elements by an extended global recovery: an introduction to e- ... approximations of the simple concept of global derivative recovery introduced in. [36] for the ...
A posteriori error estimation for extended finite elements by an extended global recovery: an introduction to e-adaptivity Marc Duflot and St´ephane Bordas
1
Introduction
In engineering, numerical methods are heavily employed in the design and optimization of increasingly complex products. Arguably, the most critical point, and certainly the most important is the measure of the error committed in the computations. In the simulation of failure, where the damage tolerance of a component is at stake and safety a natural outcome, poor understanding and management of these errors can be disastrous. This paper is concerned with the estimation of errors in computational fracture mechanics, where partition of unity (PU) enrichment is the chosen method. Finite elements had been employed for decades in the simulation of fracture, before the discovery of the concept of PU enrichment [23, 6] revolutionalized the world of computational fracture. On the basis of partition of unity enrichment, the extended finite element method (xfem) was created in 1999 [10] to model crack growth; the generalized finite element method (gfem) [29, 30] was born around the same time. Since their inception, these methods have been used very successfully to model a variety of fracture problems in two [28] and three dimensions [34, 26, 19, 32, 16], for cohesive cracks [25], crack branching [17], interface failure [33], geometrically non-linear materials [22], contact, plasticity and dynamics [27]. Remarkably, extended finite elements have been employed in industrial settings [14, 11], and open-source software is now available on the internet to promote further development of extended finite element methods [15, 18]. Two strategies exist in error estimation: the first is a priori error estimation and relies on deriving mathematically a bound for the approximation error; the second is a posteriori error estimation and concentrates on approximating the approximation error of an existing numerical solution. In the world of a posteriori error estimation, three directions can be taken: residual based estimation [5, 1, 3], recovery based estimation [36, 37, 38] and constitutive based estimation [21]. We propose in this paper a recovery based a posteriori error estimation technique for enriched finite elements and the curious reader should consult the book [4] and the review article [2] for more details on error estimation.
1
Some remarkable work has been performed on a posteriori error estimation for the generalized finite element method [8, 7, 31], and a first step taken in [35] to improve crack tip fields in the extended finite element method by statically admissible stress recovery. The authors devised an a posteriori error estimator for extended finite element methods in References [13, 12], named extended moving least squares recovery, or xmls recovery. The idea of xmls recovery is to employ an intrinsically enriched moving least squares approximation for the computation of the recovered derivatives. This requires employing an enriched approximation over the whole domain, unless blending regions are constructed to ensure continuity of the approximation between an enriched and a standard domain. Large bases in moving least squares approximation lead to high computational cost, and loss of flexibility in the recovery technique. The present paper addresses the issue of derivative recovery and error estimation in extended finite element methods through a generalization to enriched approximations of the simple concept of global derivative recovery introduced in [36] for the finite element method. The starting point of global derivative recovery is the remark that when only C0 continuity of functions in the trial space is assumed in finite element methods, the strain and stress fields are discontinuous through element boundaries. The principle presented in Reference [36] is to construct an enhanced stress field interpolated with the same ansatzt functions as the displacements, and such that the L2 norm over the whole domain of the difference between the enhanced and original finite element strains (stresses) is minimized. In xfem , the standard displacement approximation is extended (enriched) by special functions providing information about the solution sought. In linear elastic fracture mechanics (LEFM), the theoretical near-tip fields are added. We propose to follow the idea of global strain recovery described in the previous paragraph, but instead of using the same approximation as the displacement approximation to construct the enhanced strain field, we use the standard finite element approximation with which the enriched displacement field was built, and enrich it with special functions spanning the near-tip strain field (the spatial derivatives of the near-tip displacement fields). We coin this technique extended global derivative recovery (xgr). Through global minimization, we obtain an enhanced strain field, which is a better aproximant to the exact solution, which is itself unknown. Comparing the orginal (raw) xfem strains to the enhanced strains, we define a local (elementwise) error which can be used to drive adaptive strategies. The idea proposed, through its simplicity and ease of introduction in existing codes, is well-suited to engineering analysis. The paper is organized as follows: in Section 2, we present the extended global recovery in more detail and exercise it in Section 3 on two-dimensional fracture problems where we compare it to the extended moving least squares recovery (XMLS). Section 4 presents three-dimensional examples before Section 5 closes with some conclusions and future work directions.
