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

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4. Integration: A consequence of adding custom tailored enrichment functions to
the 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 taken
in numerically integrating over the elements that are intersected by the discontinuity
surface. The standard Gauss quadrature cannot be applied in elements
enriched by discontinuous terms, because Gauss quadrature implicitly assumes
a 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 the
form
uh(x) = uh
f em(x)+uh enr(x)

i ?I
Ni(x)ui+ å
j?I?
N?j (x)r(x)aj. (1.4)
For simplicity only one enrichment term is considered. The approximation consists
of a standard finite element (FE) part and the enrichment. The individual variables
stand 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 than
those of the standard part of the approximation (Ni(x)), These functions build a
partition of unity, å
j?I?
N?j (x) = 1. in elements whose nodes are all in the nodal
subset I?. In these elements, the global enrichment function can be reproduced
exactly; we call these elements reproducing elements. In elements with only
some of their nodes in I?, does not build a partition of unity, å
j?I?
N?j (x) 6= 1.
As a consequence, the global enrichment function r(x) cannot be represented
exactly in these elements. Elements with only some of their nodes in I? are
called blending elements. Several publications discuss problems arising from
blending elements.
• r(x): global enrichment function. The enrichment function r(x) carries with it
the nature of the solution or the information about the underlying physics of the
problem, for example, r(x)= H, is used to capture strong discontinuities, where
H is the Heaviside function.
• aj: unknown of the enrichment at node j

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