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0px Helvetica}It is noted that the sign- and Heaviside function lead to identical results as they span thesame approximation space. Different enrichment functions proposed in the literatureto capture strong and weak discontinuities arising in different problems in mechanics.1.2 Extended spectral element methodA modified spectral element method based on an extension of the standard spectralelement space can be used. This XSEM space has optimal approximation propertiesfor piecewise smooth functions. In practice the interface G is approximated by an interfacecapturing method like, for example, a level set method. The level set functionis discretised and (an approximation) of the zero level of this discrete level set functionis 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 polluteother variables. If the discontinuity is assumed to move freely within the domain, fittingthe mesh to the discontinuity becomes computationally very expensive. In the caseof 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 ofa strong discontinuity (the type of discontinuity considered in this thesis), the generalidea behind XFEM, in a very formal description, is to enrich the original finite elementspace of admissible functions by something discontinuous. This allows the numerics tocapture the discontinuity and achieve optimal order of convergence for functions witha lower regularity. This enrichment is achieved by adding to the original finite elementspace, a space which is spanned by discontinuous basis functions. In this thesis, weapply the method to spectral elements and hence name it, the eXtended Spectral ElementMethod (XSEM).The XSEM was first proposed by Legay et al.

33 when studying strong and weakdiscontinuities using, what they called, spectral finite elements. In that article, theauthors note that additional considerations, such as careful design of the blending elements,are required when higher-order elements are considered. In that article, theunion of the elements which contain the discontinuity, either strong or weak, was denotedWLPU and the union of the blending elements (the elements which share an edgeor node with the elements of WLPU) was denoted WB. The global enriched approximationwas given by:u(x) = åI?SNPI (x)uI + åJ?SPfJ(x)y(x)qJ (1.6)where NPI are the spectral basis functions of order P, y is an enrichment function, uIare nodal values, qJ are additional degrees of freedom, S is the set of nodes in the