Kernel Of Integral Equation Research Articles

Quantum dynamics of phonon assisted emission of carriers at deep level centers

A formalism is proposed to investigate quantum dynamics of localized states involving highly non-adiabatic time-evolution of electron–lattice systems. The effect of electron itinerancy is projected onto the dynamics of local variables through an integral kernel of Volterra's integral equation. The method is applied to the problem of thermal emission of carriers at deep level centers in semiconductors. It is shown that the real situation is in the adiabatic limit, and the probability of thermal emission of the trapped carriers is one per a single lattice oscillation, if the amplitude of the oscillation exceeds a critical value but zero if not.

A Low-Rank IE-QR Algorithm for Matrix Compression in Volume Integral Equations

A single-level matrix compression algorithm based on pivoted QR factorization with partial matrix assembling, which exploits the rank deficiency of matrix blocks for physically separated groups of basis functions, is presented for the volume integral equation solution of electromagnetic scattering from arbitrarily shaped dielectric bodies. For a system of N equations, an amount of work of the order O(N/sup 2/) has traditionally been required by the method of moments (MoM). The algorithm of the present paper reduces both computational complexity and storage requirement to O(N/sup 1.5/) with relatively less dependence on the integral equation kernel. Hence, the proposed algorithm is more practical for large-scale problems and can be implemented in a wide range of applications with few or no modifications.

Transient analysis of the grounding electrode based on the wire antenna theory

Grounding electrode is modelled as an end-driven monopole antenna horizontally buried in a lossy medium. The current distribution along the wire in frequency domain is governed by the half-space Pocklington integral equation. The influence of the air–earth interface is taken into account via the reflection coefficient appearing within the integral equation kernel. The corresponding equation is solved using the variational boundary element procedure. Frequency response of the grounding system is obtained multiplying frequency spectrum of the input impedance with Fourier transform of the input current impulse. The transient impedance is readily calculated applying inverse Fast Fourier Transform to the frequency response of the grounding electrode.

Locally corrected Nyström method for EM scattering by bodies of revolution

The Nyström method allows for high order solutions to integral equations. Modifying the Nyström method with local corrections for singular integrands allows the method to be applied to the inherently singular kernels of frequency-domain electromagnetic integral equations. Here the efficiency and high order convergence properties of the Nyström method are brought together with the body of revolution (BOR) geometry to create an efficient approach to solving this important class of problems. In this paper we summarize the locally corrected Nyström method and the magnetic field integral equation (MFIE) for a BOR scatterer. Results are presented that demonstrate high order convergence and excellent agreement with reference data.

Plates on two-parameter elastic foundations with nonlinear boundary conditions by the boundary element method

A boundary element method is developed for the bending analysis of plates having nonlinear boundary conditions and resting on two-parameter elastic foundations. The nonlinearity of the problem arises from the normal bending moment of plates which is assumed to be nonlinear function of the boundary slope recognized as a support model with nonlinear rotational restraint. Thus, the solution can be treated all cases of the boundary conditions ranging from simple support to completely fixed support. The kernels of the boundary integral equations are conveniently established which the fundamental solution for the linear plate theory is used. The surface integration of the kernels for the foundation pressure is evaluated by using the property of Dirac delta function. The system of nonlinear equations is established and solved by the Newton–Raphson iterative process. The application of high-order elements, i.e. cubic elements, for improving the solution is adopted. Numerical results of several problems are given to demonstrate the accuracy and applicability of the proposed method.

A fast boundary cloud method for exterior 2D electrostatic analysis

AbstractAn accelerated boundary cloud method (BCM) for boundary‐only analysis of exterior electrostatic problems is presented in this paper. The BCM uses scattered points instead of the classical boundary elements to discretize the surface of the conductors. The dense linear system of equations generated by the BCM are solved by a GMRES iterative solver combined with a singular value decomposition based rapid matrix–vector multiplication technique. The accelerated technique takes advantage of the fact that the integral equation kernel (2D Green's function in our case) is locally smooth and, therefore, can be dramatically compressed by using a singular value decomposition technique. The acceleration technique greatly speeds up the solution phase of the linear system by accelerating the computation of the dense matrix–vector product and reducing the storage required by the BCM. Copyright © 2002 John Wiley & Sons, Ltd.

