Surface Integral Representation Research Articles

A Single-Layer Dual-Mesh Boundary Element Method for Multiscale Electromagnetic Modeling of Penetrable Objects in Layered Media

A surface integral representation of Maxwell's equations allows the efficient electromagnetic (EM) modeling of three-dimensional structures with a two-dimensional discretization, via the boundary element method (BEM). However, existing BEM formulations either lead to a poorly conditioned system matrix for multiscale problems, or are computationally expensive for objects embedded in layered substrates. This article presents a new BEM formulation which leverages the surface equivalence principle and Buffa-Christiansen basis functions defined on a dual mesh, to obtain a well-conditioned system matrix suitable for multiscale EM modeling. Unlike existing methods involving dual meshes, the proposed formulation avoids the double-layer potential operator for the surrounding medium, which may be a stratified substrate requiring the use of an advanced Green's function. This feature greatly alleviates the computational expense associated with the use of Buffa-Christiansen functions. Numerical examples drawn from several applications, including remote sensing, chip-level EM analysis, and metasurface modeling, demonstrate speed-ups ranging from 3x to 7x compared to state-of-the-art formulations.

Hybrid Time-Domain PTD and Physical Optics Contour Integral Representations for the Near-Field Backscattering Problem

Time-domain physical optics (TDPO) contour integral representations are developed for backscattered near fields from a perfectly conducting surface illuminated by a dipole source. The proposed representations are derived from the TDPO surface integral representations by introducing two theorems: surface gradient and surface divergence theorems. The proposed representations are free from singularities for all the source/observer positions in which the dipole antenna is located. In addition, the time-domain physical theory of diffraction is introduced to modify the physical optical field considering the effect of the edge diffraction. Numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method.

A fast high order algorithm for multiple scattering from large sound-hard three dimensional configurations

We describe an efficient fast high order algorithm for simulating acoustic scattering in an unbounded three dimensional medium exterior to a configuration with a large number of sound-hard particles. Surface integral representations of the scattered and far fields facilitate reformulation of the model for unknown fields on the closed bounded surface of each particle in the configuration. Our surface integral equation based algorithm uses a multiple scattering variant of complex dense system Krylov subspace techniques combined with an efficient matrix vector product scheme. In this article we develop a memory efficient fast algorithm in three steps using appropriate expansion methods (EM). The first step is based on a radiating basis EM (RadBEM) and the third step is based on a regular basis EM (RegBEM). The second step transforms RadBEM to RegBEM and is accelerated using the well known multilevel fast multipole method. The full algorithm has linear complexity with respect to the number of particles to evaluate quadratically large multiple scattering interactions exterior to the configuration. We present numerical results obtained with our fast algorithm for simulations with up to about twenty thousand sound-hard particles and for large dense complex systems with more than twelve million unknowns. We compare our results with a recent high-order accurate accelerated direct solver by Bremer et al. (2015) and demonstrate substantial computational advantages of our RadBEM–RegBEM algorithm.

Accurate computation of gravitational field of a tesseroid

We developed an accurate method to compute the gravitational field of a tesseroid. The method numerically integrates a surface integral representation of the gravitational potential of the tesseroid by conditionally splitting its line integration intervals and by using the double exponential quadrature rule. Then, it evaluates the gravitational acceleration vector and the gravity gradient tensor by numerically differentiating the numerically integrated potential. The numerical differentiation is conducted by appropriately switching the central and the single-sided second-order difference formulas with a suitable choice of the test argument displacement. If necessary, the new method is extended to the case of a general tesseroid with the variable density profile, the variable surface height functions, and/or the variable intervals in longitude or in latitude. The new method is capable of computing the gravitational field of the tesseroid independently on the location of the evaluation point, namely whether outside, near the surface of, on the surface of, or inside the tesseroid. The achievable precision is 14–15 digits for the potential, 9–11 digits for the acceleration vector, and 6–8 digits for the gradient tensor in the double precision environment. The correct digits are roughly doubled if employing the quadruple precision computation. The new method provides a reliable procedure to compute the topographic gravitational field, especially that near, on, and below the surface. Also, it could potentially serve as a sure reference to complement and elaborate the existing approaches using the Gauss–Legendre quadrature or other standard methods of numerical integration.

