Single Boundary Integral Equation Research Articles

Galerkin boundary integral formulation for axisymmetric stokes flow

A Galerkin boundary integral formulation for 3D axisymmetric Stokes flow is presented. The singular integrals are evaluated by splitting the complicated Green’s function kernels into a singular term that can be integrated analytically, plus a term for which Gauss quadrature provides sufficient accuracy. As in a previous axisymmetric Laplace implementation, the treatment of the additional on-axis singularity is aided by employing a modified Galerkin weight function, and a similar splitting method is then employed to handle this singularity. The target application of the Stokes algorithm is to model the breakup of one viscous fluid enclosed inside a second, and this two fluid problem can be formulated in terms of a single boundary integral equation along the interface. The Galerkin form for this equation is derived herein.

An interface integral equation method for solving transient heat conduction in multi-medium materials with variable thermal properties

In this paper, a new single interface integral equation method is presented for solving transient heat conduction problems consisting of multi-medium materials with variable thermal properties. Firstly, adopting the fundamental solution for the Laplace equation, the boundary-domain integral equation for transient heat conduction in single medium is established. Then from the established integral equation, a new single interface integral equation is derived for transient heat conduction in general multi-medium functionally graded materials, by making use of the variation feature of the material properties. The derived formulation, which makes up for the lack of boundary integral equation in solving multi-medium problems, has the feature that only a single boundary integral equation is used to solve multi-medium transient heat conduction problems. Compared with conventional multi-domain boundary element method, the newly proposed method is more efficient in computational time, data preparing, and program coding. Based on the implicit backward differentiation scheme, an unconditionally stable and non-oscillatory time marching solution scheme is developed for solving the time-dependent system of differential equations. Numerical examples are given to verify the correctness of the presented method.

Modeling of fluid flow through fractured porous media by a single boundary integral equation

The objective of this work is to provide theoretical materials for modelling two-dimensional fluid flow through an anisotropic porous medium containing intersecting curved fractures. These theoretical developments are suitable for numerical simulations using boundary element method and thus present a great advantage in mesh generation term comparing to finite volume discretization approaches when dealing with high fracture density and infinite configuration. The flow is modelled by Darcy’s law in matrix and Poiseuille’s law in fractures. The mass conservation equations, at a point on the fracture and an intersection point between fractures in the presence of a source or a sink, are derived explicitly. A single boundary integral equation is developed to describe the fluid flow through both porous media and fractures, i.e. the whole domain, which includes particularly the mass balance condition at intersection between fractures. Numerical simulations are performed to show the efficiency of this proposed theoretical formulation for high crack density.

Interface integral BEM for solving multi-medium heat conduction problems

In this paper, a new and simple boundary element method, called interface integral boundary element method (IIBEM), is presented for solving heat conduction problems consisting of multiple media. In the method, the boundary integral equation is derived by a degeneration technique from domain integrals involved in varying heat conductivity problems into interface integrals in multi-medium problems. The main feature of the presented technique is that only a single boundary integral equation is used to solve heat conduction problems with different material properties. The effect of nonhomogeneity between adjacent materials is embodied in the interface integrals including the material property difference between the two adjacent materials. Comparing with conventional multi-domain boundary integral equation techniques, the presented method is more efficient in computational time, data preparing, and program coding. Numerical examples are given to verify the correctness of the presented technique.

Three-Dimensional Piezoelectric Boundary Element Method

The boundary element method is applied to problems of three-dimensional linear piezoelectricity. The continuum equations for conservation of linear momentum and charge are combined into one governing equation for piezoelectricity. A single boundary integral equation is developed from this combined e eld equation and Green’ s solution for a piezoelectric medium. Green’ s function and its derivatives are derived using the radon transform, and the resulting solution is represented by a line integral that is evaluated numerically using standard Gaussian quadrature. The boundary integral equation is discretized using eight-node isoparametric quadratic elements, resulting in a matrixsystem ofequations. Thesolution of theboundary problem forpiezoelectricmaterialsconsistsof elastic displacements, tractions, electric potentials, and normal charge e ux densities. The complete e eld solutions can be obtained once all boundary values have been determined. The accuracy of this linear piezoelectric boundary element method is illustrated with two numerical examples. The e rst involves a unit cube of material with an applied mechanical load. Thesecond exampleconsistsof a spherical holein an ine nitepiezoelectricbody loaded by a unit traction on itsboundary. Comparisonsare made to the analytical solution for the cube and an axisymmetric e nite element solution for the spherical hole. The boundary element method is shown to be an accurate solution procedure for general three-dimensional linear piezoelectric material problems. I. Introduction T HEbrothersPierreandJacquesCuriediscoveredpiezoelectricity in 1880 during their investigation of crystals. A coupling occurs between the electrical and mechanical e elds in these materials, such that application of an external mechanical loading induces achangeinpolarizationproportionaltoit.Conversely,applicationof an electric e eld produces a proportional material deformation. This phenomenon was discovered in ceramics in the 1940s, and the subsequent manufactured piezoceramic materialshave higher coupling properties than their naturally occurring predecessors. This strong piezoelectric effect has made these materials an important part of the emerging technologies of smart materials and structures. Smart structures are those that incorporate actuators and sensors that are integrated into the structure and have structural functionality. The sensors and actuators are used to ine uence the structure’ s states

Electromagnetic scattering from bounded or infinite subsurface bodies

The direct scattering problem from a stack of two rough interfaces or from a buried object with a rough interface above it is addressed. The two boundaries are separating three homogeneous and isotropic media. The profiles are assumed to be invariant in one (and the same) direction, and the probing field is time harmonic. The problem is solved with a rigorous integral method leading to a single boundary integral equation. This equation is converted into a linear system of equations, thanks to a boundary finite element method. To improve the accuracy, especially when the two boundaries are very close to each other, the singularity of the kernel is isolated and its integration is performed analytically. As shown through several examples, this versatile tool is well suited for the investigation of physical phenomena, especially the competition between surface and subsurface scattering.

Elastodynamic Thin Plate Bending Analysis by Boundary Element Method with Laplace Transform

The boundary element method(BEM) is developed for the dynamic analysis of thin elastic plate bending problems with arbitrary boundary conditions. The formulation employs Laplace-transform technique, where the boundary integral equations are obtained on the Laplace transformed domain with the fundamental solutions derived from Kelvin's functions. The accuracy of the numerical results mainly depends on those of numerical estimation of the singular integral derived from the static term of the fundamental solutions. In the present paper, an BEM formulation based on a single boundary integral equation of the deflection, which employes a source point on both the boundary and outer region, is discussed in detail. A non-singular boundary integral equation is introduced on the transformed domain, which is obtained by superposition of the analyzed field and the referenced field with a uniform gradient of deflection. Numerical results obtained by the proposed method are compared with the analytical solutions and the other numerical solutions by means of several numerical examples. These examples also serve to illustrate the use of the proposed method.