# Finite element based CAD systems (Electrical Machine)

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For designing and constructing electromagnetic devices an accurate knowledge of the field quantities inside the magnetic circuit is necessary. In many cases the air gap is of particular importance (e.g. motors, switches, relays, contactors, actuators). Here the conversion from elcctrical to mechanical energy and vice versa takes place. In the air gap the field quantities such as flux density and field strength have to be calculated very accurately in order to be able correctly to asses the operational behaviour of the device.

Although Maxwell equations have been known for more than a century, in the past the task in calculating a magnetic circuit was to find as many assumptions and simplifications as possible. Then, results could be obtained with rather low numerical efforts. Using this approach, only devices or problems with a strongly simplified geometry could be studied. It was a design following simple rules, found empirically. Physical effects were considered by correction factors applied to the existing rules. In the following period of time this design through rules has changed into another design philosophy: design analysis. Here, computer models were used to solve the field problem. Analysis means the treatment of the field problem by numerical simulation.

With the ongoing developments in computer hard- and software and numerical research, difficulties concerning computational costs and numerical problems are continuously moving to the background. Today, efficient numerical solutions can be obtained for a wide range of problems beyond the scope of analytical methods. In particular the limitations imposed by the analytical methods, their restrictions to homogeneous, linear and steady state problems can be overcome using numerical methods.

In general, the procedure for analysing an electromagnetic device can be divided into three steps:

• pre-processing

• processing and

Fig. 2.1. Field analysis steps.

• the post-processing.

In the first step, the field problem is defined and prepared to be solved. The second step delivers the numerical solution of the physical problem. During the post-processing, the obtained solution is prepared to calculate the required field quantities or to evaluate forces and other macroscopic quantities. This threefold approach of defining, solving and evaluating is typical for every analysis procedure, numerical or analytical. The different techniques, data structures or algorithms used in the individual steps, influence and/or limit the overall procedure during the analysis of a field problem (Fig, 2.1).

To define a field problem, the input data describing the geometry of the domain of interest, the material representation and the boundary conditions are always required. Even with enhanced CAD drawing techniques, most of the analysis time will have to be spent on the preprocessing. Given error bounds will support a desired accuracy of the solution. Often, the user can not influence this step. During the postprocessing, the solution must be prepared to study the local field effects. The post-processing represents an open-ended process, because the user of the analysis can evaluate the calculated solution in various ways for different aspects.

The methods and algorithms used in the single steps of the overall procedure can form an efficient analysis or design tool and determine the quality of the results of the analysis. For example a use of particular internal data structure can enable very quick search routines to obtain an efficient, fast and automated discretisation with parameterised geometries and materials. The various possible coupling mechanisms of different fields, circuit equations, methods such as FEM/BEM combinations, motion term or geometries yield into an accurate approximation of a realistic physical problem. The properties of the coefficient matrix decide which equation solver or algorithms must be used to solve the problem.

CAD systems to treat two-dimensional field problems are in common use nowadays. Developments in hard- and software have made it possible to realise user-friendly and reliable systems running on different hardware platforms such as UNIX and on PCs. Commercially available software is oriented mainly to operate inside a PC Windows™ environment. A reason for this can be seen in the price/performance development of the PC market in the past years. Unix workstations remain more expensive when compared to the PC competitor and the performance of standard PCs is already comparable to UNIX workstations. A growing demand on PC-CAD systems for magnetics can be noticed. This tendency in the market can identify several reasons. The lifetime of hard- and software is decreasing. Having user-friendly software available enables the user to change the CAD system without large training efforts more quickly. It turns out that software systems are becoming consumables. Observing recent years, the CAD software price developments have been rather calm, but the performance characteristics of the software, such as solver speed and user-friendliness, are rapidly increasing. Using commercial software can solve more and more complicated problems:

• complex geometries in 2D/3D

• complexity of the analysis increases

• external circuits including capacitances and inductivities, voltage and current driven

• motion effects

• enhanced force computation, local field quantities

• coupled field analysis.

