On the construction of approximation space to model discontinuities and cracks with linear and quadratic extended finite elements
 M. Ndeffo^{1},
 P. Massin^{1},
 N. Moës^{2},
 A. Martin^{1}Email author and
 S. Gopalakrishnan^{3}
https://doi.org/10.1186/s4032301700903
© The Author(s) 2017
Received: 26 October 2016
Accepted: 18 September 2017
Published: 11 December 2017
Abstract
This paper presents a robust enrichment strategy to model weak and strong discontinuities as well as cracks for industrial applications. First, numerical issues encountered with popular extended finite element approximation spaces are pointed out. Then, the paper gives indications on how to circumvent those issues. The very originality of the paper relies on questioning the theoretical approximation spaces with respect to numerical results and to modify accordingly their design. The relationship between the new design and the previous designs is clearly established, in order to highlight the very small implementation cost of the modifications exposed here. Hence with minimal additional computational cost, gains in accuracy can be significant as shown later in the paper.
Keywords
Introduction
Strain localization is usually an issue for conventional finite element approaches due to numerical issues in the softening regime of the stress–strain relation, when the problem becomes illposed. Apart from taking care directly of the localization by an adaptation of the mesh to the discontinuity, different methods have been used in the literature to circumvent this difficulty. Smearedcracked models were first proposed [1, 2] with perturbations of the fields across the interface. Discontinuous embedded elements appeared almost at the same time with initial papers of Ortiz et al. [3] or Dvorkin et al. [4], with arbitrary orientations through an element but independent from an element to its neighbor. A little bit later, special interface elements [5, 6] were proposed localized in between conventional elements, which require frequent remeshing and refined meshes in order to allow for crack propagation in the correct direction. The eXtented Finite Element Method XFEM [7] and the Generalized Finite Element Method GFEM [8] finally allow meshes not to respect the crack geometry while providing a continuous transfer of information from one element to the next one about the crack surface localization unlike the generalized class of embedded discontinuous finite element approaches [3, 4]. These methods with nodally based enrichments have managed to combine performance and robustness, considering nonmeshed cracks in a finite element framework. XFEM and GFEM use the Partition of Unity [9] and enrich the classical basis of shape functions with discontinuous functions [10]. The discontinuity of the displacement field across the crack surface is then introduced by a generalized Heaviside function, and adding asymptotic fields at the front crack gives good precision in linear elastic fracture mechanics [10, 11]. The main advantage of these methods in comparison to meshless methods is their easy implementation in a general finite element software, and their capabilities to be applied to various fields: large transformations [12] or plasticity [13] in the XFEM context for example... One can say that XFEM and GFEM extend the possibilities of the FEM, while keeping all its advantages. A useful amelioration has been proposed by Sukumar et al. [14] with the introduction of level set functions to represent discontinuities (cracks, voids,...). These approaches are extremely handy in 3D to treat crack propagation [15, 16].
Moreover, these methods have been implemented to solve linear elasticity fracture mechanics problems [8, 17] with better accuracy than with finite element methods. However, the extension of those methods to quadratic elements in three dimensions is a big challenge. Meanwhile, industrial numerical tools use extensively quadratic elements and simulations in three dimensions are almost the norm. In the wake of [18–20], this article attempts to close in the gap between the theoretical methods and issues related to implementation in industrial software.
Minor concerns in one dimension with linear elements turn out to be daunting challenges in 3D or with quadratic elements. Those concerns have been described in [17, 21–23] but have raised little interest since recent publications [24, 25]. Firstly, they are related to the ill conditioning of strong or weak discontinuous approximations in the general case of nonconforming interfaces and secondly, to the numerical issues related to “geometrical enrichment” techniques, near the cracktip.
The analysis will be developed more particularly in the case of strongly discontinuous approximations because a direct link with conditioning can be clearly established. For weak discontinuities involved in bimaterials for instance, conditioning issues are still present but they are coupled with the quality of the approximation space to represent continuous solutions with discontinuous gradients [14, 26–28]. A specific section will be dedicated to this analysis.
In “Strong discontinuity approximation conditioning” and “Singular enrichment space optimality and conditioning” sections, conditioning issues related to strong discontinuity and singular functions are investigated, as well as, strategies available in the literature to solve those issues. In “Approximation spaces in the literature” and “Numerical behavior of strongly discontinuous and singular approximations” sections, those strategies are benchmarked with linear and quadratic elements. Further analysis unfolds that quadratic elements emphasize conditioning and accuracy issues almost unseen with linear elements. The results exposed here are general enough, given the wide range of approximation spaces considered in the paper.
Strong discontinuity approximation conditioning

either:$$\begin{aligned} measure\left( \left\{ {\Omega _{+} \cap Supp(\varPhi _i )} \right\} \right) \ll measure\left( \left\{ {\Omega _{} \cap Supp(\varPhi _i )} \right\} \right) \end{aligned}$$(1)

or:$$\begin{aligned} measure\left( \left\{ {\Omega _{} \cap Supp(\varPhi _i )} \right\} \right) \ll measure\left( \left\{ {\Omega _{+} \cap Supp(\varPhi _i )} \right\} \right) \end{aligned}$$(2)
Practically, this means that node numbered i is either on the side \(\Omega _{} \) or on the side \(\Omega _{+} \) and does not “see” the information from the opposite side. From now on, the information from the opposite side will be called complementary information.
The criterion used here replaces the measurement of volumes [18, 24] by the measurement of distances between the nodes of the mesh and the interface on cut edges. When these criteria are associated to a relocation of the interface they are usually named fittovertex. Along each edge (see example on Fig. 2), the balance of “Heaviside information” is weighted through the comparison of lengths [\(\hbox {N}_{1}\)IP] and [\(\hbox {N}_{2}\)IP] with respect to [\(\hbox {N}_{1}\) \(\hbox {N}_{2}\)]. Levelsets (abbreviated “lsn”) are a handy tool to evaluate those lengths, since they do not require computing the coordinates of the intersection points.
Then, the levelset values of both nodes are compared in order to enrich or not node \(\hbox {N}_{1}\). A similar approach consists in changing directly the value of the levelset of the \(\hbox {N}_{2}\) node to zero. In any case, a threshold is needed to decide whether or not the enrichment needs to be modified. This threshold based on the ratio of characteristic dimensions is generally chosen between \(10^{2}\) and \(10^{3 }\) [18, 24].

Node \(\hbox {N}_{1}\) is eliminated or the level set is moved to \(\hbox {N}_{2}\) by resetting lsn \((\hbox {N}_{2})\) to zero if,$$\begin{aligned} \frac{\left {lsn(\hbox {N}_2 )} \right }{\left {lsn(\hbox {N}_1 )} \right +\left {lsn(\hbox {N}_2 )} \right }<10^{2} \end{aligned}$$(5)

Node \(\hbox {N}_{2}\) is eliminated or the level set is moved to \(\hbox {N}_{1}\) by resetting lsn \((\hbox {N}_{1})\) to zero if,$$\begin{aligned} \frac{\left {lsn(\hbox {N}_1 )} \right }{\left {lsn(\hbox {N}_1 )} \right +\left {lsn(\hbox {N}_2 )} \right }<10^{2} \end{aligned}$$(6)
On second hand, to control the condition number, one can consider an algebraic preconditioner [21], dedicated to orthogonalize shape functions and enrichment functions, with a Cholesky factorization.
Singular enrichment space optimality and conditioning
The first eigenvalue \(\lambda _{0}=0.5\) represents the less regular term of this expansion series. The associated functions \((\propto \sqrt{r})\) are responsible for a significant loss of accuracy within regular FEM [17].
Laborde et al. [17] showed that a fixed enrichment zone is needed, around the cracktip, to ensure the optimal accuracy of singular approximations (see Fig. 5). This technique uncouples the size of the enrichment zone and the size of the mesh, so that the enrichment zone does not shrink to zero with mesh refinement.

