Youssef F. Rashed

BEM for dynamic analysis using compact supported radial basis functions

Abstract This paper presents a novel dynamic modelling of structures. The main idea of the present formulation is to re-formulate the dual reciprocity method (DRM) using compact supported radial basis functions (CS RBF). The fictitious displacement and traction particular solution kernels are derived, for the first time, for four classes of CS RBF. Additional complementary solutions are derived to replace the particular solutions outside the zone of the compact radius (α). The continuity of the two solutions and their derivatives up to the third derivatives are ensured along the circumference of the compact supported edge circle. The present formulation is general, as proved validity for modelling structures bounded by interior and exterior domains (structures in infinite domains) which have never been studied previously using the DRM.

Transient dynamic boundary element analysis using Gaussian-based mass matrix

This paper presents a new boundary element formulation for transient dynamic analysis. The formulation is based on the solution of the problem using the static fundamental solution. The inertia term is approximated using particular solutions using radial bases functions. Another collocation scheme is preformed to determine suitable coefficients for the radial bases function approximation. The Gaussian radial bases function is used as it rapidly decays leading to better conditioned and sparse matrices. The necessary kernels are derived and their limiting values at the singular node are given. The formulation is implemented into a computer program that accounts for boundary and internal nodes. Two examples are presented to show the accuracy and the validity of the present formulation. A numerical discussion on using the Gaussian function with compact supported or cut-off radius is also given.

A boundary element formulation for a Reissner plate on a Pasternak foundation

Abstract A boundary element formulation for a Reissner plate on a Pasternak foundation is presented in this paper. The complete expressions for internal point kernels are given for the first time. Quadratic isoparametric boundary elements are used to discretise the plate boundary. An efficient arrangement of the boundary element system matrices is discussed. Several examples are presented to demonstrate the accuracy and the validity of the present formulation.

The annihilator method for computing particular solutions to partial differential equations

Abstract We generalize the well-known annihilator method, used to find particular solutions for ordinary differential equations, to partial differential equations. This method is then used to find particular solutions of Helmholtz-type equations when the right hand side is a linear combination of thin plate and higher order splines. These particular solutions are useful in numerical algorithms for solving boundary value problems for a variety of elliptic and parabolic partial differential equations.

The boundary element modelling of the human body exposed to the ELF electromagnetic fields

Abstract In this paper a cylindrical model of human body exposed to the extremely low frequency (ELF) electromagnetic field is presented. The analysis is based on the solution of the simplified integral equation for thick wires. The numerical solution of the integral equations is performed by the Galerkin–Bubnov variant of the boundary element method. Several numerical results for the ELF exposures are presented.

On the evaluation of the stresses in the BEM for Reissner plate-bending problems

Abstract This paper deals with the evaluation of the boundary and internal stresses in the boundary element method (BEM) for plate-bending analysis. The Reissner plate theory is employed as the general plate-bending theory for modelling both thin and thick plates. Two methods are discussed for the evaluation of boundary stresses. The first is based on the local boundary tractions and strains, and the other is based on the direct evaluation of the stress tensor using the stress integral equation. The proposed methods can be used as postprocessing procedures for the BEM solution. The evaluation of three-dimensional (3-D) internal stresses through the plate thickness is also discussed and compared with the 3-D boundary element solutions. Numerical examples are presented to demonstrate the accuracy of the proposed methods.

A Boundary Integral Transformation for Bending Analysis of Thick Plates Resting on Bi-Parameter Foundation

In this work an efficient boundary element analysis for plates resting on bi-parameter foundation is presented. The present formulation mainly deffers from previously published formulations in the way of the treatment of body forces. Domain integrals due to uniform loading are transformed to equivalent boundary integrals. The Reissner plate bending theory is used to model the bending behavior of the plate and the Pasternak (bi-parameter) foundation model is used. Unlike, common techniques of the transformation, which are based on the Green second identity, the present formulation employs the Green first identity which gives more smooth and simpler transformed kernels. The necessary particular solutions are derived and the kernels for the computation of stress resultants at the internal points are derived and given in explicit forms. A general technique for avoiding the appearance of jump terms in the transformed boundary integrals is presented. Three numerical examples are presented to demonstrate the accuracy and the validity of the present formulation.

Efficient evaluation of plate–half space interaction using contour integrals

Abstract In this paper, the domain integrals resulting from plate–half space interaction are transformed to contour integrals along the plate internal cells boundaries. The half space sub grade tractions are assumed to have cell-wise constant variation underneath the plate domain. The plate can be modeled using either the thin or the thick plate theory. Whereas, the half space is modeled using the Boussinesq–Cerruti model. Two new sets of equivalent contour integrals are derived. The first formulation is based on Greens first identity (GFI). Whereas, the second formulation is based on the multiple reciprocity method (MRM). The necessary kernels and the relevant particular solutions are derived and listed. Two numerical examples are presented to show the accuracy of the present formulation.