3D multidomain BEM for a Poisson equation

This paper deals with the efficient 3D multidomain boundary element method (BEM) for solving a Poisson equation. The integral boundary equation is discretized using linear mixed boundary elements. Sparse system matrices similar to the finite element method are obtained, using a multidomain approach, also known as the ‘subdomain technique’. Interface boundary conditions between subdomains lead to an overdetermined system matrix, which is solved using a fast iterative linear least square solver. The accuracy, efficiency and robustness of the developed numerical algorithm are presented using cube and sphere geometry, where the comparison with the competitive BEM is performed. The efficiency is demonstrated using a mesh with over 200,000 hexahedral volume elements on a personal computer with 1 GB memory.     

Integral and geometrical means in the analytical evaluation of the BEM integrals for a 3D Laplace equation

By using integral theorems and geometrical interpretations, the analytical formulas for the coefficients occurring in boundary element method (BEM) equations for a 3D Laplace equation were found for arbitrary planar polygonal boundary elements with constant approximation. Closed forms for the gradients of the coefficients were also obtained. In addition, an analytical formula for planar triangular boundary elements with a linear approximation of potential was given. The formulas obtained are appropriate especially in singular and nearly singular cases.     

3D Frequency domain BEM for solving dipolar gradient elastic problems

A three dimensional (3D) boundary element method (BEM) for treating time harmonic problems in linear elastic structures exhibiting microstructure effects is presented. These microstructural effects are taken into account with the aid of the dipolar gradient elastic theory, which is the simplest dynamic version of Mindlins generalized elastic theory. A variational statement is established to determine all possible classical and non-classical (due to gradient terms) boundary conditions of the general boundary value problem. The dipolar gradient frequency domain elastodynamic fundamental solution is explicitly derived and used to construct the boundary integral representation of the solution with the aid of a reciprocal integral identity. In addition to a boundary integral representation for the displacement, a boundary integral representation for its normal derivative is also necessary for the complete formulation of a well posed problem. Surface quadratic quadrilateral boundary elements are employed and the discretization is restricted only to the boundary. The solution procedure is described in detail. A numerical example serves to illustrate the method and demonstrate its accuracy     

A first order method for the Cauchy problem for the Laplace equation using BEM

The purpose is to propose an improved method for inverse boundary value problems. This method is presented on a model problem. It introduces a higher order problem. BEM numerical simulations highlight the efficiency, the improved accuracy, the robustness to noisy data of this new approach, as well as its ability to deblur noisy data.     

A self-regularized approach for rank-deficient systems in the BEM of 2D Laplace problems

The Laplace problem subject to the Dirichlet or Neumann boundary condition in the direct and indirect boundary element methods (BEM) sometimes both may result in a singular or ill-conditioned system (some special situations) for the interior problem. In this paper, the direct and indirect BEMs are revisited to examine the uniqueness of the solution by introducing the Fichera’s idea and the self-regularized technique. In order to construct the complete range of the integral operator in the BEM lacking a constant term in the case of a degenerate scale, the Fichera’s method is provided by adding the constraint and a slack variable to circumvent the problem of degenerate scale. We also revisit the Fredholm alternative theorem by using the singular value decomposition (SVD) in the discrete system and explain why the direct BEM and the indirect BEM are not indeed equivalent in the solution space. According to the relation between the SVD structure and Fichera’s technique, a self-regularized method is proposed in the matrix level to deal with non-unique solutions of the Neumann and Dirichlet problems which contain rigid body mode and degenerate scale, respectively, at the same time. The singularity and proportional influence matrices of 3 by 3 are studied by using the property of the symmetric circulant matrix. Finally, several examples are demonstrated to illustrate the validity and the effectiveness of the self-regularized method.     

Desingularized meshless method for solving Laplace equation with over-specified boundary conditions using regularization techniques

The desingularized meshless method (DMM) has been successfully used to solve boundary-value problems with specified boundary conditions (a direct problem) numerically. In this paper, the DMM is applied to deal with the problems with over-specified boundary conditions. The accompanied ill-posed problem in the inverse problem is remedied by using the Tikhonov regularization method and the truncated singular value decomposition method. The numerical evidences are given to verify the accuracy of the solutions after comparing with the results of analytical solutions through several numerical examples. The comparisons of results using Tikhonov method and truncated singular value decomposition method are also discussed in the examples.     

