**By R. Belmans**

**After this first more or less subjective** judgment of the various coupling mechanisms, in the following discussion the coupled problems are distinguished with respect to physical and numerical aspects. The single involved types or mechanisms of coupling the various fields are described here as sub-problems with specific properties. It will be concluded with a matrix systematic. The matrix entries distinguish between the problem, the model description, the coupling mechanism, a proposed iteration scheme and a proposed method for solving the overall field problem.

## Sub-problem extent: domain/interface

The different interacting physical phenomena described by the coupled problem are defined on partially or totally overlapping domains. For example thermo-electromagnetic problems belong to this group. For the electromagnetic problem definition the surrounding air has to be modelled. The same domain is considered in the thermal problem by special boundary conditions such as heat transfer due to convection or radiation boundaries. By using the FEM, different meshes for each sub-problem can be used. The interaction takes place through interface equations. The involved field problems can be numerically strong, i.e. on matrix level, or weak coupled, computed in a cascade algorithm.

**In a second class,** the interaction of the fields is described by interface equations of the sub-problems. A heat transfer or cooling problem with different models for a heated object and the cooling fluid belongs to the second group. The group determined by overlapping domains is sometimes referred to as "class I" and the interface group as "class II".

## Sub-problem discretisation methods: homogenous/ hybrid

**It is sometimes advantageous** to apply different discretisation methods for the involved fields. The methods used can be the FEM opposed to BEM or FEM methods with different types of elements to result in a hybrid method. Analytical models can be considered. The addition of algebraic equations originating from equivalent circuit models is possible as well. For example, a two dimensional FEM model to compute the temperature distribution inside an electrical machine can be extended by an equivalent thermal circuit model to consider the heat transfer in the axial machine direction. In this way, a quasi three-dimensional approach is obtained by the coupled methods. The combination of different FEM models with an additional analytical model is possible. External electric circuits can be coupled to consider the voltage or current-driven energy source. V

## Numerical iterative solution methods: full/cascade algorithms

Due to the nature of the physical sub-problems and the chosen discretisation method, differing numerical properties can be linked to the equations descending from the sub-problems. A variety of numerical methods can be chosen to solve the single sub-problem. Most of them can be regarded as block iterative schemes. It is possible to put all the subsystems in a single matrix, with off-diagonal blocks mathematically describing the (linearised) coupling. This can be considered as a numerically strong and thus fully coupled approach.

**On the other hand,** several blocks can be solved separately with a well-suited equation solver. Not considering a possible parallellisation, the solution of the sub-problems is usually obtained in successive steps in a "cascade" algorithm. The newly obtained part of the solution can be used immediately in the next step of the iterative process. Other suitable solution techniques are domain-decomposition (DD) algorithms.

## Classification matrix

**The above remarks on the classification** of coupled problems to build up a matrix systematic underline the difficulty of putting all the mechanisms with respect to their different nature into a single systematic. The developed matrix shows couplings between entries in the horizontal as well as in the vertical direction (Table 6.1). Bi-directional links to other entries are possible as well.

The columns of the matrix represent the mentioned differences of the considered problems with its coupling mechanism. The rows of the systematic represent the proposed types of problem to put into the appropriate columns.

**With respect to the geometry**, in the first column the studied domains have different properties, such as strong differing material characteristics. The numerical sub-problems are described by partial differential equations (PDEs) and the coupling of the systems of equations is defined by its boundary conditions or interface equations. Depending on the condition of the single sub-problems, a full coupling and weak coupling by cascade algorithms is proposed. For example a hybrid FEM/BEM can be used to solve the overall field problem or in the case of strong differences in the condition of the sub-problems domain decomposition (DD) algorithms, a weak coupling can be employed. Here, an ambivalence of the overall problem can be noticed. Using a hybrid method can be considered as a coupled method and the DD as a weak coupling of physical systems.

**The physical nature of the field sub**-problems is considered in the second column. Examples of this are coupled magnetic/thermal or other field combinations. The fields can be described either by PDEs or by a combination of PDE and algebraic equations, if equivalent circuit models are used for one of the sub-problems. The coupling is mainly performed by the exchange of the material parameters and source terms or directly by the circuit equations; for example if external electric circuits are considered. For the solution, numerically strong and weak coupled iteration schemes can be applied.

**Hybrid methods are put into the third column**. The coupled phenomena have different numerical properties. All possible coupled methods such as FEM, BEM, magnetic-, thermal-equivalent circuits as well as the classical analytical field theory coupled to modern numerical techniques, are put to this matrix entry. The model description of the overall problem can be done by coupling PDEs, circuit equations, analytical methods or other methods.

**The difference of behaviour in time of the coupled** effects considers the last column of the matrix. Here, all the transient problems can be found. Simulations with respect to the differential equation of motion, an ordinary differential equation (ODE) are put into this matrix entry. Various methods are suited to solve such in time-coupled problems.

**Table 6.1. Classification scheme for coupled field problems.**

**PDE partial differential equation**, alg algebraic equation, ODE ordinary differential equation, DD domain decomposition method