The first concern is the definition of the enrichment area and the related enrichment strategy in blending elements. In the literature, two main strategies emerge: the “cutoff” strategy of Nicaise et al. [29] and the geometrical enrichment strategy [17]. Firstly, the “cutoff” strategy adds global singular d.o.f. and an additional function to soften the transition between the enriched zone and the nonenriched zone. Secondly, the geometrical enrichment introduces local singular d.o.f. which values are set to zero outside the enrichment zone. Nicaise and et al. [29] shows that both strategies are relevant and do not alter the convergence rates.

The second concern is the swift increase of the condition number. Latest works of Chevaugeon et al. [30] and Guptaa et al. [25] give strong research directions to improve the condition number. A vectorial enrichment reduces drastically the number of d.o.f. needed to describe the singular solution. Guptaa et al. [25] suggests that vectorial enrichment could be further improved to permanently remove conditioning issues. The idea of [25] is to subtract from the enrichment function its linear interpolation on the set of elements where the singular enrichment is defined.
Approximation spaces in the literature
In the literature, there are numerous approximation spaces to model strong discontinuities and cracks within the general framework of extended finite element methods and alike. The earliest work of [7] introduced the standard XFEM approximation space of Table 1 in 1999. Then in 2000, the work of [8] extrapolated GFEM [22] to crack modeling. Although XFEM and GFEM both aimed to model cracks, GFEM was designed to deal with a wider scope of problems than XFEM. Initially, XFEM and GFEM were assumed as different approaches. XFEM introduced scalar functions to model Irwin modes at the cracktip. On the opposite, GFEM used directly Irwin modes as vector functions to model the singular behavior at the cracktip. In “Approximation spaces for a cracked domain” section, this difference is investigated. We will find out that XFEM and GFEM are closely related.
In 2004, Laborde et al. [17] paper questioned the numerical accuracy of the XFEM approximation, particularly with higher order elements. Laborde et al. [17] suggested that XFEM with one layer of enriched elements at the cracktip, is not very accurate. When the enrichment zone enlarges, accuracy improves, but conditioning deteriorates. Therefore, it suggested a new approximation space to take care of conditioning issues and with a better behavior for quadratic elements. As this new approximation space is restricted to 2D analysis, this approximation is not fitted for industrial numerical simulations. Finally, to address higher order modeling, XFEM design evolved into vectorial enrichment [30], close to GFEM.
Very recently, in 2013, even the GFEM approximation space has evolved into a new space called SGFEM [25, 31]. SGFEM addresses conditioning issues of GFEM with a new definition of the vectorial functions. Following [32] it is even defined by the fact that the condition number of the associated stiffness matrix has to be of the same order with respect to mesh refinement than the one of FEM. However, SGFEM as proposed in [25, 31] has some drawbacks that will be discussed in “Approximation spaces for a cracked domain” section.
Specifically, to model strong discontinuities, many approximations have been released in the literature. The earliest work of [7] introduced a Heaviside function to model the jump d.o.f. on nodes in the vicinity of a strongly discontinuous interface (Fig. 3). The next relevant approximation space was the one of Hansbo et al. [33] that introduces discontinuous polynomials within the partition of unity. Numerous spaces were derived from those previous approximation spaces and far too many to be studied thoroughly in this paper [19, 20, 34, 35]. Hence, we will focus, in “Strong discontinuity representation” section, on the ones of Moës et al. [7], Hansbo et al. [33] and Belytchko et al. [36], which is somehow intermediary between those of [7] and [33].
From the mathematical point of view, we will find out that the three spaces are very similar.
Strong discontinuity representation
Strong discontinuity enrichments comparison
XFEM [7]  Domain enrich. [33]  Shifted enrich. [26]  

Linear shape functions on enriched nodes 





 
Displacement approximation  \({\begin{array}{l} \left. {u^{h}} \right _{\Omega +} =\sum \limits _{j\in I} {a_j \varPhi _j +\sum \limits _{j\in I_H } {b_j \varPhi _j } } \\ \left. {u^{h}} \right _{\Omega } =\sum \limits _{j\in I} {a_j \varPhi _j \sum \limits _{j\in I_H } {b_j \varPhi _j } } \\ \end{array}}\)  \({\begin{array}{l} \left. {u^{h}} \right _{\Omega +} =\sum \limits _{i\in I/I_H } {a_i } \varPhi _i +\sum \limits _{j\in I_H } {\alpha _{j,1} \varPhi _{j,1} } \\ \left. {u^{h}} \right _{\Omega } =\sum \limits _{i\in I/I_H } {a_i } \varPhi _i +\sum \limits _{j\in I_H } {\alpha _{j,2} \varPhi _{j,2} } \\ \end{array}}\)  \({\begin{array}{l} \left. {u^{h}} \right _{\Omega +} =\sum \limits _{i\in I} {c_i } \varPhi _i \sum \limits _{j\in I_H \cap \Omega } {2d_j \varPhi _{j,2} } \\ \left. {u^{h}} \right _{\Omega } =\sum \limits _{i\in I} {c_i } \varPhi _i +\sum \limits _{j\in I_H \cap \Omega +} {2d_j \varPhi _{j,1} } \\ \end{array}}\) 
Jump approximation  \(\Delta u^{h}=\sum \limits _{j\in I_H } {2b_j \varPhi _j } \)  \(\Delta u^{h}=\sum \limits _{j\in I_H } {\alpha _{j,1} \varPhi _{j,1} \alpha _{j,2} \varPhi _{j,2} } \)  \(\Delta u^{h}=\sum \limits _{j\in I_H \cap \Omega } {2d_j \varPhi _{j,2} } \sum \limits _{j\in I_H \cap \Omega +} {2d_j \varPhi _{j,1}} \) 

\(\Omega _\mathrm{(H=+1)} =\Omega _{+}\)

\(\Omega _\mathrm{(H=1)} =\Omega _{}\) .

