On the construction of approximation space to model discontinuities and cracks with linear and quadratic extended finite elements

“Straightforward” singular enrichment class

“Bubble” enrichment class

\(\delta I_{CT} \) is defined here as the transition layer of nodes between crack-tip 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].

Numerical analysis of strongly discontinuous approximations with linear elements

Let us consider the following arbitrary numerical values: \({\alpha ={10}/9}\) and \({\beta } =10.\)

Then, the solution is computed in standard 64-bit arithmetic, with the linear solver UMFPACK, to reproduce the behavior of direct solvers.

The numerical error has to be close to zero, for any given position of the interface. The error is evaluated here in terms of the H\(^{1}\)-norm:

$$\begin{aligned} \left\| {u-u_h } \right\| _{H^{1}} =\sqrt{\int _0^3 {\left[ {\left( {u-u_h } \right) ^{2}+\left( {{u}'-{u}'_h } \right) ^{2}} \right] dx} } \end{aligned}$$

(39)

Although all three formulations represent the same approximation space, they do not have exactly the same numerical behavior (Fig. 9), at least concerning the error in H\(^{1}\)-norm. The X-FEM error increases steadily while the “shifted” formulation and the “domain” formulation levels of error stay close to machine precision (about \(2.2 \cdot 10^{-16}\)).

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 pre-conditioner is needed for the three enrichments. In the case of X-FEM, we studied Béchet et al. [21] pre-conditioner. For other formulations we used a diagonal pre-conditioner to scale the d.o.f.

The diagonal pre-conditioner allows a scaling of rows and columns through a multiplication with a diagonal matrix to the left and to the right:

$$\begin{aligned} Ku=f\rightarrow K'u'=f'\;\hbox {with}\;\left\{ {\begin{array}{l} K'=D_c KD_c \\ u=D_c u' \\ f'=D_c f \\ \end{array}} \right. \end{aligned}$$

(40)

In linear elasticity, \(D_c \) is usually defined as,

$$\begin{aligned} \left[ {D_c } \right] _{i,i} =\frac{1}{\sqrt{K_{i,i} }}\sqrt{\frac{\max (K_{i,i} )+\min (K_{i,i} )}{2}} \end{aligned}$$

(41)

Thus, on Fig. 11, condition numbers are reduced drastically. However, the error still increases in case of X-FEM. The error is clearly not only related to conditioning. Another explanation should be considered.

Numerical analysis of strongly discontinuous approximations with quadratic elements

The same one-dimensional case is considered here. The same discretization of the weak form is used with quadratic shape functions. The results are plotted on Fig. 14.

For the three approximations, conditioning worsens very quickly, at about three times the rate of linear elements. More worrying is that the error increases. X-FEM enrichment error worsens more quickly than the others.

On Fig. 15, the same pre-conditioners than in “Numerical behavior of strongly discontinuous approximations” section are considered. The wrong behavior noticed above still applies.

This numerical behavior with quadratic elements is well explained by the coupling between enrichment functions, as pictured on Fig. 16 with domain enrichment [33].

Proof

Let \(\kappa (K)\) be the condition number of the stiffness matrix:

$$\begin{aligned} \kappa (K)=\frac{\max \left( \left\{ {e^{T}Ke} \right\} _{\left\| e \right\| _2 =1} \right) }{\min \left( \left\{ {e^{T}Ke} \right\} _{\left\| e \right\| _2 =1}\right) }\ge \frac{e_1^T Ke_1}{e_2^T Ke_2}\quad \forall e_1 ,e_2/\left\| {e_1} \right\| _2 =1,\left\| {e_2} \right\| _2 =1 \end{aligned}$$

(46)

where \(\left\| \right\| _2\) is the Euclidian norm 2.

In order to estimate the lower bound of the condition number, we consider the polynomials e \(_1\) and e \(_2\) illustrated on Fig. 17.

