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model and SP ? S is the set of nodes in WLPU. The functions fJ form a local partition

of unity over WLPU. For the case of strong discontinuities, the functions fJ are

constructed from linear spectral basis functions N1

J whilst for the case of weak discontinuities,

or gradient discontinuities, the functions fJ are constructed from spectral basis

functions of one order lower than the standard approximation, i.e. from NP?1

J . The

spectral interpolant employed by Legay et al. is based on Chebyshev polynomials and

the Chebyshev-Gauss grid. Legay et al. found that for the case of strong discontinuities

no additional considerations were necessary in the blending elements. However,

for the case of weak discontinuities, Legay et al. found that higher-order terms appear

in the blending elements which need to be removed. They noted that using polynomials

of degree P?1 for the enrichment was sufficient to remove the higher-order terms

in the blending elements. However, when P = 1, the assumed strain method ? was

required to deal with the blending problems. Even though spectral basis functions were

considered by Legay et al. 33, the authors approached XSEM from the perspective

of high-order FEM. Therefore, the maximum order the authors considered was 4 and

h?type convergence (in other words, convergence with respect to mesh width) of the

method was studied. Legay et al. found that, for weak straight discontinuities, the

method obtained nearly optimal order of convergence. However, they found that for

weak curved discontinuities, the method obtained suboptimal order of convergence.

The suboptimal order of convergence was attributed to approximations in the quadrature

scheme. The quadrature scheme employed by Legay et al. involved subdividing

the spectral element containing the discontinuity into smaller elements. If one of these

smaller elements contains the discontinuity the smaller element is again subdivided

into triangles so that a linear approximation of the discontinuity takes place within the

element.

Cheng and Fries 18 studied strong and weak discontinuities with the aim of

improving the suboptimal order of convergence found when higher-order XFEM is

applied to curved discontinuities. As the approximations in the quadrature were the

source of the suboptimal order of convergence found by Legay et al., Cheng and Fries

18 proposed a different quadrature scheme where they subdivided an element containing

the discontinuity into smaller elements where one side of the subelement is

curved. This means that there are more nodes on the curved side of the element than

the straight sides. This adds additional complexity to the quadrature. In this thesis, we

use a different scheme for the quadrature. Cheng and Fries 18 also proposed using a

corrected higher-order XFEM. The standard higher-order XFEM did not obtain optimal

rates of convergence for curved weak discontinuities. However, Cheng and Fries

found that the corrected higher-order XFEM did obtain optimal rates of convergence

when equal order basis functions are used for the standard and extended parts of the

enriched approximation. They also considered a modified abs-enrichment (where the

enrichment function is an absolute value function) and found that suboptimal order of

convergence is seen for curved discontinuities.

Let W = ?1,1d with 1 ? d ? 2. We decompose our domain into K uniform, non