Abstract of Doctoral Thesis
Mejora de los Elementos de Transición en X-FEM aplicado a Mecánica de la Fractura Elástica Lineal
Doctoranda: Ana Vercher Martínez
Directores: Dr. José Enrique Tarancón Caro y Dr. Javier Fuenmayor Fernández

The Finite Element Method (FEM) is one of the most employed techniques to solve boundary problems. One of the fundamentals of the numerical approximation in FEM is the polynomial interpolant, and therefore its application to smooth solution problems gives optimal results.
      
The possible presence of cracks in the material is taken into account in the Linear Elastic Fracture Mechanic (LEFM) approach. The behaviour of the analytical solution is not smooth near these imperfections. The local character exhibited by the solution is governed by the singularity, whose intensity depends on the geometry and applied loads.
      
Usually, the FEM has been applied to LEFM problems in order to calculate the Stress Intensity Factors (SIFs), the parameters that characterize the behaviour of the solution near the singularity. The adaptive refinement of the mesh in both crack surroundings and crack tip as well as the use of special elements in the crack tip zone, have been the principal strategies to enhance the solution. In order to obtain the SIFs, the energetic or indirect methods show important advantages because they use the results obtained numerically far away from the crack tip where the solution has low discretization error.
      
The Extended Finite Element Method (XFEM) arises as an alternative to problems with different types of singularity. In XFEM is not necessary to adapt the mesh to the geometry of the singularity. The local enrichment of the solution based on the partition of unity concept are the main characteristics of the method. When XFEM is applied to LEFM problems, two classes of enrichment functions are taken into consideration, in order to represent both the discontinuous solution between crack faces and the asymptotic behaviour of the solution near the crack front.
      
The XFEM has been programmed in this Thesis including several enhancements that, during last years, have been developed with the main objetive of improving the bases of this numerical technique. The aspects treated are the following: enrichment strategies, rate of convergence, numerical integration, numerical conditioning and blending elements.
      
The error in blending elements, due to the lack of partition of unity property, has been deeply analyzed in this Thesis. A detailed study of the different methodologies that have been developed to enhance the blending elements has been performed. A new technique in XFEM to enhance these elements is proposed, based on the addition of new degrees of freedom through hierarchical shape functions. The approach offers important advantages as it has been verified with different numerical examples. The accuracy of the results increases without an important growth of the computational cost because the new degrees of freedom are suitably added, only in the blending elements. The error in different magnitudes is reduced in a significant manner not only within the crack tip elements but in distant elements. At the same time, with this approach, the condition number of the system of equations dose not increase. The proposed methodology has been extended to three-dimensional problems giving also very good results.