Short note: Boundary element method

The boundary element method (BEM) is a numerical method in which the governing differential equation is converted into an integral equation over the boundary of the domain, using a fundamental solution (Green's function). Only the boundary is divided into elements, so a 3D problem becomes a 2D surface problem.

Procedure

1. Write the governing PDE (e.g. Laplace or Navier equation) and its weighted-residual form.
2. Apply Green's theorem to move all derivatives to the weight function, which is the fundamental solution.
3. Obtain the boundary integral equation, then discretise the boundary into elements.
4. Evaluate integrals and form $[H]\{u\}=[G]\{t\}$, where $u$ is boundary displacement and $t$ traction.
5. Apply boundary conditions, solve for the unknown boundary values, and then compute interior values.

Advantages: fewer unknowns and easy data preparation; accurate for stress concentration; suited to infinite and semi-infinite domains.

Disadvantages: matrices are full and unsymmetric; a fundamental solution is needed; non-linear and non-homogeneous problems are difficult.

Applications: soil-structure interaction, tunnels, foundations, fracture mechanics, potential flow and seepage.