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.

### How BEM operates

The Boundary Element Method solves (analyses) 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.

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 is 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.

### Superior analysis of open boundary problems

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.

### BEM does error analysis

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.