This special issue contains a selection of papers presented at the renowned International Conference on Boundary Elements and other Mesh Reduction Methods (BEM) which is now in its 39th edition, having started in 1978. The meeting was organised by the University of Mississippi in the USA and the Wessex Institute, UK.

It was at Southampton University where the technique started in the mid-1970s, with the first successful development of boundary integral equations into what is now known as the Boundary Element Method. The success of the meeting has been possible by its continuous evolution. In the early 1990s it was decided to cover all types of Mesh Reduction, opening up a wide new field of technical and applied research.

The success of that policy is reflected in the continuous growth of the ranking of the Journal associated with the Conference, i.e. the one of Engineering Analysis with Boundary Elements. The Journal is now in the top third of all its categories.

A major development took place last year when WIT Press decided to make available in Open Access form all the conference papers published since 1993. This move has dramatically increased the number of citations that the papers achieve. It is part of the Wessex Institute policy of disseminating scientific and technical outputs as widely as possible.

The international researcher will find in this issue a selection of the most recent developments in the method which in the last few years has attracted the attention of a variety of industrial users.

The next 40th Conference will take place in the New Forest National Park in the UK, home of the Wessex Institute, giving the occasion to the participants to become more aware of the activities on Campus, where research is focused on the development and applications of boundary elements.

The Editors would like to thank all authors for the quality of their papers and the members of the International Scientific Advisory Committee and other colleagues for their help in reviewing the material. The support of Elsevier is gratefully acknowledged for financing the expenses associated with awarding of the 2016 George Green’s Medal.

The Editors

Siena, 2016

A review of Green’s functions for dissimilar or homogeneous elastic space containing penny-shaped or annular interfacial cracks under singular ring-shaped loading sources is presented. The solutions are based on fictitious singular loading sources and superposition of the fundamental solutions of the following two problems: (a) Dissimilar elastic solid without crack under singular source, and (b) Dis-similar elastic solid containing crack under surface tractions. The above Green’s functions have the following advantages: (i) No multi-region BE modeling for the dissimilar material is necessary, and (ii) No discretization of the crack surface is necessary. Numerical examples are presented and discussed.

A formulation is presented to perform crack propagation analyses in cohesive materials with the dual boundary element method (DBEM) using the tangential differential operator in the traction boundary-integral equations. The cohesive law is introduced in the system of equations to directly compute the cohesive forces at each loading step. A single edge crack is analyzed with the linear function to describe the material softening law in the cohesive zone, and the results are compared with those from the literature.

For the analysis of cracks in three-dimensional isotropic thermoelastic media, a temperature and displacement discontinuity boundary element method is developed. The Green functions for unit-point temperature and displacement discontinuities are derived, and the temperature and displacement discontinuity boundary integral equations are obtained for an arbitrarily shaped planar crack. Our boundary element method is based on the Green functions for a triangular element. As an application, elliptical cracks are analyzed to validate the developed method. The influence of various thermal boundary conditions is studied.

The paper reviews the influence of the variability in the morphology and the tissue properties of the human brain and eye, respectively, exposed to high-frequency (HF) radiation. Deterministic-stochastic modeling enables one to estimate the effects of the parameter uncertainties on the maximum induced electric field and Specific Absorption Rate (SAR). Surface Integral Equation (SIE) scheme applied to the brain exposed to HF radiation and hybrid boundary element method (BEM)/finite element method (FEM) scheme used to handle the eye exposure to HF radiation are discussed.

Furthermore, a simple stochastic collocation (SC), through which the relevant parameter uncertainties are taken into account, is presented. The SC approach also provides the assessment of corresponding confidence intervals in the set of obtained numerical results. The expansion of statistical output in terms of the mean and variance over a polynomial basis (via SC approach) is shown to be robust and efficient method providing a satisfactory convergence rate. Some illustrative numerical results for the maximum induced field and SAR in the brain and eye, respectively, are given in the paper, as well.

A time-dependent fully-parallelised formulation of the BEM is applied to transient thermal problems in the context of light-based medical devices. The method is initially verified against benchmark problems. The limitations of the model are discussed, particularly the singularity challenge inherent in the fundamental solution. The method is then applied for a representative 3D clinical problem, further illustrating the singularity challenge.

