# Boundary element method

The **boundary element method** (REM, English *boundary element method* , *BEM* , especially in electrical engineering also moment *method* , English *MoM* , *method of moments* ) is a discretization method for the calculation of initial and boundary value problems with partial differential equations and a numerical calculation method in engineering . Carl Friedrich Gauß is named as the father of the boundary element method .

## Areas of application

The boundary element method can be used in many areas, e.g. B. for

- electrical field problems ( electrostatics ) and electromagnetism
- thermal field problems (stationary and transient heat conduction )
- mechanical field problems (elastomechanics), solid mechanics , fracture mechanics and plasticity
- Acoustics in the frequency and time domain .

In the field of numerical fluid mechanics (CFD), the boundary element method is used less often.

The SEM developed roughly parallel to the Finite Element Method (FEM). For most questions, however, the FEM is more widespread because it has fewer restrictions with regard to the properties of the area to be described (in the case of the elasticity theory of continua these are e.g. small deformations / distortions and linear-elastic behavior ).

Because the boundary element method for the example of elastic continua is based on Green's influence functions, it represents an improved solution compared to the FE method.

The boundary element method can be coupled very efficiently and elegantly with the finite element method (SEM-FEM coupling).

## functionality

With the boundary element method, in contrast to the finite element method, only the edge or the surface of an area or a structure is viewed in a discretized manner, but not its area or volume. The unknown state variables are only on the edge.

With the help of jump relations, the partial differential equations are converted into integral equations , which represent the properties of the entire area. These integral equations are then discretized and numerically solved using a technique similar to FEM. The boundary element method uses the relationships from the integral theorems according to Green , Gauss and Stokes .

## Numerical properties

Because the boundary element method only considers the boundary and not the volume of an area, the number of discrete support points (nodes) and thus the degrees of freedom is significantly lower than with the FEM and also than with the finite difference method (FDM). However, one obtains a fully occupied, asymmetrical linear system of equations, which restricts or complicates the choice of the solution algorithm and (partially) compensates for the advantage of the lower number of degrees of freedom.

The SEM is therefore advantageously used in cases in which the FEM leads to high numerical effort, for example:

- for half-space contact problems where the half-space extends to infinity (e.g. an elastically bedded foundation )
- when solving differential equations on outskirts. A more academic example of this would be the Laplace operator's solution in an outdoor area; when using the FEM to solve this problem, additional artificial boundary conditions would have to be introduced.

## Web links

- The Boundary Element Method
- Boundary Element Template Library
- Boundary Element Resources Network (links to books, programs, journals, conferences)
- Bembel A 3D, isogeometric, higher-order, open-source BEM software for Laplace, Helmholtz and Maxwell problems
- BEM ++ , an open-source BEM software for 3D Laplace, Helmholtz and Maxwell problems
- AcouSTO Acoustics Simulation TOol, open-source parallel BEM solver for Kirchhoff-Helmholtz Integral Equation (KHIE)

## literature

- LC Wrobel, MH Aliabadi:
*The Boundary Element Method*, April 2002, ISBN 0-470-84139-7 - L. Gaul and C. Fiedler:
*Method of the boundary*elements*in statics and dynamics*, Vieweg ISBN 3528067810 - F. Hartmann: C. Katz:
*Structural Analysis with Finite Elements*, Springer-Verlag ISBN 3-540-40416-3 - C. Pozrikidis:
*A practical guide to boundary element methods with the software library*, BEMLIB, ISBN 1-58488-323-5 - W. McLean:
*Strongly Elliptic Systems and Boundary Integral Equations*, Cambridge University Press - S. Sauter, C. Schwab:
*Edge element methods. Analysis, numerics and implementation of fast algorithms*, Vieweg + Teubner - Reza Sabbagh Amirkhizi:
*EMC analysis of multilayer printed circuit boards with the moment method and hybrid*approaches Shaker Verlag 2006, ISBN 978-3-8322-5450-6