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Noniterative homogeneous solutions of integral equations for coupled open channel and coupled eigenvalue scattering
A numerical technique is developed analytically to solve a class of singular integral equations occurring in mixed boundary-value problems for nonhomogeneous elastic media with discontinuities. The approach of Kaya and Erdogan (1987) is extended to treat equations with generalized Cauchy kernels, reformulating the boundary-value problems in terms of potentials as the unknown functions. The numerical implementation of the solution is discussed, and results for an epoxy-Al plate with a crack terminating at the interface and loading normal to the crack are presented in tables.
This work implemented a 3-D Electromagnetic Scattering model to gain a better understanding in the observations of L- and P-band radar returns from the multi-layered Martian upper surface layer. The model is based on the Integral Equation Model (IEM) developed by Adrian Fung. Our model implementation treats the upper surface (regolith) layer as an inhomogeneous medium with irregular boundaries for a range of Martian surface and shallow subsurface scenarios. The model is well suited to study the near subsurface (upper 10 meters of surface sediments) ice detection capability of the L-band (32 cm wavelength) Synthetic Aperture Radar (SAR) planned for the international Mars Ice Mapping (I-MIM) mission launching later in the 2020s.
Three dimensional pressure distribution functions for lifting surfaces with trailing-edge controls based on integral equations for subsonic flow
A numerical scheme for solving two dimensional Fredholm integral equations of the second kind is developed. The proof of the convergence of the numerical scheme is shown for three cases: the case of periodic kernels, the case of semiperiodic kernels, and the case of nonperiodic kernels. Applications to the incompressible, stationary Navier-Stokes problem are of primary interest.
The time-dependent form of the Lippmann-Schwinger integral equation is used as the basis of several new wave packet propagation schemes. These can be formulated in terms of either the time-dependent wave function or a time-dependent amplitude density. The latter is nonzero only in the region of configuratiaon space for which the potential is nonzero, thereby in principle obviating the necessity of large grids or the use of complex absorbing potentials when resonances cause long collision times (leading, consequently, to long propagation times). Transition amplitudes are obtained in terms of Fourier transforms of the amplitude density from the time to the energy domain. The approach is illustrated by an application to a standard potential scattering model problem where, as in previous studies, the action of the kinetic energy operator is evaluated by fast Fourier transform (FFT) techniques.
The purpose of this study is to investigate the performance of the integral equation computations using numerical source field-panel method in a massively parallel processing (MPP) environment. A comparative study of computational performance of the MPP CM-5 computer and conventional Cray-YMP supercomputer for a three-dimensional flow problem is made. A serial FORTRAN code is converted into a parallel CM-FORTRAN code. Some performance results are obtained on CM-5 with 32, 62, 128 nodes along with those on Cray-YMP with a single processor. The comparison of the performance indicates that the parallel CM-FORTRAN code near or out-performs the equivalent serial FORTRAN code for some cases.
This report treats the Kernel function of the integral equation that relates a known or prescribed downwash distribution to an unknown lift distribution for harmonically oscillating wings in supersonic flow. The treatment is essentially an extension to supersonic flow of the treatment given in NACA report 1234 for subsonic flow. For the supersonic case the Kernel function is derived by use of a suitable form of acoustic doublet potential which employs a cutoff or Heaviside unit function. The Kernel functions are reduced to forms that can be accurately evaluated by considering the functions in two parts: a part in which the singularities are isolated and analytically expressed, and a nonsingular part which can be tabulated.
Although free space cannot generate electromagnetic waves, the majority of existing accounts of frequency-domain electromagnetic scattering by particles and particle groups are based on the postulate of existence of an impressed incident field, usually in the form of a plane wave. In this tutorial we discuss how to account for the actual existence of impressed source currents rather than impressed incident fields. Specifically, we outline a self-consistent theoretical formalism describing electromagnetic scattering by an arbitrary finite object in the presence of arbitrarily distributed impressed currents, some of which can be far removed from the object and some can reside in its vicinity, including inside the object. To make the resulting formalism applicable to a wide range of scattering-object morphologies, we use the framework of the volume integral equation formulation of electromagnetic scattering, couple it with the notion of the transition operator, and exploit the fundamental symmetry property of this operator. Among novel results, this tutorial includes a streamlined proof of fundamental symmetry (reciprocity) relations, a simplified derivation of the Foldy equations, and an explicit analytical expression for the transition operator of a multi-component scattering object.
This paper describes an accurate economical method for generating approximations to the kernel of the integral equation relating unsteady pressure to normalwash in nonplanar flow. The method is capable of generating approximations of arbitrary accuracy. It is based on approximating the algebraic part of the non elementary integrals in the kernel by exponential approximations and then integrating termwise. The exponent spacing in the approximation is a geometric sequence. The coefficients and exponent multiplier of the exponential approximation are computed by least squares so the method is completely automated. Exponential approximates generated in this manner are two orders of magnitude more accurate than the exponential approximation that is currently most often used for this purpose. Coefficients for 8, 12, 24, and 72 term approximations are tabulated in the report. Also, since the method is automated, it can be used to generate approximations to attain any desired trade-off between accuracy and computing cost.
