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It is noted that the sign- and Heaviside function lead to identical results as they span the

same approximation space. Different enrichment functions proposed in the literature

to capture strong and weak discontinuities arising in different problems in mechanics.

1.2 Extended spectral element method

A modified spectral element method based on an extension of the standard spectral

element space can be used. This XSEM space has optimal approximation properties

for piecewise smooth functions. In practice the interface G is approximated by an interface

capturing method like, for example, a level set method. The level set function

is discretised and (an approximation) of the zero level of this discrete level set function

is used as an approximation of G.

The approximation of a discontinuity, that is unfitted to the computational mesh,

using continuous polynomials can lead to spurious oscillations local to the discontinuity.

These oscillations can propagate throughout the computational domain and pollute

other variables. If the discontinuity is assumed to move freely within the domain, fitting

the mesh to the discontinuity becomes computationally very expensive. In the case

of finite elements, a method known as the eXtended Finite Element Method (XFEM)

was proposed by Moes et al. 40, 8 in an attempt to alleviate this issue. In the case of

a strong discontinuity (the type of discontinuity considered in this thesis), the general

idea behind XFEM, in a very formal description, is to enrich the original finite element

space of admissible functions by something discontinuous. This allows the numerics to

capture the discontinuity and achieve optimal order of convergence for functions with

a lower regularity. This enrichment is achieved by adding to the original finite element

space, a space which is spanned by discontinuous basis functions. In this thesis, we

apply the method to spectral elements and hence name it, the eXtended Spectral Element

Method (XSEM).

The XSEM was first proposed by Legay et al. 33 when studying strong and weak

discontinuities using, what they called, spectral finite elements. In that article, the

authors note that additional considerations, such as careful design of the blending elements,

are required when higher-order elements are considered. In that article, the

union of the elements which contain the discontinuity, either strong or weak, was denoted

WLPU and the union of the blending elements (the elements which share an edge

or node with the elements of WLPU) was denoted WB. The global enriched approximation

was given by:

u(x) = å

I?S

NP

I (x)uI + å

J?SP

fJ(x)y(x)qJ (1.6)

where NP

I are the spectral basis functions of order P, y is an enrichment function, uI

are nodal values, qJ are additional degrees of freedom, S is the set of nodes in the