The corrosion electric field around the surface of stainless steel under tensile stress is addressed through the experiment and simulation. When the stress is applied, the passive film is locally damaged on the grain boundaries causing microscopic stress and strain concentrations. In a corrosive environment, the plastic strain induced by the strain concentration breaks the passive film and generates a new surface without the passive film. This causes a galvanic corrosion between the intact surface with passive film and the damaged surface without passive film. The effect of stress on the polarization curve was observed by electrochemical and mechanical experiments, and we found that the spontaneous potential decreased as the applied stress increased. To evaluate the electrochemical property of stressed stainless steel, the electric field analysis is formulated by the boundary element method (BEM) with the damaged passive film model and the empirical polarization curve model.

In recent years significant focus has been given to the study of Radial basis functions (RBF), especially in their use on solving partial differential equations (PDE). RBF have an impressive capability of inter- polating scattered data, even when this data presents localized discontinuities. However, for infinitely smooth RBF such as the Multiquadrics, inverse Multiquadrics, and Gaussian, the shape parameter must be chosen properly to obtain accurate approximations while avoiding ill-conditioning of the interpolating matrices. The optimum shape parameter can vary significantly depending on the field, particularly in locations of steep gradients, shocks, or discontinuities. Typically, the shape parameter is chosen to be high value to render flatter RBF therefore yielding a high condition number for the ensuing interpola- tion matrix. However, this optimization strategy fails for problems that present steep gradients, shocks or discontinuities. Instead, in such cases, the optimal interpolation occurs when the shape parameter is chosen to be low in order to render steeper RBF therefore yielding low condition number for the interpolation matrix. The focus of this work is to demonstrate the use of RBF interpolation to capture the behaviour of steep gradients and shocks by implementing a blending scheme that combines high and low shape parameters. A formulation of the RBF blending interpolation scheme along with test- ing and validation through its implementation in the solution of the Burger’s linear advection equation and compressible Euler equations using a Localized RBF Collocation Meshless Method (LRC-MM) is presented in this paper.

The vibration behaviour of ships is noticeably influenced by the surrounding water, which represents a fluid of high density. In this case, the feedback of the fluid pressure onto the structure cannot be neglected and a strong coupling scheme between the fluid domain and the structural domain is necessary. In this work, fast boundary element methods (BEMs) are used to model the semi-infinite fluid domain with the free water surface. Two approaches are compared: A symmetric mixed formulation is applied where a part of the water surface is discretized. The second approach is a formulation with a special half-space fundamental solution, which allows the exact representation of the Dirichlet boundary condition on the free water surface without its discretization. Furthermore, the influence of the compressibility of the water is investigated by comparing the solutions of the Helmholtz and the Laplace equation. The ship itself is modeled with the finite element method (FEM). A binary interface to the commercial finite element package ANSYS is used to import the mass matrix and the stiffness matrix. The coupled problems are formulated using Schur complements. To solve the resulting sys- tem of equations, a combination of a direct solver for the finite element matrix and a preconditioned GMRES for the overall Schur complement is chosen. The applicability of the approach is demonstrated using a realistic model problem.

In this paper, we present briefly the derivation of the equations of motion and boundary conditions for elastic plates with functionally graded Young’s modulus and mass density of the plate subjected to transversal transient dynamic loads. The unified formulation is derived for three plate bending theories, such as the Kirchhoff–Love theory (KLT) for bending of thin elastic plates and the shear deformation plate theories (the first order – FSDPT, and the third order – TSDPT). It is shown that the transversal gradation of Young’s modulus gives rise to coupling between the bending and in-plane deformation modes in plates subject to transversal loading even in static problems. In dynamic problems, there are also the inertial coupling terms. The influence of the gradation of material coef- ficients on bending and in-plane deformation modes with including coupling is studied in numerical experiments with consideration of Heaviside impact loading as well as Heaviside pulse loading. To decrease the order of the derivatives in the coupled PDE with variable coefficients, the decomposition technique is employed. The element-free strong formulation with using meshless approximations for spatial variation of field variables is developed and the discretized ordinary differential equations with respect to time variable are solved by using time stepping techniques. The attention is paid to the stability of numerical solutions. Several numerical results are presented for illustration of the coupling effects in bending of elastic FGM (Functionally Graded Material) plates. The role of the thickness and shear deformations is studied via numerical simulations by comparison of the plate response in three plate bending theories.