2
2
Extended global recovery
2.1
Extended finite element approximation
In the extended finite element method (xfem), the standard finite element approximation, defined on the N nodes of the mesh covering domain Ω by its shape functions (φi )1≤i≤N , is enriched extrinsically with special functions. • Nodes in set J, whose support is completely cut by a crack are enriched with function h : Ω → R which takes the value +1 at points above the crack and the value -1 at points below the crack; • Nodes in set K 1 are enriched with the near-tip fields. Denoting as usual the polar coordinate system associated with a crack tip by (r, θ), the neartip functions (Fk )1≤k≤4 : Ω → R spanning the LEFM crack-tip fields may be written √ √ √ √ θ θ θ θ F= r cos , r sin , r sin sin (θ) , r cos sin (θ) . 2 2 2 2 (1) With these notations, the enriched finite element approximation for the displacement field reads
h
u (x) =
N X i=1
φi (x) ai +
X
φj (x) h (x) bj +
j∈J
X
φk (x)
k∈K
4 X
Fk (x) ckl
l
!
, (2)
where the ai , bj and ckl are scalar coefficients associated with the standard, discontinuous, and near-tip fields, respectively.
2.2
Extended global recovery of the strain field
The approximate strain field is given by the symmetric gradient of the approximate displacement field εh (x) =
1 ∇ + ∇T ⊗ uh (x) . 2
(3)
1 located within a ball centered on the crack tip (2D) or a circular cylinder around the front (3D) with radius renr . It was shown in [20, 9] that this enrichment scheme improves convergence rates. There is still no known rule on how to choose the value of this enrichment radius. There is no real optimum since the error keeps decreasing as renr increases. However, the enrichment describes the local behaviour, around a feature of interest. Far from it, the part of the solution due to this feature decays rapidly, and the role played by the enrichment decreases while that of the standard, polynomial, FEM approximation increases. It is likely that a computationally optimum enrichment radius can in general be found.
3
It is easy to show that if the functions in Equation (1) span the near-tip displacement fields, the following functions span the near-tip strain fields: θ 1 √ , cos G= 2 r θ 1 √ sin , r 2 (4) θ θ 3θ 1 √ cos sin cos , 2 2 2 r θ 1 θ 3θ √ cos sin sin . r 2 2 2 The approximation we build for the smoothed (enhanced) strain field εs has three parts: a standard, C0 part given by the standard shape functions, a discontinuous part provided by function h and a near-tip part constructed through functions G, and writes:
εs (x) =
N X
φi (x) di +
i=1
X
φj (x) h (x) ej +
j∈J
X
k∈K
φk (x)
4 X l
Gl (x) fkl
!
. (5)
To fully define the smoothed strain field, we need to evaluate the scalar coefficients di , ej and fkl , which is done by minimization of the square of the L2 norm of the difference between the xfem strain field and the smoothed strain field, over the whole domain. The associated functional writes: Z
h
ε − εs 2 dΩ. (6) Ω
Differentiating with respect to the unknown coefficients di , ej and fkl , we obtain the following system of linear algebraic equations
R R h di R Ω φI φk Gl dΩ R Ω φI ε hdΩ ej = , R Ω φJ hφk Gl dΩ R Ω φJ hε hdΩ fkl Ω φK GL φk Gl dΩ Ω φK GL ε dΩ (7) which is sufficient to compute the sought coefficients. Remark: the set of equations (7) can be solved for each component of the strain tensor successively, in which case a single matrix inversion is necessary. R R Ω φI φi dΩ R Ω φJ hφi dΩ Ω φK GL φi dΩ
2.3
R R Ω φI φj h dΩ R Ω φJ hφj h dΩ Ω φK GL φj h dΩ
Numerical integration
The matrix present in the left hand side of (7) has the structure of a mass matrix, and is computed by numerical integration. There are three typical entries in this matrix:
4
• φi φj entries are quadratic functions, their exact integration necessitates three Gauß points in each triangle and four in each tetrahedron; • φi φj h is a piecewise quadratic function since h is piecewise constant requiring element subdivision, as is standard in the xfem . The same subdivision as for the xfem stiffness matrix is used; • terms featuring the special functions G (which are not polynomial) cannot be integrated exactly using Gauß quadrature. In our implementation “many” points are used, as is commonly the case in xfem problems. We could also have built an ad hoc scheme as in [20]. Remark: it would be possible to lump the mass matrix following the same procedure as described in [24].
