Hybrid Integral Equation Research Articles

Algorithms of the hybrid projection method and integral equation method for analysis of wave focusing by Mikaelian lens in 2D case

A two-dimensional problem of plane electromagnetic wave scattering on dielectric and magneto-dielectric Mikaelian lenses is considered. A new algorithm based on the hybrid projection method is developed for solution of the problem with an option of accounting for not only variable profile of the lens dielectric permittivity as it has already been realized earlier but also variable profile of the lens magnetic permeability. There is also developed a new algorithm of the integral equation method in the case of H-polarized wave scattering on a dielectric Mikaelian lens allowing reduction of the problem to a system of linear algebraic equations of smaller order in comparison with the known cases by means of involving only one component of the magnetic field strength vector. Some numerical results characterizing both effectiveness of the developed algorithms and comparative features of wave focusing by a purely dielectric lens and magneto-dielectric lens with identical profiles of permittivity and permeability are presented and discussed.

Inversion for 3D Conductivity and Chargeability Models Using EM Data Acquired by the New Airborne TargetEM System in Ontario, Canada

This paper introduces an original approach to the joint inversion of airborne electromagnetic (EM) data for three-dimensional (3D) conductivity and chargeability models using hybrid finite difference (FD) and integral equation (IE) methods. The inversion produces a 3D model of physical parameters, which includes conductivity, chargeability, time constant, and relaxation coefficients. We present the underlying principles of this approach and an example of a high-resolution inversion of the data acquired by a new active time domain airborne EM system, TargetEM, in Ontario, Canada. The new TargetEM system collects high-quality multicomponent data with low noise, high power, and a small transmitter–receiver offset. This airborne system and the developed advanced inversion methodology represent a new effective method for mineral resource exploration.

A Burton–Miller-type boundary element method based on a hybrid integral representation and its application to cavity scattering

This study builds on a recent paper by Lai et al. (2018) in which a novel boundary integral formulation is presented for scalar wave scattering analysis in two-dimensional layered and half-spaces. The seminal paper proposes a hybrid integral representation that combines the Sommerfeld integral and layer potential to efficiently deal with the boundaries of infinite length. In this work, we modify the integral formulation to eliminate the fictitious eigenvalues by employing the integral equation of the Burton–Miller type. We also discuss reasonable parameter settings for the hybrid integral equation to ensure efficient and accurate numerical solutions. Furthermore, we extend the modified formulation for the scattering from a cavity in a half-space whose boundary is locally perturbed. To address the cavity scattering, we introduce a virtual boundary enclosing the cavity and couple the integral equation on it with the hybrid equation. The effectiveness of the proposed method is demonstrated through numerical examples.

Electromagnetic scattering analysis of large-area vegetation based on multilevel periodic fast multipole algorithm

The electromagnetic scattering of ground objects always has important application value in remote sensing, detection, anti-stealth and other fields. Among them, vegetation has received extensive attention due to its wide variety and uneven distribution. With the expansion of the scene, the theoretical model of vegetation electromagnetic scattering becomes more and more complex, the calculation amount increases sharply, and the solution accuracy decreases. It is urgent to establish a full-wave simulation method that can accurately and fast analyze the electromagnetic scattering of vegetation. To solve this problem, this paper establishes a hybrid integral equation for solving the objects with thin dielectric sheets (TDS) and dielectric bodies, and calculates the electromagnetic scattering of large-area vegetation by simulating an infinite periodic scene. In addition, the periodic Green's function (PGF) is expanded by the multilevel periodic fast multipole algorithm (MLP-FMA), which further improves the computational efficiency. The method proposed in this paper provides model and theoretical support for the electromagnetic scattering simulation of actual large-scale vegetation scenes.

