Strong Formulation Finite Element Method for Arbitrary Shaped Composite Structures

When it comes to numerical methods a huge amount of methodologies and techniques comes out. Since several researchers throughout all the past decades tried a better way to solve different problems in engineering. It can be pointed out that most of the numerical approaches hitherto proposed can be simply classified as based on two different formulations. Engineers in several branches know that physical problems can be mathematically solved, generally, using a system of partial differential equations. Since analytical solutions for these problems are usually limited to very simple cases numerical techniques must be employed in order to find an approximate solution for these problems. Historically speaking two approaches have been followed. On one hand the system is directly discretized using the so-called collocation methods based on the distribution of points upon the physical domain. On the other hand variational or weak formulations find an approximate solution evaluating the equations in their weakened form. As it is generally known collocation methods are global and use polynomials of high degree for approximating the unknown field, whereas domain decomposition techniques, such as the Finite Element Method (FEM), decompose the domain in several subdomains evaluating several local solutions which are related by the assembly. The aim of this work is to present several methods under the heading of SFEM (Strong Formulation Finite Element Method). Since a strong formulation, analogous to the collocation approach, is coupled with a domain decomposition technique, proper of the FEM. In this way the standard FEM could be termed WFEM. Due to the fact that it is based on the weak formulation and in opposition to SFEM, which is based on the strong formulation. Summarizing the main contributions are the general view in which strong formulation methods could be presented and the introduction of the mapping technique (generally connected to the FEM) for the solution of arbitrary shaped elements without introduction a variational form of the differential system, but solving it directly. Convergence, stability, accuracy and reliability of the SFEM are illustrated varying the principal parameters of this approach and comparing them to WFEM. Thus varying the number of points, the point location, the basis functions and the number of elements it is possible to observe the numerical behaviour of the SFEM.