2.4
Measure of the adequacy of the estimator
Let Ωq be an element in the mesh, the error between the raw and the smoothed solution can be measured by the norm sZ ehs Ωq =
2
Ωq
kεh (x) − εs (x)k dx.
(8)
Since the enhanced solution is different from the exact solution, this measure is only an approximate error. The global approximate error is measured by the sum of the elemental errors on the nelt elements of the mesh v uX unelt hs 2 hs e =t eΩq . (9) q=1
ehe Ωq
We then define as the following error norm between the raw xfem solution and the exact solution on element Ωq sZ ehe Ωq =
2
Ωq
kεh (x) − εexact (x)k dx,
(10)
and we call this error measure the exact error since it measures the distance between the raw xfem solution and the exact solution. Summing over the elements in the mesh, the global exact error writes v uX unelt he 2 he e =t eΩq . (11) q=1
The effectivity index of the error estimator is defined as the ratio of the approximate error to the exact error θ=
ehs . ehe 5
(12)
A good estimator has an effectivity close to unity, which means that the measured error is close to the exact error. In other words, the enhanced solution is close to the exact solution.
3 3.1
Two-dimensional fracture mechanics examples Mode I problem and comparison with extended moving least squares recovery
This section is concerned with the case of an edge crack subjected to uniform tension applied at infinity so as to create a pure mode I field around the crack tip. Traction boundary conditions representing the exact stress state due to the far field are applied on the boundary of the square domain, while the bottom left corner is restrained in both directions and the bottom right corner in the vertical (y) direction. The xfem approximation of Equation (2) can exactly reproduce the exact solution. In this two dimensional example, the results are presented through six figures. (a) raw strain field obtained directly from the xfem plotted on the whole mesh; (b) strain field recovered using the extended global recovery (xgr) technique plotted on the whole mesh; (c) y component of the xgr strain field along line x = 0, comparison with the exact strain field; (d) error indicator in the mesh from the xgr method; (e) convergence of the L2 norm of the difference between the raw, xfem strain and the xgr strain with mesh refinement; (f ) convergence of the effectivity index of the error indicator with mesh refinement. Items (e) and (f) also study the influence of the enrichment radius renr upon the convergence. 3.1.1
Structured mesh
First, we consider structured meshes of linear triangles. Figure 1(b) shows the smooth strain isolines obtained through xgr compared to the raw strain field obtained from the xfem which are represented in Figure 1(a). The C0 nature of this raw xfem field is further depicted on Figure 1(c), where we see that the recovered strain is very close to the exact strain field, even in the close vicinity of the crack tip.
6
Table 1: Mode I problem, structured mesh, convergence for various xfem enrichment radii. See also Figure 1. Close to optimal onvergence is attained for a fixed area enrichment. The results tend to show that the rate of 1. would be reached if the whole component were enriched. The xmls recovery shows slightly better convergence rates for all enrichment radii. However, this faster convergence is offset by a much higher computational cost, since in the simplest version of xmls , the mls basis is enriched in the whole domain, unless blending between enriched and non-enriched regions is employed. Enrichment radius: renr only tip element 0.1 0.2 0.3 0.4
xmls 0.46022 0.81271 0.86728 0.90644 0.91052
xgr – 0.77959 0.82568 0.86427 0.86917
