# Finite element method (Electrical Machine) Part 1

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In general, differential equations are hard to solve. The idea is to find a solution for the overall problem by substituting for the complicated problem a series of simpler ones. This means setting up the problem by a easily-solved linear system of equations.

Therefore, the problem has to be discretised in adequate sub-problems. The sub-problems are geometrically described by geometrically simple shaped elements such as triangles (Fig. 5.25) or rectangles for two-dimensional and mainly tetrahedrons for three-dimensional problems. Other element shapes are possible as well. When comparing the meshes from the FDM and the FEM model in Fig. 5.23, it is obvious that the FEM model approximates better the geometry of the studied domain. A necessary local mesh adaptation is possible in this model as well. Fig. 5.25. Minimal triangular discretisation of a two-dimensional finite element model.

These elements, forming the numerical discretisation, the mesh, are called the finite elements. On this discretisation, the problem describing differential equation is locally approximated by simple basis function. The approximated overall solution is obtained by assembling all sub-problems into a system of equations and solving this. After this procedure, the approximated potential solution is known in certain points of the discretisation.

The problem is to determine the field describing potential functions for the discrete problem, the finite element equations, and to define an adequate basis or shape function to be able to assemble the overall system of equations. Different methods can be used to determine the finite element equation from the differential equation (Fig. 5.26).

If for particular field problems a variational principle is known, the discrete problem can be obtained by using the Ritz method. In this case, the generation of the discrete problem is easy to obtain. Unfortunately,this variational principle is not known for every technical problem. In this case, for instance, the very common method of the weighted residues can be applied. The most important finite element methods are:

• various Ritz methods

• variational method

• weighted residual method (weak form of the governing equations)

• different types of Galerkin method

• approaches based on the energy-minimum functional. Fig. 5.26. The basic concept of the finite element.

## Variational approach

In this section, it is assumed that a variational equation exists for the studied field problem. Therefore, the generation of the discrete problem is easy to obtain. With is a real value depending on two functions and V is a set of differentiable functions in the field domain with on the boundary is a linear form on V.

To determine the variational equation, a Poisson equation is chosen as an example, assuming the appropriate boundary conditions, with the Laplacian and with q , a given continuous and u, a potential function.

The differential equation is multiplied with an arbitrary function and integrated over to obtain a linear form. It can be written:  is a closed and oriented surface, which includes the domain Q, R are functions of three variables defined in The partial derivative of first order from P, Q, R must exist and must be continuous. With the approach: and considering the boundary condition v = 0, the right hand side of the Gauss integral gauge can be written by: with the derived functions it can be written: With this, the boundary problem described by a Poisson equation is transferred into a variational equation. ### Discretisation of the differential equation

The variational equation is applied to the standard example of a Poisson equation:  independent linear functions out of is the set of linear combinations is called the ^-dimensional partial space from are the basis- or global shape functions (Goering et al.44, Zienkiewicz & Taylor l23).

The idea now is to find an approximated solution using the linear combination of the N functions This yields the formulation of an equivalent problem:  Equations (5.38) and (5.39) are equivalent problems and are called the discrete problems.

## Practical considerations

Equation (5.38) was the starting point of the theoretical considerations of the FEM. The initial position for the practical calculation of the approximation is eq.(5.39). Using and substituting in the discrete problem, assuming a being a bi-linear form it can be written: This represents a system of N equations with the N unknown and the coefficient matrix  with The discrete problem corresponds to a system of equations. Using the shape functions to calculate the elements of the coefficient matrix and the right hand side of and solving the system of equations (5.41) leads to the unknown Evaluating eq.(5.37) gives the approximation of the problem The practical realisation of the FEM depends strongly on the choice of the shape functions The basis functions must have some particular properties.

## Shape or basis function

The definition of the shape or basis functions is not dependent on the single FEM. The same ideas used to construct the basis functions can be taken for the weighted residual methods, the Ritz methods, the different Galerkin approaches, the method using a variational equation and the energy minimum functional as well. The following ideas are of a general application.

If the approximation u, from u must be very accurate, the number N of the basis functions must be very high. This results in a large system of equations. This is the reason why the basis functions must be chosen in such a way that the coefficient matrix contains as much as possible of zero elements. The best possibility would be to choose the shape functions such that is the unity-matrix. This is not practically possible.

In general is an integral over the domain of summations of products of the and their derivatives. If the shape functions are chosen such that they are only in a partial domain non-zero and else equal to zero, the products are only for some combinations of non-zero. This means for a FEM discretisation that for many should not have common nodes. This results in the desired sparse system of equations. The requirements to be fulfilled by the shape functions are:

• smoothness, piece-wise differentiable

• additional properties resulting from the boundary conditions must be satisfied

• the shape functions should be simple

### Construction of the basis functions

To construct a useful shape function, (Goering et al.44) the unit square (Fig. 5.27) is discretised into partial squares with the co-ordinates with   Fig. 5.27. Unit square with triangular FEM discretisation.

In this example, triangular-shaped elements are assumed, and so every partial square is represented by two triangular elements (Fig. 5.27). is the space characterised by a linear function inx and y inside each triangle.

The shape functions are chosen in such a way that one node of the triangle has the function valije 1 and the other two nodes a zero value (Fig. 5.28): Therefore, it is

Fig. 5.28. Simple linear shape function

For example triangle 1 can be written: Now the shape functions from the triangles 1, 2, …. 6 are considered together. If the linear function is applied, the requirements for the triangular regions (5.45) result in a system of equations. In terms of matrix representation: The solution of this system of equations yields and The value of the shape function for all triangles can be given by: Fig. 5.29 illustrates the properties of the shape function introduced, is a simple linear shape function with the desired properties. It fulfils the conditions to generate a sparse overall coefficient matrix. Fig. 5.29. Properties of the approximation applying linear shape functions.

Various basis functions of higher order such as quadratic, cubic etc. are possible and in common use, for example linear:  The degree of freedom (DOF) per finite element for a chosen basis function can be determined by (Kost6S):

 polynomial degree of p    2 6 1 3 4 2 6 10 3 10 20 4 15 35 5 21 56

Using polynomials of higher order generates a more accurate approximation of the exact solution. The DOF increases with rising order of the polynomial and this means that the computational expenses are increasing as well.

From the properties of the basis functions, general conclusions on the properties of the mesh discretising the domain H can be given. A mesh must consist of:

• non-overlapping elements

• nodes, corresponding to the nodes of an adjacent element (for node elements). Fig. 5.30. a) Regular and b) not regular triangular element mesh.

## Basic principle of the FEM

Three steps determine the basic principle of the FEM: • The approximated solution from • The approximated solution from 