Degenerate scale problem when solving Laplace's equation by BEM and its treatment

In this paper, Laplace problems are solved by using the dual boundary element method (BEM). It is found that a degenerate scale problem occurs if the conventional BEM is used. In this case, the influence matrix is rank deficient and numerical results become unstable. Both the circular and elliptical bars are studied analytically in the continuous system. In the discrete system, the Fredholm alternative theorem in conjunction with the SVD (Singular Value Decomposition) updating documents is employed to sort out the spurious mode which causes the numerical instability. Three regularization techniques, method of adding a rigid body mode, hypersingular formulation and CHEEF (Combined Helmholtz Exterior integral Equation Formulation) concept, are employed to deal with the rank‐deficiency problem. The addition of a rigid body term, c, in the fundamental solution is proved to shift the original degenerate scale to a new degenerate scale by a factor e^?c. The torsion rigidities are obtained and compared with analytical solutions. Numerical examples including elliptical, square and triangular bars were demonstrated to show the failure of conventional BEM in case of the degenerate scale. After employing the three regularization techniques, the accuracy of the proposed approaches is achieved. Copyright © 2004 John Wiley & Sons, Ltd.     

h-Hierarchical functions for 2D and 3D BEM

In this paper, it is shown that the idea of hierarchical BEM formulations can be extended in a systematic way to the h-method, and that the accuracy obtained with h-hierarchical shape functions is the same as for standard shape functions of the same order (i.e. the two functional spaces are identical). A rigorous way of obtaining new h-hierarchical shape functions in BEM is developed and put into effect for both two-dimensional quadratic and three-dimensional quadrilateral quadratic elements.     

On one boundary-value problem for the Laplace equation

We have derived a Fredholm-type equation of the second kind for a directional derivative problem arising in the stationary theory of heat conduction. One result of Ya. B. Lopatinskii has been refined. __________ Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 80, No. 2, pp. 197–200, March–April, 2007.     

BEM–FEM coupling for 3D fracture mechanics applications

Attention is here focused on the implementation of a coupled BEM–FEM procedure employing the BE method for the modelling of the near-crack region and the FEM for the far-field, where no singularities in the stress field are expected to arise. The symmetric variational version of the BE method is utilized, allowing to obtain a final linear system endowed with a symmetric matrix. With respect to 3D linear elastic fracture mechanics, the code developed is used to evaluate stress intensity factors for some benchmarks and simulate fracture propagation. A research grant from MURST (Cofinanziamento 2000) is acknowledged.     

3D fracture analysis by the symmetric Galerkin BEM

 The subject of this paper is the formulation and the implementation of the symmetric Galerkin BEM for three-dimensional linear elastic fracture mechanics problems. A regularized version of the displacement and traction equations in weak form is adopted and the integration techniques utilized for the evaluation of the double surface integrals appearing in the discretized equations are detailed. By using quadratic isoparametric quadrilateral and triangular elements, some example crack problems are solved to assess the efficiency and robustness of the method. Received 6 November 2000     

Gaussian quadratures for singular integrals in BEM with applications to the 2D modified Helmholtz equation

Singular integrals frequently arise in the calculation of the diagonal elements of BEM influence matrices and related multiple reciprocity applications. The underlying theory of Gaussian integration is applied to develop a general algorithm for the determination of the Gaussian parameters with the singular fundamental solution chosen as weight function. In contrast to singularities like ln(1/x), 1/x, and 1/x², which are frequently met in applied mechanics, the singularity of K₀(cx), the fundamental solution of the modified Helmholtz equation, is not treated extensively in the literature. The general algorithm is applied to the special case of this fundamental solution, and the Gaussian weights and ordinates are determined. Numerical experiments and comparisons with alternate methods of integration are carried out to assess the merit of the newly developed quadrature.     

