An Efficient Dual Application of the Adaptive Cross Approximation for Scattering and Radiation Problems

Abstract

This paper presents a computationally efficient approach for the solution of electromagnetic problems based on the combination of the Characteristic Basis Functions and Adaptative Cross Approximation, allowing an improvement of the processing time and memory requirements with respect to the conventional MoM and CBFM approaches. The presented technique has been validated using a number of cases in order to assess its accuracy and efficiency.

1. Introduction

The Method of Moments (MoM) [1] is widely recognized as the standard boundary element technique for analyzing scattering and radiation problems. However, its practical use is often limited by the strong computational demands imposed by the size and density of the impedance matrix. Over time, various strategies have been developed to address these limitations.

The Multilevel Fast Multipole Method (MLFMM) [2,3] is a very well-known acceleration technique for the matrix–vector product computations in the iterative solution process, and significantly reduces the memory usage by storing only near-field coupling terms. Despite these advances, practical challenges remain, including convergence issues due to either the scale of the problem or the resonant nature of some complex geometries. The MLFMM can be used together with complementary techniques to enhance its efficiency for larger or more intricate cases. It can be combined with methods that reduce the effective size of the problem by means of the manipulation of the existing mesh [4], the compression of the aggregation/disaggregation terms using spherical harmonics [5] or an interpolation of the aggregation terms for the monostatic analysis with close angular or frequential samples [6].

The additional strategies followed to increase the efficiency of the conventional MoM are the Complex Multipole Beam Approach (CMBA) [7], the Adaptive Integral Method (AIM) [8] or the Impedance Matrix Localization technique (IML) [9], following the strategy of a fast computation of matrix–vector products while avoiding the storage of the full coupling matrix, as in the case of the MLFMM.

Another class of efficient techniques introduces an additional stage that defines new basis functions over extended domains [10,11,12,13,14,15,16,17,18,19]. These macro-basis functions (MBFs) complement the conventional low-level basis functions typically used in the MoM. Approaches like the Characteristic Basis Function Method (CBFM) [10,11] and the Synthetic Functions Expansion (SFX) [12] method fall into this category. MBFs are essentially weighted aggregations of low-level functions within a specific supporting block, typically calculated as orthogonalized current solutions of sub-problems where the block is isolated from the rest of the geometry and subject to artificially introduced excitations. The primary advantage of these techniques is a substantial reduction in the total number of unknowns, often by an order of magnitude, extending the applicability of direct solvers to larger problems. It is possible to combine MBF-based approaches with MLFMM, matching the block and region partitioning schemes [11]. Some recent works describe the generation of Krylov sub-spaces involving MBFs, as presented in [16,17]. Additionally, ray-tracing methods can further reduce the number of unknowns by determining different thresholds for blocks depending on their position [18]. A rapid approach for generating MBFs using only the near-field components of low-level coupling matrices is presented in [14]. In [19], MBFs serve as a sparse basis for a Compressive Sensing method to analyze the bistatic radar cross-section (RCS) of 3D targets.

Some techniques make use of matrix compression algorithms based on rank reduction [20] in order to improve the efficiency of conventional approaches. The Adaptive Cross Approximation (ACA) [21] is a well-known technique used to compress rank-deficient matrices, defining them as an approximate QR decomposition where one dimension of the factors is much smaller than the other, without having to store all the coefficients of the original matrix. Multiple right-hand side vectors are efficiently compressed in [22], using the ACA. The Dual-Modified Gram Schmidt technique (Dual-MGS) [23] has been also applied for this purpose with good results. A multilevel subdivision approach applied to analyze dielectric objects and using the Multiscale Adaptive Cross Approximation in order to assemble the impedance matrix is described in [24]. A method used to generate multilevel MBFs using a recursive decomposition of the impedance matrix as well as its compression by means of a fast Adaptive Cross Approximation algorithm is described in [25]. A weighted-average ACA is proposed in [26] to mitigate the loss of error control in cases where the matrix alternates between zero and nonzero sub-blocks. A fast parallel direct solver based on the ACA is described in [27], where the off-diagonal blocks are compressed after performing an LU decomposition. The authors have previously presented in [28] an approach where the low-level near-field impedance matrix is compressed using ACA in order to speed up the generation of the reduced matrix. An additional work using a fast and efficient multilevel ACA-based technique (MLFACA) for the compression of the low-level impedance matrix is described in [29].

