**By R. Belmans**

**Using arbitrary potentials instead of physical** quantities and the associated functionals in the formulation of the equations, raises the need for a closer look at the post-processing. The user of a FEM system desires to analyse a physical system in terms of field strength, energies, forces, densities etc. The potential itself does not necessarily have a physical meaning. In some cases, such as in the electrostatic and in the thermal analysis, the potential represents the electric potential and the temperature respectively (Table 5.9). Therefore, most of the interesting quantities in the post-process are numerically derived quantities. The type and order of the shape function of the potential over an element (linear, quadratic, etc.) and the element type (nodal, edge, etc.) determine the achievable relative accuracy of numerically derived values. The accuracy of the results is influenced by the discretisation and, related to it, the choice of the error estimator for an adaptive mesh refinement, if applied. Another difficulty arises in the calculation of lumped parameters (inductances, reactances, etc.), used in non-FEM analysis procedures, such as circuit analysis. Several different definitions of these quantities may exist, as for the inductance calculation of linear and non-linear energy transducers.

The aim of this topic is to provide an overview of possible derived quantities, the necessary formulations and ways of influencing the accuracy of the results.

## Potentials

**As shown in the previous topics,** the chosen potentials for the different types of problems do not necessarily directly represent a physical quantity. The formulations for defining these potentials are chosen such that their application might impose simplifications in the formulation of the functionals or the choice of the gauges. A selection of problem types and the physical meaning of their potentials are collected in Table 5.9.

**Table 5.9. Physical meaning of selected potentials.**

The achievable accuracy of all derived values cannot be better than the accuracy of the computed potentials. The latter is determined by the choice of the element type, the shape function and the functionals used.

### Energies

To calculate particular quantities for the elements of lumped parameter models such as inductivities, reactances, resistors etc, or to compute local forces and/or torques acting on bodies present in a magnetic field, energies are used to determine such quantities. There are various definitions.

### Stored energy

**Energies are global quantities.** It has already been discussed that in the finite element method, using the variational technique, an energy term, the functional, is minimised. This energy term does not necessarily have the meaning of a physical energy, for example a stored energy. Potentials are chosen in such a way that the minimisations of the related functional approximate the solution of the partial differential equation. In the case of a Laplace equation, the functional is:

**For a Cartesian, non-linear magnetostatic problem the stored magnetic energy can be calculated by:**

or for linear materials, with the material reluctivity

Equations (5.296) and (5.297), (5.298) are similar. The total stored energy in the overall system, as an integration value is more accurate than any locally derived quantity.

**Functionals for other differential equations can have additional terms in the integrand, but the similarity with an energy formulation is still maintained. The functional for the Poisson equation includes such an additional term:**

**The second term of the integrand can be thought or as being related to the energy input from the supply. In linear systems the energy input from the supply is calculated from the integral over the coil area by:**

**For linear systems,** the stored energy equals the energy input from the supply. This only holds if all potentials of Dirichlet boundaries are set to zero, i.e. no additional flux is forced into the system at the boundaries. This last expression is useful even for non-linear problems for the calculation of flux linkages and inductances in special cases, when the behaviour with changing current is important. The functional used for problems with linear permanent magnets is:

Therefore, the third term in the integrand is related to the energy output from the permanent magnet to the system.

**Most electrostatic problems are linear. The stored energy in an electrostatic model is:**

### Co-energy

Associated with the energy is the concept of co-energy (Cartesian geometry, magnetic problem):

This integral is effectively the "surface under the BH-curve". The co-energy is useful for force calculation. In problems with linear materials and no permanent magnets, the value of the co-energy and the energy are equal.

## Local field quantities

**As shown in Table 5.9, some local field quantities are directly represented by the particular potentials. In this case, their accuracy is determined by:**

**• the simplifications made to the applied differential equation**

**• the choice of the gauges**

**• the choice of the element type**

**• the choice of the shape function**

**• the accuracy of the equation solver**

**• the quality of the discretisation.**

**Therefore, the error convergence of these quantities is of the same order as that of the related potentials. An example to illustrate this:**

**Using the 2D magnetic vector potential,** the normal component of the flux density through the edge of an element is always continuous. The flux through the edge equals the difference of the potentials at the adjacent nodes (the unit of the vector potential is Wb/m). This allows calculation of the flux through a line span between two points just by calculating the difference of the potential value at the end-points. Practically, this could be applied to the calculation of flux linkages.

