# Engineering Analysis with Boundary Elements

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ZZ-Type a posteriori error estimators for adaptive boundary element methods on a curve
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January 2014

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In the context of the adaptive finite element method (FEM), ZZ-error estimators named after Zienkiewicz and Zhu (1987)  are mathematically well-established and widely used in practice. In this work, we propose and analyze ZZ-type error estimators for the adaptive boundary element method (BEM). We consider weakly singular and hyper-singular integral equations and prove, in particular, convergence of the related adaptive mesh-refining algorithms. Throughout, the theoretical findings are underlined by numerical experiments.

A boundary integral approach to analyze the viscous scattering of a pressure wave by a rigid body

February 2007

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The paper provides boundary integral equations for solving the problem of viscous scattering of a pressure wave by a rigid body. By using this mathematical tool uniqueness and existence theorems are proved. Since the boundary conditions are written in terms of velocities, vector boundary integral equations are obtained for solving the problem. The paper introduces single-layer viscous potentials and also a stress tensor. Correspondingly, a viscous double-layer potential is defined. The properties of all these potentials are investigated.By representing the scattered field as a combination of a single-layer viscous potential and a double-layer viscous potential the problem is reduced to the solution of a singular vectorial integral equation of Fredholm type of the second kind.In the case where the stress vector on the boundary is the main quantity of interest the corresponding boundary singular integral equation is proved to have a unique solution.

A Case Study in Parallel Scientific Computing: The Boundary Element Method on a Distributed-Memory Multicomputer

February 1995

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The Boundary Element Method is a widely-used discretization technique for solving boundary-value problems in engineering analysis. The solution of large problems by this method is limited by the storage and computational requirements for the generation and solution of large matrix systems resulting from the discretization. We discuss the implementation of these computations on the IBM SP-2 distributed-memory parallel computer, for applications involving the 3DD Laplace and Helmholtz equations.

Defining an accurate MFS solution for 2.5D acoustic and elastic wave propagation

December 2009

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This paper proposes a simple methodology to assess the accuracy of the method of fundamental solutions (MFS) when applied to 2.5D acoustic and elastic wave propagation. The proposed technique is developed in the frequency domain. It copes with the precision uncertainty difficulty presented by the MFS solution through its dependency on the number and position of virtual sources and collocation points.The methodology relies on the correlation between the errors registered along surfaces, where boundary or continuity conditions are known a priori, with those obtained along the system domain. Circular cylindrical domains are modeled to illustrate the efficiency of the proposed methodology, since in this case analytical solutions are available.A numerical example is used to illustrate the application of the methodology to a more complex case. An elastic column exhibiting an embedded curved crack, with null thickness, is used to illustrate the applicability of the proposed technique. Since, there are no known analytical solutions; the results provided by the traction boundary element method (TBEM) are used as reference solutions.

Study of transient heat conduction in 2.5D domains using the boundary element method

June 2004

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This paper presents the solution for transient heat conduction around a cylindrical irregular inclusion of infinite length, inserted in a homogeneous elastic medium and subjected to heat point sources placed at some point in the host medium. The solution is computed in the frequency domain for a wide range of frequencies and axial wavenumbers, and time series are then obtained by means of (fast) inverse Fourier transforms into space–time.The method and the expressions presented are implemented and validated by applying them to a cylindrical circular inclusion placed in an infinite homogeneous medium and subjected to a point heat source, for which the solution is calculated in closed form.The boundary elements method is then used to evaluate the temperature field generated by a point source in the presence of a cylindrical inclusion, with a non-circular cross-section, inserted in an unbounded homogeneous medium. Simulation analyses using this model are then performed to study the transient heat conduction in the vicinity of these inclusions.

Applications of the dual integral formulation in conjunction with fast multipole method in large-scale problems for 2D exterior acoustics

June 2004

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In this paper, we solve the large-scale problem for exterior acoustics by employing the concept of fast multipole method (FMM) to accelerate the construction of influence matrix in the dual boundary element method (DBEM). By adopting the addition theorem, the four kernels in the dual formulation are expanded into degenerate kernels, which separate the field point and source point. The separable technique can promote the efficiency in determining the coefficients in a similar way of the fast Fourier transform over the Fourier transform. The source point matrices decomposed in the four influence matrices are similar to each other or only some combinations. There are many zeros or the same influence coefficients in the field point matrices decomposed in the four influence matrices, which can avoid calculating repeatedly the same terms. The separable technique reduces the number of floating-point operations from O(N2) to where N is number of elements and a is a small constant independent of N. To speed up the convergence in constructing the influence matrix, the center of multipole is designed to locate on the center of local coordinate for each boundary element. This approach enhances convergence by collocating multipoles on each center of the source element. The singular and hypersingular integrals are transformed into the summability of divergent series and regular integrals. Finally, the FMM is shown to reduce CPU time and memory requirement thus enabling us apply BEM to solve for large-scale problems. Five moment FMM formulation was found to be sufficient for convergence. The results are compared well with those of FEM, conventional BEM and analytical solutions and it shows the accuracy and efficiency of the FMM when compared with the conventional BEM.