## Components and modules

A regular CAD system for magnetics includes various components to be able to solve the field problem in an appropriate way. To handle the problem in the three steps of pre-processing, solving and post-processing, modules such as:

• graphical drawing tool to generate the geometry

• mesh generator

• material library manager and modeller

• problem definition tool

• different solver modules for the various field types and formulations

• post-processor tools

• visualisation modules to evaluate the solution and

• file manager tool for the data transfer to other software modules are pre-requisites of a FEM CAD system. User interfaces are recommended to have a maximum of process control for a minimum of efforts during the analysis. To obtain a high quality field analysis tool, the user-interfaces must allow sufficient interaction of the user with the process steps. The influence of the interface on the mesh, problem formulation and solution can be taken from Fig. 2.2. An open data base interface allows further manipulations of the data by other software products, for example to generate different graphical representations of the solution or to analyse FEM models generated by another software package.

Fig. 2.2. Process control of a field analysis.

## Graphical drawing tool for problem definition

To be able to model the technical device to be studied by field analysis, a graphical interface is necessary to generate a technical drawing. This user-interface must support primitives such as lines, arcs, circles and points to describe the geometry properly. If different software packages are used, interfaces to the other software should be available (IDEAS, PATRAN, AutoCAD DXF, …).

Using the generated technical drawing, boundary conditions are set and domains with different material properties are defined. Various labels represent the chosen material of particular domains and set edges of the geometry to characterise the given boundary conditions there. Commercial program packages are supporting libraries with various grades of non-linear ferromagnetic and hardmagnetic materials (Fig.2.3). To define own materials, special software modules can be used to include such data. The defined data can be controlled visually after their definition.

Fig. 2.3. Typical material representations: a) non-linear ferromagnetic and b) permanent magnet material characteristics.

If semi-automated mesh generators are used, mesh size definitions have to be given in this step of problem preparation. Therefore, lines, circles and arcs are selected and subdivided into several parts to form edges of the finite elements to be generated in the next step,

## Mesh generation in general

A mesh generation module must supply a numerical discretisation of interior regions by the finite elements. Standard triangular elements are in common use in two-dimensional models (Fig. 2.4) and tetrahedrons are regularly used to model three-dimensional field problems.

Fig. 2.4. Domain of interest of two conductors in air and the two-dimensional FEM mesh.

The mesh generation is automated with a minimum of user interaction. Several a-priori criteria can be employed to guarantee a certain quality of the discretisation. The solution accuracy is strongly dependent on this mesh. If the strategy of an automated mesh adaptation is supported to enhance the quality of the discretisation in successive computation steps, only a minimum discretisation is recommended in the first mesh. More details can be found in the section on adaptive mesh refinement.

The generation of three-dimensional FEM models is extremely time consuming. Two different strategies can be followed:

• mesh extrusion (Fig. 2.5)

• solid modelling (Fig. 2.8).

The extrusion approach works with two-dimensional meshes extruded in the third direction. Rotations of axis-symmetrical geometries are possible to form the 3D model as well. A disadvantage is that not every contour can be modelled realistically. For example, conical surfaces represent a problem (Fig. 2.7). If the scalar potential formulation is used, excitation coils and windings can be introduced into the model in a following step (Fig. 2.6).

A solid modeller works mainly in two steps. First the surface of the geometry is discretised, and after this the volume is meshed in a second step. Various suggestions to generate solid meshes can be found in the literature (Tsukerman & Plaks 1H).

Fig. 2.5. Basic idea of the extrusion technique.

Fig. 2.6. A three-dimensional model of a wire heating device.

Fig. 2.7. Comparison of a) extrusion based and b) solid modelling of a claw-pole generator.

Fig. 2.8. Solid modelled 3D mesh of the rotor of a claw-pole generator.

After the mesh generation the field problem is defined and can be solved by the equation solver. This is mainly in commercial software packages performed without or with a minimum of user interactivity. The appropriate solver has to be chosen.

## Post-processor tools

Several tools are recommended to evaluate the field solution. The potential solution has to be transformed into physical quantities such as flux density, field strength or forces. Therefore, numerical manipulations of the potential are necessary. The post-processor module must consist of a calculator to perform such manipulations.

To be able to evaluate the solution, various graphical representations of the solution can be of interest:

• colour plots of selected quantities

• plots of the lines of constant potential, flux plots

• diagrams showing quantities along defined contours.

To extract parameters out of the solution, a post-processor calculator can be used as well.