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This paper presents a boundary element formulation for transient convection‐diffusion problems employing the fundamental solution of the corresponding steady‐state equation with constant coefficients and a dual reciprocity approximation. The formulation allows the mathematical problem to be described in terms of boundary values only. Numerical results show that the BEM does not present oscillations or damping of the wave front as appear in other numerical techniques.

The purpose of this paper is to describe an extension of the boundary element method (BEM) and the dual reciprocity boundary element method (DRBEM) formulations developed for one- and two-dimensional steady-state problems, to analyse transient convection–diffusion problems associated with first-order chemical reaction.

The mathematical modelling has used a dual reciprocity approximation to transform the domain integrals arising in the transient equation into equivalent boundary integrals. The integral representation formula for the corresponding problem is obtained from the Green’s second identity, using the fundamental solution of the corresponding steady-state equation with constant coefficients. The finite difference method is used to simulate the time evolution procedure for solving the resulting system of equations. Three different radial basis functions have been successfully implemented to increase the accuracy of the solution and improving the rate of convergence.

The numerical results obtained demonstrate the excellent agreement with the analytical solutions to establish the validity of the proposed approach and to confirm its efficiency.

Finally, the proposed BEM and DRBEM numerical solutions have not displayed any artificial diffusion, oscillatory behaviour or damping of the wave front, as appears in other different numerical methods.

The axisymmetric steady‐state convective‐diffusive thermal field problem associated with direct‐chill, semi‐continuously cast billets has been solved using the dual reciprocity boundary element method. The solution is based on a formulation which incorporates the one‐phase physical model, Laplace equation fundamental solution weighting, and scaled augmented thin plate splines for transforming the domain integrals into a finite series of boundary integrals. Realistic non‐linear boundary conditions and temperature variation of all material properties are included. The solution is verified by comparison with the results of the classical finite volume method. Results for a 0.500[m] diameter Al 4.5 per cent Cu alloy billet at typical casting conditions are given.

A numerical formulation for solving homogeneous anisotropic heat conduction problems based on the use of an isotropic fundamental solution is presented in detail. The analysis is carried out assuming a generic position of the coordinate axes, which may not coincide with the principal directions of orthotropy of the material. The two primary integral equations of the method are derived from the governing differential equation of the problem. Then, the numerical procedure is developed by rewriting the internal degrees of freedom that arise from the domain discretization in terms of the boundary nodes and solving the resulting system of linear equations for the boundary unknowns only. Special attention is given to the differentiation of singular integrals which yields additional terms as well as to the evaluation of the resulting Cauchy principal value integral. The main feature of the proposed formulation is its generality, which makes possible its direct extension to solve the problem of three‐dimensional heat conduction in anisotropic media and, foremost, to three‐dimensional orthotropic and anisotropic elasticity or elastoplasticity.

This paper gives a review of the finite element techniques (FE) applied in the area of material processing. The latest trends in metal forming, non‐metal forming and powder metallurgy are briefly discussed. The range of applications of finite elements on the subjects is extremely wide and cannot be presented in a single paper; therefore the aim of the paper is to give FE users only an encyclopaedic view of the different possibilities that exist today in the various fields mentioned above. An appendix included at the end of the paper presents a bibliography on finite element applications in material processing for the last five years, and more than 1100 references are listed.

A novel numerical formulation of the two‐phase macroscopic balance equations governing the flow field in incompressible porous media is presented. The numerical model makes use of the weighted average flux method and total variation diminishing flux limiting techniques, and results in a second‐order accurate scheme. A shock tube study was carried out to examine the interaction of a normal shock wave with a thin layer of porous, incompressible cellular ceramic foam. Particular attention was paid to the transmitted and reflected flow fields. The numerical model was used to simulate the experimental test cases, and their results compared with a view to validate the numerical model. A phenomenological model is proposed to explain the behaviour of the transmitted flow field.

The purpose of this paper is to present an inverse analysis procedure based on a coupled numerical formulation through which the coefficients describing non‐linear thermal properties of blood perfusion may be identified.