Adaptive 2D IGA boundary element methods

April 2015

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We derive and discuss a posteriori error estimators for Galerkin and collocation IGA boundary element methods for weakly-singular integral equations of the first-kind in 2D. While recent own work considered the Faermann residual error estimator for Galerkin IGA boundary element methods, the present work focuses more on collocation and weighted- residual error estimators, which provide reliable upper bounds for the energy error. Our analysis allows piecewise smooth parametrizations of the boundary, local mesh-refinement, and related standard piecewise polynomials as well as NURBS. We formulate an adaptive algorithm which steers the local mesh-refinement and the multiplicity of the knots. Numerical experiments show that the proposed adaptive strategy leads to optimal convergence, and related IGA boundary element methods are superior to standard boundary element methods with piecewise polynomials.

BIEM for 2D steady-state problems in cracked anisotropic materials

July 2005

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The two-dimensional ‘in-plane’ time-harmonic elasto-dynamic problem for anisotropic cracked solid is studied. The non-hypersingular traction boundary integral equation method (BIEM) is used in conjunction with closed form frequency dependent fundamental solution, obtained by Radon transform. Accuracy and convergence of the numerical solution for stress intensity factor (SIF) is studied by comparison with existing solutions in isotropic, transversely-isotropic and orthotropic cases. In addition a parametric study for the wave field sensitivity on wave, crack and anisotropic material parameters is presented.

Hybrid extended displacement discontinuity-charge simulation method for analysis of cracks in 2D piezoelectric media

May 2009

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The extended displacement discontinuity method (EDDM) and the charge simulation method (CSM) are combined to develop an efficient approach for analysis of cracks in two-dimensional piezoelectric media. In the proposed hybrid EDD–CSM, the solution for an electrically impermeable crack is approximately expressed by a linear combination of fundamental solutions of the governing equations, which includes the extended point force fundamental solutions with the sources placed at chosen points outside the domain of the problem under consideration and the extended Crouch fundamental solutions with the extended displacement discontinuities placed on the crack. The coefficients of the fundamental solutions are determined by letting the approximated solution satisfy the conditions on the boundary of the domain and on the crack face. Furthermore, the hybrid EDD–CSM is applied to solve the problems of cracks under electrically permeable condition, as well as under semi-permeable conditions by using an iterative approach. Two important crack problems in fracture mechanics, the center cracks and the edge cracks in piezoelectric strips, are analyzed by the proposed method. The stress intensity factor and the electric displacement intensity factor are calculated. Meanwhile the effects of strip size and the electric boundary conditions on these intensity factors are studied.

Stress and velocity in 2D transient elastodynamic analysis by the boundary element method

March 1999

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This paper is mainly concerned with the development of integral equations to compute stress and velocity components in transient elastodynamic analysis by the boundary element method. All expressions required are presented explicitly. The boundary is discretized by linear isoparametric elements whereas linear and constant time interpolation are assumed, respectively, for the displacement and traction components. Time integration is carried out analytically and the resulting expressions are presented. An assessment of the accuracy of the results provided by the present formulation can be seen at the end of the article, where two examples are presented.

RBF-based meshless methods for 2D elastostatic problems

October 2004

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The work presented here concerns the use of radial basis functions (RBFs) for the analysis of two dimensional elastostatic problems. The basic characteristic of the formulation is the definition of a global approximation for the variables of interest in each problem (the deflection for the plate bending problem and the stress function for the stretching plates) from a set of RBFs conveniently placed (but not necessarily in a regular manner) at the boundary and in the domain. Depending on the type of collocation chosen, non-symmetric or symmetric systems of linear equations are obtained. Comparisons are made with other results available in the literature.

Fully discrete Alpert multiwavelet Galerkin BEM in 2D

February 2008

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A fully discrete Galerkin boundary element method (BEM) based on Alpert multiwavelets is proposed for fast solution of Laplace's boundary integral equations in two dimensions. To make it more suitable for practical use, the highest resolution levels in the boundary patches are allowed to be different from each other. New patch and level dependent cut-off parameters which can compress the nonzero entries to at most O(NlogN) (where N is the degrees of freedom) are presented. A diagonal preconditioner is utilized to improve the system matrix. To evaluate the logarithmic singular double integrals more efficiently, coordinate transformations are introduced to remove the singularities. Numerical results show that the method can achieve O(NlogN) complexity.

