## p.p1 • r(x): global enrichment function. The enrichment

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s2 {font: 12.0px Helvetica}span.s3 {font: 17.0px Helvetica}span.s4 {font: 7.0px Helvetica}4. Integration: A consequence of adding custom tailored enrichment functions tothe FE approximation basis, which are not necessarily smooth polynomial functions(for example, ?r in case of the LEFM) is that special care has to be takenin numerically integrating over the elements that are intersected by the discontinuitysurface.

The standard Gauss quadrature cannot be applied in elementsenriched by discontinuous terms, because Gauss quadrature implicitly assumesa polynomial approximation.Suppose that the domain W is discretized by nel elements, numbered from 1 to nel.I is the set of all nodes in the domain, and I? is the nodal subset of the enrichment(I? ? I).

A standard extended finite element approximation of a function u(x) is of theformuh(x) = uhf em(x)+uh enr(x)=åi ?INi(x)ui+ åj?I?N?j (x)r(x)aj. (1.4)For simplicity only one enrichment term is considered. The approximation consistsof a standard finite element (FE) part and the enrichment. The individual variablesstand for• uh(x): approximated function,• Ni(x): Standard FE function of node i,• ui: unknown of the Standard FE part at node i,• N?j (x): standard FE shape functions which are not necessarily the same thanthose of the standard part of the approximation (Ni(x)), These functions build apartition of unity, åj?I?N?j (x) = 1.

in elements whose nodes are all in the nodalsubset I?. In these elements, the global enrichment function can be reproducedexactly; we call these elements reproducing elements. In elements with onlysome of their nodes in I?, does not build a partition of unity, åj?I?N?j (x) 6= 1.

As a consequence, the global enrichment function r(x) cannot be representedexactly in these elements. Elements with only some of their nodes in I? arecalled blending elements. Several publications discuss problems arising fromblending elements.

• r(x): global enrichment function. The enrichment function r(x) carries with itthe nature of the solution or the information about the underlying physics of theproblem, for example, r(x)= H, is used to capture strong discontinuities, whereH is the Heaviside function.• aj: unknown of the enrichment at node j

x

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