Field Integral Equation Research Articles

Effective medium method in solution of the elastic and thermo-elastic homogenization problems for polycrystals with spheroidal transverse isotropic grains

Self-consistent effective medium method is used for determination of the effective elastic and thermo-elastic properties of polycrystalline materials with spheroidal transverse isotropic grains. The grains with arbitrary aspect ratios including very oblate (disks) and very prolate (fibers) spheroids are considered. The function of the Owens-March type is used for description of distribution of grains over orientations. The equations for the effective elastic stiffness tensor and the tensor of effective coefficients of thermo-expansion of polycrystals are derived using integral equations for elastic and thermo-elastic fields in heterogeneous media and the hypotheses of the effective medium method. The numerical algorithm of the method is described in detail. Numerical examples are presented for polycrystalline clay rock materials with strongly anisotropic grains and for polycrystalline zinc. Graphs of the dependences of the effective elastic constants and effective coefficients of thermo-expansion of the polycrystals on the parameters of the orientation distribution function and aspect ratios of grains are presented.

English

Optical diffraction on a periodical interface belongs to relatively lowly exploited applications of the boundary integral equations method. This contribution presents a less frequent approach to the diffraction problem based on vector tangential fields of electromagnetic intensities. The problem is formulated as the system of boundary integral equations for tangential fields, for which existence and uniqueness of weak solution is proved. The properties of introduced boundary operators with singular kernel are discussed with regard to performed numerical implementation. Presented theoretical model is of advantage when the electromagnetic field near the material interface is studied, that is illustrated by several application outputs.

Surface Integral Equation Method for Generalized Soft-and-Hard Boundary Condition

A surface integral equation (SIE) method is developed for analyzing electromagnetic scattering by objects with generalized soft-and-hard (GSH) boundary conditions. GSH boundary condition is an anisotropic impedance boundary condition, which limits the tangential electric and magnetic fields in specified directions. In the developed SIE formulation, the GSH boundary condition is expressed in a vector form with two sets of orthogonal vectors, which can be combined with the field integral equations. The resulting equation can be discretized with the standard method of moments (MoM) using triangular elements and the Rao-Wilton-Glisson (RWG) functions. By varying the GSH condition, the direction of the tangential electric and magnetic fields on the surface can be flexibly changed.

Multiple-Traces Surface Integral Equations for Electromagnetic Scattering From Complex Microstrip Structures

In this communication, a new multiple-traces Poggio–Miller–Chang–Harrington–Wu–Tsai (MT-PMCHWT) equation is proposed to simulate the electromagnetic scattering from microstrip structures. Based on the surface equivalent principle, the original object can be decomposed into two independent domains, i.e., the exterior region (free space) and interior region (microstrip). Different from the traditional EFIE-PMCHWT method, in this new multiple-traces method, dielectric surface is discretized with two pairs of meshes, and Robin transmission conditions (TCs) are enforced on the interface between the exterior and interior regions to ensure the continuity of fields. In comparison with the traditional EFIE-PMCHWT, this new MT-PMCHWT has a better iteration convergence property. Additionally, a new multiple-traces electric field/combined field integral equation with better iteration convergence is also proposed. Because the continuity of fields is ensured by TCs, the exterior and interior regions can be discretized with nonconformal meshes, which improves the flexibility and efficiency of the proposed methods substantially.

On the contributions of W. Edward Olmstead

With this article, we wish to honor the many contributions of our mentor, colleague and dear friend, Professor W. Edward Olmstead, on the occasion of his retirement from Northwestern University. Ed has spent over five decades at Northwestern University, first as a graduate student and then as a member of the faculty. During this time he completed his PhD, played a key role in the formation of the Department of Engineering Science and Applied Mathematics (ESAM), developed several courses in applied mathematics, participated in the education of numerous students, and made vast and important contributions in the field of applied mathematics. Here, we give a (mostly chronological) account of some of Ed's major research interests and contributions, primarily in the field of Integral Equations.

Simulation of Circular Cylindrical Metasurfaces Using GSTC-MoM

Modeling of circular cylindrical metasurfaces using the method of moments (MoM) based on generalized sheet transition conditions (GSTCs) is presented. GSTCs are used to link the integral equations for fields on the inner and outer contour of the circular cylindrical metasurface. The GSTC-MoM is validated by two examples: 1) anisotropic and gyrotropic metasurface capable of two field transformations; and 2) nongyrotropic metasurface capable of transforming the field generated by an infinite electric line source at origin to the field generated by a displaced electric line source. The formulations presented here can be used as a platform for deriving GSTC-MoM for 3-D spherical and conformal metasurfaces.

Scattering of electromagnetic waves from 3D multilayer random rough surfaces based on the second-order small perturbation method: energy conservation, reflectivity, and emissivity.