Interval boundary element method for 2D transient diffusion problem using the directed interval arithmetic

March 2011

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In this paper the description of an unsteady heat transfer for a two-dimensional problem is presented. It is assumed that all thermophysical parameters appearing in the mathematical model of the problem analyzed are given as directed intervals. The problem discussed has been solved using the 1st scheme of the boundary element method. The interval Gauss elimination method with the decomposition procedure has been applied to solve the obtained interval system of equations. In the final part of this paper results of the numerical computations are shown.

An adaptive element subdivision technique for evaluation of various 2D singular boundary integrals

August 2008

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In this paper, a unified algorithm is presented for the numerical evaluation of weakly, strongly and hyper singular boundary integrals with or without a logarithmic term, which often appear in two-dimensional boundary element analysis equations. In this algorithm, the singular boundary element is broken up into a few sub-elements. The sub-elements involving the singular point are evaluated analytically to remove the singularities by expressing the non-singular parts of the integration kernels as polynomials of the distance r, while other sub-elements are evaluated numerically by the standard Gaussian quadrature. The number of sub-elements and their sizes are determined according to the singularity order and the position of the singular point. Numerical examples are provided to demonstrate the correctness and efficiency of the proposed algorithm.

Efficient 2D and 3D electrical impedance tomography using dual reciprocity boundary element techniques

July 1998

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Numerical algorithms based on boundary element methods are developed for application to problems in Electrical Impedance Tomography (EIT). Two types of EIT problems are distinguished. In the first type internal boundaries of domains of constant conductivity are imaged. For such problems an algorithm based on identifying the shape of the included region is developed, and uses conventional BEM techniques. For problems where a distribution of conductivity is to be imaged algorithms that use dual reciprocity techniques are developed.The size of the inverse problem required to be solved is much reduced, offering substantial speed-ups over conventional techniques. Further, the present algorithms use simple parametrization of the unknowns to achieve efficiency. Numerical results from tests of this algorithm on synthetic data are presented, and these show that the method is quite promising.

A BEM analysis of fracture mechanics in 2D anisotropic piezoelectric solids

January 1999

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This paper presents a single-domain boundary element method (BEM) analysis of fracture mechanics in 2D anisotropic piezoelectric solids. In this analysis, the extended displacement (elastic displacement and electrical potential) and extended traction (elastic traction and electrical displacement) integral equations are collocated on the outside boundary (no-crack boundary) of the problem and on one side of the crack surface, respectively. The Green's functions for the anisotropic piezoelectric solids in an infinite plane, a half plane, and two joined dissimilar half-planes are also derived using the complex variable function method. The extrapolation of the extended relative crack displacement is employed to calculate the extended `stress intensity factors' (SIFs), i.e., KI, KII, KIII and KIV. For a finite crack in an infinite anisotropic piezoelectric solid, the extended SIFs obtained with the current numerical formulation were found to be very close to the exact solutions. For a central and inclined crack in a finite and anisotropic piezoelectric solid, we found that both the coupled and uncoupled (i.e., the piezoelectric coefficient eijk=0) cases predict very similar stress intensity factors KI and KII when a uniform tension σyy is applied, and very similar electric displacement intensity factor KIV when a uniform electrical displacement Dy is applied. However, the relative crack displacement and electrical potential along the crack surface are quite different for the coupled and uncoupled cases. Furthermore, for a inclined crack within a finite domain, we found that while a uniform σyy (=1 N m−2) induces only a very small electrical displacement intensity factor (in the unit of Cm−3/2), a uniform Dy (=1 C m−2) can produce very large stress intensity factors (in the unit of Nm−3/2).

Condition number of the BEM matrix arising from the Stokes equations in 2D

September 2008

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We study the condition number of the system matrices that appear in the boundary element method when solving the Stokes equations at a 2D domain. At the boundary of the domain we impose Dirichlet conditions or mixed conditions. We show that for certain critical boundary contours the underlying boundary integral equation is not uniquely solvable. As a consequence, the condition number of the system matrix of the discrete equations is infinitely large. Hence, for these critical contours the Stokes cannot be solved by the boundary element method. To overcome this problem the domain can be rescaled. Several numerical examples are provided to illustrate the solvability problems at the critical contours.

Application of a direct Trefftz method with domain decomposition to 2D potential problems

July 1999

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This article presents an application of a direct Trefftz method with domain decomposition method to the two-dimensional potential problem. In the direct Trefftz methods, regular T-complete functions satisfying the governing equations are taken as the weighting functions and then, the boundary integral equations are derived from the weighted residual expressions of the governing equations. Since the T-complete functions are regular, the final equations are also regular and therefore, much simpler than the ordinary boundary element methods employing the singular fundamental solutions. Their computational accuracy, however, is dependent on the condition number of the coefficient matrices of the algebraic system of equations. So, for improving the accuracy, we introduce the domain decomposition method to the direct Trefftz methods. The present method is applied to the two-dimensional potential problem in order to confirm the validity.

