Volume-surface Integral Equation Research Articles

The extended resonant modal theory and its applications

AbstractThe authors extend the resonant modal theory (RMT) developed previously for a metal object to an arbitrary source region consisting of metals, dielectrics, or a combination of both. The influences of dielectrics on the fields are replaced by equivalent volume sources through the use of the compensation theorem in electromagnetic theory. The resonant frequencies can be determined by finding the roots of the determinant of the matrix resulted from the discretisation of the real homogeneous volume–surface integral equation derived from the requirement that the difference of stored field energies in the source region vanishes. As applications of the extended RMT, three examples have been investigated. The first example is a dielectric resonator antenna and is designed by exciting the first resonant mode of the composite structure in which the dielectric cylinder is combined with a conformal metallic strip. The second example is a dual‐band dielectric‐coated metallic wire antenna. The third example studies the resonant modes of a rectangular patch antenna.

An efficient iterative method for reconstructing the refractive index in complex domains from far field data

We present a computational method for reconstructing the refractive index of an unknown complex-shaped two-dimensional medium with embedded metal inclusions from its transverse magnetic electromagnetic scattering properties. We present a novel hybrid surface-volume integral equation that generalises the Lippmann–Schwinger equation to media containing embedded scatterers. Using this hybrid equation, we show that the Frechet derivative of the forward-mapping satisfies a particular inhomogeneous wave scattering problem. We solve the inhomogeneous wave scattering problem using a novel coupled FEM-BEM formulation. Our numerical scheme is based on a thin plate spline ansatz for the refractive index, which can be constructed using a relatively small number of control points, and can be efficiently constructed and evaluated even for the complex-shaped multiply-connected domains of interest. Numerical experiments demonstrate the effectiveness of our method by reconstructing several challenging media.

VoxImp: Impedance Extraction Simulator for Voxelized Structures

An impedance extractor, called VoxImp, is proposed to compute the impedances of the structures discretized by voxels. VoxImp iteratively solves the volume-surface integral equations discretized by a carefully selected set of basis functions. During iterative solution, matrix-vector multiplications are accelerated by the fast Fourier transform, while the rapid convergence of the iterative solution at resonant frequencies is ensured by a novel sparse preconditioner. The memory requirement of the sparse preconditioner is reduced by a sparse LU decomposition obtained by a multifrontal algorithm with compressed frontal matrices. The overall memory requirement of the VoxImp is further reduced by the Tucker decomposition. VoxImp’s accuracy, efficiency, and capability are demonstrated through impedance extraction of various voxelized structures, including an RF coil array discretized by more than four million voxels and fifteen million panels and analyzed on a commodity desktop computer.

Fast and Efficient Analysis of Receiving Antennas Enclosed by Variable‐Thickness Radome

AbstractIn this paper, the full‐wave volume‐surface integral equation (VSIE), the modified surface integration (MSI) method, and the ray‐tracing process are combined to efficiently analyze the electromagnetic performance of receiving antennas enclosed by radome with variable thickness. The mutual interactions between the antenna and the radome are considered by coupling the VSIE and MSI in a nested iterative manner, in which the VSIE is applied to model the complex antenna with composite conductor‐dielectric structure, while the MSI method is used to deal with the large‐size dielectric radome. To improve modeling accuracy of the MSI method for multilayered radome with variable thickness, the ray‐tracing technique is incorporated to calculate the angles of incidence and refraction at each interface and the local thickness of the dielectric wall for each sampling ray and then the local transmission matrix. Numerical results illustrate the accuracy and effectiveness of the proposed method.

A Hybrid Volume-Surface Integral Equation Method for Rapid Electromagnetic Simulations in MRI.

We developed a hybrid volume surface integral equation (VSIE) method based on domain decomposition to perform fast and accurate magnetic resonance imaging (MRI) simulations that include both remote and local conductive elements. We separated the conductive surfaces present in MRI setups into two domains and optimized electromagnetic (EM) modeling for each case. Specifically, interactions between the body and EM waves originating from local radiofrequency (RF) coils were modeled with the precorrected fast Fourier transform, whereas the interactions with remote conductive surfaces (RF shield, scanner bore) were modeled with a novel cross tensor train-based algorithm. We compared the hybrid-VSIE with other VSIE methods for realistic MRI simulation setups. The hybrid-VSIE was the only practical method for simulation using 1 mm voxel isotropic resolution (VIR). For 2 mm VIR, our method could be solved at least 23 times faster and required 760 times lower memory than traditional VSIE methods. The hybrid-VSIE demonstrated a marked improvement in terms of convergence times of the numerical EM simulation compared to traditional approaches in multiple realistic MRI scenarios. The efficiency of the novel hybrid-VSIE method could enable rapid simulations of complex and comprehensive MRI setups.

