Boundary element method
The Boundary Element Method (BEM, english boundary element method, BEM, v. a in electrical engineering and method of moments, 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 science. As the father of the Boundary Element Method Carl Friedrich Gauss called.
Applications
The boundary element method can be applied in many fields, for example, in fluid mechanics, acoustics, heat transfer, electromagnetism, solid mechanics, fracture mechanics, plasticity, etc. The SEM has developed approximately parallel with the finite element method (FEM). For most questions, however, the EMF is more widely because it has less restrictions concerning the characteristics of the area to be described ( in the case of the continua of Elastitiztätstheorie are as small deformation / distortion and linear elastic behavior).
Because it is based on the example of elastic continua on the Green's influence functions, it juxtaposes the FE method an improved solution represents the boundary element method can turn very efficient and elegant coupled with the finite element method (SEM - FEM coupling) be.
The boundary element method is less often in the area of Computational Fluid Dynamics (CFD ) is used, but rather for electric field problems ( electrostatics ), thermal field problems ( steady state and transient heat conduction ), as well as mechanical field problems ( elastomechanics ) and acoustics in the frequency and time domain.
Operation
In the boundary element method, in contrast to the Finite Element Method, only the edge or surface of an area or structure considered discretized, but not their area or volume. The unknown state variables are located just on the edge. With the aid of step relations, the partial differential equations to integral equations are transformed to reflect the characteristics of the whole area. These integral equations are then using a technique similar to the FEM, discretised and solved numerically.
The boundary element method uses the relationships of the integral sets to Green, Gauss and Stokes. For the solution of problems which should be solved with the boundary element method, it is not necessary to know the Green's function (or Green's fundamental solution ) since the jump relations and the preparation of the solution as a single-layer, double-layer or volume potential without knowledge of these can be set up Green's function.
Numerical properties
In the SEM, the number of discrete grid points ( nodes), and thus the degrees of freedom (FHG ) is substantially lower than that of the FEM and also in the finite difference method (FDM). However, yielded a fully loaded, unbalanced system of linear equations, which limits the choice of solution algorithm or difficult, and compensates for the advantage of the lower number of FHG (partially). The SEM is advantageously used in cases where the FEM leads to high computational cost: for example, in a half-space contact problems, in which the half-space extends to infinity, or the solution of differential equations on exterior areas. An example of the former is an elastically bedded foundation. A rather academic problem would be the solution of the Laplace operator in an outdoor area. In FEM additional artificial boundary conditions must be introduced.