Those polynomials belong to the discrete space of [33] (likewise [7] and [26]). They can be used to determine a lower bound of the condition number. Functions \({e}_{1}\) and \({e}_{2}\) are expressed in the discrete space of [33] as:

$$\begin{aligned} \left\{ {{\begin{array}{l} {e_1 =-\frac{1}{\sqrt{5}}\left( {2\Phi _1+\Phi _2 } \right) } \\ {e_2 =\frac{1}{\sqrt{17}}\left( {4\chi _{\Omega ^{-}} \Phi _3 +\chi _{\Omega ^{-}} \Phi _4 } \right) } \\ \end{array} }} \right. \end{aligned}$$

(47)

In the discrete space of [33], we have:

$$\begin{aligned} \left\| u \right\| _2 =\sqrt{\sum _{i\in I/I_H } {a_i^2 } +\sum _{j\in I_H } {\alpha _{j,1}^2 +\sum _{j\in I_H } {\alpha _{j,2}^2 } } } \end{aligned}$$

(48)

so that we have \(\left\| {e_1 } \right\| _2 =\left\| {e_2 } \right\| _2 =1\).

As \(\frac{1}{2}e_{1}^{T}Ke_1\) and \(\frac{1}{2}e_2^T Ke_2\) represent the elastic energy of these solutions, we have:

$$\begin{aligned} \frac{e_1^T Ke_1}{e_2^T Ke_2}=\frac{\int \nolimits _{0}^{1} {\left( {e_1^{\prime } } \right) ^{2}dx} }{\int \nolimits _{2-\varepsilon }^{2} {\left( {e_2^{\prime } } \right) ^{2}dx} }=\frac{51}{80\varepsilon ^{3}} \end{aligned}$$

Thus,

$$\begin{aligned} \kappa (K)\ge C\varepsilon ^{-3} \end{aligned}$$

The asymptotic behavior of \(\varPhi _M (\xi )\) [at the first order of \((1-\xi )\)] is:

$$\begin{aligned} \varPhi _M (\xi )\approx -\varPhi _S (\xi )/4 \end{aligned}$$

Then, the related d.o.f. become redundant, which leads to ill conditioning. As enrichment strategies are equivalent, the redundant d.o.f. pollutes also the approximation spaces of the other formulations [7, 26].

• 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)

when \(\xi \) goes to 1, there is no coupling between Bernstein polynomials as observed with Lagrange polynomials. However, the polynomial \(\varPhi _S \) cancels out accordingly to \((1-\xi )^{2}\), which also leads to the same conditioning slope of three (Fig. 18) (as shown in the previous section).

It is noticeable that [33] has better accuracy with Bernstein polynomials than with Lagrange polynomials (Figs. 18, 19). \(\square \)

Numerical analysis of crack approximations with linear elements: crack opening in mode 1

The analytical solution in displacement corresponds to the exact mode I limited to the \(\surd r\) term in Williams series expansion [11] and is given by the first term of (15). Then, Dirichlet boundary conditions are applied on three sides. On the left side, Neumann conditions are preferred. As the left side is cut by the crack, Dirichlet conditions are more difficult to apply given the discontinuity of the displacement field (Fig. 20).

The condition number is estimated by MUMPS solver, so that the results shown here have to be taken only as rough estimates.

On Fig. 21, the condition number of X-FEM almost skyrockets as noticed in [21]. We did not consider the pre-conditioner 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].

Then, the relative error in energy norm is computed accordingly to the following formula:

$$\begin{aligned} \Vert u-u^{h}\Vert _\mathrm{energy}=\sqrt{\int \nolimits _\Omega {\left( {\sigma ^{h}-\sigma } \right) :\left( {\varepsilon ^{h}-\varepsilon } \right) }d\Omega \bigg /{\int \nolimits _\Omega {\sigma :\varepsilon }d\Omega }} \end{aligned}$$

(50)

On Fig. 22, the rate of convergence in energy norm of the X-FEM scalar enrichment is under-optimal (around 0.91) as noticed in [17]. Optimality is recovered with the vectorial enrichment, the convergence rate being then 0.989. Nonetheless, X-FEM is more accurate than the vectorial enrichment as predicted in Lemma 1.

On Fig. 25, the conditioning of the X-FEM vectorial enrichment becomes very high and reaches a steady value around \(10^{12}\). This behavior may be explained by the ill conditioning of the Heaviside enrichment. As the “Heaviside information” becomes very unbalanced, its ill conditioning overcomes the regular increase of conditioning expected with the vectorial enrichment (see Fig. 21) as explained in “Strong discontinuity approximation conditioning” section. Nonetheless, both enrichments have optimal convergence accuracy (see Fig. 26).

Numerical analysis of crack approximation with quadratic elements: crack opening in mode 1

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.

