This special issue contains papers selected from the 40th International Conference on Boundary Elements and other Mesh Reduction Methods, held in the New Forest, UK, organised by the Wessex Institute.

The annual conference on Boundary Elements and other Mesh Reduction Methods (BEM/MRM) started in 1978 and is now in its 40th version. It is well established as the recognised international forum for the latest advances in those techniques and their application in science and engineering.

The continued success of the meeting is a result of the strength of the research on boundary elements and mesh reduction techniques being carried out all over the world. The conference has continually evolved in line with the latest developments in the field since the successful development of boundary integral techniques into BEM was reported in the first meeting held in Southampton in 1978.

The objective of the research papers presented at the meetings is the further development of techniques that reduce or eliminate the type of meshes required by first generation computational methods, such as finite differences or finite elements.

This has steadily been achieved through the development of BEM as a computational tool and continues through more recent research into advanced techniques, leading to further mesh reduction aiming to produce truly meshless methods.

Also included are papers on the use of BEM and, in particular, the description of new applications.

The Editors would like to thank all authors for the quality of their papers and other colleagues for their help in reviewing the material.

The Editors

2017

The paper reviews the collocation boundary element method (BEM) exactly as it has been originally proposed on the basis of a weighted residuals statement that leads to Somigliana’s identity, but with two subtle conceptual improvements for a generally curved boundary: (a) the interpolation function for normal fluxes or traction forces (for potential or elasticity problems) must be redefined and (b) only Gauss-Legendre quadrature turns out to be required if the numerical integration issues are mathematically adequately stated. A simple, unified code is proposed – as presently shown for 2D problems – to arrive at arbitrarily high computational accuracy of the constituent matrices as well as of results at internal points independently from how convoluted a problem’s topology may be (but given the representation limitations of a discretization mesh). In fact, the higher the effect of a quasi-singularity may be, as for an internal point infinitely close to the boundary, the more accurate a result is achievable with just a few number of quadrature points. A collateral, but not less relevant, outcome of the pro- posed developments is that regularization methods, special quadrature schemes and so many methods that intend to conceptually deviate from the originally stated BEM as an attempt to offer numerical improvements are actually unnecessary (they are in most cases just misleading). Moreover, the inaccurate, albeit popular constant element is actually not simpler to deal with than high-order elements. Owing to space restrictions, most of the detailed developments as well as the hopefully very convincing numerical results deal with potential problems, although the more general problem of elasticity is adequately posed and assessed.

The paper presents a fluid-structure interaction analysis of fuel tanks with cylindrical and spherical compartments partially filled with a liquid. The compound shell of revolution is considered as a con- tainer model. The shell is supposed to be thin, so the Kirchhoff–Love linear theory hypotheses are applied. The liquid is an ideal and incompressible one. Its properties and filling levels may be different within each compartment. The shell vibrations coupled with liquid sloshing under the force of gravity have been considered. The tank structure is modelled by a finite element method, whereas liquid sloshing in the compartments is described by a boundary element method. A system of singular integral equations is obtained for evaluating the fluid pressure. At the first stage, both spherical and cylindrical fluid-filled unconnected rigid shells are considered. Different filling levels as well as small radii of free surfaces are taken into account in problems of liquid sloshing in spherical shells. The sloshing frequencies in the presence of complete or partially covered free surfaces are determined for cylindrical shells. The boundary element method has proven to be effective and accurate in all the problems considered. At the second stage, the natural frequencies and modes of the dual compartment tank are obtained including sloshing, elasticity, and gravity effects.

In general, internal cells are required to solve large deformation problems using a conventional boundary element method (BEM). However, in this case, the merit of BEM, which is the ease of data preparation, is lost. Triple-reciprocity BEM enables us to solve elastoplasticity problems with a small plastic deformation. In this study, it is shown that two-dimensional large plastic deformation problems with a friction coefficient can be solved without the use of internal cells, by the triple-reciprocity BEM. Initial stress and strain formulations are adopted and the initial stress or strain distribution is interpolated using boundary integral equations. In this method, only boundary elements are remeshed. A new computer program is developed and used to solve several problems.

We apply the Kansa–radial basis function (RBF) collocation method to two– dimensional nonlinear boundary value problems. The system of nonlinear equations resulting from the Kansa–RBF discretization is solved by directly applying a standard nonlinear solver. In a natural way, the value of the shape parameter in the RBFs employed in the approximation is included in the unknowns to be determined. The numerical results of some examples are presented and analysed.

