Comparison between Discrete and Variational Approach

A numerical code for modeling crack propagation using the Cell Method is proposed. The Leon failure surface is used to compute the direction of crack propagation, and the new crack geometry is realized by an intra-element propagation technique. Automatic remeshing is then activated. Applications in Mode I, Mode II and Mixed Mode are presented to illustrate the robustness of the implementation. 1 Comparison between Discrete and Variational Approach The use of a discrete formulation instead of a variational one is advantageous, since several problems of the variational formulation are avoided by means of the discrete formulation. With the variational formulation:Field variables f = f (x, y, z, t) are needed, which are point and instant dependent. If not directly in possession of point functions, the variational approach obtains them from the global quantities by means of the density notion and limit process. A subsequent phase of equation discretisation and integration is needed to reach the solution. The solution is not obtained for the mesh nodes directly, but extrapolated to them. It is not possible to attain convergence greater than the second order. Heterogeneities represent an obstacle. Singularities of the domain contour represent an obstacle. Punctual forces represent an obstacle. The definition of a model for treating the zone ahead of the crack edge is needed. On the contrary, with the discrete formulation:Only global variables g = g (x, y, z, L,A, V, t) are needed, which are point and instant, but also line, area and volume dependent. In a word, global functions are domain and not point functions. Balance equations are expressed using global variables, so they do not contain derivatives and do not therefore require integration. The solution is obtained for the mesh nodes directly. The convergence order is equal to the FEM one, at worst. One can reache convergences of the fourth order.