\(\left. {u^{h}} \right _{\Omega +} \) represents the approximation of the solution on the dimain \(\Omega _{+} \) extended by continuity to the domain \(\Omega _{} \) over the interface \(\Gamma \),

\(\left. {u^{h}} \right _{\Omega } \) represents the approximation of the solution on the dimain \(\Omega _{} \) extended by continuity to the domain \(\Omega _{+} \) over the interface \(\Gamma \).
Let us compare sidebyside, the XFEM representation of the displacement jump introduced by [7] with the other wellknown formulations of [33] and [26]. Even though the jump approximations have different expressions (Table 1), we will investigate next whether the approximation spaces are really different or not.
Change of variables when switching enrichment strategies. \(\forall j\in I_H\), the following changes of variables hold

Thus, from the mathematical point of view, the approximation spaces involved in the three formulations are equivalent (see Table 2). The same can be said of other formulations encountered in the literature [19, 34, 35]. We would like to stress the fact that this equivalency only holds in the case of strong discontinuity. When singular functions are injected in the approximation, the relationship between the different approximation spaces may be different as studied in the next section.
Approximation spaces for a cracked domain
The model problem is a cracked domain \(\Omega \) (Fig. 6), under a linear elasticity assumption. The material is also assumed to be homogeneous and isotropic. Dirichlet boundary conditions are applied on the boundary \(\Gamma _{\mathrm{D}}\) and Neumann boundary conditions are applied on \(\Gamma _{\mathrm{N}}\).

“Straightforward” singular enrichment class” section shows that XFEM and GFEM are “straightforward” enrichments. Moreover, GFEM is a subspace of XFEM. This statement implies that XFEM is somehow more accurate than GFEM, but both methods are nonetheless very close.

“Bubble” enrichment class” section shows that a “bubble” enrichment as the one of Gupta et al. [25] is not a “straightforward” enrichment. This statement implies that “bubble” enrichments have numerical properties not identical to “straightforward” enrichments that will be investigated in “Singular approximation at a crack tip” section.
“Straightforward” singular enrichment class
Lemma 1
Proof

\(V_{CUTOFF}^h (W_h^{2,\infty } )\subset V^h :\)

\(V^h \subset V_{XFEM}^h {:}\)

Equation (27) shows that \(\underline{K}_\alpha \) can be expressed as \(\underline{K}_\alpha =\sum \limits _{l=1}^4 {\sum \limits _{m=1}^d {k_{l,m,\alpha } F^{l}\underline{e}^{m}} } \), where \(k_{i,m,\alpha } \) is a constant tensor.
In case of a straight crack, the local basis is constant.
Thus, the local basis expresses in cartesian coordinates as, \(\underline{e}^{m}=\sum \limits _{j=1}^d {\mu _{m,j} \underline{E}_j } \), where \(\mu _{m,j} \) is a constant tensor.
Then,With a suitable change of variable,$$\begin{aligned} \underline{K}_\alpha= & {} \sum _{l=1}^4 {\sum _{m=1}^d {\sum _{j=1}^d {k_{l,m,\alpha } \mu _{m,j} F^{l}\underline{E}_j } } }\\ \sum _{\alpha =1}^d {c_{k,\alpha } \varPhi _k \underline{K}_\alpha }= & {} \sum _{\alpha =1}^d {\sum _{l=1}^4 {\sum _{m=1}^d {\sum _{j=1}^d {c_{k,\alpha } k_{l,m,\alpha } \mu _{m,j} F^{l}} } } \varPhi _k \underline{E}_j } \\= & {} \sum _{l=1}^4 {\sum _{j=1}^d {\left( {\sum _{\alpha =1}^d {\sum _{m=1}^d {c_{k,\alpha } k_{l,m,\alpha } \mu _{m,j} } } } \right) } F^{l}\varPhi _k \underline{E}_j } \\ \end{aligned}$$we have finally,$$\begin{aligned} c_{k,l}^j =\sum _{\alpha =1}^d {\sum _{m=1}^d {c_{k,\alpha } k_{l,m,\alpha } \mu _{m,j} } } \end{aligned}$$which means that:$$\begin{aligned} v^{h}=\sum _{i\in I} {\sum _{j=1}^d {a_{i,j} \varPhi _i \underline{E}_j } } +\sum _{i\in I_H } {\sum _{j=1}^d {b_{i,j} H\varPhi _i \underline{E}_j } } +\sum _{k\in I_{CT} } {\sum _{l=1}^4 {\sum _{j=1}^d {c_{k,l}^j } F^{l}\varPhi _k \underline{E}_j } } \end{aligned}$$$$\begin{aligned} v^{h}\in V_{XFEM}^h \end{aligned}$$ 
In case of a curved crack, the result of the lemma holds with a discretization of the local basis of [30]. Let us consider a discrete local basis at each node \(\left( {\underline{e}_k^1 ,\underline{e}_k^2 ,\underline{e}_k^3 } \right) \), such as,The straight crack proof still holds here, with the additional node index “k”.$$\begin{aligned} v^{h}= & {} \sum _{i\in I} {\sum _{j=1}^d {a_{i,j} \varPhi _i \underline{E}_j } } +\sum _{i\in I_H } {\sum _{j=1}^d {b_{i,j} H\varPhi _i \underline{E}_j } } +\sum _{k\in I_{CT} } {\sum _{\alpha =1}^d {c_{k,\alpha } \varPhi _k \underline{K}_{\alpha ,k}}}\\ \underline{K}_{\alpha ,k}= & {} \sum _{l=1}^4 {\sum _{m=1}^d {k_{l,m,\alpha } F^{l}\underline{e}_k^m } }\\ \underline{e}_k^m= & {} \sum _{j=1}^d {\mu _{m,j,k} \underline{E}_j } \end{aligned}$$
Remark
as a practical consequence of Lemma1, the error bound of XFEM should be lower than the ones of \(V^h \) and of the cutoff [29]. With XFEM, the optimization process performs on a larger space than with other formulations and reaches a closer infimum to the exact solution of the problem. Nevertheless, XFEM introduces more d.o.f. than GFEM and cutoff enrichments which increases its condition number.
“Bubble” enrichment class
\(\delta I_{CT} \) is defined here as the transition layer of nodes between cracktip elements and other elements of the mesh for which \(\underline{K}_\alpha \) unknowns are set to zero. This allows to connect the enriched layer with the remaining of the mesh and to avoid blending issues [26].
Lemma 2
Proof

\(V_{SGFEM}^h \not \subset V_{XFEM}^h \)

\(V_{XFEM}^h \not \subset V_{SGFEM}^h \)

\(F^{1}\underline{E}_1 \sum \limits _{k\in I_{CT} } {\varPhi _k } \) belongs to the test function space \(V_{XFEM}^h \) with \(a_{i,j} =0, b_{i,j} =0, c_{k,1}^1 =1\) and \(c_{k,\alpha }^j =0\) elsewhere (for \(\alpha \ne 1)\).

However, \(F^{1}\underline{E}_1 \sum \limits _{k\in I_{CT} } {\varPhi _k } \) does not belong to \(V_{SGFEM}^h \), which can be shown by contradiction.