Joint Inversion of Gravity and Magnetotelluric Data for the Depth-to-Basement Estimation

It is well known that both gravity and magnetolluric (MT) methods can be used for the depth-to-basement estimation due to the density and conductivity contrast between the sedimentary basin and the underlaid basement rocks. In this case, the primary targets for both methods are the interface between the basement and sedimenary rocks as well as the physical properties of the rocks (density and conductivity). The solution of this inverse problem is typically nonunique and unstable, especially for gravity inversion. In order to overcome this difficulty and provide a more robust solution, we have developed a method of joint inversion to recover both the depth to the basement and the physical properties of the sediments and basement using gravity and MT data simultaneously. The joint inversion algorithm is based on the regularized conjugate gradient method. To speed up the inversion, we use an effective forward modeling method based on the surface Cauchy-type integrals for the gravity field and the surface integral equation representations for the MT field, respectively. We demonstrate the effectiveness of the developed method using several realistic model studies.

Two Theorems About Surface-Integral Representation of Electromagnetic Force and Torque

This paper has dealt with two theoretical problems of electromagnetic force/torque calculation in anisotropic regions. The first problem considers equivalence of both Maxwell and Lorentz methods for magnetically anisotropic and conductive region. The second problem deals with the independence of force/torque value calculated by the Maxwell method from the integration surface position. It formulated the sufficient condition for this independence for anisotropic and nonconductive region.

Radome Diagnostics—Source Reconstruction of Phase Objects With an Equivalent Currents Approach

Radome diagnostics are acquired in the design process, the delivery control, and in performance verification of repaired and newly developed radomes. A measured near or far field may indicate deviations, e.g., increased side-lobe levels, but the origins of the flaws are not revealed. In this paper, radome diagnostics is performed by visualizing the equivalent surface currents on the 3-D radome body, illuminated from the inside. Three different far-field measurement series at 10 GHz are employed. The measured far field is related to the equivalent surface currents on the radome surface by using a surface integral representation. In addition, a surface integral equation is employed to ensure that the sources are located inside the radome. Phase shifts, insertion phase delays (IPD), caused by patches of dielectric tape attached to the radome surface, are localized. Specifically, patches of various edge sizes (0.5-2.0 free-space wavelengths), and with the smallest thickness corresponding to a phase shift of a couple of degrees are imaged.

Anisotropy component of electromagnetic force and torque

The paper deals with the problem of surface-integral representation of electromagnetic force/torque for magnetically anisotropic region. It is pointed out that in some anisotropic regions a component of electromagnetic force/torque appears the so-called anisotropy component. The total electromagnetic field force/torque calculated with the help of Maxwell’s and Lorentz’s methods could lead to the different values for some anisotropic medium (homogeneous, without hysteresis). The coenergy method is used to evaluate total force/torque too. Analytical calculations of force/torque for isotropic and anisotropic media in electromagnetic field are presented. The condition for surface integral representation of Lorentz’s either force or torque is formulated.

AN INTEGRAL REPRESENTATION OF MULTIPLE HURWITZ-LERCH ZETA FUNCTIONS AND GENERALIZED MULTIPLE BERNOULLI NUMBERS

A surface integral representation of a multiple generalization of the Hurwitz–Lerch zeta function is given, which is a direct analogue of the well-known contour integral representation of the Riemann zeta function of Hankel's type. From this integral representation, we derive a detailed description of the set of its possible singularities. In addition, we present two formulae for special values of the zeta function at non-positive integers in terms of generalizations of Bernoulli numbers. These results are refinements of previously known ones.

Surface-integral formulation of scattering theory

We formulate scattering theory in the framework of a surface-integral approach utilizing analytically known asymptotic forms of the two-body and three-body scattering wavefunctions. This formulation is valid for both short-range and long-range Coulombic interactions. New general definitions for the potential scattering amplitude are presented. For the Coulombic potentials, the generalized amplitude gives the physical on-shell amplitude without recourse to a renormalization procedure. New post and prior forms for the Coulomb three-body breakup amplitude are derived. This resolves the problem of the inability of the conventional scattering theory to define the post form of the breakup amplitude for charged particles. The new definitions can be written as surface-integrals convenient for practical calculations. The surface-integral representations are extended to amplitudes of direct and rearrangement scattering processes taking place in an arbitrary three-body system. General definitions for the wave operators are given that unify the currently used channel-dependent definitions.