Dual error indicators for the local boundary integral equation method in 2D potential problems

August 2006

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Three relative error measurements in the numerical solution of potential problems are firstly investigated in detail, and then an algorithm based on the proposed dual error indicators is developed for the meshless local boundary integral equation (LBIE) method. Numerical experiments show that a combined use of the two error indicators is necessary to adequately measure the error of the LBIE solutions.

Explicit formulae of the integral kernels and C-matrix in the 2D Somigliana identity for orthotropic materials

December 1995

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Simple explicit complex representations of the fundamental displacements and tractions using Lekhnitskii and Stroh theories of plane orthotropic elastic bodies are introduced. The representation of the fundamental tractions is applied to derive in a simple way new real analytical formulae of the coefficient matrix of the free term in the boundary form of the 2D Somigliana identity for orthotropic materials.

Boundary-only element solutions of 2D and 3D nonlinear and nonhomogeneous elastic problems

December 2007

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This paper presents a robust boundary element method (BEM) that can be used to solve elastic problems with nonlinearly varying material parameters, such as the functionally graded material (FGM) and damage mechanics problems. The main feature of this method is that no internal cells are required to evaluate domain integrals appearing in the conventional integral equations derived for these problems, and very few internal points are needed to improve the computational accuracy. In addition, one of the basic field quantities used in the boundary integral equations is normalized by the material parameter. As a result, no gradients of the field quantities are involved in the integral equations. Another advantage of using the normalized quantities is that no material parameters are included in the boundary integrals, so that a unified equation form can be established for multi-region problems which have different material parameters. This is very efficient for solving composite structural problems.

2D velocity vorticity based LES for the solution of natural convection in a differentially heated enclosure by wavelet transform based BEM and FEM

August 2006

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A wavelet transform based boundary element method (BEM) numerical scheme is proposed for the solution of the kinematics equation of the velocity-vorticity formulation of Navier–Stokes equations. FEM is used to solve the kinetics equations. The proposed numerical approach is used to perform two-dimensional vorticity transfer based large eddy simulation on grids with 105 nodes. Turbulent natural convection in a differentially heated enclosure of aspect ratio 4 for Rayleigh number values Ra=107–109 is simulated. Unstable boundary layer leads to the formation of eddies in the downstream parts of both vertical walls. At the lowest Rayleigh number value an oscillatory flow regime is observed, while the flow becomes increasingly irregular, non-repeating, unsymmetric and chaotic at higher Rayleigh number values. The transition to turbulence is studied with time series plots, temperature–vorticity phase diagrams and with power spectra. The enclosure is found to be only partially turbulent, what is qualitatively shown with second order statistics—Reynolds stresses, turbulent kinetic energy, turbulent heat fluxes and temperature variance. Heat transfer is studied via the average Nusselt number value, its time series and its relationship to the Rayleigh number value.

2D Simulation of composite materials using BEM

August 2004

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This paper presents the recent progress achieved by the authors' group on the simulation of 2D elastic solids containing a large number of randomly distributed inclusions using BEM. A new scheme of repeated similar sub-domain BEM is proposed in this paper. The randomly distributed inclusions investigated include identical circular inclusions, elliptical inclusions with identical size and shape but different orientation, and also inclusions with different shape, size and different material properties. The number of inclusions is approximately 100 by conventional BEM, and it can be more than 1000 by the fast multipole BEM on only one PC. The interface between the matrix and inclusion can be at the ideal interface as well as at the interface with interphase layers. The numerical results show that for such investigations, the BEM is more suitable compared to FEM and other numerical methods. The simulated effective elastic modulus are presented and compared with different theoretical results. Future investigations should be extended to simulation of 3D models and simulation on brittle failure process.

Symmetry properties of the kernels of the hypersingular integral and the corresponding regularized integral in the 2D Somigliana stress identity for isotropic materials

September 1997

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Hypersingular integral kernel Sijk in the two-dimensional Somigliana stress identity for isotropic materials and the corresponding strongly singular kernel in the regularized Somigliana stress identity are analysed. It is shown, by direct calculation of the kernels' components individually, that they are completely symmetric tensors. Interpretation of these kernels as influence functions giving stresses due to an edge dislocation dipole and an edge dislocation, together with their expressions by derivatives of the Airy stress function due to a wedge disclination, give an insight into this, not immediately evident, symmetry. Finally, symmetrical formulae of these kernels and of the pertinent hypersingular and strongly singular kernels in the traction boundary integral equations are derived.