Let us assume \(F^{1}\underline{E}_1 \sum \limits _{k\in I_{CT} } {\varPhi _k } \) belongs to \(V_{SGFEM}^h \),$$\begin{aligned} F^{1}\underline{E}_1 \sum _{k\in I_{CT} } {\varPhi _k }= & {} \sum _{i\in I} {\sum _{j=1}^d {a_{i,j} \varPhi _i \underline{E}_j } } +\sum _{i\in I_H } {\sum _{j=1}^d {\sum _{k=1}^4 {b_{i,j,k} H\psi _{i,k} \varPhi _i \underline{E}_j } } } \\&+\sum _{i\in I_{CT} } {\sum _{\alpha =1}^d {c_{i,\alpha } \varPhi _i \left( {\underline{K}_\alpha \Pi \underline{K}_\alpha } \right) } } \end{aligned}$$

Singular functions cannot be represented by discontinuous polynomials basis, so we necessarily have \(b_{i,j,k} =0\)

Singular functions cannot be represented by continuous polynomials as well, so, \(\sum \limits _{i\in I} {\sum \limits _{j=1}^d {a_{i,j} \varPhi _i \underline{E}_j } } \sum \limits _{i\in I_{CT} } {\sum \limits _{\alpha =1}^d {c_{i,\alpha } \varPhi _i \Pi \underline{K}_\alpha } } =\underline{0}\) as \(\varPhi _i \Pi \underline{K}_\alpha \) is also a polynomial.

\(\varPhi _i \Pi \underline{K}_\alpha \) introduces at least three higher order polynomials per node,

Which leads to, \(\sum \limits _{\alpha =1}^d {c_{i,\alpha } \varPhi _i \Pi \underline{K}_\alpha } =\underline{0}\),

Then, \(c_{i,\alpha } =0\),

And, \(a_{i,j} =0\) because \(\sum \limits _{j=1}^d {a_{i,j} \varPhi _i \underline{E}_j } =\underline{0}\).


Thus, \(F^{1}\underline{E}_1 \sum \limits _{k\in I_{CT} } {\varPhi _k } =\underline{0}\) which is contradictory.


Remark
As a practical consequence of Lemma2 the “bubble” class of “Bubble” enrichment class” section does not belong to the “straightforward” class of “Straightforward” singular enrichment class” section, in which XFEM is found to be the largest space. Hence, the definition of two classes of enrichment makes perfectly sense.
Numerical behavior of strongly discontinuous and singular approximations
In this section the numerical behavior of the different approximation spaces introduced previously is investigated through a couple of benchmarks. First of all, we assess the asymptotic behavior of strongly discontinuous approximations, through a one dimensional case. Secondly, we consider the convergence study of a twodimensional problem with a crack and singular approximation.
Numerical behavior of strongly discontinuous approximations
Numerical analysis of strongly discontinuous approximations with linear elements

The problem is the solution of the following differential equation:$$\begin{aligned} {u''(x)=0}\quad \forall x\in \Omega =\left[ {0,3} \right] \end{aligned}$$(30)

With arbitrary Dirichlet boundary conditions:$$\begin{aligned} {u(0)=\beta }\quad {u(3)=3\alpha +\beta } \end{aligned}$$(31)

And with continuity conditions at the interface expressed in terms of the continuity of the derivative of the displacement field (stress continuity at the interface):$$\begin{aligned} \left. {\frac{du}{dx}} \right _{x\left( {\varepsilon ^{+}} \right) } =\alpha ^{}\quad \quad \left. {\frac{du}{dx}} \right _{x\left( {\varepsilon ^{}} \right) } =\alpha ^{+}\quad \quad \alpha ^{}=\alpha ^{+}=\alpha \end{aligned}$$(32)
Let us consider the following arbitrary numerical values: \({\alpha ={10}/9}\) and \({\beta } =10.\)
Then, the solution is computed in standard 64bit arithmetic, with the linear solver UMFPACK, to reproduce the behavior of direct solvers.
Moreover, all three enrichment conditionings increase as the distance between the interface and node \(\hbox {N}_{3}\) decreases (Fig. 10). This means that all three enrichment strategies are very sensitive to the position of the interface. A preconditioner is needed for the three enrichments. In the case of XFEM, we studied Béchet et al. [21] preconditioner. For other formulations we used a diagonal preconditioner to scale the d.o.f.
Remark

For iterative methods (PCG), the error is generally higher because those solvers are very sensitive to conditioning,

For matrix factorization based solvers (UMF), the error is lower than with iterative methods,

For matrix inversion based solvers, good results can be obtained frequently.
Numerical analysis of strongly discontinuous approximations with quadratic elements
For the three approximations, conditioning worsens very quickly, at about three times the rate of linear elements. More worrying is that the error increases. XFEM enrichment error worsens more quickly than the others.
This numerical behavior with quadratic elements is well explained by the coupling between enrichment functions, as pictured on Fig. 16 with domain enrichment [33].

In case of Lagrange polynomials, we have:$$\begin{aligned} \varPhi _S (\xi )= & {} (1\xi )(12\xi ),\quad \forall \xi \in \left[ {0,1} \right] \nonumber \\ \varPhi _M (\xi )= & {} 4\xi (1\xi ),\quad \forall \xi \in \left[ {0,1} \right] \end{aligned}$$(43)
Proof

In case of Bernstein polynomials, we have:$$\begin{aligned} \varPhi _S (\xi )= & {} (1\xi )^{2},\quad \forall \xi \in \left[ {0,1} \right] \nonumber \\ \varPhi _M (\xi )= & {} 2\xi (1\xi ),\quad \forall \xi \in \left[ {0,1} \right] \end{aligned}$$(49)
It is noticeable that [33] has better accuracy with Bernstein polynomials than with Lagrange polynomials (Figs. 18, 19). \(\square \)
Numerical behavior of singular approximations at the crack tip
Numerical analysis of crack approximations with linear elements: crack opening in mode 1

Gupta et al. SGFEM kind of enrichment [25, 31] (let us recall that accordingly to [32] a genuine SGFEM enrichment is defined by the fact that the condition number of the associated stiffness matrix is of the same order—2 [38]—with respect to mesh refinement than the one of FEM) needs far too many additional Heaviside d.o.f. which are difficult to implement and may lead to conditioning issues,

Cutoff enrichment is not very convenient, as assembling global d.o.f. is out of the scope of the industrial software we used (Code_Aster). Moreover, the method does not extend properly in 3D [29].
Example 1
(Horizontal crack opening in mode I with linear elements). The domain geometry is a square defined by \(\Omega =\left[ {\,0.5,+\,0.5} \right] \times \left[ {\,0.5,+\,0.5} \right] .\) The meshing procedure subdivides the domain into regular sized cells, which edges are parallel to the crack. The size of the enrichment zone is \(r=0.1\).
On Fig. 21, the condition number of XFEM almost skyrockets as noticed in [21]. We did not consider the preconditioner of [21] here to show how both enrichment strategies, scalar and vectorial, behave without expensive treatment. Even the conditioning slope of the vectorial enrichment is far from optimal. The expected optimal conditioning slope is two [25].
Example 2
(Inclined crack opening in mode I with linear elements). We introduce a more realistic case with a chosen inclination of the crack at \(44.9{^{\circ }}\), in order to test the conditioning with both Heaviside and singular enrichments. The analytical solution is still the one of a mode I problem limited to the \(\surd r\) term in Williams series expansion given by the first term of (15) (Fig. 23). The inclined crack analytical solution is expressed through a rotation of \(44.9{^{\circ }}\) of the horizontal crack solution. The strain and stress tensors are rotated as well of the same angle. Then the same mix of Neumann and Diriclet boundary conditions are applied (Fig. 24), similarly to the case of the horizontal crack. We consider the same regular meshes as above which are not oriented accordingly to the crack surface here. The preconditioner of [21] is not applied here as in the previous case.
Numerical analysis of crack approximation with quadratic elements: crack opening in mode 1

