On the construction of approximation space to model discontinuities and cracks with linear and quadratic extended finite elements

Abstract

This paper presents a robust enrichment strategy to model weak and strong discontinuities as well as cracks for industrial applications. First, numerical issues encountered with popular extended finite element approximation spaces are pointed out. Then, the paper gives indications on how to circumvent those issues. The very originality of the paper relies on questioning the theoretical approximation spaces with respect to numerical results and to modify accordingly their design. The relationship between the new design and the previous designs is clearly established, in order to highlight the very small implementation cost of the modifications exposed here. Hence with minimal additional computational cost, gains in accuracy can be significant as shown later in the paper.

3D coupled HM–XFEM modeling with cohesive zone model and applications to non planar hydraulic fracture propagation and multiple hydraulic fractures interference

“…First, a robust procedure has been developed in [57] for the integration of functions discontinuous across arbitrary curved interfaces. It is combined with an improved XFEM enrichment technique proposed by [56] and well-suited for strong discontinuities. Second, a novel mortar formulation derived from [58] is employed for the cohesive zone model.…”

“…The quantity H j (lsn(x)) is 0 if the point x and the node j are in the same subdomain and ±2 otherwise. The coefficient 2 is introduced in order to facilitate the expression of the displacement jump along the interface [56]. This "complementary" formulation has been introduced by Ndeffo [56] and is inspired by the work of Hansbo et al [64].…”

“…The coefficient 2 is introduced in order to facilitate the expression of the displacement jump along the interface [56]. This "complementary" formulation has been introduced by Ndeffo [56] and is inspired by the work of Hansbo et al [64]. Ndeffo proved that the deterioration in the conditioning of the problem is reduced by this "complementary" formulation compared with the formulation of Moës et al [22].…”

“…This XFEM model is based on the previous numerical model developped by Faivre et al [34] but with the following distinguished features : (i) fully hydromechanical coupling based on an improved XFEM enrichment technique, (ii) fracture propagation on non-predefined paths, (iii) 3D configuration including complex nonplanar fracture geometries and (iv) multiple-crack junction. Besides the extension of our model to fluid-filled cohesive crack junctions, most of these new features borrow on recently published advances on the XFEM [56,57,58,59]. The assembly of all these state-of-the art techniques results in a more complete model which is a step forward to address the challenges mentioned above.…”

“…Alternative techniques rely on local criteria for the determination of the bifurcation angle. In order to overcome conditioning issues usually encountered with 3D XFEM elements, a robust XFEM enrichment technique is employed [56]. It is associated to a consistent integration technique adapted to curved cracks and branched discontinuities [57].…”