We present in this work a dual application of the ACA to an efficient numerical approach that combines the CBFM and MLFMM, for the solution of homogeneous electromagnetic problems in the frequency domain. This strategy allows to compute the coupling matrix between MBFs more efficiently by compressing the low-level matrices corresponding to the active and testing blocks and, on the other hand, compress the near-field reduced matrix, containing coupling terms between macro-basis functions, allowing additional CPU time and memory improvements in the solution process. Note that this work is a continuation of that presented in [28], with the main contribution being that ACA is also used to compress the near-field part of the reduced matrix and applied in the matrix–vector product operator, resulting in the use of an MLFMM-ACA-CBFM combination for the iterative solution of the global problem.

The document is arranged as follows: Section 2 presents an outline of the efficient MLFMM-CBFM combination considered in this work, emphasizing the generation of the macro-basis functions given that it is a process that can involve a considerable investment in terms of processing time, and containing a description of the application of ACA to accelerate the computation of the reduced matrix and its factorization. Some representative results are presented in Section 3 and, finally, the main conclusions and future lines of work appear in Section 4.

2. Description of the Proposed Approach

In the CBFM, the currents on the scatterer are expressed in terms of a set of pre-computed CBFS that extend over relatively large surfaces called blocks. The CBFS can be considered as MBFs, because the size of the blocks is similar to (or larger than) the wavelength. The use of these functions leads to a reduced matrix whose size is much smaller than that obtained in the conventional MoM. As a consequence, direct solvers can be applied to solve the problem of a moderate electrical size, which could only be addressed previously by relying upon iterative techniques.

As the problem size grows and the number of CBFs needed to model the currents increases substantially, the CBFM system matrix can still become large enough to require an iterative solution. Additionally, the memory required to store its coupling matrix, usually known as a reduced matrix, can become a bottleneck. It is possible in this case to mitigate these challenges by storing only the near-field terms of the matrix while using the MLFMM to compute far-field interactions. In this case, the geometry is divided into several cubical boxes, which are then clustered into larger, higher-order groups. At the initial level, these cubes contain a limited number of basis functions, and the coupling between basis functions in nearby cubes is accurately calculated and stored. The combined MLFMM-CBFM approach enables the efficient storage of only the near-field coupling matrix terms, while far-field interactions are computed dynamically during the iterative process.

The MLFMM-CBFM technique offers significant computational benefits over other conventional approaches, particularly in terms of memory usage and CPU time. The reduced matrix and the aggregation and disaggregation terms are major contributors to the total storage demands. Two primary factors drive the memory reduction: the CBFM reduces the number of unknowns, thereby decreasing the size of both the impedance matrix and the multipole aggregation and disaggregation terms, and this method also reduces the CPU time by producing a smaller system of equations, enabling faster solutions. The iterative solution process gains efficiency from the MLFMM-optimized matrix–vector multiplications.

Note that the MLFMM-CBFM approach can require substantial time for the preprocessing stage. When the total solution time is high compared to that of the preprocessing phase, using larger block sizes is advantageous, as the reduced iteration time more than offsets preprocessing demands. This is particularly beneficial for monostatic RCS problems, where multiple excitations are solved with the same reduced matrix. Conversely, for cases where iteration demands are low, such as in antenna radiation pattern computation or bistatic RCS problems, the preprocessing time may become excessive, making MLFMM-CBFM less efficient.

The main contribution of this work relies on the application of the ACA with the CBFM in order to (i) compute the CBFs efficiently and (ii) calculate the reduced matrix with an important reduction in CPU time and memory requirements.

