Transform Boundary Element Method Research Articles

On modelling three-dimensional elastodynamic wave propagation with boundary spectral element method

ABSTRACTIn this paper, a boundary spectral element method (BSEM) for solving the problem of three-dimensional wave propagation is introduced. In the new formulation, elastodynamics of structures is computed by the Laplace transformed boundary element method (BEM), and boundaries of structures are discretised into high-order isoparametric spectral elements. Three types of spectral elements – Lobatto, Gauss–Legendre and Chebyshev elements – have been implemented. With a significantly higher computational efficiency than the conventional BEM, the BSEM provides a competitive alternative for modelling high-frequency wave propagation in engineering applications.

A triple reciprocity method in Laplace transform boundary element method for three-dimensional transient heat conduction problems

In this paper, a boundary element method employing the Laplace transform is developed to solve the three-dimensional transient heat conduction problems. The fundamental solution for the modified Helmholtz equation is adopted in order to derive the basic integral equations. Due to the effects of initial temperature and heat generation, the domain integrals appearing in the integral equation will degrade the advantages of boundary element method. Therefore, a new triple reciprocity formulation in Laplace domain is proposed to convert the domain integrals into boundary integrals. The higher order fundamental solutions required in the triple reciprocity method can be obtained from the original fundamental solution by multiplying a constant term, which leads to a much simpler formulation in the Laplace domain. Then, the inverse transformation can be used to obtain the solutions in the time domain. Several numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method. In summary, the triple reciprocity formulation is a useful approach to capture the transient heat conduction responses.

Stress analysis for thin multilayered coating systems using a sinh transformed boundary element method

In this paper, the boundary element method (BEM) based on the elasticity theory is developed for two-dimensional (2-D) thin-structural problems with thickness-to-length ratio in the micro (10-6) or nano (10-9) scales. An efficient non-linear co-ordinate transformation, based on the sinh function, is developed to deal with the troublesome nearly-singular integrals arising in the BEM formulation for thin structures. The proposed BEM formulation with thin-body capabilities is also extended to the multi-domain problems and applied to the stress analysis of multilayered coating systems. Promising BEM results with only a small number of elements are obtained for thin films and coatings with the thickness-to-length ratio is as small as 10-9, which is sufficient for modeling most thin layered coating systems as used in smart materials and micro-electro-mechanical systems. The advantages, disadvantages and potential applications of the proposed method, as compared with the finite element method (FEM), are discussed in the last section.

11th International Conference on the Mechanical Behavior of Materials Analysis of Defects in Viscoelastic Solids by a Transformed Boundary Element Method

Due to the inclusion of time as an independent variable to describe the mechanical behavior of viscoelastic materials, the available analytical solutions have been obtained only for a few simplified problems. To study the mechanical behavior of viscoelastic solids, the numerical approaches such as finite element method (FEM) and boundary element method (BEM) are normally needed. The main advantages of BEM are the reduction of the problem dimension by one and the exact satisfaction of certain boundary conditions for particular problems if their associated fundamental solutions are embedded in boundary element formulation. Through the use of correspondence principle, the viscoelastic solids can be effectively treated in Laplace domain. To take advantage of the available fundamental solutions for the defects in anisotropic elastic materials, in this paper we use the transformed BEM to treat the problems of viscoelastic solids containing defects such as holes, cracks, or inclusions. By using the subregion technique, the problems with simultaneous existence of multiple holes, cracks, and inclusions can also be treated. The main feature of this proposed method is that no meshes are needed along the boundary of defects and the boundary conditions are satisfied exactly, which means that the present approach should be more efficient and correct.

Transient heat conduction in homogeneous and non-homogeneous materials by the Laplace transform Galerkin boundary element method

The Green's function for three-dimensional transient heat conduction (diffusion equation) for functionally graded materials (FGMs) is derived. The thermal conductivity and heat capacitance both vary exponentially in one coordinate. In the process of solving this diffusion problem numerically, a Laplace transform (LT) approach is used to eliminate the dependence on time. The fundamental solution in Laplace space is derived and the boundary integral equation formulation for the Laplace Transform boundary element method (LTBEM) is obtained. The numerical implementation is performed using a Galerkin approximation, and the time-dependence is restored by numerical inversion of the LT. Two numerical inversion techniques have been investigated: a Fourier series method and Stehfest's algorithm, the latter being preferred. A number of test problems have been examined, and the results are in excellent agreement with available analytical solutions.

Transient analysis of three-dimensional crack problems by the laplace transformed boundary element method

The Laplace transformed boundary element method is used to analyse the transient responses of a three-dimensional crack. The choice of Laplace inversion parameters and the calculation of dynamic stress intensity factors (SIF) are discussed. In order to verify the computation procedure and choose the discretization format, the transient displacement response of a prismatical beam under various dynamic loadings and the transient SIF response of a thick plate with a through-thickness crack under two different discretization schemes are calculated. Finally, the transient SIF response of a plate with a semi-circular surface crack under several different loading conditions is also computed, and the results are shown to be quite good.

A study on Computational Accuracy of the Laplace Transform Boundary Element Method for Unsteady Elastodynamic Problems (Two-Dimensional Cases).

In the Laplace transform boundary element analysis for unsteady elastodynamic problems, the inverse transform must be performed numerically. Computational accuracy and efficiency of the results are influenced by the computational conditions such as the sampling number and the number of boundary elements. This paper proposes a new procedure to determine the computational conditions for accurate and efficient analysis of the unsteady elastodynamic problems. The proposed procedure employs two techniques of the numerical inverse Laplace transform, that is, Hosono's technique for obtaining the reference results and Durbin's technique for main use in numerical analysis. Numerical computation is carried out for several two-dimensional problems to demonstrate the usefulness of the proposed procedure.

Dynamic analysis of 3-D structures by a transformed boundary element method

In this work, an advanced implementation of the direct boundary element method applicable to periodic (steadystate) and transient dynamic problems involving three-dimensional structures of arbitrary shape and connectivity is presented. Interior, exterior and halfspace type of problems can all be solved by the present method. The discussion first focuses on the formulation of the method, followed by material pertaining to the fundamental singular solutions and to the isoparametric boundary elements used for discretizing the surface of the problem. Subsequently, numerical integration techniques and the solution algorithm are introduced. This methodology has been incorporated in a versatile, general purpose computer program. Finally, the stability and high accuracy of this dynamic analysis technique are established through comparisons with available analytical and numerical results.