Numerical Approximation of Time Domain Electromagnetic Scattering from a Thin Wire

We present two new stable schemes for computing the current induced on the surface of a thin wire by an incident time-dependent electromagnetic field. The problem involves solving a retarded potential integral equation (RPIE). One algorithm solves the previously studied reduced kernel RPIE problem, and the other solves the more complicated exact kernel RPIE problem (for which there are no previous numerical results). Both algorithms behave stably for arbitrarily chosen values of the mesh size. Test experiments and numerically computed values of the induced current are presented.     

Modified integral current methods in electrodynamics of nonhomogeneous media

The main difficulty in numerical solution of integral equations of electrodynamics is associated with the need to solve a high-order system of linear equations with a dense matrix. It is therefore relevant to develop numerical methods that lead to linear equation systems of lower order at the cost of more complex evaluation of the coefficients. In this article we propose a method for solving linear equations of electrodynamics which is a modification of the integral current method. The main distinctive feature of the proposed method is double integration of the electric Green’s tensor in the process of algebraization of the original integral equation. The solutions of the system of linear equations are thus integral means of the electric field inside the anomaly constructed by the proposed transformation formula. We prove convergence and derive error bounds for both the solution of the integral equation and the electromagnetic field components evaluated from approximate transformation formulas.     

Temporal discretization choices for stable boundary element methods in electromagnetic scattering problems

Diverse alternative temporal discretization schemes are analyzed for stable numerical solution of the surface integral equations in obtaining the transient scattering response of arbitrarily shaped conducting bodies. Streamlined formulations for three main categories including using either the conventional time integrators or the subdomain temporal basis functions, or the entire-domain time bases are presented in conceptually similar frameworks for solving types of the electric, magnetic, and combined field integral equations. To this end, first compatible temporal interpolations with conveniently usable time integrators are introduced based on stability analysis of the delay differential equations (DDE). Detailed guidelines for effective implementation of appropriate subdomain time basis functions are then studied. It is demonstrated that since in the latter approach the time derivatives are handled analytically, the extension of the stable region tremendously enhances while approaching small time step sizes. Eventually, the orthogonal weighted Laguerre polynomials are set forth to provide unconditionally stable schemes. Besides, adaptive partitioning of triangular patches is proposed to efficiently control the precision of numerical quadratures over the surface of source distribution. Numerical results are verified through comparison with the results obtained using the finite integration technique (FIT). Convergence behaviour of the widely used schemes is also investigated.     

Application of semiorthogonal spline wavelets and the Galerkin method to the numerical simulation of thin wire antennas

The Bubnov-Galerkin method based on spline wavelets is used to solve singular integral equations. For the resulting systems of linear algebraic equations, the properties of their coefficient matrices are examined. Sparse approximations of these matrices are constructed by applying a cutting barrier. The results are used to numerically analyze thin wire antennas. Numerical results are presented.     

Decoupled field integral equations for electromagnetic scattering from homogeneous penetrable obstacles

We present a new method for the analysis of electromagnetic scattering from homogeneous penetrable bodies. Our approach is based on a reformulation of the governing Maxwell equations in terms of two uncoupled vector Helmholtz systems: one for the electric field and one for the magnetic field. This permits the derivation of resonance-free Fredholm equations of the second kind that are stable at all frequencies, insensitive to the genus of the scatterers, and invertible for all passive materials including those with negative permittivities or permeabilities. We refer to these as decoupled field integral equations.     

Pathwise Taylor schemes for random ordinary differential equations

Random ordinary differential equations (RODEs) are ordinary differential equations which contain a stochastic process in their vector fields. They can be analyzed pathwise using deterministic calculus, but since the driving stochastic process is usually only Hölder continuous in time, the vector field is not differentiable in the time variable. Traditional numerical schemes for ordinary differential equations thus do not achieve their usual order of convergence when applied to RODEs. Nevertheless, deterministic calculus can still be used to derive higher order numerical schemes for RODEs by means of a new kind of integral Taylor expansion. The theory is developed systematically here, applied to illustrative examples involving Brownian motion and fractional Brownian motion as the driving processes and compared with other numerical schemes for RODEs in the literature.     