In this paper we consider vibrations of the baffled elastic fuel tank partially filled with a liquid. The compound shell was a simplified model of a fuel tank. The shell is considered to be thin and the Kirchhoff–Love linear theory hypotheses are applied. The liquid is supposed to be an ideal and incompressible one and its flow introduced by the vibrations of a shell is irrotational. The problem of the fluid-structure interaction was solved using the reduced boundary and finite element methods. The tank structure was modeled by the FEM and the liquid sloshing in a fluid domain was described by using the multi-domain BEM. The rigid and elastic baffled tanks of different forms were considered. The dependencies of frequencies via the filling level were obtained numerically for vibrations of the fluid-filled tanks with and without baffles.

This paper is focused on the analysis of the numerical solution of flow problems in irregular domains. The numerical approach is based on the weighted least squares (WLS) approximation constructed over the local support domain, i.e. a sub cluster of computational nodes, to evaluate partial differential operators, in our case spatial derivatives up to second order. There are several possibilities for elegant formulation as well as computer implementation of such method, which are first and foremost consequence of the fact that the node has to be aware only of the distance to other nodes, i.e. no topological relation between nodes is required. The presented meshless approach is applied on the lid-driven cavity problem in randomly generated domain. It is demonstrated that using adequately wide support domains, i.e. enough support nodes with a proper weighting, provide stable results even in highly deformed domains, however, at the cost of the accuracy and computational complexity, especially in cases when the support domain changes during the computation. The optimal meshless configuration, i.e. support of 15 nodes weighted with Gaussian weight function and monomials up to second order as basis, is suggested based on experimental analyses. The results are presented in terms of comparison with already published data on regular nodal distributions, convergence analysis on regular nodal distribution and stability analysis of the solution with respect to the level of nodal irregularity and local support size.

In this paper, we derive a boundary-domain integral formulation for the energy transport equation under the assumption that the fluid properties, through which the energy is transported by diffusion and convection, are spatially and temporally changing. The energy transport equation is a second-order partial differential equation of a diffusion-convection type, with the fluid temperature as the independent variable. The presented formulation does not require a calculation of the temperature gradient, thus it is, for a known fluid velocity field, linear.

The final boundary-domain integral equation is discretized using a domain decomposition approach, where the equation is solved on each sub-domain, while subdomains are joined by compatibility conditions. The validity of the method is checked using several analytical examples. Convergence properties are studied yielding that the proposed discretization technique is second-order accurate.

The developed method is used to simulate flow and heat transfer of nanofluids, which exhibit properties that depend on the solid particle concentration. A Lagrange-Euler approach is used.

The objective of this article is to develop a boundary element numerical model to solve coupled problems involving heat energy diffusion, convection and radiation in a participating medium. In this study, the contributions from radiant energy transfer are presented using two approaches for optical thick fluids: the Rosseland diffusion approximation and the P1 approximation. The governing Navier– Stokes equations are written in the velocity–vorticity formulation for the kinematics and kinetics of the fluid motion. The approximate numerical solution algorithm is based on a boundary element numerical model in its macro-element formulation. Validity of the proposed implementation is tested on a one-dimensional test case using a grey participating medium at radiative equilibrium between two isothermal black surfaces.

Velocity-pressure coupling schemes for the solution of incompressible fluid flow problems in Computational Fluid Dynamics (CFD) rely on the formulation of Poisson-like equations through projection methods. The solution of these Poisson-like equations represent the pressure correction and the velocity correction to ensure proper satisfaction of the conservation of mass equation at each step of a time-marching scheme or at each level of an iteration process. Inaccurate solutions of these Poisson-like equations result in meaningless instantaneous or intermediate approximations that do not represent the proper time-accurate behavior of the flow. The fact that these equations must be solved to convergence at every step of the overall solution process introduces a major bottleneck for the efficiency of the method. We present a formulation that achieves high levels of accuracy and efficiency by properly solving the Poisson equations at each step of the solution process by formulating a Localized RBF Collocation Meshless Method (LRC-MM) solution approach for the approxima- tion of the diffusive and convective derivatives while employing the same framework to implement a Dual-Reciprocity Boundary Element Method (DR-BEM) for the solution of the ensuing Poisson equations. The same boundary discretization and point distribution employed in the LRC-MM is used for the DR-BEM. The methodology is implemented and tested in the solution of a backward- facing step problem.