A theoretical investigation of energy conservation, reflectivity, and emissivity in the scattering of electromagnetic waves from 3D multilayer media with random rough interfaces using the second-order small perturbation method (SPM2) is presented. The approach is based on the extinction theorem and develops integral equations for surface fields in the spectral domain. Using the SPM2, we calculate the scattered and transmitted coherent fields and incoherent fields. Reflected and transmitted powers are then found in the form of 2D integrations over wavenumber in the spectral domain. In the integrand, there is a summation over the spectral densities of each of the rough interfaces with each weighted by a corresponding kernel function. We show in this paper that there exists a "strong" condition of energy conservation in that the kernel functions multiplying the spectral density of each interface obey energy conservation exactly. This means that energy is conserved independent of the roughness spectral densities of the rough surfaces. Results of this strong condition are illustrated numerically for up to 50 rough interfaces without requiring specification of surface roughness properties. Two examples are illustrated. One is a multilayer configuration having weak contrasts between adjacent layers, random layer thicknesses, and randomly generated permittivity profiles. The second example is a photonic crystal of periodically alternating permittivities of larger dielectric contrast. The methodology is applied to study the effect of roughness on the brightness temperatures of the Antarctic ice sheet, which is characterized by layers of ice with permittivity fluctuations in addition to random rough interfaces. The results show that the influence of roughness can significantly increase horizontally polarized thermal emission while leaving vertically polarized emissions relatively unaffected.

Fast Solution of Electromagnetic Scattering From Homogeneous Dielectric Objects with Multiple-Traces EF/MFIE Method

In this letter, a novel multiple-traces electric field/magnetic field integral equation (MT-EF/MFIE) method is developed for solving electromagnetic (EM) scattering from dielectric objects. Different from the traditional Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) method, the nonconformal MT-EF/MFIE method can discretize the surface with two pairs of nonconformal meshes (exterior and interior). Meanwhile, the transmission conditions are enforced to ensure the continuity of currents, on the interface of the exterior domain and interior domain. Compared to the traditional PMCHWT method, this MT-EF/MFIE method leads to a better conditioned matrix and obviously interior memory and time consumption.

3D field theories with Chern-Simons term for large N in the Weyl gauge

Three dimensional, $U(N)$ symmetric, field theory with fermion matter coupled to a topological Chern--Simons term, in the large $N$ limit is analyzed in details. We determine the conditions for the existence of a massless conformal invariant ground state as well as the conditions for a massive phase. We analyze the phase structure and calculate gauge invariant corelators comparing them in several cases to existing results. In addition to the non-critical explicitly broken scale invariance massive case we consider also a massive ground state where the scale symmetry is spontaneously broken. We show that such a phase appears only in the presence of a marginal deformation that is introduced by adding a certain scalar auxiliary field and discuss the fermion-boson dual mapping. The ground state contains in this case a massless $U(N)$ singlet bound state goldstone boson- the dilaton whose properties are determined. We employ here the temporal gauge which is at variance with respect to past calculations using the light-cone gauge and thus, a check (though limited) of gauge independence is at hand. The large $N$ properties are determined by using a field integral formalism and the steepest descent method. The saddle point equations, which take here the form of integral equations for non-local fields, determine the mass gap and the dressed fermion propagator. Vertex functions are calculated at leading order in $1/N$ as exact solutions of integral equations. From the vertex functions, we infer gauge invariant two-point correlation functions for scalar operators and a current. Indications about the consistency of the method are obtained by verifying that gauge-invariant quantities have a natural $O(3)$ covariant form. As a further verification, in several occasions, a few terms of the perturbative expansion are calculated and compared with the exact results in the appropriate order.

Thermoelastic interaction of two offset interfacial cracks in bonded dissimilar half-planes with a functionally graded interlayer

This paper deals with the thermoelasticity problem of bonded dissimilar half-planes with a functionally graded interlayer, weakened by a pair of two offset interfacial cracks. The material nonhomogeneity in the graded interlayer is represented by spatially varying thermoelastic moduli expressed in terms of exponential functions. The cracks are assumed to be thermally insulated disturbing a steady-state uniform heat flow, and the solution is obtained within the framework of linear plane thermoelasticity. The Fourier integral transform method is employed, and the formulation of the current nonisothermal crack problem is reduced to two sets of Cauchy-type singular integral equations for temperature and thermal stress fields in the bonded system. In the numerical results, parametric studies are conducted so that the variations in mixed-mode thermal stress intensity factors are presented as a function of offset crack distance for various geometric and material combinations of the dissimilar homogeneous media bonded through the thermoelastically graded interlayer, elaborating thermally induced singular interaction of the two neighboring interfacial cracks.