Integrability and Self-Similarity in Transient Stimulated Raman Scattering

Summary. The phenomenon of stimulated Raman scattering (SRS) can be described by three coupled PDEs which define the pump electric field, the Stokes electric field, and the material excitation as functions of distance and time. In the transient limit these equations are integrable, i.e., they admit a Lax pair formulation. Here we study this transient limit. The relevant physical problem can be formulated as an initial-boundary value (IBV) problem where both independent variables are on a finite domain. A general method for solving IBV problems for integrable equations has been introduced recently. Using this method we show that the solution of the equations describing the transient SRS can be obtained by solving a certain linear integral equation. It is interesting that this equation is identical to the linear integral equation characterizing the solution of an IBV problem of the sine-Gordon equation in light-cone coordinates. This integral equation can be solved uniquely in terms of the values of the pump and Stokes fields at the entry of the Raman cell. The asymptotic analysis of this solution reveals that the long-distance behavior of the system is dominated by the underlying self-similar solution which satisfies a particular case of the third Painlevé transcendent. This result is consistent with both numerical simulations and experimental observations. We also discuss briefly the effect of frequency mismatch between the pump and the Stokes electric fields. Received December 10, 1996; second revision received October 10, 1997; final revision received January 20, 1998     

Two difference schemes for the numerical solution of Maxwell’s equations as applied to extremely and super low frequency signal propagation in the Earth-ionosphere waveguide

Two explicit two-time-level difference schemes for the numerical solution of Maxwell’s equations are proposed to simulate propagation of small-amplitude extremely and super low frequency electromagnetic signals (200 Hz and lower) in the Earth-ionosphere waveguide with allowance for the tensor conductivity of the ionosphere. Both schemes rely on a new approach to time approximation, specifically, on Maxwell’s equations represented in integral form with respect to time. The spatial derivatives in both schemes are approximated to fourth-order accuracy. The first scheme uses field equations and is second-order accurate in time. The second scheme uses potential equations and is fourth-order accurate in time. Comparative test computations show that the schemes have a number of important advantages over those based on finite-difference approximations of time derivatives. Additionally, the potential scheme is shown to possess better properties than the field scheme.     

Stability characteristics of periodic streaming fluids in porous media

We study the linear stability of a three-layer flow of immiscible liquids located in a periodic normal electric field. We consider certain porous media assumed to be uniform, homogeneous, and isotropic. We analytically and numerically simulate the system of linear evolution equations of such a medium. The linearized problem leads to a system of two Mathieu equations with complex coefficients of the damping terms. We study the effects of the streaming velocity, permeability of the porous medium, and the electrical properties of the flow of a thin layer (film) of liquid on the flow instability. We consider several special cases of such systems. As a special case, we consider a uniform electric field and solve the transition curve equations up to the second order in a small dimensionless parameter. We show that the dielectric constant ratio and also the electric field play a destabilizing role in the stability criteria, while the porosity has a dual effect on the wave motion. In the case of an alternating electric field and a periodic velocity, we use the method of multiple time scales to calculate approximate solutions and analyze the stability criteria in the nonresonance and resonance cases; we also obtain transition curves in these cases. We show that an increase in the velocity and the electric field promote oscillations and hence have a destabilizing effect.     

Mathematical modeling of the vertical magnetic dipole field in a two-dimensional nonhomogeneous medium

We consider a quasi-three-dimensional problem of remote marine sounding by a high-power stationary source located on land. A transition from the three-dimensional problem to a family of parametric two-dimensional problems is performed. The solution of the remote marine sounding problem is obtained with high accuracy after solving about 20 two-dimensional problems. The integral equations are solved by the modified integral current method, which has proved highly efficient for field computations inside a strongly conducting anomaly. The electric field amplitude is observed to increase with depth. The width of the coastal current channel is estimated by analyzing the vertical magnetic field component.     

