On the use of a mass lumping technique compatible with a fully‐coupled hydromechanical model for the case of strong discontinuities with the extended finite element method

Abstract

AbstractIn the present article, we introduce a new approach that uses a standard lumping technique, also known as the row‐sum technique, in the extended finite element method (XFEM) in order to stabilize a fully‐coupled hydromechanical model in the presence of a strong discontinuity. Typically, numerical oscillations can be observed where high gradients appear (pressure shocks) due to the violation of the maximum principle. In order to remediate this abnormal behavior, it is of common use to lump the mass matrix associated with the discretized form of the diffusion equation, a parabolic partial differential equation. Several techniques are available to diagonalize the mass matrix in XFEM in the context of dynamic crack growth. However, these techniques were designed for a wave‐like partial differential equation, a hyperbolic partial differential equation. The direct use of such techniques with a fully‐coupled hydromechanical model does not conserve the mass and leads to the violation of the maximum principle. The approach proposed in this work fulfills the features of a parabolic partial differential equation. It is also adapted to the nonlinear case for which the mass density is given by a constitutive law with a distribution that depends on the position inside the domain but which can be different on each side of the discontinuity. In order to test the robustness of our XFEM formulation, only discontinuous enrichments are considered in this article and the discontinuity is supposed impermeable in order to recover two distinct materials on each side of the boundary‐controlled interface for which we can exhibit independent analytical solutions.