It has been found that the Discrete Integral Equation (DSI)technique is a highly effective technique for the analysis of microwave circuits and devices [1,2]. The DSI is much more robust than the traditional Finite Difference Time Domain (FDTD) method in a number of ways.
The Navier-Stokes equations of motion and the equation of continuity are transformed so as to apply to an orthogonal curvilinear coordinate system rotating with a uniform angular velocity about an arbitrary axis in space. A usual simplification of these equations as consistent with the accepted boundary-layer theory and an integration of these equations through the boundary layer result in boundary-layer momentum-integral equations for three-dimensional flows that are applicable to either rotating or nonrotating fluid boundaries. These equations are simplified and an approximate solution in closed integral form is obtained for a generalized boundary-layer momentum-loss thickness and flow deflection at the wall in the turbulent case. A numerical evaluation of this solution carried out for data obtained in a curving nonrotating duct shows a fair quantitative agreement with the measures values. The form in which the equations are presented is readily adaptable to cases of steady, three-dimensional, incompressible boundary-layer flow like that over curved ducts or yawed wings; and it also may be used to describe the boundary-layer flow over various rotating surfaces, thus applying to turbomachinery, propellers, and helicopter blades.
Pitch and roll plane radiation patterns computed by use of an integral equation formulation are presented for a space shuttle annular slot communications antenna. The excellent agreement between computed patterns and experimental patterns obtained from 1/35 scale models supports the use of this method as a design tool for similar spacecraft/aircraft antenna applications.
The computational procedure and numerical results are presented for a new method to solve Kuessner's integral equation in the case of subsonic compressible flow about harmonically oscillating planar surfaces with controls. Kuessner's equation is a linear transformation from pressure to normalwash. The unknown pressure is expanded in terms of prescribed basis functions and the unknown basis function coefficients are determined in the usual manner by satisfying the given normalwash distribution either collocationally or in the complex least squares sense. The present method of solution differs from previous ones in that the basis functions are defined in a continuous fashion over a relatively small portion of the aerodynamic surface and are zero elsewhere. This method, termed the local basis function method, combines the smoothness and accuracy of distribution methods with the simplicity and versatility of panel methods. Predictions by the local basis function method for unsteady flow are shown to be in excellent agreement with other methods. Also, potential improvements to the present method and extensions to more general classes of solutions are discussed.
A novel analytical-numerical method for calculating unsteady small disturbance transonic flow over airfoils has been developed. The method uses an extended integral equation technique, based on both the velocity potential and the acceleration potential, to predict unsteady aerodynamic loading on airfoils oscillating in subcritical transonic free stream conditions. The formulation is an extension of the work of Sivaneri and Harris (1980) for steady, non-lifting flows and utilizes the linear theory of Landahl (1961) for decoupling of steady and unsteady components. The analytical-numerical procedure involves several intnegrating schemes and applies to general frequencies of oscillations. The technique is illustrated by computing the transonic flow about parabolic arc airfoils. Specific unsteady results for reduced frequencies based on semi-chord of 0.01, 0.1, 0.3, 0.4 and 0.6 are given. Comparison of results with those obtained by an ADI finite difference scheme is made.
Crack propagation in a rotating inner raceway of a high-speed roller bearing is analyzed using the boundary integral method. The model consists of an edge plate under plane strain condition upon which varying Hertzian stress fields are superimposed. A multidomain boundary integral equation using quadratic elements was written to determine the stress intensity factors KI and KII at the crack tip for various roller positions. The multidomain formulation allows the two faces of the crack to be modeled in two different subregions, making it possible to analyze crack closure when the roller is positioned on or close to the crack line. KI and KII stress intensity factors along any direction were computed. These calculations permit determination of crack growth direction along which the average KI times the alternating KI is maximum.
The novel wave-packet propagation scheme presented is based on the time-dependent form of the Lippman-Schwinger integral equation and does not require extensive matrix inversions, thereby facilitating application to systems in which some degrees of freedom express the potential in a basis expansion. The matrix to be inverted is a function of the kinetic energy operator, and is accordingly diagonal in a Bessel function basis set. Transition amplitudes for various orbital angular momentum quantum numbers are obtainable via either Fourier transform of the amplitude density from the time to the energy domain, or the direct analysis of the scattered wave packet.
The 3-D image reconstruction from cone-beam projections in computerized tomography leads naturally, in the case of radial symmetry, to the study of Abel-type integral equations. If the experimental information is obtained from measured data, on a discrete set of points, special methods are needed in order to restore continuity with respect to the data. A new combined Regularized-Adjoint-Conjugate Gradient algorithm, together with two different implementations of the Mollification Method (one based on a data filtering technique and the other on the mollification of the kernal function) and a regularization by truncation method (initially proposed for 2-D ray sample schemes and more recently extended to 3-D cone-beam image reconstruction) are extensively tested and compared for accuracy and numerical stability as functions of the level of noise in the data.