Equivalence Surface Research Articles

Application of distance field-based algorithm to adjacent spline surfaces for composite structures boundary modeling

The complex boundary modeling required for simulating composite materials was applied to a broad range of problems, including surface overlap and gap. A novel method for the simultaneous fitting of adjacent spline surfaces based on a distance field has been devised. The fitting method for spline surfaces was constructed using offline spline surface, facet simplification, and residual solid reconstruction. The distance field-based approach of simultaneous fitting of neighboring spline surfaces included producing a distance field for a spline surface and then applying that distance field to nearby spline surfaces. There would be junction and separation issues if neighboring spline surfaces were fitted individually. By simultaneously fitting the nearby spline surfaces with the aid of the equivalence surface of the distance field, it was feasible to avoid overlap and gap.

Fast Frequency and Material Parameters Sweep for the Calculation of Array Structures

In this letter, an efficient hybrid method is proposed to analyze array structures when frequency and material parameters sweep. Using the frequency and material independent reactions method, the impedance matrix is split into the product of a geometry dependent matrix and a wavenumber dependent factor. The geometry dependent matrix needs to be solved and stored only once when parameters sweep. However, the memory requirement will be very huge for electrically large objects such as array structures. To deal with this problem, the equivalence principle algorithm is used to decompose the array into subdomains and transfer unknowns on the subdomains to the unknowns on the equivalence surfaces. Furthermore, the characteristic basis functions are constructed on the equivalence surfaces to reduce the number of unknowns and matrix storage significantly. Several numerical results are shown to demonstrate the accuracy and efficiency of the proposed method.

Modeling of Multiscale Wave Interactions Based on an Iterative Scheme of MoM-PO-EPA Algorithm

Electromagnetic modeling of multiscale wave interactions is a challenging task. This is because wave physics exhibit different characteristics at different length scales that require suitable modeling algorithms. Combining these algorithms to model multiscale wave physics requires careful design and tuning. In this study, we investigated a hybrid algorithm based on the method of moment (MoM), iterative physical optics (IPO) approximation, and the equivalence principle algorithm (EPA). EPA modeled targets with details, MoM modeled targets with moderate scales, and IPO modeled electrically large scatterers. The virtual equivalence surfaces in EPA worked as interfaces between relatively large and small scatterers. An iterative scheme is used to solve the targets instead of a matrix equation with large dimensions. This algorithm achieved a good balance between accuracy and efficiency. Numerical examples of modeling the plane-wave scattering of multiscale targets verify its performance for 2D and 3D problems. The iterative scheme can become faster and need less memory usage when solving the multiscale scatterers.

A Simple Overlapping DDM Preconditioned Fast HOVSIE for Wideband Scatterings of Anisotropic Medium-Metallic Objects

In this article, a fast high-order volume surface integral equation (HOVSIE) formulas solver will be presented to fast calculate the wideband electromagnetic (EM) scattering of anisotropic medium-metallic composite objects. First, in the HOVSIE, the equivalence volume and surface current vectors are discretized by the high-order hierarchical vector basis functions (HVBFs), which can reduce unknowns and facilitate the calculation of the broadband EM scattering. Second, a fast wideband algorithm, i.e., the hybrid multilevel accelerated Cartesian expansion–multilevel fast multipole algorithm (MLACE-MLFMA), is integrated to accelerate the matrix-vector multiplication in the iterative solution. Finally, in order to improve the iterative convergence, a simple overlapping domain decomposition method (ODDM)-based preconditioner is also employed in the proposed HOVSIE enhanced by the MLACE-MLFMA. Summarily, three advantages can be found here: 1) more flexible calculation of the wideband EM scattering; 2) less memory occupation; and 3) better iterative convergence. The accuracy, efficiency, and flexibility will be demonstrated in the numerical examples.