Solution of 2D and 3D Stokes law using multiquadrics method

April 2004

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In this paper, velocity–vorticity formulation and the multiquadrics method (MQ) with iterative scheme are used to solve two (2D) and three-dimensional (3D) steady-state incompressible Stokes cavity flows. The method involves solving of Laplace type vorticity equations and Poisson type velocity equations. The solenoidal velocity and vorticity components are obtained by iterative procedures through coupling of velocity and vorticity fields. Both the Poisson type velocity equations and the Laplace type vorticity equations are solved using the MQ, which renders a meshless (or meshfree) solution. Here, the results of 2D Stokes flow problems in a typical square cavity and a circular cavity are presented and compared with other model results. Besides utilizing the MQ to solve the 3D Stokes cubic cavity flow problem, we are also obtaining promising results for the accuracy of the velocity and vorticity. The MQ model has been found to be very simple and powerful for analyzing the 2D and 3D internal Stokes flow problems.

Analytical diagonal elements of regularized meshless method for regular domains of 2D Dirichlet Laplace problems

January 2010

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This note is to present a simple approach to derive the analytical formula of the diagonal elements of the interpolation matrix of the regularized meshless method (RMM) for regular domain problems, which is a very new boundary-type numerical discretization technique. In literature, these diagonal elements are mostly calculated numerically by the desingular technique, except for the circular domain problems. Our numerical experiments show that the analytical diagonal elements can improve the solution accuracy of the RMM for some regular domain problems, and the diagonal elements are critical to the solution accuracy of the RMM. Thus, a searching process is employed to find the optimal diagonal elements for RMM.

Meshless solutions of 2D contact problems by subdomain variational inequality and MLPG method with radial basis functions

February 2005

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A subdomain variational inequality and its meshless linear complementary formulation are developed in the present paper for solving two-dimensional contact problems. The subdomain variational inequality will be defined in detail. The meshless method is based on a local weighted residual method with the Heaviside step function as the weighting function over a local subdomain and radial basis functions as trial functions for interpolation. Three different radial basis functions (RBFs), i.e. Multiquadrics (MQ), Gaussian (EXP) and Thin Plate Splines (TPS) are examined and the selection of their shape parameters is studied based on 2D solid stress problems with closed-form solutions. The developed meshless/linear complementary method is applied to solve two frictionless contact problems. For the RBFs, it has been found that the TPS shape parameter is not sensitive to nodal distance and a value of 4 is found as a good choice for TPS from this research.

Fast multipole boundary element analysis for 2D problems of magneto-electro-elastic media

November 2010

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A two-dimensional (2D) fast multipole boundary element analysis of magneto-electro-elastic media has been developed in this paper. Fourier analysis is employed to derive the fundamental solution for the plane-strain magneto-electro-elasticity. The final formulations are very similar to those for the 2D potential problems, and hence it is quite easy to implement the fast multipole boundary element method. The results are verified by comparison with the analytical solutions to illustrate the accuracy and efficiency of the approach. The numerical examples of multi-inclusion magneto-electro-elastic composites are considered to show the versatility of the proposed approach in smart structure applications.

On the preconditioners for fast multipole boundary element methods for 2D multi-domain elastostatics

July 2005

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Fast multipole method (FMM) has been successfully applied to accelerate the numerical solvers of boundary element method (BEM). However, the coefficient matrix implicitly formed by using FMM is sometimes ill-conditioned in cases when mixed boundary conditions exist, resulting in poor rate of convergence for iteration. So preconditioning is a critical part in the development of efficient FMM solver for BEM. In this paper, preconditioners based on sparse approximate inverse type are used for fast multipole BEM to deal with 2D elastostatics. Several sparsity patterns of the preconditioner are considered for single- and multi-domain problems, especially for 2D elastic body with large number of inclusions or cracks. Algorithms and cost analysis of preconditioning under different prescribed sparsity patterns are discussed. GMRES is used as the iterative solver. Numerical results show this type of preconditioner achieves satisfactory rate of convergence for fast multipole BEM and performs well for problems of fairly large sizes.

Dual reciprocity BEM analysis of 2D transient elastodynamic problems by time-discontinuous Galerkin FEM

June 2003

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This work applies the dual reciprocity boundary element method (DRBEM) to the transient analysis of two-dimensional elastodynamic problems. Adopting the elastostatic fundamental solution in the integral formulation of elastodynamics creates an inertial volume integral as well as the boundary ones. This volume integral is further transformed into a surface integral by invoking the reciprocal theorem. The analysis includes the quadratic three-noded boundary elements in the spatial domain. Importantly, the second-order ordinary differential equations in the time domain formulated by the DRBEM are solved using the time-discontinuous Galerkin finite element method. Particularly, both the displacement and velocity variables in the time domain are independently represented by quadratic interpolation functions that allow the unknown variables to be discontinuous at the discrete time levels. This method can filter out the spurious high modes and provide solutions with a fifth-order accuracy. Numerical examples are presented, confirming that the proposed method is more stable and accurate than widespread direct time integration algorithms, such as the Houbolt method.

