Treatment of nearly-singular problems with the X-FEM

Abstract

Background: In recent years, lot of research have been conducted on fictitious domain approaches in order to simplify the meshing process for computed aided analysis. The behaviour of such 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. Methods: The performance of the finite element method is evaluated in this context by means of a simple unidimensional example. Both numerical and theoretical estimates are considered in order to assess the behaviour of the numerical approximation. 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 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. Finally, the proposed strategies are extended and validated in 2D. Results: The performance of the two strategies is highlighted for both 1D and 2D examples. Both methods allow to recover proper convergence rates (optimal algebraic rate for h-convergence, exponential for p-convergence) in 1D and 2D. Conclusions: The proposed strategies allow for a very accurate solution for such solutions. The enrichment strategy is valid for both h and p refinement, whereas the mesh-design strategy is only usable for p refinement. However, such enrichment functions can be tedious to derive.

Adaptive anisotropic integration scheme for high‐order fictitious domain methods: Application to thin structures

“…3,[5][6][7][8] Thanks to their ability to use unfitted meshes, fictitious domain methods are also an answer to this objective. Albeit known for decades, 9 fictitious domain methods have been recently revisited in the context of the extended finite element method (X-FEM), [10][11][12][13][14][15] in Cartesian grids 16 for low-order approximations, and in the context of higher-order approximations with the introduction of the finite cell method. [17][18][19][20][21][22] This method can be seen as a hybrid between p-FEM and fictitious domains, retaining advantages of both (ie, exponential convergence for smooth solutions without meshing burden).…”

Prediction of apparent properties with uncertain material parameters using high‐order fictitious domain methods and PGD model reduction

“…To answer the issues raised by these two approaches, alternative methods have emerged recently with the objective to simplify the interaction between geometry and numerical computations. One can cite for example the Finite Cell method from Rank, Düster and co‐workers , and the extended finite element method (X‐FEM in the following) . The Finite Cell method is a high‐order fictitious domain method that has proved to be very efficient, thanks to the approximation properties of p‐fem .…”