In this paper, we present numerical computational methods for solving the
fracture problem in brittle and ductile materials with no prior knowledge of the topology
of crack path. Moreover, these methods are capable of modeling the crack initiation.
We perform numerical simulations of pieces of brittle material based on global
approach and taken into account the thermal effect in crack propagation. On the other
hand, we propose also a numerical method for solving the fracture problem in a ductile
material based on elements deletion method and also using thermo-mechanical behavior
and damage laws. In order to achieve the last purpose, we simulate the orthogonal