p.p1 on the curved side of the element than

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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

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