BEM Analysis of 3D Heat Conductionin 3D Thin Anisotropic Media

In this paper, the boundary integrals for treating 3D field problems are fully regularized for planar elements by the technique of integration by parts (IBP). As has been well documented in open literatures, these integrals appear to be strongly singular and hyper-singular for the associated fundamental solutions. In the past, the IBP approach has only been applied to regularize the integrals for 2D problems. The present work shows that the IBP can also be further extended to treat 3D problems, where two variables of the local coordinates are involved. The presented formulations are fully explicit and also, most importantly, very straightforward for implementation in program codes. To demonstrate their validity and our implementation, a few example cases of 3D anisotropic heat conduction are investigated by the boundary element method and the calculated results are verified using analyses by ANSYS.     

Interface integral BEM for solving multi-medium heat conduction problems

In this paper, a new and simple boundary element method, called interface integral boundary element method (IIBEM), is presented for solving heat conduction problems consisting of multiple media. In the method, the boundary integral equation is derived by a degeneration technique from domain integrals involved in varying heat conductivity problems into interface integrals in multi-medium problems. The main feature of the presented technique is that only a single boundary integral equation is used to solve heat conduction problems with different material properties. The effect of nonhomogeneity between adjacent materials is embodied in the interface integrals including the material property difference between the two adjacent materials. Comparing with conventional multi-domain boundary integral equation techniques, the presented method is more efficient in computational time, data preparing, and program coding. Numerical examples are given to verify the correctness of the presented technique.     

Galerkin boundary integral analysis for the axisymmetric Laplace equation

The boundary integral equation for the axisymmetric Laplace equation is solved by employing modified Galerkin weight functions. The alternative weights smooth out the singularity of the Green's function at the symmetry axis, and restore symmetry to the formulation. As a consequence, special treatment of the axis equations is avoided, and a symmetric‐Galerkin formulation would be possible. For the singular integration, the integrals containing a logarithmic singularity are converted to a non‐singular form and evaluated partially analytically and partially numerically. The modified weight functions, together with a boundary limit definition, also result in a simple algorithm for the post‐processing of the surface gradient. Published in 2005 by John Wiley & Sons, Ltd.     

Circular edge singularities for the Laplace equation and the elasticity system in 3-D domains

Asymptotics of solutions to the Laplace equation with Neumann or Dirichlet conditions in the vicinity of a circular singular edge in a three-dimensional domain are derived and provided in an explicit form. These asymptotic solutions are represented by a family of eigen-functions with their shadows, and the associated edge flux intensity functions (EFIFs), which are functions along the circular edge. We provide explicit formulas for a penny-shaped crack for an axisymmetric case as well as a case in which the loading is non-axisymmetric. Explicit formulas for other singular circular edges such as a circumferential crack, an external crack and a 3π/2 reentrant corner are also derived. The mathematical machinery developed in the framework of the Laplace operator is extended to derive the asymptotic solution (three-component displacement vector) for the elasticity system in the vicinity of a circular edge in a three-dimensional domain. As a particular case we present explicitly the series expansion for a traction free or clamped penny-shaped crack in an axisymmetric or a non-axisymmetric situation. The precise representation of the asymptotic series is required for constructing benchmark problems with analytical solutions against which numerical methods can be assessed, and to develop new extraction techniques for the edge flux/intensity functions which are of practical engineering importance in predicting crack propagation.     

Analytical integrations in 3D BEM: preliminaries

 This work provides a preliminary contribution in the context of analytical integrations of strongly and hyper singular kernels in boundary element methods (BEMs) in 3D. It concerns the integral of 1/r ³ over a triangle in R ³, that plays a fundamental role in BEMs in 3D, especially for the Galerkin implementation. Because the existence of the aforementioned integral depends on the position of the source point, all significant instances of the position of the source point will be considered and detailed. For its interest in the context of BEM, the integral is also considered in the more general sense of finite part of Hadamard. Received 6 August 2001     

A time domain harmonic BEM implementation for non-homogeneous 3D solids

In the present work, a Boundary Element Method implementation is presented for a specific non-homogeneous elastic media. The media under consideration is a Poisson solid (ν=0.25), which requires additionally a quadratic variation for the material parameters along one cartesian axis (e.g. depth). The 3D problem for the time-harmonic case was implemented and numerically validated for specific problems. Moreover, the static version of the present problem was used to model a real functionally graded composite of alumina–nickel.