Simple Boundary Element Method Research Articles

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.

The simple boundary element method for multiple cracks in functionally graded media governed by potential theory: a three‐dimensional Galerkin approach

AbstractThe simple boundary element method consists of recycling existing codes for homogeneous media to solve problems in non‐homogeneous media while maintaining a purely boundary‐only formulation. Within this scope, this paper presents a ‘simple’ Galerkin boundary element method for multiple cracks in problems governed by potential theory in functionally graded media. Steady‐state heat conduction is investigated for thermal conductivity varying either parabolically, exponentially, or trigonometrically in one or more co‐ordinates. A three‐dimensional implementation which merges the dual boundary integral equation technique with the Galerkin approach is presented. Special emphasis is given to the treatment of crack surfaces and boundary conditions. The test examples simulated with the present method are verified with finite element results using graded finite elements. The numerical examples demonstrate the accuracy and efficiency of the present method especially when multiple interacting cracks are involved. Copyright © 2005 John Wiley & Sons, Ltd.

A Simple Galerkin Boundary Element Method for Three-Dimensional Crack Problems in Functionally Graded Materials

This paper presents a Galerkin boundary element method for solving crack problems governed by potential theory in nonhomogeneous media. In the simple boundary element method, the nonhomogeneous problem is reduced to a homogeneous problem using variable transformation. Cracks in heat conduction problem in functionally graded materials are investigated. The thermal conductivity varies parabolically in one or more coordinates. A three dimensional boundary element implementation using the Galerkin approach is presented. A numerical example demonstrates the eáciency of the method. The result of the test example is in agreement with ßnite element simulation results.

NUMERICAL PREDICTIONS OF SOUND PROPAGATION FROM A CUTTING OVER A ROAD-SIDE NOISE BARRIER

A boundary integral equation is described for the prediction of acoustic propagation from a monofrequency coherent line source in a cutting with impedance boundary conditions onto surrounding flat impedance ground. The problem is stated as a boundary value problem for the Helmholtz equation and is subsequently reformulated as a system of boundary integral equations via Green's theorem. The numerical solution of the coupled boundary integral equations by a simple boundary element method is then described. Predictions of A-weighted insertion losses for a traffic noise spectrum are made illustrating the effects of depth of the cutting and the profile of the associated noise barrier.

A simple boundary element method for problems of potential in non‐homogeneous media

AbstractA simple boundary element method for solving potential problems in non‐homogeneous media is presented. A physical parameter (e.g. heat conductivity, permeability, permittivity, resistivity, magnetic permeability) has a spatial distribution that varies with one or more co‐ordinates. For certain classes of material variations the non‐homogeneous problem can be transformed to known homogeneous problems such as those governed by the Laplace, Helmholtz and modified Helmholtz equations. A three‐dimensional Galerkin boundary element method implementation is presented for these cases. However, the present development is not restricted to Galerkin schemes and can be readily extended to other boundary integral methods such as standard collocation. A few test examples are given to verify the proposed formulation. The paper is supplemented by an Appendix, which presents an ABAQUS user‐subroutine for graded finite elements. The results from the finite element simulations are used for comparison with the present boundary element solutions. Copyright © 2004 John Wiley & Sons, Ltd.

The simple boundary element method for transient heat conduction in functionally graded materials

This paper presents a “simple” boundary element method for transient heat conduction in functionally graded materials, which leads to a boundary-only formulation without any domain discretization. For a broad range of functional material variation (quadratic, exponential and trigonometric) of thermal conductivity and specific heat, the non-homogeneous problem can be transformed into the standard homogeneous diffusion problem. A three-dimensional boundary element implementation, using the Laplace transform approach and the Galerkin approximation, is presented. The time dependence is restored by numerically inverting the Laplace transform by means of the Stehfest algorithm. A number of numerical examples demonstrate the efficiency of the method. The results of the test examples are in excellent agreement with analytical solutions and finite element simulation results.

The simple boundary element method for transient heat conduction in functionally graded materials

This paper presents a “simple” boundary element method for transient heat conduction in functionally graded materials , which leads to a boundary-only formulation without any domain discretization . For a broad range of functional material variation (quadratic, exponential and trigonometric) of thermal conductivity and specific heat, the non-homogeneous problem can be transformed into the standard homogeneous diffusion problem . A three-dimensional boundary element implementation, using the Laplace transform approach and the Galerkin approximation, is presented. The time dependence is restored by numerically inverting the Laplace transform by means of the Stehfest algorithm. A number of numerical examples demonstrate the efficiency of the method. The results of the test examples are in excellent agreement with analytical solutions and finite element simulation results.

