Investigation of different heaviside function forms in discontinuity problems

Abstract

This study investigates the impact of different mathematical forms of the Heaviside function on the numerical solution of discontinuity problems using the extended finite element method (XFEM). The Heaviside function is commonly employed to represent displacement jumps across discontinuities, such as cracks or material interfaces, through enrichment functions. Although several formulations exist, it remains unclear whether these differences influence the accuracy or consistency of numerical results. In this work, multiple cases involving a one-dimensional bar with an internal discontinuity are analyzed using distinct Heaviside formulations. Results show that, despite variations in the enrichment terms, the computed nodal displacements and overall structural responses are identical across all cases. This confirms that the numerical solution is invariant with respect to the specific form of the Heaviside function used. The findings support the robustness of XFEM and offer practical flexibility for implementation in discontinuity modeling without compromising numerical reliability.