An Efficient Integral Equation Model-Order Reduction Method for Complex Radiation Problem

In this letter, an efficient integral equation model-order reduction method is proposed to analyze complex radiation problem. In the offline stage, the randomized singular value decomposition and discrete empirical interpolation method are used to construct a reduced-order model (ROM) for the platform. In the online stage, the equivalence principle algorithm is used to transfer unknowns on the antenna to the unknowns on the equivalence surface. The modified tap basis scheme is applied to decompose the antenna from the platform and solve the current continuity challenge when the platform is cut off by equivalence surface. When the antenna position is changed in the input region, only the tap region needs to be remeshed and resolved. The ROM of platform, antenna, and equivalence surface are reusable in the online computation, which can reduce the computational complexity significantly. Numerical results demonstrate the accuracy and efficiency of the proposed method.

Electromagnetic Modeling of Moving Mixed Conductive and Dielectric BoRs With an Effective Domain Decomposition Method

We propose an effective domain decomposition method with a spherical equivalence surface for the electromagnetic modeling and imaging of moving bodies of revolution (BoRs) from near coupling to far coupling region. The moving BoRs are modeled with a series of stationary moments with respect to the time, each object is enclosed by a spherical equivalence surface. In the near coupling region, the couplings between multiple BoRs are evaluated with the couplings between spherical equivalence surfaces. While in the far coupling region, the total scattering fields are obtained by the superposition of the scattering fields from each spherical equivalence surface. For the moving objects with different relative positions at stationary moments, the equivalence processes are evaluated only once with combined field integral equation (CFIE) and Poggio–Miller–Chang–Harrington–Wu–Tsai (PMCHWT) equations for conductive and dielectric objects, respectively. The radar images are obtained from the scattering electromagnetic fields evaluated by the proposed domain decomposition method. Both computation accuracy and efficiency are enhanced significantly. Numerical results and discussions demonstrate the validity of the proposed method for modeling and imaging of mixed conductive and dielectric multiple moving objects.

An Improved Domain Decomposition Algorithm for Modeling Dense Distribution of Bodies of Revolution

An improved algorithm is presented to overcome the shortage of the domain decomposition algorithm based on equivalence principle for multiple bodies of revolutions (EPA-MBoR). In a dense distribution of non-coaxal bodies of revolution (BoRs) containing at least two very close BoRs, no matter how to establish sphere equivalence surfaces (ESs), they are intersecting. It violates the equivalence principle and causes error results. The improvement is introduced to overcome the defect. In the improved algorithm, a sphere ES cluster is built on each BoR instead of a single sphere ES. We can freely adjust the sphere ES clusters on each BoR to avoid the intersection among them. The numerical results verify the above improvement successfully overcomes the deficiency and show the improved algorithm inherits the high efficiency of the previous algorithm.

Application of Equivalence Principle for EM Scattering From Irregular Array of Arbitrarily Oriented PEC Scatterers Using Both Translation and Rotation Addition Theorems

Electromagnetic scattering from irregular array of arbitrarily oriented PEC scatterers is investigated using a novel domain decomposition method based on equivalence principle algorithm (EPA) in conjunction with addition theorems for spherical wave harmonics (SWHs). Each element is considered to be enclosed by a spherical equivalence surface as a single domain. The primary unknowns on each element are replaced by the equivalent electric and magnetic current distributions on the equivalence sphere in terms of RWG basis functions. From these current distributions, the scattered field outside its equivalence sphere is evaluated in terms of SWHs in the local coordinates system fixed to each element body. The SWHs in rotated local coordinates are translated as a new set of SWHs in other local coordinates with the same origin but axes parallel to the global coordinate’s axes using the rotation addition theorem. Furthermore, the mutual coupling effects among the separated domains are considered using the translation addition theorem for parallel local coordinate systems. It is shown that the proposed method effectively reduces the computational costs by reducing the number of unknowns versus the conventional EPA. The efficiency and validity of the proposed method are investigated through several numerical examples.