Numerical solution of 3D problems of electromagnetic wave diffraction on a system of ideally conducting surfaces by the method of hypersingular integral equations

We construct a numerical method for solving problems of electromagnetic wave diffraction on a system of solid and thin objects based on the reduction of the problem to a boundary integral equation treated in the sense of the Hadamard finite value. For the construction of such an equation, we construct a numerical scheme on the basis of the method of piecewise continuous approximations and collocations. Unlike earlier known schemes, by using the below-suggested scheme, we have found approximate analytic expressions for the coefficients of the arising system of linear equations by isolating the leading part of the kernel of the integral operator. We present examples of solution of a number of model problems of the diffraction of electromagnetic waves by the suggested method.     

Boundary integral equations for screen problems in IR3

Here we present a new solution procedure for Helmholtz and Laplacian Neumann screen or Dirichlet screen problems in IR³ via boundary integral equations of the first kind having as unknown the jump of the field or of its normal derivative, respectively, across the screen S. Under the assumption of local finite energy we show the equivalence of the integral equations and the original boundary value problems. Via the Wiener-Hopf method in the halfspace, localization and the calculus of pseudodifferential operators we derive existence, uniqueness and regularity results for the solution of our boundary integral equations together with its explicit behavior near the edge of the screen. We give Galerkin schemes based on our integral equations on S and obtain high convergence rates by using special singular elements besides regular splines as test and trial functions.     

Iterated fast multiscale Galerkin methods for Fredholm integral equations of second kind with weakly singular kernels

We propose iterated fast multiscale Galerkin methods for the second kind Fredholm integral equations with mildly weakly singular kernel by combining the advantages of fast methods and iteration post-processing methods. To study the super-convergence of these methods, we develop a theoretical framework for iterated fast multiscale schemes, and apply the scheme to integral equations with weakly singular kernels. We show theoretically that even the computational complexity is almost optimal, our schemes improve the accuracy of numerical solutions greatly, and exhibit the global super-convergence. Numerical examples are presented to illustrate the theoretical results and the efficiency of the methods.     

Numerical method for an integral-algebraic system

We present a numerical method for solving the system of integral-algebraic equations arising in the study of the oblique derivative problem for the Laplace equation outside open curves on the plane. The problem describes the electric current in a semiconductor film with curvilinear electrodes in the presence of a magnetic field. The integral-algebraic system has singularities, and the kernel in the integral equation is represented in the form of a Cauchy integral. The numerical scheme is of the second approximation order despite the singularities.     

Transient response of dissimilar piezoelectric layers with multiple interacting interface cracks

In this study, we examine the dynamic behavior of two bonded dissimilar piezoelectric layers containing multiple interfacial cracks subjected to electro-mechanical impact loading. The problem was formulated through Fourier transformation into singular integral equations in which the unknown variables are the jumps of displacement and electric potential across the crack surface in the Laplace transform domain. The resulting integral equations together with the corresponding single-valued conditions are solved numerically for the densities of electro-elastic dislocations on a crack surface. The dynamic field intensity factors and dynamic energy release rate (DERR) history are obtained for both permeable and impermeable crack. The stress field is also obtained for the interface crack under impact loads. The results show that the field intensity factors at the crack tips and dynamic energy release rate depend on the interfacial crack geometry, electromechanical coupling and the electric boundary conditions on the crack surface.     

