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AbstarctA hierarchical multi-level fast multipole method (H-MLFMM) is proposed herein to accelerate the solutions of surface integral equation (SIE) methods. The proposed algorithm is particularly suitable for solutions of wideband and multi-scale electromagnetic problems. As documented in [1] that the multi-level fast multipole method (MLFMM) achi...
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... response from 1 MHz up to 8 GHz are calcu- lated based on one single mesh. The object adopted for this set of experiments is shown in Figure 8, where computational adversaries such as cavity, sharp corners, etc. are presented in this geometry. For the frequency range from 1000 MHz up to 8000 MHz, straightforward automatic surface refinement helps to keep the mesh size h ≈ 0.1í µí¼, i.e., kh ≈ 0.2í µí¼, where k is the wave number. ...
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... validate the accuracy of the H-MLFMM at low frequency with very small discretization size, we apply MLFMM to the same object, as shown in Figure 8, using a different discretization size, namely, hk = 0.01í µí¼. The result computed using MLFMM is used as a reference to compare against the result from the proposed H-MLFMM algorithm with a much finer discretization size. ...
Citations
... By simulating the two motion models, the real-time position and attitude of ACR can be obtained during its flight. For the calculation of RCS, high-frequency approximation methods are common employed to accomplish fast estimation of electromagnetic scattering, such as GO method [9][10][11], Shooting and Bouncing Ray (SBR) method [12][13][14], Multi Level Fast Multipole Method (MLFMM) [15][16][17], etc. Among them, the GO method has the fastest calculation speed but is prone to losing polarization information, while other methods have limited calculation speed and are difficult to achieve the effect of "real-time calculation". ...
With the spread of airborne corner reflector (ACR) in the field of shipborne equipment, radar guided anti‐ship missile is facing new challenges. In order to achieve the mastery of the jamming principle of this apparatus, the paper studied its motion properties and sequential Radar Cross Section (RCS) properties. Firstly, the motion model of the ACR was obtained under certain constraints, by deriving the centroid dynamics equation and rotation dynamics equation according to the theoretical mechanics. Then, the dynamic sequential RCS model of ACR array was established by combining the motion model with the modified geometric optics method and coherent synthesis method. Through the simulation and analysis of the motion model, it was found that the flight process of the ACR can be divided into two stages: fast falling and steady falling. In the fast falling stage, the variables of the ACR system change rapidly, while the ACR rotates slowly and falls smoothly in the steady falling stage. From simulation of dynamic sequential RCS model, the results showed that the obvious depolarisation effect is occur in the fast falling stage. Further statistical analysis showed that, the omnidirectivity of single ACR is well from the dynamic perspective, meanwhile array placement can effectively enhance RCS amplitude and improve the probability density distribution.
... The difficulty is that it is not infallible or costeffective to handle large scale GRs with universal methods used for other types of reflectors. Whether it is the analytical methods restricted by computer memory represented by the Multi-Level Fast Multipole Method (MLFMM) [14][15][16] or the high-frequency methods limited by the object's complexity represented by the Shooting and Bouncing Ray (SBR), [17][18][19] handling of GRs inevitably needs a vast amount of calculation resources and a lot of time. Nevertheless, Geometrical Optics (GO) approximation, [20][21][22] one of the underappreciated high-frequency methods never to be popularized because of its disability in analyzing polarization, provides a new possible solver for GRs. ...
The backscattering of perfectly conducting great-icosahedral-like reflectors is studied in the high-frequency domain. This particular faceted polyhedron, composed of 60 trihedral corner reflectors, is introduced to obtain closer omnidirectional backscattering. Due to the high cost of traditional methods, an estimation method for the full-polarized radar cross section is proposed, which is modified from the geometrical optics approximation method. The validity of the improved method is discussed, and its velocity is determined. The estimated results of the reflectors are studied, which lead to a conclusion that this complex structure has high-frequency properties of quasi-omnidirectivity and depolarization.
... These data structures include the near-field interaction matrices, the data structures holding the translation operator samples, and the matrices storing the far-fields of FMM groups. So far, substantial research has been performed to reduce the memory requirements of the data structures involving the near-field interactions and translation operators [5]- [10]. This study focuses on the memory saving in the data structures storing the far-field signatures of the FMM groups. ...
... The SVD technique was then leveraged in a multilevel FMM (MLFMM) scheme [12]. Apart from the SVD technique, the aggregation and disaggregation matrices were compressed by a skeleton basis in a hierarchical MLFMM algorithm [10]. All these methods proposed to reduce the far-fields' memory requirement are highly valuable and matrix decomposition-based. ...
