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The paper reports a mathematical model governing unsteady coupled moisture and heat energy transport through a green roof, e.g. the canopy (leaf cover), the soil and the structural support. The mathematical model that governs the transport phenomena in the canopy is represented by a system of nonlinear ordinary differential equations (ODEs) for the unknown temperature of the plants, and the unknown temperature and moisture content of the canopy air. A set of nonlinear partial differential equations (PDEs) describe the heat and moisture transport through the soil and structural support. Continuous field functions such as temperature and relative humidity, are considered as the driving potentials. A finite difference numerical model is used to solve the ODEs and the boundary element numerical model is used to discretize the PDEs.

The Boundary Element Method (BEM) has now become a well established numerical technique with a number of computer programmes to its credit oriented towards industrial applications. They are reliable and robust tools which stress its unique features versus finite elements, ie reduced dimensionality which makes it easier to interface it to CAD codes; better accuracy, the elegant way in which moving boundaries are dealt with and the possibility of taking into consideration infinite domains without the need of introducing artificial boundaries. An excellent paper on the early work that led to the development of boundary elements has been given by A&D Cheng [1] and from which many of the illustrations depicting famous scientists were taken. The present contribution aims to explain further how the methodology developed and consolidated towards the end of the 1970s, beginning of the 1980s.

In this paper, the domain integrals due to uniform load or self-weight that appear in the boundary element method (BEM) formulation for thick plates resting on elastic foundations are transformed to boundary integrals. The Reissner plate bending model is used to model the plate behavior, and the two-parameter Pasternak model is used to model the behavior of the foundation. The necessary particular solutions are derived, and the explict forms for the new boundary kernels are given. Two different collocation procedures are considered—external and boundary collocations. In the case of the boundary collocation and internal collocation (for computing internal functions) procedures, an additional free term is obtained, due to the discontinuity of the transformed kernels. The new boundary integrals are hypersingular integrals. However, it will be shown that these hypersingular terms vanish when integrated around a closed contour. Three numerical examples are presented with several parametric studies to demonstra...

Compactly supported radial basis functions have been used to interpolate the forcing term in the DRM. The resulting DRM matrix is sparse and our approach is very accurate and efficient. Our proposed method is especially attractive for large scale problems. Copyright © 1999 John Wiley & Sons, Ltd.

This paper deals with the evaluation of the boundary and internal stresses in the boundary element method (BEM) for plate-bending analysis. The Reissner plate theory is employed as the general plate-bending theory for modelling both thin and thick plates. Two methods are discussed for the evaluation of boundary stresses. The first is based on the local boundary tractions and strains, and the other is based on the direct evaluation of the stress tensor using the stress integral equation. The proposed methods can be used as postprocessing procedures for the BEM solution. The evaluation of three-dimensional (3-D) internal stresses through the plate thickness is also discussed and compared with the 3-D boundary element solutions. Numerical examples are presented to demonstrate the accuracy of the proposed methods.

The higher order fundamental solutions to the two-dimensional (2-D) modified Helmholtz equation have been derived. The Lth order fundamental solution has the form φ i ∗(L) = A L (kr) L K L (kr) , which satisfies the equation ∂ 2 ∂r 2 + 1 r ∂ ∂r − k 2 φ ∗(L) i +φ ∗(L−1) i =0 and the relationship of the coefficients AL is given as A L = A L−1 (2Lk 2 ) with A 0 = 1 (2π) . These solutions can be used for applying the multiple reciprocity boundary element method to the 2-D modified Helmholtz problems. The derivation procedure shown in the present paper can also be applied to obtaining higher order fundamental solutions to other engineering equations.

In this paper, axisymmetric heat conduction and thermal stress problems with three types of boundary conditions are analysed by the boundary element method. The temperature and thermal stress fields for the piston of a diesel engine are calculated using triangular finite elements and constant boundary elements, respectively, and the two results agree. However, BEM needs fewer data, less computer time (about one-sixth that of FEM) and storage volume. The advantages of BEM are sufficiently demonstrated.

The assessment of current distribution induced along complex grounding systems has been undertaken using the corresponding antenna model and the set of Pocklington integro-differential equations for curved wires. The set of Pocklington equations is numerically handled via the Galerkin-Bubnov scheme of the Indirect Boundary Element Method (GB-IBEM) featuring the isoparametric elements. Some illustrative numerical results for the current distribution are presented in the paper.