Improving the Solution Accuracy of the Volume-Surface Integral Equation for the Open PEC Structures Embedded Within Dielectrics

In the solution of volume-surface integral equation (VSIE) for open perfect electric conductor (PEC) structures embedded within dielectric substrates, how to explicitly enforce the continuity condition (CC) of electric flux on the PEC–dielectric interfaces to improve the numerical accuracy is investigated. In the traditional VSIE formed by combining the surface electric field integral equation (EFIE) and volume integral equation (VIE), the fictitious surface charge, which does not exist physically, is accumulated on the PEC–dielectric interfaces. This will lead to the inaccuracy of numerical results, which can be eliminated by the CC. To correctly enforce the CC, the equivalent surface current distribution on both sides of open PEC should be distinguished. For this purpose, the combined field integral equation (CFIE) consisting of electric field and magnetic field integral equations is used to model the open PEC part. In the proposed approach, the number of surface unknowns needs to be doubled, which can be balanced by reducing the volumetric unknowns through the enforcement of CC. Furthermore, by fully considering the numerical symmetry between both sides of open PEC surface, the filling process of the matrix as well as the excitation vector in implementing the method of moments (MoMs) can be simplified. Numerical experiment demonstrates that the enforcement of CC can improve the VSIE accuracy obviously. For example, when calculating the frequency response of frequency selective surfaces (FSSs), the results obtained from the traditional VSIE often shift to higher frequencies, while the proposed approach gives reasonably good results.

A Hybrid Method for Fast and Efficient Evaluation of Electromagnetic Performance of the Radome-Enclosed Antennas

In this article, a fast and efficient hybrid method that combines the volume-surface integral equation (VSIE) and the modified surface integration (MSI) method is proposed to evaluate the electromagnetic (EM) performance of the large-/medium-sized radome-enclosed antennas (REAs). An approximate equivalence principle is presented to formulate the physical model for the EM radiation of the REA, and then, the full-wave VSIE for the antennas and the high-frequency MSI method for the dielectric radome are coupled with each other in an iterative manner. Since mutual interactions up to the second order among the antennas and the radome are considered, the method shows good accuracy in both the near- and far-field regions of the REA system. In addition, all the computation processes involved in the method, including the method of moments (MoM) solution of the VSIE, surface integrations (SIs) in the MSI, and ray tracing for determining the local transmission matrix for the dielectric radome, are accelerated by the multilevel fast multipole algorithm (MLFMA), which makes the method highly efficient in terms of both computational time and memory requirement. Numerical results show that the proposed hybrid method can predict the EM performance, e.g., the input impedance and radiation pattern, of the REA system with good accuracy compared with the full-wave simulations.

Design of Irregularly Distributed Antenna Array towards Smart 6G Networks

The integration of sensing, computing, storage, and communications is one of the main research directions of current network development. To achieve the integration, an innovative design in the antenna array is essential. This paper presents a novel design of an irregularly distributed antenna array for smart 6G networks. Firstly, to solve the problem of fast amplitude and phase distribution of conformal arrays, the fast electromagnetic code of the multilevel fast multipole (MLFMA) based on volume surface integral equations (VSIE) is used to simulate the radiation characteristics of the irregularly distributed antenna arrays. Secondly, another stochastic global optimization algorithm, Simulated Annealing (SA), has been widely used to solve multiscale nonlinear problems. Finally, the performance of the proposed antenna array is given by simulation results and tests, to prove the effectiveness and correctness.

On the Enforcement of Electric Field Boundary Condition in the Moment Method Solution of Volume-Surface Integral Equation

When the volume-surface integral equation (VSIE) is used to model the composite objects, the electric field boundary condition (BC) can be explicitly enforced on the dielectric-perfect electric conductor (PEC) interfaces to reduce the volumetric unknowns. In the proposed BC-based VSIE, using the least-square method, one of the four coefficients of SWG basis functions belonging to a same tetrahedron terminated by a PEC triangle is directly calculated by the other three, which is no longer a volumetric unknown. The matrix equation generated by the method of moments is iteratively solved in a straightforward manner, which can be accelerated by the multilevel fast multipole algorithm. Through reliable numerical results, it is illustrated that the enforcement of the BC can reduce the memory usage and the number of iterations without compromising accuracy, in addition to good versatility.