A fast solver framework for acoustic hybrid integral equations

Reliable modeling of the scattering by acoustically large and geometrically complex objects can be achieved by means of subdomain-dependent problem formulation and a numerically rigorous solution. While the objects’ inhomogeneity has driven the development of differential equation formulations and solvers, integral equation formulations, where the object’s background is modeled via a Green’s function, are advantageous for unbounded domains. In the hybrid integral equations approach (Usner et al., 2006), the interaction of separate subdomains with external fields is described by pertinent integral equations. Their Galerkin discretization leads to a dense blocked stiffness matrix. The development of compressed representations of the matrix, which are necessary for the treatment of large systems, becomes non-trivial due to the multitude of integral equation kernels and the different geometrical and physical characteristic of the subdomains. As part of the development of a fast hybrid integral equation solver framework, we consider the case of objects composed of large inhomogeneous volumes, modeled as incompressible fluids, and of simplified solids, modeled via surface integral equations. A hybrid integral equation formulation is derived and solved numerically. The iterative solution is accelerated by employing the butterfly-compressed hierarchical representation of the stiffness matrix, recently used for acoustic volume integral equations (Kaplan, 2022).

Prediction approach for wave forces on large-scale offshore structures under linear wave action

This paper presents a numerical mothed that is implemented by applying higher-order boundary elements in the hybrid integral equation method for predicting the wave forces on arbitrarily-shaped offshore structures. In this study, the virtual cylindrical boundary (i.e., matching boundary) is set to divide the fluid region into inner and outer subdomains. The velocity potentials in the outer domain are expressed using the eigenfunction expansion method. In the inner domain, the simple Green's function is adopted as the fundamental solution and Green's second theorem is applied to derive the boundary integral equation (BIE). The velocity potentials in the two subdomains are matched on the matching boundary. The higher-order boundary elements are ultimately employed to discretize BIE and the velocity potentials in the inner domain are then solved. The wave forces on the truncated and stepped cylinders are calculated using the presented method, and the results match well with analytical results, which illustrates the validity of the presented method. The effects of element sizes and modal truncation numbers are further explored, and the interaction between waves and square columns is subsequently analyzed. Eventually, the proposed method is extended to the practical engineering application of wave diffractions from quasi-elliptic pile caps for sea-crossing bridges. The results show that the proposed method can deal with boundaries with complex geometries. In addition, satisfactory results can be obtained by setting relatively few elements and control points on boundaries, and this method can also effectively predict wave forces on the multi-column array.

Fixed point theorems for convex-power condensing operators in Banach algebra

We introduce the concept of a convex-power condensing mapping A⋅B+C in a Banach algebra relative to a measure of noncompactness as a generalization of condensing and convex-power condensing mappings. We present new fixed point theorems, and we apply these results to investigate the existence of solutions for a nonlinear hybrid integral equation of Volterra type.

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 Hybrid Volume-Surface-Wire Integral Equation for the Anisotropic Forward Problem in Electroencephalography

Solving the electroencephalography (EEG) forward problem is a fundamental step in a wide range of applications including biomedical imaging techniques based on inverse source localization. State-of-the-art electromagnetic solvers resort to a computationally expensive volumetric discretization of the full head to account for its complex and heterogeneous electric profile. The more efficient, popular in biomedical imaging circles, but unfortunately oversimplifying Boundary Element Method (BEM) relies instead on a piecewise-uniform approximation that severely curbs its application in high resolution EEGs. This contribution lifts the standard BEM contraints by treating the local anisotropies with adequate wire and thin volume integral equations that are tailored to specific structures of the fibrous white matter and the inhomogeneous skull. The proposed hybrid integral equation formulation thereby avoids the full volumetric discretization of the head medium and allows for a realistic and efficient BEM-like solution of the anisotropic EEG forward problem. The accuracy and flexibility of the proposed formulation is demonstrated through numerical experiments involving both canonical and realistic MRI-based head models.

