Boundary Element Method (BEM)
PM motor
The Boundary Element Method solves field problems by solving an equivalent source problem.
In the case of electric fields, it solves for equivalent charge, while in the case of magnetic
fields, it solves for equivalent currents.
BEM also uses an integral formulation of Maxwell's Equations, which allows for very
accurate field calculations. Unlike FEM the electric and magnetic fields are computed directly from the source. This
technique produces accuracies not attainable by Finite Element Method.
The major advantages of BEM
BEM uses only 2D elements on the surfaces which are the material interfaces or assigned boundary conditions.
Therefore, users can set up a problem quickly and easily. Since only elements on interfaces
are involved in the solution procedure, problem modifications are also easy. For example,
in motor design optimization, solutions are required for different rotor positions. Using BEM
software, only one boundary element distribution is necessary to solve all the rotor positions,
and no element reassignments are required. With FEM software, finite elements in the whole
space must be regenerated for every new rotor position. Complete 3D Finite Element Meshes
are impossible to visually represent and comprehend on a 2D drawing.
BEM allows all field variables at any point in space to be obtained very accurately. Also,
the results are more precise because the integration operation
inherently more accurate than FEM's differentiation operation. Moreover, the unknown variables
used in INTEGRATED's software are the equivalent currents or equivalent charges. These
variables have real physical meanings. By using these physical variables, global quantities
such as forces, torque, stored energy, inductance, and capacitance among others can be
accurately obtained through some very simple methods.
The analysis of unbounded structures (e.g. electromagnetic fields exterior to an electric
motor) can be solved by BEM without any additional effort because the exterior field is
calculated the same way as the interior field. The field at any point in space can be
calculated (even at infinity). Therefore, for any closed or open boundary problem, INTEGRATED's
software users need only to deal with real geometry boundaries. In contrast, open boundary
problems are problematic for FEM since artificial boundaries, which are far away from the
real structure, must be used. How to determine these artificial boundaries becomes a major
difficulty for FEM-based software users. Since most electromagnetic field problems are
associated with open boundary structures, BEM naturally becomes the best method for general
field problems.
From Green's theorem, one can show that if and only if the solution satisfies the boundary
conditions on all the boundaries, the result at any point in the solution space obtained
from the variables on the boundaries is correct. Therefore, after solving a problem with
certain element distribution, users can perform an error analysis by checking the boundary
conditions along the boundaries. One can improve the solution by simply adding more elements
on the boundary where a large error has been found. This is based on the fact that the
largest errors occur on the boundary, and that for the fields in a region the largest
contributors are the elements close to the region. We have an error detection command in all
of our 2D/RS software.