On the numerical solution of the Dirichlet initial boundary-value problem for the heat equation in the case of a torus

The initial-boundary-value problem for the heat equation in the case of a toroidal surface with Dirichlet boundary conditions is considered. This problem is reduced to a sequence of elleptic boundary-value problems by a Laguerre transformation. The special integral representation leads to boundary-integral equations of the first kind and the toroidal surface gives one-dimensional integral equations with a logarithmic singularity. The numerical solution is realized by a trigonometric quadrature method in cases of open or closed smooth boundaries. The results of some numerical experiments are presented.     

Dynamic SIF of interface crack between two dissimilar viscoelastic bodies under impact loading*

In this paper, the transient dynamic stress intensity factor (SIF) is determined for an interface crack between two dissimilar half-infinite isotropic viscoelastic bodies under impact loading. An anti-plane step loading is assumed to act suddenly on the surface of interface crack of finite length. The stress field incurred near the crack tip is analyzed. The integral transformation method and singular integral equation approach are used to get the solution. By virtue of the integral transformation method, the viscoelastic mixed boundary problem is reduced to a set of dual integral equations of crack open displacement function in the transformation domain. The dual integral equations can be further transformed into the first kind of Cauchy-type singular integral equation (SIE) by introduction of crack dislocation density function. A piecewise continuous function approach is adopted to get the numerical solution of SIE. Finally, numerical inverse integral transformation is performed and the dynamic SIF in transformation domain is recovered to that in time domain. The dynamic SIF during a small time-interval is evaluated, and the effects of the viscoelastic material parameters on dynamic SIF are analyzed.     

The mixed-type integral equations for transient elastodynamic plane crack problems

In the present paper, by use of the boundary integral equation method and the techniques of Green fundamental solution and singularity analysis, the dynamic infinite plane crack problem is investigated. For the first time, the problem is reduced to solving a system of mixed-typed integral equations in Laplace transform domain. The equations consist of ordinary boundary integral equations along the outer boundary and Cauchy singular integral equations along the crack line. The equations obtained are strictly proved to be equivalent with the dual integral equations obtained by Sih in the special case of dynamic Griffith crack problem. The mixed-type integral equations can be solved by combining the numerical method of singular integral equation with the ordinary boundary element method. Further use the numerical method for Laplace transform, several typical examples are calculated and their dynamic stress intensity factors are obtained. The results show that the method proposed is successful and can be used to solve more complicated problems.     

Trefftz Methods for Time Dependent Partial Differential Equations

In this paper we present a mesh-free approach to numerically solving a class of second order time dependent partial differential equations which include equations of parabolic, hyperbolic and parabolic-hyperbolic types. For numerical purposes, a variety of transformations is used to convert these equations to standard reaction-diffusion and wave equation forms. To solve initial boundary value problems for these equations, the time dependence is removed by either the Laplace or the Laguerre transform or time differencing, which converts the problem into one of solving a sequence of boundary value problems for inhomogeneous modified Helmholtz equations. These boundary value problems are then solved by a combination of the method of particular solutions and Trefftz methods. To do this, a variety of techniques is proposed for numerically computing a particular solution for the inhomogeneous modified Helmholtz equation. Here, we focus on the Dual Reciprocity Method where the source term is approximated by radial basis functions, polynomial or trigonometric functions. Analytic particular solutions are presented for each of these approximations. The Trefftz method is then used to solve the resulting homogenous equation obtained after the approximate particular solution is subtracted off. Two types of Trefftz bases are considered, F-Trefftz bases based on the fundamental solution of the modified Helmholtz equation, and T-Trefftz bases based on separation of variables solutions. Various techniques for satisfying the boundary conditions are considered, and a discussion is given of techniques for mitigating the ill-conditioning of the resulting linear systems. Finally, some numerical results are presented illustrating the accuracy and efficacy of this methodology.     

A boundary integral technique for the numerical modelling of the air flow for the Aaberg exhaust system

A boundary integral technique has been developed for the numerical simulation of the air flow for the Aaberg exhaust system. For the steady, ideal, irrotational air flow induced by a jet, the air velocity is an analytical function. The solution of the problem is formulated in the form of a boundary integral equation by seeking the solution of a mixed boundary-value problem of an analytical function based on the Riemann–Hilbert technique. The boundary integral equation is numerically solved by converting it into a system of linear algebraic equations, which are solved by the process of the Gaussian elimination. The air velocity vector at any point in the solution domain is then computed from the air velocity on the boundary of the solution domain.     