The coupled numerical technique involves a combination of the dual reciprocity boundary element method (DRBEM) and a genetic algorithm (GA) for the solution of the Pennes bioheat equation. Both linear and quadratic temperature‐dependent variations are considered for the blood perfusion.

The proposed DRBEM formulation requires no internal discretisation and, in this case, no internal nodes either, apart from those defining the interface tissue/tumour. It is seen that the skin temperature variation changes as the blood perfusion increases, and in certain cases flat or nearly flat curves are produced. The proposed algorithm has difficulty to identify the perfusion parameters in these cases, although a more advanced genetic algorithm may provide improved results.

The coupled technique allows accurate inverse solutions of the Pennes bioheat equation for quantitative diagnostics on the physiological conditions of biological bodies and for optimisation of hyperthermia for cancer therapy.

The proposed technique can be used to guide hyperthermia cancer treatment, which normally involves heating tissue to 42‐43°C. When heated up to this range of temperatures, the blood flow in normal tissues, e.g. skin and muscle, increases significantly, while blood flow in the tumour zone decreases. Therefore, the consideration of temperature‐dependent blood perfusion in this case is not only essential for the correct modelling of the problem, but also should provide larger skin temperature variations, making the identification problem easier.

This paper presents a boundary element formulation for transient uncoupled thermoelasticity employing a multiple reciprocity method (MRM) approximation for calculating the thermal stress field. An intermediate step, which involves curve fitting, is necessary for processing the results of the heat conduction analysis into a form suitable for the MRM. Numerical results are included which validate the present technique.

The purpose of this paper is to describe further developments on a novel formulation of the boundary element method (BEM) for inelastic problems using the dual reciprocity method (DRM) but using object-oriented programming (OOP). As the BEM formulation generates a domain integral due to the inelastic stresses, the DRM is employed in a modified form using polyharmonic spline approximating functions with polynomial augmentation. These approximating functions produced accurate results in BEM applications for a range of problems tested, and have been shown to converge linearly as the order of the function increases.

A programming class named DRMOOP, written in C++ language and based on OOP, was developed in this research. With such programming, general matrix equations can be easily established and applied to different inelastic problems. A vector that accounts for the influence of the inelastic strains on the displacements and boundary forces is obtained.

The C++ DRMOOP class has been implemented and tested with the BEM formulation applied to classical elastoplastic problem and the results are reported at the end of the paper.

An object-oriented technology and the C++ DRMOOP class applied to elastoplastic problems.

The paper gives the description of boundary element method (BEM) with subdomains for the solution of convection—diffusion equations with variable coefficients and Burgers’ equations. At first, the whole domain is discretized into K subdomains, in which linearization of equations by representing convective velocity by the sum of constant and variable parts is carried out. Then using fundamental solutions for convection—diffusion linear equations for each subdomain the boundary integral equation (in which the part of the convective term with the constant convective velocity is not included into the pseudo‐body force) is formulated. Only part of the convective term with the variable velocity, which is, as a rule, more than one order less than convective velocity constant part contribution, is left as the pseudo‐source. On the one hand, this does not disturb the numerical BEM—algorithm stability and, on the other hand, this leads to significant improvement in the accuracy of solution. The global matrix, similar to the case of finite element method, has block band structure whereas its width depends only on the numeration order of nodes and subdomains. It is noted, that in comparison with the direct boundary element method the number of global matrix non‐zero elements is not proportional to the square of the number of nodes, but only to the total number of nodal points. This allows us to use the BEM for the solution of problems with very fine space discretization. The proposed BEM with subdomains technique has been used for the numerical solution of one‐dimensional linear steady‐state convective—diffusion problem with variable coefficients and one‐dimensional non‐linear Burgers’ equation for which exact analytical solutions are available. It made it possible to find out the BEM correctness according to both time and space. High precision of the numerical method is noted. The good point of the BEM is the high iteration convergence, which is disturbed neither by high Reynolds numbers nor by the presence of negative velocity zones.