Treatment of nearly-singular problems with the X-FEM

Grégory Legrain (1) , Nicolas Moës (1)

1 Structures & Simulations

In this paper, the behaviour of non-conforming methods is studied in the case of the approximation of nearly singular solutions. Such solutions appear when problems involve singularities whose center are located outside (but close) of the domain of interest. These solutions are common in industrial structures that usually involve rounded re-entrant corners. If these structures are treated with non-conforming finite element methods such as the X-FEM (without any enrichment) or the Finite Cell, it is demonstrated that despite being regular, the convergence of the approximation can be bounded to an algebraic rate that depends on the solution. Reasons for such behaviour are presented, and two complementary strategies are proposed and validated in order to recover optimal convergence rates. The first strategy is based on a proper enrichment of the approximation thanks to the X-FEM, while the second is based on a proper mesh design that follows a geometric progression. Performances of these approaches are compared both in 1D and 2D, and enable to recover optimal convergence rates

Mots clés: X-FEM; Non-conforming; p-fem; singularity; convergence

Domaines: Mécanique des solides [physics.class-ph]; Mécanique des solides [physics.class-ph]

Grégory Legrain, Nicolas Moës. Treatment of nearly-singular problems with the X-FEM. Advanced Modeling and Simulation in Engineering Sciences, 2014, pp.1-13. ⟨10.1186/s40323-014-0013-5⟩. ⟨hal-01015607⟩