3D elastic wave propagation modelling in the presence of 2D fluid-filled thin inclusions

March 2006

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In this paper, the traction boundary element method (TBEM) and the boundary element method (BEM), formulated in the frequency domain, are combined so as to evaluate the 3D scattered wave field generated by 2D fluid-filled thin inclusions. This model overcomes the thin-body difficulty posed when the classical BEM is applied. The inclusion may exhibit arbitrary geometry and orientation, and may have null thickness. The singular and hypersingular integrals that appear during the model's implementation are computed analytically, which overcomes one of the drawbacks of this formulation. Different source types such as plane, cylindrical and spherical sources, may excite the medium. The results provided by the proposed model are verified against responses provided by analytical models derived for a cylindrical circular fluid-filled borehole.The performance of the proposed model is illustrated by solving the cases of a flat fluid-filled fracture with small thickness and a fluid-filled S-shaped inclusion, modelled with both small and null thickness, all of which are buried in an unbounded elastic medium. Time and frequency responses are presented when spherical pulses with a Ricker wavelet time evolution strikes the cracked medium. To avoid the aliasing phenomena in the time domain, complex frequencies are used. The effect of these complex frequencies is removed by rescaling the time responses obtained by first applying an inverse Fourier transformation to the frequency domain computations. The numerical results are analysed and a selection of snapshots from different computer animations is given. This makes it possible to understand the time evolution of the wave propagation around and through the fluid-filled inclusion.

Sensitivity analysis scheme of boundary value problem of 2D Poisson equation by using Trefftz method

July 2005

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This paper describes the sensitivity analysis of the boundary value problem of two-dimensional Poisson equation by using Trefftz method. A non-homogeneous term of two-dimensional Poisson equation is approximated with a polynomial function in the Cartesian coordinates to derive a particular solution. The unknown function of the boundary value problem is approximate with the superposition of the T-complete functions of Laplace equation and the derived particular solution with unknown parameters. The parameters are determined so that the approximate solution satisfies boundary conditions. Since the T-complete functions and the particular solution are regular, direct differentiation of the expression results leads to the sensitivity expressions. The boundary-specified value and the shape parameter are taken as the variables of the sensitivity analysis to formulate the sensitivity analysis methods. The present scheme is applied to some numerical examples in order to confirm the validity of the present algorithm.

Shape design sensitivity analysis of 2D anisotropic structures using the boundary element method

March 2002

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A directly differentiated form of boundary integral equation with respect to geometric design variables is used to calculate shape design sensitivities for anisotropic materials. An optimum shape design algorithm in two dimensions is developed by the coupling of an optimising technique and a boundary element stress analyser for stress minimisation of anisotropic structures. Applications of this general-purpose program to the optimum shape design of bars and holes in plates with anisotropic material properties are presented.

Vectorization and parallelization of the acoustic boundary element code BEMAP on the IBM ES/3090 VF

December 1992

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A boundary element method code for acoustic analysis, called BEMAP, is described along with the process of vectorizing and parallelizing this code for implementation on the IBM ES/3090 vector parallel computer.

Topology Optimization of 3D Elastic Structures Using Boundary Elements

July 2008

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Topological optimization provides a powerful framework to obtain the optimal domain topology for several engineering problems. The topological derivative is a function which characterizes the sensitivity of a given problem to the change of its topology, like opening a small hole in a continuum or changing the connectivity of rods in a truss.A numerical approach for the topological optimization of 2D linear elastic problems using boundary elements is presented in this work. The topological derivative is computed from strain and stress results which are solved by means of a standard boundary element analysis. Models are discretized using linear elements and a periodic distribution of internal points over the domain. The total potential energy is selected as cost function. The evaluation of the topological derivative is performed as a post-processing procedure. Afterwards, material is removed from the model by deleting the internal points and boundary nodes with the lowest values of the topological derivate. The new geometry is then remeshed using a weighted Delaunay triangularization algorithm capable of detecting “holes” at those positions where internal points and boundary points have been removed. The procedure is repeated until a given stopping criterion is satisfied.The proposed strategy proved to be flexible and robust. A number of examples are solved and results are compared to those available in the literature.