2.1. Application of the ACA to the Reduced Matrix Computation with the CBFM

The reduced matrix, containing the coupling terms between different CBFs, can be expressed as follows:

$$\left[Z^R\right]=\left(\begin{array}{cccc}{\left[Z^R\right]}_{1,1}& {\left[Z^R\right]}_{1,2}& \cdots & {\left[Z^R\right]}_{1,K}\\ {\left[Z^R\right]}_{2,1}& {\left[Z^R\right]}_{2,2}& \cdots & {\left[Z^R\right]}_{2,K}\\ ⋮& ⋮& \ddots & ⋮\\ {\left[Z^R\right]}_{K,1}& {\left[Z^R\right]}_{K,2}& \cdots & {\left[Z^R\right]}_{K,K}\end{array}\right)$$

(1)

where each [Z^R]_i,j submatrix makes reference to the coupling between the CBFs contained in the i-th testing block and the j-th source block. Each submatrix can be computed as follows:

$${\left[Z^R\right]}_{i,j}=\left(\begin{array}{cccc}\langle L\left(J_{j,1}\right),W_1^i\rangle & \langle L\left(J_{j,2}\right),W_2^i\rangle & \cdots & \langle L\left(J_{j,Mi}\right),W_1^i\rangle \\ \langle L\left(J_{j,1}\right),W_2^i\rangle & \langle L\left(J_{j,2}\right),W_2^i\rangle & \cdots & \langle L\left(J_{j,Mi}\right),W_2^i\rangle \\ ⋮& ⋮& \ddots & ⋮\\ \langle L\left(J_{j,1}\right),W_{Mi}^i\rangle & \langle L\left(J_{j,2}\right),W_{Mi}^i\rangle & \cdots & \langle L\left(J_{j,Mi}\right),W_{Mi}^i\rangle \end{array}\right)$$

(2)

where $\langle L\left(J_{j,L}\right),W_K^i\rangle$ is the inner product between the L-th CBF on the j-th block and the K-th CBF on the i-th block. Each CBF can be expressed as a linear combination of low-level (conventional) basis functions on their supporting block as follows:

$$J_K^i\left(u,v\right)={{\displaystyle \sum }}_{n=1}^{N_i}{\alpha }_{i,K}\left(n\right)T_n\left(u,v\right)$$

(3)

$$W_K^i\left(u,v\right)={{\displaystyle \sum }}_{n=1}^{N_i}{\alpha }_{i,K}\left(n\right)R_n\left(u,v\right)$$

(4)

Note that $T_n\left(u,v\right)$ and $R_n\left(u,v\right)$ are the n-th low-level basis and testing functions defined on the i-th block, and the ${\alpha }_{i,K}\left(n\right)$ coefficient makes reference to the value of the K-th CBF on the i-th block and samples at the center of the n-th low-level basis function. With these observations, the coupling between the m-th CBF defined on the i-th (testing) block and the n-th CBF defined on the j-th (source) block can be expressed as follows:

$$\langle L\left(J_{j,n}\right),W_m^i\rangle ={\displaystyle \sum }_{k=1}^{Nm}{\displaystyle \sum }_{l=1}^{N_n}{\alpha }_{j,n}\left(l\right){\alpha }_{i,m}^{*}\left(k\right)\langle T_l\left(u,v\right),R_k\left(u^{\prime }, v^{\prime }\right)\rangle$$

(5)

which can be written as follows:

$$\langle L\left(J_{j,n}\right),W_m^i\rangle ={\displaystyle \sum }_{k=1}^{Nm}{\displaystyle \sum }_{l=1}^{N_n}{\alpha }_{j,n}\left(l\right){\alpha }_{i,m}^{*}\left(k\right)Z_{k,l}$$

(6)

where Z_k,l refers to the coupling between the low-level functions k and j.

With the previous considerations, each element of the reduced matrix can be obtained in the following way:

$$Z_{m,n}=\langle L\left(J_{j,n}\right),W_m^i\rangle ={\displaystyle \sum }_{k=1}^{N_m}{\displaystyle \sum }_{l=1}^{N_n}{\alpha }_{j,n}\left(l\right){\alpha }_{i,m}^{*}\left(k\right)\langle T_l\left(u,v\right),R_k\left(u\prime ,v\prime \right)\rangle$$