XFEM conditioning skyrockets with quadratic elements, even with a small enrichment zone,

Gupta et al. SGFEM kind of enrichment [25, 31] needs far too many additional Heaviside d.o.f., so that it is difficult to implement and may lead to ill conditioning,

Cutoff with global d.o.f. assembling requires a “macro” element, which is not straightforward to implement in finite element software. Moreover the method does not extend properly in 3D [29].
Remark
a dedicated integration scheme is needed at the tip of the crack, to get an optimal rate of convergence in energy norm with quadratic elements [17]. Here, we used a GaussRadau integration rule of order 20 [39]. As the aim of the paper is not about singular integration, we did not try to optimize the procedure. Optimization of integration schemes has been well studied in [30, 40].
Improving the design of quadratic approximation spaces to deal with previous numerical issues

For strongly discontinuous approximations, we suggest a correction to the approximation spaces for quadratic elements. Note that as denoted by [17] optimal convergence rates can only be achieved if the discontinuous approximation space is of the same order than the one of the continuous space, that is to say quadratic. The correction exposed here on the quadratic form is slightly different from the ones discussed in “Strong discontinuity approximation conditioning” section (fittovertex and elimination of d.o.f.).

For singular enrichment, a reshape of the approximation space is considered. The new approximation sums up benefits of work on strong approximation spaces with the significant improvement associated to the “bubble” space as discussed in the next section.
Improvement of strongly discontinuous approximations
Partition of unity alteration
As discussed in “Numerical analysis of strongly discontinuous approximations with quadratic elements” section, the use of vertex node and middle node shape functions (resp. \(\chi _{\Omega +} \varPhi _S \) and \(\chi _{\Omega +} \varPhi _M \) in Fig. 16) leads to an incorrect asymptotic behavior of strongly discontinuous formulations, both for Bernstein and Lagrange polynomials.
When getting rid of the middle node d.o.f. \(\chi _{\Omega +} \varPhi _M\) for \(\varepsilon \) around \(10^{3}\), the condition number decreases sharply (Fig. 28). This threshold value is close to estimates of [18, 24], but we stress on the fact that only the middle d.o.f. is removed and that the position of the interface is not shifted. This alteration process allows the analysis to proceed beyond \(\varepsilon =10^{12}\) as with linear elements.

The fittovertex and volume criterion remove both the vertex node and the middle node d.o.f. (\(\chi _{\Omega +} \varPhi _S \) and \(\chi _{\Omega +} \varPhi _M\)). Then, with the fittovertex, the interface position is switched to the closest node.

With the new strategy, only the minimal information is removed from the approximation (a single d.o.f., the nearest to the interface) with a tuned threshold. The interface is not switched.
Extrapolation in 3D
Let us consider the patchtest of [24]. It is a 3D block of side 4m defined by \(\Omega =\left[ {\,2,+\,2} \right] \times \left[ {\,2,+\,2} \right] \times \left[ {\,2,+\,2} \right] \) and split by an inclined interface. The domain is meshed with regular quadratic hexahedral elements (\(4\,\times \,4\,\times \,4\) elements).

Preconditioning strategies are unchanged (Béchet et al. preconditioner [21] with XFEM and diagonal scaling with shifted enrichment),

The elimination threshold for the middle nodes, is fixed at a ratio of relative distance to the vertex of \(10^{3}\) along the edge (see on the 1D example illustrated by Fig. 16), or, could be expressed with a volume ratio criterion [24]:$$\begin{aligned} \min \left( {{\begin{array}{l} {measure(\left\{ {\Omega _{+} \cap Supp(\varPhi _M )} \right\} ),} \\ {measure(\left\{ {\Omega _{} \cap Supp(\varPhi _M )} \right\} )} \\ \end{array} }} \right) \le 10^{9}measure(\left\{ {Supp(\varPhi _M )} \right\} ) \end{aligned}$$(53)
Furthermore, the volume ratio criterion on the middle node is very satisfactory. The results improve as soon as the elimination process is activated around a node to interface distance (here \(\delta /\sqrt{3})\) of \(10^{3}\,\)m.
Synthesis
Although, all three strongly discontinuous formulations considered in the paper represent the same approximation space, they do not have the same numerical performance. From the results above, domain and shifted enrichments are less sensitive to the position of the interface than XFEM. With quadratic elements, the three formulations need a special care regarding the asymptotic behavior of the shape functions as shown in “Numerical analysis of strongly discontinuous approximations with quadratic elements” section. The XFEM jump formulation is far less accurate in 3D.

A simpler preconditioner than XFEM, which decreases the computational cost,

Less discontinuous d.o.f. than [33], which eases the implementation and improves the accuracy of the formulation in case of elimination of d.o.f. with Lagrange polynomials,

A good accuracy in 3D with quadratic elements.
Singular approximation at a crack tip
New “bubble” approximation space

\(\underline{K}_\alpha (r,\left \theta \right )\) overlaps with \(\underline{K}_\alpha (r,\theta )\) on \(\Omega _{+} \),

\(\underline{K}_\alpha (r,\left \theta \right )\) overlaps with \(\underline{K}_\alpha (r,\theta )\) on \(\Omega _{} \).

Assuming the evaluation point (gauss point) is located on \(\Omega _{+} \),$$\begin{aligned} \tilde{\Pi }^{m}(\underline{K}_\alpha )=\sum _{k\in \left\{ {I_{CT,m} \cup \delta I_{CT,m} } \right\} /I_T } {\underline{K}_\alpha (r_k ,\left {\theta _k } \right )\Gamma _{k,m}} \end{aligned}$$(55)

Assuming the evaluation point (gauss point) is located on \(\Omega _{} \),$$\begin{aligned} \tilde{\Pi }^{m}(\underline{K}_\alpha )=\sum _{k\in \left\{ {I_{CT,m} \cup \delta I_{CT,m} } \right\} /I_T } {\underline{K}_\alpha (r_k ,\left {\theta _k } \right )\Gamma _{k,m}} \end{aligned}$$(56)
Remark

Let us stress on the fact that the ghost node interpolation does not introduce additional nodes. The two branches of discontinuous functions are interpolated over the element through extrapolation of branches (Fig. 31);

The difference between the original SGFEM and the “new” bubble space \(V_{bub,m,n}^h\) lies in the behavior of the interpolation in the bandwidth of elements where the enrichment function becomes unsmooth or discontinuous. The original SGFEM uses the same interpolation operator whether the enrichment function is smooth or discontinuous. The new “bubble” space replaces the enrichment function with smooth continuous extensions in the bandwidth of elements where the enrichment function becomes discontinuous. Thanks to the ghost node interpolation, both smooth extensions are interpolated without additional node (Fig. 31).
Comparison with previous approximations
On Fig. 32, the “bubble” enrichment has a better condition number than the linear vectorial enrichment. On Fig. 33, all enrichment strategies have optimal convergence rates in energy norm. The bubble transformation of the singular function preserves the optimality of the vectorial formulation.
Nonetheless, bubble spaces \(V_{bub, m\,=\,2, n\,=\,1}^h \) and \(V_{bub,\,m\,=\,2,\,n\,=\,2}^h \) are about five times more accurate than the bubble space \(V_{bub,\,m\,=\,1,\,n\,=\,1}^h \). The bubble space \(V_{bub,\,m\,=\,1,\,n\,=\,1}^h \) has the same accuracy than XFEM vectorial enrichment. Clearly, the bubble spaces with quadratic interpolation of the singular function outperform XFEM and the linear bubble space. As a matter of fact, among the bubble spaces with quadratic interpolation of singular function denoted \(V_{bub,\,m\,=\,2,\,n\,=\,1}^h \) and \(V_{bub,\,m\,=\,2,n\,=\,2}^h \), we recommend to choose n = 1 (linear enrichment for the singular part) but m = 2 (quadratic interpolation of the singular function) to have the best accuracy and lowest condition number with an order with respect to mesh refinement around 3.25 for an optimal expected value of 2 [31, 32, 38] (see Fig. 32).
Remark

\(V_{bub,\,m\,=\,1,\,n\,=\,2}^h \) is not very interesting because it has a poor numerical behavior. The shift with a lower order of interpolation has not been considered even in the original method [25].