This paper describes a selective numerical integration scheme to compute the integrals present in a 2-D elastostatic boundary element formulation including those due to body forces. The selective integration ensures that the program runs efficiently while producing accurate results. The integrals are solved numerically using the third order polynomial transformation 9 which increases the accuracy of the classical Gaussian quadrature schemes.

A round-off error accumulation is observed in a multiple reciprocity computation for a fission neutron source iteration problem when a certain convergence condition is not satisfied. The present paper presents a reformulation of the multiple reciprocity method to remove the numerical problem. The neutron diffusion equation (NDE) is arranged using Wielandt's spectral shift technique in such a way that the convergence condition is always satisfied through source iterations. The boundary integral equation in this case requires the fundamental solution to the standard Helmholtz equation, while the fundamental solution corresponding to the correspoding to the original NDE was one to the modified Helmholtz equation. Except for this, the new multiple reciprocity formulation is identical to the original one for the fission source iteration problems. Some test calculation results indicate that a rapid and stable convergence can be realized using the new method and no round-off error accumulation is observed any more.

This paper deals with the numerical solution of contact problems between elastic bodies by using a boundary element flexibility approach formulation. As a first step, a flexibility matrix involving only the nodes which are expected to come into contact is computed by boundary elements. This produces a small system of equations which results in a procedure which is computationally very efficient. The relevant theory is described in this paper and validated by solving contact problems and comparing results with analytical or other numerical solutions. The examples comprise contact problems between elastic and rigid bodies with and without friction. The results are in good agreement with previous solutions.

This paper presents fundamentals of the multiple reciprocity method as well as its application to linear potential problems with body forces. Results of some numerical tests and comparisons with analytical solutions are presented.

Time-domain analysis of the transient current distribution along parallel wires located at different heights above a real ground is presented. The numerical solution is carried out via time-domain variant of the Galerkin–Bubnov indirect boundary element method (GB-IBEM) of solution extended to handle the case of arbitrary array of parallel wires located in a homogenous lossless medium, above a perfectly conducting (PEC) ground, or above a dielectric half-space. The mathematical formulation is based on the set of coupled space-time integral equations of the Hallen type. Some illustrative numerical results being compared to the results obtained via other methods wherever possible are presented.

This paper discusses the validity and accuracy of the results obtained using the boundary element flexibility approach for solving elastic contact problems. The method is such that the size of the flexibility matrices, which represent the relationship between tractions and displacements at the contact nodes depends only on the actual number of nodes in contact. Therefore, the approach is computationally very efficient. Moreover the contact analysis and the formation of the boundary element matrices can be carried out separately. Thus it is easy to implement the flexibility contact approach into any standard boundary element code. This paper presents a new contact criterion for the solution and carries out a series of comparisons with results obtained using other techniques. Although constant elements were employed in all cases, it is possible to extend the same approach to higher order elements if required. The flexibility approach is validated against Hertzian and non-Hertzian contact problems, including numerical solutions. The results are shown to be in agreement and the accuracy of the solution is investigated.

The present paper describes a general way of transforming domain integrals into equivalent boundary ones in boundary element formulations of fission neutron source problems. The critical eigenvalue, which is defined originally by the ratio of two different domain integrals has been transformed into a ratio involving only boundary integrals. The domain integral of fission source multiplied by the fundamental solution is converted into an equivalent boundary integral by the use of a particular solution technique, in which the source distribution is expanded into a Fourier series. The coefficients required in the expansion are determined using a boundary-only integral which has a recurrence form and is updated throughout the neutron source iteration scheme.

This paper reports on the boundary element method used to generate energy-dependent matrix-type boundary conditions along core/reflector interfaces and along baffle-plate surfaces of pressurized water reactors. This method enables one to deal with all types of boundary geometries including convex and concave corners. The method is applicable to neutron diffusion problems with more than two energy groups and also can be used to model a reflector with or without a baffle plate. Excellent eigenvalue and flux shape results can be obtained when the boundary conditions generated by this technique are coupled with core-only finite difference calculations.

This work extends the formulation developed in part I to include multiple knots and non-uniform blending functions. In this way, it is possible to lower the degree of continuity of the main variable at points of geometric discontinuity. Applications are presented for problems with corners, including the flow through an earth dam for which B-splines are specially suitable since they are able to maintain continuity of the slope between the free surface and the seepage face.