High-order, Dispersionless “Fast-Hybrid” Wave Equation Solver. Part I: O(1) Sampling Cost via Incident-Field Windowing and Recentering

This paper proposes a frequency/time hybrid integral-equation method for the time dependent wave equation in two and three-dimensional spatial domains. Relying on Fourier Transformation in time, the method utilizes a fixed (time-independent) number of frequency-domain integral-equation solutions to evaluate, with superalgebraically-small errors, time domain solutions for arbitrarily long times. The approach relies on two main elements, namely, 1) A smooth time-windowing methodology that enables accurate band-limited representations for arbitrarily-long time signals, and 2) A novel Fourier transform approach which, in a time-parallel manner and without causing spurious periodicity effects, delivers numerically dispersionless spectrally-accurate solutions. A similar hybrid technique can be obtained on the basis of Laplace transforms instead of Fourier transforms, but we do not consider the Laplace-based method in the present contribution. The algorithm can handle dispersive media, it can tackle complex physical structures, it enables parallelization in time in a straightforward manner, and it allows for time leaping---that is, solution sampling at any given time $T$ at $\mathcal{O}(1)$-bounded sampling cost, for arbitrarily large values of $T$, and without requirement of evaluation of the solution at intermediate times. The proposed frequency-time hybridization strategy, which generalizes to any linear partial differential equation in the time domain for which frequency-domain solutions can be obtained (including e.g. the time-domain Maxwell equations), and which is applicable in a wide range of scientific and engineering contexts, provides significant advantages over other available alternatives such as volumetric discretization, time-domain integral equations, and convolution-quadrature approaches.

An efficient hybrid integral-equation method for point-absorber wave energy converters with a vertical axis of symmetry

To assist in the prototyping and controller design of point-absorber wave energy converters (WECs), an easy-to-implement hybrid integral-equation method is presented for computing the frequency-domain hydrodynamic properties of bodies with a vertical axis of symmetry in waves. The current hybrid method decomposes the flow domain into two parts: an inner domain containing the body and an outer domain extending to infinity. The solution in the inner domain is computed using the boundary-element method, and the outer-domain solution is expressed using eigenfunctions. Proper matching at the domain boundary is achieved by enforcing continuity of velocity potential and its normal derivative. Body symmetry allows efficient computation using ring sources in the inner domain. The current method is successfully applied to three different body geometries including a vertical truncated floating cylinder, the McIver toroid, and the coaxial-cylinder WEC being developed in the authors’ laboratory. In particular, the current results indicate that, by replacing the flat bottom of the coaxial-cylinder WEC with the Berkeley-Wedge (BW) shape, viscous effect can be significantly reduced with only minor negative impact on wave-exciting force, thus increasing WEC efficiency. Finally, by comparing to experimental measurements, the current method is demonstrated to accurately predict the heave added mass and wave-exciting force on the coaxial-cylinder WEC with BW geometry. If a viscous damping correction factor is used, the heave motion amplitude can also be accurately computed.

The electromagnetic scattering problem by a cylindrical doubly connected domain at oblique incidence: the direct problem

We consider the direct electromagnetic scattering problem of time-harmonic obliquely incident waves by a infinitely long, homogeneous and doubly-connected cylinder in three dimensions. We apply a hybrid integral equation method (combination of the direct and indirect methods) and we transform the scattering problem to a system of singular and hypersingular integral equations. The well-posedness of the corresponding problem is proven. We use trigonometric polynomial approximations and we solve the system of the discretized integral operators by a collocation method.

Radiation and Exciting Forces of Axisymmetric Structures with a Moonpool in Waves

A highly efficient “hybrid integral-equation method” for computing hydrodynamic added-mass, wave-damping, and wave-exciting force of general body geometries with a vertical axis of symmetry is presented. The hybrid method utilizes a numerical inner domain and a semi-infinite analytical outer domain separated by a vertical cylindrical matching boundary. Eigenfunction representation of velocity potential is used in the outer domain; the three-dimensional potential in the inner domain is solved using a “two-dimensional” boundary element method with ring sources and ring dipoles to exploit the body symmetry for efficiency. With proper solution matching at the common boundary, both radiation and diffraction potentials can be solved efficiently while satisfying the far-field radiation condition exactly. This method is applied to compute the hydrodynamic properties of two different body geometries: a vertical-walled moonpool with a bottom plate that restricts the opening and a spar-like structure with a diverging bottom opening inspired by designs of floating Oscillating Water Columns. The effects of the size of the bottom opening on the hydrodynamic properties of the body are investigated for both geometries. The heave motion of the floater as well as the motion of the internal free surface under incident wave excitation are computed and studied for the spar-like structure.