Meshless local Petrov–Galerkin (MLPG) method for wave propagation in 3D poroelastic solids

April 2010

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A meshless local Petrov–Galerkin (MLPG) method is applied to solve wave propagation problems of three-dimensional poroelastic solids with Biot's theory. The Laplace transform is used to eliminate the time dependence of the field variables for the transient elastodynamic case. A weak formulation with a unit step function transforms the set of governing equations into local integral equations on local subdomains. The meshless approximation based on the radial basis function (RBF) is employed for the implementation. Unknown Laplace-transformed quantities, including displacements of solid frame and pressure in the fluid, are computed from the local boundary integral equations. The time-dependent values are obtained by Durbin's inversion technique. In addition, a one-dimensional poroelasticity analytical solution is derived in this paper and introduced for comparison. Several numerical examples demonstrate the efficiency and accuracy of the proposed method.

3D Galerkin integration without Stokes' theorem

November 2000

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A direct approach to calculating the hypersingular integral for a three-dimensional Galerkin approximation is presented. The method does not employ either Stokes' theorem or a regularization process to transform the integrand before the evaluation is carried out. Integrating two of the four dimensions analytically, the potentially divergent terms arising from the coincident and adjacent edge integrations are identified and canceled exactly. The method is presented in the simplest possible situation, the hypersingular kernel for the Laplace equation, and linear triangular elements.

The scattering of 3D sound sources by rigid barriers in the vicinity of tall buildings

October 2002

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The acoustic scattering of a three-dimensional (3D) sound source by an infinitely long rigid barrier in the vicinity of a tall building is analyzed using the boundary element method (BEM). The acoustic barrier is modeled using boundary elements, and is assumed to be non-absorbing, while the image source method is used to model the tall building as an infinite vertical barrier. A frequency domain BEM formulation is used, and time domain responses are then obtained by applying an inverse Fourier transformation.Since the geometry of the problem does not vary along one direction, the 3D solution can be calculated as the summation of a sequence of 2D problems, each solved for a different spatial wavenumber, kz. To obtain the 3D solution, a discrete form wavenumber transform is performed by considering an infinite number of virtual point sources equally spaced along the z axis. Complex frequencies are used to minimize the influence of these neighboring fictitious sources.Numerical simulations are performed using barriers of varying sizes, evaluating the attenuation of the sound pressure level in the vicinity of the building façade. The creation of shadow zones by the barriers is analyzed and time responses are presented to better understand the sound propagation around these obstacles.

3D acoustic scattering from an irregular fluid waveguide via the BEM

June 2001

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The BEM is used to calculate the variation in the pressure field generated by a dilatational point load inside a channel filled with a homogeneous fluid, in the presence of an irregular floor. The Green's functions are defined in the frequency domain and obtained by superposing virtual acoustic sources combined so as to generate the boundary conditions of the free or rigid surfaces of the channel. The responses in the time domain are obtained by means of Fourier transforms, making use of complex frequencies. The main features and spectral representation of the signals scattered by irregular floors are then described and used to elucidate the most important aspect of wave acoustics, which can provide the basis for the development of non-destructive testing and imaging methods.

Static analysis of 3D damaged solids and structures by BEM

June 2002

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The boundary element method is employed for the analysis of three-dimensional (3D) brittle solids and structures, such as those composed of concrete, rock, ceramics or masonry, under static (monotonic or cycling) loading. The mechanical behavior of these solids can be successfully described by continuum damage mechanical theories. The integral formulation of the problem contains not only boundary, but also volume integrals as well, accounting for damage effects. Thus, in addition to the boundary one, an interior discretization is necessary, which can be restricted to those parts of the structure expected to behave inelastically. Isoparametric linear quadrilateral elements are used for the surface discretization and isoparametric linear hexahedra for the interior discretization. Advanced numerical integration techniques for singular and nearly singular integrals are employed. Numerical examples involving 3D concrete type structures under static loads are presented to illustrate the method and demonstrate its advantages.

Numerical analysis of 3D electrostatics of deformable conductors using a Lagrangian approach

June 2004

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In this paper, we introduce a Lagrangian approach for 3D electrostatic analysis. In this approach, when the conductors undergo deformation or shape changes, the surface charge densities on the deformed conductors can be computed without updating the geometry of the conductors. This alleviates the computation-intensive task of remeshing the structure and regeneration of shape functions.

Particular solutions of 3D Helmholtz-type equations using compactly supported radial basis functions

September 2000

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In this paper we show how to obtain analytic particular solutions for inhomogeneous Helmholtz-type equations in 3D when the source terms are compactly supported radial basis functions (CS-RBFs) (J. Approx. Theory, 93 (1998) 258). Using these particular solutions we demonstrate the solvability of boundary value problems for the inhomogeneous Helmholtz-type equation by approximating the source term by CS-RBFs and solving the resulting homogeneous equation by the method of fundamental solutions. The proposed technique is a truly mesh-free method and is especially attractive for large-scale industrial problems. A numerical example is given which illustrates the efficiency of the proposed method.