Nowadays, a variety of numerical methods and numerical formulations exits to solve complex or coupled field problems in three dimensions. Most of them are generally applicable to nearly arbitrary kind of field problems. On the other hand, some highly optimized methods are available, which are predestined for the solution of a specific kind of problem. Especially in the case of weakly coupled multiphysics problems, a mixture of several numerical methods is very advantages to benefit from different properties of numerical methods for diverse physical sub-problems. A very promising approach for a flexible coordination of the related solution process is the application of software agents. Then, the results of one sub-problem are converted into boundary values or volume source distributions for another sub-problem and software agents choose solution methods independently for each sub-problem. Furthermore, two main aspects have to be considered in applications of numerical methods. First, the solution of a boundary value problem should be computed efficiently and second, the solution is evaluated for visualization and interpretation of obtained results. In practice, it is difficult to choose a single appropriate method, which is well suited both for the solution of a problem and its evaluation, since the demands differ in both cases. Here, a concept is presented to apply various numerical methods successfully to the solution and evaluation of complex field problems. Attention is mainly turned on the integration of boundary element methods into the concept of mixed numerical formulations.

The linear coupled stretching-bending problem for general laminates is here formulated with the mid-plane stress function and the lateral deflection as independent field variables. A mathematical similarity between the two problems is achieved by introducing a re-arranged mid-plane strain tensor as one of the dependent variables. As a step towards a genuine boundary element solution for this problem, its fundamental solutions are derived using a Fourier transform approach. First, the transforms of the solutions are obtained in terms of the transform space variables and their inverses are deduced using complex integral calculus. Through the use of these fundamental solutions, boundary integral equations of the linear coupled stretching-bending problem are formulated without the presence of any irreduc- ible domain integrals. Issues regarding the numerical implementation of this formulation are raised and discussed.

This paper presents a numerical method for topology optimisation for two-dimensional elastodynamics based on the level set method and the boundary element method (BEM) accelerated by the H-matrix method and its application to identifications of defects in an infinite elastic medium. Gradient-based topology optimisation methods require design sensitivity, which is obtained by solving some boundary value problems. The BEM is employed for this sensitivity analysis because the BEM can deal with infinite domains rigorously without any approximation. However, the computational cost in the BEM is expensive, and this is a serious drawback since we need to repeat sensitivity analysis even for a single optimisation process. In this study, the H-matrix method is used as an acceleration method of the BEM for the reduction of the computational cost of the sensitivity analysis. Also proposed is a method to improve the efficiency of the H-matrix method by exploiting a property of the kernel function of the elastodynamic fundamental solution. Some numerical examples are demonstrated, and the effectiveness of the proposed method is confirmed.

A model for numerical analysis of compound structures made of various materials is presented. The mathematical concept of solution is based on quasi-static evolution of debonding processes occurring along the interface. It is formulated in terms of energies considering the stored energy represented by the elastic energy of the structures and dissipation due to damage processes, plastic slip at the inter- face or friction. The numerical solution includes a semi-implicit time stepping procedure, relying on splitting of the whole problem at a current time step into two problems of variational nature solved recursively. The space discretisation includes Symmetric Galerkin Boundary Element Method used to obtain the stored energies, and, in combination with the variational character of the recursive problems, also to calculate its gradients to be utilized in non-linear programming algorithms for finding the time- evolving solution. Numerical results are demonstrated for a steel-concrete interface frequently met in civil engineering applications to assess the model applicability in engineering practice.

Crack propagation in a single-edge notched beam is analyzed with the three-point bending test. Two constitutive laws that describe the material softening in the cohesive zone were tested, and their results were compared. The dual boundary element method (DBEM) is employed with the traction boundary integral equation using the tangential differential operator. A constitutive law was introduced in the system of equations, and the cohesive forces were directly computed at each loading step. The results are compared with the experimental and numerical results available in the literature.

Sound transmission through thin elastic shell with different fluids on the inside and outside is simulated using the in-house program based on the coupled finite element and boundary element method. The structure dynamics is simulated using the finite element method and the acoustic fields are simulated using the boundary element method. To avoid the non-uniqueness problem existing in the exterior acoustic boundary element method, Burton and Miller formulation is employed. The hyper-singular boundary integral is dealt with a regularization relationship. To validate this approach, a case with analytical solutions is simulated.

This work investigates the mass ejected from surface perturbations as the shockwave reaches the AL-vacuum interface, which originates from unstable Richtmyer–Meshkov (RMI) impulse phenomena. The main purpose is to explore the relationships between the shockwave impulse and the geometric properties of surface perturbations, and how those relationships drive the total ejected mass, directionality and velocity distribution. We discuss in detail different types of surface geometry (sinusoidal, square-wave, chevron and semicircle), as well as the wavelengths and amplitudes of surface perturbation. The time evolutions of micro-jet ejection are simulated using a hydrodynamic Lagrangian-Remapping Eulerian method. The calculated results show that primary jetting ejection can be formed from the different shapes, and with increasing wavelength, the ejection mass keeps an increase while the jet head-velocity decreases. However, not all periodic perturbations behave similarly, and masses ejected from irregular surface cannot be normalized to its cross-sectional areas. The square-wave surface may yield pronounced, velocity-enhanced secondary jetting, which is a result of collision of primary jets.