Cauchy Fast Multipole Method for General Analytic Kernels

The fast multipole method (FMM) is a technique allowing the fast calculation of long-range interactions between $N$ points in ${\mathcal{O}}(N)$ or ${\mathcal{O}}(N \log N)$ steps with some prescribed error tolerance. The FMM has found many applications in the field of integral equations and boundary element methods, in particular by accelerating the solution of dense linear systems arising from such formulations. Standard FMMs are derived from analytical expansions of the kernel, for example using spherical harmonics or Taylor expansions. In recent years, the range of applicability and the ease of use of FMMs have been extended by the introduction of black-box and kernel independent techniques. In these approaches, the user provides only a subroutine to numerically calculate the interaction kernel. This allows changing the definition of the kernel with minimal changes to the computer program. This paper presents a novel kernel independent FMM, which leads to diagonal multipole-to-local operators. The result is a significant reduction in the computational cost, particularly when high accuracy is needed. The approach is based on Cauchy's integral formula and the Laplace transform. We will present a numerical analysis of the convergence and numerical results in the case of a multilevel one-dimensional FMM.

PT -symmetric deformations of integrable models

We review recent results on new physical models constructed as -symmetrical deformations or extensions of different types of integrable models. We present non-Hermitian versions of quantum spin chains, multi-particle systems of Calogero–Moser–Sutherland type and nonlinear integrable field equations of Korteweg–de Vries type. The quantum spin chain discussed is related to the first example in the series of the non-unitary models of minimal conformal field theories. For the Calogero–Moser–Sutherland models, we provide three alternative deformations: a complex extension for models related to all types of Coxeter/Weyl groups; models describing the evolution of poles in constrained real-valued field equations of nonlinear integrable systems; and genuine deformations based on antilinearly invariant deformed root systems. Deformations of complex nonlinear integrable field equations of Korteweg–de Vries type are studied with regard to different kinds of -symmetrical scenarios. A reduction to simple complex quantum mechanical models currently under discussion is presented.

Fundamentals of numerical analysis: spaceborne radio astronomy antennas

The field integral equations and their discretization in the form of matrix equations as applied in numerical simulations of spaceborne radio astronomy antennas are outlined. A manifold of integral equation solvers are available, from freeware up to industrial grade software, to calculate the current distribution in antenna systems, which represents the main computational burden in the solution of transmission, reception and scattering problems. A complete and straightforward approach is presented which covers all relations building the fundament of the numerical simulations. A simple example is treated in detail to illustrate how a solvable linear equation system is obtained from the underlying integral equations. Finally, it is shown how a sophisticated computer program can be used to deal with a real spaceborne antenna system, namely the antennas of the radio wave instrument onboard Solar Orbiter.

Combining self-consistent and numerical methods for the calculation of elastic fields and effective properties of 3D-matrix composites with periodic and random microstructures

The work is devoted to the calculation of static elastic fields in 3D-composite materials consisting of a homogeneous host medium (matrix) and an array of isolated heterogeneous inclusions. A self-consistent effective field method allows reducing this problem to the problem for a typical cell of the composite that contains a finite number of the inclusions. The volume integral equations for strain and stress fields in a heterogeneous medium are used. Discretization of these equations is performed by the radial Gaussian functions centered at a system of approximating nodes. Such functions allow calculating the elements of the matrix of the discretized problem in explicit analytical form. For a regular grid of approximating nodes, the matrix of the discretized problem has the Toeplitz properties, and matrix–vector products with such matrices may be calculated by the fast fourier transform technique. The latter accelerates significantly the iterative procedure. First, the method is applied to the calculation of elastic fields in a homogeneous medium with a spherical heterogeneous inclusion and then, to composites with periodic and random sets of spherical inclusions. Simple cubic and FCC lattices of the inclusions which material is stiffer or softer than the material of the matrix are considered. The calculations are performed for cells that contain various numbers of the inclusions, and the predicted effective constants of the composites are compared with the numerical solutions of other authors. Finally, a composite material with a random set of spherical inclusions is considered. It is shown that the consideration of a composite cell that contains a dozen of randomly distributed inclusions allows predicting the composite effective elastic constants with sufficient accuracy.

An Efficient Numerical Method for Calculation of Elastic and Thermo-Elastic Fields in a Homogeneous Medium with Several Heterogeneous Inclusions

The work is devoted to calculation of static elastic and thermo-elastic fields in a homogeneous medium with a finite number of isolated heterogeneous inclusions. Firstly, the problem is reduced to the solution of inte-gral equations for strain and stress fields in the medium with inclusions. Then, Gaussian approximating func-tions are used for discretization of these equations. For such functions, the elements of the matrix of the dis-cretized problem are calculated in explicit analytical forms. The method is mesh free, and only the coordi-nates of the approximating nodes are the geometrical information required in the method. If such nodes compose a regular grid, the matrix of the discretized problem obtains the Toeplitz properties. By the calcula-tion of matrix-vector products with such matrices, the Fast Fourier Transform technique may be used. The latter accelerates essentially the process of the iterative solution of the disretized problem. The results of calculations of elastic fields in 3D-medium with an isolated spherical heterogeneous inclusion are compared with exact solutions. Examples of the calculation of elastic and thermo-elastic fields in the medium with several inclusions are presented.