The good results detailed in this section with quadratic elements, are confirmed with linear elements. In this case the condition number for the bubble space \(V_{bub,\,m\,=\,1,\,n\,=\,1}^h\) evolves with an order with respect to mesh refinement around 2 which corresponds to the optimal expected value (see Fig. 34 related to Figs. 21, 22 and Fig. 35 related to Figs. 25, 26 of “Numerical analysis of crack approximations with linear elements: crack opening in mode 1” section). In addition, an accuracy study of stress intensity factors is also shown with convergence orders with respect to mesh refinement close to the theoretical value of 2 (Fig. 36).
Extension to weak discontinuities
Similarly to the analyses performed for strong discontinuities and singular enrichment, this section will describe several enrichments for weak discontinuities found in the literature. Those are used in case of bimaterials. We will show that getting a result with correct accuracy does not depend only on conditioning issues but also on the quality of the approximation space chosen to represent continuous fields with discontinuous gradients. These two separate issues are often mixed in the literature as it will be shown later in this section.
Enrichment functions for weak discontinuities
Weak discontinuity enrichments comparison
Sukumar [14]  Moës [27]  Belytschko [28]  

F(x)  \(\left {\sum \limits _{i\in I_H } {lsn_i \varPhi _i \left( \mathbf x \right) } } \right \)  \(\sum \limits _{i\in I_H } {\left {lsn_i } \right \varPhi _i \left( \mathbf x \right) } \left {\sum \limits _{i\in I_H } {lsn_i \varPhi _i \left( \mathbf x \right) } } \right \)  \(\left {\sum \limits _{i\in I_H } {lsn_i \varPhi _i \left( \mathbf x \right) } } \right \left {lsn_j } \right \) 
We analyze in the following the influence of these choices on the overall behavior of the solution with respect to the satisfaction of simple patch tests convergence and accuracy properties.
Remark
Note also that the zero of the level set term \(\sum \limits _{i\in I_H } {lsn_i \varPhi _i \left( \mathbf x \right) } \) which appears in [14, 27, 28] represents a discretization of the interface. In the following we took the same orders of interpolation for the functions F(x) and the level sets, which seemed to be quite natural.
Numerical analysis of weakly discontinuous approximation
In order to see if conditioning issues are also relevant for weak discontinuities we performed a numerical analysis similar to the one of “Numerical behavior of strongly discontinuous approximations” section, relying on a one dimensional bimaterial problem. We will show that even if these conditioning issues can be prevented by a change of formulation, convergence issues can still be observed, due to the fact that simple patch tests cannot be represented correctly for linear or quadratic elements.
Analytical solution for the one dimensional bimaterial problem

For the one dimensional bimaterial bar, the solution verifies the following differential equation$$\begin{aligned} \frac{d^{2}u}{dx^{2}}= & {} \hbox {0},\quad \forall x\in \Omega =\left[ {{0}, x_0 } [\cup ] {x_0 ,\hbox {L}} \right] ,\hbox { so that},\\ u(x)= & {} \left\{ {\begin{array}{ll} \alpha ^{}x+\beta ,&{}\quad if\,\, x\le x_{\mathrm{0}} \\ \alpha ^{+}x+\beta ',&{}\quad if\,\, x\ge x_{\mathrm{0}} \\ \end{array}} \right\} , \hbox {where } x_{\mathrm{0}}\hbox { is the location of the interface}, \end{aligned}$$

With arbitrary Dirichlet boundary conditions we choose to impose on the central element nodes for the sake of simplicity,$$\begin{aligned} u(0)=0,\quad u({L})={\overline{u}} \end{aligned}$$

Material properties are given by:$$\begin{aligned} E_i (x)=\left\{ {\begin{array}{ll} E_1 ,&{}\quad if\,\, x\le x_{\mathrm{0}} \\ E_2 ,&{}\quad if\,\, x\ge x_{\mathrm{0}} \\ \end{array}} \right\} , \hbox {where }x_{\mathrm{0}} \hbox { is the location of the interface}, \end{aligned}$$

Assuming small strains, the expected solution satisfying the boundary conditions above is:$$\begin{aligned} u(x)=\left\{ {\begin{array}{l@{\quad }l} \frac{E_2 {\overline{u}} }{E_1 L+(E_2 E_1 )x_{\mathrm{0}} }x,&{} \hbox {if}\,x\le x_{\mathrm{0}} \quad \hbox {so that}\;\alpha ^{}=\frac{E_2 {\overline{u}} }{E_1 L+(E_2 E_1 )x_{\mathrm{0}} } \\ \frac{E_1 x+(E_2 E_1 )x_{\mathrm{0}} }{E_1 L+(E_2 E_1 )x_{\mathrm{0}} }{\overline{u}} ,&{} \hbox {if}\,x\ge x_{\mathrm{0}} \quad \hbox {so that}\;\alpha ^{+}=\frac{E_1 {\overline{u}} }{E_1 L+(E_2 E_1 )x_{\mathrm{0}} } \\ \end{array}}\right. \end{aligned}$$(60)
Numerical analysis of weakly discontinuous approximations with linear elements
In the numerical analysis of “Numerical analysis of weakly discontinuous approximations with linear elements” section, to ease the calculations, we consider \(\Omega =\left[ {\,1,2} \right] \) with the following Dirichlet boundary conditions \(u\left( {\hbox {\,1}} \right) =\hbox {0,}\; u\left( 2 \right) =\bar{{u}}\) so that \(L = 3\) m. The interface is \(\Gamma =\left\{ {x_0 } \right\} \).
The weak form of the problem becomes:
Ability of the enrichment to catch piecewise linear and quadratic solution
First of all, we would like to test that the proposed enrichments (see Table 4) are able to capture piecewise linear and quadratic solutions, so that the enrichment function for the material interface problem preserves the equivalence between the FEM and XFEM discretized spaces. For that purpose, we will consider the previous problem of the onedimensional bimaterial bar for the linear case [14]. Then to capture piecewise quadratic displacements we adapt the bimaterial volumic load test of [27] using a constant volumic load, instead of a quadratic one.
Expressions of F(x) for weakly discontinuous enrichments with linear and quadratic approximations resulting from Table 3 for the one dimensional bimaterial bar problem
Sukumar [14]  Moes [27]  Belytschko [28]  