A Green's Function for Variable Density Elastodynamics under Plane Strain Conditions by Hormander's Method

A free-space Green's function for problems involving time-harmonic elastic waves in variable den- sity materials under plane strain conditions is developed herein by means of Hormander's method in the con- text of matrix algebra formalism. The challenge when solving problems involving inhomogenous media is that the coefficients appearing in the governing equations of motion are position-dependent. Furthermore, an addi- tional difficulty stems from the fact that these govern- ing equations are vectorial, which implies that coordinate transformation techniques that have been successful with scalar waves can no longer be used. Thus, the present work aims at establishing the necessary background that will allow for construction of Green's functions for a gen- eral class of inhomogeneous media that is not necessar- ily restricted to the variable density case. These func- tions, besides being useful in their own right, are also important within the context of boundary integral formu- lations, where they appear as kernels in the underlying integral equations. Finally, a numerical example serves to illustrate the proposed methodology and to quantify the influence of a variable density profile on the propaga-

A new universal numerical technique for the solution of boundary integral equations

A new numerical technique for the solution of boundary integral equations is presented, This technique leads to uniquely solvable discretized equations of any mesh and its convergence and rate of convergence do not depend on singularities of kernels of integral equations or boundary geometries, In this sense, this technique is universal. The unique solvability of discretized equations, convergence, and the rate of convergence of the numerical technique are established under only one natural condition of unique solvability of boundary integral equations for any right-hand side.

An alternative domain/boundary element technique for analyzing plates on two-parameter elastic foundations

This paper presents an alternative domain/boundary element technique for analyzing plates on two-parameter elastic foundations, in which the fundamental solution for the linear plate theory is used in the formulation. The main advantages of using this technique are that the kernels of the boundary integral equations are conveniently established and can be used for solving plate problems with various boundary conditions as well as mixed boundary conditions. The surface integration of the kernels for the foundation pressure is evaluated using a numerical procedure presented herein instead of the conventional Gauss–Legendre method, resulting in the reduction of computing time. The application of higher-order elements, i.e. cubic elements, for improving the solution is adopted. Numerical results of several problems with various boundary conditions are given to demonstrate the accuracy and validity of the method.

Reconstruction of a Pair Integral Operator of the Convolution Type

For an arbitrary operator, we pose a general reconstruction problem inverse to the problem of finding solutions. For the pair operator considered, this problem is reduced to the equivalent problem of reconstruction of the kernels of the pair integral equation of the convolution type that generates this operator. In the cases investigated, we prove theorems that characterize the reconstruction of the corresponding kernels, which are constructed in terms of two functions from different Banach algebras of the type L1(−∞, ∞) with weight.

Electromagnetic modeling of finite length wires buried in a lossy half-space

The horizontal buried wire of a finite length is analyzed by using the scattering theory. The current distribution along the wire due to an external electromagnetic interference (EMI) source is governed by the half-space Pocklington integral equation. The effect of the lower half-space is taken into account via the transmission coefficient (TC) appearing within the integral equation kernel. The principal advantage of the TC approach vs rigorous Sommerfeld integral approach is the formulation simplicity and reduced computational cost. The variational boundary element procedure is used for solving the Pocklington integral equation and the current induced along the buried wire due to the transmitted plane wave or current source excitation is obtained.