Let us precise the differences between the removal of the middle node and straightforward criteria from “Strong discontinuity approximation conditioning” section:

• The fit-to-vertex 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 fit-to-vertex, 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.

However, the alteration of the partition of unity has a noticeable impact on enrichment [33] with Lagrange polynomials (Fig. 28): the H1-error increases sharply after the alteration, instead of decreasing as observed with other formulations. This behavior might be explained by the partition of unity and the nature of the solution represented here:

• The exact solution is continuous,

• In [7, 26] discontinuous functions are added to continuous functions, so that the removal of the discontinuous middle node d.o.f. does not alter the partition of unity of continuous space functions [9] and consequently the representation of continuous functions.

This is in contrast with [33], as the removal of the enriched middle node d.o.f. prevents the representation of continuous functions on some parts of the domain where the partition of unity is broken.

Extrapolation in 3D

Let us consider the patch-test 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).

A multi-axial loading is applied. Given the elastic behavior, the stress is homogeneous within the block at a value of 10 MPa, for any given position of the interface. The interface equation is parameterized with \(\delta \) as:

$$\begin{aligned} x+y+z+\delta =0 \end{aligned}$$

(51)

The error is analyzed with the energy norm:

$$\begin{aligned} \Vert u-u^{h}\Vert _\mathrm{energy}=\sqrt{\int \nolimits _\Omega {\left( {\sigma ^{h}-\sigma } \right) :\left( {\varepsilon ^{h}-\varepsilon } \right) }d\Omega \bigg /{\int \nolimits _\Omega {\sigma :\varepsilon }d\Omega }} \end{aligned}$$

(52)

As the Hansbo et al. [33] enrichment performs as well as the “shifted” enrichment, only the “shifted” enrichment is tested here.

The strategies discussed in 1D are extrapolated in 3D (an example is given Fig. 29):

• Preconditioning strategies are unchanged (Béchet et al. pre-conditioner [21] with X-FEM 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)

On (Fig. 30), the shifted formulation is more accurate than the X-FEM one, as observed in the 1D case. The difference in accuracy reaches a \(10^{10}\) factor: accuracy issues observed in 1D with X-FEM are heightened in 3D.

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 X-FEM. 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 X-FEM jump formulation is far less accurate in 3D.

New “bubble” approximation space

We introduce a simpler space than the one of Gupta et al. [25, 31], to reduce its too many additional Heaviside d.o.f. As a matter of fact Gupta et al. approach requires up to 12 Heaviside d.o.f. per node instead of 3 so that it is difficult to implement and may lead to ill conditioning.

Comparison with previous approximations

The mode I problem of “Numerical behavior of singular approximations at the crack tip” section is considered here, with a similar enrichment radius of \(r=0.1\). In this section, we focus on the convergence analysis carried out with quadratic elements.

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 X-FEM vectorial enrichment. Clearly, the bubble spaces with quadratic interpolation of the singular function outperform X-FEM 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).

Analytical solution for the one dimensional bi-material problem

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:

On considering the \(4\,\times \,4\) sub-matrix that is free from boundary conditions, the diagonal pre-conditioner can easily scale the formulation of Moës et al. [27] as entire rows (symmetrically columns) vanishes when \(\varepsilon = 0 \) or \(\varepsilon = 1\) (see Fig. 37). For the enrichments of Sukumar et al. [14] and Belytschko et al. [28], no conditioning issues can be observed, in case of linear approximation. However, we will see in the next paragraph that these latest two formulations are far from optimal with respect to convergence issues, being unable to represent simple patch tests.

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 X-FEM discretized spaces. For that purpose, we will consider the previous problem of the one-dimensional bi-material bar for the linear case [14]. Then to capture piecewise quadratic displacements we adapt the bi-material volumic load test of [27] using a constant volumic load, instead of a quadratic one.

For Sukumar and Belytschko formulations, the X-FEM 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 bi-material 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 X-FEM with an enrichment that does not allow to catch the discontinuity.