Tucker decomposition is proposed to reduce the memory requirement of the far-fields in the fast multipole method (FMM)-accelerated surface integral equation simulators. It is particularly used to compress the far-fields of FMM groups, which are stored in three-dimensional (3-D) arrays (or tensors). The compressed tensors are then used to perform fast tensor-vector multiplications during the aggregation and disaggregation stages of the FMM. For many practical scenarios, the proposed Tucker decomposition yields a significant reduction in the far-fields' memory requirement while dramatically accelerating the aggregation and disaggregation stages. For the electromagnetic scattering analysis of a 30{\lambda}-diameter sphere, it reduces the memory requirement of the far-fields more than 87% while it expedites the aggregation and disaggregation stages by a factor of 15.8 and 15.2, respectively, where {\lambda} is the wavelength in free space.
... The interpolation decomposition method (ID) [17] is introduced for scalar problems, the equivalence points on the axillary surfaces are introduced at the interface of the near and far region, the dominant skeletons can be found with the ID compression of the source points against the equivalence points. In [18]- [20], the vector equivalence basis functions are introduced on the equivalence spheres for the analysis of low frequency overly dense meshing dynamic problems. In this contribution, we introduced the spherical distributed axillary RWGs [14] to be the equivalence vector basis functions; the skeleton RWGs are obtained by compressing the source RWGs against the equivalence RWGs with ACA [4], [5]; these skeleton RWGs are effective for all the far region, as we can set a predetermined threshold of ACA, as opposed to the random decomposition in ID. ...
... To find the correct skeletons for each group and its far interaction lists is much important for the low rank approximation precision. In [17], [18], the random sampling is employed in the ID. In this work, the ACA is chosen to find the optimal skeleton RWGs τ t in each groups, by compressing the source group against its far interaction lists with predetermined threshold. ...
We propose a mixed “skeleton” and “equivalence” nested approximation method to compress the impedance matrix of the method of moments (MoM) for the wideband multiscale simulations. We first introduce a nested skeleton approximation, where the impedance matrix is expressed recursively by sampling the dominant basis functions (e. g. skeletons) with a fully algebraic implementations from the original basis functions. The idea is to introduce the automatically constructed test surface around the interface between near- and far-field regions, for each group the dominant RWGs are sampled recursively with the adaptive cross approximation (ACA) to compress the matrix against the test surface. Secondly, we introduce a mixed-form algorithm of “skeleton” and “equivalence” nested approximation method, at low levels, the nested skeleton approximation is employed, and it is smoothly transferred to standard wideband nested equivalence approximation (WNESA) at high levels. Accurate number of skeletons can be always found with predetermined threshold at low level, which will improve computation efficiency, with respect to WNESA. The computational complexity of the proposed algorithm is O(N logN), N is the number of unknowns. Numerical wideband multiscale simulations demonstrate the efficiency of the proposed algorithm.
... When the variety in the element size is large, conventional implementations often suff er from inaccuracy, instability, and/or ineffi ciency issues due to numerical breakdowns in the context of discretization, expansion, and/or matrix solution. As a natural consequence, there is an enormous collective eff ort in the literature [1][2][3][4][5][6][7][8][9] to develop accurate, stable, and effi cient numerical solvers for multi-scale problems involving nonuniform discretizations. ...
... • Expansion of interactions: In MLFMA, the standard diagonalization using plane waves fails as some boxes become very small in comparison to the operation wavelength. For sub-wavelength boxes, alternative wave-expansion schemes [3,4,6,13] that are stable for short distances are therefore needed if the complexity of MLFMA is desired to be kept at linearithmic levels. ...
In computational electromagnetics, nonuniform discretizations with a large variety in the sizes of the discretization elements have always been challenging to handle. Such problems are inherently multi-scale, where different regimes coexist, as small elements are used to model tiny details while large elements are used on suitable parts comparable to the wavelength. When the variety in the element size is large, conventional implementations often suffer from inaccuracy, instability, and/or inefficiency issues due to numerical breakdowns in the context of discretization, expansion, and/or matrix solution. As a natural consequence, there is an enormous collective effort in the literature [1–9] to develop accurate, stable, and efficient numerical solvers for multi-scale problems involving nonuniform discretizations.