Linear F(x)  \(\left {xx_0 } \right \)  \(x+x_0 2\frac{xx_0 }{L}\left {xx_0 } \right \)  \(\left {xx_0 } \right \left {x_J x_0 } \right \) 
Quadratic F(x)  \(\left {xx_0 } \right \)  \(\bullet \,\,\hbox {If }\hbox {L}/2\ge x_{0}\) \(x_0 +\frac{x}{L}\left( {L6x_0 } \right) +4x_0 \left( {\frac{x}{L}} \right) ^{2}\left {xx_0 } \right \)  \(\left {xx_0 } \right \left {x_J x_0 } \right \) 
\( \bullet \,\,\hbox {If }\hbox {L}/2\le x_{0}\) \(x_0 +\frac{x}{L}\left( {2x_0 3L} \right) +4\left( {Lx_0 } \right) \left( {\frac{x}{L}} \right) ^{2}\left {xx_0 } \right \) 
For Sukumar and Belytschko formulations, the XFEM blending elements which are adjacent to those completely cut by the interface have a quadratic interpolation [even with linear elements, due to the interpolation (58) and the expression of F(x) in Table 4] or cubic interpolation [with quadratic elements and the expressions (58) and F(x)] with just one active \({b}_{i}\) degree of freedom (see Fig. 8, for the left and right blending elements). Since the patch test is linear, these degrees of freedom must be equal to zero which cancels out all discontinuity. Hence linear patch tests cannot be satisfied and are just approximated by these formulations, which will affect the energy norm as noticed by [14]. Note that Moës formulation with the same order of interpolation for F(x) and the shape functions \(\varPhi (x)\) is able to catch the linear patch test solution, and that Moës formulation with the linear interpolation of F(x) is also able to catch the solution for quadratic or linear shape functions. Table 5 gives a summary of the principal results obtained for the linear patch test. Table 6 represents the relative error in energy norm for this bimaterial bar under traction for different positions of the interface. One can see that the enrichments of Sukumar and Belytschko, even when quadratic (see Table 7), do not yield proper results due to the transition of F(x) in blending elements [26], while the problem is avoided with Moës enrichment.
Moreover, as we cannot recover the exact linear solution for this patch test, an order of convergence of 0.5 in energy norm with respect to mesh refinement is obtained for Sukumar and Belytschko enrichments (Fig. 38), which is consistent with the previous work of Moës et al. [44] extended to XFEM with an enrichment that does not allow to catch the discontinuity.
Analytical solution for the volumic load problem
coefficients used to capture the solution with the finite element approximation for the linear patch test
Sukumar [14]  Moës [27]  Belytchko [28]  

Linear \(\phi (x)\) and linear F(x)  The analytical solution cannot be represented. Incompatible with blending elements.  \(\left\{ \begin{array}{ll} a_1 &{}= 0\\ a_2 &{}= \bar{u}\\ b_1 &{}= b_2 = \frac{(E_2  E_1)\bar{u}}{2(E_1 L + (E_2  E_1)x_0)}\end{array}\right. \)  The analytical solution cannot be represented. Incompatible with blending elements. 
Quadratic \(\phi (x)\) and quadratic F(x)  The analytical solution cannot be represented. Incompatible with blending elements.  \(\bullet \) if \(L/2 \ge x_0 \left\{ \begin{array}{ll}a_1 &{}= 0\\ a_2 &{}= \frac{\bar{u}}{2} + \frac{(E_2  E_1)x_0\bar{u}}{2(E_1 L + (E_2  E_1)x_0)} \\ a_3 &{}= \bar{u} \\ b_1 &{}= b_2 = b_3 = \frac{(E_2  E_1)\bar{u}}{2(E_1 L + (E_2  E_1)x_0)} \end{array}\right. \)  The analytical solution cannot be represented. Incompatible with blending elements. 
\(\bullet \) if \(L/2 \le x_0 \left\{ \begin{aligned} a_1&= 0\\ a_2&= \frac{\bar{u}}{2} + \frac{(E_2  E_1)(L  x_0)\bar{u}}{2(E_1 L + (E_2  E_1)x_0)} \\ a_3&= \bar{u} \\ b_1&= b_2 = b_3 = \frac{(E_2  E_1)\bar{u}}{2(E_1 L + (E_2  E_1)x_0)} \end{aligned} \right. \)  
Quadratic \(\phi (x)\) and linear F(x)  The analytical solution cannot be represented. Incompatible with blending elements.  \(\left\{ \begin{aligned} a_1&= 0\\ a_2&= \frac{(E_2  E_1)x_0\bar{u} + E_1L\bar{u}}{2(E_1 L + (E_2  E_1)x_0)} = \frac{\bar{u}}{2}\\ a_3&= \frac{(E_2  E_1)x_0\bar{u} + E_1L\bar{u}}{(E_1 L + (E_2  E_1)x_0)}= \bar{u}\\ b_1&= b_2 = b_3 = \frac{(E_2  E_1)\bar{u}}{2(E_1 L + (E_2  E_1)x_0)} \end{aligned}\right. \)  The analytical solution cannot be represented. Incompatible with blending elements. 
Comparison of relative error in energy norm for different interface locations between the different enrichment functions for linear elements used in the bimaterial bar problem (results obtained with E1 = 2.05 MPa, E2 = 20.5 MPa, L = 25 m and \(\bar{u} =3 \cdot 10^{6} \)m)
Interface location  Sukumar  Moës  Belytschko 

12.3  2.38688E−01  3.26389E−15  2.38688E−01 
12.4  2.42403E−01  4.30236E−15  2.42403E−01 
12.5  2.45626E−01  2.82096E−15  2.45626E−01 
12.6  2.48370E−01  5.19794E−15  2.48370E−01 
12.7  2.50650E−01  6.25056E−15  2.50650E−01 
Comparison of relative error in energy norm for different interface locations between the different enrichment functions for quadratic elements used in the bimaterial bar problem (results obtained with E1 = 2.05 MPa, E2 = 20.5 MPa, L = 25 m and \(\bar{u} =3 \cdot 10^{6} \)m)
Interface location  Sukumar  Moës  Belytschko 

12.3  8.37168E−02  2.19329E−14  8.37168E−02 
12.4  8.02193E−02  2.72702E−14  8.02193E−02 
12.5  7.62151E−02  3.39596E−12  7.62151E−02 
12.6  7.17862E−02  1.85735E−14  7.17862E−02 
12.7  6.70550E−02  1.43340E−14  6.70550E−02 
coefficients used to capture the solution with the finite element approximation for the quadratic patch test
Sukumar [14]  Moës [27]  Belytchko [28]  

