high-order, error-controllable boundary element framework for generating referencequality radar cross-section (RCS) benchmarks of perfectly electrically conducting (PEC) targets with smooth and sharp-edged geometries. The proposed approach combines a Chebyshev-based boundary integral equation (CBIE) discretization of the magnetic field integral equation (MFIE) with hierarchical matrix (H-matrix) acceleration and distributed parallelism. The resulting solver, referred to as H-CBIE, is designed to deliver high-accuracy solutions with predictable error–cost trade-offs, making it well suited for benchmarking and validation of computational electromagnetics (CEM) solvers. A key feature of the framework is its reliance on high-order discretization and p-refinement, which exploits the O ( h p ) convergence of the CBIE formulation to dramatically reduce the number of unknowns required for a prescribed accuracy. In contrast to traditional low-order RWG-MoM approaches, whose computational cost grows exponentially with increasing accuracy demands, the proposed high-order strategy yields only quadratic growth in wall time with respect to additional digits of precision. This behavior is rigorously supported through an error-level-dependent complexity analysis that accounts for near-field integration, H-matrix assembly, and H-LU factorization. The framework incorporates adaptive hp-refinement to accurately resolve current singularities near edges and junctions while maintaining high-order efficiency on smooth surface regions. Geometry is represented using NURBS models and quadrilateral Bézier patches, enabling accurate CAD-to-analysis transfer and controlled geometry error. Robust iterative convergence is ensured through an H-matrix-based preconditioning strategy, allowing large-scale problems to be solved efficiently on modern parallel architectures. Extensive numerical results demonstrate the accuracy, scalability, and benchmarking capability of the proposed method. Validation is performed using analytically solvable sphere scattering problems, canonical benchmark targets such as the NASA almond, and electrically large, complex geometries including a multi-million-unknown bunny model and a B-2-like aircraft. Comparisons with Mie solutions, experimental benchmark data, and commercial RWG-MoM–MLFMM solvers show that H-CBIE achieves one to two orders of magnitude reduction in unknown count, memory usage, and runtime for comparable or superior accuracy. Overall, this work establishes a robust, high-order computational framework capable of producing reference-level RCS solutions for complex PEC targets. The methodology provides a practical foundation for future benchmarking efforts and sets a high standard for accuracy-controlled electromagnetic scattering simulations.

This research examines the use of heterogeneous tolerance strategies within the H -matrix framework to optimize solver performance for integral equations. The H-matrix method, widely employed in computational electromagnetics, is valued for reducing memory and computational complexity. A key aspect of this technique is managing block-wise accuracy, which can be adjusted across the matrix structure. This study introduces distinct tolerances for different block types in the block cluster tree. Specifically, leaf admissible blocks are assigned looser tolerances compared to non-leaf admissible blocks, reflecting their differing contributions to the solution process. Leaf blocks, often representing finer matrix levels, can operate with lower accuracy, while non-leaf blocks require higher precision to maintain overall solution accuracy. The research highlights the impact of these heterogeneous tolerances on solver efficiency. By allocating computational resources more effectively, the approach accelerates the solution process while preserving accuracy. A case study demonstrates the advantages of applying separate tolerances to leaf and non-leaf blocks, showcasing improved computational speed and solution precision.

In this work, we present a framework for efficiently solving sharp-edge geometries in integral equations without explicitly capturing singularities at the edges. The framework is designed for point-based discretization methods, particularly the mixed-order Locally Corrected Nyström (LCN) method and Chebyshev-based Nyström (CBIE) discretization. A hybrid approach is introduced: low-order solutions with local h-refinement are employed at edge elements to maintain accuracy, while high-order LCN or CBIE methods are applied to non-edge elements. This strategy ensures both accuracy and computational efficiency, mitigating the numerical difficulties posed by sharp edges. The results validate that this framework enhances solution accuracy while maintaining numerical stability in sharp-edge geometries.

This work introduces a fast direct computational framework based on the high-order Chebyshev Nyström Boundary Integral Equation (CBIE) method accelerated by hierarchical matrices (H-matrix). The recently introduced CBIE offers substantially faster matrix fill than the popular Locally Corrected Nystrom (LCN) approach. Combination of the methods results in a first-in-class fast direct computational framework which is error-controllable, fast, and insensitive to poor conditioning of the pertinent matrix equations. Matrix fill, factorization, and solution times are reported for a metallic sphere, and more complicated examples such as scattering from a curvilinear NURBS-based airplane CAD model and a spiral antenna are presented to demonstrate the solver's versatility.