(7)

and this expression can be written in terms of the conventional low-level impedance coefficients contained in the i-th and j-th blocks, as follows:

$${\mathrm{Z}}_{\mathrm{m},\mathrm{n}}{=\langle \mathrm{L}(\mathrm{J}}_{\mathrm{j},\mathrm{n}}{),\mathrm{W}}_{\mathrm{m}}^{\mathrm{i}}\rangle ={{\displaystyle \sum }}_{\mathrm{k}=1}^{{\mathrm{N}}_{\mathrm{m}}}{{\displaystyle \sum }}_{\mathrm{l}=1}^{{\mathrm{N}}_{\mathrm{n}}}{\mathsf{\alpha }}_{\mathrm{j},\mathrm{n}}{\left(\mathrm{l}\right)\mathsf{\alpha }}_{\mathrm{i},\mathrm{m}}^{*}{\left(\mathrm{k}\right)\mathrm{z}}_{\mathrm{k},\mathrm{l}}$$

(8)

Due to the reduction in the number of unknowns, which is generally an order of magnitude less than that produced by the MoM, the reduced matrix becomes more manageable, but for large problems, additional approaches can be necessary in order to reduce the computational cost associated to the computation of this matrix. This work proposes a factorization using the Adaptive Cross Approximation with the purpose of expressing the matrix as follows:

$${\left[{\tilde{\mathrm{Z}}}_{\mathrm{k},\mathrm{l}}\right]}_{{\mathrm{N}}_{\mathrm{m}}{\times \mathrm{N}}_{\mathrm{n}}}={\left[{\mathrm{Q}}_{\mathrm{k},\mathrm{l}}\right]}_{{\mathrm{N}}_{\mathrm{m}}\times \mathrm{r}}{\left[{\mathrm{R}}_{\mathrm{k},\mathrm{l}}\right]}_{{\mathrm{r}\times \mathrm{N}}_{\mathrm{n}}}$$

(9)

where r is much smaller than either N_m or N_n, and the computation of these matrices does not require the storage of the original matrix or the calculation of all its coefficients.

The ACA is exclusively based on algebraic manipulations, and does not depend on the integral equation kernel. It requires the dynamic generation of rows and columns of the original matrix without having them stored in advance in order to build the Q and R factors, making use of a tolerance parameter ε that allows to control the degree of accuracy in the compression of the matrix:

$$\Vert {\left[R_{k,l}\right]}_{N_m\times N_n}\Vert =\Vert {\left[Z_{k,l}\right]}_{N_m\times N_n}-{\left[{\tilde{Z}}_{k,l}\right]}_{N_m\times N_n}\Vert \le \epsilon \Vert {\left[Z_{k,l}\right]}_{N_m\times N_n}\Vert$$

(10)

where [R_k,l] is the error matrix, and $\Vert ·\Vert$ represents the Frobenius norm. Substituting (9) into (8), and using the matrix representation of a sub-block of the reduced matrix, we obtain the following:

$${\left[{\tilde{Z}}_{m,n}^{\left(R\right)}\right]}_{i,j}={\left[A_k\right]}^H\left[Q_{k,l}\right]\left[R_{k,l}\right]\left[A_l\right]$$

(11)

where each column of $\left[A_l\right]$ contains the coefficients of the low-level basis functions required to represent each of the CBFs contained in the j-th block as follows:

$$\left[A_l\right]=\left[\begin{array}{ccc}{\alpha }_{1,l}\left(1\right)& \dots & {\alpha }_{M_l,l}\left(1\right)\\ ⋮& \ddots & ⋮\\ {\alpha }_{1,l}\left(N_l\right)& \dots & {\alpha }_{N_l,l}\left(N_l\right)\end{array}\right]$$

(12)

considering that N_l is the number of CBFs in the block.