Linear \(\phi (x)\) and linear F(x)  The analytical solution cannot be represented. Incompatibility for the central element.  
Quadratic \(\phi (x)\) and quadratic F(x)  The analytical solution cannot be represented. Incompatible with blending elements.  No solution. Incompatibility for the central element.  The analytical solution cannot be represented. Incompatible with blending elements. 
Quadratic \(\phi (x)\) and linear F(x)  The analytical solution cannot be represented. Incompatible with blending elements.  \(\left\{ \begin{aligned} a_1&= 0\\ a_2&= \frac{f}{4E_1}\left( \frac{5}{2}x_0 L  x_0^2\right) + \\&\frac{f}{4E_2}\left( \frac{3}{2}L^2  \frac{5}{2}x_0 L + x_0^2\right) \\ a_3&= \frac{f}{2E_1}x_0(2L  x_0) + \frac{f}{2E_2}(x_0  L)^2\\ b_1&= \frac{f}{4}\left( \frac{1}{E_1}  \frac{1}{E_2}\right) (2L  x_0)\\ b_2&= \frac{f}{8}\left( \frac{1}{E_1}  \frac{1}{E_2}\right) (3L  2x_0)\\ b_3&= \frac{f}{4}\left( \frac{1}{E_1}  \frac{1}{E_2}\right) (L  x_0) \end{aligned}\right. \)  The analytical solution cannot be represented. Incompatible with blending elements. 
Figure 40 shows the convergence rates on the relative error in terms of the energy norm with respect to mesh refinement for the quadratic patch test, using linear shape functions \(\varPhi (x)\) and linear F(x) and quadratic shape functions \(\varPhi (x)\) and quadratic F(x). Using linear elements we obtain the optimal convergence rates of 1. Using quadratic elements we obtain 1.5. In this later case, a convergence rate of 2 would be expected since the analytical solution does not lie in the finite element space (cf. Table 8). We will not prove formally that the obtained convergence rates are in fact the expected ones, nevertheless we can give some arguments that those rates are indeed the expected ones.
In the quadratic patch test case, the first and second derivatives of the analytical solution are discontinuous. The finite element space with enrichment we use with quadratic F(x) and quadratic shape functions \(\varPhi (x)\) does not allow to capture this solution. However it allows to capture the linear patch test solution with a discontinuous first derivative. We can deduce from this analysis that its rate of convergence on energy will be greater than 1 but lower than 2. Then the loss of 0.5 in convergence rate observed numerically for quadratic elements, with a value of 1.5 observed numerically instead of an expected 2, is coherent with the analysis for linear elements (0.5 instead of 1 when elements do not catch the discontinuity on the first derivative) performed by Moës et al. [44].
From this analysis, we recommend to use Moës formulation with quadratic elements and linear F(x).
Convergence analysis for weak discontinuities
Volumic load problem with straight interface
The test we propose in the following was already investigated by Moës et al. [18]. Let us consider again the squared plate of Fig. 39 with side \(L=\hbox {2 m}\) separated in its middle by two materials \(E_{\mathrm{1}} = 1\) MPa and \(E_{\mathrm{2}} = 10\) MPa and with \(\nu _{\mathrm{1}} =\nu _{\mathrm{2}} = 0 \). This time the variable volumic force is \(\left\ \vec {f} \right\ =f\left( {\frac{x}{L}} \right) ^{\mathrm {2}}\) imposed on the plate along the \(\vec {x}\) direction.
Circular inclusion with imposed displacement
The domain of computation is a square \(\Omega _{2}\) of side L = 2 m. We considered an unstructured mesh with 5, 10, 20, 40, 80 and 160 triangular elements by side. Figure 42 shows the convergence rates on the relative error in terms of the energy norm with respect to mesh refinement for circular inclusion, using linear shape functions \(\varPhi (x)\) and linear F(x), quadratic shape functions \(\varPhi (x)\) and linear F(x) and quadratic shape functions \(\varPhi (x)\) and quadratic F(x). Using linear elements we obtain the optimal convergence rate. Using quadratic elements we obtain 1.5. In the case of quadratic F(x), this behavior arises from the discontinuity of the second derivative of the analytical displacement. The case of linear F(x) is slightly different, due to its link with the geometrical approximation of the level sets and more particularly of the interface established at the end of “Enrichment functions for weak discontinuities” section. Ferté et al. [45, 46] showed for strong discontinuities that the approximation of the level set impacts the expected theoretical convergence rates, in the case of quadratic elements: 2 is expected if a quadratic approximation of the level set is used (equivalently quadratic F(x)), while 1.5 is expected if a linear representation of the level set is used (equivalently linear F(x)). To illustrate this behavior, we also considered two models based on the enrichment designed for strong discontinuities, using Lagrange multipliers to enforce the continuity of the displacement through the interface: one with quadratic shape functions and a linear approximation of the level set and one with quadratic shape functions and a quadratic approximation of the level set. The convergence rates obtained for these two models are also shown on Fig. 42.
Finally for weak discontinuities, optimal rates of convergence in energy norm are difficult to obtain for quadratic elements. If linear approximations of the level sets (equivalently linear F(x)) are used associated to a quadratic approximation of the enrichment, convergence rates are limited due to errors on the geometry. If quadratic approximations of the level sets (equivalently quadratic F(x)) are used associated to a quadratic approximation of the enrichment, convergence rates are limited due to errors on the approximation spaces. It appears that both convergence rates end up being the same, with a loss of 0.5 in energy norm with respect to optimal orders of convergence of 2. Optimal orders of convergence are recovered when using a strong discontinuity approximation space with lagrangian multipliers to represent weak discontinuities.
Conclusion
This paper outlines numerical issues around popular approximation spaces to model weak discontinuities or strong discontinuities and cracks, particularly with quadratic elements. We analyzed those issues on a few benchmarks. Those benchmarks revealed that popular enrichment strategies do not behave well with asymptotic configurations (when the discontinuity gets close to a node of the mesh, typically for a distance below 10% of the length of the cut edge) for strongly discontinuous enrichments and that the singular enrichment could be improved with respect to conditioning and accuracy.
Instead of working on complex and expensive strategies to solve those issues for strongly discontinuous enrichments, we preferred to apply a slight reshape of the approximation spaces, as it is in our sense, the most effective approach to deal with those issues for industrial applications. In fact, the new approximation spaces get through the proposed benchmarks, with significant improvement of the numerical behavior and accuracy of the formulations studied.
Nonetheless, more theoretical work is needed to understand why very unsmooth “bubble” functions tend to give better results than “straightforward” singular functions. At least, our results confirmed the numerical properties of “bubble” spaces denoted in the literature [25, 31, 32]. Moreover, the integration procedure needs to be improved, particularly in 3D with quadratic elements, to further the analysis on complex crack surface and crack front geometries.
At last, in case of weakly discontinuous enrichments, it was shown that most approximation spaces of the literature did not exhibit conditioning issues. However, for quadratic elements, orders of convergence in energy norm were 0.5 lower than those obtained with an equivalent strongly discontinuous enrichment, so that we recommend using the latest approach even in the case of weak discontinuities, associated to the strategy exposed in the present work to avoid conditioning issues.
Declarations
Author's contributions
MN conducted the numerical implementation of strong discontinuities approximations exposed in the paper. PM contributed to draft the manuscript and helped assessing the numerical properties of approximations spaces. NM helped strengthening the theoretical background and he provided numerous ideas to shape new approximations spaces. AM and SG participated in the study of weak discontinuities implementation and benchmarks. All authors read and approved the final manuscript.
Acknowledgements
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
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Consent for publication
Not applicable.
Ethics approval and consent to participate
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Funding
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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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