- Research Article
- Open Access
Treatment of nearly-singular problems with the X-FEM
© Legrain and Moës; licensee Springer. 2014
Received: 15 April 2014
Accepted: 26 June 2014
Published: 18 July 2014
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.
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.
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.
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.
Industrial structures usually involve re-entrant corners with possibly small fillets. Accurate stress analysis for such structures requires the proper treatment of these geometrical features. The size of these fillets is highly dependent on the manufacturing process, and depending on the quantity of interest and the size of the fillets with respect to global scale of the structure, it may be neglected in the definition of the mathematical model which is used for the computation. The problem is that such areas with high curvature necessitates the use of very small elements, unless blending mapping  or Nurbs-Enhanced FEM , are used. These very small elements have a high impact of the computational cost of the analysis, so that these geometrical features are usually discarded in the analysis and replaced by acute corners. In this case, the numerical solution is not consistent anymore with the mathematical model: the verification of the model is no longer possible, especially if the quantity of interest are related to stresses or strains in the fillet’s area. The use of non-conforming approaches such as the X-FEM  or fictitious domain  can be considered in order to solve this mesh-density issue. Indeed, conforming meshing can be avoided, the price being paid in the integration process. In addition, one has to take care of the correct geometrical description: for example, the use of a level-set for representing the geometry won’t allow to obtain an accurate geometry, unless a mesh with a density of the order of the curvature radius is used. Otherwise, one has to consider “sub-grid” level-sets as advocated in -, a fine pixelized representation, as in , or the so called Nurbs-Enhanced X-FEM . By means of these strategies, the size of the computational mesh does not have to be related to the size of the geometrical features. The solution being regular, optimal convergence rates are expected. However, albeit being regular, mechanical fields can be very rough in the fillet area. As highlighted in the following, this quasi-singularity prevents an optimal convergence of the solution when using non conforming “engineering” meshes i.e. meshes with a moderate number of elements. The objective of this contribution concerns the quantification of strategies for improving the convergence rate of low and high-order non-conforming finite element methods. Two paths can be followed: (i) using the partition of unity  and enrich the finite element approximation with adapted functions ; or (ii) using p-fem strategies that are based on non-conforming meshes with proper grading near the singularities . These two strategies are first motivated in a one-dimensional settings, then validated and compared in a 2D setting.
This work is organized as follows: first, mechanical fields near fillets are presented, and their nearly singular behaviour highlighted. In a second part, the eXtended Finite Element Method is introduced together with some recent improvement in the field of high-order approximations. Then, a 1D model problem is introduced in order to highlight the influence of nearly singular fields on the convergence properties of the finite element method. A close study in the error contribution of the elements of the mesh enables us to propose strategies in order to improve the convergence. These strategies are extended in the 2D setting, and validated by means of various numerical examples. Finally, performances of these strategies are compared before concluding.
Near-fillets mechanical fields
It can be seen that both modes produce singular stresses at the apex of the corner when λ i <1, which causes a loss of convergence for finite elements, as discussed in sections `Discussion’ and `Convergence for nearly singular problems’. In particular, for 2α=π/2, one finds λ1=0.5448 and λ2=0.9085: Mode I is more singular than mode II.
where ϵ1 and ϵ2 are also given in . All the coefficients presented before can be evaluated from the knowledge of the geometry of the rounded corner (i.e. its opening angle 2α and its radius of curvature ρ). Finally, it is possible to obtain the displacement field associated with this asymptotic stress field, by means of the constitutive law and proper integration.
In order to discuss the expected behaviour of the asymptotic solution derived in the previous section, let us consider the terminology coined in . This classification, based on the features of the analytical solution of a problem, enables to predict the behaviour of the finite element method. Solutions can be separated in three classes:
Category A If the solution is analytic everywhere in the domain (including boundaries);
Category B If the solution is analytic everywhere, except at a finite number of singular points (and edges in 3D);
Category C If the solution does not belong to the previous categories (material interfaces for example);
Practical problems usually belong to category B. Note however that the solution is not necessarily singular near singular points: it depends on the eigenvalues of the expansion of the solution. If the eigenvalues are strictly smaller than one, then the solution is singular and the problem is said strongly in category B. Otherwise, it is qualified as weakly in category B.
As stated in section `Sharp corner’, the stress field associated with the sharp corner eqn. (3) is singular and problems involving these geometrical features belong strongly to category B. In this case, the convergence of the finite element method is bounded by the order of the singularity of the solution i.e. min(λ1,λ2). Note however that if 2α≥π, then λ i ≥1, and the problem becomes regular (weakly in category B). The convergence is thus bounded by the polynomial order of the approximation (for h finite elements). On the contrary, the stress field related to the rounded corner eqn. (8) is regular, although it can be very rough if the radius of curvature ρ is small. The problem is then always weakly in category B, but if ρ is small then it tends to be strongly in category B. As the solution is regular, one would expect h convergence rates associated with the order p of the polynomial approximation (i.e. in in the energy norm). This is the case asymptotically, but not necessary for “engineering meshes” (meshes with a moderate number of elements), as illustrated in section `1D model problem’.