On the pseudo-differential operators in the dual boundary integral equations using degenerate kernels and circulants

The spectral properties for the six kernels (influence matrices) in the dual boundary integral equations (dual BEM) are investigated for the Laplace and Helmholtz equations of a circular domain. Based on the two-point functions for the six kernels of single layer, double layer, normal derivatives of single and double layer potentials, tangent derivatives of single and double layer potentials, they can be expressed in degenerate kernels. Using the analytical properties of circulants, the spectral properties are studied exactly in a discrete system for a circular cavity when a uniform constant element scheme is adopted. After considering the number of degrees of freedom for the discrete system to be infinite for continuous system, the spectral properties of continuous system can be obtained. The relation for the influence matrices between the interior and exterior problems is addressed. Also, the condition number for the matrices and the orders of the pseudo-differential operators are examined. Finally, the properties of Calderon projector in discrete formulation are derived and are demonstrated analytically by an example of circular domain. Also, numerical results using the dual BEM program are performed to check the identities for the Calderon projector.

Can optimized effective potentials be determined uniquely?

Local (multiplicative) effective exchange potentials obtained from the linear-combination- of-atomic-orbital (LCAO) optimized effective potential (OEP) method are frequently unrealistic in that they tend to exhibit wrong asymptotic behavior (although formally they should have the correct asymptotic behavior) and also assume unphysical rapid oscillations around the nuclei. We give an algebraic proof that, with an infinity of orbitals, the kernel of the OEP integral equation has one and only one singularity associated with a constant and hence the OEP method determines a local exchange potential uniquely, provided that we impose some appropriate boundary condition upon the exchange potential. When the number of orbitals is finite, however, the OEP integral equation is ill-posed in that it has an infinite number of solutions. We circumvent this problem by projecting the equation and the exchange potential upon the function space accessible by the kernel and thereby making the exchange potential unique. The observed numerical problems are, therefore, primarily due to the slow convergence of the projected exchange potential with respect to the size of the expansion basis set for orbitals. Nonetheless, by making a judicious choice of the basis sets, we obtain accurate exchange potentials for atoms and molecules from an LCAO OEP procedure, which are significant improvements over local or gradient-corrected exchange functionals or the Slater potential. The Krieger–Li–Iafrate scheme offers better approximations to the OEP method.

On some one-speed neutron transport problems revisited and reformulated

The solution of a number of one-speed neutron transport problems involving infinite media have been re-considered in the light of a transformation first used by Wallace (Wallace, P.R., 1944a. Boundary Conditions at Thin Absorbing Shells and Plates I. Canadian National Research Council Report MT-34; Wallace, P.R., 1944b. On the Thermal Utilisation of Plates in the Presence of Linear Anisotropic Scattering. Canadian National Research Council Report MT-63). The outcome of this transformation is that the infinite medium problem can be reduced to one in terms of an integral equation involving finite regions only. For example, in the case of an infinitely reflected slab, the infinite reflector is removed and its presence transferred to the kernel of a new integral equation. These kernels turn out to be the point or plane kernels of the corresponding infinite medium problem in the pure reflector material. In this paper the method is extended to slabs with arbitrary anisotropic scattering in slab and reflector; it is also applied to reflected spheres. In this case however, there is a limitation that the total mean free path in sphere and reflector be the same. Finally, we comment on the physical meaning of the standard anisotropic formalism and show that a more realistic eigenvalue exists which is directly related to the isotropic fission source. Some numerical results are given to illustrate our conclusions.

The Green Matrix for Strictly Hyperbolic Systems with Second-Order Derivatives

Mathematical modeling of dynamic problems of continuum mechanics in domains of complicated geometry for various types of boundary conditions usually leads to boundary value problems for hyperbolic systems. The method of boundary integral equations is a convenient tool for solving such problems; it allows one to reduce the original differential boundary value problem in a domain to a system of boundary integral equations on the boundary, thus reducing the dimension of the equations, improving the stability of numerical solution methods, and so on. Now this method is widely used for a large class of problems in mechanics and mathematical physics. The construction of fundamental solutions and kernels of boundary integral equations of the system is a key point of the method of boundary integral equations. The fundamental solutions of classical equations of elliptic and hyperbolic types have been studied quite comprehensively [1, 2]. Less is known about fundamental solutions of systems of partial differential equations. Related papers mainly deal with specific equations of continuum mechanics [3–7]. In the present paper, we consider strictly hyperbolic systems of M equations with second-order derivatives in an (N + 1)-dimensional space. For such systems, we construct the Green matrix and analyze its properties. We also study systems invariant under the orthogonal group.

