Boundary element method

The boundary element method (BEM) is a numerical computational method of solving linear partial differential equations (PDEs) arising in engineering and mathematical modeling. It is similar to the more widely used finite element method, in that it breaks down the object of study into a series of points in space and then iteratively applies calculations at the points.

BEM differs from FEM (and similar methods) in that it allows the values in interior or exterior points to be calculated using a calculation based on the values a defined limit, often the outer surface of an object or similar boundary. This is accomplished by re-formulating the PDEs as integral equations (i.e. in boundary integral form). This reduces the calculation of the larger volume to calculations based on the solutions at the boundary, which greatly reduces the total amount of calculations that need to be solved. For instance, to consider the transmission of sound through a lake, an FEM would require an enormous volume of points to be calculated, whereas BEM can reduce this to a calculation of the conditions on the exterior surface of the emitter.

BEM is widely used in stress analysis, fluid mechanics, acoustics, and similar fields. It is also very common in the study of electromagnetism, especially the performance of radio antennas. In this particular case, it is so widely used that it is generally referred to by another term, the method of moments (MoM).

The boundary element method is also well suited for analyzing fracture mechanics and cracks in solids and there are several BEM approaches for crack problems. One such approach is to formulate the conditions on the cracks in terms of hypersingular boundary integral equations, see (Ang 2013) and contact mechanics.

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