# A STABLE AND IMPROVED VERSION OF THE GFEM FOR THE ANALYSIS OF PROBLEMS IN ELASTIC LINEAR FRACTURE

## Palavras-chave:

GFEM, Flat-top PU, Trigometric PU, Scaled Condition Number, Strong discontinuities## Resumo

Currently the Generalized Finite Element Method (GFEM) has been widely applied in the

modeling of localized solids failures. Its main advantage consists of the expansion of the Finite Element

Method (FEM) approach space by inserting functions (known as enrichment functions) that best locally

represent the behavior of the searched solution. Such functions may have specific characteristics or even

be generated numerically. On the one hand, the GFEM provides optimal convergence, however, it is

prone to introduce linear dependencies into its system of equations, making the matrix ill-conditioned or

even singular. The so-called stable version of the Generalized Finite Element Method (SGFEM) explores

a modification in the enrichment functions to improve the conditioning of the stiffness matrix. However,

such a modification leads to loss of precision in problems such as strong discontinuities. In order to rec-

oncile the incompatibility between the solution precision and the system matrix conditioning, this work

addresses a new modification of the space of GFEM shape functions associated with enrichment: flat-top

functions as Partition of Unit (PU) and a new PU based on trigonometric functions, these are used exclu-

sively in the construction of enriched shape functions. This new version of the GFEM presents a system

matrix conditioning almost insensitive to the mesh / discontinuity configuration, even if the crack path

approaches the element nodes. In addition, for flat-top PU with a small stabilization parameter, this ver-

sion is almost of the same precision as the GFEM. Since only the PU is modified, the presented proposal

can be easily implemented in pre-existing GFEM codes. Several representative numerical simulations of

benchmark tests are presented to validate the proposal, considering both the accuracy of the solution and

the conditioning of the system matrix.