By R. Belmans
In general, it is possible to distinguish between the coupled problem in two ways, in its physical or its numerical nature. Very often a coupled problem is called either
• strong, or
• weak.
In the physical sense, the strong coupling describes effects that are physically strongly coupled and the phenomena can not numerically be treated separately. If numerical formulations exist, the coupling can be found in the governing differential equations due to the coupling terms. The weak coupling describes a problem where the effects can be separated. The problem with this definition is obvious: If coupled problems are studied, it is not very well known how strong or weak they are physically coupled; this is the desired answer expected from the analysis of the overall problem. For example if the material property describing parameters are non1 inearly dependent on the field quantities, the coupling, (strong/weak) can even change with varying field quantities and the field quantities are the result of the analysis. Therefore, the definition of strong/weak coupling should be chosen according to the numerical aspects instead of their physical nature. Choosing for the numerical aspects, it is possible to have a combined strong/weak coupling of field problems. This means that the strategy of coupling can vary, and thus the methods/models, while solving the problem.
Numerical strong coupling is the full coupling of the problem describing equations on matrix level. The equations of all involved and modelled effects are solved simultaneously. This implies that the coupling terms are entries in the coefficient matrix as well.
The numerical weak coupled problem is understood as a cascade algorithm, where the considered field problems are solved in successive steps and the coupling is performed by updating and transferring the field dependent parameters to the other field definition before solving again.
Since the problems cannot be distinguished by means of elimination, a bidirectional influence exists. The sensitivity of a subproblem to changes of the variables of the studied problem can differ strongly. It is difficult to quantify a threshold for separation of both groups, and therefore the separation may be considered as somewhat subjective. In this respect, the time constants of the subproblems play an important role. Usually the thermal and mechanical time constants are several orders larger than the electromagnetic time constants. So, on a short term, the problem with a larger time constant can be considered as weak coupled. But this is not true if the stationary solution is of interest.
FEM coupling of two fields
In this section the strong coupling FEM equation system of a magnetic/thermal problem is derived. For simplicity it is assumed that both field problems are defined on the same mesh. For a more realistic coupling, projection methods can be applied to enable the field definitions on different meshes. This approach results in additional coupling terms in the final coefficient matrix. For further simplicity, the material’s properties v and k are assumed to be independent of A and T respectively. The coupling of the fields causes the remaining nonlinearity by the loss mechanism.
The magnetic/thermal coupled problem is modelled by a set of two equations:
It is assumed that the source term of the thermal equation consists only of joule losses:
The first term will appear on the righthand side of the system. The second term, the eddy current losses, have to be linearised and represent the coupling term with a nonlinear coefficient:
Written in matrix/vector notation eq.(6.4) is rewritten as:
There is a coupling present through the coefficients, although there is a zero entry in the offdiagonal of the magnetic equation. Applying the Galerkin approach results in an integral per element of the form:
For twodimensional first order elements this yields six algebraic equations:
The first three equations are complex, the last three real. The entries marked with an * are the same terms that would be found in the decoupled problem. The terms marked with a + result from the eddy current heat source term.
Nonlinear iteration
A method of handling the remaining nonlinearity, is the Piccard iteration or successive substitution. The block iteration scheme can be given by:
stepl 
solve die magnetic equation (with relaxation) 

step2 
calculate the heat source terms 
step3 
solve the thermal equation (with relaxation) 

step4 
check convergence 
Every loop involves the solution of two systems of equations in successive steps. For the calculation of the heat sources, the magnetic solution from step I can be employed (GaussSeidellike algorithm) or the magnetic solution of the previous loop (a slower Jacobilike algorithm). The relaxation of the iterations proves very important. An adaptive relaxation parameter can reduce the number of iterations significantly. A faster convergence can be expected applying Newton iterations.