José Domínguez

Boundary elements : an introductory course

Boundary concepts are introduced and immediately applied in simple - but useful - computer codes. These codes facilitate the comprehension of boundary elements. This text also discusses basic concepts, potential problems, elastostatics and other topics of interest to engineers.

Boundary element methods for potential problems

Abstract The boundary element formulation of potential problems is presented using weighted residual techniques. The paper emphasizes the simplicity of the boundary methods and the way in which they can be applied in engineering. The advantage of using this method in preference to finite elements is discussed in the applications.

Dynamic analysis of cracks using boundary element method

Abstract This paper presents a procedure for dynamic stress intensity factor computations using traction singular quarter-point boundary elements. Cracks in a complete space, a half-space and a finite body loaded by steady state waves are studied. Curves for elastodynamic stress intensity factors vs frequency are presented. Transient stress intensity factors are computed by means of Fourier transform. The results are compared with other authors and shown to be accurate in all cases. The dynamic stress intensity factors are computed in a very direct and easy way to implement. This versatile procedure allows for the study of problems with complex geometry that include one or several cracks.

Flux and traction boundary elements without hypersingular or strongly singular integrals

The present paper deals with a boundary element formulation based on the traction elasticity boundary integral equation (potential derivative for Laplaces problem). The hypersingular and strongly singular integrals appearing in the formulation are analytically transformed to yield line and surface integrals which are at most weakly singular. Regularization and analytical transformation of the boundary integrals is done prior to any boundary discretization. The integration process does not require any change of co-ordinates and the resulting integrals can be numerically evaluated in a simple and efficient way. The formulation presented is completely general and valid for arbitrary shaped open or closed boundaries. Analytical expressions for all the required hypersingular or strongly singular integrals are given in the paper. To fulfil the continuity requirement over the primary density a simple BE discretization strategy is adopted. Continuous elements are used whereas the collocation points are shifted towards the interior of the elements. This paper pretends to contribute to the transformation of hypersingular boundary element formulations as something clear, general and easy to handle similar to in the classical formulation. Copyright

A comparative study of three boundary element approaches to transient dynamic crack problems

Abstract Using displacement integral representations for time dependent elastic problems, three different boundary element (BE) approaches can be considered. One is based on the time domain formulation, a second one on the frequency domain formulation, and the third on the dual reciprocity formulation. In this paper, the three BE approaches are applied and compared when used for the computation of stress intensity factors (SIF) of stationary cracks under dynamic loading. In the three cases a quadratic discretization in space and a singular quarter-point (SQP) element are used. Several problems are solved and the results obtained by the three approaches compared. The dual reciprocity approach for dynamic SIF computation is presented in this paper. Conclusions and recommendations on the capabilities of the three approaches in dynamic fracture mechanics are presented.

Three-dimensional fracture analysis in transversely isotropic solids

Abstract In this paper a boundary element formulation for three-dimensional crack problems in transversely isotropic bodies is presented. Quarter-point and singular quarter-point elements are implemented in a quadratic isoparametric element context. The point load fundamental solution for transversely isotropic media is implemented. Numerical solutions to several three-dimensional crack problems are obtained. The accuracy and robustness of the present approach for the analysis of fracture mechanics problems in transversely isotropic bodies are shown by comparison of some of the results obtained with existing analytical solutions. The approach is shown to be a simple and useful tool for the evaluation of stress intensity factors in transversely isotropic media.

A singular element for three-dimensional fracture mechanics analysis

Abstract In this article, a singular boundary element for three-dimensional fracture mechanics analysis is presented. It is a nine-node quadratic element with plane geometry. These nodes are located at one quarter of the distance between two opposite sides of the element. Shape functions with a 11 √r singularity at the crack front are used to represent the tractions. The Stress Intensity Factors are computed as system unknowns appearing (except for a constant) as traction nodal values. Special attention is paid to the development of a simple and accurate integration approach for this singular element. The accuracy of the results obtained with the proposed element is demonstrated by solving several crack problems including edge and embedded cracks with different geometries. The element can be easily implemented and incorporated into existing quadratic boundary element codes. In a companion paper the element is formulated and used for fracture mechanics problems in transversely isotropic materials. Extension to other fields for which boundary element formulations exist, is quite simple.

General BE approach for three-dimensional dynamic fracture analysis

Abstract A general mixed boundary element approach for three-dimensional dynamic fracture mechanics problems is presented in this paper. A mixed traction-displacement integral equation formulation in the frequency domain is used. The hypersingular and strongly singular kernels are regularized by analytical transformations yielding an easy to implement BE approach. Nine-node quadrilateral and six-node triangular continuous quadratic elements are used for external boundaries and crack surfaces. The crack front elements have their mid node at one quarter of the element length allowing for a proper representation of the crack surface displacement. The present approach is intended for the frequency domain analysis of fracture mechanics problems of any general 3D geometry; i.e. boundless or bounded regions, single or multiple, surface or internal cracks. Transient dynamic problems are studied using the FFT algorithm. The numerical results presented show the robustness and accuracy of the approach which requires a reasonable number of elements and degrees of freedom.

Boundary elements for the analysis of the seismic response of dams including dam-water-foundation interaction effects. II

Abstract A Boundary Element frequency domain approach that takes into account the dynamic interaction between solid and fluid regions, which was presented in a separate paper (part I), is used to study the seismic response of concrete gravity dams. Results for a particular dam geometry, two different soil profiles, and full and empty reservoir conditions are shown. The agreement with corresponding published results obtained by other authors using Finite Elements is very good. The BE approach is shown to be capable of modelling a larger variety of soil profiles than any of the previous FE approaches. The effects of several parameters of the model on the seismic response of the dam are studied to show the importance of the joint dam-water-foundation interaction effects and the importance of the correct representation of each one of the three regions involved. Particular attention is paid to the aspects related to the BE modelling of the problem. The use of artificial boundaries, where far-field boundary conditions are prescribed, for cases where the radiation conditions are not satisfied, has been shown to be effective.

Numerical behavior of time domain BEM for three-dimensional transient elastodynamic problems

The present paper deals with the time domain formulation of the boundary element method for three-dimensional elastodynamic problems and its actual implementation for the solution of transient problems relative to bounded domains with any geometry. Particular attention is paid to stability and accuracy of the computed solutions. The numerical approach is based on a constant velocity prediction algorithm and the combination of the integral representation for several time steps. Quadratic boundary elements are considered. A subdivision of the elements is introduced for integration in order to preserve the causality condition and to improve the accuracy of the solution. As opposite to previously published techniques, the present approach is tested for finite 3D bodies with actual values of the elastic constants (non-zero Poissons ratio). The obtained results show the stability and accuracy of the approach for a wide enough range of time step size and non-uniform meshes.