Boundary integral equation formulation for the generalized micro-polar thermoviscoelasticity

A formulation of the boundary integral equation method for generalized linear micro-polar thermoviscoelasticity is given. Fundamental solutions, in Laplace transform domain, of the corresponding differential equations are obtained. The initial, mixed boundary value problem is considered as an example illustrating the BIE formulation. The results are applicable to the generalized thermoelasticity theories: Lord-Shulman with one relaxation time, Green-Lindsay with two relaxation times, Green-Naghdi theories, and Chandrasekharaiah and Tzou with dual-phase lag, as well as to the dynamic coupled theory. The cases of generalized linear micro-polar thermoviscoelasticity of Kelvin-Voigt model, generalized linear thermoviscoelasticity and generalized thermoelasticity can be obtained from the given results.     

A symmetric boundary integral formulation for cohesive interface problems

An incremental symmetric boundary integral formulation for the problem of many domains connected by non-linear cohesive interfaces is here presented. The problem of domains with traction-free cracks and/or rigid connections are particular instances of the proposed cohesive formulation. The numerical approximation of the considered problem is achieved by the symmetric Galerkin boundary element method.     

Volume integration in the hypersingular boundary integral equation

The evaluation of volume integrals that arise in conjunction with a hypersingular boundary integral formulation is considered. In a recent work for the standard (singular) boundary integral equation, the volume term was decomposed into an easily computed boundary integral, plus a remainder volume integral with a modified source function. The key feature of this modified function is that it is everywhere zero on the boundary. In this work it is shown that the same basic approach is successful for the hypersingular equation, despite the stronger singularity in the domain integral. Specifically, the volume term can be directly evaluated without a body-fitted volume mesh, by means of a regular grid of cells that cover the domain. Cells that intersect the boundary are treated by continuously extending the integrand to be zero outside the domain. The method and error results for test problems are presented in terms of the three-dimensional Poisson problem, but the techniques are expected to be generally applicable.     

Degenerate scale for the analysis of circular thin plate using the boundary integral equation method and boundary element methods

In this paper, the degenerate scale for plate problem is studied. For the continuous model, we use the null-field integral equation, Fourier series and the series expansion in terms of degenerate kernel for fundamental solutions to examine the solvability of BIEM for circular thin plates. Any two of the four boundary integral equations in the plate formulation may be chosen. For the discrete model, the circulant is employed to determine the rank deficiency of the influence matrix. Both approaches, continuous and discrete models, lead to the same result of degenerate scale. We study the nonunique solution analytically for the circular plate and find degenerate scales. The similar properties of solvability condition between the membrane (Laplace) and plate (biharmonic) problems are also examined. The number of degenerate scales for the six boundary integral formulations is also determined. Tel.: 886-2-2462-2192-ext. 6140 or 6177     

Thermal analysis of carbon-nanotube composites using a rigid-line inclusion model by the boundary integral equation method

The boundary integral equation (BIE) method is applied for the thermal analysis of fiber-reinforced composites, particularly the carbon-nanotube (CNT) composites, based on a rigid-line inclusion model. The steady state heat conduction equation is solved using the BIE in a two-dimensional infinite domain containing line inclusions which are assumed to have a much higher thermal conductivity (like CNTs) than that of the host medium. Thus the temperature along the length of a line inclusion can be assumed constant. In this way, each inclusion can be regarded as a rigid line (the opposite of a crack) in the medium. It is shown that, like the crack case, the hypersingular (derivative) BIE can be applied to model these rigid lines. The boundary element method (BEM), accelerated with the fast multipole method, is used to solve the established hypersingular BIE. Numerical examples with up to 10,000 rigid lines (with 1,000,000 equations), are successfully solved by the BEM code on a laptop computer. Effective thermal conductivity of fiber-reinforced composites are evaluated using the computed temperature and heat flux fields. These numerical results are compared with the analytical solution for a single inclusion case and with the experimental one reported in the literature for carbon-nanotube composites for multiple inclusion cases. Good agreements are observed in both situations, which clearly demonstrates the potential of the developed approach in large-scale modeling of fiber-reinforced composites, particularly that of the emerging carbon-nanotube composites.     

On solvability of a boundary integral equation of the first kind for Dirichlet boundary value problems in plane elasticity

The solution of a Dirichlet boundary value problem of plane isotropic elasticity by the boundary integral equation (BIE) of the first kind obtained from the Somigliana identity is considered. The logarithmic function appearing in the integral kernel leads to the possibility of this operator being non-invertible, the solution of the BIE either being non-unique or not existing. Such a situation occurs if the size of the boundary coincides with the so-called critical (or degenerate) scale for a certain form of the fundamental solution used. Techniques for the evaluation of these critical scales and for the removal of the non-uniqueness appearing in the problems with critical scales solved by the BIE of the first kind are proposed and analysed, and some recommendations for BEM code programmers based on the analysis presented are given.     