2-D Scattering Integral Field Equation Solution Through an IMLS Meshless-Based Approach

In this work, we apply a meshless-based method to a set of integral equations arising in electromagnetic wave propagation and scattering. The objective is not only to solve these equations through a meshless-based method, but also to find a way to build shape functions that could work for any cross-sectional geometry. We have found that the Moving Least Squares (MLS) approximation is not able to provide useful shape functions in every situation. This technique relies on matrix inversions and, according to the geometry, singular matrices can occur. In order to avoid this problem, we have taken the Improved Moving Least Squares (IMLS) approximation, that does not depend upon matrix inversions and then applied it to a number of cross-sectional geometries.

Generalized fast multipole method

The fast multipole method (FMM) is a technique allowing the fast calculation of long-range interactions between N points in O(N) or O(N ln N) steps with some prescribed error tolerance. The FMM has found many applications in the field of integral equations and boundary element methods, in particular by accelerating the solution of dense linear systems arising from such formulations. Original FMMs required analytical expansions of the kernel, for example using spherical harmonics or Taylor expansions. In recent years, the range of applicability and the ease of use of FMMs has been extended by the introduction of black box [1] or kernel independent techniques [2]. In these approaches, the user only provides a subroutine to numerically calculate the interaction kernel. This allows changing the definition of the kernel with minimal change to the computer program. In this talk we will present a novel kernel independent FMM, which leads to diagonal multipole-to-local operators. This results in a significant reduction in the computational cost [1], in particular when high accuracy is needed. The approach is based on Cauchy's integral formula and the Laplace transform. We will present a numerical analysis of the convergence, methods to choose the parameters in the FMM given some tolerance, and the steps required to build a multilevel scheme from the single level formulation. Numerical results are given for benchmark calculations to demonstrate the accuracy as a function of the number of multipole coefficients, and the computational cost of the different steps in the method.

Backscattering From a Two Dimensional Rectangular Crack Using FIE

The (numerical) solution of a rectangular crack in a perfectly conducting surface is appropriate for non-destructive testing (NDT) applications to model faults. The paper presents a direct modeling technique for determining the H and E-polarized backscattering signatures of a two-dimensional crack in a metallic surface that is suitable for inverse scattering problem. The governing field integral equations (FIE) with logarithmic and hyper singular kernels is first discretized and solved by a collocation method based on Chebyshev polynomials. By using ad hoc quadrature rules, the integral equation is then reduced to a linear system of algebraic equations. This approach does not have the size or frequency limitations of the regular techniques such as modal expansion and quasi-static manners. The results are in good agreement with the entirely numerical and non-reversible solution of a finite element method.

Approximation of solution operators of elliptic partial differential equations by $${\mathcal{H}}$$ - and $${\mathcal{H}^2}$$ -matrices

We investigate the problem of computing the inverses of stiffness matrices resulting from the finite element discretization of elliptic partial differential equations. Since the solution operators are non-local, the inverse matrices will in general be dense, therefore representing them by standard techniques will require prohibitively large amounts of storage. In the field of integral equations, a successful technique for handling dense matrices efficiently is to use a data-sparse representation like the popular multipole method. In this paper we prove that this approach can be generalized to cover inverse matrices corresponding to partial differential equations by switching to data-sparse $${\mathcal{H}}$$- and $${\mathcal{H}^2}$$-matrices. The key results are existence proofs for local low-rank approximations of the solution operator and its discrete counterpart, which give rise to error estimates for $${\mathcal{H}}$$- and $${\mathcal{H}^2}$$-matrix approximations of the entire matrices.

Particles versus fields in % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBqj3BWbIqubWexLMBb50ujbqegm0B % 1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqr % Ffpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0F % irpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaa % GcbaWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgaiuaacqWF % pepucqWFtepvaaa!46A4! $$ \mathcal{P}\mathcal{T} $$ -symmetrically deformed integrable systems

We review some recent results on how PT-symmetry, that is a simultaneous time-reversal and parity transformation, can be used to construct new integrable models. Some complex valued multi-particle systems, such as deformations of the Calogero-Moser-Sutherland models, are shown to arise naturally from real valued field equations of non-linear integrable systems. Deformations of complex non-linear integrable field equations, some of them even allowing for compacton solutions, are also investigated. The integrabilty of various systems is established by means of the Painleve test