Analytical solution for the volumic load problem

The exact displacements in the two materials are given by:

$$\begin{aligned} \vec {u}_{\mathrm{1}} (x)= & {} -\frac{f}{E_{\mathrm{1}} }\left( {\frac{x^{\mathrm {2}}}{ 2 }-Lx} \right) \vec {x}\quad \hbox {if}\,\, x\le x_0 \\ \vec {u}_{\mathrm{2}} (x)= & {} -\left[ {\frac{f}{E_{\mathrm{2}} }\left( {\frac{x^{\mathrm {2}}}{ 2 }-Lx} \right) +f\left( {\frac{x_0^2 }{ 2 }-Lx_0 } \right) \left( {\frac{ 1 }{E_{\mathrm{1}} }-\frac{ 1 }{E_2 }} \right) } \right] \vec {x}\quad \hbox {if}\,\, x\ge x_0 \\ \end{aligned}$$

No linear element with linear F(x) is able to capture this quadratic solution directly. For Sukumar and Belytschko formulations with quadratic elements, the xfem blending elements which are adjacent to those completely cut by the interface give a cubic interpolation (linear F(x)) or order four interpolation (quadratic F(x)) with just one \(\hbox {b}_{i}\) degree of freedom (see Fig. 8, for the left and right elements). Since the patch test is quadratic, these degrees of freedom must be equal to zero which cancels out all discontinuity. Hence quadratic 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 quadratic shape functions \(\varPhi (x)\) and linear F(x) is able to catch the quadratic patch test solution, and that Moës formulation with quadratic shape functions \(\varPhi (x)\) and quadratic F(x) is not able to catch it. Table 7 gives a summary of the principal results obtained 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).

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.

The exact displacements in the two materials are:

$$\begin{aligned} {\vec {u}}_{\mathrm{1}} (x)= & {} \frac{f}{ 3 E_{\mathrm{1}} L^{\mathrm {2}}}\left( {L^{\mathrm {3}}x-\frac{x^{\mathrm {4}}}{ 4 }} \right) \vec {x} \\ {\vec {u}}_{\mathrm{2}} (x)= & {} \left[ {\frac{f}{ 3 E_{\mathrm{2}} L^{\mathrm {2}}}\left( {L^{\mathrm {3}}x-\frac{x^{\mathrm {4}}}{ 4 }} \right) +\frac{ 31 f\hbox {L}^{\mathrm {2}}}{ 192 }\left( {\frac{ 1 }{E_{\mathrm{1}} }-\frac{ 1 }{E_{\mathrm{2}} }} \right) } \right] \vec {x} \end{aligned}$$

We considered an unstructured mesh with 5, 10, 20 and 40 triangular elements on each side. Figure 41 shows the convergence rates on the relative error in terms of the energy norm with respect to mesh refinement for the volumic load problem, 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 or quadratic elements and linear enrichment function we obtain the optimal convergence rates. Using quadratic elements and quadratic enrichment function we obtain 1.5. Note that the second derivative of the analytical displacement is discontinuous in this case too.

Circular inclusion with imposed displacement

To investigate the behavior of weak discontinuity enrichments in presence of curved interfaces, Moës et al. [18] suggested the following test case. A circular inclusion \(\Omega _{1}\) of radius \(a= 0.4\) m is placed in the center of a material \(\Omega _{2}\). A linear displacement is imposed on the boundary \(\Gamma _{2}\) (\(r = b =\) 2 m). The same material parameters are chosen: \({E}_{1} = 1\hbox { MPa}, \nu _{\mathrm{1}} =\hbox {0.25}\) and \({E}_{2} = 10\) MPa, \(\nu _{\mathrm{2}} = 0.3\). This problem has an analytical solution provided in [18]. The analytical displacement reads:

$$\begin{aligned} u_r \left( r \right) =\left\{ {{\begin{array}{l} {\left[ {\left( {1-\frac{b^{2}}{a^{2}}} \right) } \right] \alpha +\frac{b^{2}}{a^{2}},\quad \hbox {if}\,\, {0\le r\le a}} \\ {\left( {r-\frac{b^{2}}{a^{2}}} \right) \alpha +\frac{b^{2}}{a^{2}},\quad \hbox {if}\,\, a\le r\le b} \\ \end{array} }} \right. , \end{aligned}$$

with:

$$\begin{aligned} \alpha =\frac{\left( {\lambda _1 +\mu _1 +\mu _2 } \right) b^{2}}{\left( {\lambda _2 +\mu _2 } \right) a^{2}+\left( {\lambda _1 +\mu _1 } \right) \left( {b^{2}-a^{2}} \right) +\mu _2 b^{2}} \end{aligned}$$

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.