It can be important to highlight that since we are considering a compression based on a reduced rank approximation, the diagonal blocks are not included in the factorization process using ACA. The original matrix is, therefore, expressed as the sum of two matrices where one of them contains the coupling coefficients where the testing and source blocks are the same, as seen in the following:

$$\left[Z\right]=[Z^{dia}\left]+[Z^{far}\right]$$

(13)

with $[Z^{dia}\left]=[Z_{i,i}\right]$. Note that this matrix is different from the $[Z^{near}]$ near-field coupling matrix defined by the MLFMM, since the latter also includes coupling coefficients between neighboring blocks. The matrix being excluded from the ACA processing therefore is smaller than that being excluded from the computation of far-field elements using the MLFMM by means of the aggregation, translation and disaggregation stages. Note that the low-level coupling matrix can be discarded after applying the processing given by (11), including its factorized form. Figure 1 shows an illustrative scheme of the block structure of the impedance matrix of a problem involving a 2λ plate (top) with a block size of λ and a 4λ plate (bottom) with a block size of 0.25 lambda. In the top plot, we have surrounded the diagonal near-field blocks with a red line and the off-diagonal near-field blocks with a blue line, while the far-field blocks are represented as white squares. In each of these plots, nz represents the total number of coupling coefficients contained in the near-field impedance matrix.

The previous development has been applied to the submatrix resulting from the interactions between a source block and an observation block, and can be extended to compress submatrices that include multiple source and/or observation blocks by applying the ACA approximation to the submatrix that captures the interactions among all their elements. In this case, the matrix to be compressed will be larger, allowing for greater compression.

Regarding the CPU time required for the factorizing of the coupling matrix, using a submatrix that contains multiple sub-blocks reduces the number of submatrices to which the ACA method must be applied, as well as the total number of rows and columns that need to be calculated from the original coupling matrix. This results in a significant reduction in CPU time. In terms of memory requirements, although the compression is more effective when using multiple blocks, the individual size of each factored matrix will be larger than when applying the factorization to a single source and observation block. Since these factorized matrices are discarded once the section of the reduced matrix that captures the coupling between the CBFs of the affected blocks has been calculated, the memory required for this phase does not add up to that of the solution process. The size of the submatrices to be factorized can therefore be adjusted depending on the available memory. This degree of flexibility allows us to adapt to the resources available by selecting a submatrix size for each application of the ACA method.

In our experience, a reasonable compromise is given by applying ACA to submatrices including the elements of a single testing block with all the corresponding source blocks, or vice versa. Once the described process is complete, the CBFM-reduced matrix is obtained, and the remaining steps can be followed as usual. This allows combining CBFM with other techniques to accelerate the subsequent phases (for example, techniques that speed up the solution when using a direct method, or techniques like MLFMM, used with an iterative solver).

2.2. Dual Use of the ACA in the Calculation of the Reduced Matrix with ACA-CBFM

In this paper, we are considering the dual application of the ACA for computing the reduced matrix of the CBFM (ACA-CBFM). As shown in (8), the computation of the reduced matrix requires elements from the low-level coupling matrix. In (9) and (11), this coupling matrix is factorized using the ACA method, ultimately yielding the reduced matrix ${\left[{\tilde{Z}}_{m,n}^{\left(R\right)}\right]}_{i,j}$. This matrix can also be compressed using ACA, allowing it to be expressed as follows:

$${\left[Q_{m,n}^{\left(R\right)}\right]}_{i,j}{\left[R_{m,n}^{\left(R\right)}\right]}_{i,j}={\left[A_k\right]}^H\left[Q_{k,l}\right]\left[R_{k,l}\right]\left[A_l\right]$$

(14)

where ${\left[Q_{m,n}^{\left(R\right)}\right]}_{i,j}$ and ${\left[{\mathrm{R}}_{\mathrm{m},\mathrm{n}}^{\left(\mathrm{R}\right)}\right]}_{\mathrm{i},\mathrm{j}}$ represent the matrices that result from the application of the ACA compression over the m and n CBFs of the i-th and j-th blocks. In this way, the reduced matrix is compressed and factorized. Both the coupling matrix and the reduced matrix have been factorized using a dual application of the ACA method. This process is simplified if the domains used for the dual application of the compression method are the same.

The factorized form of the reduced matrix is used in the solution process and, unlike the factorized coupling matrix, it cannot be discarded, so it is essential to minimize its storage requirements. As mentioned before, this can be achieved by applying the ACA algorithm to a subset of matrices that encompass multiple blocks. As previously indicated, the diagonal sub-matrices of the reduced matrix ${\left[{\mathrm{Z}}^{\left(\mathrm{R}\right)}\right]}_{\mathrm{m},\mathrm{m}}$ are not included in this ACA factorization process.