Figure 1(e) shows that the L2 norm of the difference between the xgr strain and the raw xfem strain tends vanishes upon mesh refinement. More importantly, Figure 1(f), shows that the effectivity index of the error indicator converges toward unity upon mesh refinement. This proves that the approximate error converges to the exact error, and, therefore, that the error indicator is indeed a correct measure of the error. Note also that the larger the xfem enrichment radius, the closer the convergence rate of Figure 1(e) is to 1. This corroborates earlier findings in the context of the xmls recovery technique [13, 12] and is explained by the fact that larger enrichment radii lead to more accurate solutions, thus more accurate recovered solutions, and therefore an approximate error which is close to the exact error. In Table 1, the convergence rates of the xgr and xmls methods are compared for different values of the xfem enrichment radius. We notice that the xmls provides slightly higher convergence rates at the expense of computational efficiency, at least in the version of Reference [13, 12]. Comparing the converged values of the effectivity index shown in Figure 1(f) to the converged effectivities published in [13, 12], we note that the xgr effectivities converge between 93 and 96%, whereas the xmls effectivities are in the vicinity of 99%. For the whole range of mesh sizes, the xmls effectivities are better than the xgr effectivities, this is due to the fact that the xmls approximation is C2 where as the xgr approximation is only C1 . We also notice that the xmls results are less sensitive to the value of the enrichment radius than the xgr results. This is not surprising, since the xmls recovery is built with a global intrinsic enrichment of the mls approximation, whereas the enrichment used for the strain recovery in xgr is only active in a small2 ball (cylinder) around the crack tip (front). 2 with
respect to the domain size
7
(a) raw strain
(b) xgr strain
2
εh εs exact
1.8
ε
1.6
εyy
1.4 1.2 1 0.8 0.6 0.4 -1
-0.5
0
0.5
1
y
(d) distribution of ehs Ωq of Equation (8)
(c) ǫyy (x = 0, y)
1
renr=0.1 renr=0.2 renr=0.3 renr=0.4
0.96
0.1 Effectivity index
Strain difference norm
0.98
0.01 0.02
renr=0.1 renr=0.2 renr=0.3 renr=0.4 0.04
0.06
Element size
(e) strain difference L2 norm: 0.77959 0.82568 0.86427 0.86917
0.94 0.92 0.9 0.88 0.86 0.84 0.82
0.08
0.1
0.8 0.02
0.04
0.06
0.08
0.1
Element size
slopes = (f) effectivity indices versus mesh refinement
Figure 1: Recovery results for the near-tip problem and structured meshes. See also Table 1 for a comparison with xmls . 8
Table 2: Mode I problem, comparison of the convergence of the L2 norm of the difference between the raw xfem strain field and the recovered xgr and xmls strain fields, for structured (s) and unstructured (u) meshes, for various xfem enrichment radii, renr . See also Figure 2. Enrichment radius: renr only tip element 0.1 0.2 0.3 0.4
3.1.2
xmls (s) 0.46022 0.81271 0.86728 0.90644 0.91052
xgr (s) – 0.77959 0.82568 0.86427 0.86917
xgr (u) 0.75262 0.81658 0.84776 0.84559
Unstructured mesh
Second, we consider unstructured meshes of linear triangular elements. The same remarks as for the structured case apply for Figures 2(a-d). Note also that the zone of highest error is about the same size as in the structured case, approximately one third of the crack length. We notice from Figures 2(e) and (f), compared to Figure 1(e) and (f), that the convergence is more erratic than in the structured mesh case, which is expected, since convergence is strictly defined for uniform refinement of structured meshes. Additionally, in an unstructured mesh, the location of the crack tip relative to the nodes of the tip element vary a lot, which may explain the presence of the oscillation: when the tip is close to a node, the tip element stiffness matrix may be badly conditioned and negatively alter the accuracy. Moreover, the convergence rates are slightly lower than for structured meshes, as summarized in Table 2.
3.2
Inclined crack
We now consider the case of an inclined crack under uniaxial tension. Again, by comparing Figures 3(a) and (b), we note that the xgr strain field, which is C1 is much smoother than the raw xfem field, which is only C0 . The xgr results are quite satisfactory, and show that the L2 norm of the difference between the xgr strain and the raw xfem strain tends to zero at a rate becoming closer to unity as the xfem enrichment radius is increased (Figure 3(d)). As in the previous examples, we also note that increasing the enrichment radius decreases the overall error. The error distribution shown in Figure 3(c) shows that the “hot” points are located around the crack tips, which corroborates the intuitive refinement schemes that might have been deduced from the fringes in Figure 3(b).
