Nonlinear perturbations on a free surface induced by a submerged body: A boundary integral approach

The Boundary Integral scheme developed by Dold and Peregrine¹ has been extended and modified in order to allow for the investigation of non-periodic (spatial) disturbances on a free surface of water produced by a submerged body. The problem is treated as two dimensional: the body is shaped like a horizontal long cylinder with an arbitrary smooth contour, (a twice differentiable closed curve), moving in a direction perpendicular to its generators; the cylinder can execute any twice differentiable, (with respect to time), prescribed motion. The numerical code has been successfully implemented and produces accurate results with a comparatively small computational effort. The method has several features which enhance efficiency and precision. Among these: there is an economical high order discretization of the system of integral equations; the resulting system of linear algebraic equations is solved by a Gauss-Seidel iterative scheme which converges in few iterations, less than 10 usually; a fifth order Taylor series is used to march in time. Numerical results are presented; for a weir set in an otherwise uniform current, radiation of waves produced by simple oscillatory movements of a cylinder, and the interaction of an incoming solitary wave with a fixed submerged cylinder are shown.     

From a boundary integral formulation to a vortex method for viscous flows

For the numerical solution of flow problems past a solid body it is worth to consider boundary integral techniques for their inherent capability to manage efficiently the far-field boundary conditions as well as the approximation of the solid body contour. However, for the analysis of large Reynolds number flows, of major interest in the applications, several computational difficulties appear when using the integral representation for the velocity or for the vorticity field in its classical form with interpolating functions (BEM). In particular, the evaluation of the volume integrals is a serious drawback while the steepness of their kernel introduces artificial diffusion in the calculation. To satisfy the opposite requirements of the advective and of the diffusive part of the Navier-Stokes equations, we adopt an operator splitting scheme according to the Chorin-Marsden product formula (Chorin et al. 1978), together with a proper vorticity generation scheme at the solid boundary. A solution procedure based on the approximation of the vorticity field by a finite number of point vortices (PVM) follows as a natural evolution of the boundary integral formulation.The numerical results given by the two methods for the merging of two like-signed vortices in free space reveal the excessive numerical diffusion of BEM. The better accuracy of PVM is also established through the evaluation of some first integrals of motion. Several results are also reported for flows in presence of solid boundaries where the vorticity generation is crucial. In this case accurate solutions are only obtained with PVM, while BEM is even less satisfactory than in free space. Finally, the proposed vortex-like method (PVM) is tested on the classical problem of the wake behind a cylinder, in comparison with other well established techniques.This work was partly supported by the Italian Ministry for Scientific Research through a MURST grant and by C.N.R. through Progetto Finalizzato Trasporti II.     

A symmetric boundary integral approach to transient poroelastic analysis

 The problem of the transient quasi-static analysis of a poroelastic body subjected to a history of external actions is formulated in terms of four boundary integral equations, using time-dependent Green's functions of the “free” poroelastic space. Some of these Green's functions, not available in the literature are derived “ad hoc”. The boundary integral operator constructed is shown to be symmetric with respect to a time-convolutive bilinear form so that the boundary solution is characterized by a variational property and its approximation preserving symmetry can be achieved by a Galerkin boundary element procedure. Communicated by S. N. Atluri, 1 July 1996     

On the boundary integral equations approach to a semi-infinite strip investigation

Summary An orthotropic semi-infinite strip under arbitrary boundary conditions is considered. By means of Fourier transforms, boundary integral relations of special type with moving and motionless singularities of the Cauchy type are obtained. These relations lead to a system of singular integral equations corresponding to the various mixed boundary value problem. The power of singularities at the corner points, stresses and stress intensity factors are calculated for different loads and various material properties.     

Application of the boundary element method to elastic wave scattering by irregular defects

A time-harmonic boundary element formulation for elastic wave scattering in 3D is adapted to ultrasonic NDE. Defect classes addressed are volumetric voids and inclusions, and crack-like elliptical voids. For axisymmetric flaws, comparisons are made with method of optimal truncation (MOOT) and transition-matrix calculations. Comparison to experiment is made for more general shapes. For crack-like voids, comparisons are made with the Kirchhoff, geometric theory of diffraction (GTD), and quasistatic asymptotic approximations. The efficiency and usefulness of the boundary element method (BEM) in finding the bounds of applicability of these approximate theories are demonstrated. An example of a flaw characterization technique based on intermediate frequency scattering data simulated by BEM is given. The ability of BEM to handle nonplanar incident fields, as described by a transducer beam model, is shown. Other computational and modeling efficiencies of the BEM are noted.     

Numerical resolution of the boundary integral equations for elastic scattering by a plane crack

The problem of wave scattering by a plane crack is solved, either in the case of acoustic waves or in the case of elastic waves incidence using the boundary integral equation method. A collocation method is often used to solve that equation, but here we will use a variational method, first writing the problem of Fourier variables, and then writing the associated integrals in the sesquilinear form with weak singularity kernels. This representation is used in the numerical approach, made with a finite element method in the surface of the crack. Numerical tests were made with circular and elliptical cracks, but this method can be extended to other shapes, with the same convergence profiles. Extensive results are given concerning the crack opening displacement, the scattering cross-section, the back-scattered amplitude and far-field patterns.     