Transient response of a multiple wire configuration in the presence of a two-media configuration excited by a voltage source (antenna mode) or illuminated by an incident field is analysed using the boundary element method (BEM). The analysis is based on the solution of the corresponding set of the coupled space-time Hallen integral equation and it is carried out directly in the time domain. The influence of a two media configuration is taken into account via the space time reflection coefficient. The corresponding integral equation set is handled via the time domain variant of the Galerkin–Bubnov indirect boundary element method (GB-IBEM). Some illustrative numerical results for both antenna and scattering mode are presented in the paper.

Electromagnetic– thermal analysis of human exposure to base station antennas radiation is presented in this article. The formulation is based on a simplified cylindrical representation of the human body. Electromagnetic analysis involves incident and internal field dosimetry, while the thermal model deals with the bio-heat transfer phenomena in the body. The electric field induced in the body is determined from the axial current distribution inside the body. This current distribution is obtained along the body obtained by solving the Pocklington integral equation via the Galerkin– Bubnov Boundary Element Method. Once the internal electric field and related total absorbed power in the human body is obtained, a related temperature rise in the body due to the Global system Mobile exposure is calculated. This temperature rise is determined by solving the bio-heat transfer equation via the dual reciprocity boundary element method, and is found to be rather negligible.

The evaluation of boundary element influence integrals often involve numerical complexities due to the inherent singular nature of the integrals. A common practice to avoid some of the difficulties in treating these singularities is based on a simple technique commonly known as RowSum) '2 This technique takes advantage of certain relationships between influence terms that must hold in order to satisfy conditions of constant potential, rigid body displacement, (Brebbia l) or rigid body rotations (Niku & Brebbia 2) which are based on the physical interpretation of the problem. The RowSum technique is simple to implement and produces accurate results, provided of course that the regular influence terms are evaluated accurately. However, their application is limited to problems for which the conditions of constant potential or rigid body displacement exist and also those which produce a single singular term. In this paper a technique based on particular solutions is presented. The technique is simple and versatile, it can be applied to any problem and be used to evaluate as many influence terms as required.

A solution of steady-state Darcy’s flow in heterogeneous porous media is developed by means of the Dual Reciprocity Boundary Element Method (DRBEM). The governing equation is reduced to a Poisson-type equation involving a nonlinear non-homogeneous term. The Dual Reciprocity Method (DRM) is used for converting the resulting domain integral into equivalent boundary integrals. The present formulation is general as it allows the solution of Darcy’s flow with any hydraulic conductivity variation. Numerical examples are provided to demonstrate the validity of the procedure.

Up to now, all applications of the Dual Reciprocity Method considered is this book were related to the Laplace operator. This means that the governing partial differential equations were initially recast as some kind of Poisson’s equation and the fundamental solution of Laplace’s equation employed to obtain an equivalent integral formulation. The resulting domain integral was then approximated in a DRM fashion, and a boundary integral equation obtained and solved without resorting to internal discretization or integration.

Preface. Solving heat transfer problems by the dual reciprocity BEM. Transient problems using time-dependent fundamental solutions. Solving linear heat conduction problems by the multiple reciprocity method. Solving nonlinear heat transfer problems using the boundary element method. Coupled conduction-convection problems. Solving coupled problems involving conduction, convection and thermal radiation. Advanced thermoelastic analysis. Integral equation analyses of natural convection problems in fluid flow. Improperly posed problems in heat transfer. Acknowledgments. References.

It is generally agreed that the modern definition of Boundary Elements dates from 1978 when the notation used nowadays was accepted, partly due to the publication of the Proceedings of the first international conference on Boundary Elements [1] and partly as a result of the first book on the topic by the Computational Mechanics research group at Southampton [2]. Other books by the Southampton group have been published since, all of them using the same notation [3,4,5] which, with minor changes, will be that used in the present book.

The impressive success of the Boundary Element Method [1],[2] in the last decade has been mainly due to the fact that the technique became applicable to a wide range of engineering problems e.g. [3–8]. The boundary value problems of heat transfer are among those which can be effectively solved by BEM. For many practical cases, e.g. linear and non-linear steady-state heat conduction, some problems of thermal radiation and many others, BEM can result in a simple and elegant formulation. However, some heat transfer problems lead to an integral equation with domain integrals. The presence of these integrals diminishes the efficiency of the method and makes the data preparation needed for the computer codes much more difficult.

