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.

Minimal triangular discretisation of a two-dimensional finite element model.

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.

The basic concept of the finite element.

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.

tmp10469_thumb

Withtmp10470_thumbis a real value depending on two functionstmp10471_thumband V is a set of differentiable functions in the field domaintmp10472_thumbwithtmp10473_thumb on the boundarytmp10474_thumbis a linear form on V.

To determine the variational equation, a Poisson equation is chosen as an example, assuming the appropriate boundary conditions,

tmp10480_thumb

with the Laplacian

tmp10481_thumb

and with q , a given continuous and u, a potential function.

The differential equation is multiplied with an arbitrary function tmp10482_thumband integrated overtmp10483_thumbto obtain a linear form. It can be written:

tmp10486_thumb

The Gauss-integral gauge transfers a volume integraltmp10489_thumbinto a surface integraltmp10490_thumbtmp10491_thumb

tmp10492_thumbis a closed and oriented surface, which includes the domaintmp10493_thumb Q, R are functions of three variables defined intmp10494_thumbThe partial derivative of first order from P, Q, R must exist and must be continuous. With the approach:

tmp10498_thumb

and considering the boundary condition v = 0, the right hand side of the Gauss integral gauge can be written by:

tmp10502_thumb

with the derived functions

tmp10503_thumb

it can be written:

tmp10504_thumb

With this, the boundary problem described by a Poisson equation is transferred into a variational equation.

tmp10505_thumb

Discretisation of the differential equation

The variational equationtmp10506_thumbis applied to the standard example of a Poisson equation:

tmp10508_thumb

tmp10509_thumbindependent linear functions out oftmp10510_thumbis the set of linear combinationstmp10511_thumbis called the ^-dimensional partial space fromtmp10512_thumbare the basis- or global shape functions (Goering et al.44, Zienkiewicz & Taylor l23).

An approximated solution fromtmp10513_thumbis a functiontmp10514_thumb

The idea now is to find an approximated solutiontmp10515_thumbusing the linear combination of the N functions

tmp10523_thumb

with the unknowntmp10524_thumb

This yields the formulation of an equivalent problem:

tmp10525_thumb

This is a projection of the problemtmp10526_thumbin V into a problem intmp10527_thumbIf this is true for alltmp10528_thumband for alltmp10529_thumbit can be written:

 

tmp10538_thumb

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 approximationtmp10539_thumbis eq.(5.39). Using

tmp10541_thumb

and substituting in the discrete problem,

tmp10542_thumb

assuming a being a bi-linear form it can be written:

tmp10543_thumb

This represents a system of N equations with the N unknowntmp10544_thumband the coefficient matrixtmp10545_thumb

tmp10550_thumb

with

tmp10551_thumb

The discrete problem corresponds to a system of equations. Using the shape functionstmp10552_thumbto calculate the elements of the coefficient matrix tmp10553_thumband the right hand side oftmp10554_thumband solving the system of equations (5.41) leads to the unknowntmp10555_thumbEvaluating eq.(5.37) gives the approximation of the problemtmp10556_thumbThe practical realisation of the FEM depends strongly on the choice of the shape functionstmp10557_thumbThe 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 matrixtmp10564_thumbcontains as much as possible of zero elements. The best possibility would be to choose the shape functions such thattmp10565_thumbis the unity-matrix. This is not practically possible.

In generaltmp10566_thumbis an integral over the domaintmp10567_thumbof summations of products of thetmp10568_thumband their derivatives. If the shape functions are chosen such that they are only in a partial domaintmp10569_thumbnon-zero and else equal to zero, the productstmp10570_thumbare only for some combinations oftmp10571_thumbnon-zero. This means for a FEM discretisation that for manytmp10572_thumbshould 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

• a good approximationtmp10573_thumbfromtmp10574_thumbshould be obtained.

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-ordinatestmp10575_thumbwithtmp10576_thumb tmp10577_thumb

 

 

 

 Unit square with triangular FEM discretisation.

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).

tmp10593_thumbis thetmp10594_thumbspace characterised by a linear function inx and y inside each triangle.

The shape functionstmp10595_thumbare 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):

tmp10599_thumb

Therefore, it is

tmp10600_thumbSimple linear shape function

Fig. 5.28. Simple linear shape function

For example triangle 1 can be written:

tmp10605_thumb

Now the shape functions from the triangles 1, 2, …. 6 are considered together. If the linear function

tmp10606_thumb

is applied, the requirements for the triangular regions (5.45) result in a system of equations.

tmp10607_thumb

In terms of matrix representation:

tmp10608_thumb

The solution of this system of equations yieldstmp10609_thumbandtmp10610_thumbThe value of the shape function for all triangles can be given by:

 

tmp10615_thumb

Fig. 5.29 illustrates the properties of the shape function introduced, tmp10616_thumbis a simple linear shape function with the desired properties. It fulfils the conditions

tmp10619_thumb

to generate a sparse overall coefficient matrix.

Properties of the approximation applying linear shape functions.

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:

tmp10621_thumb

quadratic:

tmp10622_thumb

The degree of freedom (DOF) per finite element for a chosen basis function can be determined by (Kost6S):

polynomial degree of p

tmp10-623 tmp10-624
tmp10-625 tmp10-626

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).

 a) Regular and b) not regular triangular element mesh.

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:

• Choose N shape functionstmp10628_thumbsuch thattmp10629_thumbis only in a partial regiontmp10630_thumbfrom the domaintmp10631_thumbnon-zero.

• Calculate thetmp10636_thumband solve the system of equations to obtaintmp10637_thumb

tmp10640_thumb

• The approximated solution from

tmp10641_thumb

• The approximated solution from

tmp10642_thumb