A Boundary Element Method Without Internal Cells for Two-Dimensional and Three-Dimensional Elastoplastic Problems

In this paper, a new and simple boundary element method without internal cells is presented for the analysis of elastoplastic problems, based on an effective transformation technique from domain integrals to boundary integrals. The strong singularities appearing in internal stress integral equations are removed by transforming the domain integrals to the boundary. Other weakly singular domain integrals are transformed to the boundary by approximating the initial stresses with radial basis functions combined with polynomials in global coordinates. Three numerical examples are presented to demonstrate the validity and effectiveness of the proposed method.

A complex variable boundary element method for elliptic partial differential equations in a multiple-connected region

A simple boundary element method based on the Cauchy integral formulae is proposed for the numerical solution of a class of boundary value problems involving a system of elliptic partial differential equations in a multiple-connected region of infinite extent. It can be easily and efficiently implemented on the computer.

NOISE PROPAGATION FROM A CUTTING OF ARBITRARY CROSS-SECTION AND IMPEDANCE

A boundary integral equation is described for the prediction of acoustic propagation from a monofrequency coherent line source in a cutting with impedance boundary conditions onto surrounding flat impedance ground. The problem is stated as a boundary value problem for the Helmholtz equation and is subsequently reformulated as a system of boundary integral equations via Green's theorem. It is shown that the integral equation formulation has a unique solution at all wavenumbers. The numerical solution of the coupled boundary integral equations by a simple boundary element method is then described. The convergence of the numerical scheme is demonstrated experimentally. Predictions of A-weighted excess attenuation for a traffic noise spectrum are made illustrating the effects of varying the depth of the cutting and the absorbency of the surrounding ground surface.

TRANSIENT SOUND FIELDS FROM IMPACTED PLATES: COMPARISON BETWEEN NUMERICAL SIMULATIONS AND EXPERIMENTS

A comparison between measured and simulated transient acoustic fields generated by a thin rectangular impacted steel plate is reported. The plate vibration is modelled in the time domain using a finite difference scheme of second-order in time and fourth-order in space (2-4 scheme) and the radiated acoustic pressure is obtained by solving numerically the Rayleigh integral equation by using a simple Boundary Element Method (trapezoidal rule). The simulated results are evaluated in the form of the spatial pressure distribution at times short-after the impact excitation. The measurements are performed using an optical method: pulsed two-reference-beam holographic interferometry. Results in the form of 2-D projections of the acoustic pressure field are obtained and compared with the simulations. A high degree of similarity is found between the measured and simulated sound fields.

Sound radiation by baffled and constrained plates

The coupling between the acoustic waves in a half-space above a thin plate and the vibration of the plate is studied. For the sake of simplicity, the geometry is such that the problem considered is two-dimensional. The plate equation is then reduced to a beam equation. Two different problems are considered: the baffled plate, which is free to vibrate within a finite region but is rigidly clamped outside this region; and the constrained plate, which is free to vibrate both within and outside the finite region but is constrained (clamped) on the boundary between them. In each case the plate is excited by a harmonic force located in the finite region and then radiates an acoustic wave. Four topics are considered. First, a formulation and a numerical solution, obtained by using a simple boundary element method (the collocation method), of a boundary integral equation for the baffled plate case is presented. Secondly, an analytical method for the constrained case is developed. Comparison of the plate displacement in the two cases shows that the displacement of the baffled plate can be approximated by that of the constrained plate, restricted to the domain limited by the constraints. Then, formulae for the far acoustic field behaviour of the constrained plate and for the exact and approximate (deduced from the constrained case) solutions for the baffled plate are presented. Numerical examples show that, in the case treated here (a steel plate in contact with water), the exact solution and the approximate one are in agreement.

Acoustic Scattering By a Near-Surface Obstacle in a Rigid Porous Medium

A boundary integral equation method is described for the prediction of the acoustic field due to a point source in a homogeneous quiescent atmosphere, above a homogeneous rigid porous half-space containing a smooth rigid obstacle. The problem is initially stated as a boundary value problem and is subsequently reformulated as a boundary integral equation via Green's second theorem. It is shown that the boundary value problem and the boundary integral equation are equivalent. The numerical solution of the boundary integral equation by a simple boundary element method is then described. The solution method, which reduces to a system of linear equations with a block circulant coefficient matrix, is applicable to any obstacle which is axisymmetric about a vertical axis. The numerical solution shows good agreement with a known analytical solution—that of scattering of an incident plane wave by a rigid sphere in an infinite homogeneous medium, and with solutions for scattering due to a rigidly backed layer.

Simple boundary element method for three-dimensional magnetostatic problems

For numerical solution of three-dimensional magnetostatic problems the applications of finite-difference (FDM) and finite-element (FEM) methods require long calculating time and large storage capacity. In some cases these requirements can be reduced by the use of boundary element methods (BEM). A simple boundary element solution, and its application to the calculation of magnetization and stray fields of magnetic bodies, is described. The results are compared with experimental stray-field measurements performed using a vibrating pick-up loop magnetometer with high geometrical resolution [14].