**By R. Belmans**

## Coupled field problems

**The term "coupled problem"** is used in many numerical approaches and applications. Various coupling mechanisms in a different context, such as field problems with electrical circuits, methods in a geometrical or physical sense, couplings in time and/or coupled methods to solve a field problem, are meant with this term. For a proper classification of these problems and related solution methods a systematic definition is proposed. It can be used in the evaluation and comparison of solution methods for various problems.

**A coupled system or formulation is defined** on multiple domains, possibly coinciding, involving dependent variables that cannot be eliminated on the equation level (Zienkiewicz 123). In the literature, this notion is often linked to a distinguishing context of various physical phenomena or methods, without further specification. This paper proposes a classification scheme in which the numerical models meeting the proposed definitions can be put. This may lead to the definition of a series of test problems for specified coupled problems and solution algorithms. A classification scheme can simplify the comparison of the various examples and approaches out of the literature that solve such coupled problems.

**Next to "coupled problems"** the terms "weak-" or "strong-coupled" will be discussed to propose a more homogenous terminology.

To start with a definition of standards or a classification of technical physical problems, the properties and the interdependencies of such phenomena must be considered and discussed.

**A general and simplified structure** of considered field problems is drawn in Fig. 6.1. Here, the link between the single fields is determined by material properties depending on the corresponding field quantities. If the field blocks represent numerical methods to solve the single problem in two dimensions, further couplings to external equations such as electrical circuits, magnetic or thermal equivalent circuit models are possible to complete the scheme.

**The link between the drawn blocks is,** in the context of coupled problems and its numerical solution, a computer model or method. The following question is in which way the physical phenomena have to be considered in an overall solution. From the idea of how to link the effects numerically, a classification of the methods in this sense can be performed.

**The coupling of magnetic field equations**, described by a partial differential equation (PDE) and the electrical circuit equations providing algebraic expressions for the electrical current densities, can be considered as a special type of coupled problem.

**Fig. 6.1. Simplified structure of coupled fields.**

**In general,** more than one independent physical field variable is involved. The field variables for stationary problems are present in a set of PDEs, or in the transient case in ordinary differential equations (ODE). The coupling is often non-linear and this results in a complicated numerical solution process.

**Fields can be described by differential equations.** A general form of a differential equation has to be studied to understand the parameter coupling between equations. Equation (6.1) represents the general form of a differential equation with its possible coefficients in the particular terms. In coupled fields problems, such coefficients are field dependent and represent the link between the various field types, such as magnetic/thermal etc.

The first term characterises the equation being parabolic. Stationary equations do not contain this termIn Laplace’s equation terms 2 and 5 are present. To obtain the Helmholtz equation, term 4 can be added. For these two types of equations, a variational formulation exists. The 3rd term is typical for problems considering motion effects, eq.(5.284).

**The coefficients in (6.1)** are usually derived from given material characteristics. For example, temperature dependent material properties of permanent magnet material can be used to define a coupled magnetic/thermal field problem. Within a field problem definition, the characteristics vary locally.