ANALYSIS OF THREE-DIMENSIONAL TRANSIENT ACOUSTIC WAVE PROPAGATION USING THE BOUNDARY INTEGRAL EQUATION METHOD

This paper describes a boundary integral equation (boundary element) method for the solution of a variety of transient acoustic problems. The spatial and temporal discretization employs quadratic isoparametric elements with high-order Gauss quadrature, and the ensuing equations are implicit. The implicit formulation both eliminates the instabilities reported in explicit treatments, and permits a freedom of choice of timestep which can reduce costs dramatically. The accuracy of the approach is demonstrated by comparison with the analytical solution for a sphere. Results for more demanding sphere–cone–sphere geometries extending to seven wavelengths long are presented, and compared to those obtained from a related frequency domain approach.     

Simply supported plates by the boundary integral equation method

Plates governed by Kirchhoff's equation have been analysed by the boundary integral equation method using the fundamental solution of the biharmonic equation. In the case of supported plates, the boundary conditions permit the uncoupling of the field equation into two harmonic equations that originate, due to the nature of the fundamental solution, easier integration kernels and a simpler system of equations. The calculation of bending and twisting moments and transverse shear force can be formed, combining derivatives of the integral equation which defines the expression of the deflection on any point of the plate. The uncoupling of the biharmonic equation into two Poisson's equations involves the discretization of the domain of the studied problems. Nevertheless, the unknown quantity of the problem does not appear in the domain integrations for which a refined discretization is unnecessary. In the paper, however, a numerical alternative is considered to express the domain integral by means of boundary integrals. In this way, we need only discretize the boundary of the plate, making it necessary to solve a supplementary system of equations in order to calculate the coefficients of the approximation carried out.     

Numerical solution of a Cauchy problem for Laplace equation in 3-dimensional domains by integral equations

A numerical method based on integral equations is proposed and investigated for the Cauchy problem for the Laplace equation in 3-dimensional smooth bounded doubly connected domains. To numerically reconstruct a harmonic function from knowledge of the function and its normal derivative on the outer of two closed boundary surfaces, the harmonic function is represented as a single-layer potential. Matching this representation against the given data, a system of boundary integral equations is obtained to be solved for two unknown densities. This system is rewritten over the unit sphere under the assumption that each of the two boundary surfaces can be mapped smoothly and one-to-one to the unit sphere. For the discretization of this system, Weinert’s method (PhD, Göttingen, 1990) is employed, which generates a Galerkin type procedure for the numerical solution, and the densities in the system of integral equations are expressed in terms of spherical harmonics. Tikhonov regularization is incorporated, and numerical results are included showing the efficiency of the proposed procedure.     

Three-dimensional structural vibration analysis by the Dual Reciprocity BEM

The dual reciprocity boundary element method (DR/BEM) is employed for the analysis of free and forced vibrations of three-dimensional elastic solids. Use of the elastostatic fundamental solution in the integral formulation of elastodynamics creates an inertial volume integral in addition to the boundary ones. This volume integral is transformed into a surface integral by invoking the reciprocal theorem. A general analytical method is described for the closed form determination of the particular solutions of the displacement and traction tensors corresponding to any radial basis function employed in the transformation process. The simple but effective 1+r radial basis function is used in the applications of this paper. Quadratic continuous and discontinuous 9-noded boundary elements are used in the analysis. Free vibrations are studied by solving the corresponding eigenvalue problem iteratively. Harmonic forced vibration problems are solved directly in the frequency domain. Transient forced vibration problems are solved by integrating the equations of motion stepwise with the aid of various algorithms. Interior collection points are also used for assessing the accuracy of the method. Two numerical examples involving free and forced vibrations of a sphere and a cube are presented in detail.     