A volume-perturbation approach to the problem of Rayleigh wave scattering at a rectangular groove

The problem of scattering of Rayleigh waves at a rectangular groove is addressed. Grooves are known to excite bulk waves upon scattering and, hence, are potential sources (albeit secondary) in bulk-acoustic-wave (BAW) devices. The groove is formulated as a volume perturbation of the geometry. A modal method is used, and the results of Rayleigh wave reflection as well as bulk wave radiation are obtained and compared with the results available in the literature. The method is compared with the boundary perturbation formulation. The equivalence of the boundary perturbation method and the volume perturbation method is shown.     

Hypersingular integral formulation of elastic wave scattering

A hypersingular boundary integral formulation for calculating two dimensional elastic wave scattering from thin bodies and cracks is described. The boundary integral equation for surface displacement is combined with the hypersingular equation for surface traction. The difficult part in employing the traction equation, the derivation of analytical formulas for the hypersingular integral by means of a limit to the boundary, is easily handled by means of symbolic computation. In addition, the terms containing an integrable logarithmic singularity are treated by a straightforward numerical method, bypassing the use of Taylor series expansions. Example wave scattering calculations for cracks and thin ellipses are presented.     

Boundary conditions in an integral approach to scattering

Scattering of electromagnetic radiation by an object of arbitrary shape or a structured surface, infinite in extent, is considered. When radiation is incident on an interface separating vacuum from a material medium, a current density is induced in the bulk and a surface current density may appear on the boundary surface. The electromagnetic field is then the sum of the incident field and the field generated by the current densities. This concept leads to expressions for the electric and magnetic fields that can easily be shown to be exact integrals of Maxwell's equations both in the vacuum and in the medium. At the boundary surface, the electric and magnetic fields must be discontinuous, with the discontinuity determined by the surface charge and current densities. This is usually referred to as boundary conditions for Maxwell's equations. We show that the integrals for the electric and magnetic fields automatically satisfy these boundary conditions, no matter the origin of the current densities.     

The scattering of a scalar wave by a thin oblate body of revolution

The scattering of a scalar incident wave by a thin oblate body of revolution is studied. The scattering wave is represented as a distribution of ring singularities in a disk inside the body. Boundary conditions result in one-dimensional integral equations for singularity distributions. The asymptotic solution of these equations for thin bodies furnishes the singularity distribution as well as the radius α of the disk containing ring singularities. Example of the plane wave axially incident upon an oblate ellipsoid is presented. The total scattering cross-section is computed.     

A pure boundary element method approach for solving hypersingular boundary integral equations

This paper is concerned with discretization and numerical solution of a regularized version of the hypersingular boundary integral equation (HBIE) for the two-dimensional Laplace equation. This HBIE contains the primary unknown, as well as its gradient, on the boundary of a body. Traditionally, this equation has been solved by combining the boundary element method (BEM) together with tangential differentiation of the interpolated primary variable on the boundary. The present paper avoids this tangential differentiation. Instead, a “pure” BEM method is proposed for solving this class of problems. Dirichlet, Neumann and mixed problems are addressed in this paper, and some numerical examples are included in it.     

A simplified approach for imposing the boundary conditions in the local boundary integral equation method

A simplified approach for imposing the boundary conditions in the local boundary integral equation (LBIE) method is presented. The proposed approach employs an integral equation derived using the fundamental solution and the Green’s second identity when the collocation node is at the boundary of the solution domain (global boundary). The subdomains for the nodes placed at the global boundary preserve their circular shapes; avoiding in this way any integration over the global boundary. Consequently, the difficulties related to evaluation of singular integrals and determination of intersection points between the global and local circular boundaries are avoided. So far, attempts to avoid these issues have focused on using schemes based on meshless approximations. The downside of such schemes is that the weak formulation is abandoned. In this study the interpolation of field variables over the boundaries of the subdomains is carried out using the radial basis function approximation. Numerical examples show that the proposed approach despite its simplicity, achieves comparable accuracy to the classical treatment of the boundary conditions in the LBIE.     

Three-dimensional boundary element analysis of electromagnetic wave scattering by penetrable bodies

A frequency domain boundary element methodology of solving three dimensional electromagnetic wave scattering problems by dielectric particles is reported. The method utilizes a computationally attractive surface integral equation containing only weakly and strongly singular integrals in the contrast to most formulations involving not only strongly singular but hypersingular integrals as well. The main advantage of this integral equation is the fact that its strongly singular part is similar to the one appearing in the corresponding integral equation of dynamic elasticity. Thus, well known advanced integration techniques used successfully in elastic scattering problems can be directly applied to the present analysis. Both continuous and discontinuous quadratic elements are employed in order to accurately treat dielectric scatterers with smooth and piecewise smooth boundaries. Numerical examples dealing with three dimensional electromagnetic wave scattering problems demonstrate the accuracy and efficiency of the proposed boundary element formulation.     