In this paper, we derive a boundary-domain integral formulation for the vorticity transport equation under the assumption that the viscosity of the fluid, through which the vorticity is transported by diffusion and convection, is spatially changing. The vorticity transport equation is a second order partial differential equation of a diffusion-convection type.

The final boundary-domain integral representation of the vorticity transport equation is discretized using a domain decomposition approach, where a system of linear equations is set-up for each sub-domain, while subdomains are joint 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 for constant and variable viscosity cases.

In this paper, we present a fast boundary element method (BEM) algorithm for the solution of the velocity-vorticity formulation of the Navier-Stokes equations. The Navier-Stokes equations govern incompressible fluid flow, which is inherently nonlinear and when discretizised by BEM requires the discretization of the domain and calculation of domain integrals. The computational demands of such method scale with O(N2), where N is the number of boundary nodes. To accelerate the solution process and reduce the computational demand, we present two different approaches, the subdomain method and an approximation procedure with hierarchical structure. Several approximation techniques exist, such as multipole approximation methods FMM (fast multiple method), SVD (singular value decomposition method), wavelet transform method and a cross approximation method. In this paper, we present the cross approximation method in combination with the hierarchical H-structure. The cross approximation method can reduce the computational demands from O(N2) to O(N log N). There are many forms of the cross approximation, like the algebraic cross approximation and the hybrid cross approximation. Here, we applied the algebraic cross approximation form. The main advantage is that we did not need to evaluate the integral and then to change it with a degenerate kernel function. The cross approximation algorithm was used to solve the kinematics equation for unknown boundary vorticity values. Results show that an increasing of the compression rate has a negative influence on the solution accuracy. On the other hand, the solution accuracy increases with computational grid density. Tests were performed using the 3D lid-driven cavity test case with Reynolds numbers up to 1000. Solution accuracy was similar for all Reynolds numbers considered. In conclusion, the tests showed that our implementation of the algebraic cross approximation for the acceleration of the solution of the kinematics equation can be applied to decrease the computational demands and to accelerate the BEM.

Phononic crystals have been extensively studied, and their capacity to attenuate the propagation of sound waves at specific frequency bands is well known and documented in the literature. However, few studies exist concerning the behaviour of such structures in the context of elastic media, with the purpose of attenuating the transmission of vibrations. Applying this concept can be quite interesting, and may allow new vibration control devices to be developed, tailored at specific applications. As an example, buried periodic structures may be used to control elastic wave propagation in the ground, and thus to help reducing the vibrations that can reach sensible structures.

In this work, the authors make use of a 2.5D numerical model based on the Method of Fundamental Solutions (MFS) to analyse this complex problem, considering the case of arrays of elastic inclusions buried in a homogeneous medium, fully considering the complete elastodynamic interaction between the inclusions and the host medium. Due to the geometric periodicity of the analysed problem, the numerical formulation can be simplified, particularly in what concerns the calculation of the system matrix, and significant computational gains can be obtained. The results of a numerical study concerning the behaviour of a sequence of embedded inclusions within an elastic material, when subject to the incidence of waves with different frequencies, is here presented, and the interpretation of the involved phenomena is described in order to clarify the main wave propagation features in the presence of multiple elastic inclusions. The computed results are promising, clearly revealing the existence of band gaps where large attenuation occurs, although limitations related to the existence of guided waves traveling along the inclusions are also identified.

The dynamics of multi-connected thermoelastic semiplane with the non-stationary power source and thermal effects by using of a model of coupled thermoelasticity is investigated. Green’s tensor in the space of the Laplace transforms in time describes the displacements of medium under the effect of the impulse concentrated power and thermal sources. The generalized solution of the problem of the dynamics of thermoelastic semiplane with the free boundary under the effect of arbitrary mass forces and thermal sources in 2D-case is built.