This article addresses a specific type of boundary conditions in plane elastic boundary value problems, BVP. An elastic plane composed of two dissimilar isotropic materials is considered. It is assumed that the displacement vector orientations are known on both sides of the contour that separates the entire plane into interior and exterior domains. The stress vector is assumed to be continuous across the contour. The aim of this study is the investigation of solvability of this BVP and the development of appropriate numerical methods for solving the corresponding singular integral equation. The latter is necessary as the integral equation is homogeneous. It is shown that depending on the behaviour of the displacement vector orientations the solution of the problem may include a certain number of arbitrary linear parameters. A numerical approach is proposed based on the solution of the homogeneous Riemann BVP to form a non-homogeneous right hand side of the integral equation.

In this paper, we apply the recently proposed fast block-greedy algorithm to a convergent kernel-based collocation method. In particular, we discretize three-dimensional second-order elliptic differential equations by the meshless asymmetric collocation method with over-sampling. Approximated solutions are obtained by solving the resulting weighted least squares problem. Such formulation has been proven to have optimal convergence in H2. Our aim is to investigate the convergence behaviour of some three dimensional test problems. We also study the low-rank solution by restricting the approximation in some smaller trial subspaces. A block-greedy algorithm, which costs at most O(NK2) to select K columns (or trial centers) out of an M × N overdetermined matrix, is employed for such an adaptivity. Numerical simulations are provided to justify these reductions.

In boundary element method (BEM), one encounters linear system with a dense and non-symmetric square matrix which might be so large that inverting the linear system is too prohibitive in terms of cpu time and/or memory. Each usual powerful treatment (Fast Multipole Method, H-matrices) developed to deal with this issue is optimized to efficiently perform matrix vector products. This work presents a new technique to adequately and quickly handle such products: the Sparse Cardinal Sine Decomposition. This approach, recently pioneered for the Laplace and Helmholtz equations, rests on the decomposition of each encountered kernel as series of radial Cardinal Sine functions. Here, we achieve this decompo- sition for the Stokes problem and implement it in MyBEM, a new fast solver for multi-physical BEM. The reported computational examples permit us to compare the advocated method against a usual BEM in terms of both accuracy and convergence.

The Dual Reciprocity BEM (DRBEM) and the Time-Dependent BEM (TDBEM) are considered in the context of radiative and time-dependent thermal transport, respectively. In order to achieve sensible solution times for realistic 3D problems with large meshes, a range of optimisation techniques are considered, and a number of parallelisation techniques applied: shared memory using multi-core threading, Graphics Processing Unit (GPU) acceleration using CUDA, and distributed memory on a high performance cluster using MPI. Particular consideration is given to practical methods to invert large dense matrices.

The boundary element method (BEM) is a widely used engineering tool in acoustics. The major disadvantage of the three-dimensional boundary element method (3D-BEM) is its computational cost, which increases with the size of the simulated obstacle and the simulated wave number. Thus, the geometrical details of the obstacle and the simulated frequency range are limited by computer speed and memory. The computational cost for simulating large obstacles like noise barriers is often reduced by applying the two-dimensional boundary element method (2D-BEM) on three-dimensional obstacles. However, the 2D-BEM limits the geometry of the boundary to obstacles with a one-dimensionally constant profile. An interesting compromise solution between the 2D-BEM and the 3D-BEM is the quasi-periodic boundary element method (QP-BEM). The QP-BEM allows the simulation of periodically repetitive complex three-dimensional structures and periodic sound fields while keeping the computational cost at a reasonable level. In this study, first, the QP-BEM was implemented and coupled with the fast multipole method. Second, the QP-BEM was used to simulate the sound field radiated by a simple geometric object, i.e., a uniformly vibrating cylinder. Results were compared to an analytic solution, for the evaluation of the numerical accuracy of our QP-BEM implementation. For the demonstration of some use cases, third, the QP-BEM was used to simulate the sound field scattered by a sonic crystal noise barrier and a noise- barrier top element.