Implementation of the Equivalence Principle Algorithm for Potential Integral Equations

A novel implementation of the equivalence principle algorithm (EPA) employing potential integral equations (PIEs) is presented. EPA is generalized to be compatible with PIEs that are used to formulate inner problems inside equivalence surfaces. Based on the stability of PIEs, the resulting EPA-PIE implementation is suitable for low-frequency problems involving dense discretizations with respect to wavelength. Along with the formulation and a clear demonstration of the EPA-PIE mechanism, high accuracy, stability, and efficiency of the implementation are presented on canonical problems.

Reduced-Order Model With Equivalence Surface for Scattering Problems

A reduced-order model for analyzing scattering problems with surface integral equation method is proposed in this letter. An equivalent surface (ES) is introduced to enclose the scatterer and the corresponding interaction matrix between the ES and the scatterer is generated. A series of eigenvectors is obtained by performing randomized singular-value decomposition to the matrix. According to the equivalence principle, the wave input–output relation of the scatterer is encapsulated by a limited number of eigenvectors with a given truncation error, which can be precomputed and reused when the location of radiator changes but the scatterer does not. In addition, discrete empirical interpolation method is applied to further accelerate the calculation of the interaction between the radiator and the background. Numerical examples are provided to demonstrate the efficiency and accuracy of the proposed scheme.

Electromagnetic Multiple PEC Object Scattering Using Equivalence Principle and Addition Theorem for Spherical Wave Harmonics

A novel domain decomposition procedure is presented to analyze the electromagnetic wave multiple scattering among separate PEC objects. The method is based on the equivalence principle algorithm (EPA) that transfers the primary unknowns on the objects to the unknowns on the equivalence surfaces. Each equivalence surface is considered to be a spherical surface encompassing a PEC object. Inside the equivalence spheres, the analysis is performed based on the Rao–Wilton–Glisson (RWG) basis functions where the electromagnetic interactions outside them are considered by means of the addition theorem for spherical wave harmonics (SWHs). The translation procedures are presented to translate the RWG basis functions to SWHs and vice versa. The reduction of unknowns in the presented method versus the conventional EPA (which uses translation operators among equivalence surfaces) shows its efficiency. Some numerical examples are presented to evaluate the efficacy of the proposed approach. The acceptable consistency among the results of the proposed method and the conventional EPA as well as the direct method of moments validates the proposed method.

Improved Generalized Single-Source Tangential Equivalence Principle Algorithm with Contact-Region Modeling Method for Array Structures

An improved generalized single-source tangential equivalence principle algorithm (GSST-EPA) is proposed for analyzing array structures with connected elements. In order to use the advantages of GSST-EPA, the connected array elements are decomposed and computed by a contact-region modeling (CRM) method, which makes that each element has the same meshes. The unknowns of elements can be transferred onto the equivalence surfaces by GSST-EPA. The scattering matrix in GSST-EPA needs to be solved and stored only once due to the same meshes for each element. The shift invariant of translation matrices is also used to reduce the computation of near-field interaction. Furthermore, the multilevel fast multipole algorithm (MLFMA) is used to accelerate the matrix-vector multiplication in the GSST-EPA. Numerical results are shown to demonstrate the accuracy and efficiency of the proposed method.

Volume integral equation equivalence principle algorithm domain decomposition with body of revolution equivalence surface

A novel equivalence principle algorithm (EPA) domain decomposition method previously for the surface integral equation is explored for volume integral equation in this work. The whole object is decomposed into several subdomains, the equivalence spheres are automatically produced to enclose each subdomain. The Rao–Wilton–Glisson (RWG) and body of revolution (BoR) basis functions are defined on the equivalence sphere surface. The equivalence currents defined on the RWG basis functions are evaluated with multilevel fast multipole algorithm, and then the currents defined on the RWGs can be projected onto the BoR basis functions. Compared with surface integral equation EPA-BoR algorithm, the ratio of the number of average original unknowns in each subdomain with respect to the equivalence unknowns is relatively larger, which leads to significant reduction of computations. Numerical simulations validate the accuracy and efficiency of the proposed method.