Numerical operational methods for time-dependent linear problems

A general and systematic discussion on the use of the operational method of Laplace transform for numerically solving complex time-dependent linear problems is presented. Application of Laplace transform with respect to time on the governing differential equations as well as the boundary and initial conditions of the problem reduces it to one independent of time, which is solved in the transform domain by any convenient numerical technique, such as the finite element method, the finite difference method or the boundary integral equation method. Finally, the time domain solution is obtained by a numerical inversion of the transformed solution. Eight existing methods of numerical inversion of the Laplace transform are systematically discussed with respect to their use, range of applicability, accuracy and computational efficiency on the basis of some framework vibration problems. Other applications of the Laplace transform method in conjunction with the finite element method or the boundary integral equation method in the areas of earthquake dynamic response of frameworks, thermaliy induced beam vibrations, forced vibrations of cylindrical shells, dynamic stress concentrations around holes in plates and viscoelastic stress analysis are also briefly described to demonstrate the generality and advantages of the method against other known methods.     

An alternative method for degenerate scale problems in boundary element methods for the two-dimensional Laplace equation

The boundary integral equation approach has been shown to suffer a nonunique solution when the geometry is equal to a degenerate scale for a potential problem. In this paper, the degenerate scale problem in boundary element method for the two-dimensional Laplace equation is analytically studied in the continuous system by using degenerate kernels and Fourier series instead of using discrete system using circulants [Engng Anal. Bound. Elem. 25 (2001) 819]. For circular and multiply-connected domain problems, the rank-deficiency problem of the degenerate scale is solved by using the combined Helmholtz exterior integral equation formulation (CHEEF) concept. An additional constraint by collocating a point outside the domain is added to promote the rank of influence matrix. Two examples are shown to demonstrate the numerical instability using the singular integral equation for circular and annular domain problems. The CHEEF concept is successfully applied to overcome the degenerate scale and the error is suppressed in the numerical experiment.     

Analytical study and numerical experiments for degenerate scale problems in boundary element method using degenerate kernels and circulants

For a potential problem, the boundary integral equation approach has been shown to yield a nonunique solution when the geometry is equal to a degenerate scale. In this paper, the degenerate scale problem in boundary element method (BEM) is analytically studied using the degenerate kernels and circulants. For the circular domain problem, the singular problem of the degenerate scale with radius one can be overcome by using the hypersingular formulation instead of the singular formulation. A simple example is shown to demonstrate the failure using the singular integral equations. To deal with the problem with a degenerate scale, a constant term is added to the fundamental solution to obtain the unique solution and another numerical example with an annular region is also considered.     

Boundary element analysis of dynamic coupled thermoelasticity problems

Some possibility of numerical analysis of coupled dynamic problems of linear elastic heat conductors on classical thermoelasticity theory by using the boundary element method is shown in this paper. The boundary integral equation formulation and its numerical implementation of the two-dimensional problem are developed in the manner by the newly derived fundamental solution for the coupled equations of elliptic type in the transformed space and the numerical inversion of Laplace transformation. The boundary element unsteady solutions of the first and second Danilovskaya problems and the Sternberg and Chakravorty problem in the half-space are demonstrated through comparison with the existing solutions.     

Radial integration boundary integral and integro-differential equation methods for two-dimensional heat conduction problems with variable coefficients

This paper presents new formulations of the radial integration boundary integral equation (RIBIE) and the radial integration boundary integro-differential equation (RIBIDE) methods for the numerical solution of two-dimensional heat conduction problems with variable coefficients. The methods use a specially constructed parametrix (Levi function) to reduce the boundary-value problem (BVP) to a boundary-domain integral equation (BDIE) or boundary-domain integro-differential equation (BDIDE). The radial integration method is then employed to convert the domain integrals arising in both BDIE and BDIDE methods into equivalent boundary integrals. The resulting formulations lead to pure boundary integral and integro-differential equations with no domain integrals. Numerical examples are presented for several simple problems, for which exact solutions are available, to demonstrate the efficiency of the proposed methods.     

Application of the boundary element method to three-dimensional potential problems in heterogeneous media

The present work discusses a solution procedure for heterogeneous media three-dimensional potential problems, involving nonlinear boundary conditions. The problem is represented mathematically by the Laplace equation and the adopted numerical technique is the boundary element method (BEM), here using velocity correcting fields to simulate the conductivity variation of the domain. The integral equation is discretized using surface elements for the boundary integrals and cells, for the domain integrals. The adopted strategy subdivides the discretized equations in two systems: the principal one involves the calculation of the potential in all boundary nodes and the secondary which determines the correcting field of the directional derivatives of the potential in all points. Comparisons with other numerical and analytical solutions are presented for some examples.     