9
(a) raw strain
(b) xgr strain
2
εh εs exact
1.8
ε
1.6
εyy
1.4 1.2 1 0.8 0.6 0.4 -1
-0.5
0
0.5
1
y
(d) distribution of ehs Ωq of Equation (8)
(c) ǫyy (x = 0, y)
1
renr=0.1 renr=0.2 renr=0.3 renr=0.4
0.96
0.1 Effectivity index
Strain difference norm
0.98
0.01 0.02
renr=0.1 renr=0.2 renr=0.3 renr=0.4 0.04
0.06
0.92 0.9 0.88 0.86 0.84 0.82
0.08
0.8 0.02
0.1
Element size
(e) strain difference L2 norm: 0.75262 0.81658 0.84776 0.84559
0.94
0.04
0.06
0.08
0.1
Element size
slopes = (f) effectivity indices versus mesh refinement
Figure 2: Recovery results for the near-tip problem, and unstructured meshes. 10
(b) xgr strain field
Strain difference norm
(a) xfem strain field
0.1
0.01 0.02
renr=0.1 renr=0.2 renr=0.3 renr=0.4 0.04
0.06
0.08
0.1
Element size
(c) distribution of ehs Ωq of Equation (8)
(d) convergence of the L2 norm of the difference between the raw xfem strain field and the xgr strain field, slopes = 0.72283 0.78853 0.79003 0.81865
Figure 3: Inclined crack under uniaxial tension.
11
4
Three-dimensional fracture mechanics example
In this section, we tackle the three-dimensional example of a quarter-circular crack emanating from a hole in a cylindrical shell subjected to a uniform internal pressure. The elements are linear tetrahedral elements. In this problems, elements in a cylinder centered on the crack front and with radius renr are enriched with the near-tip fields. The enrichment radius is varied, and the influence on the error distribution is analysed. First, we assume that the enrichment radius is equal to the radius of the quarter-penny crack. In Figure 4, we compare the raw xfem strain field (a) to the xgr strain field (b), and show the convergence of the L2 norm of the difference between the raw xfem strain field and the recovered xgr strain field with mesh refinement, in Figure 4(c). As in the two-dimensional cases convergence happens at a rate close to the optimal value of 1. Let us now examine the influence of the xfem enrichment radius on the error distribution by plotting, in Figure 5, the error distribution for renr = rcrack (a) and renr = rcrack /5 (b). The results are very interesting. They show that increasing the enrichment radius from rcrack /5 (roughly one element size) to rcrack decreases the error, and reduces the size of the peak error zone. This corroborates our findings in two dimensions, as well as the conclusions drawn in References [20, 9]. More importantly, the remarks made in the previous paragraph suggest a strategy for h−adaptivity in enriched finite element methods, and hint at a new approximation adaptation scheme specifically taylored to enriched approximations. Indeed, it is clear that the error should be minimized by first evaluating the optimal enrichment radius renr 3 and, second, if the overall or/and local errors are still above the tolerance specified for the analysis at hand, proceed to h− or/and p− refinement, while keeping the enrichment radius constant. This procedure of estimating the optimal enrichment radius can be seen as a generalization of h− and p− adaptivity to encompass the non-polynomial functions present in the xfem approximation. This new adaptivity could be coined enrichment-adaptivity, or e-adaptivity, and is subject to on-going research.
5
Closure
In this paper, an extension to enriched finite element methods of Zienkiewicz and Zhu’s global derivative recovery [36] is proposed; the resulting technique is coined extended global derivative recovery (xgr). Based on these recovered derivatives, an error estimator is defined. We exercise our method on two and three dimensional linear elastic fracture mechanics problems, and show that the effectivity index of the estimator tends to unity upon mesh refinement, which 3 this optimal enrichment radius is problem dependent. In our experience, it is situated in the vicinity of the length of the crack.
12
(a) raw xfem strain field
(b) xgr strain field
500
Strain difference norm
450 400 350
300
250 0.6
0.7
0.8 0.9 Element size
1
1.1
1.2
(c) convergence of the L2 norm of the difference between the xfem strain field and the xgr strain field: the slope is ????????
Figure 4: Quarter-circular crack emanating from a hole in a cylindrical shell under internal pressure; the enrichment radius renr is equal to the radius of the quarter-penny crack. The slope of the convergence curve of the L2 norm of the difference between the raw xfem strain field and the xgr strain field is ??????????