Entropic depletion in colloidal suspensions and polymer liquids: role of nanoparticle surface topography.

We employ a hybrid Monte Carlo plus integral equation theory approach to study how dense fluids of small nanoparticles or polymer chains mediate entropic depletion interactions between topographically rough particles where all interaction potentials are hard core repulsion. The corrugated particle surfaces are composed of densely packed beads which present variable degrees of controlled topographic roughness and free volume associated with their geometric crevices. This pure entropy problem is characterized by competing ideal translational and (favorable and unfavorable) excess entropic contributions. Surface roughness generically reduces particle depletion aggregation relative to the smooth hard sphere case. However, the competition between ideal and excess packing entropy effects in the bulk, near the particle surface and in the crevices, results in a non-monotonic variation of the particle-monomer packing correlation function as a function of the two dimensionless length scale ratios that quantify the effective surface roughness. As a result, the inter-particle potential of mean force (PMF), second virial coefficient, and spinodal miscibility volume fraction vary non-monotonically with the surface bead to monomer diameter and particle core to surface bead diameter ratios. A miscibility window is predicted corresponding to an optimum degree of surface roughness that completely destroys depletion attraction resulting in a repulsive PMF. Variation of the (dense) matrix packing fraction can enhance or suppress particle miscibility depending upon the amount of surface roughness. Connecting the monomers into polymer chains destabilizes the system via enhanced contact depletion attraction, but the non-monotonic variations with surface roughness metrics persist.

Bistatic electromagnetic scattering from a three-dimensional perfect electric conducting object above a Gaussian rough surface based on the Kirchhoff–Helmholtz and electric field integral equation

A hybrid integral equation is developed to solve the problem of electromagnetic (EM) scattering from a three-dimensional (3D) perfect electric conducting (PEC) object above a two-dimensional (2D) PEC or dielectric Gaussian rough surface. Firstly, the Kirchhoff–Helmholtz (KH) equation is adopted to describe the wave reflection on the rough surface; only one integral operation on the rough surface is needed, and the scattering from the object can be described by solving the electric field integral equation (EFIE) on the surface of the object. Moreover, according to scattering theory, the KH equation and the EFIE are coupled together (KH-EFIE) to describe wave propagation between the object and the rough surface. Then method of moments (MoM) is adopted to solve the KH-EFIE, and the current is obtained to calculate the scattering field. Finally, compared with other methods, the accuracy of the proposed approach is validated, and its efficiency is proved to be much higher than numerical solutions. Furthermore, by calculating the statistic composite radar cross-section (RCS) of the object/surface and the difference radar cross-section (DRCS) of the object, the influence of the rough surface root mean square (rms) height, the correlation length, the medium permittivity, the shape of the object, and the altitude of the object on the scattering characteristic is investigated.

Effects of the ionic size-asymmetry around a charged nanoparticle: unequal charge neutralization and electrostatic screening