An immersed finite element method with integral equation correction

We propose a robust immersed finite element method in which an integral equation formulation is used to enforce essential boundary conditions. The solution of a boundary value problem is expressed as the superposition of a finite element solution and an integral equation solution. For computing the finite element solution, the physical domain is embedded into a slightly larger Cartesian (box‐shaped) domain and is discretized using a block‐structured mesh. The defect in the essential boundary conditions, which occurs along the physical domain boundaries, is subsequently corrected with an integral equation method. In order to facilitate the mapping between the finite element and integral equation solutions, the physical domain boundary is represented with a signed distance function on the block‐structured mesh. As a result, only a boundary mesh of the physical domain is necessary and no domain mesh needs to be generated, except for the non‐boundary‐conforming block‐structured mesh. The overall approach is first presented for the Poisson equation and then generalized to incompressible viscous flow equations. As an example of fluid–structure coupling, the settling of a heavy rigid particle in a closed tank is considered. Copyright © 2010 John Wiley & Sons, Ltd.     

Large deformation of shear-deformable plates by the boundary-element method

Boundary-integral equations for large deformation of shear-deformable plates are presented. Two different methods are used to calculate the derivatives of the nonlinear terms in the domain integral. The first approach requires the evaluation of a hypersingular domain integral. The second approach avoids the calculation of a hypersingular integral by utilizing radial basis functions to approximate the integrand. Quadratic isoparametric boundary-elements are used to discretise the boundary, and constant cell elements are used to discretise the domain. For the solution of a nonlinear problem four methods are presented. They include: total incremental method, cumulative-load incremental method, Euler method and nonlinear system method. Several examples are presented and comparisons with analytical results and other numerical results are made to demonstrate the accuracy of the proposed methods.     

A meshless solution to two-dimensional convection–diffusion problems

An integral equation domain decomposition method has been implemented in a meshless fashion. The method exploits the advantage of placing the source point always in the centre of circular sub-domains in order to avoid singular or near-singular integrals. Three equations for two-dimensional (2D) or four for three-dimensional (3D) potential problems are required at each node. The first equation is the integral equation arising from the application of the Green's identities and the remaining equations are the derivatives of the first equation in respect to space coordinates. Radial basis function interpolation is applied in order to obtain the values of the field variable and partial derivatives at the boundary of the circular sub-domains, providing this way the boundary conditions for solution of the integral equations at the nodes (centres of circles). Dual reciprocity method (DRM) has been applied to convert the domain integrals into boundary integrals, though the approach is general and can be applied without the DRM. The accuracy and robustness of the method has been tested on a convection–diffusion problem. The results obtained using the current approach have been compared with previously reported results obtained using the finite element method (FEM), and the DRM multi-domain approach (DRM-MD) showing similar level of accuracy.     

On integral equation approaches to the mechanics of fibre-crack interaction

The paper examines the problem of a penny-shaped crack which is formed by the development of a crack in both the fibre and the matrix of a composite consisting of an isolated elastic fibre located in an elastic matrix of infinite extent. The composite region is subjected to a uniform strain field in the direction of the fibre. The paper presents two integral-equation based approaches for the analysis of the problem. The first approach considers the formulation of the complete integral equations governing the associated elasticity problem for a two material region. The second approach considers the boundary integral equation formulation of the problem. Both methods entail the numerical solution of the governing integral equations. The solutions to these integral equations are used to evaluate the stress intensity factor at the boundary of the penny-shaped crack.     

Integral equation formulations with the primitive variables for incompressible viscous fluid flow problems

New integral equation formulations for steady and unsteady flow problems of an incompressible viscous fluid are presented. The so-called direct approach in which the velocity vector and the pressure are inclued as unknowns is employed in this paper. The nonlinear boundary value, and the initial-boundary value problems described with the Navier-Stokes equations are transformed into integral equations by the method of weighted residuals. Fundamental solutions of the Stokes approximate equations are used as the weight function. The fundamental solution tensors are presented for the steady-state and unsteady-state problems. For the unsteady-state problem, we derive not only the time-dependent fundamental solution tensor but also the one using the finite difference approximation for the time derivative. A numerical example of the two-dimensional driven cavity flow is given to show the validity and effectiveness of the method.