Elemental Enriched Spaces for the Treatment of Weak and Strong Discontinuous Fields

Abstract

This paper presents a finite element that incorporates weak, strong and both weak plus strong discontinuities with linear interpolations of the unknown jumps for the modeling of internal interfaces. The new enriched space is built by subdividing each triangular or tetrahedral element in several standard linear sub-elements. The new degrees of freedom coming from the assembly of the sub-elements can be eliminated by static condensation at the element level, resulting in two main advantages: first, an elemental enrichment instead of a nodal one, which presents an important reduction of the computing time when the internal interface is moving all around the domain and second, an efficient implementation involving minor modifications allowing to reuse existing finite element codes. The equations for the internal interface are constructed by imposing the local equilibrium between the stresses in the bulk of the element and the tractions driving the cohesive law, with the proper equilibrium operators to account for the linear kinematics of the discontinuity. To improve the continuity of the unknowns on both sides of the elements on which a static condensation is done, a contour integral has been added. These contour integrals named inter-elemental forces can be interpreted as a “do nothing” boundary condition (Coppola-Owen and Codina, 2011) published in another context, or as the usage of weighting functions that ensure convergence of the approach as proposed by J.C. Simo (Simo and Rifai, 1990). A series of numerical tests for scalar unknowns as a simple representation of more general numerical simulations are presented to illustrate the performance of the enriched elemental space.