### Numerically derived local field quantities

**Most local field quantities,** as well as other derived quantities such as force, require numerical derivatives of the potentials. Using nodal elements, the potentials are known at each node as a result of the approximate solution of the partial differential equation. The change of the potential inside one element is determined by the choice of the shape function:

Knowing the potentials at the nodes of the elements, the coefficients a, b and c can be calculated using this basis function. The definition of the potential now determines the required mathematical operations yielding the required local field value. In two-dimensional magnetostatic problems, the vector potential A is defined by:

**Using such linear shape functions to approximate the vector potential, the x- and ^-components of the flux density inside a finite element are calculated as follows:**

The flux density B inside an FEM model is piecewise constant (5.306) if a continuous distribution of the vector potential is assumed. Accounting for this and assuming a small value of h as the maximum characteristic diameter of a finite element, the FEM is convergent towards the exact solution of order q+\. The constant q describes the polynomial order of the elements used. With e as the global error, the order of convergence for the potential solution is

The factor C is independent of the size h of the elements and depends only on the

**• type of discretisation**

**• choice of shape function**

**Equation (5.307)** identifies the convergence problem transferred into the approximation problem. Usingfirst order linear shape functions the rate of convergence is of orderDeriving the field quantities from the potential formulation numerically results, in a rate of convergence for those quantities, i.e. a loss in accuracy of one order compared to the potential solution. Using these field quantities this inherent inaccuracy influences the results of force calculations. This fact identifies the difficulty in obtaining accurate field quantities as a problem of the order of convergence of the numerical method used. To illustrate this fact, consider a domain containing a single linear material. By applying Dirichlet boundary conditions of different values to the left and the right domain border, a constant flux is imposed.

**Fig. 5.95. a) Continuous vector potential; b) piece wise constant flux density.**

The loss of one order of accuracy due to the numerical differentiation is inherent and effects all quantities based on such values.

**There are three possibilities for possibly increasing the accuracy of local field quantities for the end-user of an FEM program package:**

**• Compute the model** with higher order (shape function) elements.

**• Increase the quality** of the discretisation (adaptive mesh refinement).

**• Lower the error** bound where the equation solver stops.

**The latter point is listed for practical reasons.** Especially for eddy current and non-linear problems, a low error bound is absolutely essential. Therefore, it is a good choice to set a stopping margin close to the machine accuracy. If a high accuracy of the local field quantity is required, the relative error of the desired quantity should be monitored. The actions listed in the first two points of the above list can help to achieve this.

**• If higher order elements are available in the code,** rerun the computation by increasing the order of the elements in each step. Plot the desired quantity versus the number of steps. The relative accuracy can be judged from the convergence of the value towards a stable value.

**• Monitor the convergence of the desired quantity** over several steps of adaptive mesh refinement. Particular attention has to be paid to the choice of the error estimator. Some error estimators might have advantages for global quantities, but may not be appropriate for a local field value. The error estimator has to have effect in the region of interest.

**• The points listed** above can be combined.

**Another possible way to increase** the accuracy of local field quantities is discussed in the next section: re-calculation of the field distribution in parts of the domain by a local post-process.

**Two further points concerning the** accuracy of local field values must be mentioned.

**• Field values in the vicinity** of singularities have a large error.

**Adaptive mesh refinement minimises** the effect of these regions with respect to the global solution, but the problem does not vanish.

**• Smoothing techniques must be applied very carefully**. They are popular as they seem to establish a principle in nature: field distributions are smooth. The danger lies in the fact that almost all of the smoothing techniques are based on geometric algorithms rather then on the underlying field equations. Smoothing may lead to just the opposite of what is intended: a loss of local field information.

## Forces and torques

Analysing electromagnetic actuators such as electrical machines, the aim is often to find next to the field quantities the electromagnetic forces generated by the studied device. Various methods are in common use. Different methods, their application and limits are discussed.