13
(a) renr = rcrack
(b) renr = rcrack /5
Figure 5: Influence of the xfem enrichment radius on the distribution of the error ehs Ωq defined in Equation (8). proves its adequacy. Moreover, the recovery technique is compared to the extended moving least squares recovery, presented earlier [13, 12], and, despite the lower continuity of the recovered fields, found to yield only slightly lower convergence rates and effectivity indices, while providing a lower computational cost and larger flexibility. The three-dimensional examples showed that in order to minimize the error and the size of the “high-error zone” the xfem enrichment radius has to be sufficiently large. Assuming that the local/global error is to be less than a given upper bound, first, an optimal value for the enrichment radius should be found: this step is called “e-enrichment ”. Then, if the error obtained is still too large, h- and p-enrichment can be carried out, while keeping the enrichment radius constant. Studies around these lines are being carried out and will be the subject of coming communications. Compared to the previously published xmls recovery, we believe that the simplicity and flexibility of the proposed approach make it an ideal candidate for industrial applications.
References [1] M. Ainsworth and A. W. Craig. A posteriori error estimators in the finite element method. Numerische Mathematik, 60:429–463, 1992.
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[2] M. Ainsworth and J. T. Oden. A posteriori error estimation in finite element analysis. Computer Methods in Applied Mechanics and Engineering, 142:1– 88, 1997. [3] M. Ainsworth and T. Oden. A unified approach to a posteriori error estimation using element residual methods. Numerische Mathematik, 65(1):23–50, December 1993. [4] Mark Ainsworth and J. Tinsley Oden. A posteriori error estimation in finite element analysis. Wiley, 2000. ISBN: 0-471-29411-X. [5] I. Babuˇska and W. C. Rheinboldt. Error estimates for adaptive finite element computations. SIAM Journal of Numerical Analysis, 15(4):736–754, August 1978. [6] I. Babuˇska and I. Melenk. Partition of unity method. International Journal for Numerical Methods in Engineering, 40(4):727–758, 1997. [7] F. B. Barros, S. P. B. Proenca, and C. S. de Barcellos. International Journal for Numerical Methods in Engineering, 60:2373–2398, 2004. [8] F. B. Barros, S. P. B. Proenca, and C. S. de Barcellos. Generalized finite element method in structural nonlinear analysis - a p adaptive strategy. Computational Mechanics, 33:95–107, 2004. [9] E. B´echet, H. Minnebo, N. Mo¨es, and B. Burgardt. Improved implementation and robustness study of the x-fem for stress analysis around cracks. International Journal for Numerical Methods in Engineering, 2005. [10] T. Belytschko and T. Black. Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering, 45(5):601–620, 1999. [11] S. Bordas, J.G.C. Conley, B. Moran, J. Gray, and E. Nichols. A simulationbased design paradigm for complex cast components. Engineering with Computers, 2006. in press, online first. [12] S. Bordas and M. Duflot. Derivative recovery and a posteriori error estimate for extended finite elements. Computer Methods in Applied Mechanics and Engineering, 2007. under review. [13] S. Bordas, M. Duflot, and P. Le. A simple error estimator for extended finite elements. Communications in Numerical Methods in Engineering, 2007. under review. [14] S. Bordas and B. Moran. Enriched finite elements and level sets for damage tolerance assessment of complex structures. Engineering Fracture Mechanics, 73:1176–1201, 2006.
