**By R. Belmans**

**Many scientific engineering or physical** problems lead to boundary value problems. The describing differential equations have to be solved in a volume satisfying particular conditions on its boundary T (Fig. 3.2). Therefore, the definition of a boundary value problem is necessary and evident. The proper definition of a numerical model is important to obtain correct results and assumes a good understanding of the underlying physical background of the field problem.

**Fig. 3.2. Boundary value problem.**

**For obtaining the solution of the boundary value problem, it can be formulated in the form:**

The boundary value problem is defined by a differential equation a(u, v) feasible in the volumeu is the exact solution of the problem that has to be found.

V is a set of continuously differentiate functions inwith for exampleat the boundaryThe type of functions v and differential equation that can be employed is subject to the method or approximation used to solve the field problem.

Assuming that the appropriate differential equation for a particular physical problem is known, the definition of the numerical model is dependent on the correct choice of boundary conditions.

**If it is considered that the** desired field solution is a local potential distributionin a local co-ordinate system and time depending

different types are possible and can be defined.

## Initial and boundary conditions

**Boundary conditions have to be applied** to a field problem to ensure a well posed problem with a unique solution. Particular attention has to be paid to the Dirichlet and Neumann boundary conditions. Applying these boundaries in an appropriate way reduces the size of the field problem significantly. On the one hand, therefore, the accuracy of the solution can be improved with the same computational expenses; on the other hand an enlarged domain can be studied resulting in the same solution accuracy.

**Mainly finite** element program packages are limited in the number of elements to approximate the geometry of the problem or the computational efforts must be limited in order to obtain acceptable computation times. Therefore, the correct and appropriate application of the boundary conditions is the key to defining field problems and to allow an accurate solution in an efficient way.

**Starting conditions** When a differential problem covers the time domain, the starting conditions are quantities valid at simulation start-time, for example the velocity, flux couplings with windings in electrical machines, field exciting currents or voltages that have to be set and defined in order to find the solution of a transient problem formulation.

**Dirichlet boundary condition** A Dirichlet boundary condition sets the unknown function to a known function on the boundary of the differential problem.

**Fig. 3.4 shows in a typical** example the application of the Dirichlet boundary condition of an electromagnetic problem. A ferromagnetic circuit is shown consisting of a U-shaped permanent magnet, an air gap and ferromagnetic back iron. Lines of constant vector potential represent the flux lines. Physically, the field is assumed to be zero at a sufficiently large distance from the magnetic circuit. Therefore, a Dirichlet boundary condition, the potential set to zero, is applied at the entire boundary of the problem. Due to eq.(3.4) it is impossible that flux lines can cross the boundary T. Fig. 3.4b shows the potential lines of an electrostatic field with Dirichlet boundary. Here, the potential distribution excited by the charged plates of a capacitor is computed.

**Fig. 3.3. Dirichlet boundary condition applied to the FEM model of an electrical machine, a) Mesh of the model accounting for flux relief, b) the flux lines, c) the mesh neglecting the flux relieve of the machine, d) its flux plot, e) the mesh applying the Kelvin transformation, f) its flux plot.**

**The Dirichlet boundary is an essential boundary** condition as it does not characterise the space V . It is sometimes called the boundary condition of first kind.

**Fig. 3.4. Dirichlet boundary condition a) for an electromagnetic and b) for an electrostatic problem.**

**For the analyst of a field problem,** a crucial question is how far away the Dirichlet boundary condition has to be applied from the field exciting sources to restrict the field-domain on the one hand and to have an accurate overall solution of the near and far fields respectively on the other hand. When analysing electromagnetic fields in the presence of ferromagnetic material and small air gaps, such as in electrical machines, an outer diameter of roughly 20% above the characteristic diameter of the device can be applied to compute the field inside the device accurately (Fig. 3.3a). Flux lines can not pass the Dirichlet boundary. If the flux relief inside a ferromagnetic core due to saturation can be neglected, the outer diameter of the back iron yoke of electrical machines can be represented by the Dirichlet boundary condition (Fig. 3.3c). If the flux outside the machine yoke can be neglected, the number of elements in the numerical model and thus the computation time, decreases.