### Lorentz force

**A frequently encountered problem is that of a current-carrying conductor in an external magnetic field. The differential force equation may be written:**

whereis the elementary length in the direction of the current Equation (5.308) is derived from the fundamental force relationship between two moving charges. It represents the magnetic part of the Lorentz force. If the conductor is straight and the field is constant along its length, the differential force may be integrated. In a two-dimensional magnetostatic finite element model, the field components are located in the plane, whereas the current is oriented perpendicularly to it. In this case (5.308) can be simplified to the following expression for the conductor of length

Those force equations are theoretically valid only for a conductor in a magnetic field. However, in practice it might be used even for the force calculation in electrical machines with many slots containing current, provided that B is the average value in the air gap. This simplification already indicates a loss of accuracy, as local information about the field is not taken into account. This approach combines analytical and numerical field analysis at a rather simplified level.

### Virtual work

One of the most popular methods for the calculation of forces is based on the spatial rate of change of the stored co energy in the model. The component of the forcein the direction of the displacement s is:

The accuracy of the co-energy calculation is rather high because, as mentioned before, the energy is computed very accurately. It can be expected that the force calculation based on the co-energy should be accurate as well, provided the following requirements are met.

• The method is valid for differential small displacements, which must be translated in terms of the dimensions of the model.

• It is assumed that the magnetic flux remains constant in the two FEM models necessary to compute

The disadvantage of this method is the need for two finite element computations to obtain a single force value.

**A corresponding expression for the torque T associated with an angular rotation****is useful for the electrical machine analysis:**

### Maxwell stress tensor

**Probably the most common approach** to determine electromagnetic forces is known as the Maxwell stress tensor method. In contrast to the virtual work method, based on the energy, the Maxwell stress tensor method describes the forces directly in terms of the magnetic field strength. This method is advantageous, as forces can be determined with only one FEM-solution,

The Maxwell stress approach computes the local stress at all points of a bounding surface and then sums the local stresses (using a surface integral) to find the overall force. The expression for the Maxwell stress tensor can be derived from (5.308). In three dimensions the force is a surface integral:

where the surface vector dS is taken as the outward normal on S. In two dimensions, this reduces to a line integral with the magnetic stress tensor T written by:

**The expression given above may be rewritten in terms of the normal and tangential components of flux density at each point on the closed contour C along which the line integral has to be evaluated. Therefore the associated components of force for an axial length****is:**

These expressions assume the following notations for the directions offor a contour parallel to the >>-axis, and traversed in the direction of increasing y,Also the component values of

(5.313) have units of stress; they do not necessarily give correct local stress values. However their closed line integral has the physical meaning of the total force on the enclosed object. The contour must be entirely in air and not pass through any other material. In many cases the contour does not need to be closed. Parts of the closed contour may be skipped if their integral value is negligible. The expressions for the force computed on a single straight line are:

**The torque on an arc of radius****is similarly given by:**

Similar expressions may be used for electric field problems by substituting E for B andfor

**The advantage of the** Maxwell stress tensor method over the virtual work method recommending only one FEM-solution is lost when the accuracy of the results is compared. As the method is based on derived quantities, particular attention has to be paid. The loss of one order of accuracy compared to the potential solution can lead to large errors, especially when computing the tangential component(5.315) and the torque T in (5.316). In electrical machines the reason for this is the huge difference of the field quantities in magnitude when comparing normal and tangential component. The normal components can differ some decades. This yields large truncation errors in the computed force.

The practical implementation of the algorithm introduces additional error sources. The stress values have to be evaluated at specified points along the contour, usually equidistantly distributed. If such a point is positioned exactly at the edge between two elements, the numerically derived B is double-valued (piecewise constant B for first-order shape function).

**The problem of accuracy of the** Maxwell stress tensor method and possible ways of improving it, have been extensively discussed in literature. Most of the proposed enhancements are based on smoothing algorithms or on different integration schemes. One of the most common methods proposes calculation of the force using different contours and averaging the result. This method can help to evaluate the margin of error, but it does not give any absolute error bounds or even an enhancement of accuracy.