Boundary-domain integral method using a velocity-vorticity formulation

The paper deals with the numerical solution of fluid dynamics (transport phenomena in incompressible fluid flow) using a boundary-domain integral method. A velocity-vorticity formulation of the Navier-Stokes equations is adopted, where the kinematic equation is written in its parabolic version.     

The computation of the J integral with the traction boundary integral equation

The solutions of the displacement boundary integral equation (BIE) are not uniquely determined in certain types of boundary conditions. Traction boundary integral equations that have unique solutions in these traction and mixed boundary cases are established. For two‐dimensional linear elasticity problems, the divergence‐free property of the traction boundary integral equation is established. By applying Stokes' theorem, unknown tractions or displacements can be reduced to computation of traction integral potential functions at the boundary points. The same is true of the J integral: it is divergence‐free and the evaluation of the J integral can be inverted into the computation of the J integral potential functions at the boundary points of the cracked body. The J integral can be expressed as the linear combination of the tractions and displacements from the traction BIE on the boundary of the cracked body. Numerical integrals are not needed at all. Selected examples are presented to demonstrate the validity of the traction boundary integral and J integral. Copyright © 2004 John Wiley & Sons, Ltd.     

Galerkin formulation and singularity subtraction for spectral solutions of boundary integral equations

A new spectral Galerkin formulation is presented for the solution of boundary integral equations. The formulation is carried out with an exact singularity subtraction procedure based on analytical integrations, which provides a fast and precise way to evaluate the coefficient matrices. The new Galerkin formulation is based on the exact geometry of the problem boundaries and leads to a non-element method that is completely free of mesh generation. The numerical behaviour of the method is very similar to the collocation method; for Dirichlet problems, however, it leads to a symmetric coefficient matrix and therefore requires half the solution time of the collocation method. © 1998 John Wiley & Sons, Ltd.     

A new boundary integral equation method for cracked 2-D anisotropic bodies

By using integration by parts to the traditional boundary integral formulation, a traction boundary integral equation for cracked 2-D anisotropic bodies is derived. The new traction integral equation involves only singularity of order 1/r and no hypersingular term appears. The dislocation densities on the crack surface are introduced and the relations between stress intensity factors and dislocation densities near the crack tip are induced to calculate the stress intensity factors. The boundary element method based on the new equation is established and the singular interpolation functions are introduced to model the singularity of the dislocation density (in the order of ) for crack tip elements. The proposed method can be directly used for the 2-D anisotropic body containing cracks of arbitrary geometric shapes. Several numerical examples demonstrate the validity and accuracy of BEM based on the new boundary integral equation.     

Boundary element method based on a new second kind integral equation formulation

A boundary integral equation (BIE) formulation for elasticity problems with mixed boundary conditions, proposed by Parton and Perlin (Mathematical Methods of the Theory of Elasticity, Mir, Moscow, 1984), is implemented in this paper using quadratic boundary elements. The formulation is specialised to Stokes flow problems by setting the Poisson ratio to 0·5 in the relevant kernels. The implementation is used to analyse non-trivial three dimensional problems in elasticity and Stokes flows. The results compare well with those obtained by a direct boundary element method. An outline of the extension of the formulation to non-linear problems is also given.     

Radial integration boundary integral and integro-differential equation methods for two-dimensional heat conduction problems with variable coefficients

This paper presents new formulations of the radial integration boundary integral equation (RIBIE) and the radial integration boundary integro-differential equation (RIBIDE) methods for the numerical solution of two-dimensional heat conduction problems with variable coefficients. The methods use a specially constructed parametrix (Levi function) to reduce the boundary-value problem (BVP) to a boundary-domain integral equation (BDIE) or boundary-domain integro-differential equation (BDIDE). The radial integration method is then employed to convert the domain integrals arising in both BDIE and BDIDE methods into equivalent boundary integrals. The resulting formulations lead to pure boundary integral and integro-differential equations with no domain integrals. Numerical examples are presented for several simple problems, for which exact solutions are available, to demonstrate the efficiency of the proposed methods.     

A fast spectral Galerkin method for hypersingular boundary integral equations in potential theory

This research is focused on the development of a fast spectral method to accelerate the solution of three-dimensional hypersingular boundary integral equations of potential theory. Based on a Galerkin approximation, the fast Fourier transform and local interpolation operators, the proposed method is a generalization of the precorrected-FFT technique to deal with double-layer potential kernels, hypersingular kernels and higher-order basis functions. Numerical examples utilizing piecewise linear shape functions are included to illustrate the performance of the method. The US Government retains a nonexclusive royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for US Government purposes.     

