**By R. Belmans**

## Introduction

**An original design and the step** by step optimization of physical technical devices is in practice often a trial and error process. During the design and construction of a device several expensive prototypes have to be built to monitor and check the mathematical approximations and the physical reality. This procedure is time consuming and expensive. Successful industrial developments demand shorter cycle times to fix or improve the economical competition of particular companies. To effectively compete in the market place nowadays, developed products of higher quality, improved efficiency and better functionality arc recommended, leading to devices with very complex geometries. Furthermore, custom designs are becoming very important. The added value of standard mass production devices is far lower. To solve the techno-economical demands, the idea is to replace the expensive prototyping by numerical simulations.

**If an appropriate simulation model is found,** various operating points can be simulated on a computer. Even the behavior of the device for hazardous situations that cannot be measured inside a laboratory and the use of arbitrary even future materials can be studied. The appropriate choice of a calculation technique for an electromagnetic device is always closely linked to the complexity of the problem.

**To develop a technical product, parasitic effects such as:**

**• ferromagnetic** saturation

**• increased** leakage flux

**• high operating** temperatures

**• irreversible flux** losses by using permanent magnet materials at elevated temperatures

**• coupling between** different effects such as thermal-magnetic-mechanical-flow field problems and

**• induced currents due to motion** effects have to be considered in the calculations accounting for sufficient accuracy. In devices with complex geometries, those effects can not be treated by a classical analytical approach. Results with a high accuracy are required to predict the behavior of the technical product. In this case the simulation of the electromagnetic fields and their effects by numerical models is suitable as an appropriate engineering tool. Using computer models and the appropriate numerical algorithms solves the physical problem. The numerical method has to fulfill specified demands such as:

**• reliability **

**• robustness **

**• application range **

**• accuracy **

**• performance.**

**To see where the numerical** simulation finds its place in the analysis of technical devices, Fig. 1.1 shows the links between the real technical device, the classical physical theory and the numerical simulation. This figure makes obvious that the numerical simulation is a connecting element between reality, measurements, and theoretical predictions. As a consequence, all numerical computations represent realistic activities in a fictive laboratory. This means that simulation results should be theoretically measurable in practice. The numerical simulation is in fact an experiment performed on the computer as a fictitious laboratory, where the engineer is using numerical tools to perform the experiments instead of measurement devices such as current, voltage, power, temperature and force meters.

**Fig. 1.1. Theory, experiment and simulation.**

**The numerical simulation influences** the analytical theory where sometimes rough approximations or constants are used to consider physical effects such as ferromagnetic saturation or hysteresis. The verification of numerical solutions and results obtained by the analytical theory can lead to improved analytical models and vice versa. Both numerical simulation and analytical theory help to understand the physical reality and to improve technical predictions.

**In Fig. 1.2,** the solution process for a system of partial differential equations is outlined.

**Fig. 1.2. Solution process for a system of partial differential equations.**

**The fields are described by differential equations**. Assumptions concerning boundary conditions, material properties such as isotropy, dependencies in time, etc. have to be made before a computation of a field can be performed. For example in magneto-static fields, the time derivative is assumed to be zero and therefore no induced currents can be considered.

**The choice of the potentials is based** on these simplifications. For each problem type, the choice of an appropriate potential is different. The choice of a gauge is necessary to obtain a regular system of equations. Using the finite element method, the choice of the gauge also determines the choice of the element type. However, the user of a CAD program package that simulates magnetic, electric or thermal fields is usually not involved in choosing for such basic numerical properties.

The numerical method to solve the partial differential equation is understood as a solution criterion.

The appropriate solution method depends on the type of equation, such as parabolic, hyperbolic or elliptic.

**For example,** the choice of the elements for the finite element method depends on the differential equation, the potential formulation, and the solution method.

**In a two-dimensional magnetostatic problem the** unknowns are node potentials. Here, the magnetic vector potential is chosen because the nodal unknowns have only a single component Az. In this two-dimensional field problem, the Coulomb gauge is satisfied automatically.

**The choice of method for solving a system** of linear equations is dependent of the differential problem and its formulation. For example the magneto-static problem is an elliptic differential problem. The Laplace operator is symmetrically adjoint and positive definite. A system of equations with such properties can be solved by a conjugate gradient method.

**Fig. 1.3. Solution processes during a field computation session.**

**To focus on the active parts performed by an design engineer,** in principle, field computation is performed in three major steps: preprocessing, processing and postprocessing. Fig. 1.3 shows a typical pattern for the FEM approach. The first step consists of the definition of the geometry of the electromagnetic device. Material properties, electrical current densities and boundaiy conditions are defined. All the activities have to be performed by the design engineer. Therefore, the preprocessing is time consuming. The estimated time expenditure for a two-dimensional problem is given in Fig. 1.3. The processing, i.e. the solution of a very large system of equations is automatically done in the second step. Only parameters to control the solution process have to be defined by the design engineer. In the last part of the FEM procedure, the interesting field quantities are computed from the solution out of the processing. If the geometrical data can be parameterised, the pre- and post-processing can be automated as well. This represents an important prerequisite for the possibility of the combination of field computation and numerical optimisation.