Characteristic Mode Formulations of Composite PEC and Lossy Dielectric Objects

Lossy dielectric substrate affects the performance of a nearby perfect electric conductor (PEC) antenna. This letter presents a set of formulae for characteristic mode analysis (CMA) of composite PEC and lossy dielectric objects. It is based on the volume-surface integral equation and substructure treatment that separates the radiation and dissipation powers of the composite structure. A narrow PEC plate (that represents a dipole antenna) is placed in proximity of a thick, lossy dielectric material, and CMA of this structure is presented as an example. The separation gap between the PEC and dielectric objects affects the resonance and radiation efficiency of the antenna, and it is shown that CMA accurately captures those effects. The influence of dielectric loss on the modal orthogonality is discussed. Since the PEC item dominates over the coupling one in the substructure operator, the computed modes maintain good orthogonality with low-loss dielectrics. Owing to the physical insights of CMA, it has many potential applications associated with lossy materials.

Solution of Volume–Surface Integral Equation Accelerated by MLFMA and Skeletonization

A fast algorithm for solving large multi-scale scattering problems by the volume–surface integral equation (VSIE) is proposed. The sub-matrices of the near-field interactions (NFIs) in terms of the multi-level fast multipole algorithm (MLFMA) are compressed by the skeletonization after the theoretical and numerical investigations on their low-rankness are carried out. A block-diagonal preconditioner (BDP) based on the recursive skeletonization (RS) is developed to improve the conditioning of the impedance matrix of the VSIE. The rules to select hyperparameters in the proposed method are given. Numerical experiments conducted on scattering problems show the performance of the proposed algorithm.

Efficient EM Scattering Analysis of Uncertain Inhomogeneous Medium

An efficient solver based on volume-surface integral equation is proposed for the electromagnetic (EM) scattering of inhomogeneous medium targets with uncertain geometrical shapes and dielectric constants. On the one hand, the nonuniform ration B-spline surfaces can be used to constructed the uncertain geometrical shapes with a number of geometrical variables. On the other hand, the uncertain dielectric constant can also be derived as the permittivity variables. Then these random variables can be expanded by the first-order Taylor series, which represent the uncertain geometrical shapes and dielectric constant. At last, the mean and variance of the radar cross section (RCS) are acquired for the uncertain inhomogeneous medium targets. The numerical results can be compared with the traditional Monte Carlo (MC) method, which can indicate the accuracy and effectiveness of the method in this letter.

A Spurious-Free Characteristic Mode Formulation Based on Surface Integral Equation for Patch Antenna Structures

Conventional surface integral equation (SIE)-based characteristic mode formulation for the patch antenna structure with a finite substrate is susceptible to the spurious (nonphysical) modes due to the dielectric part. To avoid the contamination of spurious modes, we propose a novel generalized eigenvalue formulation based on the electric field integral equation coupled Poggio-Miller-Chang-Harrington-Wu-Tsai (EFIE-PMCHWT) equation. In this formulation, the real and imaginary parts of the exterior integral operators are chosen to construct the finalized weighting matrices, to connect radiated power of the characteristic current. Compared with other SIE-based methods, this equation doesn't need additional post-processing since it can effectively avoid spurious modes. Numerical results compared with the volume-surface integral equation (VSIE) are investigated to validate the accuracy.

Low-Frequency Analysis of Lossy Interconnect Structures Based on Two-Region Augmented Volume-Surface Integral Equations

Real conductors of interconnect structures are lossy and their skin depth becomes large at low frequencies. The traditional one-region formation with the approximation of perfectly electric conductors (PECs) or surface impedances may not be valid anymore, and two-region integral equation formations are needed in the integral equation approach. Also, the electric field integral equation (EFIE) tends to break down at low frequencies, and augmented electric field integral equation (AEFIE) has been proposed to remedy the problem. In this work, we treat lossy conductors as penetrable objects and propose two-region augmented hybrid field integral equations (AHFIEs) for low-frequency analysis. The HFIEs consist of the EFIE of describing the exterior of a conductor and the magnetic field integral equation (MFIE) of describing its interior. Since the magnetic current density appears in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathcal{ L}}$ </tex-math></inline-formula> operator in the HFIEs, we select the magnetic charge density as a new unknown function to be solved and introduce the continuity equation of magnetic current density as an extra equation. By incorporating the volume integral equations (VIEs) of describing the substrate with arbitrary penetrable media in the interconnect structures, the two-region augmented volume-surface integral equations (AVSIEs) are formulated for entire structures. The AVSIEs are solved by the method of moments (MoM) where the Rao–Wilton–Glisson (RWG) and Schaubert–Wilton–Glisson (SWG) basis functions are used to represent the surface current densities of AHFIEs and volume current densities of VIEs, respectively, while a pulse basis function is employed to represent the charge densities of AHFIEs. Numerical examples are presented to illustrate the approach and good results have been obtained.

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.