Tachyon optics: Kirchhoff identities and superluminal Bragg diffraction

The diffraction of superluminal radiation fields in crystal lattices is studied. The negative mass-square of the tachyonic wave modes affects the modulation function of diffraction gratings and the scattering amplitude. The Bragg condition for tachyon diffraction as well as the longitudinal and transversal cross sections are derived. Scalar and vectorial Kirchhoff identities for superluminal Proca fields are obtained from Sommerfeld’s dipole functionals, in analogy to electromagnetic theory. These surface-integral representations of the tachyon potential and the tachyonic field strengths are used to calculate the asymptotic diffracted modes and the intensity ratios. The dependence of the primary and secondary intensity peaks on the tachyon mass is analyzed in the reciprocal lattice, and the conversion of transversal into longitudinal radiation by way of Bragg scattering is explained. Specifically, tachyonic spectral fits are performed to the TeV spectra of three active galactic nuclei, H2356−309, 1ES 1218+304, and 1ES 1101−232, obtained with the imaging atmospheric Cherenkov detectors HESS, MAGIC, and VERITAS. The curvature in the spectral maps of these blazars is shown to be intrinsic, generated by ultra-relativistic electron populations in the galactic nuclei rather than by intergalactic absorption, and is reproduced by a tachyonic cascade fit.

An integral representation of the Mordell-Tornheim double zeta function and its values at non-positive integers

A surface integral representation of the Mordell-Tornheim double zeta function is given, which is a direct analogue of a well-known integral representation of the Riemann zeta function of Hankel’s type. As an application, we investigate its values and residues at integers, where generalizations of a generating function of Bernoulli numbers naturally appear.

Current and Charge Integral Equation Formulations and Picard's Extended Maxwell System

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Important connection between computational and mathematical electromagnetics is presented. The newly developed well-conditioned electromagnetic frequency domain surface integral equation formulations, the current and charge integral equations, are shown to be related to Picard's extended Maxwell system, an extended partial differential equation system that has the correct static behavior. Electromagnetic surface integral representations are derived in this paper for traditional surface integral equation formulations and for the Picard system using the fundamental solution approach, i.e., from the definition of Dirac's delta function. The surface integral representations are constructed with proper solid angle coefficients starting from the scalar Helmholtz equation. The traditional surface integral equation formulations are shown to be derived from Maxwell's curl equations and are thus lacking the contribution of the divergence equations at zero frequency. It is shown that the new current and charge formulations can be derived from the surface integral representation of the Picard system. </para>

A planar triangular monopole antenna for UWB communication

A planar triangular monopole antenna (PTMA) is presented for high-frequency structure simulator ultra-wideband (UWB) communication. The high-frequency structure simulator three-dimensional electromagnetic solver is employed for design simulation. A printed PTMA has been realized by using the FR-4 printed circuit board substrate. The measured voltage standing wave ratio is less than 3 from 4 to 10 GHz. In the UWB communication frequency range, the measured phase distribution of the input impedance is quite linear and the H-plane patterns are almost omni-directional. The Kirchhoff's surface integral representation was adopted in the developed finite-difference time-domain code to compute the far field distributions from the near filed ones in time-domain. This is to investigate the radiated power density spectrum (PDS) shaping to comply with FCC emission limit mask. The effect of various source pulses (first-order Rayleigh pulses with /spl sigma/ of 20, 30, and 50 ps) on the radiated PDS shaping is also studied.

RECONSTRUCTION OF EQUIVALENT CURRENTS USING A NEAR-FIELD DATA TRANSFORMATION - WITH RADOME APPLICATIONS

Knowledge of the current distribution on a radome can be used to improve radome design, detect manufacturing errors, and to verify numerical simulations. In this paper, the transformation from near-field data to its equivalent current distribution on a surface of arbitrary material, i. e., the radome, is analyzed. The transformation is based on the scalar surface integral representation that relates the equivalent currents to the near-field data. The presence of axial symmetry enables usage of the fast Fourier transform (FFT) to reduce the computational complexity. Furthermore, the problem is regularized using the singular value decomposition ( SVD). Both synthetic and measured data are used to verify the method. The quantity of data is large since the height of the radome corresponds to 29 - 43wavelengths in the frequency interval 8.0 - 12.0 GHz. It is shown that the method gives an accurate description of the field radiated from an antenna, on a surface enclosing it. Moreover, disturbances introduced by copper plates attached to the radome surface, not localized in the measured near. eld, are focused and detectable in the equivalent currents.