In this study, we investigate the distribution of eigenfrequencies of boundary integral equations (BIEs) of two-dimensional elastodynamics. The corresponding eigenvalue problem is classified as a nonlin- ear eigenvalue problem. We confirm that the Burton-Miller formulation can properly avoid fictitious eigenfrequencies. The boundary element method (BEM) is expected as a powerful numerical tool for designing sophisticated devices related to elastic waves such as acoustic metamaterials. However, the BEM is known that it loses its accuracy for certain frequencies, called as fictitious eigenfrequencies, for problems defined in the infinite domain. Recent researches It has also been revealed that not only the real-valued eigenfrequencies but also the complex-valued ones may affect the accuracy of the BEM results. We examine the distribution of complex eigenvalues obtained by BIEs for time-harmonic elas- todynamic problems with the help of the Sakurai-Sugiura method which is applicable to nonlinear eigenvalue problems. We also examine its relation to the accuracy of the BEM numerical results. We also discuss an appropriate choice of the coupling parameter from a viewpoint of the distribution of fictitious eigenfrequencies.

A computational tool that integrates a Radial basis function (RBF)-based Meshless solver with a Lumped Parameter model (LPM) is developed to analyze the multi-scale and multi-physics interaction between the cardiovascular flow hemodynamics, the cardiac function, and the peripheral circulation. The Meshless solver is based on localized RBF collocations at scattered data points which allows for automation of the model generation via CAD integration. The time-accurate incompressible flow hemodynamics are addressed via a pressure-velocity correction scheme where the ensuing Poisson equations are accurately and efficiently solved at each time step by a Dual-Reciprocity Boundary Element method (DRBEM) formulation that takes advantage of the integrated surface discretization and automated point distribution used for the Meshless collocation. The local hemodynamics are integrated with the peripheral circulation via compartments that account for branch viscous resistance (R), flow inertia (L), and vessel compliance (C), namely RLC electric circuit analogies. The cardiac function is modeled via time-varying capacitors simulating the ventricles and constant capacitors simulating the atria, connected by diodes and resistors simulating the atrioventricular and ventricular-arterial valves. This multi-scale integration in an in-house developed computational tool opens the possibility for model automation of patient-specific anatomies from medical imaging, elastodynamics analysis of vessel wall deformation for fluid-structure interaction, automated model refinement, and inverse analysis for parameter estimation.

Based on the recently developed finite integration method (FIM) for solving one and two dimensional ordinary and partial differential equations, this paper extends FIM to both stationary and transient heat conduction inverse problems for anisotropic and functionally graded materials with high degree of accuracy. Lagrange series approximation is applied to determine the first order of integral and differential matrices, which are used to form the system equation matrix for two and three dimensional problems. Singular Value Decomposition (SVD) is applied to solve the ill-conditioned system of algebraic equations obtained from the integral equation, boundary conditions and scattered temperature measurements. The convergence and accuracy of this method are investigated with two examples for anisotropic media and functionally graded materials.

It is well known that the original 3D elasticity problem in plate structures subjected to transversal loading can be converted to a 2D problem. In addition to in-plane displacements, we need to introduce the deflection and/or rotation field variables in the plate mid-plane, in order to describe displacements and deformations within the plate structure. Thus, one can develop unified formulation for bending and in- plane deformation modes within the classical Kirchhoff-Love theory for bending of thin elastic plates and the shear deformation plate theories (the first order – FSDPT, and the third order - TSDPT). In this paper, we extend the derivation of the 2D formulation for coupled problems of thermoelasticity in plate structures. Three material coefficients play the role in stationary problems, namely the Young modulus, coefficient of linear thermal extension and the heat conduction coefficient. The influence of continuous gradation of these coefficients on the response of the plate subjected to thermal loadings is investigated in numerical simulations. The element-free strong formulation with using meshless approximations for spatial variation of field variables is developed.

This study investigates solvability of boundary value problems of plane elasticity formulated in terms of principal directions of the stress tensor and the orientations of the displacement vector. The analysis of solvability is performed by using the following approach. Firstly, boundary values of the complex potentials are represented by the Cauchy-type integrals with unknown density. Then a system of singular integral equations is obtained by satisfying particular boundary conditions. This system is further reduced to the system of the Riemann boundary value problems for the determination of sectionally holomorphic functions. Solvability of the Riemann problems is investigated by calculating their indexes. This allows one to determine the number of linearly independent solutions and hence the number of arbitrary parameters entering into the general solution.

Two novel formulations have been investigated for the case of elastic half-planes. In both cases the initial system of equations has been reduced to the form that allow for successive solution of its equations.

The paper revisits the use of a surface equivalence theorem in deriving the surface integral equation (SIE) based formulation for a homogeneous bio-electromagnetics problem. The vector analog of Green’s 2nd identity is used to obtain the expression for the electric field representing the mathematical foundation of the equivalence theorem. The particular emphasis is put on the treatment of boundary integral when the observation and source points, respectively, coincide. The boundary conditions at infinity are taken into account via the Sommerfeld radiation conditions. The derived coupled SIE set can be used in problems involving biological body exposed to electromagnetic field radiation.