# Finite elements or boundary elements &ndash; which is better?

Paul Boughton

The most popular method of simulating real world problems is the finite element method (FEM). It has been used extensively (and successfully) in all the major areas of engineering including mechanical, thermal, and electromagnetic problems.

For some classes of problems the boundary element method (BEM) can be quite appealing. These include open region problems in electromagnetics, acoustics, and thermal radiation.

The two methods are quite different. For FEM the entire 3D space is normally divided into bricks or tetrahedral elements, although other elements exist. For the BEM only the surface of the parts being simulated need to be descretised. The surfaces can be divided into triangles or rectangles. Thus descretising a problem using BEM is trivial compared to FEM where the entire 3D space (or volume) must be meshed as opposed to meshing a 3D surface.

For most thermal problems, and all mechanical problems, the volume of interest is the internal volume of the part or the assembly under consideration. Thus only the interior of the part needs to be descretised. When using FEM for many electromagnetic problems not only does the interior of the part have to be descretised but the air space around the part has to be divided into finite elements as well. This is typically done by enclosing the part in a box and generating a mesh up to the box. The size of the box can have a significant effect on the results. This can make some problems intractable for FEM.

A Linear problem is where BEM is ideally suited. As elements, or unknowns, are only placed on the surface, both the interior volume and exterior volume of the part are solved without a need to descretise either volume. This is a huge advantage. In addition the accuracy that can be obtained from a BEM solution far exceeds that of a FEM solution. Solution accuracies of parts per million is not uncommon.

So why not use BEM all the time? The answer is simple. For most problems extreme accuracy is not required. In addition, BEM has one major drawback; dealing with nonlinear problems. For these classes of problems a 3D mesh is generally required in the nonlinear volumes. This can radically increase solution time. As well problems involving anisotropy, transients, and motion can be difficult to deal with using BEM.

Another approach to the initial question is neither is better. FEM is suited better for some problems and BEM for others. The software that offers both is best.

Integrated Engineering Software is based in Winnipeg, Canada. www.integratedsoft.com

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