We study the consistent inclusion of ionic size-asymmetry for a wide range of macroparticle charges in the primitive model of an electrical double layer around a spherical colloid using (1) Monte-Carlo simulations, (2) the hybrid integral-equation formalism of hypernetted-chain (HNC) and mean-spherical approximation (MSA), and (3) the Gouy–Chapman theory modified for unequal ionic radii. In our simulations, for a weakly charged macroion, we observe surface charge amplification from adsorption of like-charged ions, as well as charge reversal due to overcompensation of the bare nanoparticle charge by counterions. When the nanoparticle charge increases, we detect both asymmetric neutralization and asymmetric electrostatic screening that depend on the sign of the macroion's valence. Specifically, there exists a higher reduction of the original bare charge and a smaller electrostatic potential for the case of negative nanoparticles with positive small counterions, versus the case of positive nanoparticles with negative large counterions. These coarse-grained results are in agreement with the predictions of asymmetric charge renormalization (P. Gonzalez-Mozuelos and M. Olvera de la Cruz, Phys. Rev. E, 2009, 79, 031901), in which the aqueous solvent is explicitly taken into account. Results from the Gouy–Chapman theory modified for unequal ionic radii differ notably from our obtained Monte-Carlo data, while good agreement exists between simulation results and HNC/MSA-treatment findings.

Emission and Susceptibility Modeling of Finite-Size Power-Ground Planes Using a Hybrid Integral Equation Method

A novel method based on hybrid integral equation for electromagnetic emission and susceptibility modeling of power-ground planes with a finite size is proposed in this paper. First, the entire computational domain is divided into the internal subdomain including the substrate sandwiched between the power and ground planes and the external subdomain surrounding the power-ground planes. The internal subdomain is modeled by using the integral equation with dyadic Green's functions of the rectangular cavity, while the free-space Green's function is used to model the outside subdomain. These two kinds of integral equations are coupled through the equivalent electric and magnetic currents that are placed on the interfaces. The key advantage of this proposed method is its ability to model both emission and susceptibility problems. This method can accurately simulate the impedance of the power-ground planes and the radiated field. In particular, it is able to predict the induced electric currents inside the power-ground planes that result from external interference sources. Therefore, this technique is able to provide a complete solution for the electromagnetic compatibility analysis of the system-level package.

공기챔버형 부소파제의 투과 및 산란파 해석

The transmission and reflection coefficiencies of a BBDB-type floating breakwater in water of finite depth are studied taking account of fluctuating air pressure in the air chamber. The wave potential is calculated by a hybrid integral equation consisting of a Green integral equations associated with the Rankiue Green function inside the BBDB and the Kelvin Green function outside. The transmission and reflection coefficients of the breakwater are obtained directly from the potential solution in the outer region.

Scattering and Radiation Modeling Using Hybrid Integral Equation Approach and Mixed Mesh Element Discretization

A mixed mesh algorithm is developed for the integral equation modeling of scattering by three dimensional complex radar targets that consist of conducting surfaces and radar absorbing materials. The integral equation combines the surface integral equation (for conducting surfaces) and the volume integral equation (for dielectric material regions). The mixed mesh element scheme is proposed to model the complex scatterers. In this scheme, the surface is modeled by a set of triangles and quadrangles, and the dielectric volume is modeled by a set of tetrahedrons, prisms of triangle base, and prisms of quadrangle base (the hexahedron). The simulation shows that as long as the average edge length remains the same, the results from single element mesh and combined element mesh are of the same order, but the later uses less number of unknowns, less computing resources, and sometimes less iteration numbers.

A Study of the Condition Number of Various Finite Element Matrices Involved in the Numerical Solution of Maxwell's Equations

We consider the solution of the time-harmonic Maxwell's equations inside a bounded domain on the boundary of which various conditions are prescribed, including a perfectly matched layer (PML) termination. This problem arises when, e.g., the electromagnetic fields scattered from an inhomogeneous penetrable structure are computed by using a hybrid finite element (FE) and integral equation method in conjunction with a domain decomposition technique. In each of the subdomains, the discretization process leads to a linear system, and an iterative solver may be advantageously utilized when the number of unknowns is large. In this case, the number of iterations and, hence, the computational time required to achieve a given numerical accuracy are known to increase with the condition number /spl kappa/ of the FE matrix. In this paper, we attempt to draw the rules that govern the behavior of /spl kappa/. To this effect, an eigenmodes technique is proposed that allows to dissociate the influence of the FE mesh and FE basis functions from the one of the actual physical cavity. Numerical examples are provided for one- and three-dimensional problems that illustrate the results so obtained.