A weakly singular boundary integral equation in elastodynamics for heterogeneous domains mitigating fictitious eigenfrequencies

A boundary element formulation applied to dynamic soil–structure interaction problems with embedded foundations may give rise to inaccurate results at frequencies that correspond to the eigenfrequencies of the finite domain embedded in an exterior domain of semi-infinite extent. These frequencies are referred to as fictitious eigenfrequencies. This problem is illustrated and mitigated modifying the original approach proposed by Burton and Miller for acoustic problems, which combines the boundary integral equations in terms of the displacement and its normal derivative using a complex coupling parameter . Hypersingular terms in the original boundary integral equation are avoided by replacing the normal derivative by a finite difference approximation over a characteristic distance h, still leading to an exact boundary integral equation. A proof of the uniqueness of this formulation for small h and a smooth boundary is given, together with a parametric study for the case of a rigid massless cylindrical embedded foundation. General conclusions are drawn for the practical choice of the dimensionless coupling parameter and the dimensionless distance      

Hypersingular integral formulation of elastic wave scattering

A hypersingular boundary integral formulation for calculating two dimensional elastic wave scattering from thin bodies and cracks is described. The boundary integral equation for surface displacement is combined with the hypersingular equation for surface traction. The difficult part in employing the traction equation, the derivation of analytical formulas for the hypersingular integral by means of a limit to the boundary, is easily handled by means of symbolic computation. In addition, the terms containing an integrable logarithmic singularity are treated by a straightforward numerical method, bypassing the use of Taylor series expansions. Example wave scattering calculations for cracks and thin ellipses are presented.     

Identification of cracks and cavities using the topological sensitivity boundary integral equation

The purpose of this communication is to present a novel approach to compute the so called Topological Sensitivity (TS) of any variable or functional in elasticity using Boundary Integral Equations (BIEs), and its use as a tool for identification of defects, by itself or in conjunction with zero-order methods, like Genetic Algorithms. The TS of a cost functional provides a measure of the susceptibility of a defect being at a given location. The main contributions are summarized in the following points:     

Initial value problem formulation of time domain boundary element method for electromagnetic microwave simulations

To aim to obtain more stable solutions and wider area applications for the Time Domain Boundary Element Method (TDBEM), initial value problem formulation of the TDBEM is newly introduced for microwave simulations. The initial value problem formulation of the TDBEM allows us to solve transient microwave phenomena as interior region problems, which gives us well matrix property and interior resonance free solutions. This paper concentrates on applying the initial value problem formulation of the TDBEM to wake field phenomena in particle accelerator cavities.     

An adaptive fast multipole boundary element method for three-dimensional acoustic wave problems based on the Burton–Miller formulation

The high solution costs and non-uniqueness difficulties in the boundary element method (BEM) based on the conventional boundary integral equation (CBIE) formulation are two main weaknesses in the BEM for solving exterior acoustic wave problems. To tackle these two weaknesses, an adaptive fast multipole boundary element method (FMBEM) based on the Burton–Miller formulation for 3-D acoustics is presented in this paper. In this adaptive FMBEM, the Burton–Miller formulation using a linear combination of the CBIE and hypersingular BIE (HBIE) is applied to overcome the non-uniqueness difficulties. The iterative solver generalized minimal residual (GMRES) and fast multipole method (FMM) are adopted to improve the overall computational efficiency. This adaptive FMBEM for acoustics is an extension of the adaptive FMBEM for 3-D potential problems developed by the authors recently. Several examples on large-scale acoustic radiation and scattering problems are presented in this paper which show that the developed adaptive FMBEM can be several times faster than the non-adaptive FMBEM while maintaining the accuracies of the BEM.     

Symmetric Galerkin boundary integral formulation for interface and multi-zone problems

Domains containing an ‘internal boundary’, such as a bi-material interface, arise in many applications, e.g. composite materials and geophysical simulations. This paper presents a symmetric Galerkin boundary integral method for this important class of problems. In this situation, the physical quantities are known to satisfy continuity conditions across the interface, but no boundary conditions are specified. The algorithm described herein achieves a symmetric matrix of reduced size. Moreover, the symmetry can also be invoked to lessen the numerical work involved in constructing the system of equations, and thus the method is computationally very efficient. A prototype numerical example, with several variations in the boundary conditions and material properties, is employed to validate the formulation and corresponding numerical procedure. The boundary element results are compared with analytical solutions and with numerical results obtained with the finite element method. © 1997 John Wiley & Sons, Ltd.     

Comparison of the basic and the discontinuity formulation of the 3D-dual boundary element method

The dual boundary element method (Dual BEM) has established as a numerical approach for solving arbitrary 3D-crack problems in linear elastostatics. In the case of symmetrically loaded cracks — especially traction-free cracks — often the more efficient displacement discontinuity method (DDM) is used, because one obtains a reduced system of algebraic equations.

In our paper we will show that the discontinuity method is just a special formulation of the basic Dual BEM and can be applied to arbitrary boundary value problems on the crack. We will present a numerical example for an unsymmetrically loaded crack and discuss which combinations of boundary conditions on the crack surfaces lead to a reduced system of algebraic equations. The savings in memory and computing time compared to the basic formulation of the Dual BEM will be quantified and illustrated by the numerical simulation of 3D crack propagation.