A quadratic GFEM formulation for fracture mechanics problems

Resumo

The Generalized Finite Element Method (GFEM) is a Galerkin approach that generates numerical ap-
proximations belonging to a space obtained by augmenting FEM spaces with enrichment functions capable of

representing well local behaviors of the problem solution. The method has already proved to accurately solve
different classes of problems, including those within the linear elastic fracture mechanics context. For these
problems, GFEM shape functions can represent both the discontinuous and singular behaviors of cracks by a
convenient choice of enrichment functions. Regarding convergence and conditioning aspects, recent works have
proposed well-conditioned and optimally convergent first-order approximations based on GFEM enrichments. In
this work, an initial version of a well-conditioned quadratic GFEM for problems of fracture mechanics is presented.
The methodology consists of using a quadratic Partition of Unity (PoU) to combine local approximation spaces.
A two-dimensional (2-D) numerical experiment with a linear elastic fracture mechanics problem is presented to

demonstrate that the proposed formulation delivers optimal convergence and well-conditioned systems of equa-
tions. Moreover, the robustness of the proposed approach is also demonstrated by showing that the stiffness matrix

conditioning is preserved even for some critical situations regarding the relative position between the mesh and the
crack line.