than by the original CPM. Then the adaptivity approach is introduced which includes a review of different error estimators and refinement strategies. After that, the J-integral and different methods to calculate SIFs are included. Finally, some 2D crack problems are used to demonstrate the performance of the proposed methodology.
3.2
Crack modelling by the CPM
Crack discontinuities are introduced into the CPM either by extrinsic enrichment or in- trinsic enrichment. In the former, discontinuous terms are added in the basis functions similar to the XFEM (Section 1.5.1), while the latter is through modifying the support of weight functions.
3.2.1
The visibility criterion and the diffraction criterion
Before introducing the CPM, previous strategies in the EFGM for crack modelling are firstly explained, which are also used in many other MMs. In the EFGM, for nodes adjacent to a crack, supports are adjusted to avoid them having influence on the opposite side of the crack, by which the displacement jump at cracks is obtained. Two methods have been widely used: the visibility criterion [2] and the diffraction criterion [226].
The visibility criterion comes from the idea of discontinuity opaqueness. The support of a node i is modified to the area receiving “light” from the centre, and the “shadow” area caused by the discontinuity is excluded, as shown in Figure 3.1. An issue of this approach is that an artificial discontinuity is introduced into the weight function so the resulting shape function is not even C0 continuous, although convergence can still be
reached [124]. Significant errors and oscillations can occur around the crack tip especially when large supports are used [226]. Due to the simplicity of the visibility criterion, this approach is applied to all numerical examples in this thesis.
By contrast in the diffraction method, a small part of the support around the crack tip is included following the law of “ray diffraction”. A more moderate truncation can be applied and oscillations around the tip are therefore reduced. The input parameter r =kx − xik/r0 for the weight function w(r) of the node i is modified correspondingly as
r = kx − xik + kxc− xik kx − xik γ kx − xik/r0, (3.1)
Figure 3.1: Visibility and diffraction criterion.
where xc is the crack tip coordinate, r0 is the support size and γ is a constant which is
1 or 2 as suggested in [226]. The diffraction method provides better accuracy than the visibility criterion, but cannot avoid high computational complexity, especially when used for either non-planar cracks in 3D or multiple cracks in 2D and 3D.
3.2.2
A new strategy of cracking particles
A literature review of the CPM was included in Section 1.5.6. The CPM provides an approximation of a crack path by using a set of discontinuous segments centred at parti- cles, so updating the crack pattern is easily achieved by editing cracking particles. The development of the method is described below.
In the original CPM proposed by Rabczuk et al. (2004) [202, 203], the discontinuity in displacements at cracks is obtained by using discontinuous extrinsic enrichment functions. All particles are divided into two groups, namely normal particlesN and cracking particles Nc. The displacement in the problem domain is approximated by
uh(x) =X i∈N Φi(x)ui+ X j∈Nc Ψj(x)H(x)bj, (3.2)
where H(x) is the sign function with value 1 on one side of the crack and -1 on the other side, and bj are extra unknowns. Shape functions including the normal part Φi and the
enrichment part Ψj are evaluated by the MLS approximation but with different support
sizes, where the support sizes for calculating shape functions in the enrichments are usually larger than for the normal part. A crack path is approximated by a group of discontinuous segments as in Figure 3.2 (a), and the orientations of these segments control the values of the sign function and therefore the crack discontinuity. Crack opening, described by the
3.2. Crack modelling by the CPM 57
Figure 3.2: The development of the CPM.
operator [[·]], can be obtained by the enrichment functions, as
[[u(x)]] =ku(xS+)− u(xS−)k = 2 X
j∈Nc
Ψj(x)bj, (3.3)
where S+ and S− are two opposite sides of crack paths. However, additional unknowns
bj are introduced to the variational formulation and bring extra cost to solve the weak
form equations. An alternative presented by Rabczuk et al. (2010) [204] is by splitting cracking particles into two subparts belonging to the two sides of the crack, Figure 3.2 (b). No additional terms are required in the displacement approximation, and crack opening is obtained by the relative displacements of cracking particles on the two sides of the crack, i.e. [[u(x)]] = X i∈S+ Φi(x+)ui− X i∈S− Φi(x−)ui (3.4)
The CPM in Figure 3.2 (a-b) suffers from spurious crack results as mentioned in [204, 213], when these crack segments do not align well especially for curved cracks. A new strategy to define the influence domain of cracking particles is proposed in this thesis (and as published in [227]), where bilinear segments are used instead of straight segments to cut the support as shown in Figure 3.2 (c). The orientations of crack segments are modified according to surrounding cracking particles, by which these segments are aligned to continuous crack paths. The support of a given cracking particle i is split into two sectors complying with the crack pattern, and the two segment arms are directed by angles θi
1, θi2 where the angle should satisfy θ∈ (−π, π], as
θ1i = arctan(yi+1− yi xi+1− xi
), θi2 = arctan(yi−1− yi xi−1− xi
), i∈ Nc. (3.5)
Figure 3.3: The two segment arm directions of a cracking particle.
(a) (b)
Figure 3.4: Comparison between the original and the modified CPM: (a) original CPM; (b) modified CPM.
equivalent). The crack tip particle has influence on both sides of the crack path and is not itself split.
One advantage of the proposed CPM over the original method is illustrated in Figure 3.4. In the original CPM, cracking particles are defined as each carrying a straight discon- tinuous segment of the crack face, splitting the supports as in Figure 3.4 (a). If there is a “bad” overlap between those segments, spurious cracking occurs. In the modified CPM proposed for the first time here [227], this problem is solved by using bilinear segments where the crack path can change its direction at each cracked particle. All discontinuous segments are aligned and “good” overlaps are obtained, as shown in Figure 3.4 (b).