| Matrix or Vector | Shape Functions | Integration Points |
|---|---|---|
|
Magnetic Potential Coefficient Matrix or Thermal Conductivity Matrix |
|
None on the boundary element IJK itself, however, 16-point 1-D Gaussian quadrature is applied for some of the integration on each of the edges IJ, JK, and KI of the infinite elements IJML, JKNM, and KILN (see Figure 13.11: A Semi-infinite Boundary Element Zone and the Corresponding Boundary Element IJK) |
Reference: Kaljevic', et al.([130])
This boundary element (BE) models the exterior infinite domain of the far-field magnetic and thermal problems. This element is to be used in combination with 3-D scalar potential solid elements, and can have magnetic scalar potential (MAG), or temperature (TEMP) as the DOF.
The formulation of this element is based on a first order triangular infinite boundary element (IBE), but the element can be used as a 4-node quadrilateral as well. For unbounded field problems, the model domain is set up to consist of an interior volumetric finite element domain, ΩF, and a series of exterior volumetric BE subdomains, ΩB, as shown in Figure 13.11: A Semi-infinite Boundary Element Zone and the Corresponding Boundary Element IJK. Each subdomain, ΩB, is treated as an ordinary BE domain consisting of five segments: the boundary element IJK, infinite elements IJML, JKNM and KILN, and element LMN; element LMN is assumed to be located at infinity.
The approach used here is to write BE equations for ΩB, and then convert them into equivalent load vectors for the nodes I, J and K. The procedure consists of four steps that are summarized below (see (Kaljevic', et al.[130]) for details).
First, a set of boundary integral equations is written for ΩB. To achieve this, the potential (or temperature) and its normal derivatives (fluxes) are interpolated on the triangle IJK (Figure 13.11: A Semi-infinite Boundary Element Zone and the Corresponding Boundary Element IJK) by linear shape functions:
 | (13–85) |
 | (13–86) |
where:
| φ = potential (or temperature) |
 |
| NI, NJ, NK = linear shape functions defined earlier |
| φI, φJ, φK = nodal potentials (or temperatures) |
| qnI, qnJ, qnK = nodal normal derivatives (or fluxes) |
| n = normal to the surface IJK |
Over an infinite element, such as IJML (Figure 13.12: Infinite Element IJML and the Local Coordinate System), the dependent variables, i.e., potentials (or temperatures) and their normal derivatives (fluxes) are respectively assumed to be (Figure 13.12: Infinite Element IJML and the Local Coordinate System):
 | (13–87) |
 | (13–88) |
where:
|
| qτI, qτJ = nodal (nodes I and J) normal derivatives for infinite element IJML |
| s = a variable length from node I towards node J |
| LIJ = length of edge IJ |
| ρ = radial distance from the origin of the local coordinate system O to the edge IJ |
| r = radial distance from the edge IJ towards infinity |
| β = variable angle from x-axis for local polar coordinate system |
| τ = normal to infinite elements IJML |
The boundary integral equations for ΩB are now written as:
 | (13–89) |
where:
| c(ξ) = jump term in boundary element method |
 |
|
| (x,ξ) = field and source points, respectively |
| r = distance between field and source points |
 |
The integrations in Equation 13–89 are performed in closed form on the boundary element IJK. The integrations on the infinite elements IJML, JKNM and KILN in the 'r' direction (Figure 13.12: Infinite Element IJML and the Local Coordinate System) are also performed in closed form. However, a 16-point Gaussian quadrature rule is used for the integrations on each of the edges IJ, JK and KI on the infinite elements.
Second, in the absence of a source or sink in ΩB, the flux q(r) is integrated over the boundary ΓB of ΩB and set to zero:
 | (13–90) |
Third, geometric constraint conditions that exist between the potential φ
(or temperature) and its derivatives
and
at the nodes I, J and K are
written. These conditions would express the fact that the normal derivative
qn at the node I, say, can be decomposed into components
along the normals to the two infinite elements IJML and KILN meeting at I
and along OI.
Fourth, the energy flow quantity from ΩB is written as:
 | (13–91) |
This energy flow is equated to that due to an equivalent nodal force vector {F} defined below.
The four steps mentioned above are combined together to yield, after eliminating qn and qτ,
 | (13–92) |
where:
| [K] = 3 x 3 equivalent unsymmetric element coefficient matrix |
| {φ} = 3 x 1 nodal degrees of freedom, MAG or TEMP |
| {F}eqv = 3 x 1 equivalent nodal force vector |
The coefficient matrix [K] multiplied by the nodal DOF's {φ} represents the equivalent nodal load vector which brings the effects of the semi-infinite domain ΩB onto nodes I, J and K.
As mentioned in the beginning, the INFIN47 can be used with magnetic scalar potential elements to solve 3-D magnetic scalar potential problems (MAG degree of freedom). Magnetic scalar potential elements incorporate three different scalar potential formulations (see Electromagnetic Field Fundamentals) selected with the MAGOPT command:
If there is no “iron” in the problem domain, the reduced scalar potential formulation can be used both in the FE and the BE regimes. In this case, the potential is continuous across FE-BE interface. If there is “iron” in the FE domain, the reduced potential formulation is likely to produce “cancellation errors”.
If there is “iron” and current in the FE region and the problem domain is singly-connected, we can use the difference potential formulation in order to avoid cancellation error. The formulation consists of two-step solution procedures:
Preliminary solution in the air domain (MAGOPT, 2)
Here the first step consists of computing a magnetic field {Ho} under the assumption that the magnetic permeability of iron is infinity, thereby neglecting any saturation. The reduced scalar potential φ is used in FE region and the total scalar potential ψ is used in BE region. In this case, the potential will be discontinuous across the FE-BE interface. The continuity condition of the magnetic field at the interface can be written as:
(13–93)
where:
{τ} = tangent vector at the interface along element edge {Hs} = magnetic field due to current sources Integrating the above equation along the interface, we obtain
(13–94)
If we take ψ = φ at a convenient point po on the interface, then the above equation defines the potential jump at any point p on the interface. Now, the total potential ψ can be eliminated from the problem using this equation, leading to the computation of the additional load vector,
(13–95)
where:
[K] = coefficient matrix defined with Equation 13–92 Total solution (air and iron) (MAGOPT, 3)
In this step the total field, {H} = {Ho} -
ψ, is computed where {H} is the actual field and
{Ho} is the field computed in step 1 above. Note that
the same relation given in Equation 5–39 uses φg
in place of ψ. The total potential ψ is used in both FE and BE regimes.
As a result, no potential discontinuity exists at the interface, but an additional
load vector due to the field {Ho} must be computed.
Since the magnetic flux continuity condition at the interface of air and
iron is: 
(13–96)
where:
μo = magnetic permeability of free space (air) μI = magnetic permeability of iron The additional load vector may be computed as
(13–97)
where:
{N} = weighting functions
If there is iron and current in the FE domain and the domain is multiply-connected, the generalized potential formulation can be used. It consists of three different steps.
Preliminary solution in the iron domain (MAGOPT, 1). This step computes a preliminary solution in the iron only. The boundary elements are not used for this step.
Preliminary solution in the air domain (MAGOPT, 2). This step is exactly the same as the step 1 of the difference potential formulation.
Total solution (air and iron) (MAGOPT, 3) . This step is exactly the same as the step 2 of the difference potential formulation.