The operator equations of Lippmann–Schwinger type for acoustic and electromagnetic scattering problems in L 2

This article is concerned with the scattering of acoustic and electromagnetic time harmonic plane waves by an inhomogeneous medium. These problems can be translated into volume integral equations of the second kind – the most prominent example is the Lippmann–Schwinger integral equation. In this work, we study a particular class of scattering problems where the integral operator in the corresponding operator equation of Lippmann–Schwinger type fails to be compact. Such integral equations typically arise if the modelling of the inhomogeneous medium necessitates space-dependent coefficients in the highest order terms of the underlying partial differential equation. The two examples treated here are acoustic scattering from a medium with a space-dependent material density and electromagnetic medium scattering where both the electric permittivity and the magnetic permeability vary. In these cases, Riesz theory is not applicable for the solution of the arising integral equations of Lippmann–Schwinger type. Therefore, we show that positivity assumptions on the relative material parameters allow to prove positivity of the arising volume potentials in tailor-made weighted spaces of square integrable functions. This result merely holds for imaginary wavenumber and we exploit a compactness argument to conclude that the arising integral equations are of Fredholm type, even if the integral operators themselves are not compact. Finally, we explain how the solution of the integral equations in L ² affects the notion of a solution of the scattering problem and illustrate why the order of convergence of a Galerkin scheme set up in L ² does not suffer from our L ² setting, compared to schemes in higher order Sobolev spaces.     

The well-posedness of the integral equations for thin wire antennas

In this paper the author shows that the Pocklington and Hallenintegral equations for the current induced on a thin wire byan incident harmonic electromagnetic field are well-posed. Itis also shown that the solutions must lie in certain Sobolevspaces of absolutely continuous functions     

The behavior of plasma with an arbitrary degree of degeneracy of electron gas in the conductive layer

We obtain an analytic solution of the boundary problem for the behavior (fluctuations) of an electron plasma with an arbitrary degree of degeneracy of the electron gas in the conductive layer in an external electric field. We use the kinetic Vlasov–Boltzmann equation with the Bhatnagar–Gross–Krook collision integral and the Maxwell equation for the electric field. We use the mirror boundary conditions for the reflections of electrons from the layer boundary. The boundary problem reduces to a one-dimensional problem with a single velocity. For this, we use the method of consecutive approximations, linearization of the equations with respect to the absolute distribution of the Fermi–Dirac electrons, and the conservation law for the number of particles. Separation of variables then helps reduce the problem equations to a characteristic system of equations. In the space of generalized functions, we find the eigensolutions of the initial system, which correspond to the continuous spectrum (Van Kampen mode). Solving the dispersion equation, we then find the eigensolutions corresponding to the adjoint and discrete spectra (Drude and Debye modes). We then construct the general solution of the boundary problem by decomposing it into the eigensolutions. The coefficients of the decomposition are given by the boundary conditions. This allows obtaining the decompositions of the distribution function and the electric field in explicit form.     

Regularized Combined Field Integral Equations

Summary. Many boundary integral equations for exterior boundary value problems for the Helmholtz equation suffer from a notorious instability for wave numbers related to interior resonances. The so-called combined field integral equations are not affected. However, if the boundary is not smooth, the traditional combined field integral equations for the exterior Dirichlet problem do not give rise to an L²()-coercive variational formulation. This foils attempts to establish asymptotic quasi-optimality of discrete solutions obtained through conforming Galerkin boundary element schemes.This article presents new combined field integral equations on two-dimensional closed surfaces that possess coercivity in canonical trace spaces. The main idea is to use suitable regularizing operators in the framework of both direct and indirect methods. This permits us to apply the classical convergence theory of conforming Galerkin methods.     

Nonnegativity preserving convergent schemes for the thin film equation

Summary. We present numerical schemes for fourth order degenerate parabolic equations that arise e.g. in lubrication theory for time evolution of thin films of viscous fluids. We prove convergence and nonnegativity results in arbitrary space dimensions. A proper choice of the discrete mobility enables us to establish discrete counterparts of the essential integral estimates known from the continuous setting. Hence, the numerical cost in each time step reduces to the solution of a linear system involving a sparse matrix. Furthermore, by introducing a time step control that makes use of an explicit formula for the normal velocity of the free boundary we keep the numerical cost for tracing the free boundary low. Received June 29, 1998 / Published online June 21, 2000