Global solutions by analytic parameter continuation for a class of non-linear singular integral equations
Solution of non-linear singular integral equations from hydroelasticity and electroelasticity by analytic parameter continuation
Sloss, J. M.↗
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Numerical quadrature methods for integrals of singular periodic functions and their application to singular and weakly singular integral equations
High accuracy numerical quadrature methods for integrals of singular periodic functions are proposed. These methods are based on the appropriate Euler-Maclaurin expansions of trapezoidal rule approximations and their extrapolations. They are used to obtain accurate quadrature methods for the solution of singular and weakly singular Fredholm integral equations. Such periodic equations are used in the solution of planar elliptic boundary value problems, elasticity, potential theory, conformal mapping, boundary element methods, free surface flows, etc. The use of the quadrature methods is demonstrated with numerical examples.
Sidi, A.↗
Solution of an integral equation encountered in studies on radiative transfer in completely absorbing leaf canopies
The leaf normal distribution function is related to the extinction coefficient of direct solar radiation by a Fredholm integral equation of the first kind. The integral equation is solved by the method of constrained least squares. The solutions are stable and correct only in extreme cases and, in general, it is not possible to recover an arbitrary leaf normal distribution function.
Myneni, Ranga B.↗
An interative solution of an integral equation for radiative transfer by using variational technique
An effective iterative technique is introduced to solve a nonlinear integral equation frequently associated with radiative transfer problems. The problem is formulated in such a way that each step of an iterative sequence requires the solution of a linear integral equation. The advantage of a previously introduced variational technique which utilizes a stepwise constant trial function is exploited to cope with the nonlinear problem. The method is simple and straightforward. Rapid convergence is obtained by employing a linear interpolation of the iterative solutions. Using absorption coefficients of the Milne-Eddington type, which are applicable to some planetary atmospheric radiation problems. Solutions are found in terms of temperature and radiative flux. These solutions are presented numerically and show excellent agreement with other numerical solutions.
Yoshikawa, K. K.↗
Computational technique for compressible vortex flows using the integral equation solution
The steady full-potential equation is written in the form of Poisson's equation, and the solution for the velocity field is expressed in terms of an integral equation. The integral solution consists of two surface integrals and one volume integral. The solution is obtained through successive iteration cycles. Each cycle of iteration consists of two sub-cycles, an inner cycle for wake relaxation and an out cycle for the strength of the source distribution integrals representing the flow compressibility. The density gradients in the source distribution is computed by using a type-differencing scheme of the Murman-Cole type. The method is applied to delta wings and the numerical examples show that a curved shock is captured on the wing suction side beneath the leading edge vortex sheet. Recently, a modified version of the scheme was applied to rectangular wings. In this modified scheme, the surface integral terms were computed by using a bilinear distribution of vorticity on triangular vortex panels which represent the wing and its wake. The results were compared with the available experimental data and they are in good agreement.
Kandil, Osama A.↗
An exterior Poisson solver using fast direct methods and boundary integral equations with applications to nonlinear potential flow
A general method is developed combining fast direct methods and boundary integral equation methods to solve Poisson's equation on irregular exterior regions. The method requires O(N log N) operations where N is the number of grid points. Error estimates are given that hold for regions with corners and other boundary irregularities. Computational results are given in the context of computational aerodynamics for a two-dimensional lifting airfoil. Solutions of boundary integral equations for lifting and nonlifting aerodynamic configurations using preconditioned conjugate gradient are examined for varying degrees of thinness.
Young, D. P.↗
Modeling Antenna Performance with an Efficient Hybrid Finite Element--Integral Equation--Waveguide Mode Matching technique
An efficient hybrid finite element-integral equation technique has recently being developed to model EM scattering from three-dimensional inhomogeneous penetrable bodies of arbitrary shape, and is currently being extended to treat radiation problems.
hybrid finite element-integral equation technique↗
Integral equation for small perturbations of irrotational flows
An investigation is conducted concerning the validity of analytical methods which are based on deriving an integral equation, taking into account small perturbations in the case of a nonuniform but irrotational flow. The results obtained apply to a wide Mach number range, but are restricted to small amplitude motions and to nonviscous flows. It is shown that the integral equation relating the unknown velocity potential to the known normal flow velocity can be derived from the appropriate Green's identity.
Haviland, J. K.↗
Moment method for solution of the schwarzschild-milne integral equation.
Moment method solution of schwarzschild-milne integral equation, with application to nongray atmospheres
MOMENT METHOD↗
Application of boundary integral equations to elastoplastic problems
The application of boundary integral equations to elastoplastic problems is reviewed. Details of the analysis as applied to torsion problems and to plane problems is discussed. Results are presented for the elastoplastic torsion of a square cross section bar and for the plane problem of notched beams. A comparison of different formulations as well as comparisons with experimental results are presented.
Mendelson, A.↗
Quadrature methods for periodic singular and weakly singular Fredholm integral equations
High-accuracy numerical quadrature methods for integrals of singular periodic functions are proposed. These methods are based on the appropriate Euler-Maclaurin expansions of trapezoidal rule approximations and their extrapolations. They are subsequently used to obtain accurate quadrature methods for the solution of singular and weakly singular Fredholm integral equations. Throughout the development the periodic nature of the problem plays a crucial role. Such periodic equations are used in the solution of planar elliptic boundary value problems such as those that arise in elasticity, potential theory, conformal mapping, and free surface flows. The use of the quadrature methods is demonstrated with numerical examples.