The paper proposes a new approach to the fast solution of matrix equations resulting from boundary element discretization of integral equations. By hybridizing fast iterative H-matrix solvers with a fast direct H-matrix preconditioner factorization, we create a framework that can be tuned between the extremes of a direct solver and a unpreconditioned iterative solver. This tuning is largely achieved using a single numerical parameter representing the preconditioner tolerance. A more complicated scheme involving two different tolerances is also briefly considered. The proposed framework is demonstrated on a high-order accurate Locally Corrected Nystr & ouml;m solution of surface integral equations for PEC targets. Examples consider various scattering problems including those featuring strong physical resonances. We show that appropriately choosing the preconditioner tolerance achieves the prescribed solution accuracy with minimal CPU time. Expanding from one to two tolerance parameters further enhances the framework by providing the flexibility to dynamically adjust tolerance, enabling higher compression while maintaining accuracy and fast convergence. This adaptive strategy offers significant potential for optimizing the balance between memory usage and CPU time in the future.

In this study, we introduce a fast, error controllable and parallel iterative technique for dealing with the Combined Field Integral Equation (CFIE) using the Locally Corrected Nyström (LCN) method. We use a less precise H-matrix, derived from the coefficient matrix, as a preconditioner in a linear iterative method. This inaccurate preconditioner, which includes less precision in both constructing the H-matrix and inverting the resulting matrix, significantly speeds up the process of solving the system. However, it's important to note that this accelerated approach has notable effects on the residual and accuracy of the problem, which introduces a challenge in effectively addressing the complexities associated with solving the CFIE.

In the present study, we discuss a fast, parallel iterative solution of the Combined Field Integral Equation (CFIE) through the discretization with the Locally Corrected Nystrom (LCN) method. A less precise H-matrix, derived from the coefficient matrix, is employed as a preconditioner in the context of a linear iterative method such as GMRES. The employment of a less accurate preconditioner constructed with lower precision both at the stage of computing the H-matrix and its subsequent factorization is shown to effectively accelerate the iterative matrix equation solution. The effect of the numerical tolerance in formation of the preconditioner H-matrix and its factorization on the rate of convergence is studied.

A parallel, fast, direct, high-order solution of the Locally Corrected Nystrom (LCN) method discretization of the combined field integral equation (CFIE) is presented for solving scattering problems involving perfect electric conductors (PECs) of arbitrary shape. The discrete LCN operator is represented using the hierarchical matrix (H-matrix) framework to accelerate the filling and solving processes, while consuming a fraction of the memory conventionally required for the dense system. The solver is validated for an exact parametrization of the surface of a sphere with quadrilateral patches (i.e., a mapped sphere). The accuracy is also studied for high-order solutions of arbitrary shapes, demonstrating a O(hp) convergence. Results from this direct solver are indicative that the H-matrix-accelerated LCN method will provide a flexible error-controllable preconditioner for general scattering problems.

In this research, a new method is presented for addressing the Combined Field Integral Equation (CFIE) using the Locally Corrected Nyström (LCN) technique, focusing on speed, error control, and parallel processing features of the implementation. The method involves utilizing a less precise preconditioner derived from the coefficient matrix, which accelerates the whole process despite its reduced accuracy. However, this fast approach has some challenges in trading-off between solution speed and accuracy since we are approximating H-LU decomposition as well. This is particularly affecting the residual and precision of the solution when dealing with CFIE problems. The method implemented uses MPI parallelization for coefficient filling, and it uses HLibPro, which is OpenMP parallelized, to construct H-Matrices and perform H-LU factorization. To reach the desired compression, ACA method is used to compress the matrices to the tolerance required. Giving larger tolerance to preconditioner matrix and H-LU factorization, makes the problem hard to converge to the desired accuracy. As a reason, it is necessary always to trade-off between accuracy, the speedup and compression of the matrices. To illustrate its efficacy, consider Nasa Almond with 84,000 unknowns (where $l=240 \mathrm{~m}$) exposed to an z-polarized dipole located in the x direction in the far-field. The operating frequency is set at 3 MHz, with geometric dimensions spanning two wavelengths. As it is shown in Fig. 1, the Compression Rate (CR) achieved for the preconditioner H-matrix is $91.56 \%$, while for the coefficient matrix, it is $85.44 \%$. The relative error of the computational problem, $|A x-b|$, is quantified at $4.1023 \mathrm{e}-09$, which converges in four iterations. Moreover, the CPU time required to build both H-matrices is 9555.36 seconds, while it takes 4130.66 seconds to do H-LU factorization with 40 cores working in parallel.

In this paper, we present a novel algorithm for determining the 3-D scattering centers of a target that cause an increase in radar observability or radar cross-section (RCS) of the platform. The scattering centers on a given platform are calculated using the current density induced on the surface for the given direction of illumination and frequency of operation. The current density may be computed using a full-wave or an approximate electromagnetic solver. For a given point on the surface of the scatterer, the relative RCS contribution is computed by evaluating the radiation integral within a specific neighborhood. This RCS contribution value is then assigned to the corresponding point on the surface of the target, forming a color map of RCS contributions on the 3-D model of the scatterer. This approach can be further used as feedback mechanism to reduce the RCS in an iterative manner.