15
[15] S. Bordas, V.P. Nguyen, C. Dunant, H. Nguyen-Dang, and A. Guidoum. An extended finite element library. International Journal for Numerical Methods in Engineering, 2006. in press. [16] D. L. Chopp and N. Sukumar. Fatigue crack propagation of multiple coplanar cracks with the coupled extended finite element/fast marching method. Int. J. Eng. Sci., 41(8):865—869, 2003. [17] C. Daux, N. Mo¨es, J. Dolbow, N. Sukumar, and T. Belytschko. Arbitrary branched and intersecting cracks with the extended finite element method. International Journal for Numerical Methods in Engineering, 48(12):1741—1760, 1999. [18] C. Dunant, P. Nguyen, M. Belgasmia, S. Bordas, A. Guidoum, and H. Nguyen-Dang. Architecture trade-offs of including a mesher in an objectoriented extended finite element code. European Journal of Computational Mechanics, 2007. in press. [19] A. Gravouil, N. Mo¨es, and T. Belytschko. Non-planar 3d crack growth by the extended finite element and level sets. part II: level set update. International Journal for Numerical Methods in Engineering, 53:2569–2586, 2002. [20] P. Laborde, J. Pommier, Y. Renard, and M. Salaun. High order extended finite element method for cracked domains. International Journal for Numerical Methods in Engineering, 190(47):6183—6200, 2004. [21] P. Ladev`eze and D. Leguillon. Error estimate procedure in the finite element method and applications. SIAM Journal of Numerical Analysis, pages 485– 509, 1983. [22] G. Legrain, N. Mo¨es, and E. Verron. Stress analysis around the crack tips in finite strain problems using the extended finite element method. International Journal for Numerical Methods in Engineering, 63:290—314, 2005. [23] J. M. Melenk and I. Babuˇska. The partition of unity finite element method: Basic theory and applications. Computer Methods in Applied Mechanics and Engineering, 139:289–314, 1996. [24] T. Menouillard, J. R´ethor´e, A. Combescure, and H. Bung. Efficient explicit time stepping for the eXtended Finite Element Method (X-FEM). International Journal for Numerical Methods in Engineering, 2006. [25] N. Mo¨es and T. Belytschko. Extended finite element method for cohesive crack growth. Engineering Fracture Mechanics, 69:813–834, 2002. [26] N. Mo¨es, A. Gravouil, and T. Belytschko. Non-planar 3d crack growth by the extended finite element and level sets. part I: Mechanical model. International Journal for Numerical Methods in Engineering, 53:2549–2568, 2002. 16
[27] J. Rethore, A. Gravouil, and A. Combescure. An energy-conserving scheme for dynamic crack growth using the extended finite element method. International Journal for Numerical Methods in Engineering, 2005. [28] M. Stolarska, D. L. Chopp, N. Mo¨es, and T. Belytschko. Modelling crack growth by level sets and the extended finite element method. International Journal for Numerical Methods in Engineering, 51(8):943–960, 2001. [29] T. Strouboulis, I. Babuˇska, and K. Copps. The design and analysis of the generalized finite element method. Computer Methods in Applied Mechanics and Engineering, 181:43–69, 2000. [30] T. Strouboulis, K. Copps, and I. Babuˇska. The generalized finite element method : An example of its implementation and illustration of its performance. International Journal for Numerical Methods in Engineering, 47(8):1401–1417, 2000. [31] T. Strouboulis, L. Zhang, D. Wang, and I. Babuˇska. A posteriori error estimation for generalized finite element methods. Computer Methods in Applied Mechanics and Engineering, 195(9-12):852–879, February 2006. [32] N. Sukumar, D. L. Chopp, and B. Moran. Extended finite element method and fast marching method for three-dimensional fatigue crack propagation. Engineering Fracture Mechanics, 70(1):29—48, 2003. [33] N. Sukumar, Z. Y. Huang, J.-H. Pr´evost, and Z. Suo. Partition of unity enrichment for bimaterial interface cracks. International Journal for Numerical Methods in Engineering, June 2003. accepted for publication. [34] N. Sukumar, N. Mo¨es, T. Belytschko, and B. Moran. Extended Finite Element Method for three-dimensional crack modelling. International Journal for Numerical Methods in Engineering, 48(11):1549–1570, 2000. [35] Q. Z. Xiao and B. L. Karihaloo. Improving the accuracy of xfem crack tip fields using higher order quadrature and statically admissible stress recovery. International Journal for Numerical Methods in Engineering, 66, 2006. [36] O. C. Zienkiewicz and J. Z. Zhu. A simple error estimator and adaptive procedure for practical engineering analysis. International Journal for Numerical Methods in Engineering, 24:337–357, 1987. [37] O. C. Zienkiewicz and J. Z. Zhu. The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique. International Journal for Numerical Methods in Engineering, 33:77:1331–1364, 1992. [38] O. C. Zienkiewicz and J. Z. Zhu. The superconvergent patch recovery and a posteriori error estimates. Part 2: error estimates and adaptivity. International Journal for Numerical Methods in Engineering, 33:77:1365– 1382, 1992.
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