A new weight function approach using indirect boundary integral method

A new weight function approach to determine SIF (stress intensity factor) using the indirect boundary integral method has been presented. The crack opening displacement field was represented by one boundary integral in the form of a single-layer potential whose kernel was modified from the fundamental solution. The proposed method enables the calculation of SIF using only one SIF formula without any modification of the crack geometries symmetric in a two-dimensional plane, e.g. a center crack in a plate with or without an internal hole, double edge cracks, circumferential crack or radial cracks in a pipe. The application procedure for this variety of crack geometries is very simple and straghtforward with only one SIF formula. The necessary information in the analysis is two reference SIFs. The analysis results, using several examples, verified that the present closed-form solution was in good agreement with those of the literature and applicable to various crack geometries.     

A new weight function approach using indirect boundary integral method

A new weight function approach to determine SIFs (stress intensity factors) using the indirect boundary integral method has been presented. The crack opening displacement field was represented by one boundary integral term in the form of a single-layer potential whose kernel was modified from the fundamental solution. The proposed method enables the calculation of SIFs using only one SIF solution, without any modification for the crack geometries symmetric in the two-dimensional plane, e.g. a center crack in a plate with or without an internal hole, double edge cracks, circumferential cracks or radial cracks in a pipe. The application procedure for this variety of crack geometries is very simple and straightforward with only one SIF solution. The necessary information in the analysis is two reference SIFs. The analysis results using several examples verified that the present closed-form solution was in good agreement with those of the literature and applicable to various crack geometries.     

A Duhamel integral based approach to one-dimensional wave propagation analysis in layered media

In this paper, a new method for analysing one-dimensional wave propagation in a layered medium is presented. It is based on Duhamel integrals in combination with the convolution quadrature method (CQM) [9, 10]. The CQM is a technique which approximates convolution integrals, in this case the Duhamel integrals, by a quadrature rule whose weights are determined by Laplace transformed fundamental solutions and a multi-step method. Duhamel integrals are used to ensure equilibrium between the layers. The methodology is closely related to structural engineering and should be more familiar to engineers in practice than the usual boundary element method. In order to investigate the accuracy and the stability of the proposed algorithm, two benchmark problems are studied. The method is presented for one-dimensional problems, namely rods, but it can be readily extended to two- or three-dimensional dynamic interaction problems, e.g., dynamic soil-structure interaction. The results are very stable with respect to time step size and they are in very good agreement with analytical solutions.     

Method of integral functionals for electromagnetic wave scattering from a double-periodic magnetodielectric layer

A numerical method in the frequency domain is developed for analyzing three-dimensional gratings using the concept of a double-periodic magnetodielectric layer. The method is based on the three-dimensional volume integral equations for the equivalent electric and magnetic polarization currents of the assumed periodic medium. The integral equations are solved by using the integral functionals related to the polarization current distributions and the technique of double Floquet-Fourier series expansion. Once the integral functionals are determined, the scattered fields outside the layer are calculated accordingly. The unit cell of the layer comprises several parallelepiped segments of materials characterized by the complex-valued relative permittivity and permeability of step function profiles. The arbitrary profiles of three-dimensional dielectric or metallic gratings can be flexibly modeled by adjusting the material parameters and sizes or locations of the parallelepiped segments in the unit cell. Numerical examples for various grating geometries and their comparisons with those presented in the literature demonstrate the accuracy and usefulness of the proposed method.     

Finite-element time-domain modelling of plane wave scattering by a periodic structure using an exact boundary condition

A two-dimensional singly-periodic structure, at plane wave oblique incidence, is modelled using the finite-element time-domain method. Over the non-periodic boundaries of the computational domain, an exact Floquet modal absorbing boundary condition is developed, in detail, for the finite element method in the time domain. The proposed formulation is validated by comparing its results with that of the finite-element frequency-domain method and with independently obtained published results. Very accurate numerical results, over a wide frequency range and incident angle range, are obtained for both transverse electric and transverse magnetic polarised plane wave illuminations     

Accurate treatment of lubrication forces between rigid spheres in viscous fluids using a traction-corrected boundary element method

Traditional boundary element methods cannot accurately resolve lubrication forces in the interstitial regions between nearly touching particles in viscous flows. In many cases, the interstitial tractions are underestimated and the relative particle velocities are overestimated resulting in significant errors in predicting particle trajectories. In order to accurately treat the lubrication forces between nearly touching particles, a traction-corrected boundary element method (TC-BEM) for multiple particles is developed by combining the analytical asymptotic solution for the tractions in the interstitial regions with the boundary element method. An adaptive meshing algorithm is developed to provide appropriate meshes on surfaces of particles with close interactions. The numerical method also employs an efficient parallelization scheme to make possible prediction of long-time behavior of particles suspended in viscous flow fields. The results of the TC-BEM are benchmarked by comparisons to analytical results for two particles in a linear shear flow and by considering the reversibility of three particles in a circular Couette flow. It is shown that the TC-BEM is able to correctly resolve the lubrication forces between nearly touching particles, thus enabling the accurate analysis of particles suspended in nonlinear shear flows.