2.3. Combination of ACA and MLFMM with the CBFM

The MLFMM calculates the interactions of distant elements without storing them in the matrix, instead using an aggregation, translation and disaggregation procedure within the matrix–vector product in the iterative process. Therefore, the reduced matrix of the CBFM can be divided into the following two submatrices:

$$Z_{m,n}{=Z}_{m,n}^{\mathit{nf}}{+Z}_{m,n}^{\mathit{ff}}$$

(15)

where $Z_{m,n}^{\mathit{nf}}$ y $Z_{m,n}^{\mathit{ff}}$ contain, respectively, the near- and far-field contributions. Only the elements of $Z_{m,n}^{\mathit{nf}}$ are computed and stored, considering only the $N_m^{\mathit{nf}}=N_n^{\mathit{nf}}$ near-field coefficients, as follows:

$$Z_{m,n}^{\mathit{nf}}={{\displaystyle \sum }}_{k=1}^{N_i^{\mathit{nf}}}{{\displaystyle \sum }}_{l=1}^{N_j^{\mathit{nf}}}{\mathsf{\alpha }}_{j,n}{\left(l\right)\mathsf{\alpha }}_{i,m}^{*}{\left(k\right)z}_{k,l}$$

(16)

Typically, the near-field elements considered in the MLFMM are those that belong to the same box and adjacent ones, and, therefore, distant coupling elements are not stored in the reduced matrix. Consequently, the procedure described above is applied only to the near-field matrix $Z_{m,n}^{\mathit{nf}}$, which is smaller, and the reduction achieved using the ACA is smaller than that obtained when not using the MLFMM. Nevertheless, the techniques described to use the ACA across multiple blocks and the dual calculation of the near-field matrix are applicable in this method.

Compared to the ACA-CBFM approach, more memory is required when combining it with CBFM due, on one hand, to the lower reduction when applying the method to a smaller matrix, and also because to account for distant couplings, it is necessary to store the Fourier transforms of the CBFs and distant coupling terms to perform the aggregation, translation and disaggregation.

In terms of CPU time, combining ACA-CBFM with the MLFMM reduces the solution time when the electrical size of the problem is very large, as the aggregation and translation of very distant elements require fewer operations than the matrix–vector product of the submatrices obtained by applying the ACA approach.

3. Results

This section contains a number of test cases selected in order to show the accuracy and efficiency of the approach described in the previous section. All the simulations presented here have been executed on a computer with an 11th Gen Intel (R) Core (TM) i7-11850H processor at 2.50 GHz, and with 32 GB of RAM. These cases have been analyzed using 16 MPI processes.

The first test case covers the monostatic RCS computation of two PEC spheres, shown in Figure 2, with a radius of 1λ and separated by a distance of 2λ between the centers. An angular observation cut at θ = 90°, with φ ranging from 0° to 180° in 0.5° steps, has been considered. The total number of low-level unknowns is 4984, and this number is reduced to only 738 by applying the CBFM. Figure 3 compares the results obtained using MLFMM-MoM, the conventional MLFMM-CBFM technique and the proposed ACA-CBFM. A good agreement can be observed between the three solutions provided. The restarted Biconjugate Gradient Stabilized method (BiCGSTAB) [30] has been used to solve the complete system in all the examples provided, as well as the Sparse Approximate Inverse preconditioner described in [31].

Considering that this work addresses the computational efficiency improvement of the use of the presented approaches as compared with traditional techniques,

Table 1. CPU time and memory usage comparison for the solution of the two spheres case.

	MLFMM-MoM	MLFMM-CBFM	ACA-CBFM
Pre-processing CPU time	13 s	72 s	59 s
Solution CPU time	180 s	20 s	18 s
Total CPU time	193 s	92 s	77 s
Memory requirements	584 Mb	123 Mb	109 Mb

provides the processing time and memory usage for the solution of the problem using different techniques. Note that the pre-processing CPU time includes the CPU time required for the computation of the high-level basis functions, the computation of the impedance matrix (near-field coupling matrix in the techniques applying MLMM and factorized matrix using ACA) and the Fourier Transforms in the techniques that use the MLFMM.