Since its origin in 1978, the International Conference on Boundary Element Methods has provided the recognized and established forum for innovations in boundary element research. Practically all new ideas on boundary ele ments have been presented at these conferences and the resulting papers can be found in the published books. The conference brings together the most renowned scientists and engineers working on boundary element research throughout the world. A unique feature of these meetings is that the participation of younger researchers is actively encouraged by the organizers in an effort to .bring forward to the attention of the international community an ever expanding range of new ideas. This book contains the edited version of the papers presented at the XIIIth BEM Conference held in Tulsa, Oklahoma in August of 1991. The meeting attracted a large number of participants and many excellent contributions which have been divided into nineteen different sections, i.e. Potential Prob lems; Diffusion and Convection Problems; Fluid Mechanics; Fluid Flow; Wave Propagation; Groundwater Flow; Heat Transfer; Electrical Problems; Geomechanics; Plates and Shells; Inelastic Problems; Damage Tolerance; Contact Mechanics; Industrial Applications; Design Sensitivity and Opti mization; Inverse Problems; Special Techniques; Numerical Aspects and Computational Aspects."

In recent years there has been a growing interest in the prediction of the behaviour of engineering problems involving galvanic effects. This class of problem includes galvanic corrosion, cathodic protection methods and the inverse problem of electrodeposition commonly used in manufacturing. This article will concern how a computer software system based on boundary elements can be used to accurately simulate this class of problem.

The Boundary Element Method [1,2,3] is now well established as a very powerful engineering tool. There is a variety of practical boundary value problems which can efficiently and very accurately be solved by boundary elements.

A method of numerical evaluation of two-dimensional principal value integrals is proposed. The numerical technique is used to evaluate the singular integrals of the boundary element method applied to plasticity problems. An application is presented, and results are compared with analytical and finite element solutions.

Several types of nonlinearities are considered, using the boundary element method, with emphasis on geomechanical applications. Numerical algorithms to model ‘no-tension’ plastic, viscoplastic and viscoelastic responses are presented. Extension of the method to two-dimensional piecewise homogeneous problems is shown. The overlay technique is adapted for the boundary element formulation to model complex plastic and viscoplastic responses. In particular, the technique is shown to be extremely useful to model time-dependent behaviour. It has also proved suitable for the Bauschinger-effect representation in elastoplastic analysis. Tunnel examples are presented and are shown to be very well suited to solution by boundary elements. The method deals with infinite domains without requiring the definition of an artificial outer boundary. As a result little discretization is needed.

The application of the boundary element method for solving boundary value problems with body forces, time dependent effects or certain class of non-linearities generally leads to an integral equation which contains domain integrals (e.g. Ref. 1). Although these integrals do not introduce any new unknowns they detract from the elegance of the formulation and affect the efficiency of the method as integrations over the whole volume are required. Hence, a substantial amount of research has been carried out to find a general and efficient method of transforming volume integrals into equivalent boundary ones. The approaches which have so far been proposed can be divided into three groups. The first approach started in 1982 and is called the Dual Reciprocity Method, This approach was developed by Nardini and Brebbia 2'3 and afterwards extended for a variety of problems by Brebbia and WrobeP and by Loeffler and Mansur 5. It has been proved that the Dual Reciprocity Method (e.g. Refs 6, 7, 8) permits to solve a wide range of problems and is accurate. Another group of approaches is based on the expansion of the source term into the Fourier series. The method was proposed by Tang 9 to deal with potential and elasticity problems and has also been successfully applied for solving neutron diffusion problems by Itagaki and Brebbia 1 o. The third class of methods is the use of particular solutions which can convert domain integrals into boundary integrals. This technique offers some advantages versus the previous approaches in some specific applications as demonstrated by Azevedo and Brebbial 1. It has also been pointed out in Ref. 11 that the use of global particular solutions can be thought of as a special case of Dual Reciprocity. In this paper a new method of transforming domain integrals into boundary integrals is proposed. The Multiple Reciprocity Method (MRM) can be thought of as an extension of the idea of Dual Reciprocity. However, instead of approximating the source term by the set of coordinate functions, a sequence of functions related to the fundamental solution is introduced. These constitute a set of higher order fundamental solutions which permit the second Green's identity to be applied to each term of

A boundary element method for the analysis of free vibrations in solid mechanics is developed using a non-standard boundary integral approach. It is shown that, utilizing the statical fundamental solution and employing a special class of coordinate functions, the algebraic eigenvalue problem results. The method has been realized numerically for the two-dimensional elastodynamics, and a number of examples demonstrating its accuracy have been included.

