By R. Belmans
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 easilysolved linear system of equations.
Therefore, the problem has to be discretised in adequate subproblems. The subproblems are geometrically described by geometrically simple shaped elements such as triangles (Fig. 5.25) or rectangles for twodimensional and mainly tetrahedrons for threedimensional 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 twodimensional 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 subproblems 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 energyminimum 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.
Withis a real value depending on two functionsand V is a set of differentiable functions in the field domainwith on the boundaryis 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 overto obtain a linear form. It can be written:
The Gaussintegral gauge transfers a volume integralinto a surface integral
is a closed and oriented surface, which includes the domain Q, R are functions of three variables defined inThe 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 equationis applied to the standard example of a Poisson equation:
independent linear functions out ofis the set of linear combinationsis called the ^dimensional partial space fromare the basis or global shape functions (Goering et al.44, Zienkiewicz & Taylor l23).
An approximated solution fromis a function
The idea now is to find an approximated solutionusing the linear combination of the N functions
This yields the formulation of an equivalent problem:
This is a projection of the problemin V into a problem inIf this is true for alland for allit can be written:
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 approximationis eq.(5.39). Using
and substituting in the discrete problem,
assuming a being a bilinear form it can be written:
This represents a system of N equations with the N unknownand the coefficient matrix
with
The discrete problem corresponds to a system of equations. Using the shape functionsto calculate the elements of the coefficient matrix and the right hand side ofand solving the system of equations (5.41) leads to the unknownEvaluating eq.(5.37) gives the approximation of the problemThe practical realisation of the FEM depends strongly on the choice of the shape functionsThe 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 matrixcontains as much as possible of zero elements. The best possibility would be to choose the shape functions such thatis the unitymatrix. This is not practically possible.
In generalis an integral over the domainof summations of products of theand their derivatives. If the shape functions are chosen such that they are only in a partial domainnonzero and else equal to zero, the productsare only for some combinations ofnonzero. This means for a FEM discretisation that for manyshould not have common nodes. This results in the desired sparse system of equations. The requirements to be fulfilled by the shape functions are:
• smoothness, piecewise differentiable
• additional properties resulting from the boundary conditions must be satisfied
• the shape functions should be simple
• a good approximationfromshould 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 coordinateswith
Fig. 5.27. Unit square with triangular FEM discretisation.
In this example, triangularshaped elements are assumed, and so every partial square is represented by two triangular elements (Fig. 5.27).
is thespace characterised by a linear function inx and y inside each triangle.
The shape functionsare 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 yieldsandThe 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:
quadratic:
The degree of freedom (DOF) per finite element for a chosen basis function can be determined by (Kost^{6S}):
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:
• nonoverlapping 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:
• Choose N shape functionssuch thatis only in a partial regionfrom the domainnonzero.
• Calculate theand solve the system of equations to obtain
• The approximated solution from
• The approximated solution from