An advanced 3D boundary element method for characterizations of composite materials

June 2005

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Some recent developments in the modeling of composite materials using the boundary element method (BEM) are presented in this paper. The boundary integral equation for 3D multi-domain elasticity problems is reviewed. Difficulties in dealing with nearly-singular integrals, which arise in the BEM modeling of composite materials with closely packed fillers or of thin films, are discussed. New and improved techniques to deal with the nearly-singular integrals in the 3D elasticity BEM are presented. Numerical examples of layered thin films and composites with randomly distributed particles and fibers are studied. The advantages and limitations of the BEM approach in modeling advanced composites are also discussed. The developed BEM with multi-domain and thin-body capabilities is demonstrated to be a promising tool for simulations and characterization of various composite materials.

Preconditioned multi-zone boundary element analysis for fast 3D electric simulation

September 2004

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For fast 3D electric simulation, the multi-zone collocation boundary element analysis (BEA) with iterative solver like GMRES algorithm is required. In this paper, we present a scheme of equation assembly with ordering unknowns and collocation points, and a matrix storing structure. A group of easily computed preconditioners based on the mesh neighbor method are then proposed to remarkably quicken the convergence of GMRES iteration, and demonstrate at least 30% time reduction than using the diagonal preconditioner. Compared with the equation assembly in Merkel et al. [Engng Anal Bound Elem 22 (1998) 183], where two unknowns on same interfacial node are arranged subsequently, and two storing schemes in Li et al. and Araujo et al. [Sys Engng Electron 21 (1999) 10; J Chin Inst Engrs 23 (2000) 269], the proposed method generates fewest non-zero blocks and facilitates the matrix–vector multiplication remarkably. Numerical experiments verify the analysis and show a fast iterative multi-zone BEA simulator for actual very large-scale integration interconnects with a large amount of zones.

Iterative coupling of FEM and BEM in 3D transient elastodynamics

August 2005

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A domain decomposition approach is presented for the transient analysis of three-dimensional wave propagation problems. The subdomains are modelled using the FEM and/or the BEM, and the coupling of the subdomains is performed in an iterative manner, employing a sequential Neumann–Dirichlet interface relaxation algorithm which also allows for an independent choice of the time step length in each subdomain. The approach has been implemented for general 3D problems. In order to investigate the convergence behaviour of the proposed algorithm, using different combinations of FEM and BEM subdomains, a parametric study is performed with respect to the choice of the relaxation parameters. The validity of the proposed method is shown by means of two numerical examples, indicating the excellent accuracy and applicability of the new formulation.

On computing boundary limiting values of boundary integral with strongly singular and hypersingular kernels in 3D BEM for elastostatics

December 1994

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A few original or recently described numerical methods are presented for direct computation of boundary limits of strongly singular and hypersingular boundary integrals appearing in three-dimensional BEM applied to elastostatics. The methods differ from each other in dealing with integrand singularity: Kutt's formulae in the radial direction, analytical integration in the radial direction of the leading homogeneous term of the integrand expansion and regularizations using Stokes' integral theorem. The advanced unified approach for their implementations is based on the polar coordinate transformation in the parameter plane of the singular element. Due to various groups of performed numerical tests for surfaces discretized by flat or curved boundary elements with linear, quadratic and cubic shape functions, it is possible to compare the methods for their stability and rate of convergence to exact values.

Analytical integrations for the approximation of 3D hyperbolic scalar boundary integral equations

November 2010

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The present note deals with 3D hyperbolic scalar problems modeled by integral equations . It aims at showing analytical integrations for collocation in time  as well as “variational”  approximation schemes. Numerical solution is achieved adopting polynomial shape functions of arbitrary degree (in space and time) on a trapezoidal flat tiling of a polygonal domain. Analytical integrations are performed both in space and time for Lebesgue integrals, working in a local coordinate system. For singular integrals, both a limit to the boundary as well as the finite part of Hadamard approach have been pursued. Outcomes and computational issues are presented. Extension to elastodynamics will be the subject of forthcoming publications.

A 3D boundary element optimization approach based on genetic algorithms and surface modeling

November 2004

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The optimization of three dimensional shapes is one of the most common problems addressed in engineering. This work concerns the use of genetic algorithms (GAs) and β-spline-surface modeling for the optimization of boundary element models. The paper summarizes the genetic optimization process. The model boundary is discretized by using the Boundary Element Method (BEM), and selected parts of the boundary are modeled by using β-spline surfaces, in order to allow an easy re-meshing and adaptation of the boundary to external actions. Two numerical examples are presented and discussed in detail, showing that the proposed technique is able to optimize the shape of domains requiring a minimum of computational effort. The reduction in the model volume is significant, without violating the restrictions imposed on the model.

3D multidomain BEM for a Poisson equation

May 2009

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