Boundary integral equations for thin bodies

A boundary integral equation formulation for thin bodies which uses CBIE (conventional BIE) only is well known to be degenerate. A mixed formulation for a thin rigid scatterer which combines CBIE and HBIE (hypersingular BIE) is motivated by examining the discretized form of the integral equations, and this formulation is shown to be non-degenerate for thin non-rigid inclusion problems. A near-singular integration procedure, useful for singular integrals as well, is presented. Finally, numerical examples for acoustic wave scattering from rigid and soft scatterers are presented.     

Boundary integral equations in elastodynamics of interface cracks

The paper concerns the validation of a method for solving elastodynamics problems for cracked solids. The proposed method is based on the application of boundary integral equations. The problem of an interface penny-shaped crack between two dissimilar elastic half-spaces under harmonic loading is considered as an example.     

Boundary integral equations for notch problems in an anisotropic half-space

A crack or a hole embedded in an anisotropic half-plane space subjected to a concentrated force at its surface is analyzed. Based on the Stroh formalism and the fundamental solutions to the half-plane solid due to point dislocations, the problem can be formulated by a system of boundary integral equations for the unknown dislocation densities defined on the crack or hole border. These integral equations are then reduced to algebraic equations by using the properties of the Chebyshev polynomials in conjunction with the appropriate transformations. Numerical results have been carried out for both crack problems and hole problems to elucidate the effect of geometric configurations on the stress intensity factors and the stress concentration.     

Boundary element analysis of Navier-Stokes equations

As arrangements, the fundamental solutions of anisotropic convective diffusion equations of transient incompressible viscous fluid flow and boundary elements analysis of the diffusion equation are presented. Secondly, by considering that convective diffusion equations and Navier-Stokes equations are mathematical formulations of mass and momentum conservation law respectively, and that consequently, both physical contents and equation styles are analogous, boundary integral formulations for Navier-Stokes equations are proposed on the basis of formulation of diffusion equations.     

Theoretical analysis for three-dimensional integral imaging systems with double devices

By adoption of double-device systems, integral imaging can be enhanced in image depth, viewing angle, or image size. Theoretical analyses are done for the double-image-plane integral imaging systems. Both ray optics analysis and wave optics analysis confirm that the double-device integral imaging systems can pick up and display images at two separate image planes. The analysis results are also valuable in the understanding of the conventional integral imaging systems for image positions off the central depth plane.     

Boundary integral equations for 2D thermoelectroelasticity of a half-space with cracks and thin inclusions

The paper presents a rigorous and straightforward approach for obtaining the 2D boundary integral equations for a thermoelectroelastic half-space containing holes, cracks and thin foreign inclusions. It starts from the Cauchy integral formula and Stroh orthogonality relations to obtain the integral formulae for the Stroh complex functions, which are piecewise-analytic in the complex half-plane with holes and opened mathematical cuts. Further application of the Stroh formalism allows derivation of the Somigliana type integral formulae and boundary integral equations for a thermoelectroelastic half-space. The kernels of these equations correspond to the fundamental solutions of heat transfer, electroelasticity and thermoelectroelasticity for a half-space. It is shown that the difference between the obtained fundamental solution of thermoelectroelasticity and those presented in literature is due to the fact, that present solution additionally accounts for extended displacement and stress continuity conditions, thus, it is physically correct. Obtained integral equations are introduced into the boundary element approach. Numerical examples validate derived boundary integral equations, show their efficiency and accuracy.     

Boundary integral equations for the computational modeling of three-dimensional groundwater flow problems

Boundary integral equations for the computational modeling of three-dimensional groundwater flow problems are derived. They follow from appropriate volume and surface source-type integral representations for the pressure and the flow velocity. The numerical handling of the integral equations is discussed in some detail, especially as far as the evaluation of singular integrals is concerned. Arbitrary anisotropy in the resistivity of the fluid-saturated soil is taken into account.     

Boundary integral equation solution of the single sided linear induction pump problem

This paper utilizes the Boundary Integral Equation Method to analyse the single sided linear induction pump. The advantage offered by the method is that transverse edge effects are not neglected. The equations necessary for the electromagnetic field and force analysis of the pump are developed and the numerical solution of the equations is described. Particular attention is given to means of calculating the electromagnetic fields within the molten metal secondary. Typical field and force distributions are presented.     

Boundary integral equations for the scattering of elastic waves by elastic inclusions with thin interface layers

Elastic waves are scattered by an elastic inclusion. The interface between the inclusion and the surrounding material is imperfect: the displacement and traction vectors on one side of the interface are assumed to be linearly related to both the displacement vector and the traction vector on the other side of the interface. The literature on such inclusion problems is reviewed, with special emphasis on the development of interface conditions modeling different types of interface layer. Inclusion problems are formulated mathematically, and uniqueness theorems are proved. Finally, various systems of boundary integral equations over the interface are derived.     

Boundary integral equations as applied to an oscillating bubble near a fluid-fluid interface

A new method is presented to describe the behaviour of an oscillating bubble near a fluid-fluid interface. Such a situation can be found for example in underwater explosions (near muddy bottoms) or in bubbles generated near two (biological) fluids separated by a membrane. The Laplace equation is assumed to be valid in both fluids. The fluids can have different density ratios. A relationship between the two velocity potentials just above and below the fluid-fluid interface can be used to update the co-ordinates of the new interface at the next time step. The boundary integral method is then used for both fluids. With the resulting equations the normal velocities on the interface and the bubble are obtained. Depending on initial distances of the bubble from the fluid-fluid interface and density ratios, the bubbles can develop jets towards or away from this interface. Gravity can be important for bubbles with larger dimensions.     