The eXtended finite element method
in this expression, the first term corresponds to the classical finite element approximation while the second one corresponds to the enrichment. It involves , the scalar shape function associated with the partition of unity, the scalar enriched dof and φ β (x) the βestvectorial enrichment function. Note that the number of vectorial shape functions n remains unchanged with respect to (14), and that the number of scalar shape functions is smaller than n. More precisely, in the case where and N share the same polynomial order, we have with d the spatial dimension of the problem. It reflects the different nature of these shape functions (vectorial and scalar). This difference has also an influence on the number of enriched dofs: it is reduced by a factor d if (15) is used ( rather than n×n e ).
In , the resulting conditioning number evolution was shown to increase in O(1/h2) for a model problem, which is the same rate as classical linear finite elements. This improvement in the conditioning number is of great interest in practice, as is allows to use the so-called geometrical enrichment which has been proved to be optimal in term of convergence. This aspect becomes fundamental when high-order shape functions are used, as the conditioning number increases with the polynomial order.
1D model problem
In this expression, k, γ and θ are positive constants that depend on the exact solution.
Convergence for nearly singular problems
Evolution of r 2 for the elements of a regular mesh ( ε =10 -5 , 20 elements)
Strategies for recovering optimal convergence
Two strategies are proposed in order to recover a proper convergence in the case of nearly-singular problems. The first one is based on an enrichment of the approximation, using the Partition of Unity method , see eqn. (14). The second one is based on a proper mesh design, which is close to the approaches that are classically used in the context of p-Fem.
Enrichment of the approximation
The idea consists in the enrichment of the approximation in order to capture the steep gradients of the exact solution. The enrichment function considered is x α as only this term is singular in (18), and a “geometrical” enrichment strategy is considered, as it has been shown in practice that it was leading to better convergence properties ,. Such an approach can be used for both h and p Fem.
Suitable mesh design
Equation (25) states that in the case of a quasi-uniform mesh, the length of the first elements must have the same order of magnitude as ε. This condition is very restrictive in practice, as a quasi-uniform mesh of this type is unusable for real problems. Note that the numerical results from Figure 8 are consistent with this estimate, as exponential convergence is noticeable for ε<10-3 which is close to h/33≃1.6 10-3.
From this study one can see that in practice (i.e. q≥0.15), the exponential convergence is ensured no matter the value of the geometrical progression.
Results and discussion
1D Numerical examples
The two strategies presented above are now appraised considering α=0.55 and ε=10-5 (i.e. for the most unfavourable case). The energy norm of the error is monitored with respect to the number of degrees of freedom.
As a conclusion for this section, one can see that it is possible to recover regular rates of convergence in the case of nearly-singular solutions if one of the proposed strategies is used. The enrichment of the approximation seems to be the more versatile approach, as it can be applied for both h and p refinement. In addition, it has been demonstrated on the proposed example that it was performing more efficiently than the mesh-based approach. However, note that the geometrical mesh approach is less prone to conditioning and integration issues, and that is remains applicable even when the asymptotic behaviour is not precisely known.
Extension to 2D
We now discuss the extension to 2D of the proposed strategies.
Enrichment of the approximation
2D numerical examples
Effect of the enrichment
Effect of mesh refinement
Comparison of the two strategies
In this contribution, the behaviour of non-conforming h and p finite elements has been studied in the case of nearly-singular solutions (re-entrant corners with a fillet here). In particular, it has been shown in both 1D and 2D that despite being regular, the convergence rate was algebraic and limited by the order of the singularity. Therefore, it is not possible to use fictitious domain methods such as the X-FEM without enrichment or the Finite-Cell Method if high accuracy is needed in near the fillet. Thanks to the study of a 1D model problem, it has been possible to highlight the reasons for such a behaviour. Two strategies have been proposed in order to overcome this convergence bound. The first one is based on the enrichment of the approximation near the fillet, and is usable for both h and p methods. The second one is based on the use of a mesh with a geometrical progression towards the center of the singularity, and is restricted to p methods. The performances of these two strategies have been compared in both 1D and 2D: the enrichment method is the more efficient, but can lead to conditioning issues with high-order bases unless proper preconditioning strategies are used -. Moreover, the enrichment function is problem dependent, and can be tedious to obtain. In the present application, the enrichment functions are limited to stress-free bidimensional corners, and may be not be valid in the case where Dirichlet or non-homogeneous Neumann boundary conditions are applied. This kind of limitations was also present in the work of Wagner et al.  who considered rigid particles in Stokes flow, and used an enrichment function which is only valid for rigid and not too-close particles. The actual improvement with such not-fully adapted enrichment functions should be investigated further. On the contrary, exponential convergence is ensured in the case of the use of a geometrical mesh, no matter the progression, which makes it more versatile in case of high-order methods (only). The penalty of constructing such a mesh is greatly alleviated as this mesh does not need to conform the geometry, which is a contribution of the paper.
a In (14), the vectorial nature of the field is handled by the shape functions, and not the dofs that are just coefficients. This notation facilitates the writing of the discrete operators.
b For ε=10-2 (Figure 9(b)), the maximum value of r2 is 0.51, which means that p convergence is obtained for the first element for any p>2.
Appendix A: Asymptotic displacement fields
GL worked on the algorithms, performed all the computations and drafted the manuscript. NM worked on the algorithms and carried out detailed revision. All authors read and approved the final manuscript.
The support of the ERC Advanced Grant XLS no 291102 is gratefully acknowledged
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