The resonance frequencies of the system of dielectric objects are investigated with a previously constructed rigorous algorithm based on the Analytical Regularization Method (ARM) which is widely used to construct well-conditioned algebraic equation systems of the second kind by using some problem-dependent left and right hand- side regularizers. In this paper the previous algorithm is extended to handle arbitrary boundaries and the integral equation system is constructed in a way that eliminates the inner resonances of the perfectly conducting object of the same shape. The algebraic equation system resulting from the discretization of the boundary integral equation system is a first-kind one and does not allow to search of the eigenvalues of the matrix numerically. That is why through the operators of ARM, this algebraic equation system is transformed to a second kind one for which the matrix entries are convenient for the search of eigenvalues numerically. The numerical results show that the ARM-based algorithm allows finding the eigenvalues of the system of a dielectric object accurately whereas the first kind of system does not because of numerical overflow during the root search algorithm.

The Gaussian Progress Regression (GPR) is one of the most popular techniques in supervised machine learning regression algorithms. Recently it has been started to use for the prediction of the RCS of the conducting objects since it can help to decrease the number of computation points or widen the sample size of the frequency interval under investigation. It is known that the kernel of the GPR has a significant impact on the error behaviour of the algorithm. It has been shown in the literature that composing a priori knowledge of the physical solution of the problem with the kernel may improve the accuracy of the algorithm. In this study, we perform some numerical research on the effect of the kernels on the accuracy of the predicted radar cross section (RCS) of the NASA almond calculated by the classical method of moments (MoM). In numerical results, we show that different kernels result in different root mean square error (RMSE) and in contrast to the published results the best model consists of the product of non-periodic kernels. We also show that with this model the data regarding some frequencies that are not included in the training dataset can be predicted very accurately thanks to the GPR being an effective method in sparse regression.

The perfectly electrically conducting cylinder of arbitrary cross section is illuminated by a plane wave with no electric/magnetic field component along the longitudinal direction obliquely impinging onto the obstacle with a non-zero angle from its surface normal. The electric and magnetic field integral equations modeling the phenomenon are elaborated. The super algebraically convergent algorithms are exploited for calculation of their kernels. Magnetic field integral equations lead to well-conditioned corresponding algebraic systems naturally while electric field integral equations lead to well-conditioned corresponding algebraic systems after their Analytical Regularization which will be outlined, where beforehand they are ill-conditioned. Solutions of both equations verify each other. After post processing these solutions, the accuracy as well as the condition of the algebraic systems will be investigated for each equation having incidence angle as a parameter.

A new method extending the scope of the superalgebraically convergent 2-D wave diffraction algorithms is proposed. The principal tool is the construction of an infinitely smooth approximation to the boundary contour. The construction is based on the ideas of Tikhonov's regularization to the stable summation of the Fourier series and the evaluation of unbounded operators. Both the algorithms for smoothing and the nonsaturated solutions for integral equations together provide the superalgebraic (faster than any algebraic) convergence. The comparison of the algorithms using the original (noninfinitely smooth) parameterization and the infinitely smooth one demonstrates the essential advantage of the latter over the former with respect to accuracy and time consumption.

We review guidelines to obtain fast and accurate solutions—based on integral equations—of the Helmholtz equation with mixed boundary values in two dimensions, a crucial issue when modeling electromagnetic phenomena in complex photonic media. We solve the electric- and magnetic-field integral equations (EFIE and MFIE) to treat scattering of electromagnetic waves from transverse magnetic (TM)- and transverse electric (TE)-excited impedance cylinders represented with smoothly parameterized cross-section contours. We show that it is possible to obtain superalgebraic convergence with accurate calculations of the kernels of the integral equations whose singularities vary from weak to hypersingular. These Fredholm equations of the second kind are subject to stable discretization procedures. However, for various values of impedance, numerical stability can be maintained only via analytical regularization. Finally, we provide numerical results that support our conclusions.

In this paper, the application of the Analytical Regularization Method (ARM) to a dielectric cylinder is presented. At first, the matrix of the algebraic system is constructed as a 2 × 2 block matrix where the off-diagonal blocks consist of the Fourier coefficients of non-singular operators and the entries of the diagonal blocks involve the associated singularities. Then the left- and right-hand side operators of the ARM built up on these singularities are applied to the algebraic system, transforming it to a second kind one in form of $(I + H)y=b$ where $I$ the identity operator and $H$ is a compact operator in space $l_{2}$ . The algorithm is applied to a dielectric cylinder illuminated by a TM -z polarized plane wave. The formulation of TE-z polarization is almost the same except the field functions will be in terms of magnetic field and its normal derivative. The numerical results are validated by the solution of separation of variables for a circular dielectric cylinder and then the superiority of the second kind system to the first kind one is shown by means of the condition number with respect to the truncation number of the algebraic system.