A review of techniques, advances and outstanding issues in numerical modelling for rock mechanics and rock engineering

Boundary element methods in solid mechanics

In this paper, a simple and robust method, called the radial integration method, is presented for transforming domain integrals into equivalent boundary integrals. Any two- or three-dimensional domain integral can be evaluated in a unified way without the need to discretize the domain into internal cells. Domain integrals consisting of known functions can be directly and accurately transformed to the boundary, while for domain integrals including unknown variables, the transformation is accomplished by approximating these variables using radial basis functions. In the proposed method, weak singularities involved in the domain integrals are also explicitly transformed to the boundary integrals, so no singularities exist at internal points. Some analytical and numerical examples are presented to verify the validity of this method.

The radial integration method for evaluation of domain integrals with boundary-only discretization

We survey some recent applications of radial basis functions (rbfs) for the BEM and related algorithms such as the method of fundamental solutions. Among these are the use of alternatives to the traditional 1+ r function in the dual reciprocity method such as thin plate splines, multquadrics and the recently discovered compactly supported positive definite rbfs, and convergence proofs of the DRM for Poisson’s equation. Newly discovered particular solutions for Helmholtz-type operators are discussed and applied to give efficient mesh free algorithms for the diffusion equation. In addition, a number of proposals are given for future applications of rbfs such as the use of surface rbfs for interpolation and the solution of boundary integral equations and the application of Kansa’s method to develop new rbf based coupled domain-boundary approximation methods.

Some recent results and proposals for the use of radial basis functions in the BEM

The development of the method of fundamental solutions (MFS) and related methods for the numerical solution of scattering and radiation problems in fluids and solids is described and reviewed A brief review of the developments and applications in all areas of the MFS over the last five years is also given Future possible areas of applications in fields related to scattering and radiation problems are identified

https://inside.mines.edu/~pamartin/ref-paps/R080_EABEw.pdf

The method of fundamental solutions for scattering and radiation problems

For thermoelasticity and elasticity with a body force, the regular boundary element method involves a domain integral. Special techniques have been devised to eliminate the domain integral by either exact treatment, or approximation using radial basis functions. This paper gives a unified presentation of these techniques. Body forces of gravity, centrifugal, gradient, harmonic, and arbitrary types are investigated. In the approximation by radial basis function, existing work is extended to include higher order basis functions of conical, spline, and polynomial types.

BEM for theomoelasticity and elasticity with body force : a revisit

A ‘mesh-fre’ and ‘time-free’ numerical method, based on the method of fundamental solutions, the dual reciprocity method ( DRM) for the modified Helmholtz operator, and the Laplace transform, has been developed lo solve diffusion-type and diffusion-reaction problems. Our proposed method has achieved both excellent accuracy and efficiency, as illustrated by two numerical examples, in two dimensions and three dimensions. Another attractive feature of our proposed method is the easy implementation for the 3D problems.

A mesh-free method for linear diffusion equations

Evaluation of thin plate spline based particular solutions for Helmholtz-type operators for the DRM

The method of external source collocation is used to solve a discretised boundary value problem (BVP) ∇2U=0, where U is the potential in a two-dimensional simply connected region D, subject to a mixture of Neumann and Dirichlet boundary conditions. We clarify previous discussion on how many sources are required to give sufficient accuracy and where they should be placed. A general proof shows that a sequence of solutions to the BVP converges as the sources are moved further from the boundary, provided that certain conditions are satisfied. Numeric computations provide guidelines on how many sources should be used, and where they should be placed, in order to achieve sufficient accuracy.

Youssef F. Rashed

Papers

BEM for theomoelasticity and elasticity with body force : a revisit

A mesh-free method for linear diffusion equations

Evaluation of thin plate spline based particular solutions for Helmholtz-type operators for the DRM

Convergence and stability of the method of meshless fundamental solutions using an array of randomly distributed sources

Hypersingular boundary element formulation for Reissner plates