Sidi, Avram↗
Elastic interactions of a fatigue crack with a micro-defect by the mixed boundary integral equation method
In this paper, the mixed boundary integral equation method is developed to study the elastic interactions of a fatigue crack and a micro-defect such as a void, a rigid inclusion or a transformation inclusion. The method of pseudo-tractions is employed to study the effect of a transformation inclusion. An enriched element which incorporates the mixed-mode stress intensity factors is applied to characterize the singularity at a moving crack tip. In order to evaluate the accuracy of the numerical procedure, the analysis of a crack emanating from a circular hole in a finite plate is performed and the results are compared with the available numerical solution. The effects of various micro-defects on the crack path and fatigue life are investigated. The results agree with the experimental observations.
Lua, Yuan J.↗
On the Kernel function of the integral equation relating the lift and downwash distributions of oscillating finite wings in subsonic flow
This report treats the Kernel function of an integral equation that relates a known prescribed downwash distribution to an unknown lift distribution for a harmonically oscillating finite wing in compressible subsonic flow. The Kernel function is reduced to a form that can be accurately evaluated by separating the Kernel function into two parts: a part in which the singularities are isolated and analytically expressed and a nonsingular part which may be tabulated. The form of the Kernel function for the sonic case (Mach number 1) is treated separately. In addition, results for the special cases of Mach number of 0 (incompressible case) and frequency of 0 (steady case) are given. The derivation of the integral equation which involves this Kernel function is reproduced as an appendix. Another appendix gives the reduction of the form of the Kernel function obtained herein for the three-dimensional case to a known result of Possio for two-dimensional flow. A third appendix contains some remarks on the evaluation of the Kernel function, and a fourth appendix presents an alternate form of expression for the Kernel function.
Watkins, Charles E↗
An integral equation approach to the strip problem
A semi-infinite strip held rigidly on its short end is considered. Loads in the strip at infinity (far away from the fixed end) are prescribed. Integral transform technique is used to provide an exact formulation of the problem in terms of a singular integral equation. Stress singularity at the strip corner is obtained from the singular integral equation which is then solved numerically. Stresses along the rigid end are determined and the effect of the material properties on the stress intensity factor is presented. The method can also be applied to the problem of a laminate composite with a flat inclusion normal to the interfaces.
Gupta, G. D.↗
An integral equation approach to the semi-infinite strip problem.
A semi-infinite strip held rigidly on its short end is considered. Loads in the strip at infinity (far away from the fixed end) are prescribed. The integral transform technique is used to provide an exact formulation of the problem in terms of a singular integral equation. The stress singularity at the strip corner is obtained from the singular integral equation, which is then solved numerically. Stresses along the rigid end are determined, and the effect of the material properties on the stress-intensity factor is presented. The method can also be applied to the problem of a laminate composite with a flat inclusion normal to the interfaces.
Gupta, G. D.↗
Radiative transfer in a sphere illuminated by a parallel beam - An integral equation approach
The problem of multiple scattering of nonpolarized light in a planetary body of arbitrary shape illuminated by a parallel beam is formulated using the integral equation approach. There exists a simple functional whose stationarity condition is equivalent to solving the equation of radiative transfer and whose value at the stationary point is proportional to the differential cross section. The analysis reveals a direct relation between the microscopic symmetry of the phase function for each scattering event and the macroscopic symmetry of the differential cross section for the entire planetary body, and the interconnection of these symmetry relations and the variational principle. The case of a homogeneous sphere containing isotropic scatterers is investigated in detail. It is shown that the solution can be expanded in a multipole series such that the general spherical problem is reduced to solving a set of decoupled integral equations in one dimension. Computations have been performed for a range of parameters of interest, and illustrative examples of applications to planetary problems as provided.
Shia, R.-L.↗
Finite-volume and integral-equation techniques for transonic and supersonic vortex-dominated flows
Two computational techniques are developed to calculate the compressible vortex-dominated flows. The first technique is a finite-volume Euler Solver which uses four-Stage Runge-Kutta time stepping with second- and fourth-order dissipation terms. The technique is applied to supersonic conical and three-dimensional flows about sharp- and round-edged delta wings. Attached and separated-flow solutions have been obtained depending on the values of damping coefficients. The second technique is an integral-equation solver of the full potential equation which uses a volume-integral term in addition to the classical surface-integral terms. The technique is applied to transonic three-dimensional flows about sharp-edged delta wings. A hybrid technique which combines the finite-volume and the integral-equation solvers is also presented.
Kandil, O. A.↗
Plane elasto-plastic analysis of v-notched plate under bending by boundary integral equation method
A method of solution is presented, which, when applied to the elasto-plastic analysis of plates having a v-notch on one edge and subjected to pure bending, will produce stress and strain fields in much greater detail than presently available. Application of the boundary integral equation method results in two coupled Fredholm-type integral equations, subject to prescribed boundary conditions. These equations are replaced by a system of simultaneous algebraic equations and solved by a successive approximation method employing Prandtl-Reuss incremental plasticity relations. The method is first applied to number of elasto-static problems and the results compared with available solutions. Good agreement is obtained in all cases. The elasto-plastic analysis provides detailed stress and strain distributions for several cases of plates with various notch angles and notch depths. A strain hardening material is assumed and both plane strain and plane stress conditions are considered.
Rzasnicki, W.↗