Surface equivalence principle and Surface Integral Equation (SIE) revisited for bioelectromagnetics application

Surface equivalence principle and Surface Integral Equation (SIE) revisited for bioelectromagnetics application

Surface equivalence principle and Surface Integral Equation (SIE) revisited for bioelectromagnetics application

Surface equivalence principle and Surface Integral Equation (SIE) revisited for bioelectromagnetics application

Analysis of connected structures using equivalence principle algorithm with source reconstruction method

ABSTRACTA new domain decomposition scheme based on equivalence principle algorithm (EPA) is proposed for analyzing electromagnetic scattering from connected structures. The EPA transforms the unknown of inside scatterers to the unknown on the equivalence surfaces which enclose the scatterers. The challenge arises when the equivalence surface intercepts the connected scatterers, because the current will be singular at the connection that is hard to model. In this paper, a new scheme based on source reconstruction method (SRM) is introduced to deal with the connected problems. This scheme can avoid the current singularity and keep the good convergence of EPA equations. Furthermore, generalized EPA (GEPA) is proposed to improve the theory of EPA and the ability to solve real life problems. Numerical results demonstrate the accuracy and efficiency of the proposed method.

A novel relaxed hierarchical equivalent source algorithm (RHESA) for electromagnetic scattering analysis of dielectric objects

A novel relaxed hierarchical algorithm based on the surface equivalence principle for the volume integral equations (VIEs) is proposed in this work. The equivalent sources residing on relaxed spherical equivalence surfaces are established by exact integral formulations. The equivalence surfaces are exploited hierarchically to accelerate matrix–vector product in iterative solutions for the VIEs. The computation time and memory cost complexity of the proposed algorithm for dielectric scatters scale as and , respectively. Numerical examples validate the accuracy and capability of the proposed algorithm.

Equivalence Principle Algorithm With Body of Revolution Equivalence Surface for the Modeling of Large Multiscale Structures

We propose a multiscale solver of equivalence principle algorithm with a body of revolution (BoR) equivalence surface (ES). First, the whole object is decomposed into subdomains; the ESs are defined as proper BoRs (e.g., spheres in this work) to enclose each subdomain. The Rao–Wilton–Glisson (RWG) and BoR basis functions are defined on each sphere, respectively. Second, the octree is constructed in each nonzero subdomain; the multilevel fast multipole algorithm (MLFMA) is employed to solve the equivalent currents on the ES individually, and then the equivalent currents are projected from RWG onto the BoR basis functions. The couplings between neighboring subdomains are evaluated directly using MLFMA, and the separated subdomains are substituted by the ES-to-ES couplings, and are evaluated efficiently by the BoR method of moments (BoR–MoM). To solve the equation system, a hybrid inner and outer iterative solver is employed, where the inner iteration is used to solve each local subdomains and the outer iteration is used to update the global solutions by collecting all the local solutions. Numerical results and discussions demonstrate the validity of the proposed work.

Performance Enhancement of Equivalence Principle Algorithm

The conventional equivalence principle algorithm (EPA) exploits cubical surfaces to support equivalent sources. However, it introduces strong discontinuities due to sharp edges and corners. In this letter, a spherical surface is employed as the equivalence surface to avoid singularities from nonsmoothness. To achieve better accuracy, the meshless sampling scheme is used for integral operators on the spherical equivalence surface. The numerical examples are provided to validate the performance enhancement of EPA through the proposed method.

Error analysis of equivalence principle algorithm on different equivalence surfaces

Equivalence principle algorithm( EPA) is a domain decomposition algorithm based on the Huygens' equivalence principle to solve large scale scattering problems. The errors of the equivalence principle operator( EPO) in EPA using cubic,spherical and smooth cubic equivalence surfaces were analyzed to find the source of errors. The sources of the error were investigated and a simple scheme to improve its accuracy was developed. From the numerical examples,it shows that the error of equivalent magnetic current is mainly concentrated on geometrical discontinuity of the equivalence surface,and affected by both the discontinuity of the normal vector on the equivalence surface and the choice of basis functions. Therefore it is suggested to choose smooth equivalence surfaces if possible rather than to use cubic ones directly.