... MLFMA [1], [2] and its diverse low-frequency implementations that employ alternative factorizations and diagonalizations are well known in the literature [3]- [14]. In addition, various broadband MLFMA implementations, which effectively combine low-frequency and high-frequency techniques for efficient and stable computations of interactions in different electromagnetic regimes, have been developed [15]- [25]. On the other hand, algorithmic routines, such as efficiently organizing near-field and far-field interactions, especially on nonuniform discretizations that arise in multiscale structures, have not been considered in sufficient depth [26]. ...
... Apart from the EPA, some other methods based on the hierarchical MLFMA [16], multiresolution basis functions [17], and accelerated Cartesian expansion [18] have also been used to attack multiscale problems. In various advanced computing [19], chemistry [20], biology [21], and physics [22] applications, especially in N-body problems, a modified version of the FMM, namely, the adaptive FMM, has been used. ...
An efficient and versatile broadband multilevel fast multipole algorithm (MLFMA), which is capable of handling large multiscale electromagnetic problems with a wide dynamic range of mesh sizes, is presented. By invoking a novel concept of incomplete-leaf tree structures, where only the overcrowded boxes are divided into smaller ones for a given population threshold, versatility of using variable-sized boxes is achieved. Consequently, for geometries containing highly overmeshed local regions, the proposed method is always more efficient than the conventional MLFMA for the same accuracy, while it is always more accurate if the efficiency is comparable. Furthermore, in such a population-based clustering scenario, the error is controllable regardless of the number of levels. Several canonical examples are provided to demonstrate the superior efficiency and accuracy of the proposed algorithm in comparison with the conventional MLFMA.
... It requires that the surface currents in each independent subdomain be radiated to all other subdomains. The computation is accomplished with two mathematical ingredients: (i) a hierarchical multi-level fast multipole method (H-MLFMM) [49,50], which leads to the seamless integration of multi-level skeletonization technique [51] into the FMM framework and results in an effective matrix compression for non-uniform DG discretizations; (ii) a primal-dual octree partitioning algorithm for separable subdomain coupling [52]. Namely, instead of partitioning the entire computational domain into a single octree as in the traditional FMM, we have created independent octrees for all subdomains. ...
This work investigates an adaptive, parallel and scalable integral equation solver for very large-scale electromagnetic modeling and simulation. A complicated surface model is decomposed into a collection of components, all of which are discretized independently and concurrently using a discontinuous Galerkin boundary element method. An additive Schwarz domain decomposition method is proposed next for the efficient and robust solution of linear systems resulting from discontinuous Galerkin discretizations. The work leads to a rapidly-convergent integral equation solver that is scalable for large multi-scale objects. Furthermore, it serves as a basis for parallel and scalable computational algorithms to reduce the time complexity via advanced distributed computing systems. Numerical experiments are performed on large computer clusters to characterize the performance of the proposed method. Finally, the capability and benefits of the resulting algorithms are exploited and illustrated through different types of real-world applications on high performance computing systems.
... The algorithm fills the rows and columns iteratively until the convergence of the approximate relative error. Recently, skeleton based techniques have been investigated deeply [4][5][6][7][8] which try to seek the dominant elements to represent the original obese couplings. Among them, Randomised pseudo-skeleton approximation (RPSA) [7] is based on the PSA [8] of the matrix. ...
In this paper, the localized pseudo-skeleton approximation (LPSA) method for electromagnetic analysis on electrically large structures is presented. The proposed method seeks the low rank representations of far-field coupling matrices by using pseudo-skeleton approximations (PSA). By using PSA, only part of the original matrix is needed to be calculated and stored which is very similar to the adaptive cross approximation (ACA). Moreover, rank approximation and index finding schemes are given to improve the performance of the method in this paper. Several numerical results are given to demonstrate that the proposed method performs better than the randomized pseudo-skeleton approximation (RPSA) and ACA.
We propose and demonstrate a multiple-precision arithmetic framework applied to the inherent hierarchical tree structure of the multilevel fast multipole algorithm (MLFMA), dubbed the multiple-precision arithmetic MLFMA (MPA-MLFMA) that provides an unconventional but elegant treatment to both the low-frequency breakdown and the efficiency limitations of MLFMA for electrically large problems with fine geometrical details. We show that a distinct machine precision can be assigned to each level of the tree structure of MPA-MLFMA, which in turn enables controlled accuracy and efficiency over arbitrarily large frequency bandwidths. We present the capabilities of MPA-MLFMA over a wide range of broadband and multi-scale scattering problems. We also discuss the implications of a multiple-precision framework implemented in software and hardware platforms.