In this paper the boundary element method is applied to analyse homogeneous isotropic plates in bending using Reissner's theory. This formulation involves three coupled integral equations and can be applied not only to moderately thick plates, but also to some usually analysed by classical Kirchhoff-Love thin plate theory. It is shown that the boundary element method gives accurate results and can be applied to cases when traction discontinuity may occur or even in the presence of singularities, using double nodes or simple discontinuous elements of the type described in Ref. 1. Numerical results are compared against analytical or finite element solutions to illustrate the accuracy of the boundary element formulation.

In this paper the boundary element method is applied to analyse plates resting on Winkler-type elastic foundations. The formulation presented involves two coupled integral equations and can be applied to plates of arbitrary shape and boundary conditions. The results of the numerical method are presented and compared with the respective analytical solutions, to illustrate the accuracy and applicability of the boundary element method.

An integral formulation for solving two- and three-dimensional problems governed by the scalar wave equation is presented. The three-dimensional case which has already been the subject of many analytical and numerical works is briefly discussed. The paper concentrates on finding an integral expression for the two-dimensional case, involving source density, initial and boundary conditions, which can be used for time-stepping numerical analysis.

This paper describes the application of a new technique called boundary elements, to elastostatics problems. The method is well suited to solve three-dimensional problems, as it reduces them to boundary solutions, i.e. only elements on the boundary need to be defined. This results in a much smaller system of equations and considerable reduction in the data required to solve a problem. The method is also of great interest for soil-structure interaction problems because it permits the analysis of the infinite and semi-infinite domains, such as the soil. In this paper, the boundary elements are presented as a special case of weighted residuals and as such they are not difficult to relate to finite elements or other domain methods. Thus, boundary elements can be combined with finite elements in general-purpose programs. This paper describes three-dimensional elastostatic applications of the method using constant elements.

This paper presents a boundary element formulation employing the penalty function technique for steady viscous flow problems. By regarding the convective terms in Navier-Stokes equations as body forces in elastostatics, the standard elastostatics analysis can be extended to solve viscous flow problems. In particular, several investigations were carried out for evaluating the convective terms. These include a finite difference scheme (employing both upwind and central approximations) and a boundary integral formulation. Finally, some numerical results are shown in order to demonstrate the validity of the present method and its accuracy in the evaluation of convective terms.

This paper describes the application of the boundary element method in elasticity starting by discussing the basic weighted residual expressions. Boundary elements of different orders are presented and discussed, together with the numerical integration schemes. The relationships are presented for three dimensional elasticity and specialized to the two dimensional case. Results are presented and discussed, specially in relation to finite elements and other “domain” type techniques.

This paper presents the application of the boundary element method for computing wave forces on offshore structures of constant section throughout the depth of water. The paper compares results obtained using constant, linear and quadratic elements and draws some original conclusions regarding their numerical accuracy and convergence. Examples studied include the vertical circular cylinder, the square caisson and the elliptical cylinder for which the boundary element solutions are compared against analytical or experimental results. The paper also shows how symmetry conditions can be introduced into the problem to reduce the computer storage and time required to solve the problems.

SUMMARY This paper presents a boundary element formulation employing a penalty function technique for twodimensional steady thermal convection problems. By regarding the convective and buoyancy force terms in Navier-Stokes equations as body forces, the standard elastostatics analysis can be extended to solve the Navier-Stokes equations. In a similar manner, the standard potential analysis is extended to solve the energy transport equation. Finally, some numerical results are included, for typical natural convection problems, in order to demonstrate the efficiency of the present method.

The boundary element formulation of potential problems is presented using weighted residual techniques. The paper emphasizes the simplicity of the boundary methods and the way in which they can be applied in engineering. The advantage of using this method in preference to finite elements is discussed in the applications.

A complete formulation for the boundary element method applied to rate dependent material problems is presented. The only domain approximation involved is for the inelastic strains which are introduced into the analysis in initial stress form using piecewise linear interpolation functions. The proper procedure for obtaining the correct integral equation for stresses at internal points is reviewed and demonstrated. A series of examples is presented to outline the accuracy and applicability of the formulation, including elastoplastic, creep and viscoplastic problems.