Finite Element Modelling with Transformation Techniques

Transformation methods are a very powerful tool in finite element modelling. In many cases, an adequate mapping transforms the problem into an easier one or allows advantage to be taken of the symmetries. This paper demonstrates that any mapping can be handled automatically provided the classical vector analysis approach is given up for the benefit of a differential geometry approach. As a first example, it is shown that axisymmetrical problems need no more a particular treatment provided the mapping of the cylindrical coordinates on the cartesian ones is considered as it is. Furthermore, a novel axisymmetrical formulation is proposed which relies on one further transformation and improves considerably the quality of the interpolated field. Transformation methods are also of great help to model the infinite space by means of finite elements. Many authors have presented such transformations which are often instances of the same general shell transformation that is presented here.