A VSIE based time domain discontinuous Galerkin method for composite conducting dispersive dielectric objects

To analyze the transient electromagnetic scattering of the composite conducting dispersive dielectric objects, a discontinuous Galerkin time-domain method based on volume surface integral equation (VSIE) is investigated. Since triangular basis functions are used as the temporal basis functions, the dispersion convolution relations between the electric field and the volume polarization current can be handled analytically. Then in the spatial domain, the current on the non-conformal surface of perfect electrically conducting (PEC)objects is expanded by half-RWG basis functions, while the volume current in the non-conformal dielectric objects is expanded by the half-SWG basis functions. Several numerical examples are demonstrated with the proposed method to validate its flexibility and efficiency.

Compression of volume-surface integral equation matrices via Tucker decomposition for magnetic resonance applications.

In this work, we propose a method for the compression of the coupling matrix in volume-surface integral equation (VSIE) formulations. VSIE methods are used for electromagnetic analysis in magnetic resonance imaging (MRI) applications, for which the coupling matrix models the interactions between the coil and the body. We showed that these effects can be represented as independent interactions between remote elements in 3D tensor formats, and subsequently decomposed with the Tucker model. Our method can work in tandem with the adaptive cross approximation technique to provide fast solutions of VSIE problems. We demonstrated that our compression approaches can enable the use of VSIE matrices of prohibitive memory requirements, by allowing the effective use of modern graphical processing units (GPUs) to accelerate the arising matrix-vector products. This is critical to enable numerical MRI simulations at clinical voxel resolutions in a feasible computation time. In this paper, we demonstrate that the VSIE matrix-vector products needed to calculate the electromagnetic field produced by an MRI coil inside a numerical body model with 1 mm3 voxel resolution, could be performed in ~ 33 seconds in a GPU, after compressing the associated coupling matrix from ~ 80 TB to ~ 43 MB.

Solving the Surface Current Distribution for Open PEC-Dielectric Objects Using the Volume Surface Integral Equation

For the open perfect electric conductor (PEC)-dielectric objects, a novel type of volume surface integral equation (VSIE) is proposed to solve the current distribution on both sides of the open PEC surfaces, which cannot be obtained by using the traditional VSIE. In the proposed method, both the electric and magnetic field integral equations are established on both sides of the PEC surfaces, whereas the basic functions are also defined at the borders of PEC surfaces to consider the current flowing over the edges of PEC surfaces. Compared with the traditional VSIE, due to the asynchronous solution processes of the surface current, the added CPU time for the proposed method is quite limited, whereas the memory usage is the same. Numerical results verify the validity and efficiency of the proposed method.

A Modified Hybrid Integral Equation to Electromagnetic Scattering from Composite PEC-Dielectric Objects Containing Closed-Open PEC Junctions

To efficiently analyze the electromagnetic scattering from composite perfect electric conductor (PEC)-dielectric objects with coexisting closed-open PEC junctions, a modified hybrid integral equation (HIE) is established as the surface integral equation (SIE) part of the volume surface integral equation (VSIE), which employs the combined field integral equation (CFIE) and the electric field integral equation (EFIE) on the closed and open PEC surfaces, respectively. Different from the traditional HIE modeled for the objects whose closed and open PEC surfaces are strictly separate, the modified HIE can be applied to the objects containing closed-open junctions. A matrix equation is obtained by using the Galerkin’s method of moments (MoM), which is augmented with the spherical harmonics expansion-based multilevel fast multipole algorithm (SE-MLFMA), improved by the mixed-potential representation and the triangle/tetrahedron-based grouping scheme. Because in the improved SE-MLFMA, the memory usage for storing the radiation patterns of basis functions is independent of the SIE type in the VSIE, it is highly appropriate for the fast solution of the VSIE that contains the HIE. Various numerical experiments demonstrate that during the calculation of composite objects containing closed-open PEC junctions, the application of the modified HIE in the VSIE can give reliable results with fast convergence speed.

A Tensor Train Compression Scheme for Remote Volume-surface Integral Equation Interactions

We propose a memory compression scheme for the coupling matrices appearing in volume-surface integral equation formulations. When there is some distance between the surface and the volumetric scatterers, the low-rank properties of the coupling matrix, allow us to reshape it into a set of four-dimensional tensors, which can be heavily compressed with the tensor train decomposition. The associated matrix-vector products can be rapidly performed with the aid of a graphical processing unit. We achieved a compression of more than 8 thousand times with a relative error around 1e – 5, for the calculation of the electromagnetic field generated by a radiofrequency coil inside an object at 1 mm voxel isotropic resolution. In this case, the matrix-vector product could be executed in less than a second.