On the retarded potentials of inhomogeneous ellipsoids and sources of arbitrary shapes in the three-dimensional infinite space

We analyze the infinite space solutions of the three-dimensional inhomogeneous wave equation (the ‘retarded potentials’ or ‘causal propagators’) for ellipsoidal sources and for sources of arbitrary shapes. The ‘short-time characteristics’ of the retarded potential for a spatially inhomogeneous source density of δ-shaped time profile is considered. It is found that, the short-time characteristics is governed by the spatial inhomogeneity of the source density in the immediate vicinity of a spacepoint. Surface integral representations are derived for spatial inhomogeneous source regions of ellipsoidal symmetry. For spherical sources these integral representations yield closed form solutions for the retarded potentials. We find that the wave field inside a spherical source consists of an incoming and outgoing spherical wave package, whereas the external wave field consists of an outgoing spherical wave package only. Characteristic runtime and superposition effects are discussed. Moreover, a numerical technique based on Gauss quadrature is applied to generate the wave field for a cubic source. The integral representations derived for the retarded potentials of inhomogeneous ellipsoidal sources are consistent with results previously derived by the authors for the Helmholtz potentials of homogeneous ellipsoids and ellipsoidal shells [Michelitsch, T.M., Gao, H., Levin, V.M., 2003. On the dynamic potentials of ellipsoidal shells. Q. J. Mech. Appl. Math. 56 (4), 629]. The derived solutions are crucial for many problems of wave propagation and diffraction theory as they may occur in materials science. As an example we give a formulation for the solution of the retarded Eshelby inclusion problem due to spatially and temporally varying eigenfields in the elastic isotropic infinite medium.

Ultrasonic wave propagation in an anisotropic cladding with a wavy interface

The propagation of ultrasonic waves in a thick plate with a cladding is investigated in the two-dimensional case. The surfaces of the plate are traction-free except where an ultrasonic probe is attached and emits waves into the plate. The plate is made of two different materials, the base material and the cladding, and these are both allowed to be anisotropic. The interface between the base material and the cladding is assumed to be wavy (sinusoidal) and this is common in practice for welded claddings. The null field approach is used to solve the wave propagation problem. Thus the starting point is the (surface) integral representations in the two materials. The Green’s tensors are chosen as the half space Green’s tensors as only the integrals along the interface then remain. The Green’s functions are expanded in Fourier transforms along the interface and the surface fields are likewise expanded. Applying the periodicity of the interface a discretized set of equations remains. For the sinusoidal interface all integrals can be computed analytically which leads to an efficient numerical scheme. Some numerical results show the influence of the anisotropy and the wavy interface.

Tectonomagnetic modeling based on the piezomagnetism: a review

Development of tectonomagnetic modeling on the basis of the piezomagnetic effect is reviewed for the period since the early 1990's. First, the basic theory is briefly summarized, in which the representation theorem or the surface integral representation for the piezomagnetic potential and the Green's function method are presented. In the 1990's, several field observations in earthquakes and volcanoes were interpreted with the aid of analytic solutions based on the Green's function method. A general formula was developed for an inclined rectangular fault with strike-slip, dip-slip and tensile faulting. The surface integral method has been applied to 2D and 3D models, as well as to fault models in the inhomogeneously magnetized crust. When the magnetic field is measured within a bore hole, the effect of magnetic poles around the hole should be taken into account. As a result, tectonomagnetic signals are much enhanced in a bore hole compared with on the ground surface. Finally, piezomagnetic field changes associated with the Parkfield fault model are introduced and the new aspect of the model is discussed.

Surface integral method to determine guided modes in uniaxially anisotropic embedded waveguides

A surface integral method is presented to calculate the eigenmodes of an uniaxially anisotropic embedded channel waveguide. The electromagnetic field components are expressed with electric and magnetic vector potentials which are parallel with the optic axis and for which the scalar Helmholtz-equations hold using surface integral representation and the Green's functions of the vector potentials. The single component vector potentials are expanded in Fourier-series on the internal side of the dielectric interface. The Fourier-coefficients and the corresponding eigenvalues are obtained by minimizing the quadratic difference of the longitudinal field components in the cladding and core along the dielectric interface. The convergence of the series expansion is examined numerically and the eigenvalues obtained with the surface integral method agree with those obtained by Finite Element Method up to 5 significant digits.

Time-domain Green dyadics for temporally dispersive, simple media

Time-domain fundamental solutions and Green dyadics for temporally dispersive, simple media are introduced. Second forerunner approximations to the electromagnetic fields from an electric point dipole in unbounded dispersive materials are obtained. Numerical results for two frequently used material models are presented. Moreover, surface integral representations of the electromagnetic fields in dispersive media are obtained, and, finally, surface integral equations are derived for impenetrable and permeable scatterers.