**If the far field is analysed,** a diameter of up to 5 or 6 times the characteristic dimensions of the device should be used or special transformations, such as the Kelvin-transformation, an open boundary condition (Fig. 3.3e/f), can be employed to terminate the field in the transformed infinite distance.

**Neumann boundary condition** The next important boundary condition is the Neumann boundary condition. Here, the known value of the derivative of the unknown function in the normal direction of the boundaryis prescribed.

If the derivative in normal direction is constant, lines of constant potential can pass the outer boundaryof the studied domain

The most important property of this type of boundary condition is that by knowing symmetries of the field, and applying the Neumann boundaries there, the numerical model can be reduced to obtain the same solution of the problem. In this way, the problem size, the time to prepare the field problem and the computational efforts can be reduced significantly. On the other hand, if less than the complete geometry has to be defined and discretised, a higher accuracy is achievable for the overall solution of the problem without extra effort. Therefore, particular attention must be paid to this boundary condition.

**Fig. 3.5a shows the electromagnetic field problem** with applied Neumann boundary at the symmetry line of the U-shaped permanent magnet. This is the only symmetry inside this model and yields a problem reduction of 50% with respect to the accuracy of the problem solved in Fig. 3.4.

**The Dirichlet** boundary remains of course at the outer diameter of the domain studied.

**Fig. 3.5. Neumann boundary condition applied at the line of symmetry.**

**By looking at the electrostatic example of the capacitor in Fig. 3.5,** an additional symmetry in the potential distribution attracts attention. Knowing the potentials at the electrodes of the capacitance, +100 V at one side and -100 V at the other, the Dirichlet boundary with a constant potential of 0 Volt can be applied, reducing the problem size a second time (Fig. 3.6).

**Fig. 3.6. Lines of symmetry with appropriate boundary conditions.**

The Neumann boundary is a natural boundary condition as it does not influence the definition of the space V. It is automatically satisfied at the boundary and is sometimes called the boundary condition of second kind (Zienkiewicz & Taylor122).

**Mixed boundary condition** A mixed boundary is a combination of the two last boundary conditions (Dirichlet and Neumann).

It is called a Robin or Cauchy boundary condition or boundary condition of the third kind. This type of boundary condition (Comini et al.27) can define convective boundaries in heat conduction problems. There, the heat flux as function of the temperature is prescribed at T and the temperature of the surrounding medium is known.

**Binary or periodic boundary conditions** Until now only symmetries in the geometry were considered to lead to the application of the Dirichlet or Neumann boundary conditions. Especially in cylinder symmetric devices, such as rotating electrical machines, not only symmetries in the geometry but also in the magnetic field distribution are present. Under load conditions the air gap field of an electrical machine repeats periodically every double pole pitch. At no-load operation, it repeats itself every pole pitch. This field periodicity can be used to define another type of boundary condition to reduce the size of the numerical model. The local potentials in such boundaries depend on the solution of the field problem and thus inherently occur always in pairs. One boundary is computed and the opposite one is linearly linked to this value.

**Fig. 3.7. Periodic boundary condition applied to a 4-pole induction motor model.**

**In Fig. 3.7 the** dotted lines indicate the pairs of the boundaries. Obviously, the numerical discretisation of the model at those boundaries must be identical. This type of periodic boundary condition has the form:

Ifand k equals 1 or -1 this boundary is called binary boundary condition.

**Far-field boundary condition** The differential formulation of the field equations in the finite element method has its disadvantages for computing open and unbounded physical fields. The whole field domain, theoretically until infinity from the field sources, must be discretised to be able to compute the far field. For example, if the electromagnetic field in the vicinity of a high voltage transmission line is analysed, the air and ground have to be modelled. To model the infinity, a Kelvin-transformation can be used to map the infinite space to a finite space, forcing their solution to be identical (Fig. 3,8). Using this technique reduces the problem size and computational expenses significantly. Fig. 3.8 shows the circular domain of interest of an electrostatic problem with a high voltage tower in its centre. The small circle above is the FEM model that approximates the infinite space. There, one of the centre nodes is set to zero potential whereas the nodes at its circumference have the same potential as the diameter nodes of the large circle. The lines connecting both circles illustrate this link.

**Fig. 3.8. Open or far-field boundary condition applied to the simulation of the electromagnetic field of a high voltage transmission line.**