XFEM based fictitious domain method for linear elasticity model with crack

TL;DR

This paper introduces a novel XFEM-based fictitious domain method for simulating cracks in elastic media, reducing computational costs by avoiding re-meshing and ensuring accuracy and robustness through stabilization techniques.

Contribution

It adapts a fictitious domain approach to crack problems with Neumann conditions, providing a stable, accurate, and efficient method for crack simulation in elasticity.

Findings

The method accurately captures crack-induced displacements.

It maintains robustness across complex geometries.

A realistic volcano fracture simulation demonstrates practical applicability.

Abstract

Reduction of computational cost of solutions is a key issue to crack identification or crack propagation problems. One of the solution is to avoid re-meshing the domain when the crack position changes or when the crack extends. To avoid re-meshing, we propose a new finite element approach for the numerical simulation of discontinuities of displacements generated by cracks inside elastic media. The approach is based on a fictitious domain method originally developed for Dirichlet conditions for the Poisson problem and for the Stokes problem, which is adapted to the Neumann boundary conditions of crack problems. The crack is represented by level-set functions. Numerical tests are made with a mixed formulation to emphasize the accuracy of the method, as well as its robustness with respect to the geometry enforced by a stabilization technique. In particular an inf-sup condition is theoretically proven for the latter. A realistic simulation with a uniformly pressurized fracture inside a volcano is given for illustrating the applicability of the method.