The next test case deals with the computation of the monostatic RCS of the PEC almond shown in Figure 4 at the frequency of 3 GHz using 16 processors. The dimensions of the object are 2500 mm length, 966.683 mm width and 322.23 mm height. This test case was proposed in the 2006 JINA EM Workshop [32], and its geometry is defined by the following parametric expressions:

$$\begin{array}{c}x=dt\\ y=0.193333d\left[\sqrt{1-{\left({\displaystyle \frac{t}{0.416667}}\right)}^2}\right]\mathit{cos}\psi \\ z=0.06444d\left[\sqrt{1-{\left({\displaystyle \frac{t}{0.416667}}\right)}^2}\right]\mathit{sin}\psi \\ -\pi <\psi <\pi \end{array},\hspace{0.33em}\mathrm{for} -0.416667<t<0$$

(17)

$$\begin{array}{c}x=dt\\ y=4.833450d\left[\sqrt{1-{\left({\displaystyle \frac{t}{2.083350}}\right)}^2}-0.96\right]\mathit{cos}\psi \\ z=1.611148d\left[\sqrt{1-{\left({\displaystyle \frac{t}{2.083350}}\right)}^2}-0.96\right]\mathit{sin}\psi \\ -\pi <\psi <\pi \end{array},\hspace{0.33em}\mathrm{for} 0<t<0.583333$$

(18)

where d is the length of the almond, 2.5 m in this case. This almond structure is illuminated from θ = 90° and ϕ ranges from ϕ = 0° to ϕ = 180°. The size of the blocks used to analyze this structure has been 1 λ. For this geometry, there is good agreement between the results obtained using MLFMM-MoM and the proposed approach, as shown in Figure 5.

Due to the large dynamic range in the scattered field of this simulation, we consider that the agreement between these approaches is good and the differences are due to the intrinsic approximations that are considered when applying the MLFMM, CBFM and ACA approaches. A computational analysis of the memory and CPU time required for this simulation is shown in

Table 2. CPU time results related to the computational analysis.

	MLFMM-MoM	MLFMM-CBFM	ACA-CBFM
Number of unknowns	80,961	18,634	18,634
Pre-processing CPU time	89 s	257 s	197 s
Solution CPU time	562 s	245 s	211 s
Total CPU time	651 s	502 s	408 s
Memory requirements	1196 MB	935 MB	654 MB

.

The last test case is the analysis of the PEC aircraft depicted in Figure 6, obtaining the monostatic RCS at a frequency of 0.5 GHz. The incident plane for this analysis is ϕ = 270° covering 181 directions, from θ = 0° to 180° in 1° steps. The total number of unknowns has been 676,361 applying MLFMM-MoM, and that number is reduced to 88,408 unknowns using MLFMM-CBFM (MoM-based CBFs) and the proposed approach.

The results obtained are shown in Figure 7, where we can observe a good agreement between three different techniques: MLFMM-MoM, taken as the reference given that it is a mature solver and has been extensively validated, (ii) MLFMM-CBFM and (iii) the proposed approach. The CPU time applying MLFMM-MoM, MLFMM-CBFM (MoM-based CBFs), and the proposed technique (ACA-CBFM) is outlined in

Table 3. CPU time results related to the analysis of the airplane.

	MLFMM-MoM	MLFMM-CBFM	ACA-CBFM
Pre-processing CPU time	6139 s	10,359 s	8632 s
Solution CPU time	19,802 s	7184 s	5043 s
Total CPU time	25,939 s	17,543 s	13,675
Memory requirements	18.874 GB	13.604 GB	8.731 GB

.

4. Conclusions

A novel approach for the application of the ACA algorithm to the Characteristic Basis Functions has been developed in this work, focusing on the computational efficiency improvements obtained. An efficient technique for the computation of the reduced matrix based on the dual application of the ACA algorithm is presented. Some representative examples show good accuracy, a noticeable efficiency improvement and a great decrease in the memory footprint compared to conventional techniques.