The following boundary conditions can be specified at outward and inner boundaries of the region.
Dirichlet condition specifies a known value of vector magnetic potential A0 at the vertex or at the edge of the model. This boundary condition defines normal component of the flux density vector. It is often used to specify vanishing value of this component, for example at the axis of symmetry or at the distant boundary. QuickField also supports the Dirichlet condition with a function of coordinates, it has the form
A = a + b·x + c·y |
- for planar problems; |
rA = a + b·z·r + c·r2/2 |
- for axisymmetric problems. |
Parameters a, b and c are constants for each edge, but can vary from one piece of the boundary to another. This approach allows you to model an uniform external field by specifying non zero normal component of the flux density at arbitrary straight boundary segment.
Let α be an elevation angle of the segment relative to the horizontal axis (x in planar or z in axisymmetric case). Then in both plane and axisymmetric cases the normal flux density is
Bn = c·sin(α) + b·cos(α).
Here we assume right-hand direction of positive normal vector.
Choice of constant terms a for different edges has to satisfy the continuity conditions for function A0 at all edges' junction points.
Note. For problem to be defined correctly the Dirichlet condition has to be specified at least at one point. If the region consists of two or more disjoint subregions, the Dirichlet conditions have to be specified at least at one point of the each part. Zero Dirichlet condition is defaulted at the axis of rotation for the axisymmetric problems.
Neumann condition has the following form
Ht = σ |
- at outward boundaries, |
Ht+ - Ht- = σ |
- at inner boundaries, |
where Ht is a tangent component of magnetic field intensity, "+" and "–" superscripts denote quantities to the left and to the right side of the boundary and σ is a linear density of the surface current. If σ value is zero, the boundary condition is called homogeneous. This kind of boundary condition is often used at an outward boundary of the region that is formed by the plane of magnetic antisymmetry of the problem (opposite sources in symmetrical geometry). The homogeneous Neumann condition is the natural one, it is assumed by default at all outward boundary parts where no explicit boundary condition is specified.
Note. Zero Dirichlet condition is defaulted at the axis of rotation for the axisymmetric problems.
If the surface electric current is to be specified at the plane of problem symmetry and this plane forms the outward boundary of the region, the current density has to be halved.
Zero flux boundary condition is used to describe superconducting materials that are not penetrated by the magnetic field. Vector magnetic potential is a constant within such superconducting body (rA = const in axisymmetric case), therefore superconductor's interior can be excluded from the consideration and the constant potential condition can be associated with its surface.
Note. If the surface of a superconductor has common points with any Dirichlet edge, the whole surface has to be described by the Dirichlet condition with an appropriate potential value.