Singular Weight Function Research Articles

Limited‐aperture migration

I establish a general theory on the true‐amplitude migration with limited aperture. The key point of this theory is to find the true‐amplitude weight function, eliminating the contribution from the boundary of the migration aperture, or edge effect. I show that the true‐amplitude weight function is a singular generalized function defined by the migration aperture. If it is used without regularization, the singular weight function reduces a diffraction stack to the distribution of the amplitude at a given traveltime to every point along a corresponding isochron. After regularization, I obtain a formula for the true‐amplitude weight function as a weak solution having the same global properties as the fundamental solution to the governing equation derived here. In comparison with other published weight functions that disregard migration aperture, this theory contains a taper function that makes the obtained weight function suitable for the migration with limited aperture. If the migration aperture goes to infinity, my weight function reduces to the one cited in the literature. Furthermore, I present a theoretical guideline for designing both the taper function and the migration aperture. Specifically, an optimal migration aperture should be larger than or equal to the region confined by the outer boundary of the second transformed Fresnel zone. The exact size of the optimal migration aperture is obtained when an equation describing the global property of the true‐amplitude weight function is satisfied. Using the true‐amplitude weight function developed, together with a first‐order perturbation of the traveltime surfaces of point‐diffracted rays, I also obtain an asymptotic formula for the depth‐migrated image. According to the formula, diffraction stack results in frequency scaling in the depth domain. This frequency scaling gives physical insight into the problem of pulse stretching in depth migration. If the formula is expanded into a Taylor series and if the Taylor series is truncated at its linear term, the formula published in the literature can be derived. Therefore, this paper also presents a general theory on the depth‐migrated image.

A three‐dimensional element‐free Galerkin elastic and elastoplastic formulation

A small strain, three-dimensional, elastic and elastoplastic Element-Free Galerkin (EFG) formulation is developed. Singular weight functions are utilized in the Moving-Least-Squares (MLS) determination of shape functions and shape function derivatives allowing accurate, direct nodal imposition of essential boundary conditions. A variable domain of influence EFG method is introduced leading to increased efficiency in computing the MLS shape functions and their derivatives. The elastoplastic formulations are based on the consistent tangent operator approach and closely follow the incremental formulations for non-linear analysis using finite elements. Several linear elastic and small strain elastoplastic numerical examples are presented to verify the accuracy of the numerical formulations. Copyright © 1999 John Wiley & Sons, Ltd.

A three-dimensional element-free Galerkin elastic and elastoplastic formulation

A small strain, three-dimensional, elastic and elastoplastic Element-Free Galerkin (EFG) formulation is developed. Singular weight functions are utilized in the Moving-Least-Squares (MLS) determination of shape functions and shape function derivatives allowing accurate, direct nodal imposition of essential boundary conditions. A variable domain of influence EFG method is introduced leading to increased efficiency in computing the MLS shape functions and their derivatives. The elastoplastic formulations are based on the consistent tangent operator approach and closely follow the incremental formulations for non-linear analysis using finite elements. Several linear elastic and small strain elastoplastic numerical examples are presented to verify the accuracy of the numerical formulations. Copyright © 1999 John Wiley & Sons, Ltd.

Complex variable approach to the BEM for multiple crack problems

A boundary element method for straight multiple center and edge crack problems is developed in this paper. The method is constructed upon the systematic use of the elastic singularity solutions in complex variables. The crack opening is represented by the continuous distribution of dislocation dipoles and the effect of the non-crack boundary by the continuous distributions of point forces and dislocation dipoles. The crack-tip singularity is embedded into the interpolation using orthogonal polynomials (i.e. Chebyshev and Jacobi) and their associated singular weight functions. The proposed analytical integration procedure of the Cauchy-type integrals defined over the crack eliminates the need for the quadrature formulae for numerical integration, streamlines, and enhances the accuracy of the traditional singular integral equation method for crack problems. The stress intensity factors for the fifteen problems analyzed in this paper have been accurate enough to substitute those given in stress intensity factor handbooks. Since non-crack boundary of arbitrary shape can be introduced at will the method is expected to give accurate stress intensity factors for complex real life problems.

Carleman estimates and unique continuation for the Schroedinger equation

We derive an inequality of Carleman's type for solutions to Schroedinger's equation, with singular weight function. As a consequence, we obtain a new unique continuation result for solutions in a bounded domain.

Quadrature rules by parameter optimisation II

The basis for the construction of optimum quadrature rules has been given in Cohen and Gismalla [1]. In this paper the ideas are extended to the construction of optimal Lobatto type formulae and the construction of optimal rules for where w(x) is a singular symmetric weight function.

On the use of generator coordinates in scattering problems

In the generator coordinate approach to scattering problems the use of both real and purely imaginary generator coordinates is proposed, in order to avoid the introduction of a singular weight function and the difficulties connected with its numerical treatment.

𝐿^{𝑝} behavior of power series with positive coefficients

In the analogous theorem for Fourier series the added condition an+l 0 but only for some LP spaces with a singular weight function and the question for LP without a weight seems difficult. For further information see [2]. We first show that if an > 0, then (1) implies (2). We may assume 1 E J2 Eakxk dx n-1 1-1/n k=O