Convergence and error estimation Convergence means that for increasing density of meshes, the FE numerical solution should come closer to the appropriate continuum solution. For convergent finite elements, this convergence is according to Fig.8-1, which shows that the numerical model generally underestimates the displacements of real structure. Fig.8-1 FE convergence Certain conditions must be fulfilled by the elements and their shape functions to obtain convergent behaviour. They are generally fulfilled by the elements used in large commercial FE systems. But even convergent element types can produce a non- convergent solution if they are not properly combined in a FE mesh. The basic rules FE solution continuum Mesh density Displacements

that should be kept in mind are as follows: 1. Only elements with the same number of nodes on their sides should be coupled 2. The nodes must have the same number and type of deformation parameters 3. One element side must come into contact with only one opposite neighbour side 4. In 3D, element face may contact only one neighbour face of the same shape, number of nodes and deformation parameters Error estimation Fig.8.1 shows the difference between numerical solution and a “precise”solution of an appropriate continuum problem. This error is sometimes called as discretisation error. To estimate this error even for complicated problems, where no analytical solution is known, is very important from practical point of view. The estimation is based on the difference of primary FE stress field  i, which is not continuous over the element borders, and continuous stress approximation  a, usually generated by postprocessors. Let us denote the difference between them as . For the simple example of 1D bar in tension (chapter2) the situation is in Fig.8-2.

Fig.9-2 Error estimation in bar element Energy error of each element can be evaluated from where D is the material matrix. Total error for the whole mesh of N r elements is obtained form summation

The error is usually normalised to total elestic energy U and evaluated in % : Using this estimation, looking for the solution of predefined level of precision can be fully automatised. This procedure is called adaptive solution and consists in iterative solution of a problem on a sequence of automatically generated meshes. Their density is modified according to global error value E and local distribution of e (Fig.9-3). Besides the mesh density (h-convergence), also the approximation polynomials can be modified during the iterative solution (p-convergence). An illustration of the adaptive solution is given in the Example 8.1.Example 8.1. Fig.9-3 Sequence of meshes in adaptive solution