Boundary integral equation fracture mechanics analysis of plane orthotropic bodies

In this paper, a quick and efficient means of determining stress intensity factors, K _I and K _II, for cracks in generally orthotropic elastic bodies is presented using the numerical boundary integral equation (BIE) method. It is based on the use of quarter-point singular crack-tip elements in the quadratic isoparametric element formulation, similar to those commonly employed in the BIE fracture mechanics studies in isotropic elasticity. Analytical expressions which enable K _Iand K _II to be obtained directly from the BIE computed crack-tip nodal traction, or from the computed nodal displacements, of these elements are derived. Numerical results for a number of test problems are compared with those established in the literature. They are accurate even when only a very modest number of boundary elements are used.     

Generation of protein lattices by fusing proteins with matching rotational symmetry

The self-assembly of supramolecular structures that are ordered on the nanometre scale is a key objective in nanotechnology. DNA and peptide nanotechnologies have produced various two- and three-dimensional structures, but protein molecules have been underexploited in this area of research. Here we show that the genetic fusion of subunits from protein assemblies that have matching rotational symmetry generates species that can self-assemble into well-ordered, pre-determined one- and two-dimensional arrays that are stabilized by extensive intermolecular interactions. This new class of supramolecular structure provides a way to manufacture biomaterials with diverse structural and functional properties.     

Boundary integral equations from Hamilton's principle for surface acoustic waves under periodic metal gratings

The procedure describes the derivation of boundary integral equations for surface acoustic waves propagating under periodic metal strip gratings with piezoelectric films. It takes into account the electrical and mechanical perturbations, including the effects of mass loading caused by the gratings with an arbitrary shape. First, an integral equation is derived with line integrals on the boundaries within one period. This derivation is based on Hamilton's principle and uses Lagrange's method of multipliers to alleviate the continuous conditions of the displacement and the electric potential on the boundaries. Second, boundary integral equations corresponding to each substrate, piezoelectric film, metal strip, and free space region are obtained from the integral equation using the Rayleigh-Ritz method for admissible functions. With this procedure, it is not necessary to make any assumptions for separation of the boundary conditions between two neighboring regions. Consequently, we clarify the theoretical basis for the analytical procedure using boundary integral equations for longitudinal LSAW modes.     

Boundary integral equations of the first kind for planar vector fields in multiply connected domains

Summary The boundary integral equations of the first kind arising from the Poincaré-Weyl representation of a2D vector fieldu(x) in multiply connected domains of ² are investigated using a variational approach in the energy space at the boundary. Existence, uniqueness and stability of the (weak) solution are discussed, and an application to flows past several lifting bodies in aerodynamics is indicated.     

Boundary element analysis of an integral equation for three-dmensional crack problems

In another paper, the authors proposed an integral equation for arbitrary shaped three-dimensional cracks. In the present paper, a discretization of this equation using a tensor formalism is formulated. This approach has the advantage of providing the displacement discontinuity vector in the local basis which varies as a function of the point of the crack surface. This also facilitates the computation of the stress intensity factors along the crack edge. Numerical examples reported for a circular crack and a semi-elliptical surface crack in a cylindrical bar show that one can obtain good results, using few Gaussian points and no singular elements.     

Meshless boundary integral equations with equilibrium satisfaction

A known feature of any mixed interpolation boundary integral equations (BIE)-based methods is that equilibrium is not generally guaranteed in the numerical solution. Here, a complete meshless technique, based on the boundary element-free method (BEFM) with complete equilibrium satisfaction for 2D elastostatic analysis is proposed. The BEFM adopted is a meshless method based on boundary integral equations such as local boundary integral equation (LBIE) method and boundary node method (BNM), differing from them with respect to the integration domain and the approximation scheme.     

Time marching analysis of boundary integral equations in two-dimensional elastodynamics

The present paper is an improvement of the existing time-domain formulation for solution of the boudary-initial problems in two-dimensional elastodynamics by the boundary element method. Two different time-marching schemes are applied to these problems and the boundary integral equations are made free of any singularities.     

Cauchy-type integrals and integral equations with logarithmic singularities

Summary The Gauss-type quadrature methods with a logarithmic weight function can be extended to the evaluation of Cauchy-type integrals and to the solution of Cauchy-type integral equations by reduction of the latter to a linear system of algebraic equations. This system is obtained by applying the integral equation at properly selected collocation points. The poles of the integrand lying in the integration interval were treated as lying outside this interval. The efficiency of the method, both in evaluating integrals and solving integral equations, is exhibited by a numerical example. Finally, an application of the method to a crack problem of plane elasticity is made.     

Boundary integral equation solution of viscous flows with free surfaces

Summary solutions of the biharmonic equation governing steady two-dimensional viscous flow of an incompressible Newtonian fluid are obtained by employing a direct biharmonic boundary integral equation (BBIE) method in which Green's theorem is used to reformulate the differential equation as a pair of coupled integral equations which are applied only on the boundary of the solution domain.An iterative modification of the classical BBIE is presented which is able to solve a large class of (nonlinear) viscous free surface flows for a wide range of surface tensions. The method requires a knowledge of the asymptotic behaviour of the free surface profile in the limiting case of infinite surface tension but this can usually be obtained from a perturbation analysis. Unlike space discretisation techniques such as finite difference or finite element, the BBIE evaluates only boundary information on each iteration. Once the solution is evaluated on the boundary the solution at interior points can easily be obtained.