Finite element method (Electrical Machine) Part 2

By

Weighted residuals

With the exact solution u of a boundary problem it can be written

tmp10643_thumb_thumb

The solution of the problem obtained by an approximation

tmp10644_thumb[2]

is not exact and we must consider a residual R.

tmp10645_thumb[2]

Assuming that the basis functionstmp10646_thumb[2]satisfy the boundary conditions, in the example it wastmp10647_thumb[2]the N unknown can be determined by choosing N points inside the domaintmp10648_thumb[2]For these points, the residuum is forced to be zero. Out of this, N equations to determinetmp10649_thumb[2] are obtained. This method is called the point-collocation method (Binns etal. ,3).

A better approximation is obtained by averaging with an arbitrary functiontmp10650_thumb[2]over the domain of interesttmp10651_thumb[2]

tmp10658_thumb[2]

Transferred to the example of a Poisson equationtmp10659_thumb[2]this method yields:

tmp10662_thumb[2]

The method is called the method of weighted residuals. Various weighting functions are in common use. Using, for example, the Dirac-Delta function fortmp10663_thumb[2]results in the point-collocation method as a special case of the method of the weighted residuals. Usually linear weighting functions are chosen. They are easier to implement when compared to higher order functions and already deliver a sufficiently accurate approximation.

By choosing the shape functions, introduced in the last section, to be the weighting functions,

tmp10665_thumb[2]

we obtain the local Galerkin method. The combination of weighted residuals and Galerkin method is universal applicable.

Continuity

Particular attention must be paid to the choice of basis function where the second derivative is present in the differential equation. This might cause difficulties with the integral at element boundaries. To avoid singular integrands, a continuous function must be chosen with continuous first derivative. This leads to the requirement of a defined continuity for the form functions.

If only the continuity of the linear form function is required, we have atmp10666_thumb[2]

tmp10668_thumb[2]

is continuous on the element’s boundaries.

For a function of second order, the derivative at the element’s boundary is continuous (Fig. 5.31). If in generaltmp10669_thumb[2]is required it can be written:

tmp10672_thumb[2]

 

 

 

continuity for the solution u(x) considered at the boundary from finite element

Fig. 5.31,tmp10674_thumb[2]continuity for the solution u(x) considered at the boundary from finite elementtmp10675_thumb[2]

Green’s formula

We have seen so far that we would need atmp10678_thumb[2] continuity for the basis functions to solve the example of a Poisson equation. It is desirable to have simple linear basis functions withtmp10679_thumb[2] continuity. Applying the first Green’s formula:

tmp10684_thumb[2]

removes the second derivative in the equation and yields for:

tmp10685_thumb[2]

This leads to the weak form of the Poisson equation. Weak form means that weak requirements concerning the continuity are demanded on their solution when compared to the solution of the previous differential equation. Therefore, the form of the previous differential equation is called strong.

tmp10686_thumb[2]

As desired, atmp10687_thumb[2]continuity of the basis function is now sufficient to solve the problem. A little disadvantage of this approach is the fact that now the weighting functiontmp10688_thumb[2]must havetmp10689_thumb[2]continuity as well. Constant weighting functions are not possible with this approach. The use of the first Green’s formula reduces the continuity requirements of the basis functions but increases efforts concerning the weighting functions.

It is obvious from this approach, that the solution of the strong form is always a solution of the weak form. The other way around, this does not always hold.

Energy-minimum functional

The principle of minimum energy requires that the potential distribution corresponds to the minimum of stored field energy. For several electrotechnical problems, this equivalent minimisation problem is known.

tmp10693_thumb[2]

The quantitytmp10694_thumb[2]is the total energy of the functiontmp10695_thumb[2]which

is the sum of the internal energytmp10698_thumb[2]and the load potential

tmp10702_thumb[2]

The minimum energy functional yields the same results obtained by the local Galerkin method (Kost65, Eriksson et al.38).

Types of elements

The domain of interest can be discretised by various types of finite elements. In this section the most commonly used element types for two-and three-dimensional FEM meshes will be introduced. It is the aim to use simple geometries for the elements. Cross-sectional elements such as triangles, quadrangles and rectangles are used for two-dimensional models and volume elements such as tetrahedrons and cuboids for the three-dimensional FEM models.

Table 5.3. Standard FEM element types.

Standard FEM element types.

The outer boundaries of the geometries are mainly approximated by a polygon. In general triangular or tetrahedral shapes can best approximate such geometries. A complicated geometry can be approximated by a large number of such simple-shaped elements.

The FEM model can be built up with element types with different properties. Nodal and edge elements can be distinguished. The most common types of elements will be briefly introduced in the next section. For further details on special types such as non-conform elements, please refer to the literature (Goering et al. Eriksson et al.38, Kost65, Binns et al. 13). Line elements are not considered here.

Nodal elements

The triangular nodal element is the most commonly used element type for two-dimensional problem formulations. This element shape is the most adaptable to complicated geometries. Therefore, it has advantages concerning an adaptive local mesh refinement to enhance the quality of the approximated solution.

2D triangular nodal element with linear shape function.

Fig. 5.32. 2D triangular nodal element with linear shape function.

The unknown values of the approximated function are defined at the nodes of element. By using first order basis functions,tmp10705_thumb[2]equals 1 at one node of the domain and 0 at the other two nodes of the triangular sub-domaintmp10706_thumb[2](Fig. 5.32).

tmp10709_thumb[2]

The equivalent element type for three-dimensional models is the nodal tetrahedron with the same advantage of being able to adapt complicated geometries very accurately.

tmp10710_thumb[2]

Edge elements

By using edge elements, the unknowns are referred to the edges of this element type. This is advantageous in three-dimensional problem definitions. Therefore, this type of element is in common use for three-dimensional FEM problems.

The basis functionstmp10711_thumb[2]are defined by:

tmp10712_thumb[2]

Edge elements

By using edge elements, the unknowns are referred to the edges of this element type. This is advantageous in three-dimensional problem definitions. Therefore, this type of element is in common use for three-dimensional FEM problems.

The basis functionstmp10714_thumb[2]are defined by:

tmp10713_thumb[2]

For a three-dimensional element the course of the basis function of an edge element is plotted in Fig. 5.33.

First order three-dimensional edge element.

Fig. 5.33. First order three-dimensional edge element.

Facet elements

The basis functionstmp10716_thumb[2]are defined by:

 

tmp10718_thumb[2]

 

 

 

First order three-dimensional facet element.

Fig. 5.34. First order three-dimensional facet element.

Fig. 5.34 shows the course of a linear basis function of a three-dimensional facet element. The function value in one node is zero. The course of the basis function value is linear between the nodes of the tetrahedron.

FEM element properties

The basis functionstmp10721_thumb[2]andtmp10722_thumb[2]own special properties (Table 5.4).

Table 5.4. Properties of finite elements.g

type

element properties

tmp10-723

the value of the basis function at a node is 1 at the node n and 0 at all other nodes

tmp10-724

the value of the line integral over an edge is 1 at the edge e and 0 at all other edges

tmp10-725

the value of the surface integral over a facet is 1 in the plane f and 0 at all other planes