About the approximate solution of the usual and generalized Hilbert boundary value problems for analytical functions

Abstract In this article the methods for obtaining the approximate solution of usual and generalized Hilbert boundary value problems are proposed. The method of solution of usual Hilbert boundary value problem is based on the theorem about the representation of the kernel of the corresponding integral equation by τ = t and on the earlier proposed method for the computation of the Cauchy‐type integrals. The method for approximate solution of the generalized boundary value problem is based on the method for computation of singular integral of the form proposed by the author. All methods are implemented with the Mathcad and Maple.

Singular potentials and double-force solutions for anisotropic laminates with coupled bending and stretching

Exact solutions are obtained in the framework of the classical theory of laminates subjected to the action of normal moments, double forces, double moments or momentless double dipoles. Seven cases of such loads are considered and completed by considering the case of given transversal discontinuity of normal deflection. It is shown that, in contrast to the case of infinite straight dislocations in a pure in-plane problem, the energy of this eighth solution depends on the discontinuity orientation. Some numerical examples are presented. Besides the formal value, the obtained double-force and double-moment solutions, as well as dimensionless double dipoles, can be used to construct kernels of additional boundary integral equations (BIE). Due to the coupling phenomena in the BIE system for the region with a corner point, additional variable such as corner forces appear and require the mentioned equation.

Criterion for formation of the first bound state of an optical polaron

The ground state of an optical polaron has been analyzed for an arbitrary force of the electron-phonon coupling. The eigenvalue differential equation is transformed into the Fredholm integral equation. With the trace method applied to the kernel of integral equation, it is found that the first bound state of the optical polaron appears only provided that the dimensionless coupling constant obeys the inequality αc≥2.8.

ON THE PAPER “INVERSE METHODS FOR ANALYZING AEROSOL SPECTROMETER MEASUREMENTS: A CRITICAL REVIEW”

In this paper several statements of Kandlikar and Ramachandran concerning the one-to-one correspondence between the spectrometric channels and aerosol size classes, overlaps of the kernels of integral equation, errors of the algebraic systems’ solutions and some inversion algorithms are discussed. It is shown that expression A −1 e ( A -linear algebraic system matrix, e -vector of the experimental errors) cannot be interpreted as the standard deviation of random errors and its adequate interpretation is given. The expressions for standard deviation of the system's solution are derived using the error propagation formula. It is indicated that known inequality for the errors of linear systems’ solutions including condition number of matrix cond is not applicable to the random errors, and the new inequality is given. The approach of smooth continuous size distribution function concept is shown to meet with difficulties since its actual realization is discrete, stochastic and has only two values: 0 and 1.

Be analysis of dynamics of rigid foundations embedded in transversely isotropic soils

The dynamic response of rigid massless cylindrical and hemispherical foundations embedded in transversely isotropic elastic soils is investigated in this study. The axis of symmetry of the foundation is assumed to be parallel to the material axis of symmetry of soil. The foundations are subjected to time‐harmonic forces acting in the vertical and horizontal directions and a time‐harmonic moment. The problems under consideration are solved by using the boundary element method. The kernels of the boundary integral equation correspond to elastodynamic Green's functions of a transversely isotropic elastic half space subjected to buried ring loads. Analytical solutions for Green's functions are used in the analysis. The solutions for impedances obtained from the present study are compared with those reported in literature for foundations embedded in isotropic soils. Selected numerical solutions are presented to portray the influence of material anisotropy, frequency of excitation and foundation geometry on impedances.