Unlike the Boundary Element Method (BEM), the Finite Element Method (FEM) is a numerical technique for solving models in differential form. For a given design, the FEM requires the entire design, including the surrounding region, to be modeled with finite elements. A system of linear equations is generated to calculate the potential (scalar or vector) at the nodes of each element. Therefore, the basic difference between these two techniques is the fact that BEM only needs to solve the unknowns on the boundaries, whereas FEM solves for a chosen region of space and requires a boundary condition bounding that region.

BEM and FEM have become the two dominant numerical techniques in computer-aided engineering (CAE). Both techniques have merits and restrictions. In fact, Integrated Engineering Software uses both methods as well as the Hybrid Method. Our conclusion is that BEM is overall the superior solution technique for electromagnetic designs. Historically we have therefore focussed on BEM and have been strongly associated with it in the marketplace.

### The major advantages of BEM

##### Ease of Use

Unlike FEM, which must use a 3D finite element mesh in the whole space, 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 re-generated for every new rotor position. Complete 3D Finite Element Meshes are impossible to visually represent and comprehend on a 2D drawing.

##### More accurate results

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 smoother, making BEM 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 solves non-linear problems

BEM can readily solve non-linear problems. Many people, including some who are quite knowledgeable about BEM initially, believed that BEM could not solve non-linear problems. However, the truth is that not many people know how to use BEM for solutions to non-linear problems. Later on we will discuss in more detail how BEM is used to solve non-linear problems. For now it is sufficient to point out that BEM routinely solves non-linear 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.

### Advantages of FEM

While BEM can solve nonlinear problems, the nonlinear contribution requires a volume mesh. Putting a volume mesh in begins to diminish the benefits of BEM listed above. In fact, for a saturating nonlinear magnetic problem, the saturatation characteristic is best solved with FEM. Also, the BEM benefit of solving open regions very well is not always relevant.